Rock Fragment Characteristics, Patterns and Processes on
Natural and Artificial Mesa Slopes
Zhengyao Nie
Thesis submitted for the Master of Science Degree
The University of Western Australia
School of Earth and Environment Faculty of Natural and Agricultural Sciences
In the Discipline of Soil Science and Geomorphology
2011
Supervisors: Professor Christoph Hinz and Dr. Gavan McGrath
ii
Declarations
My supervisors Prof. Christoph Hinz and Dr. Gavan McGrath contributed to the
conceptualisation and method development of the experimental and modelling study
and constructive reviews of the thesis. UWA Honour students Tia Byrd and Erin
Poultney contributed to the sampling of rock fragments and the rainfall simulation
experiment in Chapter 2 and Chapter 4. Rowan Jenner contributed to the methods of
statistical analysis in Chapter 2. Contributions to the thesis are acknowledged in each
chapter. Besides, the work presented is entirely my own unless stated otherwise.
Student: Zhengyao Nie Coordinating Supervisor: Christoph Hinz
(on behalf of all supervisors)
iii
Abstract
Rock fragments on hillslopes interact with fine soil and vegetation by affecting infiltration,
runoff, erosion and evaporation, and therefore have an important function in arid
ecohydrological systems. Previous studies have described rock fragments and their spatial
patterns using simple measures such as mean or median size. This study presents research
conducted on three natural mesa slopes and a post-mining waste rock dump, in the Great Sandy
Desert, Western Australia, to characterize the statistical properties of rock fragments and
interrelationships between particle shape and size. Digital images of surface rocks were
collected along transects placed on each hillslope. A total of 112,142 rock fragments from 263
locations were recorded. From these images perimeter, area, Feret’s diameter and circularity of
rock fragments were determined.
On natural mesas, mean Feret’s diameter, and similarly area and perimeter decreased while
mean circularity increased downslope. The results indicated that larger and more angular rock
fragments occurred on the top of these hills. From a suite of probability distributions tested,
lognormal distribution was found to describe the Feret’s diameter best. Furthermore, both the
location and scale parameters of the lognormal distribution decreased approximately linearly
with distance down each transect. None of the probability distribution functions tested
sufficiently characterised the distributions of circularity.
Transport process such as overland flow has been the predominant explanation for the observed
particle sorting on rock armoured slopes. It has been suggested as a mechanism that selectively
washes fine material away, leaving coarser particles on steeper part of hillslopes. In order to
evaluate whether a weathering phenomenon could instead cause rock size sorting, a dynamic
model of particle fragmentation was used to reproduce the changes in rock fragment size
distributions downslope. With the initial condition for the model taken as the particle size
distribution at the top of the hill, the two parameter fragmentation model reproduced observed
trends in particle size distributions and did even better than the linear regressions. Preferential
fragmentation of larger sized particles was observed. The sorting phenomenon and preferential
fragmentation were found to be analogous with abrasion phenomena in rivers. The similarities
suggest the potential for a more general principle underpinning physical weathering of rock
particles.
Following the analysis on rock fragments on natural mesa slopes, rock characteristics were also
assessed on a waste rock dump at the rehabilitated mine site that was designed to mimic the
iv
shape and surface rock cover of natural mesa slopes, for a comparison between the two
contrasting environment. With similar surface cover, rock fragments on the artificial slope were
generally smaller and more angular. Lognormal distribution was found to describe rock size.
Unlike rock shape on natural mesas, the circularity of mined rock fragments was well described
by beta distributions. No distinct spatial organisation of size and shape were observed on the
artificial slope; lognormal and beta distribution parameters were randomly distributed in space.
