Unit of geography
Department of Geosciences
University of Fribourg
Master Thesis
Technical improvement of the 4-phase model to better
assess the ice, water and air content estimation in
permafrost substrates
Case Study : Stockhorn, Valais, Switzerland
Python Samuel
Rte du Bugnon 37
1752 Villars-sur-Glâne
Supervisor : Prof. Christian Hauck
Fribourg, May 2015
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Abstract
With subsurface temperatures only a few degrees below 0°C, the mountain permafrost
is very sensitive to climate change (Harris et al., 2009). However, the subsurface
thermal regime is not directly connected to the atmospheric conditions. Snow, surface
and subsurface characteristics as well as the ice content and the topography may alter
the heat and mass transfer from the atmosphere into the soil (PERMOS, 2013). Without
direct data, such as borehole temperatures or ice cores, it is difficult to assess the
structure of the subsurface. A good alternative to these methods are the Geophysical
measurements. Thanks to this approach, it is possible to quantify the volumetric fraction
of ice, water and air using relations between the measured data (electrical resistivity and
seismic velocity) and the physical properties of the soil. The so-called 4-phase model
(4PM) was developed to this end (Hauck et al., 2008a; Hauck et al., 2011). This study
focuses on the improvement of the 4PM and on its application for the Stockhorn site. A
new porosity model and new resistivity equations have been implemented for this
purpose. Modelling results show a good concordance between the 4PM and the 1D-
physically based CoupModel (Jansson, 2014) concerning the general patterns of the ice
and water content, but with some differences in absolute content. The main process
influencing the spatial distribution of ice and water at Stockhorn is the topography. The
southern slope is more exposed to solar radiation. This provokes a lateral heat flux
decreasing to the north (Gruber et al., 2004). Besides, the topography of the plateau
canalises the snow melt water into small streams during the summer. The latter
accumulates then in a natural reservoir in the southern part of the plateau. This process
induces a consequent transport of latent heat to the south.
Key words: Alpine permafrost, 4-phase model, CoupModel, heat flux
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Acknowledgements
I would like to thank all the people who have helped me to achieve this master thesis:
- Professor Christian Hauck for his precious advice and his supervision.
- Cécile Pellet for the organisation of the field campaign 2014, for her crucial help
with the geoelectrical, seismic and soil moisture measurements and for her
explanations concerning the development of the 4PM.
- Jutta Heinonen for helping and motivating me during the field campaign 2014
and the writing of my thesis.
- Antoine Marmy for his time-saving explanations on the CoupModel and for the
simulation of the Stockhorn site.
- Christin Hilbich for her advice and explanations of the geophysical
measurements and the 4PM.
- Susanne Dängeli for her time and expertise during the campaign 2014.
- My family and friends for the moral support they have given me all throughout
my studies.
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Table of Contents
1. Introduction .............................................................................................................. 1
2. Study site .................................................................................................................. 7
2.1. Surface characteristics and geology ................................................................... 7
2.2. Meteorological data ........................................................................................... 9
2.3. Geophysical profiles ........................................................................................ 10
3. Methods .................................................................................................................. 13
3.1. Electrical Resistivity Tomography .................................................................. 14
3.2. Refraction Seismic Tomography ..................................................................... 18
3.3. Improved Four Phase Model (4PM) ................................................................ 21
3.3.1. Theory behind the model .......................................................................... 21
3.3.2. Integration of existing features into a Graphical user Interface ............... 26
3.3.3. New features of the version 7 ................................................................... 28
3.4. CoupModel ...................................................................................................... 32
3.4.1. Main equations ......................................................................................... 33
3.4.2. CoupModel calibration ............................................................................. 34
4. Results and interpretation ....................................................................................... 39
4.1. Comparison of the 4PM with the CoupModel ................................................. 41
4.1.1. CoupModel Calibration for the borehole 6000 (100m) ............................ 42
4.1.2. CoupModel Calibration for the borehole 6100 (17m) .............................. 48
4.1.3. Comparison with 4PM .............................................................................. 53
4.1.4. Conclusion of the comparison of the 4PM with the CoupModel ............. 58
4.2. Application of the 4PM at Stockhorn .............................................................. 59
4.2.1. Interpretation examples of an ERT and RST profile ................................ 59
4.2.2. 4PM calibration ........................................................................................ 62
4.2.3. 4PM results ............................................................................................... 65
4.3. Interpretation of the results .............................................................................. 73
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4.3.1. Analysis of CoupModel parameters ......................................................... 73
4.3.2. Spatial distribution of ice .......................................................................... 77
4.3.3. Spatial distribution of water in the active layer ........................................ 81
4.3.4. Temporal evolution of water content........................................................ 82
5. Discussion of the uncertainties ............................................................................... 87
5.1. Potential calibration biases in CoupModel ...................................................... 87
5.1.1. Meteorological data in CoupModel .......................................................... 87
5.1.2. Uncertainties introduced by CoupModel parameters ............................... 88
5.2. Calibration of Archie’s law in 4PM ................................................................. 89
5.2.1. Calibration of m, n and ρw with a multi-run procedure ............................ 89
5.2.2. Epsilon factor for rock resistivity ............................................................. 91
5.2.3. Improvement of Archie’s law ................................................................... 93
5.3. Solutions restrictions in the 4PM ..................................................................... 94
6. Conclusion .............................................................................................................. 97
7. Bibliography ........................................................................................................... 99
8. Appendix .............................................................................................................. 105
8.1. Field campaign protocol: Stockhorn, August 2014 ....................................... 105
8.2. List of major 4PM improvements since version 5.0 ...................................... 108
8.3. 4PM structure and coding examples .............................................................. 110
8.4. Four Phase Model Tutorial ............................................................................ 123
8.5. Sensitivity of soil moisture and temperature to CoupModel parameters ....... 129
8.6. Updated 4PM Results of previous field campaigns ....................................... 134
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Table of figures
Figure 1: Schematic representation of the permafrost in the Alps ................................... 1
Figure 2: Map of all the PERMOS study sites in Switzerland (PERMOS, 2013) ........... 3
Figure 3: North-South profile of the study site................................................................. 7
Figure 4: Situation on the northern part of Stockhorn plateau in August 2014................ 8
Figure 5: Situation on the southern part of Stockhorn plateau in August 2014. .............. 8
Figure 6: Reconstructed air temperature at Stockhorn, VS .............................................. 9
Figure 7: Geophysical profiles of the 2014 campaign .................................................... 10
Figure 8: Schematic figure of the interaction between all the methods ......................... 13
Figure 9: Current path between electrodes (Mussett and Khan, 2009: 185) .................. 14
Figure 10: Illustration of electrodes array types ............................................................. 15
Figure 11: Wave movement in two layers coming from a seismic source ..................... 18
Figure 12: Travel-time diagram ...................................................................................... 19
Figure 13: Graphical user interface of 4-phase model 7 ................................................ 27
Figure 14: Example of smoothing area if the parameter is equal to 2. ........................... 30
Figure 15: Diagram of CoupModel layers ...................................................................... 32
Figure 16: Measured temperatures at the B100m between 2002 and 2015.................... 40
Figure 17: Measured temperatures at the B17m between 2002 and 2015...................... 40
Figure 18: Near-surface soil temperatures (0.8m) at the B100m ................................... 43
Figure 19: Simulated snow depth at the B100m............................................................. 43
Figure 20: Near-surface soil resistivity (0.8m) at the B100m ........................................ 44
Figure 21: Temperature for the permafrost table (3.3m) at the B100m ......................... 45
Figure 22: Soil resistivity for the permafrost table (3.3m) at the B100m ...................... 45
Figure 23: Temperature under the permafrost table (9.3m) at the B100m ..................... 46
Figure 24: Soil resistivity under the permafrost table (9.3m) at the B100m .................. 47
Figure 25: Temperature near the surface (0.8m) at the B17m........................................ 48
Figure 26: Simulated snow depth at the B17m............................................................... 49
Figure 27: Resistivity near the surface (0.8m) at the B17m ........................................... 49
Figure 28: Temperature for the permafrost table (3m) at the B17m .............................. 50
Figure 29: Soil resistivity for the permafrost table (3m) at the B17m ........................... 50
Figure 30: Temperature under the permafrost table (9m) at the B17m .......................... 51
Figure 31: Soil resistivity under the permafrost table (9m) at the B17m ....................... 52
Figure 32: Comparison between CoupModel and 4PM for B100m, 22 August 2006. .. 54
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Figure 33: Comparison between CoupModel and 4PM for B100m, 29 July 2011 ........ 55
Figure 34: Comparison between CoupModel and 4PM for B17m, 22 August 2006 ..... 56
Figure 35: Comparison between CoupModel and 4PM for B17m, 29 July 2011 .......... 57
Figure 36: ERT result for the Longitudinal South Profile (LSP) ................................... 60
Figure 37: RST results for the Longitudinal South Profile (LSP) .................................. 61
Figure 38: Influence of Archie's parameters on the pore contents ................................. 64
Figure 39: ERT and RST results for the Cross Monitoring Profile. ............................... 66
Figure 40: 4PM results for the Cross Monitoring Profile ............................................... 66
Figure 41: ERT and RST results for the Cross New Profile .......................................... 68
Figure 42: 4PM results for the Cross New Profile ......................................................... 68
Figure 43: ERT and RST results for the Longitudinal North Profile ............................. 70
Figure 44: 4PM results for the Longitudinal North Profile ............................................ 70
Figure 45: ERT and RST results for the Longitudinal South Profile ............................. 72
Figure 46: 4PM results for the Longitudinal South Profile ............................................ 72
Figure 47: Influence of CoupModel parameters on the mean temperature .................... 74
Figure 48: Influence of CoupModel parameters on the mean water content ................. 74
Figure 49: Interpretation of the situation the 30 July 2011 at CMP by Dängeli (2013) . 79
Figure 50: Interpretation of the situation the 27 August 2014 at CMP .......................... 79
Figure 51: 3D representation of the water circulation at Stockhorn............................... 80
Figure 52: Measured temperatures at both boreholes for the same depth (0.8m) .......... 82
Figure 53: Detailed measured temperatures at both boreholes....................................... 82
Figure 54: Total and unfrozen water content simulated by CoupModel for B100m ..... 83
Figure 55: Total and unfrozen water content simulated by CoupModel for B17m ....... 83
Figure 56: Water content measured at 50cm depth, Stockhorn soil moisture station .... 84
Figure 57: Water content measured at 50cm depth, Stockhorn soil moisture station .... 85
Figure 58: Multi-Run results for the Stockhorn Longitudinal South Profile ................. 90
Figure 59: Effect of epsilon on the pore content for values between 0 and 0.1 ............. 92
Figure 60: Effect of Epsilon on pore content for every p-wave velocity and resistivity. 93
Figure 61: Range of 4PM solutions for a porosity of 50% with Archie’s law ............... 95
Figure 62: Range of 4PM solutions for a porosity of 50% with Somerton equation ..... 96
Figure 63: Functions called when the button Run 4PM is pushed ............................... 110
Figure 64: Functions called when the button Multi-Run 4PM is pushed ..................... 110
Figure 65: Effect of the critical depth of snow cover on the temperature .................... 129
Figure 66: Effect of the critical depth of snow cover on the unfrozen water content .. 129
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Figure 67: Effect of the ground water flow on the temperature ................................... 130
Figure 68: Effect of the ground water flow on the unfrozen water content. ................ 130
Figure 69: Effect of the temperature coefficient for snow melt on the temperature .... 131
Figure 70: Effect of the temperature coefficient for snow melt on the water content .. 131
Figure 71: Effect of the m-factor for the water retention on the temperature .............. 132
Figure 72: Effect of the m-factor for the water retention on the water content ............ 132
Figure 73: Effect of the hydraulic conductivity on the temperature ............................ 133
Figure 74: Effect of the hydraulic conductivity on the unfrozen water content ........... 133
Figure 75: ERT and RST results for the Cross Monitoring Profile in 2006 ................. 134
Figure 76: 4PM results for the Cross Monitoring Profile in 2006 ............................... 134
Figure 77: ERT and RST results for the Cross Monitoring Profile in 2011 ................. 135
Figure 78: 4PM results for the Cross Monitoring Profile in 2011 ............................... 135
Table of tables
Table 1: Measurement information for all the campaign 2014 profiles ......................... 10
Table 2: Range of resistivity for different materials....................................................... 15
Table 3: Parameters tested in the ERT multiple inversion procedure ............................ 17
Table 4: P-wave velocity in different materials ............................................................. 19
Table 5: Parameters tested in the RST multiple inversion process ................................ 20
Table 6: Parameter restrictions implemented in the 4PM .............................................. 30
Table 7: Coup Model calibration values ........................................................................ 36
Table 8: Porosity calibration in CoupModel .................................................................. 54
Table 9: 4PM calibration for the 2014 campaign at Stockhorn. ..................................... 62
Table 10: Comparison of simulated water content of both models with measured data 90
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Table of abbreviations (by order of apparition)
SNF Swiss National Science Foundation
TEMPS Temporal Evolution of Mountain Permafrost in Switzerland
4PM 4-phase model
3PM 3-phase model (without ice)
CoupModel Coupled Heat and Mass Transfer Model
PERMOS Swiss Permafrost Monitoring Network (http://www.permos.ch/)
B100m Borehole 6000 (100m)
B17m Borehole 6100 (17m)
CM(P) Cross Monitoring (Profile)
CN(P) Cross North (Profile)
LN(P) Longitudinal North (Profile)
LM(P) Longitudinal Middle (Profile)
LS(P) Longitudinal South (Profile)
ERT Electrical Resistivity Tomography
RST Refraction Seismic Tomography
GUI Graphical User Interface
R2 Coefficient of determination for the linear regression equation
RMSE Root Mean Square Error
ALT Active Layer Thickness
ZC Zero Curtain (effect or regime)
AL Active Layer (regime)
FR Freezing (regime)
SM Snow Melt (regime)
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1. Introduction
The occurrence of permafrost, defined as a “ground (soil, rock, included ice and organic
material) that remains at or below 0°C for at least two consecutive years” (IPA, 2005), is
strongly influenced by the climatic conditions at the surface and in the atmosphere, like snow
characteristics and air temperature (Engelhardt et al., 2010). Thus, the permafrost is mainly
found in high latitude or high altitude regions. In the Swiss Alps, it usually occurs in locations
not covered by glaciers and above approximately 2500m a.s.l., depending on the slope
orientation (Nötzli & Gruber, 2005 cited by Staub et al., 2015). This major element of the
cryosphere can be continuous (with more than 90% of the ground underlain by permafrost) or
discontinuous (IPA, 2005). If individual areas of permafrost are surrounded by unfrozen
ground, they are considered as sporadic permafrost, usually with less than 35% of the ground
underlain by permafrost (IPA, 2005).
A permafrost soil is usually composed of an active layer. The active layer thickness (ALT),
usually a few meters in the Alps (PERMOS, 2013), corresponds to the part of the soil that
thaws in summer and freezes again in winter. Under the active layer, the permafrost core is
always frozen with temperatures below 0°C. The temperature of the permafrost increases with
depth due to the earth geothermal heat flux. The depth at which the temperature reaches 0°C
is called the permafrost base. Under it, the ground is unfrozen and there is no more
permafrost. Figure 1 represents a schematic situation of the permafrost in the Alps with a
typical temperature profile.
Figure 1: Schematic representation of the permafrost in the Alps with a typical temperature profile. The blue line
represents the minimal temperatures in winter, and the red line represents the maximal temperatures in summer
(Nötzli & Gruber, 2005).
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Permafrost landforms in the Alps
With subsurface temperatures only a few degrees below 0°C, the mountain permafrost is even
more sensitive to climate change than the permafrost of many arctic regions (Harris et al.,
2009). The permafrost degradation can provoke different natural hazards depending on the
permafrost landform (Watson & Haeberli, 2004). The four main permafrost landforms present
in the Alps are rock glaciers, talus slopes, rock walls and crests (PERMOS, 2013).
A rock glacier is defined as “a mass of rock fragments and finer material, on a slope, that
contains either interstitial ice or an ice core and shows evidence of past or present movement”
(IPA, 2005). Debris is constantly transported by this landform if it is active, creating potential
starting zones of debris flows (Hoelzle et al., 1998). As such, this mixture of water, rock and
mud may cause considerable amount of damage in the villages present under the rock glacier,
as it is the case in Mattertal (VS, Switzerland) (Delaloye et al., 2013).
A talus slope is an accumulation of coarse debris in a slope (Scapozza et al., 2011). Due to
high porosity and temperature differences between the atmosphere and the subsurface, a
winter ascending and summer descending air circulation occurs throughout the talus slope
(Delaloye & Lambiel, 2005). This air circulation, called chimney effect, provokes an
overcooling of the lower part of talus slope and leads to the presence of permafrost in this
landform. In general, talus slope do not represent a danger for the population due to their
location far from the populated areas and to their relative stability. But the scientific interest
for the understanding of the chimney effect is significant (Delaloye & Lambiel, 2005; Staub et
al., 2015).
The rock walls and the crests represent an important part of the permafrost in Switzerland
(Gruber & Haeberli, 2007). In these areas, fractured bedrocks are cemented by ice. Thus, the
permafrost degradation in rock walls and crests can provoke slope instabilities leading to rock
falls events (Watson & Haeberli, 2004; Gruber & Haeberli, 2007). This is particularly
dangerous when ski stations or restaurants are built on crests, as it is the case on Schilthorn
(BE, Switzerland).
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Permafrost in Switzerland
Mountain permafrost measurements in Switzerland started in 1987 with the installation of a
borehole in the active rock glacier Murtèl (GR, Switzerland) (PERMOS, 2013; Haeberli et al.,
1998). Many researches and projects followed this study (Hoelzle et al., 1998; Harris, et al.,
2001). Within this context, a Permafrost Monitoring Network for Switzerland (PERMOS) was
created in 2000 (PERMOS, 2013) to coordinate the installation and the maintenance of
boreholes and meteorological stations with all the research institutions (see Figure 2 for a map
of all the actual PERMOS study sites). The case study selected for this work is located at the
PERMOS site Stockhorn, a rock plateau on the crest near the Gornergrat (VS, Switzerland).
In 2011, a SNF (Swiss National Science Foundation) Project called TEMPS (Temporal
Evolution of Mountain Permafrost in Switzerland) was also launched to assess the sensitivity
of the permafrost to climate change as well as the potential impact on different mountain sites
within the PERMOS network.
Figure 2: Map of all the PERMOS study sites in Switzerland (PERMOS, 2013). The sites with a borehole are
represented in blue and the kinematics sites (rock glaciers) are represented in red. The case study selected for this
work is circled in green.
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Assessment of permafrost substrates composition
As mentioned earlier, the permafrost is strongly influenced by climatic conditions at the
surface and in the atmosphere, like snow characteristics and air temperature (Engelhardt et al.,
2010). However, the subsurface thermal regime is not directly connected to the atmospheric
conditions. Snow, surface and subsurface characteristics as well as the ice content and the
topography can alter the heat and mass transfer from the atmosphere into the soil (PERMOS,
2013). But without direct data, such as borehole temperature or ice cores, it is difficult to
assess the structure of the subsurface and thereby the sensitivity of permafrost to climate
changes. A good alternative to these methods is the use of numerical models. In the context of
the TEMPS project, two models are used to simulate the permafrost substrate: the CoupModel
and the 4-phase model (4PM).
The 1D-physically based CoupModel (Jansson, 2014) allows for the description of the heat
and water circulation into the soil using two coupled differential equations respecting the law
of conservation of mass and energy. The model can simulate the temporal evolution of the
subsurface properties and snow cover. To reproduce the interface between the atmosphere and
the ground, many parameters and processes, such as snow conditions, precipitations and
evaporation, are implemented. The model structure is a vertical profile composed of several
layers with heat and water exchange in-between. Even if the model was not built originally to
simulate permafrost conditions, two case studies were conducted successfully on Schilthorn
crest (BE, Switzerland) and Murtèl rock glacier (GR, Switzerland) by Scherler et al. (2013).
The results showed that the permafrost degradation would start around 20 years after present
day for Schilthorn and 60 years for Murtèl according to the mean of all the regional climate
models (RCM) used for this study. Another study was also conducted with the CoupModel at
Lapires talus slope (VS, Switzerland) to analyse the variation in surface and near-surface
temperature due to the chimney effect (Staub et al., 2015).
Another alternative to simulate the permafrost substrate is the Geophysical measurements.
With this approach, it is possible to quantify the volumetric fraction of ice, water and air in
permafrost substrates using relations between the measured geophysical data and the physical
properties of the soil. The so-called 4-phase model (4PM) was developed in that sense (Hauck
et al., 2008a; Hauck et al., 2011). This Matlab-based model uses the second Archie’s law and
Timur’s equation to relate respectively the measured electrical resistivity and seismic velocity
with electrical and seismic properties of the different media. In the contrary of the
CoupModel, the 4PM allows for a 2D representation of the soil ice, water and air content, but
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the dynamic of the subsurface is not included in the model. Case studies using this
methodology were already conducted on Schilthorn crest (VS, Switzerland), on Murtèl rock
glacier (GR, Switzerland) and on Stockhorn rock plateau (VS, Switzerland) (Hilbich, 2009;
Schneider et al., 2013; Dängeli, 2013). Some potential calibration biases are present in the
4PM versions used in the previous studies. Many site- and material- specific properties and
parameters such as the porosity, the pore water resistivity and the Archie’s parameters m, n
have to be known exactly to run the model. Besides, the model calibration needs to be done
directly inside the Matlab code, which requires a certain programming knowledge. In
addition, Archie’s law does not include the resistivity of rock. Thus, the presence of
inhomogeneous conductive features, as it is the case on Stockhorn, gives inconsistent 4PM
results.
Research questions and structure of the thesis
Starting from this point, the main aim of this thesis is to develop new calibration methods for
the 4PM in order to transform this model into a robust, rapid and efficient tool for mountain
permafrost study. A graphical user interface (GUI), new resistivity equations and a more
complete porosity setting are implemented to reduce the calibration uncertainties. To check
the validity of these new calibration methods, two secondary objectives are defined. One of
them concerns the comparison of the 4PM with the CoupModel. The other one focuses on the
application of the 4PM to assess the spatial and temporal repartition of ice and water content
on the Stockhorn plateau. These aims lead to the following research questions:
How can the 4-phase model be improved to better assess the ice, water and air content
of the subsurface?
In what way may the 4-phase model be compared with the CoupModel?
Which processes influence the spatial and temporal repartition of the ice, water and air
content on Stockhorn plateau?
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The structure of this thesis follows the process to obtain the expected results. The chapter two
introduces the study site Stockhorn (VS), where a meteorological station and two boreholes
have been installed by the EU-PACE project (Harris et al., 2001) and maintained by
PERMOS (2013) to provide necessary data for this study. In the third chapter, the basic
principles of the 4PM and the CoupModel are developed and explained, along with a
description of the 4PM improvements. The next chapter focuses on the calibration of the
CoupModel and its utilisation as a comparative method for the 4PM. In addition, the
methodology is applied within a case study Stockhorn, where the 4PM and CoupModel
results are analysed to know which parameters influence the most the spatial and temporal
repartition of the ice, water and air content in permafrost substrates. The study is completed
by a discussion of the calibration uncertainties and a conclusion.
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2. Study site
As mentioned earlier, the study site selected to test the 4PM and to illustrate some examples
of its use is the PERMOS-site Stockhorn, a rock-plateau located on a crest between the
Gornergrat and the Stockhorn summit (at 3532m altitude), near Zermatt (VS, Switzerland).
With an altitude of 3410m, low air temperatures and a relatively easy access by ski lift in
winter and by the Gornergrat Bahn in summer, the plateau is an appropriate site to study long-
term permafrost conditions in the Swiss Alps. Within the context of the “Permafrost and
Climate in Europe” (PACE) project, two boreholes were installed at Stockhorn with thermal
sensors for long-term permafrost monitoring (Gruber et al, 2004; Hilbich, 2009). The first
deep borehole of 100m depth (B100m) is located north of the plateau. The second borehole of
31m depth (B17m) is located more to the south but the thermistor chain is only 17m long. In
this chapter, the surface characteristics and the geology of the site are quickly described.
Then, the available meteorological and geophysical data are presented.
2.1. Surface characteristics and geology
The Stockhorn plateau leans by 8 degrees to the south. It is surrounded to the north by a cliff
and a small glacier, to the south by a three meters cliff and then a steep slope of
approximately 25 degrees, to the east by the Stockhorn summit and to the west by a ski lift
terminus station and a crest. Figure 3 shows the topographic setting of the site and the two
boreholes.