In addition to measurements of rock fragments, rill erosions were measured on the artificial
slope. A rainfall simulation experiment was conducted in laboratory for assessing the initial
development of the surface rock armour. Surface rock armour was likely to develop quickly as
fines were washed out, and converge in surface cover through a self-organizing pattern,
preventing soils against. The results potentially indicated a similar self-organization pattern of
surface rock armour through different processes on natural and artificial slope, and how to
achieve long term stability on artificial mine waste dumps.
v
Table of Contents
Declarations ...................................................................................................................... ii
Abstract ............................................................................................................................ iii
List of Figures ................................................................................................................. vii
List of Tables .................................................................................................................... ix
Acknowledgments ............................................................................................................. x
Chapter 1: Overview ......................................................................................................... 1
Chapter 2: Spatial patterns of rock fragments along mesa hillslopes in the Great Sandy
Desert, Australia ................................................................................................................ 8
2.1 Introduction .......................................................................................................... 8
2.2 Materials and Methods ....................................................................................... 11
2.2.1 Study area ................................................................................................. 11
2.2.2 Field sampling .......................................................................................... 13
2.2.3 Image analysis .......................................................................................... 13
2.2.4 Statistical analysis .................................................................................... 14
2.3 Results and discussion ....................................................................................... 16
2.3.1 Spatial patterns in rock fragment coverage .............................................. 16
2.3.2 Spatial patterns in rock fragment size and shape ..................................... 18
2.3.3 Relationship between rock fragment size and shape................................ 21
2.3.4 Spatial trends in probability distributions ................................................ 22
2.3.5 Implications for processes contributing to the formation of rock armour 24
2.4 Conclusion ......................................................................................................... 25
Chapter 3: Does fragmentation weathering explain rock particle sorting on arid hills? . 31
3.1 Introduction ........................................................................................................ 31
3.2 Methods .............................................................................................................. 33
3.2.1 Site description and sampling .................................................................. 33
3.2.2 Modelling fragmentation .......................................................................... 35
3.3 Results ................................................................................................................ 37
3.4 Discussion .......................................................................................................... 39
3.5 Conclusion ......................................................................................................... 41
Chapter 4: Self-organisation of rock fragment cover on engineered and natural mesa
slopes ............................................................................................................................... 46
vi
4.1 Introduction ........................................................................................................ 47
4.2 Method ............................................................................................................... 48
4.2.1 Study site .................................................................................................. 48
4.2.2 Rock fragments ........................................................................................ 49
4.2.3 Field evidence of erosion ......................................................................... 51
4.2.4 Rainfall simulation ................................................................................... 51
4.3 Result and discussion ......................................................................................... 52
4.3.1Descriptive data of rock fragments ........................................................... 52
4.3.2 Probability distributions ........................................................................... 53
4.3.3 Spatial patterns of rock fragments ........................................................... 54
4.3.4 Rill erosion in the field ............................................................................. 56
4.3.5 Rainfall simulation ................................................................................... 57