Figure 3: North-South profile of the study site. The two boreholes are represented in black (Gruber et al. 2004)
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Regarding the lithology, the plateau is mainly composed of Albit-Muskovit schists (Gruber et
al., 2004) and is covered with fine debris, whereas the southern slope is covered with coarser
debris. Borehole temperature time series show the presence of permafrost with an active layer
of approximately 3-4m (PERMOS, 2013). In summer, snow patches may still cover the
northern part of the plateau and ice may be visible sometimes at the surface. Figure 4 and
Figure 5 illustrate the situation on Stockhorn plateau in August 2014.
Figure 4: Situation on the northern part of Stockhorn plateau in August 2014.
Figure 5: Situation on the southern part of Stockhorn plateau in August 2014.
B100m
Meteo
station
Cross New
Profile (CN)
B17m
Soil
moisture
station Longitudinal
South Profile
(LS)
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2.2. Meteorological data
In June 2002, a climatic station was installed at Stockhorn to measure the air temperature, the
snow height, the wind speed and direction, the incoming and outgoing radiation (longwave
and shortwave) and the relative humidity (Gruber et al., 2004). As for many PERMOS sites,
there are temporal data gaps in the various time series due to technical instrument failure at
these high altitudes. A first data reconstruction was made by Berthod (2012). For her work,
she used data from two close climatic stations (Zermatt and Gornergrat) to complete the
temporal data gaps for the temperature, the wind speed and direction, the relative humidity
and the incoming radiations measured at Stockhorn climatic station. In the framework of the
TEMPS project, data have been further reconstructed using an analysis of correlation in daily
temperature and precipitation series (Rajczak et al., 2015). It was then possible to extend the
period of the time series from 1982 to present day. For this work, this second reconstruction is
used. Figure 6 shows the reconstructed air temperature time series for Stockhorn. Concerning
snow height data, their reconstruction is very difficult because of their very high spatial
variability. Thus, they have not been used for this work.
Figure 6: Reconstructed air temperature in the framework of the TEMPS project at Stockhorn, VS. The actual
measurements started only in 2002. Even if it is difficult to see any trend for the past 30 years due to the annual
amplitude of approximately 20°C, higher mean annual air temperatures are expected for the last 10 years.
The mean annual air temperature of this 30-year time series at Stockhorn is -5.5°C, with a
variation of the minimum temperatures between -20°C and -10°C in winter and the maximum
temperatures between 0°C and 5°C in summer. Negative values may also be observed for
most of the year. With an annual amplitude of approximately 20°C, it is difficult to see any
trend for the past 30 years. However, higher mean annual air temperatures are expected for
the last 10 years.
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2.3. Geophysical profiles
To assess the repartition of ice and water contents at Stockhorn, four measurement profiles
were conducted during a field campaign in August 2014, in addition to the monitoring profile
already in place (see Figure 7 and Table 1). The complete protocol of the campaign is
presented in appendix 1.
Figure 7: Geophysical profiles of the 2014 campaign. The 3400m isoline represents approximately the surface of the
plateau (modified after Swisstopo, 2015). The longitudinal profile of Dängeli (2013) is located on LN and it continues
further to the east. The bedrock at the surface indicated at the southern extremity of the cross profiles on the map is
probably a small cartography error, as coarse debris and big blocks were present instead in august 2014.
Table 1: Measurement information for all the campaign 2014 profiles
Geoelectric
measurements
Number of
Electrodes
Elect.
Spacing
Seismic
Measur.
Number of
Geophones
Geophones
Spacing
Relative
Position
CM
Wenner (27.08.2014,
end at 12:00)
Schlumberger (28.08.214)
55 2.0m
27.08.2014
(18:15 –
19:45)
24 4.0m
Geophone 1
at
Electrode 9
CN Wenner – Schlumberger
(28.08.2014, end at 10:30) 43 2.0m
28.08.2014
(12:30 –
13:30)
24 4.0m
Geophone 1
at
Electrode 2
LN Wenner – Schlumberger
(29.08.2014, end at 11:30) 48 1.5m
29.08.2014
(12:00 –
13:00)
24 3.0m
Geophone 1
at
Electrode 1
LM Wenner – Schlumberger
(29.08.2014, end at 14:00) 24 1.5m Not done – – –
LS Wenner – Schlumberger
(28.08.2014, end at 16:30) 48 1.5m
28.08.2014
(17:30 –
18:30)
24 3.0m
Geophone 1
at
Electrode 1
~ 11 ~
Cross profiles (CM and CN)
A monitoring profile was installed in summer 2005 near the two boreholes (PERMOS, 2013).
Approximately half of the electrodes are located on the plateau, and the rest is located along
the southern slope. Since 2007, seven more electrodes are present in the north facing rock
face, for a total of 55 electrodes. During the 2014 campaign, a refraction seismic measurement
was conducted to allow for the utilisation of the 4PM. To match the seismic profile, the first 8
electrodes of the northern slope are not used for the inversion process of the geoelectric data.
A parallel profile to the existing monitoring profile was installed during the 2014 campaign at
exactly 5m distance in the direction of the boreholes in order to have a closer look at the area
of the boreholes. All the profiles except CM were measured with the Syscal Junior device,
from Iris instruments. The proximity of the bedrock at the surface and the presence of bigger
blocks made the installation very difficult. The five last electrodes were not used as the
connection was impossible even after many attempts were made.
Longitudinal profiles (LN, LS and LM)
Three other profiles were measured on the plateau in 2014, perpendicular to the monitoring
one, in order to assess the spatial repartition of ice, water and air. All the profiles were on
finer debris except for the beginning of LN and LS that was on the apparent bedrock.
Soil moisture data
In addition to the geophysical data, soil moisture measurements were also performed along
the profiles during the 2014 campaign. The instrument selected for these measurements,
ThetaProbe ML2x, uses the variation in the apparent dielectric constant to calculate the water
content from DC voltage measurements (ΔT, 1999). With this method, the probe may have an
error of approximately ± 5% (ΔT, 1999). In the Framework of the SNF Project
SOMOMOUNT (2012), a permanent soil moisture station has also been installed with data
taken every 30 minutes since the 27 August 2014. Only one of the seven SMT100 sensors
available on the site (SOMOMOUNT, 2012) is used for this work to observe water content
evolution at 50cm with an accuracy of ± 3% (TRUEBNER, 2012).
~ 12 ~
~ 13 ~
3. Methods
To answer the research questions, different interconnected methods are needed. In the first
part of this chapter, the Electrical Resistivity Tomography (ERT) is presented. The basic
concept of resistivity surveying is developed, followed by the processing of geoelectric data.
The second part focuses on the Refraction Seismic Tomography (RST). The refraction
seismology is explained, as well as the inversion process. The third part of this chapter deals
with the 4PM and its new features. As shown in the Figure 8, the two first methods are needed
to obtain the 4PM input data. Finally, the 1-Dimentional physically based CoupModel is
introduced in the fourth part of this chapter. This model serves as a comparison with the 4PM
and it adds dynamic information on the permafrost substrate content and temperature.
Figure 8: Schematic figure of the interaction between all the methods. The ERT and RST are needed to obtain the
4PM input data. The CoupModel serves as a comparison with the 4PM and it adds dynamic information on the
permafrost substrate content and temperature.
~ 14 ~
3.1. Electrical Resistivity Tomography
One possible method to assess the underground composition is the Electrical Resistivity
Tomography (ERT). Geoelectric methods use Ohm’s law to put in relation the resistivity of
the subsurface, the tension and the current:
𝜌 = ∆𝑉
𝐼∗
𝐴
𝐿 (1)
𝜌 = resistivity of the material (Ωm)
∆𝑉 = difference of potential between two electrodes (volt)
𝐼 = current flowing between two electrodes (ampere)
𝐴 = section of the material (m2)
𝐿 = length of the material (m)
With equation 1, it is possible to determine the resistivity of the subsurface by generating a
direct current (DC) between two pairs of electrodes and by measuring the resulting tension.
Figure 9 shows the current path. Using four electrodes instead of two allows for the reduction
of the polarization effects, due to current injection, at the electrodes where the potential
difference is measured (Musset and Khan, 2009: 185).
Figure 9: Current path between electrodes (Mussett and Khan, 2009: 185)
When the material structure is complex and three-dimensional, the fraction A/L is replaced by
a geometric factor 𝑘 depending on how the four electrodes are arranged (Hauck and Kneisel,
2008: 4).
𝜌 = 𝑘 ∗∆𝑉
𝐼 (2)
Equation 2 gives the apparent resistivity of the soil. If the ground is homogeneous, this
apparent resistivity also corresponds to the specific resistivity. But if the subsurface is
heterogeneous, an inversion process must be performed to obtain the specific resistivity. It is
then possible to assess the nature of the subsurface as every material has a different resistivity
(See Table 2).
~ 15 ~
Table 2: Range of resistivity for different materials (after Hauck and Kneisel, 2008: 5)
Material Range of resistivity (Ωm)
Rock
Clay 1 – 100
Sand 100 – 5’000
Gravel 100 – 400
Granite 5’000 – 1’000’000
Gneiss 100 – 1’000
Schist 100 – 10’000
Water Groundwater 10 – 300
Ice Glacier ice 1’000’000 – 100’000’000
Mountain permafrost, ground ice 1’000 – 1’000’000
Air Air ∞
Depending on the measurement purpose and the soil structure at the site, different electrode
configurations or electrode arrangements may be used. For this study, the Wenner and
Wenner-Schlumberger array types were selected (see Figure 10). The Wenner configuration is
one of the fastest, but the investigation depth is moderate and the resolution is only good for
horizontal structures (Hauck and Kneisel, 2008: 7). The Wenner-Schlumberger array is an
interesting compromise between Wenner and Dipole-Dipole with a moderate measurement
time and a good resolution for horizontal and vertical structure (Hauck and Kneisel, 2008: 7).
A detailed description of the electrode configurations is available in different books and
papers (Loke et al., 2013; Hauck and Kneisel, 2008; Mussett and Khan, 2009).
Figure 10: Illustration of electrodes array types. A) The Wenner configuration uses electrodes C1 and C2 as current
electrodes and the potential difference is measured at the two electrodes P1 and P2. The distance is the same between
all the electrodes B) The Wenner Schlumberger configuration uses the same principle but the distance between
electrodes C and P can vary by a factor n (Hauck and Kneisel, 2008; Loke et al., 2013).
~ 16 ~
Geoelectric data processing in Res2Dinv
As mentioned earlier, the geoelectric measurements represent the apparent resistivity of the
substrates. But if the subsurface is heterogeneous, data must be processed to obtain a 2D
profile of the specific resistivity. This data processing, called inversion routine, is performed
for this study with the Res2Dinv software version 3.59 (GEOTOMO, 2010). For this inversion
routine, a first model of specific resistivity is built. Then, a simulated repartition of apparent
resistivity is calculated from this model and compared to the measured resistivity (Loke et al.,
2013). The aim of the inversion routine is to repeat this comparison with several models in
order to reduce the difference between the calculated and the measured resistivity
(GEOTOMO, 2010). This difference is represented mathematically by the root-mean-squared
error (RMSE). The inversion algorithm of the Res2Dinv software, based on the smoothness
constrained least-squares method, is described in detail by Loke et al. (2013).
For this study, the steps to follow for the data processing with Res2Dinv software are based on
the tutorial by Hilbich & Barandum (2013). First, the data points that have inconsistent
resistivity values must be deleted. A resistivity value is considered as inconsistent if it is
completely different from the value of the neighbouring data points with no similar pattern
visible in the adjacent layers (GEOTOMO, 2010). This selection is done with the Exterminate
bad datum points option of Res2Dinv. Then, some inversion options such as the resolution,
the damping and the smoothing of the specific resistivity model may be adjusted. The
software default parameters values are assumed to be a good starting point for an inversion
procedure (Hilbich & Barandum, 2013). For this study, some parameters, presented in Table
3, are varied within a defined range to find the best value corresponding to a small RMSE
between the calculated and the measured resistivity. However, large and unrealistic variations
may appear in the model resistivity values if the RMSE is too small. Thus, personal
observations of model structures are also required as criterion for the selection of the best
inversion. A complete description of all Res2Dinv options is available in the software tutorial
(GEOTOMO, 2010).
~ 17 ~
Table 3: Parameters tested in the ERT multiple inversion procedure. The last column represents the best value
according to the RMS error and personal observations of the resistivity model.
Parameter Minimum value
tested
Maximum value
tested
Value of Best
inversion
Initial Damping factor 0.04 0.36 0.12
Minimal Damping factor 0.010 0.050 0.015
Damping factor of the first layer 2.5 6.0 4.0
Layer thickness increase factor 1.05 1.50 1.05
Damping factor gradient 1.05 1.25 1.05
Data inversion robust constraint 0.01 0.10 0.05
Model inversion robust constraint 0.001 0.010 0.005
Extended model No Yes Yes
Model refinement (resolution) 1/2 electrode
spacing
1 electrode spacing 1/2 electrode
spacing
Mesh refinement None Finest Finer
~ 18 ~
3.2. Refraction Seismic Tomography
Another potential method to assess the underground composition is the Refraction Seismic
Tomography (RST). It uses the propagation of seismic waves into the soil to establish the
composition of different ground layers. For the refraction seismic surveys, detection
instruments called geophones are placed along the measurement line. Then, a seismic source,
usually a hammer shot, is activated at some points of the profile (called shot points) and the
resulting waves are registered by the geophones. The measurement is usually repeated several
times for each shot points (approximately 15 times during the field campaign 2014) in order
to reduce the effect of the surrounding noise and increase the quality of the results.
There are four main types of waves corresponding to different rock deformations: the
longitudinal wave (also called p-wave), the transversal wave (also called s-wave), the
Rayleigh wave and the love wave (Musset and Khan, 2009). As the rock deformation is not
the same for all the wave types, the velocity of these wave changes too. The p-waves are the
fastest. Thus, they are detected first by the geophones (Musset and Khan, 2009).
The waves can take different paths to the geophones. Some might arrive directly at the
geophone by taking the shortest path to the surface (in blue in Figure 11), while others might
penetrate underground. When the waves reach a boundary between two layers with different
densities, they may be reflected back to the surface or be refracted in the second layer and
continue their way until they reach the next layer boundary. One exception occurs when the
waves reach the layer boundary with a critical angle 𝜃𝑖𝑐. The latter depends on the ratio
between v1 (p-wave velocity in layer 1) and v2 (p-wave velocity in layer 2) in Snellius law
illustrated by the equation 3 (Mussett and Kahn, 2009: 66-67). In that case, the waves follow
their path along the boundary between two layers and produce secondary waves continuously
refracted to the surface (see red paths in Figure 11).
sin(𝜃𝑖𝑐) =𝑣1
𝑣2 (3)
Figure 11: Wave movement in two layers coming from a seismic source (i.e. a hammer shot). The critical distance (in
green) corresponds to the distance where a critical angle occurs. The cross over distance (in black) corresponds to the
distance where the refracted and direct waves arrive simultaneously (modified after Mussett and Khan, 2009: 67).
~ 19 ~
Consequently, the direct p-waves are the first to reach the geophones near the quake source.
But further from the shot point, the refracted p-waves are first. The critical distance and the
cross over distance, represented respectively in green and in black in Figure 11 and Figure 12,
illustrate this statement.
Figure 12: Travel-time diagram. The critical distance (in green) corresponds to the distance where a critical angle
occurs. The cross over distance (in black) corresponds to the distance where the refracted and direct waves arrive
simultaneously (modified after Mussett and Khan, 2009: 67).
With the information collected by the geophones, it is possible to assess the seismic velocity
along the profile with a certain depth (generally one third of the profile length) by doing an
inversion with appropriate software. As for the resistivity, the velocity is different for every
material and the latter may be deduced for the different layers. Table 4 shows some examples
of p-waves velocities.
Table 4: P-wave velocity in different materials (modified after Mussett and Kahn, 2009; Hauck and Kneisel, 2008).
Material Range of seismic velocity (m/s)
Rock
Clay 1’000 – 2’500
Sand, dry 200 – 1’000
Sand, saturated 1’500 – 2’000
Gravel 150 – 2’000
Talus deposit 550 – 2’500
Magmatic rock 2’400 – 5’100
Metamorphic rock 3’000 – 5’800
Water Groundwater 1’500
Ice Glacier ice 3’100 – 4’500
Permafrost 2’400 – 4’300
Air Air 330
~ 20 ~
Refraction seismic data processing in Reflexw
Before they can be used in the 4PM, seismic data must be processed. The inverted tomogram
was obtained with the Reflexw software (Sandmeier, 2014). First, the p-waves first arrival is
picked for each geophone and each shot point. The result of this process, called picking, is a
travel-time model through the profile. Then, the same inversion principle as for the ERT is
applied in Reflexw. A first model of seismic wave velocities is built. Then, travel-time of each
wave is calculated from this model and compared to the travel-time model created from the
picking. The aim of the inversion routine is to repeat this comparison with several models in
order to reduce the difference between the calculated and the measured travel-time
(Sandmeier, 2014). This difference is also represented mathematically by the root-mean-
squared error (RMSE).
For this study, the software default parameters values proposed by Hilbich et al. (2014) are a
good starting point for the inversion procedure. Then, some parameters such as the resolution,
the initial p-wave velocity and smoothing factors of the seismic model are varied within a
defined range to find the best value corresponding to a small RMSE between the calculated
and the measured travel-time (see Table 5 for the complete list of parameters). However, large
and unrealistic variations may appear in the model seismic velocity values if the RMSE is too
small. Thus, personal observations of model structures are also required as criterion for the
selection of the best inversion. A complete description of all Reflexw options is available in
the software tutorial (Sandmeier, 2014).
Table 5: Parameters tested in the RST multiple inversion process. The last column represents the best value according
to the RMS error and personal observations of the seismic model.
Parameter Minimum value
tested
Maximum value
tested
Value of Best
inversion
Space increment (resolution) (in
m)
0.5 1 0.5
Average smoothing 0 5 2
d(p-wave velocity)/dz (in 1/s) 100 320 250
p-wave initial velocity (in m/s) 300 900 800
Convergence search (in %) 1 20 5
Maximum change per iteration
(in %)
50 550 500
Data variance 0.005 0.050 0.010
~ 21 ~
3.3. Improved Four Phase Model (4PM)
The Four Phase Model is a Matlab-based model that allows for a combination of Electrical
Resistivity Tomography (ERT) and Refraction Seismic Tomography (RST) data to calculate
ice, water and air content of the subsurface (Hauck et al., 2008a; Hauck et al., 2011). One of
the main objectives of this work is to improve this model. First, the theory of the 4PM is
presented in part 3.3.1, followed by a presentation of the graphical user interface (GUI) in part
3.3.2. Then, the new features added in the 4PM version 7 are presented in part 3.3.3.
3.3.1. Theory behind the model
The original 4PM uses Archie’s law (1942) and Timur’s equation (1968) extended with a
term for air (Hauck et al., 2011).
The Timur’s equation put the p-wave velocity and the volumetric fraction in relation:
1
𝑣=
𝑓𝑤
𝑣𝑤+
𝑓𝑟
𝑣𝑟+
𝑓𝑖
𝑣𝑖+
𝑓𝑎
𝑣𝑎 (4)
𝑓𝑤 + 𝑓𝑟 + 𝑓𝑖 + 𝑓𝑎 = 1 (5)
With: 𝑓𝑤,𝑟,𝑖,𝑎 = volumetric fraction of water, rock, ice and air (in %)
𝑣𝑤,𝑟,𝑖,𝑎 = p-wave velocity of water, rock, ice and air (in m/s)
The Archie’s law is given by: 𝜌 = 𝑎 𝜌𝑤Φ−𝑚𝑆𝑤−𝑛 (6)
With: 𝜌 = bulk resistivity (in Ωm) 𝜌𝑤 = pore water resistivity (in Ωm)
Φ = Porosity (in %) 𝑆𝑤 = Saturation with water
m = cementation index n = saturation exponent a = Archie factor
These three equations use the principle that water, rock, ice and air have different resistivity
and p-wave velocities, as it has been shown in section 3.1 and 3.2. It is then possible to
deduce the fraction of air, water and ice in the ground by prescribing the porosity, the pore
water resistivity, the p-wave velocity in each material and Archie’s factors m, n and a
according to the site lithology and geomorphology (see chapter 4.2.2 for a parameterization
example).
~ 22 ~
As four unknowns are present in the equations 4, 5 and 6, the porosity Φ = 1 − fr must be
prescribed. Then, the fraction of air, water and ice can be calculated with 𝑠𝑤 = 𝑓𝑤 (1 − 𝑓𝑟)⁄ :
𝑓𝑤 = (𝑎 𝜌𝑤(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚 )1/𝑛
(7)
𝑓𝑖 =𝑣𝑖𝑣𝑎
𝑣𝑎−𝑣𝑖 [
1
𝑣−
𝑓𝑟
𝑣𝑟−
1−𝑓𝑟
𝑣𝑎+ (
𝑎 𝜌𝑤(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚 )1/𝑛
(1
𝑣𝑎−
1
𝑣𝑤)] (8)
𝑓𝑎 =𝑣𝑖𝑣𝑎
𝑣𝑖−𝑣𝑎 [
1
𝑣−
𝑓𝑟
𝑣𝑟+
1
𝑣𝑖(𝑓𝑟 − 1) − (
𝑎 𝜌𝑤(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚)
1/𝑛
(1
𝑣𝑤−
1
𝑣𝑖)] (9)
If the fraction of ice is assumed to be equal to zero, a 3-phase model (3PM) may be run. The
base equations for 3PM are the same as for the 4PM with 𝑓𝑖 = 0. Timur’s equation without
ice is now given by:
1
𝑣=
𝑓𝑤
𝑣𝑤+
𝑓𝑟
𝑣𝑟+
𝑓𝑎
𝑣𝑎 (10)
𝑓𝑤 + 𝑓𝑟 + 𝑓𝑎 = 1 (11)
As ice is not present in Archie’s law, the equation does not change. With one unknown, 𝑓𝑖, not
present anymore, the porosity (1 − 𝑓𝑟) does not need to be prescribed and a single equation
for fr can be isolated directly from the equations 6, 10 and 11:
− 1
𝑣 +
1
𝑣𝑎 −
𝑓𝑟
𝑣𝑎 +
𝑓𝑟
𝑣𝑟+
(1−𝑓𝑟)𝑛−𝑚
𝑛 (𝑎 𝜌𝑤
𝜌)
1/𝑛 (𝑣𝑎−𝑣𝑤)
𝑣𝑎 𝑣𝑤= 0 (12)
Because of the 𝑛−𝑚
𝑛 power for fr, this equation cannot be solved analytically. In consequence,
a numerical approximation must replace the analytical solution. The “Newton Method” is
used for this purpose (Rappaz and Picasso 2010, p125), which sues three steps:
1) The equation must be expressed as a function ℱ of the unknown (𝑓𝑟 in this case).
ℱ(𝑓𝑟) = − 1
𝑣 +
1
𝑣𝑎 −
𝑓𝑟
𝑣𝑎 +
𝑓𝑟
𝑣𝑟+
(1−𝑓𝑟)𝑛−𝑚
𝑛 (𝑎 𝜌𝑤
𝜌)
1/𝑛 (𝑣𝑎−𝑣𝑤)
𝑣𝑎 𝑣𝑤 (13)
2) Then the objective is to find the “zeros” of the function, i.e. the values of 𝑓𝑟 for which
ℱ(𝑓𝑟) = 0. To do so, an iterative approximation of 𝑓𝑟 is done using the newton
approximation formula.
𝑓𝑟,𝑘+1 = 𝑓𝑟,𝑘 −ℱ(𝑓𝑟,𝑘)
ℱ′(𝑓𝑟,𝑘) (14)
With 𝑓𝑟,𝑘 = approximated fraction of rock after k iterations
ℱ′(𝑓𝑟,𝑘) = derivative of ℱ(𝑓𝑟,𝑘)
~ 23 ~
3) This process may be stopped when there is no significant change between two
iterations. With any initial value 𝑓𝑟,0 between 0.1 and 0.9, this situation is reached
after the fourth iteration with an error of less than 0.0001% for 𝑓𝑟. The following
iterative equation for the fraction of rock, may then be deduced:
𝑓𝑟,𝑘+1 = 𝑓𝑟,𝑘 −−
1
𝑣 +
1
𝑣𝑎 −
𝑓𝑟,𝑘𝑣𝑎
+ 𝑓𝑟,𝑘
𝑣𝑟+
(1−𝑓𝑟,𝑘)
𝑛−𝑚𝑛 (
𝑎 𝜌𝑤𝜌
)1/𝑛
(𝑣𝑎−𝑣𝑤)
𝑣𝑎 𝑣𝑤
− 1
𝑣𝑎 +
1
𝑣𝑟 −
(1−𝑓𝑟,𝑘)
𝑛−𝑚𝑛
−1 (𝑛−𝑚) (
𝑎 𝜌𝑤𝜌
)1/𝑛
(𝑣𝑎−𝑣𝑤)
𝑛 𝑣𝑎 𝑣𝑤
(15)
After having obtained an approximation of 𝑓𝑟 with an error of less than 0.0001%, the fraction
of water 𝑓𝑤 and air 𝑓𝑎 may be obtained as follows:
𝑓𝑤 = (𝑎 𝜌𝑤(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚)
1/𝑛
(16)
𝑓𝑎 = 𝑣𝑎 (1
𝑣 −
𝑓𝑤
𝑣𝑤−
𝑓𝑟
𝑣𝑟) (17)
Archie’s law with Brandt rock resistivity factor
In the new 4-phase model version 7, two other resistivity equations are implemented in
addition to Archie’s law: Somerton (1992) random model and Archie’s law including a factor
for rock resistivity (after I. Brandt, TU Denmark; Sen et al., 1988). One inconvenient of
Archie’s law is the absence of the resistivity of rock, air and ice in the formula. This is
explained by the fact that these media have a much higher resistivity than the pore water.