4.4 Conclusion ......................................................................................................... 58
Chapter 5: Summary ....................................................................................................... 62
Appendices ...................................................................................................................... 66
Appendix A. The R script: Assessment of gamma, Weibull and lognormal
distributions for particle size .................................................................................... 66
Appendix B. Results of the probability distribution assessments ............................ 69
Appendix C. The R script: Searching for best model parameters (Mesa 1 as an
example) ................................................................................................................... 72
Appendix D. Further results of rock fragment analysis on the artificial waste dump
slope as a comparison to its natural analogues ........................................................ 76
Appendix E. Field evidence of erosion .................................................................... 80
vii
List of Figures
Figure 2.1. Photograph of Mesa 1 with the hard cap formed by secondary crust formation
and rock fragments with patchy vegetation covering the slope. ................................. 12
Figure 2.2. Examples of image processing procedure: original image (left), rectified image
using Turbo-Reg within ImageJ, binary image with hand-traced rock fragments and an
example of fragment characteristics as determined by the particle size analyser in
ImageJ (after [Byrd, 2008]). ......................................................................................... 14
Figure 2.3. Example profile from transect M1T1 showing the original elevation data
obtained from the GPS records, together with the fitted power regression line. ......... 14
Figure 2.4. Trends in rock fragment coverage (%) with respect to distance from capping on
(a) Mesa 1, (b) Mesa 2 and (c) Mesa 3. ....................................................................... 17
Figure 2.5. Trends in rock fragment coverage (%) with respect to gradient on (a) Mesa 1
and (b) Mesa 2. ............................................................................................................ 17
Figure 3.1. Results of best fit parameters (α = -87, β = 0.31) on Mesa 2 as an example of
systematic examination of model parameters by minimizing root mean square (RMS)
error between modeled distributions and observed ones. ............................................ 36
Figure 3.2. Changes in the size distribution of rock fragments with position on each hill,
with the frequency histogram of observed data (bars), truncated lognormal probability
density functions (pdf) fitted to the data (line), and modeled initial (solid circle) and
predicted (open circles) size distributions. The top is 0 m, the middle 30 m (Mesa 1
and 2), 27 m (Mesa 3) and the bottom 60 m (Mesa 1 and 2) and 51 m (Mesa 3) from
the duricrust cap. .......................................................................................................... 37
Figure 3.3. Changes in particle size lognormal distribution location μ and scale σ
parameters (Eq. 3.5) as a function of distance from duricrust cap. Shown are the
empirical fits to observed data (crosses), fitted linear regressions and 95% confidence
intervals (lines) and predicted distribution parameters by the fragmentation model
(circles) given the initial particle size distribution, emphasized by the large point. .... 38
Figure 3.4. Changes in cumulative mass fractions of (a) empirical mass data converted
from observed particle size distribution; (b) best fit from model prediction (α < 0); (c)
model prediction while α = 0; and (d) model prediction while α > 0. ......................... 40
Figure 4.1. The digial elevation model (DEM) of designed waste rock dump and sample
points on five selected transects. .................................................................................. 49
Figure 4.2. Relationship between lognormal distribution parameter μ and σ from all sites on
(a) natural mesa slopes and (b) the waste rock dump. The dark grey, grey and light
grey solid circles are the combination of distribution parameters on the top, middle and
viii
bottom of a typical transect on each system – natural mesas and the waste rock dump,
respectively as shown in Figure 4.4. ............................................................................ 53
Figure 4.3. Spatial changes of rock fragment surface coverage on (a) Mesa 1; (b) Mesa 2;
(c) Mesa 3 and (d) the waste rock dump. ..................................................................... 54
Figure 4.4. Lognormal distribution parameters of particle size change with respect of
distance are shown as changes of (a) μ and (b) σ down a typical mesa transect and (f) μ
and (g) σ down a typical waste rock dump transect; probability density of Feret’s
diameter with fitted lognormal distribution line changes on the mesa transect from (c)
the top at 0 m from cap, (d) middle at 30 m, to (e) bottom at 60 m, and on the waste
rock dump from (h) the top at 0m, (i) middle at 40m, to (j) bottom at 80m. ............... 55
Figure 6.1. Graphical residual analysis for the fitted regression lines for the area data from
Mesa 1. ......................................................................................................................... 71
Figure 6.2. Boxplots of all Feret’s diameter data on three natural mesa slopes and the waste
rock dump. .................................................................................................................... 76
Figure 6.3. Boxplots of all circularity data on three natural mesa slopes and the waste rock
dump. ............................................................................................................................ 76
Figure 6.4. (a) relationship between beta distribution parameters β1 and β2, with (b), (c) and
(d) density circularity histograms corresponding to solid circles of different
combination of β1 and β2, and the fitted beta distribution line. .................................... 77
Figure 6.5. Inter-relationship between (a) mean circularity and mean Feret’s diameter; (b)
rock surface coverage and mean Feret’s diameter and (c) rock surface coverage and
mean circularity on five transects on the waste rock dump. ......................................... 78
Figure 6.6. Spatial changes of (a) surface rock cover; (b) mean Feret’s diameter and (c)
mean circularity along distance on five transects on the waste rock dump. ................. 79
Figure 6.7. Changes of Feret’s diameter orienting downslope on top, middle and bottom of
a typical mesa transect and a typical waste rock dump transect. .................................. 79
Figure 6.8. Photographs looking downslope from the top of (a) Transect B (in Treatment 2)
and (b) Transect D (in Treatment 1). ............................................................................ 80
ix
List of Tables
Table 2.1. A summary of previous studies of spatial distribution of rock fragments in arid
and semi-arid regions. .................................................................................................. 10
Table 2.2. General information of three mesas. ................................................................... 13
Table 2.3. Descriptive statistics of rock fragments for all samples points on all transects
from all three Mesas. ................................................................................................... 16
Table 3.1. Linear regression coefficients; best fit model parameters and root mean square
(RMS) errors. For the linear regressions x denotes distance in meters from the top of
each transect. ................................................................................................................ 38
Table 4.1. Descriptive statistics of rock fragment characteristics on the waste rock dump
and natural mesa slopes. .............................................................................................. 52
Table 4.2. Results of rill erosion and the corresponding surface cover of rock fragments on
each transect on the waste rock dump. ........................................................................ 56
Table 4.3. Changes in rock fragment surface coverage for each replicate of each volume
ratio during rainfall simulation. ................................................................................... 58
Table 6.1. Hartigan's Diptest Results. Please note that the distance here in the “Transect and
Location” column is the distance from the bottom of the transect. ............................. 69
Table 6.2. Results of fitted distributions (lognormal, gamma and weibull) to Feret’s
diameter of rock fragments on three mesas. ................................................................ 70
x
Acknowledgments: This research was supported under the Australian Research Council
Linkage Projects (project no. LP0774881 in conjunction with Newcrest Mining Ltd. and
Minerals and Energy Research Institute of Western Australia) funding schemes.
I thank my supervisors Prof. Dr. Christoph Hinz and Dr. Gavan McGrath for their incredible
support and sharing ideas and feedback;
Newcrest Mining Ltd. for giving me the opportunity doing this research;
Tia Byrd and Erin Poultney for doing an excellent job in data collection;
Rowan Jenner for the committed support in statistics and surviving together from the ‘not so fun’
data analysis;
Martha Orozco, Basu Dev Regmi, Deborah Lin, Daniel Dempster for the happy times during
work;
Ziwen Zhang for the loving care and sharing in this journey;
And all the people whose supports and comments made it a better thesis.
1
Chapter 1: Overview
Rock armoured hillslopes are common in many arid and semiarid environments around the
world. Cooke et al. [1993] provided an elaborate review of stone pavements and their evolution
in deserts. The presence of surface rock armour results mainly from the preferential removal of
fine materials by deflation or overland flow, leaving the coarse materials behind [Cooke et al.,
1993]. These surface rock fragments have drawn researchers’ interests for several reasons.
Importantly, rock fragments play a role in controlling erosion, especially in arid land areas with
sparse vegetation [Simanton et al., 1984; Cooke et al., 1993]. Secondly, downslope fining of
surface armoured rock fragments has been observed extensively on arid and semiarid hillslopes,
and this sorting phenomenon could have eco-hydrological implications [Dury, 1966; Parsons,
1988; Abrahams et al., 1990; Cooke et al., 1993]. Therefore, a number of studies have
investigated spatial patterns of rock fragment characteristics, such as size and cover, in relation
to geomorphic features of the landscape such as slope type, gradient, curvature and aspect
[Abrahams et al., 1984; Abrahams et al., 1985; Abrahams et al., 1990; Simanton et al., 1994;
Poesen et al., 1998; Canfield et al., 2001; Li, 2007; Zhu and Shao, 2008].