Thus, they may be neglected as it is done in Archie’s law. In case of areas where the profile
has a very conductive rock matrix, this simplification might not be valid anymore. The new
equation with Brandt rock resistivity factor considers the fact that there may be
inhomogeneous features in the permafrost, giving inconsistent 4PM results. Some
improvements have been suggested to add the effect of clay content on the resistivity (Sen et
al., 1988), i.e.:
𝜌 = 𝑎 Φ−𝑚𝑆𝑤
−𝑛
𝜎𝑤(𝜎𝑤+𝐶𝑄𝑣+𝐴𝑄𝑣
𝜎𝑤+𝐶𝑄𝑣) (18)
With: 𝜎 = bulk conductivity (in 1/Ωm) 𝜎𝑤 = conductivity of pore water (in 1/Ωm)
𝑄𝑣 = function of cation exchange capacity (in 𝑚𝑜𝑙 𝑙⁄
1 𝑚⁄)
A = 3.8 𝑆 𝑚⁄
𝑚𝑜𝑙 𝑙⁄ 𝐶 = 0.7/𝑄𝑣
~ 24 ~
As clay is not always present at permafrost field sites, equation 18 has been reworked to
obtain a more general form:
𝜌 = 𝑎 Φ−𝑚𝑆𝑤
−𝑛
𝜎𝑤(𝜎𝑤+𝜀
𝜎𝑤)
= 𝑎 𝜌𝑤
1+𝜀 𝜌𝑤Φ−𝑚𝑆𝑤
−𝑛 (19)
With: 𝜀 = rock resistivity factor (in 1/Ωm)
With the factor 𝜀, it is possible to include the effect of the rock resistivity for specific zones in
the profile. The fraction of air, water and ice can be calculated similarly to equations 7 to 9:
𝑓𝑤 = (𝑎
𝜌𝑤1+𝜀 𝜌𝑤
(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚 )
1/𝑛
(20)
𝑓𝑖 =𝑣𝑖𝑣𝑎
𝑣𝑎−𝑣𝑖 [
1
𝑣−
𝑓𝑟
𝑣𝑟−
1−𝑓𝑟
𝑣𝑎+ (
𝑎 𝜌𝑤
1+𝜀 𝜌𝑤(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚 )
1/𝑛
(1
𝑣𝑎−
1
𝑣𝑤)] (21)
𝑓𝑎 =𝑣𝑖𝑣𝑎
𝑣𝑖−𝑣𝑎 [
1
𝑣−
𝑓𝑟
𝑣𝑟+
1
𝑣𝑖(𝑓𝑟 − 1) − (
𝑎 𝜌𝑤
1+𝜀 𝜌𝑤(1−𝑓𝑟)𝑛
𝜌 (1−𝑓𝑟)𝑚 )
1/𝑛
(1
𝑣𝑤−
1
𝑣𝑖)] (22)
The solution for the 3PM is very similar to the formulation with classical Archie’s law (eq. 15
to 17) with the epsilon factor added to 𝜌𝑤:
𝜌𝑤 =𝜌𝑤,0
1+𝜀 𝜌𝑤,0 (23)
With: 𝜌𝑤,0 = initial value for the resistivity of pore water, as present in Archie’s law.
Somerton random model
The third resistivity equation implemented in the new 4PM is the so-called random model
(e.g. Somerton, 1992), based on the geometric mean of the different material volume
fractions. In the deviation of the approach, the volume fractions are considered to be
randomly distributed and arbitrary shaped (Glover, 2010). This random model is given by
(Somerton, 1992):
𝜌 = 𝜌𝑤𝑓𝑤 ∗ 𝜌𝑎
𝑓𝑎 ∗ 𝜌𝑟𝑓𝑟 ∗ 𝜌𝑖
𝑓𝑖 (24)
With: 𝜌 = bulk resistivity (in Ωm) 𝜌𝑎 = resistivity of air (in Ωm)
𝜌𝑟 = resistivity of rock (in Ωm) 𝜌𝑖 = resistivity of ice (in Ωm)
~ 25 ~
The cementation index m and the saturation exponent n of Archie’s law are no longer present,
but the resistivity of all the media present in the profile must be prescribed. With Timur’s
equation, the fraction of ice, air and water are now given as:
𝑓𝑖 =
1−𝑓𝑟−log(𝜌)
log(𝜌𝑤)+
𝑓𝑟 log(𝜌𝑟)
log(𝜌𝑤)+𝑣𝑎 [
log(𝜌𝑎)
log(𝜌𝑤)−1] [
1𝑣
−𝑓𝑟𝑣𝑟
−log(𝜌)
𝑣𝑤 log(𝜌𝑤)+
𝑓𝑟 log(𝜌𝑟) 𝑣𝑤 log(𝜌𝑤)
1−𝑣𝑎 log(𝜌𝑎) 𝑣𝑤 log(𝜌𝑤)
]
1−log(𝜌𝑖)
log(𝜌𝑤)−𝑣𝑎 [
log(𝜌𝑎)
log(𝜌𝑤)−1] [
𝑣𝑎 log(𝜌𝑖)
𝑣𝑤 log(𝜌𝑤)−
𝑣𝑎 𝑣𝑖
1−𝑣𝑎 log(𝜌𝑎) 𝑣𝑤 log(𝜌𝑤)
]
(25)
𝑓𝑎 =
𝑣𝑎𝑣
−𝑓𝑟 𝑣𝑎
𝑣𝑟−
𝑓𝑖 𝑣𝑎
𝑣𝑖−
𝑣𝑎𝑣𝑤 log(𝜌𝑤)
[log(𝜌)−𝑓𝑖 log(𝜌𝑖)−𝑓𝑟 log(𝜌𝑟)]
1− 𝑣𝑎 log(𝜌𝑎)
𝑣𝑤 log(𝜌𝑤)
(26)
𝑓𝑤 =log(𝜌)
log(𝜌𝑤)− 𝑓𝑖
log(𝜌𝑖)
log(𝜌𝑤)− 𝑓𝑟
log(𝜌𝑟)
log(𝜌𝑤)− 𝑓𝑎
log(𝜌𝑎)
log(𝜌𝑤) (27)
If no ice is present, the same principle as in the 3PM above is applied to the random model
with 𝑓𝑖 = 0:
𝜌 = 𝜌𝑤𝑓𝑤 ∗ 𝜌𝑎
𝑓𝑎 ∗ 𝜌𝑟𝑓𝑟 (28)
It gives the following solutions for the rock, air and water fraction:
𝑓𝑟 =
1−log(𝜌)
log(𝜌𝑤)−
𝑣𝑎
(1−𝑣𝑎 log(𝜌𝑎) 𝑣𝑤 log(𝜌𝑤)
) [
1
𝑣−
log(𝜌)
𝑣𝑤 log(𝜌𝑤)+
log(𝜌𝑎)
log(𝜌𝑤) 𝑣−
log(𝜌𝑎) log(𝜌)
log(𝜌𝑤) 𝑣𝑤 log(𝜌𝑤)]
𝑣𝑎
(1−𝑣𝑎 log(𝜌𝑎) 𝑣𝑤 log(𝜌𝑤)
) [
log(𝜌𝑟)
𝑣𝑤 log(𝜌𝑤)−
1
𝑣𝑟+
log(𝜌𝑎)
log(𝜌𝑤) 𝑣𝑟−
log(𝜌𝑎) log(𝜌𝑟)
log(𝜌𝑤) 𝑣𝑤 log(𝜌𝑤)]−
log(𝜌𝑟)
log(𝜌𝑤)
(29)
𝑓𝑎 =
𝑣𝑎𝑣
−𝑓𝑟 𝑣𝑎
𝑣𝑟−
𝑣𝑎𝑣𝑤 log(𝜌𝑤)
[log(𝜌)−𝑓𝑟 log(𝜌𝑟)]
1−𝑣𝑎 log(𝜌𝑎)
𝑣𝑤 log(𝜌𝑤)
(30)
𝑓𝑤 =log(𝜌)
log(𝜌𝑤)− 𝑓𝑟
log(𝜌𝑟)
log(𝜌𝑤)− 𝑓𝑎
log(𝜌𝑎)
log(𝜌𝑤) (31)
~ 26 ~
3.3.2. Integration of existing features into a Graphical user Interface
One of the main objectives of this work is to improve the 4PM. In the version 5.0, the
insertion of ERT and RST data as well as the calibration of the porosity, the seismic velocity
and Archie’s parameters needed to be done directly inside the code for each site, which
required certain knowledge about Matlab. Besides, only few options were available to
prescribe the porosity and to get values for Archie’s parameters. All the changes made since
version 5.0 are listed in appendix 2 and every calculation step implemented in the model is
commented directly into the code (see appendix 3 for some coding examples). The main
improvement of the 4-phase model version 7 is the creation of a graphical user interface
(GUI). This allows for the calibration of the model without changing directly the code. The
GUI has been built with GUIDE, a Matlab tool for buttons creation and management without
the writing of all the base code. When a new GUI feature is added, a corresponding function
is automatically created in the main Matlab file. Then, the code can be modified to integrate
the new button in the model. The files containing the 4PM are separated from the main one
where the buttons are managed to avoid any manipulation error. The 4PM version 7 has been
built with eight modules corresponding to the different parts of the program, as shown in
Figure 13 (see the tutorial in appendix 4 for more information about how to use and handle
the model).
Data Insertion
In this section, it is possible to enter the Site Code and a Date for the each profile. Both will
be part of title plots. Then, the data filenames with the extension (.xyz or .txt for ERT data
and .ASC for RST data) must be inserted (see tutorial in appendix 4). Finally, the RST and
ERT RMS error can be inserted to be displayed in the graphs. With the new model, it is not
necessary anymore to include a file for the topography. The latter is calculated automatically
by taking the top of the RST profile as reference for the topography.
~ 27 ~
Figure 13: Graphical user interface of 4-phase model 7
Matching of ERT and RST
As the ERT and RST profiles do not necessarily start at the same point, a matching of both
profiles on the same grid must be done. In the 4PM version 7, it is possible to perform an
auto-calibration for the matching of ERT and RST data. With this option, the model analyses
the input ERT and RST files to set the X shift and Z shift, considering that the first electrode is
at the same position than the first geophone. If it is not the case, a manual calibration is still
possible afterward. First, the ERT horizontal (X) and vertical (Z) coordinates can be inverted
if needed with Reverse X and Reverse Z check box. If the beginning of ERT data corresponds
to the end of RST data, the option Reverse X should be activated. If the depth values are
negative for ERT data and positive for RST data (or vice-versa), the option Reverse Z should
be activated. Besides, ERT data can be shifted in the 4PM to fit exactly the RST data with X
shift and Z shift.
~ 28 ~
Other improvements are added in addition to the GUI. In the 4PM version 5, the resolution is
restricted to one meter due to interpolation issues of the ERT data. In the new 4PM version 7,
it is now possible to use different resolutions calculated automatically according to the space
increment used in RST. Besides, empty columns present at the beginning or at the end of RST
file are automatically deleted if there is any.
Virtual Boreholes
In this section it is possible to insert virtual boreholes with a specific name, X position and
depth. The latter can then be displayed in the results and ice, water and air contents can be
extracted along a vertical profile. These features were already implemented in 4PM version
5.0. By consequence, they will not be explained further in this work.
3.3.3. New features of the version 7
In the 4PM version 7, other improvements have been added in addition to the GUI. It is now
possible to choose the resistivity equation, to create a complex porosity matrix, to calibrate
Archie’s parameters using soil moisture or ice core data and to compare two data sets for a
same profile.
Choice of the resistivity equation
In the 4PM 7, it is now possible to change the resistivity equation that will be part of the 4PM
model equations. Those three different electrical mixing rules are introduced in section 3.3.1.
If the option compare with the other equations is selected, the difference in ice, water and air
content between the equations will be calculated and displayed. The parameters for all the
equations must be setup before using this option.
~ 29 ~
Complex porosity model
In the 4PM, the porosity of the soil must be prescribed to solve the equations. However, the
pore content usually decreases with depth and the subsurface can be very heterogeneous at
some sites such as Stockhorn. It is then necessary to include the possibility to build a complex
porosity model in the 4PM.
As a first step, a main decreasing porosity must be defined for the whole profile. To do it, an
initial value (Phi ini between 0.1 and 1) is set for the surface. Then, the porosity decreases
with depth following the topography, as a function of a gradient (Phi Grad between 0 and 0.2
pixel-1
). When a defined threshold is reached (Phi Threshold between 0.05 and 1), the
gradient is divided by 10. In addition, the porosity cannot go under a specified minimum
value (Phi min between 0.005 and 0.5). This first step already existed in version 5 of the 4PM.
In the 4PM version 7, two optional approaches may be combined to better fit the
characteristics of the field site. First, a 3-phase model (3PM) can be run in those regions
where the ice fraction is assumed to be equal to zero in order to calibrate automatically the
porosity for these pixels. A restriction is implemented to save the porosity value only for the
area with less than a certain percentage of ice (Tolerance Threshold) according to the 4PM. If
the Auto-Calibration with 3PM option is selected, a porosity file is automatically created in
the Data folder. It is then possible to use the same porosity for a different date by entering the
porosity filename. With the Gradient Porosity with 3PM activated, the porosity continues to
decrease from the automatic calibration values instead of the main model.
With the second optional approach, it is possible to create small zones with a specific
prescribed porosity, based on the same principle than the main model. The Activated Trigger
allows for the activation (1) or deactivation (0) of a zone. The Structure Trigger defines if the
zone must follow the topography (1) or must stay horizontal (0). Xmin/Xmax and
DepthMin/DepthMax give the position and the size of the zone. The initial porosity (Phi_ini),
the gradient (Phi_grad) and the threshold (Threshold) are the same parameters as for the main
porosity model.
Finally, a smoothing parameter can be applied on the entire porosity matrix including the
optional approaches. The value of each pixel is an average of all surrounding pixels within a
specific range (1, 2, 3, 4 or 5), as illustrated in Figure 14. Even if the smoothing parameter can
technically go beyond 3, a higher value can strongly reduce or erase some porosity patterns
created by the optional approaches used to calibrate the porosity model. Thus, the smoothing
should be applied with care.
~ 30 ~
Figure 14: Example of smoothing area if the parameter is equal to 2.
Model parameters
In this section, the resistivities and p-wave velocities of the different media, as well as the
Archie parameters can be prescribed. It is also possible to choose which plots will be
displayed. Finally, some blanks due to the absence of consistent solutions for the equations
may be filled in with the fill gaps option. However, the results with this option activated must
be interpreted with caution. Besides, they should be compared with the results without the fill
gaps option. For Archie with epsilon and Somerton equations, it is possible to define a zonal
parameter to represent a variable rock resistivity. The zones are built identically to the
porosity zones. A minimum and maximum value is suggested for all the parameters (see
Table 6 below for the values). The range of resistivity and seismic velocity for different
materials corresponds to the values shown in Table 2 and Table 4. The range of value for Rw
is extended to consider extremes cases such as salty pore water in Svalbard permafrost
substrate, Norway. Different simulations were run to assess the range of Brandt’s epsilon
factor (see section 5.2.2.). Finally, the values for Archie’s parameters are suggested according
to Schön (2004).
Table 6: Parameter range implemented in the 4PM. The values for the resistivity and the wave velocity come from the
Table 2 and Table 4. The range of value for Rw is extended to consider extremes cases such as salty pore water in
Svalbard permafrost substrate, Norway. Different simulations were run to assess the range of Brandt’s epsilon factor
(see section 5.2.2.). Finally, the values for Archie’s parameters are suggested according to Schön (2004).
Parameter Short description Minimum Maximum Default value
Vw P-wave velocity in water (m/s) 1’000 2’000 1500
Va P-wave velocity in air (m/s) 300 350 330
Vi P-wave velocity in ice (m/s) 2’000 5’000 3500
Vr P-wave velocity in rock (m/s) 100 7’500 6000
Rw Resistivity of pore water (Ωm) 0.1 2’000 100
Ra Resistivity of air (Ωm) 100’000 50’000’000 100’000
Ri Resistivity of ice (Ωm) 1’000 1’000’000 20’000
Rr Resistivity of rock (Ωm) 100 1’000’000 3’000
Epsilon Brandt factor for Rr (Ωm-1
) 0 1 0
n Archie factor n 1.0 10.0 2.0
a Archie factor a 0.5 2.0 1.0
m Archie factor m 1.0 3.0 2.0
~ 31 ~
Multi-run procedure to test Parameters set
The Archie parameter m and n can be determined for each site if the lithology and the profile
structure are known. But for some sites, it may be hard to assess these values, even if
information is available. Besides, the pore water resistivity must also be prescribed. In the
4PM version 7, it is possible to test different parameter sets to improve the model calibration
procedure. A 4PM (or 3PM if no ice is present) is run for parameter combinations defined by
the user and the result is compared to water or ice content measurements taken along or inside
the profile. A root square error (equation 32) is used to check the compatibility of both data:
𝐸𝑟𝑟𝑜𝑟 = √∑ (𝑓𝑐𝑎𝑙𝑐−𝑓𝑚𝑒𝑎𝑠)2𝑛
𝑖=1
𝑛 (32)
With: 𝑓𝑐𝑎𝑙𝑐 = calculated fraction of water or ice
𝑓𝑚𝑒𝑎𝑠 = measured fraction of water or ice
𝑛 = number of ice or water content points in the file
The best runs (i.e. those giving the smallest error between measured and calculated values)
are then plotted in histograms. The numbers of best runs plotted in histograms are defined by
the GUI parameter # of accepted runs. The multi-run procedure is discussed in more detail in
section 5.2.1.
Comparison with another date or data set:
In the new version of the model, it is possible to see the evolution of the ice, water and air
content by comparing two measurement dates or two data sets for a same profile. Such
comparisons have already been done in previous studies (Dängeli, 2013; Hauck et al., 2008a;
Hilbich, 2009). Nevertheless, it has never been implemented in a previous version of the
4PM. After running the model for both data sets, the difference in ice, water and air content is
plotted with a different colour scale. Positive changes are represented in blue and negative
ones in red.
~ 32 ~
3.4. CoupModel
One secondary aim of this master thesis is to compare 4PM water, air and ice contents with
the same outputs in the 1D-physically based CoupModel (Jansson, 2014). The latter model
allows for the description of the heat and water circulation into the soil using two coupled
differential equations. To simulate the interface between the atmosphere and the ground,
many parameters and processes, such as snow conditions, precipitation and evaporation, are
implemented. The equations are built with two physical assumptions (Jansson, 2014):
1) “The law of conservation of mass and energy”
2) “Flows occur as a result of gradients in water potential (Darcy’s Law) or temperature
(Fourier’s law)”.
As illustrated in Figure 15, the model structure is a vertical profile composed of several layers
with heat and water exchange in-between.
Figure 15: Diagram of CoupModel layers. a) The mass balance depends mainly on the evaporation, the precipitation,
the surface runoff and the ground water inflow and outflow. b) The conduction and the convection are the main
components of ground heat flow (Jansson et al., 2013a).
~ 33 ~
3.4.1. Main equations
Soil heat processes
The general heat flow equation used in the CoupModel is a combination of the conduction,
the convection and the energy conservation law (Jansson et al., 2013):
𝜕(𝐶𝑇)
𝜕𝑡− 𝐿𝑓𝜌
𝜕𝜃𝑖
𝜕𝑡=
𝜕
𝜕𝑧(𝑘
𝜕𝑇
𝜕𝑧) − 𝐶𝑤𝑇
𝜕𝑞𝑤
𝜕𝑧− 𝐿𝑣
𝜕𝑞𝑣
𝜕𝑧− 𝑠ℎ (33)
With: 𝐶 = heat capacity (J/ ° C) 𝑇 = soil temperature (° C)
𝐿𝑓 = freezing latent heat (J/kg) 𝜌 = density (kg/m3)
𝜃𝑖 = volumetric ice content (%) 𝑡 = time (sec)
𝑧 = depth (m) 𝑘 = conductivity (W/m/ ° C)
𝐶𝑤 = water heat capacity (J/ ° C) 𝑞𝑤/𝑞𝑣 = water flux / vapour flux (1/m/day)
𝐿𝑣 = vapour latent heat (J/kg) 𝑠ℎ = soil heat pump (source/sink term) (J/m2/day)
The lower and upper boundary conditions may then be deduced from the energy conservation
law, as well as the soil surface temperature and the influence of the snow cover. See Jansson
et al. (2013) for a more detailed description of the various processes.
Soil water processes
The general water flow equation used in the CoupModel is a combination of the water flow, a
source term and the mass conservation law (Jansson and Karlberg, 2013):
𝜕𝜃
𝜕𝑡= −
𝜕
𝜕𝑧[−𝑘𝑤 (
𝜕𝜓
𝜕𝑧− 1) − 𝐷𝑣
𝜕𝑐𝑣
𝜕𝑧+ 𝑞𝑏𝑦𝑝𝑎𝑠𝑠] + 𝑠𝑤 (34)
With: 𝜃 = water content (%) 𝑐𝑣 = vapour concentration (kg/m3)
𝜓 = water tension (cm) 𝑠𝑤 = source term (mm/day)
𝑘𝑤 = water conductivity (mm/day) 𝐷𝑣 = vapour diffusion coefficient (m5/kg/day)
𝑞𝑏𝑦𝑝𝑎𝑠𝑠 = bypass flow in macro pore (flow that traverses some layers) (mm/day)
The hydraulic conductivity function is the main parameter of the water flow between the
layers.
~ 34 ~
Soil resistivity
The soil resistivity is calculated using Archie’s second law as it is implemented in the 4PM
(see equation 6). But in the CoupModel, the water resistivity 𝜌𝑤 can vary in addition with
temperature and salt concentration:
𝜌𝑤 =1000
10 𝑆𝑎𝑙𝑡𝐶𝑜𝑛𝑐
𝑅𝑒𝑠𝐼𝑜𝑛(1+𝑅𝑒𝑠𝑆𝑒𝑛𝑠𝑇(𝑇−𝑅𝑒𝑠𝑁𝑜𝑟𝑚𝑇))
(35)
With: 𝑆𝑎𝑙𝑡𝐶𝑜𝑛𝑐 = salt concentration in the layer (in mg/l)
𝑅𝑒𝑠𝐼𝑜𝑛 = coefficient for ions resistivity (in g/mol)
𝑅𝑒𝑠𝑆𝑒𝑛𝑠𝑇 = coefficient for the sensitivity of resistivity to temperature (in °C-1
)
𝑅𝑒𝑠𝑁𝑜𝑟𝑚𝑇= reference temperature for the resistivity (in °C)
Even if the soil resistivity implemented in the CoupModel has been already used successfully
(Lundmark and Jansson, 2008), it has not been tested in permafrost conditions yet. As the
pore water resistivity is constant in the 4PM, the sensitivity factor to the temperature is set to
0 in the CoupModel.
3.4.2. CoupModel calibration
For this work, the CoupModel is calibrated with the borehole data from Stockhorn and
specific resistivity values from the ERT inversion models. Reconstructed data from the
meteorological station are taken as input. As the CoupModel is quite complex and its
calibration can be quite time consuming, a pre-calibrated simulation by Antoine Marmy
(University of Fribourg, TEMPS project), is taken as starting point for this work. It includes,
amongst others, values for slope, latitude and some known soil and snow properties (Marmy
et al. 2015).
Multi-Run procedure
If some calibration parameters are unknown, it is possible to select a range of values to test
for each parameter. In the CoupModel, there are two possibilities for this: The bayesian
calibration or the General Likelihood Uncertainty Estimation (GLUE) method (Jansson and
Kasmaei, 2013).
~ 35 ~
The Bayesian calibration is based on Markov Chain Monte Carlo (MCMC) algorithms. This
approach tries to find the best run by conducting a random walk to reach the region of highest
probability (Juston, 2010).
The GLUE calibration, a random sampling of the parameter values, is based on the
equifinality principle (Beven, 2006). It is defined as an “inability to meaningfully distinguish
one single best parameter set given inherent uncertainties and errors in available data and
model structures” (Juston, 2010:4). In other terms, several parameters sets may lead to the
same results and residuals. The best run must then be selected according the user experience.