Rock fragments are defined as strongly cemented particles with a diameter greater than sand and
smaller than a pedon. In some studies, rock fragments are defined as being either greater than 2
mm [Poesen and Lavee, 1994] or greater than 5 mm [Li et al., 2007; Poesen et al., 1998]. These
studies commonly focus on the soil surface, in particular in semi-arid and arid environments,
but also on the rock fragment distribution within soil profiles [Bunte and Poesen, 1993;
Brakensiek and Rawls, 1994; Poesen and Lavee, 1994]. In soil science literature the fine earth
fraction (< 2mm) have received considerable attention, however much less effort has been
invested in understanding the coarse fraction such as stones and rock fragments. This has partly
been due to the higher surface area to volume ratio of the fine fraction which controls to large
extent physical, chemical as well as biological properties of the material. However rock
fragments in particular in arid and semi-arid environments exert significant effects on
hydrological processes by influencing the soil water balance which in turn affects erosion
[Poesen and Lavee, 1994]. On the one hand surface rock fragments stabilize soil by shielding
the soil beneath the surface from raindrop impact and runoff detachment, and on the other hand
rock fragments entrap splashed sediment. In addition, the presence of rock fragment on the soil
surface increases roughness which largely determines infiltration rates, surface runoff and
erosion [van Wesemael et al., 1996a]. A high cover of surface rock fragments promotes
infiltration by inhibiting surface sealing, but the effects also depend on rock size, shape and
position [Cerdà, 2001; Poesen and Ingelmo-Sanchez, 1992; Poesen and Lavee, 1994; Valentin,
1994]. In conclusion, rock fragments in soil, especially on soil surfaces, protect against erosion
2
and physical degradation in general [Poesen and Lavee, 1994].
Weathering of rock fragments is extensive in desert by thermal fracture, salt crystallization and
biochemical weathering processes [Cooke et al., 1993]. The combination of weathering,
hydrological processes such as runoff and erosion will then affect the spatial distribution of rock
fragments in the soil and on the soil surface. Spatial patterns of rock fragment characteristics
including coverage, mean and/or median size along arid hillslopes have been studied by a
number of researchers. Among these studies, Simanton et al. [1994] reported a logarithmic
relationship between surface rock fragment cover and slope gradient in Nevada and Arizona,
and later Simanton and Toy [1994] developed a hyperbolic equation to predict surface rock
fragment cover in relation to a soil-slope factor (SSF), which combines effects of slope gradient
and soil profile rock content. Abrahams et al. [1985] found particle size positively related to
slope gradient, independent of distance from the divide in the Mojave Desert. They calculated
regression coefficients of the size-slope relations and the changes in coefficient values seemed
to relate to different slope platform shapes (plan-concave and plan-convex). Poesen et al. [1998]
observed an increase in rock fragment size and coverage with gradient in cultivated areas in
semiarid southeast Spain. They also found the best fit regression of this increase as a function of
slope gradient and the cover percentage of rock fragments are controlled by lithology and the
aspect of the slope. All the above studies proposed that spatial patterns of rock fragments on
hillslopes were largely controlled by surface runoff, which selectively transports smaller rocks
faster downslope in comparison to large rocks. However, spatial trends in size sorting were also
observed for large rocks, too large to be expected to be transported as bedload [Abrahams et al.,
1990]. Runoff creep, which is a slow creep process of large blocks resulting from selective
removal of fines that supports rocks by runoff has been hypothesised, but has yet to be
confirmed as a sorting mechanism [de Ploey and Moeyersons, 1975; Abrahams et al., 1990]. In
some undisturbed desert pavements on the Eastern Libyan Plateau, Egypt however, Adelsberger
and Smith [2009] found no spatial relationship between pavement characteristics and local
geomorphic features. Desert pavement surfaces in this study region could have developed
without significant influence from transport mechanisms such as overland flow, but rather was
more influenced by in-situ mechanical breakdown and pedogenesis.
Fragmentation in this thesis refers to a physical weathering process whereby one particle breaks
down into two or more smaller particles while preserving the mass. Probability size
distributions of rock particles are proposed to be indicators of breakdown processes and have
been explored in various areas of research including fluvial geomorphology of river systems and
material processing in the mining industries [Krumbein and Tisdel, 1940; Friedman, 1962;
Grady and Kipp, 1985]. Turcotte’s [1986] fragmentation model has been applied to describe
3
power-law particle size distributions in soil [Bittelli et al., 1999; Perfect, 1997]. It assumes that
fragmentation occurs as an instantaneous cascade, with the probability of breakage independent
of size [Turcotte, 1986; 1992]. Instead of an instantaneous cascade, multiple breakage events
have been shown to lead to a lognormal particle size distribution [Kolmogorov, 1941 as cited by
Dacey and Krumbein, 1979]. However, the concept of using spatial changes of particle size
distributions to infer processes contributing to the formation of rock pavements have not been
used to date. Dunkerley [1995] pointed to the significance of understanding the rock particle
size distribution in an empirical study of surface stone size. Surface and subsurface water flow
as well as erosion could vary significantly with different types of rock size distributions, even
where the mean sizes are the same [Dunkerley, 1995].
There is a need to quantify the size distributions of rock fragments on debris mantled
slopes and to assess how they change spatially.
The research described in this thesis was conducted near Telfer, located in the south centre of
the Paterson Range on the edge of the Great Sandy Desert, Western Australia [Parker, 2006].
Newcrest Mining Ltd. mines gold and copper at Telfer [Parker, 2006]. Stability of waste rock
dumps is an issue faced by the mining industry as surface and tunnel erosion occurred
frequently on the traditional berm-and-bench waste rock dump design. As Hancock et al. [2008]
suggested, slope shape and soil properties (e.g., rock fragment content) are largely responsible
for the slope stability. As a result a waste rock dump was designed and constructed at mine site
to mimic the concave shape and rock armour cover of natural mesa hillslopes in the Telfer
region. As slope surface develop, a number of processes may take place during the landform
evolution, including water wash, vertical sorting of rock fragments, creep and surface
weathering [Cooke, 1970]. It is likely that wash off is the dominating mechanism shaping the
two year old engineered slope, while surface weathering is more important on the natural mesas.
However, experimental evidence of the performance of engineered mesa-shaped slopes in
improving rehabilitation is missing.
This thesis consists of a series of three papers, aiming (1) to investigate characteristics and
spatial patterns of both rock size and shape on arid mesa slopes in Telfer region, and further
quantify probability distribution which could provide us with a better understanding of rock
pavement evolution; (2) to assess the hypothesis that in-situ fragmentation is a possible
mechanism explaining the change in size distributions along the natural mesa hillslopes; and (3)
to investigate rock characteristics and spatial trends on the mesa shaped rock dump at
Newcrest’s Telfer gold mine, and compare these with those of the natural mesas. As the thesis is
presented as a series of three papers, some parts of the introductions and methodology sections
4
in various chapters will appear repeatedly.
5
References
Abrahams, A. D., A. J. Parsons, and R. U. Cooke, and R. W. Reeves (1984), Stone movement on
hillslopes in the Mojave Desert, California: A 16 year record, Earth Surf. Process. Landforms,
9(4), 365-370.
Abrahams, A. D., A. J. Parsons, and P. J. Hirshi (1985), Hillslope gradient-particle size relations:
evidence for the formation of debris slopes by hydraulic processes in the Mojave Desert, J.
Geol., 93(3), 347-357.
Abrahams, A. D., N. Soltyka, and A. J. Parsons (1990), Fabric analysis of a desert debris slope:
Bell Mountain, California, J. Geol., 98(2), 264-272.
Adelsberger, K. A., and J. R. Smith (2009), Desert pavement development and landscape
stability on the Eastern Libyan Plateau, Egypt, Geomorph., 107(3-4), 178-194.
Bittelli, M., G. S. Campbell and M. Flury (1999) Characterization of particle-size distribution in
soils with a fragmentation model, Soil Sci. Soc. Am. J., 63(4), 782-788.
Brakensiek, D. L., and W. J. Rwals (1994), Soil containing rock fragments: effects on
infiltration, Catena, 23(1-2), 99-110.
Bunte, K., and J. Poesen (1993), Effects of rock fragment covers on erosion and transport of
noncohesive sediment by shallow overland flow, Water Resour. Res., 29(5), 1415-1424.