As the uncertainty for the measurements and the model structure cannot be easily evaluated, a
simple GLUE calibration is used for this work (Jansson and Kasmaei, 2013). This method is
described in more detail by Marmy et al. (2015).
For the multi-run procedure, Marmy et al. (2015) selected the soil and snow parameters that
are influencing the most the permafrost temperature. For this study, the water content and the
resistivity should also match between the measured and simulated values. Thus, some other
parameters influencing the most these new features were added. The best simulation was
selected amongst 46’000 runs for the B100m and 48’000 runs for the B17m. The Table 7
shows all the multi-run calibration values. Those values are analysed in detail in section 4.3.1.
To select the best run, two statistical indicators were used. The first is the coefficient of
determination for the linear regression equation R2, calculated with equation 36. It represents
the variance of the data for all the time steps. In other terms, this factor calculates the
matching between the variation of the simulated and measured values with time. R2 varies
from 0, for no correlation at all, to 1 for a perfect match of the temporal variation.
𝑅2 =1
𝑛−1∗ ∑ (
[𝑂(𝑡)−𝑂] [𝑆(𝑡)−𝑆]
√∑[𝑂(𝑡)−𝑂]2
√∑[𝑆(𝑡)−𝑆]2)
2
(36)
With: 𝑂(𝑡) = Observed value at a specific time step
𝑂 = Mean of all the observed values
S(𝑡) = Simulated value at a specific time step
𝑆 = Mean of all the simulated values
n = Number of observations
~ 36 ~
The second statistical indicator is the Root Mean Square Error (RMSE, eq. 37). It represents
the error between simulated and observed values. It is more important at larger depths where
the annual variation is less pronounced.
𝑅𝑀𝑆𝐸 = √𝑂(𝑡)−𝑆(𝑡)
𝑛 (37)
Table 7: CoupModel calibration values. These parameters were selected because of their significant influence on the
temperature, water content and resistivity.
Parameter (from Antoine Marmy)
Description (Jansson and
Karlberg, 2013) Min Max
Best Run
B100m
Best Run
B17m
SnowPack
CritDepthSnowCover
“Thickness of mean snow height
that corresponds to a complete
cover of the soil” (m)
0.1 1.0 0.70 0.66
SnowPack
SThermalCondCoef
“Thermal conductivity
coefficient for snow” (W m5
°C
-1 kg
-2)
1x10-7
3x10-6
1.34x10-7
1.31x10-7
SnowPack
OnlyRainPrecTemp
“Above this temperature all
precipitation is rain.” (°C) 0.1 4 1.29 1.20
SnowPack
OnlySnowPrecTemp
“Below this temperature all
precipitation is snow.” (°C) -4 0 -2.07 -1.78
SnowPack
DensityOfNewSnow
“Density of new snow”
(Kg/m3)
60 200 111 63
SnowPack
MeltCoefAirTemp
“Temperature coefficient in
the empirical snow melt
function” (Kg °C-1
m-2
day-1
)
0.5 4 1.18 2.24
SnowPack
MeltCoefGlobRad
“Global radiation coefficient in
the empirical snow melt
function” (Kg/J)
0 3x10-6
1.85x10-6
2.21x10-6
Radiation properties
AlbedoWet “Albedo of a wet soil” (%) 5 25 8.1 16.8
Radiation properties
AlbedoDry “Albedo of a dry soil” (%) 10 40 12.0 13.6
Soil evaporation
EquilAdjustPsi
“Factor to account for
differences between water
tension in the middle of top
layer and actual vapour
pressure at soil surface.”
0 1 0.39 0.40
Soil Hydraulic
Hydraulic conductivity
(0-2m)
Matrix and total hydraulic
conductivity (mm/day) 100 10
5 21’474 4’785
Soil Hydraulic
Hydraulic conductivity
(2-5m)
Matrix and total hydraulic
conductivity (mm/day) 10 10
4 5’966 939
Soil Hydraulic
Hydraulic conductivity
(5-100m)
Matrix and total hydraulic
conductivity (mm/day) 1 10
3 861 40
Soil Hydraulic
m-value (0-2m)
factor for the van Genuchten
water retention function 0.1 2 1.56 0.17
Soil Hydraulic
m-value (2-5m)
factor for the van Genuchten
water retention function 0.1 2 0.99 1.72
Soil Hydraulic
m-value (5-100m)
factor for the van Genuchten
water retention function 0.1 2 1.46 0.63
~ 37 ~
Parameter (added for this thesis)
Description (Jansson and
Karlberg, 2013) Min Max
Best Run
B100m
Best Run
B17m
Salt Tracer
Archie M Archie’s parameter m 1.5 2.5 2.41
(kept as
calibrated
for B100m)
Salt Tracer
Archie N Archie’s parameter n 1.5 2.5 1.67
(kept as
calibrated
for B100m)
Salt Tracer
Water resistivity
(indirectly through Salt
Concentration) (Ωm)
Calculated Water resistivity
(Ωm) with 𝑅𝑒𝑠𝐼𝑜𝑛 = 25 and
𝑅𝑒𝑠𝑆𝑒𝑛𝑠𝑇 = 0
50 500 95
(kept as
calibrated
for B100m)
Drainage and deep
percolation
GWSourceFlow
Constant rate of water source
(at the first layer) (mm/day) 0 0.5 0.45 0.39
Soil Hydraulic
Residual Water (0-2m) Residual water content (%) 2 8 3.84 3.63
Soil Hydraulic
Residual Water (2-5m) Residual water content (%) 3 9 6.89 5.60
Soil Hydraulic
Residual Water (5-
100m) Residual water content (%) 4 10 7.71 7.84
Soil Hydraulic
Saturation (0-2m) (%) Porosity of the soil (%) 40 50 49.9 40.2
Soil Hydraulic
Saturation (2-5m) (%) Porosity of the soil (%) 25 40 36.0 25.8
Soil Hydraulic
Saturation (5-100m) (%) Porosity of the soil (%) 10 25 15.4 20.4
~ 38 ~
~ 39 ~
4. Results and interpretation
Now that the context and the methods of this study are defined, the results may be presented.
As mentioned in the previous chapter, the 4PM uses geophysical methods to simulate a 2D
one-time repartition of ice, water and air in permafrost substrate. On the other hand, the
CoupModel uses meteorological and borehole data to simulate a 1D temporal repartition of
ice, water and temperatures in permafrost substrate. Thus, the aim of this chapter is to
combine the results of the 4PM and the CoupModel to obtain a spatial and temporal
repartition of ice, water and temperatures in permafrost substrate. Both models have inherent
uncertainties, mostly regarding the calibration of the unknown free parameters, such as the
porosity (see chapter 5 for the analysis of these uncertainties). Using two models with
different calibration procedures may then help to assess the value of those parameters (see
section 3.3.3 and 3.4.2).
The first part of this chapter concerns the comparison between the 4PM and the CoupModel.
An analysis of the CoupModel calibration will be exposed, followed by a comparison
between both models. The second part focuses on the application of the 4PM at Stockhorn.
Interpretation examples of ERT and RST are described shortly. Then, the model calibration is
introduced before the presentation of 4PM results for each profile. The last part of this chapter
concerns the interpretation of the results. The 4PM and CoupModel results are combined to
assess the spatial and temporal repartition of ice and water at Stockhorn.
As mentioned earlier, two boreholes, B100m and B17m, are installed at Stockhorn. Even if
they are only 28m apart, their respective thermal regimes are very different (see Figure 16 and
Figure 17) and they should be analysed in more detail. The annual amplitude of the
temperatures is higher at B100m than at B17m. Thus, the decoupling of the soil temperature
from the atmosphere is more pronounced at B17m. The active layer thickness (ALT) is
usually between 3.0m and 3.5m at B100m, between 4.0m and 4.5m at B17m and it is reached
in October. Around 9m depth, the annual temperature fluctuations are reduced with a mean
value of -2.3°C at B100m and -1.0°C at B17m. The temperature differences raised in this
paragraph are discussed in more detail in chapter 4.3.
~ 40 ~
Figure 16: Measured temperatures at the borehole 6000 (B100m) between 2002 and 2015. The active layer thickness
(ALT) is usually between 3.0m and 3.5m and it is reached around the beginning of October.
Figure 17: Measured temperatures at the borehole 6100 (B17m) between 2002 and 2015. The active layer thickness
(ALT) is usually between 4.0m and 4.5m and it is reached around middle of October.
~ 41 ~
4.1. Comparison of the 4PM with the CoupModel
The CoupModel of Per-Erik Jansson (2014), selected for the comparison with the 4PM, has
shown to be well suited for the representation of permafrost conditions in the Alps
(Engelhardt et al., 2010; Scherler et al., 2013) and for the use of soil resistivity (Lundmark
and Jansson, 2008). Besides, this model uses also Archie’s law as it is the case in the 4PM.
Thus, the water resistivity and the factors m and n may be defined identically in both models
to reduce the potential calibration bias. The 1-dimensionnal vertical CoupModel is applied at
the two boreholes present along the Stockhorn monitoring profile. The aim of this validation
is to see if the 4PM, a 2D simple model with 3 equations, can provide a similar assessment of
the pore water and ice contents as a complex 1D physically based model. First, the
CoupModel simulation is discussed for the boreholes 6000 (B100m) and 6100 (B17m) to see
the quality of the calibration with the resistivity. Then, the results are compared to a five
pixels horizontal mean of the 4PM at the location of the two boreholes.
Three vertical layers, corresponding to the position of thermistors in the boreholes, are
selected to calibrate the CoupModel and to check the matching of the CoupModel simulation
and the temperature measurement in the boreholes:
- The first layer is situated near the surface between 0.5m and 1.1m. The effect of the air
temperature can clearly be observed at this depth.
- The second, between 2.1m and 4.5m, is at the permafrost table. At Stockhorn, the
ALT varies between 3.5m and 4.5m (PERMOS, 2013). Thus, this thick layer may
induce some imprecisions in the CoupModel results. The annual variations are still
important at this depth.
- The last layer is located below the permafrost table, between 7.3m and 11.3m. At this
depth, the soil is permanently frozen and a delay of approximately six months for the
heat transfer from the surface to this depth can be observed (Harris et al., 2009;
PERMOS 2013).
~ 42 ~
4.1.1. CoupModel Calibration for the borehole 6000 (100m)
At Stockhorn, the 100m borehole 6000 was drilled in the northern part of the plateau, in a big
block surrounded by finer materials. The maximal investigation depth of geophysical survey
depends on the total length of the profile. With profiles of 70m to 95m length and with the
steep topography at Stockhorn, the 4PM can only assess the pore content of this area for the
first 18m. Thus, the deeper part of the borehole is not taken into account for this work.
Temperature and resistivity near the surface
In the near-surface zone, the effect of the atmospheric temperature is strongly present with an
annual variation of approximately 20°C between summer and winter, even if the snow and
soil properties create already a decoupling of the temperature between the surface and the
atmosphere (Ekici et al., 2014).
A systematic underestimation of the temperature during the summer period can be observed
for the whole simulated period (see Figure 18). This bias is accentuated for the record summer
of 2003 and for 2004. This means that the heat cannot penetrate into the soil as much as it
should in reality in summer. This isolation effect could be explained by an underestimation of
heat conductivity of the matrix or the pores. Another difference may be observed during
winters 2002-2003 and 2011-2012. The temperature is overestimated with a difference of
almost 10°C for the former period. The soil stays at 0°C for part of the winter instead of going
down as it should be. This can be explained by a too strong Zero-Curtain (ZC) effect in the
simulations. In other terms, the phase transition of water to ice that should occur in winter is
delayed. In both cases the water freezes at the beginning of the winter, and the temperature
starts to fall. But then, a significant amount of water arrives at this layer. This provokes a
continuous release of latent heat into the soil leading to this ZC effect in the middle of the
winter. This water inflow near the surface may come from the snow melt water. Figure 19
shows a much deeper snow cover during winters 2002-2003 and 2011-2012 than for the other
years. A strong wind effect on Stockhorn plateau might limit the snow cover depth to
approximately 0.5m at B100m in reality, but this phenomenon might not be represented
correctly by the CoupModel. In addition, this overestimation of the snow cover also increases
its insulation effect. To represent the circulation of snow melt water on the plateau, a constant
water source, called GWSourceFlow, is added explicitly in the CoupModel calibration. This
might also add another water inflow stronger than it is in reality, during this period.
~ 43 ~
Figure 18: Near-surface soil temperatures (0.8m) at Borehole 6000 (100m). The measured temperatures are in blue
and the simulated temperatures are in green.
Figure 19: Simulated snow depth at borehole 6000 (B100m). The higher snow depth during winters 2002-2003 and
2011-2012 correspond to the overestimation of the temperature (see Figure 18)
Even if data are available for only six days, the resistivity of the soil can still be calibrated at
least for the summer period, when resistivity measurements were conducted. The specific
CoupModel parameters used to calibrate the resistivity are a constant water source flow, the
residual water content, the pore water resistivity and Archie’s parameter m and n. Figure 20
shows a good match between simulated and measured values at this depth. As the electrical
conductivity of water is very high compared to the other phases present in the subsurface, an
overestimation of the soil resistivity implies that the water content is probably underestimated.
This situation may be observed for 2012. It is probably due to stronger water drainage than it
is in reality or to specific conditions in the water circulation at the time of the measurement.
This water circulation will be analysed in more detail in section 4.3.3.
~ 44 ~
Figure 20: Near-surface soil resistivity (0.8m) at Borehole 6000 (100m). The specific resistivities obtained from the
inversion are in blue and the simulated resistivities are in green.
Temperature and resistivity at the permafrost table
At the permafrost table, the heat penetration time already provokes a delay of approximately
two months with the surface temperature, according to the borehole data at Stockhorn.
A large difference may be observed in winter 2002-2003, where the temperature is strongly
overestimated with a difference of 4°C (see Figure 21). The soil stays close to 0°C instead of
going down as it should be. This situation is the consequence of the strong ZC described for
the first layer. A significant amount of water arrives at the upper layers. This provokes a
continuous release of latent heat into the soil and partially prevents the cold from penetrating
to a depth of 3.3m. The winter 2011-2012 is less concerned by this effect, but an
overestimation of the temperature is still visible. Another difference concerns the active layer
thickness (ALT). The measurements clearly indicate temperatures above 0°C for warm
summers (2003, 2004, 2011 and 2012), but the simulation does not represent this situation. As
the ALT varies between 2.88m and 4.28m between 2002 and 2010 (PERMOS, 2013), the
punctual measurements of the thermistor at 3.3m can detect this fluctuation. But as the
amplitude of the ALT stays inside the simulated layer going from 2.1m to 4.5m depth, the
latter cannot correctly represent these variations.
~ 45 ~
Figure 21: Temperature for the permafrost table (3.3m) at Borehole 6000 (100m). The measured temperatures are in
blue and the simulated temperatures are in green.
Figure 22 shows a good match between simulated and measured resistivities at this depth,
except for 2008. As this mismatch only concerns one date with a much higher resistivity, this
is probably due to specific conditions on the day of the measurements or simply to an error in
the measurements, respectively in the inversion process.
Figure 22: Soil resistivity for the permafrost table (3.3m) at the Borehole 6000 (100m). The specific resistivities
obtained from the inversion are in blue and the simulated resistivities are in green.
~ 46 ~
Temperature and resistivity under the permafrost table
At 10m depth, a delay of approximately six months may be expected for the heat penetration
(PERMOS, 2013). As the temperatures are constantly below 0°C, the ice never melts and
there is no strong effect of latent heat flux at this depth. Besides, annual temperature
fluctuations are reduced and their effect is negligible as long as the temperatures do not reach
0°C. Thus, the mean annual temperature should match in the best possible way between
measured and simulated values and the fluctuations become a secondary priority. Statistically
speaking, the R2 error between measured and simulated values becomes less important for the
CoupModel calibration, but the RMSE should be reduced to the minimum.
Borehole data indicate values between -3°C and -2°C for the complete period, but the
CoupModel simulates a layer 1°C warmer, as shown in Figure 23. This overestimation of the
temperature is also visible during winter time at lower depth (see Figure 21). It can then be
assumed that this difference comes directly from the discrepancies observed in the upper
layers. However, it is not possible to find a better fit of the temperature at the surface and at
depth with the current state of the CoupModel calibration.
Figure 23: Temperature under the permafrost table (9.3m) at Borehole 6000 (100m). The measured temperatures are
in blue and the simulated temperatures are in green.
~ 47 ~
At 10m depth, the resistivity values are largely overestimated in the simulation with values
almost ten times higher than it is actually measured (see Figure 24). The ERT data shows a
rapid decrease of the resistivity with depth for the 100m borehole, but this is not considered in
the CoupModel. As the pore spaces should be filled mostly by ice under the permafrost layer,
the resistivity should be around 50’000 Ωm, and not 5.000 Ωm as it is the case with the
geoelectrical measurements. One possibility explaining such low values is that the pores could
be filled mostly with unfrozen water, which is only possible with a high concentration of ions
that prevents the latter from a complete freezing even below 0°C. One more realistic solution
would be the presence of a highly conductive rock such as iron, as already mentioned by
Dängeli (2013). As the bedrock has been represented as very compact sand in the
CoupModel, rocks with different resistivities cannot be simulated. Even if the calibration
seems to be completely wrong at this depth, it is not directly due to a mismatch in water
content. By consequence, it should still be possible to use this simulation to compare it with
4PM results.
Figure 24: Soil resistivity under the permafrost table (9.3m) at Borehole 6000 (100m). The specific resistivities
obtained from the inversion are in blue and the simulated resistivities are in green.
~ 48 ~
4.1.2. CoupModel Calibration for the borehole 6100 (17m)
The second borehole was drilled on the southern part of the plateau, closer to the edge than
the deeper one, in a small block also surrounded by finer materials. The maximum depth of
the 4PM corresponds this time to the measured data of the borehole around 17m. However,
this calibration was much more difficult to perform, as illustrated in the next paragraphs.
Temperature and resistivity near the surface
For the borehole 6100, the CoupModel is calibrated for the same layers than for the previous
one. The first zone where the simulated and measured values of temperature and resistivity
should match is again the near-surface zone.
The correspondence of temperature between the simulation and the measurements is good, as
shown in Figure 25, except for some details. In both cases, the temperature is oscillating
between 5°C in summer and -3°C in winter, with a maximum of 10°C for the warm summer
of 2003. The simulated winter of 2005-2006 is much colder than the other ones, but no data
from the boreholes are available to confirm that. Nevertheless, the three winters preceding this
one were relatively cold (see Figure 6 in section 2.2 for the temperatures). Thus, a cooling
effect from the surface cannot be excluded.
The same overestimation of the temperatures than for the B100m can be observed during
winters 2002-2003 and 2011-2012. Figure 26 shows a much deeper snow cover during those
winters as it is the case for the B100m. Thus, the simulation of the snow as one layer in the
CoupModel might produce imprecisions in the calibration process.
Figure 25: Temperature near the surface (0.8m) at Borehole 6100 (17m). The measured temperatures are in blue and
the simulated temperatures are in green.
~ 49 ~
Figure 26: Simulated snow depth at borehole 6100 (B17m). The higher snow depth during winters 2002-2003 and
2011-2012 correspond to the overestimation of the temperature (see Figure 25).
The near-surface soil resistivity of B17m is much smaller than for B100m at the same depth
with values around 5’000 Ωm, as it can be observed in Figure 27. This can be interpreted as a
layer with a very high water content. Simulated and measured resistivities show a good match
at this depth, with a small underestimation of the simulation. This might come from an
overestimation of the simulated water content at this depth.
Figure 27: Resistivity near the surface (0.8m) at Borehole 6100 (17m). The specific resistivities obtained from the
inversion are in blue and the simulated resistivities are in green.
~ 50 ~
Temperature and resistivity at the permafrost table
At the 17m borehole, the permafrost table is not reached yet at 3m, but the small variation and
the prolongation of the zero curtain effect show that it should not be far. This was expected
considering the warmer temperatures measured at this depth. Even if the matching seems to
be good at first sight for the summer in Figure 28, the simulated temperatures never go below
0°C. This means that the ZC lasts all winter. In other terms, the water present at this depth
cannot freeze completely because its total available latent heat is too high. One possible cause
of this effect may be an overestimation of the unfrozen water content at the freezing front. An
overestimation of the snow height at the surface might also act as an insulator from the cold in
winter. These processes are analysed in more detail in chapter 4.3.
Figure 28: Temperature for the permafrost table (3m) at Borehole 6100 (17m). The measured temperatures are in
blue and the simulated temperatures are in green.
An underestimation of the resistivity may be observed in Figure 29. This might also induce an
overestimation of the water content at this depth as it is the case for the other layers.
Figure 29: Soil resistivity for the permafrost table (3m) at Borehole 6100 (17m). The specific resistivities obtained
from the inversion are in blue and the simulated resistivities are in green.
~ 51 ~
Temperature and resistivity under the permafrost table
The calibration difficulties for this borehole have asked for some compromises. It was not
possible to obtain a close match of the simulated and the measured temperatures for both the
near-surface and under the permafrost table. To be sure that the relation between the
atmosphere and the near-surface temperatures was well represented, the matching of the
simulation and the measurements has been partially put aside for deeper layers.
Thus, an overestimation of approximately 1°C may then be observed in Figure 30. Besides,
the seasonal variation of the temperature is not represented at all and it increases constantly
along the years. This shows that the CoupModel includes a process in the upper part of the
profile that isolates the soil from the seasonal effect. This may be due to a wrong
parametrization of the heat conductivity of the soil. Too high simulated liquid water content at
the freezing front might also induce an isolation and warming effect. These hypotheses are
discussed more in detail in the section 5.1. With these discrepancies, the ice and water
contents calculated at this depth should be interpreted with care.
Figure 30: Temperature under the permafrost table (9m) at Borehole 6100 (17m). The measured temperatures are in
blue and the simulated temperatures are in green.
~ 52 ~
At 9m depth, the resistivity values are expected to be largely overestimated in the simulation
as it is the case for the deeper borehole, due to the presence of a conductive rock matrix (see
Figure 24). However, this pattern is not observed in any multi-run simulations and the
observed and measured temperatures are mismatching in any case (see Figure 31). To fit the
simulated and measured temperature at the surface, the CoupModel overestimates the water
content in order to compensate the lateral heat flux coming from the southern slope (Gruber et
al., 2004). Thus, the good fit between the simulated and measured resistivities is due to an
overestimation of the water content that compensate the effect of the conductive rock matrix.
Figure 31: Soil resistivity under the permafrost table (9m) at Borehole 6100 (B17m). The specific resistivities obtained
from the inversion are in blue and the simulated resistivities are in green.
~ 53 ~
4.1.3. Comparison with 4PM
Now that the CoupModel calibration is analysed, the comparison with the 4PM may be done.
One aim of this work is to assess the values of the porosity, the pore water resistivity and
Archie’s parameters m and n using the best CoupModel run for Borehole 6000 (B100m),
according to the RMS error and R2 mentioned in section 3.4.2. Thus, the cementation index m
is prescribed to 2.41, the saturation exponent n to 1.67, the factor a to 1 and the pore water
resistivity 𝜌𝑤 to 94.9 Ωm. As these values are physically consistent, they are also used in the
CoupModel calibration for B17m, but the porosity is calibrated again for each layer in the
multi-run procedure (see Table 7). Then, the porosity and Archie parameter values calibrated
in the CoupModel are used in the 4PM to facilitate the interpretation of the ice and water
content used for this comparison. This also includes having a different porosity for each
borehole and each layer in the 4PM as it is the case in the CoupModel. In addition, a small
smoothing parameter is added in the 4PM porosity because of the finer model resolution. As
mentioned earlier, the values from the 4PM correspond to a horizontal mean of the five pixels
around the position of both boreholes. This method allows for more robustness in the results.
Borehole 6000 (100m)
According to the CoupModel calibration with the resistivity for this borehole, the porosity
was set to 50% for the first two meters, 36% between 2.5 and 5m and 15% for the deeper part.
As shown in Table 8, this is much higher than the porosity values found by Marmy et al.
(2015), at least between 2m and 5m. With the resistivity added in the calibration process, the
CoupModel might try to compensate some water related processes by increasing the porosity.
The first date selected for the comparison is 22 August 2006. According to the borehole
temperature, the freezing front can be estimated to be around 3m of depth. It means that the
soil should be unfrozen above, as the temperature is positive. Below, the ice content should
start to increase.