Bureau of Meteorology (2011), Climate statistics for Australian locations - TELFER AERO,
Commonwealth of Australia, http://www.bom.gov.au/climate/averages.
Canfield, H. E., V. L. Lopes, and D. C. Goodrich (2001), Hillslope characteristics and particle
size composition of surficial armoring on a semiarid watershed in the southwestern United
States, Catena, 44(1), 1-11.
Cerdà, A. (2001), Effects of rock fragment cover on soil infiltration, interrill runoff and erosion,
Eur. J. Soil. Sci., 52(1), 59-68.
Cooke, R. U. (1970), Stone pavements in deserts, Ann. of the Ass. of Am. Geog., 60(3), 560-577.
Cooke, R. U., A. Warren, and A. Goude (1993), Surface particle concentrations: stone
pavements, pp 68-76, in Desert Geomorphology, ULC Press, London, United Kingdom.
Dacey, M. F., and W. C. Krumbein (1979), Models of breakage and selection for particle size
distributions, J. Int. Assoc. Math. Geol., 11(2), 193-222.
Dury, G. H. (1966), Pediment slope and particle size at Middle Pinnacle, near Broken Hill, New
South Wales, Aust. Geogr. Stud., 4(1), 1-17.
de Figueiredo, T., and J. Poesen (1998), Effects of surface rock fragment characteristics on
interrill runoff and erosion of a silty loam soil, Soil and Tillage Res., 46(1-2), 81-95.
de Ploey, J., and J. Moeyersons (1975), Runoff creep of coarse debris: Experimental data and
some field observations, Catena, 2, 275-288.
Dunkerley, D. L. (1995), Surface stone cover on desert hillslopes; parameterizing characteristics
6
relevant to infiltration and surface runoff, Earth Surf. Process. Landforms, 20(3), 207-218.
Friedman, G. M. (1962), On sorting, sorting Coefficients, and the lognormality of the grain-size
distribution of sandstones, J. Geol., 70(6), 737-753.
Grady, D. E., and M. E. Kipp (1985), Geometric statistics and dynamic fragmentation, J. Appl.
Phys., 58(3), 1210-1222.
Hancock, G. R., D. Crawter, S. G. Fityus, J. Chandler, and T. Wells (2008), The measurement
and modelling of rill erosion at angle of repose slopes in mine spoil, Earth Surf. Process.
Landforms, 33(7), 1006-1020.
Krumbein, W. C., and F. W. Tisdel (1940), Size distribution of source rocks of sediments, Am. J.
Sci., 238(4), 296-305.
Li, X. Y., S. Contreras, and A. Sole-Benet (2007), Spatial distribution of rock fragments in
dolines: A case study in a semiarid Mediterranean mountain-range (Sierra de Gádor, SE
Spain), Catena, 70(3), 366-374.
Parker, R. M. (2006), Understanding controls of landform stability of mesa systems to design
slopes of rock waste dumps at Telfer Goldmine, Pilbara WA, Perth, Australia.
Parsons, A. J. (1988), Hillslope form and climate, in Hillslope Form, pp. 47-67, Routledge, New
York.
Perfect, E. (1997), Fractal models for the fragmentation of rocks and soils: a review, Engineer.
Geol., 48(3-4), 185-198.
Poesen, J., and F. Ingelmo-Sanchez (1992), Runoff and sediment yield from topsoils with
different porosity as affected by rock fragment cover and position, Catena, 19(5), 451-474.
Poesen, J., and H. Lavee (1994), Rock Fragments in Top Soils - Significance and Processes,
Catena, 23(1-2), 1-28.
Poesen, J. W., D. Torri, and K. Bunte (1994), Effects of rock fragments on soil erosion by water
at different spatial scales: a review, Catena, 23(1-2), 141-166.
Poesen, J. W., B. van Wesemael, K. Bunte, and A. S. Benet (1998), Variation of rock fragment
cover and size along semiarid hillslopes: a case-study from southeast Spain, Geomorph.,
23(2-4), 323-335.