~ 54 ~
Table 8: Porosity calibration in CoupModel
Calibrated porosity with
resistivity (this work)
Calibrated porosity without
resistivity (Marmy et al., 2015)
0m - 2m 50% 45%
2m - 5m 36% 11%
5m - 100m 15% 5%
The water content is almost identical for both models, except for the first meter (see Figure
32). This is unexpected as the CoupModel calibration failed partially concerning the water
content due to the presence of a highly conductive rock matrix and a lateral heat flux in
plateau. The difference in the first meter is due to more accumulated water at the freezing
front simulated in the CoupModel, as mentioned in the section 4.1.1.
For the ice content, both models show a freezing front between 1.5m and 2m with a rapid
increase of ice content to reach a maximum between 4m and 6m. The small disparity with the
expected value for the limit of unfrozen soil may come from the different layer thicknesses in
the models. The 4PM has a constant resolution of 0.5m for the profiles at Stockhorn, but the
CoupModel has a layer thickness of 2.4m around 4m depth to respect the position of the
thermistors. Quantitatively, a maximum difference of 10% may be observed at 4m and below.
The CoupModel considers the soil as saturated with ice and residual water when the
temperature goes below 0°C, but it is not the case for the 4PM. This will be discussed in the
section 5.1.
Figure 32: Comparison between the CoupModel and the 4PM for the borehole 6000 (B100m), 22 August 2006.
~ 55 ~
The second date selected for the comparison is 29 July 2011. According to the borehole
temperature, the freezing front may be estimated to be located at a depth between 2.7m and
2.9m. The water content fits for both models with the same difference in the first meter (see
Figure 33). For the ice, both models show a peak due to water accumulation at the freezing
front, but the ice content is 10% higher for the peak and 5% lower below 7m with the 4PM.
One possible explanation can be that the measurements used to calibrate the CoupModel are
taken directly in the bedrock, but the 4PM data come from a profile two or three meters away
in fine sediments. It is then possible that both models indicate correct values and the spatial
variability is very strong. The accuracy of both models might also be a reason for the
discrepancies, as some uncertainties are present in the calibration procedure of the 4PM and
the CoupModel.
Figure 33: Comparison between the CoupModel and the 4PM for the borehole 6000 (B100m), 29 July 2011.
~ 56 ~
Borehole 6100 (17m)
According to the CoupModel calibration with the resistivity for this borehole, the porosity
was set to 40% for the first two meters, 26% between 2.5 and 5m and 20% for the deeper part.
This borehole is not used by Marmy et al. (2015). Thus, a comparison of the porosity is not
possible. With the borehole temperature of 22 August 2006, the freezing front may be
estimated to be at a depth of around 3m. A large difference in water content may be observed
in Figure 34 for the first two meters with 26% for the CoupModel and 12% for the 4PM. This
is again due to more accumulated water at the freezing front simulated in the CoupModel, but
the pattern is present for both models. The gap between the borehole and the geoelectrical
profile might again explain this difference. But an overestimation of the simulated water
content at the B17m in the CoupModel is probably the main reason of this contrast.
For the ice content, it appears near the expected freezing front for both models with a rapid
increase. Here again, a maximum difference of 10% can be observed at 8m and deeper. This
comes from the CoupModel that considers the soil as saturated with ice and residual water
when the temperature goes below 0°C.
Figure 34: Comparison between the CoupModel and the 4PM for the borehole 6100 (B17m), 22 August 2006.
~ 57 ~
According to the borehole temperature of 29 July 2011, the freezing front can be estimated to
be at a depth between 1.6m and 5m. This large uncertainty is due to temperatures oscillating
between 0.002°C and 0.04°C for more than 3m. The uncertain freezing front position seems
to be represented by the CoupModel with oscillating ice content. This pattern is also visible in
the 4PM with ice content stagnation between 3m and 5m. Under the permafrost table, both
models are matching with an ice content difference of only 2-3%. As a lot of water
accumulates at this place, the soil is more easily saturated. The 4PM also shows a clear peak
for the water accumulation, but with 15% instead of 25%. High CoupModel water content at
low depth may still be observed in Figure 35. This is probably due to the overestimation of
the simulated water content observed in the section 4.1.2.
Figure 35: Comparison between the CoupModel and the 4PM for the borehole 6100 (B17m), 29 July 2011.
~ 58 ~
4.1.4. Conclusion of the comparison of the 4PM with the CoupModel
Even if there are some discrepancies in the pore content between the 4PM and the
CoupModel, the variation with depth matches very well. The depth of the freezing front is
similar for both models and the accumulation of water is always detected, if there is any. As
both models are very different in their conception, it is difficult to identify precisely the cause
of the discrepancies. For the first 3m near the surface, the CoupModel is probably more
robust, except for the B17m where the water content is overestimated. On the other hand, the
4PM can more easily consider some specific and local features directly from the ERT and
RST data that are not detected by CoupModel. Besides, the longer distance between the
meteorological station and the smaller borehole 6100 (B17m) may also induce more
imprecisions in the CoupModel for this specific area.
For the deeper part, the heterogeneity of the soil with the presence of fractures leads to the
conclusion that the ice content is probably between the simulated values of 4PM and
CoupModel. The only solution to know for certain which model better assesses the absolute
ice content at Stockhorn would be to compare the results with a drill core. The quality of both
models and the reasons for discrepancies will be discussed further in chapter 5.
~ 59 ~
4.2. Application of the 4PM at Stockhorn
The aim of this chapter is to see how the improvements of the 4PM and the utilisation of
spatially distributed profiles can improve the understanding of the spatial distribution of ice
and water for the specific case study Stockhorn (VS, Switzerland). First, an ERT and an RST
profile are interpreted as example. Then, the 4PM calibration is presented. Finally, the results
of the 2014 campaign are analysed shortly.
4.2.1. Interpretation examples of an ERT and RST profile
Even if the 4PM allows for the quantification of ice, water and air content in the subsurface, it
is important to be able to assess qualitatively these elements by interpreting directly the ERT
and RST profile. The Longitudinal South Profile (LSP) is taken as example for the
interpretation.
Interpretation of an ERT profile
As it can be observed in Figure 36, the repartition of the resistivity is markedly heterogeneous
with vertical and horizontal variations. This is the sign of a complex terrain with different
processes interacting with each other. Thus, it demonstrates the importance of analysing this
kind of profile qualitatively and quantitatively to better understand these processes. Five areas
are defined for the analysis of Figure 36:
1) The first three or four meters show some areas with resistivity up to 100’000Ωm. (see
1a and 1b in Figure 36). This corresponds to the presence of a significant amount of
air in the rock matrix, acting as an electrical insulator. As bedrock is present at the
surface in the area 1a, lower resistivity might be expected, as well as a deeper freezing
front. Thus, this bedrock is probably highly fractured.
2) Some areas with low resistivity between 1’000 Ωm and 4’000 Ωm may be observed
near the surface (see 2a and 2b in Figure 36). It indicates the presence of very
conductive and ice free material (Hauck and Kneisel, 2008). At this depth, it probably
corresponds to water accumulation at the freezing front.
~ 60 ~
3) Below the water accumulation, a small area with resistivity between 20’000 Ωm and
50’000 Ωm shows the presence of ice.
4) However, resistivity around 10’000 Ωm may be observed under the freezing front (see
zone 4 in Figure 36). Considering the fact that the soil is mostly frozen at this depth,
the relatively low resistivity is most probably due to a more conductive matrix.
5) The resistivity reduction due to conductive rock is even more pronounced on both
sides of the profile (see 5a and 5b in Figure 36). In this case, a correction must be done
in the 4PM to avoid the interpretation of these areas as filled with unfrozen water.
Figure 36: ERT result for the Longitudinal South Profile (LSP). The repartition of the resistivities is markedly
heterogeneous with high resistivities at the surface (1a and 1b) corresponding to high air content. Low resistivity near
the surface (2a and 2b) represents probably areas with a consequent amount of water. At depth, zones with a low
resistivity may be observed. As the soil should be frozen, it is probably due to conductive bedrock.
Interpretation of an RST profile
The RST profile is more homogeneous than the ERT profile, with a relatively regular increase
of the wave velocity. Three areas are defined for the analysis of the Figure 37.
6) Wave velocity of approximately 330m/s to 500m/s may be observed near the surface
for the whole profile in Figure 37. According to that, no ice is present in the first two
meters. Besides, this low p-waves velocity in the area 6a confirms the presence of
fractures in the bedrock at the surface. The areas with a resistivity between 20’000 Ωm
and 50’000 near the surface (see 3 in Figure 36) may then be due to an area with a
lower porosity or to a little of water that increases the soil conductivity. Even if it is
less pronounced with the RST, the higher ALT in the apparent bedrock is also visible
(see 6a in Figure 37).
~ 61 ~
7) Even if the profile is relatively homogeneous, small areas with lower wave velocities
are present at around 10m depth. It may be due to more air content in the fractures (see
7a, 7b and 7c in Figure 37).
8) On the other side, zones with high velocities may be observed deeper. There, the
porosity is probably smaller and the relative ice content is higher.
Figure 37: RST results for the Longitudinal South Profile (LSP). Even if the wave velocity increase is relatively
homogeneous, some small patterns may still be observed.
In conclusion, the ERT and RST are two complementary ways to investigate the structure of
the subsurface and some processes are better represented by one or another method. For
example, the ERT can better detect the presence of highly conductive materials like liquid
water. On the other hand, the RST can easily detect the boundary between air and ice (Harris
et al., 2009). By consequence, combining both ERT and RST in a new model may
considerably improve the assessment of ice, water and air content of the soil.
~ 62 ~
4.2.2. 4PM calibration
To see if the CoupModel multi-run procedure might lead to physically consistent values for
Archie’s parameters m, n and ρw, the latter are let free. But even if the CoupModel calibration
gives physically consistent results for the porosity and Archie’s parameters, the values do not
correspond to the expected situation at Stockhorn, especially for the pore water resistivity that
is underestimated according the measurements of GO4ICE (2011). Thus, the 4PM has been
run again with a new calibration. The consequence of this new set-up and the related
uncertainties are discussed in the next paragraphs. The Table 9 resumes all the parameters
values for Stockhorn profiles used in this study.
Table 9: 4PM calibration for the 2014 campaign at Stockhorn.
Porosity Model
(calibration with 3PM)
Resistivity Equation
(Archie with rock res.) P-wave velocities
Initial
porosity
50 % (debris)
30 % (apparent
bedrock)
Pore water
resistivity 500 Ωm
Velocity in
ice 3500 m/s
Porosity
gradient
6 %/m (0.6 %/m
under the threshold) Epsilon
0.15 from 5m depth with a
5 pixels smoothing factor
Velocity in
water 1500 m/s
Threshold
value 20% m 1.8
Velocity in
air 300 m/s
Minimal
value 10% n 2.1
Velocity in
rock 6000 m/s
Porosity of the profiles
With the new 4PM version 7, a complex porosity model can be built (see section 3.3.3). First,
a 3PM is run for all the profile where no ice is assumed to be present. Then, the main porosity
model is set up with an initial value of 50%, a gradient of 6% / m and a threshold at 20%. This
corresponds to the 3PM results obtained previously in most of the areas. Finally, a zonal
porosity with an initial value of 30% is defined for parts of the profile with apparent bedrock.
This new set-up is relatively similar to the previous set-up used for the comparison between
the 4PM and CoupModel, with a more regular decrease of the porosity.
~ 63 ~
Choice of the resistivity equation
Considering the presence of highly conductive zones on the plateau, the basic Archie’s law
cannot represent properly the situation. Besides, Somerton equation did not give sufficient
physically consistent solution to be considered. Thus, Archie’s law with rock resistivity has
been selected for the 4PM as it was for the previous set-up used for the comparison between
the 4PM and CoupModel.
Choice of the parameters
The prescribed p-wave velocities have been kept to known values for each medium shown in
Table 4 as it was the case for the previous set-up used for the comparison between the 4PM
and CoupModel. On the other side, Archie’s parameters are much more difficult to assess.
One aim of the new 4PM version 7 was to calibrate them using surface water content as a
comparison. After several tests on all the profiles, the multi-run built for this purpose in the
new version of the 4PM (see section 3.3.3) has proved to be inefficient. The reason for the
failure of the multi-run calibration is discussed in more detail in section 5.2.1. Thus, the m
and n factors are determined with values found in the literature according to the site
specificities (Schön, 2004). Considering the possible fractures in the bedrock, the cementation
exponent m is set to the relatively low value of 1.8 even if the bedrock itself is highly
cemented. The saturation exponent n depends on “the rock texture, wetting properties and
saturation history caused by capillary effect” (Schön, 2004:423). As the distribution of
conducting water in the pore space is not free because of the ice, the value of n should be
relatively high. Thus, it is set to 2.1 for Stockhorn. Archie’s parameters m and n are now quite
different from the previous set-up used for the comparison between the 4PM and the
CoupModel. But the effect of the pore water resistivity is much more consequent, as
illustrated in Figure 38. The assessment of the pore water resistivity ρw is often problematic
due to the lack of measured data. Besides, ρw may vary during the year. Even if the water
content is probably overestimated in the CoupModel, the difference with the 4PM in the near-
surface area is important. Thus, the 4PM may underestimate the water content with a pore
water resistivity of 95Ωm. The pore water resistivity has been set to 500 Ωm according to
measured data from the GO4ICE (2011) project. It corresponds to a broad mean between 417
Ωm and 588 Ωm measured respectively in august 2007 and 2008. Finally, the epsilon factor
for rock conductivity is set to 0.15 from 5m depth with a 5 pixels smoothing factor for most of
the profiles.
~ 64 ~
Figure 38: Influence of Archie's parameters on the pore contents. The influence of the new calibration for m, n and ρw
is represented in green for the air content, in blue for the ice content and in red for the water content. As shown by
this figure, the pore water resistivity is the most influencing parameter for a measured resistivity of 5’000 Ωm and a
p-wave velocity of 3000m/s.
~ 65 ~
4.2.3. 4PM results
To catch the local specificities, all the profiles are described one after another for the ice,
water and air content. For this analysis, the terrain is separated in three parts. The first one
includes the north of the plateau with the LNP and the upper part of the CMP and the CNP.
The second one considers the south of the plateau with the LSP and the middle of the CMP
and the CNP. The last part concerns the southern slope with the rest of the CMP and the CNP.
Monitoring Cross Profile
This profile was installed in summer 2005, at approximately six meter distance to the
boreholes. The ERT data shows a zone at depth with a very low resistivity (Figure 39),
interpreted as a very conductive rock (Dängeli, 2013: 98).
For this profile, high ice contents up to 30% can be observed for the northern part of the
plateau (see 1 in Figure 40) with some frozen parts up to the surface. Deeper, the ice is
reduced to 20% (see 2 in Figure 40). However, those values should be interpreted with care as
the rock conductivity factor epsilon plays a large role at this depth. Near the cliff with bedrock
at the surface, the soil is unfrozen for the first 5 meters (see 3 in Figure 40). The results for the
southern slope show maximum ice content of 15% with a thick ice free layer of almost 10m
for the lower part of the slope (see 4 in Figure 40).
The water is mainly concentrated near the surface with values up to 15% for the southern part
of the plateau (see 5 in Figure 40). Deeper, residual content of maximum 5% is present with
some zones with higher values, probably due to the very low resistivity measured for this
profile.
The air is mostly present near the cliff where the relative content can reach 100% (see 6 in
Figure 40). Deeper, residual air, around 5%, is present. This remaining content may be due air
trapped in fractures or in the ice, or simply to model inaccuracies.
~ 66 ~
Figure 39: ERT and RST results for the Cross Monitoring Profile. The area with supposedly conductive rock is
clearly visible in the ERT profile. High wave velocities in the first meters show the presence of ice near the surface,
especially on the plateau.
Figure 40: 4PM results for the Cross Monitoring Profile. High ice content can be observed on the northern part of the
plateau. A water accumulation is present near the B17m where almost no ice is present near the surface.
1
2
4
3
5
6
~ 67 ~
Cross New Profile
This profile is parallel to the CMP, five meters closer to the boreholes. The ERT shows a zone
with a very low resistivity at depth (Figure 41), interpreted as a very conductive rock.
For this profile, high ice content up to 30% may also be observed for the northern part of the
plateau (see 1 in Figure 42), even at the surface. For the southern part of the plateau, the 4PM
results show ice content between 10% and 15% (see 2 in Figure 42). Again, those values
should be interpreted with care because of the rock conductivity factor epsilon. Here, the
surface is unfrozen in the three first meters. The freezing front can even reach five meters in
the small cliff, due to the bedrock present at the surface. Concerning the southern slope, it
contains 25% of ice and the surface is still frozen (see 3 in Figure 42). This is much more than
the values for CMP. The high variability of ice content in the southern slope is discussed in
the section 4.3.2.
The water is mainly concentrated near the surface with values between 10% for the northern
part of the plateau and up to 15% for the southern part (see 4 in Figure 42). The accumulation
observed for the CMP is also visible for the CNP. Deeper, residual content of approximately
5% is present with some zones with higher values. The latter are probably due to the very low
resistivity measured for this profile that are not corrected with the epsilon factor.
The air is mostly present near the cliff where the relative content can reach 100% (see 5 in
Figure 42). Deeper, residual air, around 5%, is present.
~ 68 ~
Figure 41: ERT and RST results for the Cross New Profile. The area with supposedly conductive rock is clearly
visible in the ERT profile.
Figure 42: 4PM results for the Cross New Profile, five meters closer to the boreholes than CMP. As expected, the
values are relatively similar to the CMP. Thus, using the latter to compare it with borehole data should not pose a
major issue even if it is 5 meters away.
1 3
2
4
5
~ 69 ~
Longitudinal North Profile
This profile is situated on the northern part of the plateau. A large snow patch was present
nearby all along the measurement line. The ERT and RST data show a surface with a
relatively low resistivity and p-wave velocities around 1500m/s (see 1 in Figure 43) which
point out the presence of water in the first meters. Near the middle of the profile, p-wave
velocities around 3000m/s also show that ice is present near the surface (see 2 in Figure 43).
The 4PM results confirm this hypothesis with ice content between 20% and 30% at the
surface in the middle of the profile (see 1 in Figure 44). On the western part of the plateau
(area 2 in Figure 44), the bedrock is visible at the surface and no ice is present at least in the
three first meters. With a lower porosity, this area is more sensible to summer heat flux.
Deeper, the 4PM reveals decreasing ice content from 20% to 10%, corresponding to relative
pore content between 80% and 90%. Again, those values should be interpreted with care
because of the rock conductivity factor epsilon.
Concerning the water content, high values may be noticed in the first meter where fine
materials are present (see 3 in Figure 44). This accumulation just above the freezing front is
typical of the snow melt process in permafrost conditions (Hinkel et al., 2001). Horizontal
variations may be pointed out depending on melt water flow paths at the freezing front. For
the rest of the profile, only residual water may be observed.
The air is mostly present near the surface, as expected. The relative content can reach 100%
where the bedrock is visible at the surface (see 4 in Figure 44). The horizontal variations
correspond to the water flow paths mentioned above and to the presence or not of big blocks
at the surface. Deeper, almost no air is present, except for a large area in the middle of the
profile (see 5 in Figure 44). The presence of air trapped in fractures may explain this pattern.
However, a wrong calibrated porosity may also induce this kind of effect. The high variability
of ice and air contents in the southern slope is discussed in more detail in section 4.3.2.
~ 70 ~
Figure 43: ERT and RST results for the Longitudinal North Profile. The areas near the surface with low resistivity
and wave velocities around 1500m/s show the presence of water, probably from the snow melt.
Figure 44: 4PM results for the Longitudinal North Profile. The high water content near the surface confirms the
observations made in the Figure 43.
1
2 1
1
2 1
1
2
3
4
5
~ 71 ~
Longitudinal South Profile
This profile is situated on the southern part of the plateau and the bedrock is visible at the
surface for the western part of the profile. The ERT and RST results are analysed as example
in section 4.2.1.
The 4PM results indicate an ice-free layer of two meters depth in the middle of the profile
(see 1 in Figure 46). On the western part of the plateau, the bedrock is visible at the surface
and the ice-free layer can reaches four meters (see 2 in Figure 46). As it may be observed in
Figure 44 and Figure 46, the freezing front is deeper for the LSP than for the LNP. This
horizontal variation of the freezing front depth will be developed in more detail in section
4.3.2. Deeper, the 4PM reveals ice content around 15%, corresponding to relative pore
content between 80% and 90%. Again, those values should be interpreted with care because
of the rock conductivity factor epsilon.
Concerning the water content, high values may be noticed in the first meter, but it is
constrained mainly to one 10m large area (see 3 in Figure 46). This strong accumulation just
above the freezing front may indicate the presence of a water pool. For the rest of the profile,
only residual water may be observed.
The air is mostly present near the surface, where the relative content can reach 90% (see 4 in
Figure 46). Deeper, the air content decreases rapidly to reach almost 0% at 15m.
~ 72 ~
Figure 45: ERT and RST results for the Longitudinal South Profile. The area near the surface with a very low
resistivity shows the presence of water probably coming from the northern part of the plateau.
Figure 46: 4PM results for the Longitudinal South Profile. As expected with the ERT, the water content is very high
near the B17m.
2
3
4
1
~ 73 ~
4.3. Interpretation of the results
Now that the profiles have been analysed, the results may be interpreted. In this chapter, the
spatial distribution of ice and water on the field study will be assessed based on the 4PM
results, CoupModel simulations of both boreholes and soil moisture measurements from the
SNF-project SOMOMOUNT (2012). A previous interpretation was already performed in a
master thesis by Susanne Dängeli (2013). Thus, the aim of this chapter is not to redo
completely the analysis, but to see how the improvements of the 4PM and the utilisation of
spatially distributed profiles can improve the understanding of the spatial distribution of ice
and water at Stockhorn. For a better comparison between the years, the 4PM is run again for
2006 and 2011 with the geophysical data used by Dängeli (2013), with a better resolution and
with the same parameters as for 2014 (see Figure 76 and Figure 78 in appendix 6).
4.3.1. Analysis of CoupModel parameters
At Stockhorn, the temperature and the soil moisture are very different between both
boreholes. This spatial contrast on the plateau is analysed in more detail in sections 4.3.2,
4.3.3 and 4.3.4. But before interpreting the situation at Stockhorn, it is useful to know which
processes influence the soil moisture and temperature. To do so, the CoupModel parameters,
determined by the Stockhorn calibration process (Table 7), are analysed to compare the
situation at both boreholes. As shown in the Figure 47 and Figure 48, the sensitivity of the
temperature and the soil moisture varies for each parameter. But even if some of them do not
seem to have a strong influence on the mean soil moisture and the temperature, represented by
the RMS error in the CoupModel calibration (eq. 37 in section 3.4.2), their importance can be
found in the annual changes of these variables. This is for example the case for the residual
water. However, these annual variations, represented by the R2 in the CoupModel calibration
(eq. 36 in section 3.4.2), are not analysed in this work due to a very time-consuming process
to extract the data for each year. For the analysis, the parameters are separated into three
categories: the snow properties (in red in the Figure 47 and Figure 48), the water inflow (in
blue) and the water outflow (in green). Archie’s parameters are not analysed here as they have
a direct influence only on the resistivity, and not on the water content itself.
~ 74 ~
Figure 47: Influence of CoupModel parameters on the mean temperature of the period 2002 – 2012 for the B100m.
See the appendix 5 for a detailed sensitivity to some important parameters. The snow properties are in red/brown, the
water inflow in blue and the water outflow in green.
Figure 48: Influence of CoupModel parameters on the mean water content of the period 2002-2012 for the B100m. See
the appendix 5 for a detailed sensitivity to some important parameters. The snow properties are in red/brown, the
water inflow in blue and the water outflow in green.
~ 75 ~
Snow properties
The most influencing factor of this category is the critical depth for the decoupling effect of
the snow between the atmosphere and the soil, CritDepthSnowCover. A higher critical depth
reduces the isolation effect and the cold, as well as the heat, can penetrate more easily into the
soil (Engelhardt et al., 2010). Figure 47 and Figure 48 show a very high influence of this
phenomenon on the temperature and water content (see Figure 65 and Figure 66 in appendix 5
for detailed sensitivity curve of CritDepthSnowCover). The parameter was set up
automatically to 70cm for the B100m and to 66cm for the B17m by the CoupModel
calibration. This is more than twice the maximum value of 30cm normally authorized in the
CoupModel until the new version of January 2015. With a maximum simulated snow depth
around 50cm for almost all the winters, the soil at Stockhorn is considered as never decoupled
completly from the atmosphere (see Figure 19 in section 4.1.1 and Figure 26 in section 4.1.2).
In any case, the values are very similar for both boreholes. Thus, this process does not explain
why the temperature and the water content is higher at B17m than at B100m.
Other snow parameters might explain this spatial contrast on the plateau. But the density of
new snow, the temperature limit between snow and rain and the albedo of the snow do not
have a strong impact on the mean temperature and soil moisture or their values are almost
identical for B100m and B17m as it is the case for the coefficient for snow thermal
conductivity SThermalCondCoef.
Water inflow
As the Stockhorn plateau is in high altitude, most of the precipitations are in form of snow.
Thus, the main source of water at the boreholes is the snow melt. The first influencing
parameter is the external source flow, GWSourceFlow, added to the CoupModel calibration to
represent the percolation of water coming from the snow melt in the upper part of the plateau
(see Figure 67 and Figure 68 in appendix 5 for detailed sensitivity curve). Surprisingly, the
temperature seems to be more sensitive to this process than the soil moisture. This might be
explained by a high water inflow at the B100, bringing a certain amount of latent heat,
followed by a rapid evacuation of the water to the south. This percolation process is much
likely considering the position of the B100m directly in bedrock. With a value of 0.45 for the
B100m and 0.39 for the B17m, this parameter is not directly responsible for the temperature
difference between both boreholes.
~ 76 ~
The second influencing process is the effect of the radiation and the temperature on the snow
melt. It is represented respectively by two CoupModel parameters, MeltCoefGlobalRad and
MeltCoefAirTemp. Both of them have a medium effect on temperature and soil moisture.
However, the MeltCoefAirTemp is very different between both boreholes with a value of 1.18
Kg °C-1
m-2
day-1
for the B100m and 2.24 Kg °C-1
m-2
day-1
for the B17m (see Figure 69 and
Figure 70 in appendix 5 for detailed sensitivity curve). It means that the snow melts faster at
the B17m for a given positive temperature in the CoupModel simulation. This might be a
compensation of the fact that the B17m receives more radiation than the B100m due to a
lateral heat flux from the southern slope (Gruber et al., 2004). This process is developed in
more detail in section 4.3.2.
Water outflow
The water outflow depends on the available pore space, the residual water, the evaporation
and the hydraulic conductivity. Some parameters such as the saturation, the residual water and
the evaporation factor EquilAdjustPsi do not have a strong impact on the mean temperature
and soil moisture. The sensitivity is much more visible with the hydraulic conductivity and
the m-value in the retention curve (see Figure 71, Figure 72, Figure 73 and Figure 74 in
appendix 5 for detailed sensitivity curve). The former is almost five times smaller for the
B17m with a value of 4’785 mm/day against 21’474mm/day for the B100m.The same
observation can be made for the latter with 0.17 for the B17m and 0.56 for the B100m. This
explains the much higher water content in B17m than in B100m.
In conclusion, a very high retention of water may be observed for the smaller borehole B17m.
This can easily lead to a strong accumulation. Besides, the northern part of the plateau is
subjected to a fast drainage. However, the CoupModel calibration shows that this retention
might be too strong for the B17m, leading to an overestimation of the water content (see
section 4.1.2.). Now, the next step is to see if the 4PM results confirm this analysis.
~ 77 ~
4.3.2. Spatial distribution of ice
The ice content is one of the main factors influencing the sensitivity of a specific permafrost
site to climate change (Scherler et al., 2013). With a lot of processes involved, the Stockhorn
plateau shows large variations of ice content. It is then important to understand the causes of
these contrasts. As shown in the Figure 49, the spatial distribution of ice was already analysed
in more detail by Susanne Dängeli (2013). Thus, this part should be seen as a complement to
confirm or disprove the previous interpretations.
High ice content on the plateau
On the northern part of the plateau, ice content around 30% is calculated between 2m and 7m
depth for CMP, CNP and LNP (see Figure 50). Susanne Dängeli (2013) made the same
observation for 2006 and 2011. The first reason she mentioned is the influence of the
topography (Gruber, 2004; Noetzli & Gruber, 2009). The southern slope is more exposed to
solar radiation and the resulting lateral heat flux does not affect the northern part of the
plateau as much as for the rest. The second reason is the presence of snow patches on the
northern part of the plateau, even during late summer. This was also observed in 2014. This
snow layer increases the albedo and the resulting reflexion reduces the effect of the incoming
radiation. Besides, a considerable amount of energy is needed to melt the snow and the water
is evacuated rapidly to the south. Considering the results of 2014, the effect of this second
process might be more important than expected by the previous studies (Dängeli, 2013). Thus,
it will be further analysed in chapter 4.3.3. With ice patches still present at the surface, it is
difficult to assess precisely the freezing front at Stockhorn from the data of 2014. However, a
freezing front around three meters for the B100m and four meters for the B17m may be
estimated at the end of August according to the measured borehole temperatures. With the
longitudinal profiles, it appears that the high ice content only concerns the zone with finer
material at the surface. On the western part of the plateau, the bedrock is visible at the surface
and no ice is present in the first meters according to the 4PM. On the contrary, the maximum
thickness of the thaw depth in August 2014 seems to reach at least 4m for this area. This can
be explained by lower pore space for ice. Besides, the exposition of the bedrock to the air
removes the isolation effect of finer material.
~ 78 ~
Highly conductive area on the plateau
Under the zone with high ice content on the plateau, an area with low resistivity may be
differentiated from the rest. Two possible reasons for these values are explained by Dängeli
(2013). The first one is a high concentration of ions in the water that prevents it to freeze
completely even below 0°C. The second one is the presence of highly conductive rock. It is
impossible to know which solution is correct only with ERT and RST. However, the
geological map presented by Dängeli (2013) does show an anomalous layer near the surface
at Stockhorn that can be holding iron. In any case, the relative water content up to 80%
calculated with the classical Archie’s law is nearly impossible, except for extreme situations.
To take this low resistivity into account, an epsilon factor is added in classical Archie’s law
(see eq. 19). With an epsilon factor value around 0.15 for the deeper part of the Stockhorn
plateau, the 4PM results indicate a total ice content around 20% in this area (see Figure 50).
But as it was mentioned earlier, this number should be interpreted with care because no
practical values can be found in the literature for epsilon and the latter plays a large role at
this depth in the 4PM.
High variability in the southern slope
As mentioned earlier, a new profile, CNP, was installed in 2014 parallel to the CMP but five
meters closer to the boreholes. The main objective of this operation is to see whether there is a
visible effect of the metallic structures around the B17m and B100m boreholes on the
geoelectrical measurements. This effect is not clearly visible in the ERT results (see Figure 39
and Figure 41 in section 4.2.3). However, both profiles brought to light other differences,
particularly on the southern slope (see Figure 40 and Figure 42 in section 4.2.3). At the CNP,
ice is present near the surface all along the southern slope in 2014. But five meters nearby at
the CMP, an ice free layer up to 10m may be observed at the end of the southern slope. In her
work, Dängeli (2013) got an even deeper ice-free layer of at least 15m (see Figure 49). With
the new 4PM calibration for 2006 and 2011, this strange structure is not visible anymore at
depth (see Figure 76 and Figure 78 in appendix 6) and the situation is similar to the 4PM
results for 2014, at least for 2011. One explanation for these spatial variations along the
southern slope would be the presence of bigger blocks along the surface. Those blocks are
only a little in contact with the ground and the large air masses in between prevent heat
conduction. This structure may induce a cold air circulation process underneath (Gruber &
Hoelzle, 2008) which results in a cooling effect. For Dängeli (2013), the presence of finer
~ 79 ~
material in the lower part of the southern slope interrupts this system for the specific area with
less ice. However this would imply that this cooling effect is active for all the southern part of
CNP and CMP except at one specific place. Another explanation proposed by Dängeli (2013)
is an error in the porosity calibration. The bedrock could be much closer to the surface where
finer material may be observed. This would then be a similar situation to the bedrock at the
surface on the plateau or for the cliff.
Figure 49: Interpretation of the situation the 30 July 2011 at the CMP by Susanne Dängeli (2013). The values along
the surface correspond to ground temperature measurements.
Figure 50: Interpretation of the situation the 27 August 2014 at the CMP (with the maximal extend of the Active
Layer). The water flow from the snow melt is represented in blue.
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Figure 51: 3D representation of the water circulation at Stockhorn for the end of August 2014. The blue arrows
represent the melt water flow at the freezing front (2-3m below the surface).
~ 81 ~
4.3.3. Spatial distribution of water in the active layer
Water flow on the plateau
Considering the 4PM results of 2014, the spatial distribution of water seems to play a
considerable role in the heat repartition on the plateau. The snow patch present at the end of
August provokes a typical accumulation just above the freezing front, as seen for the LNP in
Figure 44 (Hinkel et al., 2001). Horizontal variations may also be pointed out depending on
melt water flow paths. With a slope of 8% on the plateau, the water is evacuated rapidly to the
south. Besides, the topography canalises the water into small streams. The latter accumulates
then in a natural reservoir with ice and bedrock acting like a small dam. This phenomenon
may be observed at the CMP, CNP and LSP with water content up to 30% (see Figure 40,
Figure 42 and Figure 46). When the reservoir is full, the water flows over the bedrock and fall
from the cliff in small cascades. This situation is illustrated in the Figure 51.
Resulting heat flow on the plateau
Gruber et al. (2004) explains the temperature difference between both boreholes with
topography, i.e. the orientation of B17m more to the south. The southern slope is more
exposed to solar radiation and the resulting lateral heat flux does not affect the northern part
of the plateau as much as for the rest. This affirmation is confirmed by ground temperature
measurements (Dängeli, 2013) and the freezing front in the 4PM is indeed deeper for the
southern slope and the ridge, than for the northern part of the plateau. However, the water
circulation at the freezing front induces an important transfer of latent heat (Scherler et al.,
2010). A considerable amount of energy is needed to melt the snow on the northern part and
this heat is brought to the south by the water. In any case, it is difficult to quantify which
process, the topography or the latent heat flux, influences the most the spatial difference of
temperature on the plateau. One solution to test if the lateral heat flux effect postulated by
Gruber et al. (2004) is sufficient to create the higher temperatures to the south by itself would
be to include a heat source in the CoupModel for the B17m. In the meantime, a qualitative
assessment is possible by analysing the annual variation of temperature and water content
between both boreholes.
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4.3.4. Temporal evolution of water content
The spatial repartition of the water observed at the end of August 2014, with active snow
patches and water streams, is very specific from this period of the year. The objective of this
part is to interpret CoupModel and 4PM results with soil moisture measurements to explain
the annual variation of water in the subsurface. To do so, the year is separated into specific
periods for heat and mass transfer according to Hinkel et al. (2001): The Active Layer Regime
(AL), the Zero Curtain Regime (ZC), the Freezing Regime (FR) and the Snow and Ice Melt
Regime (SM).
Figure 52: Measured temperatures at both boreholes for the same depth (0.8m) between 2002 and 2012. Every year is
separated in four regimes: AL (Active Layer Regime), ZC (Zero Curtain Regime), FR (Freezing Regime) and SM
(Snow Melt Regime).
Figure 53: Detailed measured temperatures at both boreholes for the same depth (0.8m) between 2013 and March
2015. As the meteorological data are not reconstructed for this period (see section 2.2), these data are not included in
the CoupModel calibration. The measured temperature at the soil moisture station (50cm) is similar to the measured
temperature at B17m (80cm).
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Figure 54: Total and unfrozen water content simulated by the CoupModel for the B100m. The annual variation is
visible and the values stay between 1% and 5% for the unfrozen content.
Figure 55: Total and unfrozen water content simulated by the CoupModel for the B17m. Here, the annual variation is
clearly visible with total water content up to 30% and with a long Zero Curtain Regime and Snow/Ice Melt Regime.
Active Layer Regime (AL)
The Active Layer Regime usually occurs in summer when all the snow has melted and the
thickness of the unfrozen layer at the surface is increasing slowly. During this period, the
subsurface is more sensible to air temperature fluctuations (Hinkel et al., 2001). At Stockhorn,
this regime starts around the end of June for the B100m and between July and August for the
B17m, as it can be seen with positive fluctuating temperatures in Figure 52 and Figure 53. At
0.8m all the ice is supposed to be melted. Thus, the AL corresponds to the period with
unfrozen and total water content equals (see Figure 54 and Figure 55). However, snow or ice
patches may persist until the summer as it was observed for the northern part of the plateau in
August 2014. With maximum temperatures of 4°C - 5°C some days, the snow melts and a lot
of water is introduced into the soil. At this period, the freezing front of the permafrost can
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easily be saturated with unfrozen water. As the latter has a thermal conductivity 3-4 times
lower than ice, an accumulation of liquid at the base of the thawed zone may act as a
temporary heat transfer insulator between the permafrost and the atmosphere (Hinkel et al.,
2001). Besides, the heat capacity of the water makes it an efficient agent for ice thawing.
Thus, an incorrect estimation of the water content in the models may have a considerable
effect on the subsurface thermal regime.
The daily variation of water content is also to be considered for the analysis. As snow melts
generally with positive temperatures, the circulation process is mainly active during the day.
Besides, a delay may be observed at Stockhorn between the beginning of the snow melt on the
north and the moment the cascade appears along the cliff on the south of the plateau, as
observed in 2014. Figure 56 illustrates perfectly the peak appearing during the afternoon for
the water content. The data are taken at the SOMOMOUNT (2012) soil moisture station near
the meteorological station (see Figure 51 for its exact location). Thus, the time when the
geoelectric measurements are taken is important. Figure 56 shows that the LSP may show up
to 10% higher water content than the other profiles only because the measurements were
taken during the peak. However, this pattern is not clearly visible with water content around
25% in both LSP (Figure 42) and CNP (Figure 46). In addition, these fluctuations are not
considered by the daily means used for the simulation in the CoupModel, which may be a
source of uncertainty for the comparison with 4PM results.
Figure 56: Water content measured at 50cm depth, Stockhorn soil moisture station (SOMOMOUNT, 2012). The daily
variation is clearly visible. The time of each geophysical measurement is marked in red (ERT) or blue (RST). The CM
geoelectric profile was measured the 27 August 2014 in the afternoon, before the installation of the soil moisture
station.
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Zero Curtain Regime (ZC)
In October, the active layer starts to freeze and the snow layer increases due to the low
temperatures (Scherler et al., 2010). During the Zero Curtain Regime, a considerable amount
of latent heat is released by the fusion of water into ice. Consequently, the temperature is
stabilized around 0°C until most of the water has been frozen. The duration of this regime
depends mostly on the water available for the freezing. As mentioned earlier, the topography
provokes a drainage on the north and an accumulation in a reservoir next to the B17m. Thus,
between 2.5% and 3.0% of water is available for the B100m (see Figure 54), but this value is
almost ten times higher for the B17m (see Figure 55). With such a difference in initial water
content, the ZC lasts much longer at the reservoir and the thermal regimes diverge
completely. The disparity is clearly visible in the Figure 52.
To see the strong spatial variation of the thermal regime at Stockhorn, data taken at the
SOMOMOUNT (2012) soil moisture station are also analysed in comparison to simulated
values for the boreholes. The situation at this place is between the two extremes observed at
the boreholes, as shown in Figure 57. As water streams are crossing this area (see Figure 51),
the effect of the late snow patch during the AL is clearly visible. The ZC is also well
represented in the first half of November. However, the station is not directly in contact with
the reservoir. The water content is then half of the one simulated for the B17m. The ZC is also
shorter.
Figure 57: Water content measured at 50cm depth, Stockhorn soil moisture station (SOMOMOUNT, 2012). The Zero
Curtain Regime (ZC) and the effect of late snow patch during the Active Layer Regime (AL) is clearly visible.
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Freezing Regime (FR)
The Freezing Regime occurs in November for the B100m and around the end of December
for the B17m. It stops with the beginning of the snow melt between April and May. During
the FR, the soil is completely frozen and it is usually covered by snow that acts as an
insulator. Above a certain snow height, the subsurface may be considered as decoupled from
the atmosphere. In the CoupModel, this process is defined by the parameter
CritDepthSnowCover. With snow height around 0.5m in winter, this decoupling is not active
for both boreholes.
The ice may also regulate the variation of temperature coming from the high heat capacity of
the water. Besides, high ice content may reduce markedly the pore space available for the
water and air flow through the soil layers (Stähli et al., 1999). Thus, the heat transfer coming
from this mass circulation is also affected. With much higher ice content in winter (see Figure
54 and Figure 55), the B17m is then less sensible than the B100m to the cold temperatures.
This observation matches the observed differences between both boreholes (see Figure 52 and
Figure 53).
Snow Melt regime (SM)
The Snow Melt Regime occurs in spring, usually between April and June. During this period,
a considerable amount of energy is used to melt the snow and the ice. With the topography,
the melt water from the north is evacuated to the south. Thus, the water flow and the
accumulation at the freezing front is essential to understand the soil heat transfer during the
snow melt regime (Scherler et al., 2010; Hansson et al., 2004; Hinkel et al., 2001). Added to
the lateral heat flux effect postulated by Gruber et al. (2004), the SM creates a clear
disequilibrium of heat between the north and the south of the plateau. With this circulation of
latent heat, the snow cover and the ice near the B17m receive more energy. However, very
high ice content at the reservoir makes the ice melt last much longer at the B17m, as shown in
Figure 52.
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5. Discussion of the uncertainties
Now that the results of this study have been presented and interpreted, some elements should
be discussed. The first part of this chapter concerns the potential calibration biases in the
CoupModel and the position of the meteorological station. The second part focuses on the
problems caused by the calibration of Archie’s law. At first, the implementation of Archie’s
law in CoupModel and 4PM is compared. Then, the calibration of Archie’s parameters and
the epsilon factor for rock resistivity are discussed, followed by possible improvements.
Finally, an analysis of the 4PM restrictions for inconsistent solutions is quickly presented in
the last part of this chapter.
5.1. Potential calibration biases in CoupModel
The complexity of the CoupModel makes it an efficient tool for the simulation of the heat and
mass transfer into the soil. But the more a model takes into account all the factors involved in
these physical processes, the more it requires input data and other information about the
environment. With more than one hundred possible parameters to tune, the CoupModel has
become difficult to calibrate correctly with physically consistent parameters. Some potential
biases may then appear even if the calibrated and simulated temperatures are matching.
5.1.1. Meteorological data in CoupModel
The first source of uncertainty is the data from the meteorological station. The latter was
installed in June 2002 near the deeper borehole 6000 (100m) (Gruber et al., 2004). Thus,
some measured variables might not correspond to the situation at the smaller borehole. The air
temperature, the relative humidity of the air, the wind direction and the wind speed should not
be subjects to strong spatial changes. The processes involved are indeed occurring at a scale
much larger than the 28m separating the two boreholes. On the other hand, the snow cover
height may vary locally. The wind, the slope and the air and water circulation near the surface
may considerably influence the snow deposition and thawing (Scherler et al., 2013). This is
one of the reasons why the snow height is not used directly as input in the CoupModel.
Concerning the incoming and outgoing short-wave and long-wave radiation, they should not
vary along the plateau. Nevertheless, the small borehole is closer to the southern slope. The
~ 88 ~
diffusion of the radiation from this slope into the soil may add a consequent heat transfer that
the CoupModel does not take into account. One solution to improve the calibration and to
include this effect could be then to add a heat source in the simulated profile or to adapt the
thermal conductivity (Scherler et al., 2013; Marmy et al., 2015), but this requires a time-
consuming process to know which parameters are the best to represent this situation. The
daily air and water circulation near the surface and the short refreezing during the night must
also be considered. These processes occur at a small time scale and they cannot be well
represented in the general heat and mass transfer of the CoupModel. It might partially explain
the difference in water content near the surface between both models.
5.1.2. Uncertainties introduced by CoupModel parameters
Even if a lot of physical processes are taken into account in the CoupModel, some limitations
may still be observed especially for the representation of a frozen rock plateau like at
Stockhorn. One main source of uncertainty is the simulation of the composition of subsurface
pore matrix. In the CoupModel, the best way to represent bedrock is to consider a very dense
and relatively homogeneous sandy soil with porosity around 5%. But, it does not take into
account the possible fractures, ice lenses, etc. present under the surface. Thus, the macro-pore
flow is not well represented. To include this effect, the porosity must be raised and then the
bedrock is not represented precisely. It is then harder to calibrate precisely the water content if
the bedrock is simulated as sand, because of unrealistic water pressure head. This problem
also affects the simulation of air content into the soil. In a sandy soil, the hydraulic
conductivity concerns a homogeneous soil and it allows for a filling of the pores with ice. But
in fractured bedrock, the ice may obstruct the fractures and the air can be more easily trapped
into the subsurface.
Another uncertainty introduced in the CoupModel is the compensation effect of some
parameters. As explained in section 3.4.2., the GLUE calibration used for this work is a
method subject to the equifinality principle (Beven, 2006). This implies that two parameters
may compensate eachother and thus, a good matching between measured and simulated data
may be obtained with two completely different parameter sets. Thus, values should be
interpretated with care, even if the model is well calibrated.
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5.2. Calibration of Archie’s law in 4PM
The calibration of Archie’s equation is one of the main sources of uncertainty in the 4PM.
Finding the correct values of the parameters is always an issue that should be treated
carefully. First, the failure of the 4PM multi-run procedure for Stockhorn study site is
analysed. Then, the calibration of the Archie parameters and the epsilon factor for rock
resistivity are discussed, followed by possible improvements.
5.2.1. Calibration of m, n and ρw with a multi-run procedure
One aim of the new 4PM version 7 is to build a multi-run procedure in order to calibrate
Archie’s parameters using surface water content as a comparison (see section 3.3.3).
However, the application of this procedure is not conclusive for Stockhorn profiles of 2014.
For all the profiles, the measured water content with the ThetaProbe ML2x is much higher
than in the 4PM multi-runs results. It follows logically that all Archie’s parameters are
overestimated when the best run is searched, as shown in the Figure 58. This figure shows the
RMSE of all single run with different values of Archie’s parameters m, n and the pore water
resistivity (Figure 58, on the left hand side) and the parameter values for the best run, i.e. the
run with the lowest RMSE (Figure 58, on the right hand side). Several reasons may explain
the inability to find conclusive results. First, the instrument used for the measurement in 2014,
ThetaProbe ML2x, can have an error of approximately ± 5% (ΔT, 1999). Besides, modelled
values correspond to mean water content in the first 50cm, but the measurements are taken
right at the surface. Then, it was difficult to find a place to insert the ThetaProbe with the
abundance of bedrock and coarse blocks along the profiles. Considering the high spatial
variability of the soil moisture at Stockhorn, this is not representative of the situation around
and under the ThetaProbe. By consequence, data concern only specific areas without big
blocks and air in-between. With all these sources of errors, it is possible to affirm that using
soil moisture data at the surface is not an efficient way for the calibration of Archie’s
parameters, at least in a very heterogeneous soil such as a rock plateau. In his master thesis at
Albert-Ludwig-Universität Freiburg, Benjamin Mewes (2014) used the fuzzy logic to reduce
the uncertainty linked to Archie’s parameters. This solution might be included in a new
version of the model. In the meantime, a manual calibration is still needed with values found
in the literature according to the site characteristics.
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Figure 58: Multi-Run results for the Stockhorn Longitudinal South Profile. With a much higher measured water
content than the 4PM results, all Archie’s parameters are overestimated. On the left hand side: RMSE of all single
run with different values of Archie’s parameters m, n and the pore water resistivity. On the right hand side:
parameter values for the best run, i.e. the run with the lowest RMSE.
To see which model gives the best representation of water content near the surface, one
solution would have been to compare the CoupModel and 4PM values to soil moisture data
taken at the surface during the field campaign of August 2014. But with all these sources of
imprecisions, the data must be considered with care and a comparison may not be considered
as a reliable validation.Table 10 shows the simulated water content of both models and the
measured soil moisture at both boreholes.
Table 10: Comparison of the simulated water content of both models with measured data. For the models, the value
corresponds to the maximal water content in the first meter. The measurements are taken at the surface.
4PM
Aug 2006
4PM
July 2011
COUP
Aug 2006
COUP
July 2011
Measurement
Aug 2014
Maximal Water content in the
first meter (Borehole 100m) 2% 3% 4% 1.5% 10.5%
Maximal Water content in the
first meter (Borehole 17m) 11% 10% 26% 25% 25%
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For the borehole 6000 (B100m), both models indicate a water content between 2% and 4%
and the measured data give a soil moisture of 10.5%. With the abundance of bedrock visible
at the surface in this region, it was difficult to find a place to insert the ThetaProbe. Thus, the
difference of 6% to 8% may be due to much higher porosity (60%-80%) just where the
instrument was planted, which is not representative of the situation around. For the borehole
6100 (B17m), the measured soil moisture is very close to the CoupModel values. But again,
the measured water content might be overestimated due to the difficulty to find a place to
insert the ThetaProbe.
5.2.2. Epsilon factor for rock resistivity
An improvement of the 4PM is the use of the extended Archie’s law including an epsilon
factor parameterising a rock that is conductive (after I. Brandt, TU Denmark). The 4PM with
classical Archie’s law interprets areas with small resistivity and medium wave velocity as
mostly filled with water. But it may not detect if the low resistivity comes from the matrix or
from the pores. This is especially the case in the Stockhorn ERT monitoring profile where a
high water content is simulated under the permafrost table with temperatures below 0°C (see
Figure 39). With the Brandt epsilon factor, it is now possible to make a distinction between
pore and matrix low resistivity and to interpret those areas as filled mostly with ice and not
with water (see Figure 40). As this feature has never been treated before, no value for epsilon
can be found in the literature to assess the impact of the rock resistivity according to the
geology of the site. To assess the range of epsilon, a first test is run to see its impact on the
pore content for the areas such as those observed at Stockhorn. Thus, the resistivity is set to
10’000 Ωm, the wave velocity to 3’500 m/s and the other parameters to the same values as in
the 4PM calibration. Figure 59 shows the result of this investigation. The impact on pore
content is already considerable with an epsilon of 0.01. For a value of 0.1, the relative water
content falls from 95% to 15% and the relative ice content rises from 0% to 70%. A small
increase of 10% for the air may also be observed.
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Figure 59: Effect of epsilon on the pore content for values between 0 and 0.1. For epsilon>0.1, the pore contents
become more stable.
Now that the range of epsilon has been defined, the second test concerns the effect of epsilon
for different values of wave velocity and resistivity. Figure 60 indicates that, for a porosity of
20% and the other parameters set identically to the 4PM calibration (see Table 9), the
influence of epsilon is more pronounced in a certain range of resistivity around 10’000 Ωm.
However, the relative water content changes by at least 20% in most of the cases. Thus, it is
important to use the rock conductivity factor only if the matrix conductivity is higher than
usually for some areas.
In conclusion, it is difficult to quantify the accuracy of the pore content when using Archie’s
law with rock conductivity considering the uncertainty of epsilon value. In any case, a
possible error of at least 5% should be considered for the interpretation of the results.
Concerning the reasons for discrepancies in air content between the CoupModel and the 4PM,
it does not come from this new feature, regarding the fact that the epsilon factor has an impact
mostly on water and ice.
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Figure 60: Effect of Epsilon on pore content for every seismic velocity and resistivity. The other parameters were set
identically to the 4PM calibration.
5.2.3. Improvement of Archie’s law
One possible solution to improve the calibration of Archie’s law would be to consider the
relationship between resistivity and temperature applied to the pore water:
𝜌𝑤 =𝜌𝑤,0
1+𝛼 (𝑇−𝑇0) (39)
With: 𝜌𝑤 = pore water resistivity (Ωm) 𝑇 = Temperature (°C)
𝜌𝑤,0 = pore water resistivity at 𝑇0 (Ωm) 𝑇0 = Reference temperature (°C)
𝛼 = Temperature coefficient of resistivity, usually around 0.025 °C-1
This relation, described by Keller and Frischknecht (1966), may be parameterized in the
CoupModel. But as it is not the case in the 4PM, the temperature coefficient of resistivity 𝛼
was set to 0. Besides, this equation only concerns positive temperatures, which is mostly not
the case in permafrost areas. If boreholes data would be used to include this process in the
4PM, the latter would then gain in accuracy.
~ 94 ~
5.3. Solutions restrictions in the 4PM
One other source of uncertainty in the 4PM is the absence of physically consistent solutions
for some resistivities and wave velocities. If the value of ice, water or air content for a pixel is
below zero or above the porosity, the solutions are considered as inconsistent and they are
replaced by blanks. To avoid too many of these blanks in the profile, the possibility to extend
the range of solutions has been added in the GUI. With this option activated, negative values
of ice, water or air content are considered to be equal to 0. In that case, the other two pore
contents are normed to obtain a total pore content equal to the porosity. The corresponding
Matlab code, implemented in the 4PM file calc_restrictions.m is presented in appendix 3. But
the problem of this extension method is that different restrictions may be implemented to fill
the blanks. For example, values of ice, water or air content above the porosity could be
considered to be equal to the porosity. In that case, the other two pore contents are set to 0. In
conclusion, uncertainties coming from the choice of the restriction method should also be
considered for the interpretation of the results.
As shown in Figure 61 and Figure 62, the 4PM cannot find any correct content values if the
seismic velocities are too small. Unfortunately, this is often the case near the surface when a
lot of air is present. For this reason, the extended solutions have been applied to the 4PM for
the comparison with the CoupModel. As it concerns mainly the uppermost three meters, the
results near the surface are subject to imprecisions due to this extension.
In addition, only few combinations of resistivity and p-wave velocity have consistent solution
with Somerton random model compared to Archie’s law. Thus, the use of the extended model
is almost compulsory with the former.
~ 95 ~
Figure 61: Range of 4PM solutions for a porosity of 50% with Archie’s law. a) Physically correct solutions. b)
Extension of the solutions by assuming a content of 0% for negative values.
~ 96 ~
Figure 62: Range of 4PM solutions for a porosity of 50% with Somerton equation. a) Physically correct solutions. b)
Extension of the solutions by assuming a content of 0% for negative values. Without the extension, only few solutions
exist for the model.
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6. Conclusion
The first aim of the thesis was to examine how the 4-phase model may be improved to better
assess the ice, water and air content of the subsurface. In the 4PM version 7, a topography file
is not needed anymore and a higher resolution may be used for more accurate results. Besides,
it is now possible to easily change all free parameters using a graphical user interface (GUI).
The porosity can be defined more precisely and two alternative equations to Archie can be
selected to consider the conductivity of rock. Soil moisture or ice core data can also be used to
calibrate the model with a multi-run. However, for the presented case study at Stockhorn, the
comparison with data taken at the surface with a ThetaProbe ML2x does not work due to the
heterogeneity of the soil and the accuracy of the instrument.
The second aim of the thesis was to see in what way the 4PM may be compared with the
CoupModel. Even if there are some discrepancies in pore content between the 4PM and the
CoupModel, the variation with depth matches very well. The location of the freezing front is
similar for both models and the accumulation of water is always detected, if there is any. As
both models are different in their conception, it is difficult to identify precisely the cause of
the discrepancies. For the first three meters near the surface, the effect of the extended 4PM
solution might provoke imprecisions and the CoupModel is probably more robust. On the
other hand, the 4PM can more easily consider some specific and local features directly from
the ERT and RST that are not detected by CoupModel. The distance between the
measurements used in the models may also play a major role. Even though only 3-4 meters
separate them, the geophysical profile is installed mostly in fine sediments and the boreholes
are drilled in apparent bedrock. For the deeper part, the inhomogeneity of the soil with the
presence of fractures leads to the conclusion that the ice content values at Stockhorn are
probably between the simulated values of the 4PM and the CoupModel, i.e. around 13% for
the B100m and 15% for the B17m at 10m depth. The only solution to know for certain which
model better assesses the absolute ice content at Stockhorn would be to compare the results
with ice core data, which do not exist so far. In conclusion, the comparison between the
CoupModel and the 4PM shows some conclusive results, but further validations are needed to
be done with measured ice contents, as it will be within the master thesis of David Schwery at
the University of Fribourg for field sites in Svalbard.
~ 98 ~
The last aim of the thesis focused on the spatial and temporal repartition of ice and water
content on the Stockhorn plateau. The main process influencing this repartition is hereby the
topography. The southern slope is more exposed to solar radiation and the resulting lateral
heat flux does not affect the northern part of the plateau in the same way as for the southern
part. In addition, the topography of the plateau canalises the snow melt water into small
streams during the summer. The latter accumulates then in a natural reservoir next to the
B17m with ice and bedrock acting like a small dam. This process induces a consequent
transport of latent heat and it also acts as an insulator against the summer heat. With such
spatially heterogeneous hydraulic processes influencing the repartition of latent heat on the
plateau, the models are very complicated to calibrate. In any case, the water content should
not be neglected for the calibration of the CoupModel.
Other improvements should be considered for future versions of the 4PM. First, the resistivity
and seismic equations could be reconsidered as it was already the case with Archie’s law.
Many possibilities are proposed in different papers (see Glover, 2010). The calibration of
Archie’s parameters is still problematic. Other solutions are also considered to solve this
problem. The system based on fuzzy logic proposed by Mewes (2014) could be included in a
new version of the model. In the meantime, the 4PM version 7 is now fully functional and it
can be applied and tested for various data sets and applications.
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7. Bibliography
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Berthod, N. (2012). Exploration et complétion des données de la station climatique du
Stockhorn. Bachelor Thesis, Fribourg, Geosciences Department, University of
Fribourg.
Beven KJ. (2006). A manifesto for the equifinality thesis. Journal of Hydrology, 320,
18-36.
Dängeli, S. (2013). Eisgehalt in alpinen Permafrostregionen. Master Thesis, Fribourg,
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Delaloye, R., & Lambiel, C. (2005). Evidence of winter ascending air circulation
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Delaloye, R., Morard, S., Barboux, C., Abbet, D., Gruber, V., Riedo, M. & Gachet, S.
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Gesellschaft, 29, 21-31.
ΔT (1999). ThetaProbe soil moisture sensor type ML2x user manual. Cambridge,
Delta-T Devices.
Ekici, A., Chadburn, S., Chaudhary, N., Hajdu, L. H., Marmy, A., Peng, S., ... & Beer,
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~ 104 ~
~ 105 ~
8. Appendix
8.1. Field campaign protocol: Stockhorn, August 2014
27.08.2014
Arrival on site at 9am. The first two hours on the site were used to install the material in the
cabin and to explore the terrain. The terrain was inconvenient to walk through, especially
when we need to pass the 4m cliff with the cross profile. Besides, some rocks were slippery in
the southern slope (Jutta slightly hurt her leg when she was climbing up). Until the end of the
day, we were able to finish the Seismic for the monitoring profile (CM) with one ERT
measurement. We also installed the New Cross Profile (CN) and did an electric measurement.
After a manipulation mistake on the computer, the first ERT data (Wenner) for the CN profile
were deleted. We then decided to do only Wenner-Schlumberger measurements instead of one
Wenner and one Dipole-Dipole as planned at the beginning.
28.08.2014
In the morning, we did the ERT measurement again for CN and we installed the seismic
profile. Then, we prepared the Longitudinal South (LS) profile and did both electric and
seismic measurements until the end of the day. We also did another electric measurement for
the fix profile.
29.08.2014
The last day, we did a measurement for the fix profile. Then, we installed the Longitudinal
North (LN) Profile. We started the electric measurement for LN 5 minutes during the electric
measurement of the fix profile and we stopped it when we realized that the fix profil was still
running. After the fix profile was finished we started again the measurement for LN, and did
also the seismic measurements .Finally, we did an electric measurement for a half profile
(LM) between the two other longitudinal. The helicopter came back at 2pm.
~ 106 ~
Cross Monitoring Profile (CM):
A monitoring profile was installed in summer 2005 near the two boreholes. Approximately
half of the electrodes were on the plateau, and the rest was on the southern slope. An
extension of 7 electrodes to the north (for a total of 55) was made two years later. During the
2014 campaign, a refraction seismic measurement was conducted to allow for the utilisation
of the 4-Phase-Model. To match the RST profile the first 8 electrodes of the northern slope
were not used for the inversion process
Cross New Profile (CN):
As the monitoring profile was at 6m (STO_6100) or 7m (STO_6000) from the boreholes, we
decided to put a first profile parallel to the monitoring profile (at exactly 5m in direction of
the boreholes) in order to have a closer look to the boreholes area. Then, the objective was to
install a second cross profile on the other side of the boreholes to have a 3D profile of the
boreholes (not done by lack of time and by a too difficult terrain on the southern slope)
Most of the electrodes on the plateau were good (RS-Check 20-150 kΩ). Four of them had to
be corrected (successfully). On the southern slope, most of the electrodes were not detected.
Then, we put two electrodes and as many sponges at each point to increase the conductivity.
We got a RS-Check of less than 250 kOhm for all the electrodes except for the 5 last that we
were not able to connect. These 5 electrodes have been deleted from the profile.
- Electrode 1 in melted ice (we dug into the ice to reach the soil)
- Electrodes 24 and 25 (Geophone 12) at the edge of the cliff
- Electrodes 26 and 27 (Geophone 13) on the cliff
- Electrodes 28 (Geophone 14) at the base of the cliff
- Electrodes 44,45,46,47,48 not used (unable to detect the electrodes)
- Passing Borehole 100m at electrodes 8-9 (at 2.5m)
- Passing Borehole 17m at electrodes 19-20 (at 0.5m)
- Crossing LN at electrode 7-8
- Crossing LS at electrode 19
~ 107 ~
Longitudinal South Profile (LS):
LS was delimited to pass next to the 17m borehole, perpendicular to the cross profiles and
parallel to the small cliff, but at a reasonable distance. This profile was quite fast to install
compared to CN, but still with RS-Check values between 100 kΩ and 150kΩ.
- Passing Borehole 17m at electrodes 15 (at 0.5m)
- Crossing CN at electrode 15-16
- At approximately 10m from small cliff
Longitudinal North Profile (LN):
LN was delimited to pass next to the 100m borehole, perpendicular to the cross profiles and at
the position of Susanne Daengeli’s Longitudinal Profile. (Old electrodes were found on the
site for the first half of LN Profile). This profile was also quite fast to install compared to CN,
but still with RS-Check values between 100 kΩ and 150kΩ. A lot of point had two electrodes
and sponges.
- Electrode 1 to 5 on a big block
- Passing Borehole 100m at electrodes 15 (at 0.5m)
- Crossing CN at electrode 14 (19m) and crossing CM at electrode 17 (24m)
- At 8m from the big north cliff at electrode 1
- At 23m from the big north cliff at electrode 48
Longitudinal Middle Profile:
This profile was delimited to pass between LN and LS and perpendicular to CM and CN.
Three electrodes were more difficult to install. By lack of time, the seismic measurements
have not been done.
- Electrode 24 at 9.5 m to electrode 33 of LN
- Crossing CN at electrode 5-6 and crossing CM at electrode 9
- Electrode 1 to 3 at the base of a big block
~ 108 ~
8.2. List of major 4PM improvements since version 5.0
Version 5.2
- Creation of a 3PM independent from the 4PM
o Integration of the 3PM in the 4PM to calculate porosity where there is no ice
(according to a certain depth).
o Numerical solver of 3PM equation to allow a parameterization with different m
and n parameters
o Test to compare analytical and numerical solutions (former tests showed an
maximal error of 10-12
% for rock content after 4 iterations
- Possibility to include data with 0.5m resolution
Version 6.0
- Integration of a graphical user interface
- Correction of the porosity calculation using 3PM results.
- Integration of the RMS error in the plots
- Minor bugs correction (old 3PM parameters disturbed the run, quit without saving, ...)
Version 6.3
- Integration of a menu system
- Integration of handles instead of normal parameters (Handles parameters can be more
easily transported between the functions)
- Correction of minor bugs (restriction of porosity for negative values, ...)
Version 7.0
- The parameters for each site/profil are now stored in .FPM files.
- GUI management
o Separation of the GUI management and model functions.
o The file Main_User_Interface.m is only used to manage GUI buttons.
o Display/Hide some parts of the GUI according to options.
o Integration of error messages in case of incorrect values inserted in the GUI
- Porosity model
o Integration of a smoothing parameter and a minimal porosity in the main
porosity model
o The values in the zonal model are now in meters and not in pixels ( ==> no
changes if resolution = 0.5m)
o The porosity matrix is not reversed anymore.
o The position of the surface is defined according to the surface in RST data.
- Integration of new parameters for the new equations
- Integration of a multi-run procedure to test different parameter set.
o Possibility to include water content or ice content data for validation
- Integration of the possibility to include a second data set for a temporal comparison.
- Some variables appear again in the main workspace to export the results if needed.
- The GUI resizes itself according to the screen resolution.
~ 109 ~
Version 7.1
- The Brandt and Somerton equations are implemented
- The 3PM calibration is included in the multi-run procedure
- The colormap for differential plots is different from the classical one.
- The two data sets can be switched
- The restrictions are corrected.
- The resolution and the presence of empty columns are automatically calculated.
- The topography is calculated automatically (the topo file is not needed anymore)
- The parameters values are restricted between a min and a max.
- The FPM files are no longer the same (but older files can still be read).
Version 7.2
- More comments are added to the code to help the users
- The restrictions are corrected.
- ERT and RST data can have different dates.
- The correction of empty columns at the beginning of the RST data is included.
- A first automatic matching of ERT and RST data is implemented
- The interval for the parameter values is not restricted anymore, but it is kept as a
suggestion.
~ 110 ~
8.3. 4PM structure and coding examples
As the Matlab code became consequent with the addition of the new features, the functions
have been decomposed and reorganised in several files. The Figure 63 and Figure 64 below
show which functions are called when the button Run 4PM and Multi-Run 4PM are pushed
and the link between those functions.
Figure 63: Functions called when the button Run 4PM is pushed. The functions in grey are called only if a specific
option is activated
Figure 64: Functions called when the button Multi-Run 4PM is pushed. The functions in grey are called only if a
specific option is activated
Examples of some new Matlab functions coded in the 4PM version 7 (model core, complex
porosity model, 3PM calculation, 4PM calculation, restrictions and 4PM multi-run) are
presented in the next pages.
~ 111 ~
Model Core (new features)
% calc_ModelCore.m
% Function where all the basis functionalities of the model are done.
% This function is always called first by the model.
% 1a. Data insertion and model resolution
% 1b. Data insertion for comparison
% 2. Parameters to display the results
% 3. Definition of the Boreholes
% 4. Other functions called in the model core
function [handles]=calc_ModelCore(handles)
evalin('base', 'clear all'); % Clear the base workspace.
% ********************************************
% 1a. Data insertion and model resolution
% ********************************************
% Definition of the variable to catch some errors in the 4PM.
handles.error_found = 0;
% ******* Data insertion with error management *******
try % Check if inserting the data return an error.
ERT1=load(handles.metricdata.ERTin); % Extraction of ERT data
RST1=load(handles.metricdata.RSTin); % Extraction of seismic data
catch err % if an error occurs when inserting the data, the "catch" section is called
handles.error_found = 1;
if (strcmp(err.identifier,'MATLAB:load:numColumnsNotSame')) % Display an error message if
the ERT file is still complete.
errordlg('The ERT file does not have the same number of columns for each line. Forgot
to keep only the wanted part of ERT file ?','Impossible to read the ERT file');
elseif(exist(handles.metricdata.ERTin,'file')==0) % If the name of the file is not valid,
display an error message
errordlg('The ERT file was not found. (Forgot the .xyz or .txt extension ?)','File not
found');
elseif(exist(handles.metricdata.RSTin,'file')==0) % If the name of the file is not valid,
display an error message
errordlg('The RST file was not found. (Forgot the .ASC extension ?)','File not
found');
else
rethrow(err);
end
return
end
% ******* Model Resolution *******
ModRes = RST1(2,2)-RST1(1,2); % Calculate the model resolution
handles.ModRes = ModRes; % Define the model resolution for all the functions
if (handles.ModRes~=1 && handles.ModRes~=0.5) % Error message if the resolution is not valid
errordlg('The model resolution used in RST is not valid. Please use a 0.5 or 1.0 space
increment','Invalid resolution');
handles.error_found = 1;
return
end
…
~ 112 ~
% ******* Extraction of the topography *******
txx = zeros(size(vel1,2),1);
tzz = zeros(size(vel1,2),1);
for i=1:size(vel1,2)
b = find(~isnan(vel1(:,i)), 1 ,'first'); % Calculation of ground surface position in the
RST matrix
txx(i,1)=i*handles.ModRes; % create x dimension for topography
tzz(i,1)=b*handles.ModRes-0.5; % create z dimension for topography
end
% ******* Creation of handles variables *******
handles.x = x;
handles.Z = Z;
handles.txx = txx; % X-Values for the topography
handles.tzz = tzz; % Z-Values for the topography
handles.Nz = Nz;
handles.vel1 = vel1; % Seismic data
handles.res1 = res1; % ERT data interpolated on RST grid and cut where no RST data exists
handles.RES1 = RES1; % ERT data interpolated on RST grid.
handles.multi_run_on = 0; % variable used to know if the multi-run is active
handles.restrictions_on = 0; % variable used to know if the restriction function is active
% ******* Creation of workspace variables *******
assignin('base', 'Resolution', handles.ModRes);
assignin('base', 'X', handles.x);
assignin('base', 'Z', handles.Z);
assignin('base', 'RST_data', vel1);
assignin('base', 'ERT_data', res1);
assignin('base', 'ERT_raw_data', RES1);
…
% ********************************************
% 2. Parameters to display the results
% ********************************************
handles.axisdimMatch=[x(1) x(size(x,1)) -DepthMax-2 6]; % axis dimension for all subplots
handles.axisdim=[x(1) x(size(x,1)) -DepthMaxVel-2 6]; % axis dimension for all subplots
…
% ********************************************
% 4. Other functions
% ********************************************
[handles] = calc_RrComplex(handles); % function for complex resistivity of rock
return
~ 113 ~
Porosity function
% Function to build the porosity model
function [handles] = calc_porosity(handles,SelectedModel)
…
% Creation of a porosity matrix
Phi_grad = handles.vel1;
[row,col]=find(~isnan(Phi_grad));
for ii=1:size(row); Phi_grad(row(ii),col(ii))=1; end;
% 3PM run in case of calibration with 3PM
if (handles.metricdata.CheckAutoCal3PM == 1)
[TPMfr,TPMfw,TPMfa,handles]=calc_3PM(handles,handles.res1,handles.vel1,SelectedModel);
Td3phi_1 = handles.TPMphi;
end
% Extraction of calibration file in case of calibration with 3PM
if (handles.metricdata.CheckCalFile == 1)
try
Phi3PM = dlmread(handles.metricdata.CalFilName); % Extraction of porosity file
catch err
…
if (strcmp(err.identifier,'MATLAB:load:numColumnsNotSame'))
errordlg('The porosity file does not have the same number of columns for each
line. The file has been ignored.','Impossible to read the file');
elseif(exist(handles.metricdata.WaterContData,'file')==0) % If the name of the file is
not valid, display an error message
errordlg('The porosity file was not found. (Forgot the .txt extension ?). The file
has been ignored.','File not found');
else
rethrow(err);
end
return
end
end
% Definition of a fix depth for horizontal zones and integration of model resolution
for k = 1:size(Zones,1)
FixDepth(k)=-100;
Zones(k,3)=round(handles.metricdata.ComplexPorosity(k,3)/handles.ModRes);
Zones(k,4)=round(handles.metricdata.ComplexPorosity(k,4)/handles.ModRes);
Zones(k,5)=round(handles.metricdata.ComplexPorosity(k,5)/handles.ModRes);
Zones(k,6)=round(handles.metricdata.ComplexPorosity(k,6)/handles.ModRes);
if (handles.metricdata.ComplexPorosity(k,6) == 0)
Zones(k,6)=Nz;
end
end
% Beginning of horizontal loop (each i correspond to each column of the matrix (X Position) )
for i = 1:size(Phi_grad,2)
% Calculation of ground surface position in the RST matrix
b = find(~isnan(Phi_grad(:,i)), 1 ,'first'); % Calculation of ground surface position in
the RST matrix
phi_cur0 = phi_ini + grad; %Reinitialization of main porosity value for every column
~ 114 ~
%Definition of min depth and max depth for every zone
for k = 1:size(Zones,1)
if (Zones(k,1) == 1 && i >= Zones(k,3) && i <= Zones(k,4))
phi_cur(k) = Zones(k,7) + Zones(k,8);
DEPTHmin(k) = Zones(k,5) + b; DEPTHmax(k) = Zones(k,6) + b; %Depths definition
if (Zones(k,2) == 0 && i >= Zones(k,3) && i <= Zones(k,4)) % Condition for
horizontal zones
if (FixDepth(k) == -100) FixDepth(k) = DEPTHmin(k); %Fix depth defintion for
horizontal zones
else DEPTHmin(k) = FixDepth(k); DEPTHmax(k) = FixDepth(k) + (Zones(k,6)-
Zones(k,5)); %Depths for horizontal zones
end
end
end
end
% Beginning of vertical loop (each j correspond to a different depth)
for j = b:size(Phi_grad,1)
% Main Porosity Model
if (phi_cur0 <= PhiMin)
phi_cur0 = PhiMin; % No decrease under the minimum value
elseif (phi_cur0 > thres)
phi_cur0 = phi_cur0-grad; % Rapid decrease of the porosity above the threshold
else
phi_cur0 = phi_cur0-grad*0.1; % Low decrease of the porosity below the threshold
end
Phi_grad(j,i) = phi_cur0;
% 3PM Porosity Model
if (handles.metricdata.CheckCalFile && size(Phi3PM,2)>=i && size(Phi3PM,1)>=j &&
isnan(Phi3PM(j,i))==false)
Phi_grad(j,i)=Phi3PM(j,i);
if (handles.metricdata.checkgrad3pm == 1) phi_cur0 = Phi3PM(j,i); end
elseif (handles.metricdata.CheckAutoCal3PM == 1 && isnan(Td3phi_1(j,i))==false)
Phi_grad(j,i)=Td3phi_1(j,i);
if (handles.metricdata.checkgrad3pm == 1) phi_cur0 = Td3phi_1(j,i); end
end
% Zonal Porositiy Model
for k = 1:size(Zones,1)
if (Zones(k,1) == 1 && i >= Zones(k,3) && i <= Zones(k,4) && j >= DEPTHmin(k) && j
<= DEPTHmax(k))
if (phi_cur(k) > Zones(k,9))
Phi_grad(j,i) = phi_cur(k) - Zones(k,8);
else Phi_grad(j,i) = phi_cur(k) - Zones(k,8)*0.1;
end
phi_cur(k) = Phi_grad(j,i);
…
% Smoothing of the porosity matrix
% The smoothing code comes from the smooth2a file by Greg Reeves (2009).
% The complete file is available at the following address:
% http://www.mathworks.com/matlabcentral/fileexchange/23287-smooth2a
…
~ 115 ~
4PM calculation
% calc_4PM.m
% Function to calculate volumetric fractions of rock, water, ice and air
function [FPMfr,FPMfw,FPMfi,FPMfa]=calc_4PM(handles,res,vel,Res_equ)
% Extraction of the GUI variables
…
% 4PM calculation according to the resistivity equation
switch Res_equ
case 'Archie'
FPMfr = 1-(phi);
FPMfw = ((rw*a)^(1/n)./(1-FPMfr).^(m/n-1))./(res.^(1/n));
FPMfi = ((vi./vel)-(vi*a^(1/n)*rw^(1/n)./(vw.*(1-FPMfr).^(m/n-1)))./(res.^(1/n))+(-
vi.*FPMfr./vr-vi/va+vi.*FPMfr./va)+(vi*a^(1/n)*rw^(1/n)./(va.*(1-FPMfr).^(m/n-
1)))./(res.^(1/n)))*(1/(1-vi/va));
FPMfa = (va./vel-(va*rw^(1/n)*a^(1/n)./(vw.*(1-FPMfr).^(m/n-1)))./(res.^(1/n))+(-
va.*FPMfr./vr-va/vi+va.*FPMfr./vi)+(va*rw^(1/n)*a^(1/n)./(vi.*(1-FPMfr).^(m/n-
1)))./(res.^(1/n))).*(1/(1-va/vi));
case 'Brandt'
rwB = rw./(1+rw.*epsilon);
FPMfr = 1-(phi);
FPMfw = ((rwB.*a).^(1/n)./(1-FPMfr).^(m/n-1))./(res.^(1/n));
FPMfi = ((vi./vel)-(vi*a^(1/n).*rwB.^(1/n)./(vw.*(1-FPMfr).^(m/n-1)))./(res.^(1/n))+(-
vi.*FPMfr./vr-vi/va+vi.*FPMfr./va)+(vi*a^(1/n)*rwB.^(1/n)./(va.*(1-FPMfr).^(m/n-
1)))./(res.^(1/n)))*(1/(1-vi/va));
FPMfa = (va./vel-(va.*rwB.^(1/n).*a^(1/n)./(vw.*(1-FPMfr).^(m/n-1)))./(res.^(1/n))+(-
va.*FPMfr./vr-va/vi+va.*FPMfr./vi)+(va*rwB.^(1/n)*a^(1/n)./(vi.*(1-FPMfr).^(m/n-
1)))./(res.^(1/n))).*(1/(1-va/vi));
case 'Somerton'
Lres = log10(res);
Lra = log10(ra);
Lri = log10(ri);
Lrw = log10(rw);
Lrr = log10(rr);
xxx = va*(Lra/Lrw-1);
yyy = 1-va*Lra/vw/Lrw;
FPMfr = 1-(phi);
FPMfi = (1-FPMfr-Lres./Lrw+FPMfr.*Lrr./Lrw+xxx./yyy.*(1./vel-FPMfr./vr-
Lres./vw./Lrw+FPMfr.*Lrr./vw./Lrw)) ./ (1-Lri/Lrw-xxx./yyy.*(Lri/vw/Lrw)-1/vi);
FPMfa = 1./yyy.*(va./vel-FPMfr.*va./vr-FPMfi.*va./vi-(va/vw/Lrw).*(Lres-FPMfi.*Lri-
FPMfr.*Lrr));
FPMfw = (1/Lrw).*(Lres-FPMfi.*Lri-FPMfr.*Lrr-FPMfa.*Lra);
otherwise
FPMfr = 1-phi;
FPMfw = phi-phi;
FPMfi = phi-phi;
FPMfa = phi-phi;
end
% Restrictions
[FPMfr,FPMfw,FPMfa,FPMfi]=calc_restrictions('4PM',handles,vel,res,Res_equ,FPMfr,FPMfw,FPMfa,FP
Mfi);
return
~ 116 ~
3PM calculation
% calc_3PM.m
% 3PM function to calculate volumetric fractions of rock, water and air
function [TPMfr,TPMfw,TPMfa,handles]=calc_3PM(handles,res,vel,Res_equ)
% Extraction of the GUI variables
vw = handles.metricdata.TPvw; % Prescribed seismic velocity of water
va = handles.metricdata.TPva ; % Prescribed seismic velocity of air
vr = handles.metricdata.TPvr; % Prescribed seismic velocity of rock
…
% 3PM calculation according to the resistivity equation
switch Res_equ
case 'Archie'
fr0=vel-vel+0.7; % Creation of a rock fraction matrix with an initial value = 0.7
fr1=fr0-(-1./vel+1/va-fr0./va+fr0./vr+((1-fr0).^((n-m)/n).*((a*rw)./res).^(1/n).*(va-
vw))./(va*vw))./(-1/va+1/vr-((1-fr0).^((n-m)/n-1).*(n-m).*(a.*rw./res).^(1/n).*(va-
vw))./(n*va*vw));
fr2=fr1-(-1./vel+1/va-fr1./va+fr1./vr+((1-fr1).^((n-m)/n).*((a*rw)./res).^(1/n).*(va-
vw))./(va*vw))./(-1/va+1/vr-((1-fr1).^((n-m)/n-1).*(n-m).*(a.*rw./res).^(1/n).*(va-
vw))./(n*va*vw));
fr3=fr2-(-1./vel+1/va-fr2./va+fr2./vr+((1-fr2).^((n-m)/n).*((a*rw)./res).^(1/n).*(va-
vw))./(va*vw))./(-1/va+1/vr-((1-fr2).^((n-m)/n-1).*(n-m).*(a.*rw./res).^(1/n).*(va-
vw))./(n*va*vw));
TPMfr=fr3-(-1./vel+1/va-fr3./va+fr3./vr+((1-fr3).^((n-
m)/n).*((a*rw)./res).^(1/n).*(va-vw))./(va*vw))./(-1/va+1/vr-((1-fr3).^((n-m)/n-1).*(n-
m).*(a.*rw./res).^(1/n).*(va-vw))./(n*va*vw));
TPMfw = ((a*rw)./res.*(1-TPMfr).^(n-m)).^(1/n); % Water content
TPMfa = va.*(1./vel-TPMfw./vw-TPMfr./vr); % Air content
phi4 = 1-TPMfr;
case 'Brandt'
rwB = rw./(1+rw.*epsilon);
fr0=vel-vel+0.7; % Creation of a rock fraction matrix with an initial value = 0.7
fr1=fr0-(-1./vel+1/va-fr0./va+fr0./vr+((1-fr0).^((n-
m)/n).*((a.*rwB)./res).^(1/n).*(va-vw))./(va*vw))./(-1/va+1/vr-((1-fr0).^((n-m)/n-1).*(n-
m).*(a.*rwB./res).^(1/n).*(va-vw))./(n*va*vw));
fr2=fr1-(-1./vel+1/va-fr1./va+fr1./vr+((1-fr1).^((n-
m)/n).*((a.*rwB)./res).^(1/n).*(va-vw))./(va*vw))./(-1/va+1/vr-((1-fr1).^((n-m)/n-1).*(n-
m).*(a.*rwB./res).^(1/n).*(va-vw))./(n*va*vw));
fr3=fr2-(-1./vel+1/va-fr2./va+fr2./vr+((1-fr2).^((n-
m)/n).*((a.*rwB)./res).^(1/n).*(va-vw))./(va*vw))./(-1/va+1/vr-((1-fr2).^((n-m)/n-1).*(n-
m).*(a.*rwB./res).^(1/n).*(va-vw))./(n*va*vw));
TPMfr=fr3-(-1./vel+1/va-fr3./va+fr3./vr+((1-fr3).^((n-
m)/n).*((a.*rwB)./res).^(1/n).*(va-vw))./(va*vw))./(-1/va+1/vr-((1-fr3).^((n-m)/n-1).*(n-
m).*(a.*rwB./res).^(1/n).*(va-vw))./(n*va*vw));
TPMfw = ((a*rwB)./res.*(1-TPMfr).^(n-m)).^(1/n); % Water content
TPMfa = va.*(1./vel-TPMfw./vw-TPMfr./vr); % Air content
phi4 = 1-TPMfr;
~ 117 ~
case 'Somerton'
Lres = log10(res);
Lra = log10(ra);
Lrw = log10(rw);
Lrr = log10(rr);
yyy = 1-va*Lra/vw/Lrw;
TPMfr = (1 - va./vel./yyy + va.*Lres./vw./Lrw./yyy - Lres./Lrw -
Lra.*va./Lrw./vel./yyy + Lra.*va.*Lres./Lrw./vw./Lrw./yyy ) ./ (-va./vr./yyy +
va.*Lrr./vw./Lrw./yyy - Lrr./Lrw + Lra.*va./Lrw./vr./yyy -
Lra.*va.*Lrr./Lrw./vw./Lrw./yyy );
TPMfa = ( va./vel - TPMfr.*va./vr - va./vw./Lrw.*(Lres-TPMfr.*Lrr) )./yyy;
TPMfw = Lres./Lrw - TPMfr.*Lrr./Lrw - TPMfa.*Lra./Lrw;
phi4 = 1-TPMfr;
otherwise
TPMfr = vel-vel;
TPMfw = vel-vel;
TPMfa = vel-vel;
phi4 = 1-TPMfr;
end
% Restrictions
[TPMfr,TPMfw,TPMfa,~]=calc_restrictions('3PM',handles,vel,res,Res_equ,TPMfr,TPMfw,TPMfa,0);
% Restrictions if the 3PM is used to calibrate the porosity.
if (handles.metricdata.CheckAutoCal3PM==1 && handles.restrictions_on == 0 &&
handles.multi_run_on == 0)
…
[~,~,FPMfi,~]=calc_4PM(handles,handles.res1,handles.vel1,Res_equ);
for k = 1:size(TPMfr,2)
for l = 1:size(TPMfr,1)
if (FPMfi(l,k) > handles.metricdata.TTolThresh/100 || isnan(FPMfi(l,k)) == 1 ||
phi4(l,k)>0.8)
TPMfr(l,k) = NaN;
TPMfw(l,k) = NaN;
TPMfa(l,k) = NaN;
phi4(l,k) = NaN;
end
end
end
handles.TPMphi=phi4;
% Creation of the 3PM porosity file.
filename=([handles.metricdata.SiteCodeString '_' num2str(handles.metricdata.DateRST1)
'_Porosity_Calibration_3PM.txt']);
dlmwrite(filename,phi4);
movefile(filename,'./Data');
end
return
~ 118 ~
Restrictions
% calc_restrictions
% Function to eliminate the unconsistent solutions of the 4PM and 3PM
function [fr,fw,fa,fi]=calc_restrictions(Type,handles,vel,res,SelectedModel,fr,fw,fa,fi)
% Extraction of restriction level from GUI
Fill=handles.metricdata.checkFillGap;
if (handles.metricdata.checkSupersat == 1 && handles.metricdata.checkFillGap == 1)
Sat=1; else Sat=0; end
% Restrictions for 3PM
…
% Restrictions for 4PM
if (strcmp(Type,'4PM'))
for i = 1:size(fi,1)
for j = 1:size(fi,2)
% Restrictions for small resistivity or velocity
if vel(i,j) <= 175 || res(i,j) <= 175 %if res or vel values below threshold, set
fi=fa=fw=NaN
fi(i,j)=NaN;
fa(i,j)=NaN;
fw(i,j)=NaN;
end
if vel(i,j) <= 700 && res(i,j) <= 3000 % fi=fa=fw=NaN for values below threshold.
fi(i,j)=NaN;
fa(i,j)=NaN;
fw(i,j)=NaN;
end
if vel(i,j) <= 1100 && res(i,j) <= 700 % fi=fa=fw=NaN for values below threshold.
fi(i,j)=NaN;
fa(i,j)=NaN;
fw(i,j)=NaN;
end
% Restrictions when the water content is negative
if (fw(i,j) <= 0)
if (Fill == 0) % If the "fill gaps" option is not activated, fi=fa=fw=NaN
fw(i,j) = NaN;
fi(i,j) = NaN;
fa(i,j) = NaN;
else
fw(i,j) = 0;
Somme=fa(i,j)+fi(i,j);
if (Sat == 1 && Somme > 1)
fa(i,j) = fa(i,j)/Somme*1;
fi(i,j) = fi(i,j)/Somme*1;
else
fa(i,j) = fa(i,j)/Somme*handles.Phi_grad(i,j);
fi(i,j) = fi(i,j)/Somme*handles.Phi_grad(i,j);
end
end
end
~ 119 ~
% Restrictions when the ice content is negative
if (fi(i,j) <= 0)
if (Fill == 0) % If the "fill gaps" option is not activated, fi=fa=fw=NaN
fw(i,j) = NaN;
fi(i,j) = NaN;
fa(i,j) = NaN;
else
fi(i,j) = 0;
Somme=fw(i,j)+fa(i,j);
if (Sat == 1 && Somme > 1)
fa(i,j) = fa(i,j)/Somme*1;
fw(i,j) = fw(i,j)/Somme*1;
else
fa(i,j) = fa(i,j)/Somme*handles.Phi_grad(i,j);
fw(i,j) = fw(i,j)/Somme*handles.Phi_grad(i,j);
end
end
end
% Restrictions when the air content is negative
if (fa(i,j) <= 0)
if (Fill == 0) % If the "fill gaps" option is not activated, fi=fa=fw=NaN
fw(i,j) = NaN;
fi(i,j) = NaN;
fa(i,j) = NaN;
else
fa(i,j) = 0;
Somme=fw(i,j)+fi(i,j);
if (Sat == 1 && Somme > 1)
fi(i,j) = fi(i,j)/Somme*1;
fw(i,j) = fw(i,j)/Somme*1;
else
fi(i,j) = fi(i,j)/Somme*handles.Phi_grad(i,j);
fw(i,j) = fw(i,j)/Somme*handles.Phi_grad(i,j);
end
end
end
% Restrictions when the total pore content is above the porosity
if (Sat == 0 && fw(i,j)+fi(i,j)+fa(i,j)>handles.Phi_grad(i,j))
Somme=fw(i,j)+fi(i,j)+fa(i,j);
if (Sat == 1 && Somme > 1)
fa(i,j) = fa(i,j)/Somme*1;
fi(i,j) = fi(i,j)/Somme*1;
fw(i,j) = fw(i,j)/Somme*1;
elseif (Fill == 1)
fa(i,j) = fa(i,j)/Somme*handles.Phi_grad(i,j);
fi(i,j) = fi(i,j)/Somme*handles.Phi_grad(i,j);
fw(i,j) = fw(i,j)/Somme*handles.Phi_grad(i,j);
elseif (fw(i,j)>handles.Phi_grad(i,j) || fi(i,j)>handles.Phi_grad(i,j) ||
fa(i,j)>handles.Phi_grad(i,j))
fw(i,j) = NaN;
fi(i,j) = NaN;
fa(i,j) = NaN;
end
end
end
end
end
return
~ 120 ~
Multi-Run 4PM
% calc_4PM_multi
% Function to calculate the RMS Error between measured data and calculated values for
different parameters
function [Multi_RMS,handles]=calc_4PM_multi(handles,Res_equ)
% Extraction and setting of data
…
% Creation of different matrix for Multi-Run
% Interpolation of validation data position in the RST grid.
bMinus = zeros(size(MESfw,1),1); % Definition of ground surface position in the RST matrix
bPlus = zeros(size(MESfw,1),1); % Definition of ground surface position in the RST matrix
for i = 1:size(MESfw,1)
MESfw(i,1)=round((MESfw(i,1)-min(handles.x))/handles.ModRes);
bMinus(i,1) = find(~isnan(handles.vel1(:,MESfw(i,1))), 1 ,'first'); % Calculation of
ground surface position in the RST matrix for the left of the calculated point
bPlus(i,1) = find(~isnan(handles.vel1(:,MESfw(i,1)+1)), 1 ,'first'); % Calculation of
ground surface position in the RST matrix for the right of the calculated point
MESfw(i,2)=round(MESfw(i,2)/handles.ModRes)+bMinus(i,1);
end
% Extraction of velocities where validation data exist
vel = zeros(size(MESfw,1),1);
for i = 1:size(MESfw,1)
vel1 = handles.vel1(MESfw(i,2),MESfw(i,1));
vel2 = handles.vel1(MESfw(i,2)-bMinus(i,1)+bPlus(i,1),MESfw(i,1)+1);
vel(i,1)=(vel1+vel2)/2; % Average of the value on the left and on the right of the
measured point (which is between two pixels)
end
% Extraction of resistivity where validation data exist
...
% Extraction of rock resistivity where validation data exist
...
% Extraction of epsilon factor (Brandt) where validation data exist
...
% Extraction of porosity where validation data exist
…
% Definition of the parameters changing for each equation
switch Res_equ
case 'Archie'
TableMulti = handles.metricdata.TableMulti;
P(1,1) = handles.metricdata.TPn;
P(2,1) = handles.metricdata.TPm;
P(3,1) = handles.metricdata.TPRw;
P(4,1) = 0;
handles.NP1 = 'n';
handles.NP2 = 'm';
handles.NP3 = 'Rw';
handles.NP4 = '';
…
~ 121 ~
% Creation of a progression bar
wait = waitbar(0,'Multi-Run ongoing. Please wait...',...
'CreateCancelBtn',...
'setappdata(gcbf,''canceling'',1)');
setappdata(wait,'canceling',0)
try
% Multi-Run loops over all parameters set
for i = TableMulti(1,1):TableMulti(1,3):TableMulti(1,2)
for j = TableMulti(2,1):TableMulti(2,3):TableMulti(2,2)
for k = TableMulti(3,1):TableMulti(3,3):TableMulti(3,2)
for l = TableMulti(4,1):TableMulti(4,3):TableMulti(4,2)
% Variable to know how many multi-run are made
nMulti_RMS = nMulti_RMS + 1;
…
% Update the progression bar for multi-run
if getappdata(wait,'canceling') % Check if the button cancel was clicked
delete(wait);
handles.MultiSelect = zeros (5,handles.metricdata.AcceptedRun);
return
end
% Report current estimate in the waitbar's message field
waitbar(nMulti_RMS / handles.metricdata.NMulti,wait);
% Calculate 4PM
[~,FPMfw,FPMfi,~]=calc_4PM(handles,res,vel,Res_equ);
% Calculate 4PM with complex porosity model
if (handles.metricdata.CheckAutoCal3PM == 1)
[TPMfr,~,~,handles]=calc_3PM(handles,res,vel,Res_equ);
for q=1:(size(MESfw,1))
if(FPMfi(q,1) < handles.metricdata.TTolThresh/100 && isnan(TPMfr(q,1))
== 0 && TPMfr(q,1) >=0.2)
phi(q,1) = 1-TPMfr(q,1);
end
end
handles.phi = phi;
[~,FPMfw,FPMfi,~]=calc_4PM(handles,res,vel,Res_equ);
end
% RMS calculation
RMS=0;
nRMS=0;
for q=1:(size(MESfw,1))
if (handles.metricdata.Checkicedata == 1) % Distinction between ice
content data and water content data
if (~isnan(FPMfi(q,1)) && ~isnan(MESfw(q,3))) % Check for NaN values
RMS = RMS + (FPMfi(q,1)-MESfw(q,3))^2; % RMS calculation
nRMS = nRMS + 1; % Number of valid RMS calculated
end
else
if (~isnan(FPMfw(q,1)) && ~isnan(MESfw(q,3)))
RMS = RMS + (FPMfw(q,1)-MESfw(q,3))^2;
nRMS = nRMS + 1;
end
end
end
~ 122 ~
RMS = sqrt(RMS/nRMS);
Multi_RMS(nMulti_RMS,1) = i;
Multi_RMS(nMulti_RMS,2) = j;
Multi_RMS(nMulti_RMS,3) = k;
Multi_RMS(nMulti_RMS,4) = l;
Multi_RMS(nMulti_RMS,5) = RMS;
end
end
end
end
delete(wait);
catch err
handles.error_found = 1;
errordlg('Something went wrong with the Multi-Run loop. (Impossible parameter set
?)','Multi-Run Error');
delete(wait);
rethrow(err);
end
% Best runs
% Sort the result from min RMS to MAX RMS
[Y,I]=sort(Multi_RMS(:,5));
Multi_RMS = Multi_RMS(I,:); %use the column indices from sort() to sort all columns.
% Selection of the best runs
MultiSelect = zeros (handles.metricdata.AcceptedRun,5);
for i = 1:handles.metricdata.AcceptedRun
MultiSelect(i,:) = Multi_RMS(i,:);
end
handles.MultiSelect = MultiSelect';
…
return
~ 123 ~
8.4. Four Phase Model Tutorial
~ 124 ~
~ 125 ~
~ 126 ~
~ 127 ~
~ 128 ~
~ 129 ~
8.5. Sensitivity of soil moisture and temperature to CoupModel
parameters
Figure 65: Effect of the critical depth of snow cover (for a thermal decoupling between the atmosphere and the
ground) on the mean temperature between 2002 and 2012 (B100m). A higher critical depth reduces the isolation effect
and the cold can penetrate more easily.
Figure 66: Effect of the critical depth of snow cover (for a thermal decoupling between the atmosphere and the
ground) on the mean unfrozen water content between 2002 and 2012 (B100m). The strong increase of water content
with a critical depth below 0.3 is probably due to positive temperatures observed in Figure 65.
~ 130 ~
Figure 67: Effect of the ground water flow on the mean temperature between 2002 and 2012 (B100m). The additional
water income brings a considerable amount of latent heat. This is clearly visible with the increase of the temperature.
Figure 68: Effect of the ground water flow on the mean unfrozen water content between 2002 and 2012 (B100m). The
stability of the water content is due to a very low retention.
~ 131 ~
Figure 69: Effect of the temperature coefficient for snow melt on the mean temperature between 2002 and 2012
(B100m). A higher parameter accelerates the snow melt in spring, resulting in temperature increase.
Figure 70: Effect of the temperature coefficient for snow melt on the mean unfrozen water content between 2002 and
2012 (B100m). A higher parameter increases the water content, but for the B100m, a strong drainage occurs.
~ 132 ~
Figure 71: Effect of the m-factor for the van Genuchten water retention function on the mean temperature between
2002 and 2012 (B100m). A threshold is visible. It probably corresponds to a limit pressure head that reduces the water
retention if the m-value is above 1.3.
Figure 72: Effect of the m-factor for the van Genuchten water retention function on the mean unfrozen water content
between 2002 and 2012 (B100m). The threshold is also visible for m=1.3. A smaller m-factor induces higher retention
and the water content increases. The effect is more visible near the surface (red points) where the water is not frozen
in summer.
~ 133 ~
Figure 73: Effect of the hydraulic conductivity on the mean temperature between 2002 and 2012 (B100m). With a
smaller conductivity, the water can liberate more latent heat.
Figure 74: Effect of the hydraulic conductivity on the mean unfrozen water content between 2002 and 2012 (B100m).
With a smaller conductivity, the water can accumulate more easily.
~ 134 ~
8.6. Updated 4PM Results of previous field campaigns
Figure 75: ERT and RST results for the Cross Monitoring Profile in 2006. The area with supposedly conductive rock
is clearly visible in the ERT profile. Low wave velocities in the first meters show the absence of ice near the surface.
Figure 76: 4PM results for the Cross Monitoring Profile in 2006. Very large ice free areas can be observed in the
southern slope, which does not correspond to the results in 2014. This can come from very warm condition in 2006, or
from errors in measurement or post-processing. The water accumulation near the B17m does not seem to be active at
the time of the measurements.
~ 135 ~
Figure 77: ERT and RST results for the Cross Monitoring Profile in 2011. The area with supposedly conductive rock
is still visible in the ERT profile. Higher wave velocities in the first meters show the presence of ice near the surface.
Figure 78: 4PM results for the Cross Monitoring Profile in 2011. This time, the results are much more similar than in
2014 with the presence of ice on the plateau, the same repartition of ice in the southern slope and with a small
accumulation of water near the B17m.