Simanton, J. R., K. G. Renard, C. M. Christiansen, and L. J. Lane (1994), Spatial-distribution of
surface rock fragments along catenas in semiarid Arizona and Nevada, USA, Catena, 23(1-2),
29-42.
Simanton, J. R., and T. J. Toy (1994), The relation between surface rock-fragment cover and
semiarid hillslope profile morphology, Catena, 23(3-4), 213-225.
Turcotte, D. L. (1986), Fractals and Fragmentation, J. Geophys. Res., 91(B2), 1921-1926.
Turcotte, D. L. (1992), Fragmentation, in Fractals and Chaos in Geology and Geophysics, pp.
20-34, Cambridge University Press, New York, USA.
Valentin, C. (1994), Surface Sealing as Affected by Various Rock Fragment Covers in West-
7
Africa, Catena, 23(1-2), 87-97.
van Wesemael, B., J. Poesen, T. de Figueiredo, and G. Govers. (1996a), Surface roughness
evolution of soils containing rock fragments, Earth Surf. Process. Landforms, 21(5), 399 -
411.
van Wesemael, B., J. Poesen, C. S. Kosmas, N. G. Danalatos, and J. Nachtergaele (1996b),
Evaporation from cultivated soils containing rock fragments, J. Hydrol., 182(1-4), 65-82.
Zhu, Y. J., and M. A. Shao (2008), Spatial distribution of surface rock fragment on hillslopes in
a small catchment in wind-water erosion crisscross region of the Loess Plateau, Sci. China
Ser. D, 51(6), 862-870.
8
Chapter 2: Spatial patterns of rock fragments along mesa
hillslopes in the Great Sandy Desert, Australia
Zhengyao Nie1, Christoph Hinz1, Gavan S. McGrath1, Rowan Jenner1 and Tia Byrd1
1. School of Earth and Environment, The University of Western Australia, Western Australia,
Australia.
Abstract
Rock mantled hillslopes are common in arid and semiarid environments. Investigating the size
and shape of the mantled rock fragments and their spatial distributions has implications for
geomorphic processes. In this study, rock fragments on three mesa hillslopes were characterized
by size (area, Feret’s diameter, and perimeter) and shape (circularity) using image processing of
digital photographs taken of the slope surface. A total of 93415 individual rock fragment
measurements were obtained from 222 positions along the mesa slopes. Particle size and
circularity were negatively related. Spatial patterns of these rock fragments were assessed by
relating mean particle size and shape to geomorphic features including slope gradient and
surface distance from the hill top. Distance was found to be a better independent variable
describing the changes in rock size and shape along the slope then gradient. On two mesas,
mean particle size decreased nonlinearly with distance from the cap, while mean circularity
increases linearly. The other mesa exhibited a weaker spatial trend of rock particles. In addition
to simple descriptive statistics, a set of probability distributions were tested to describe the
distribution of size and shape of rock fragments along each hillslope. Truncated lognormal
distribution proved to describe the particle size distribution well, with both distribution
parameters decreasing linearly with distance downslope on all three mesas. None of the three
distributions beta, Weibull and logit-normal consistently described the shape parameter.
Keywords: rock fragments; mesa slope; distance; spatial patterns; lognormal distribution
2.1 Introduction
Rock fragments have various geomorphic and ecological functions in arid ecosystems. One of
them is stabilizing landforms by forming rock armours over potentially erodible materials
[Cooke et al., 1993]. The important role of rock fragments has been recognized in the soil water
balance by reducing bare soil evaporation and increasing soil water storage, and preventing
erosion and physical degradation [Cooke et al., 1993; Poesen and Lavee, 1994; Valentin, 1994;
Poesen et al., 1998; van Wesemael et al., 2000].
9
The distribution of rock fragments in space is a reflection of landform evolution processes under
geomorphic and hydrological controls. Investigations of spatial patterns of rock fragments often
relate their size and/or surface coverage to geomorphic features of the landscape [Abrahams et
al., 1985; Simanton et al., 1994; Poesen et al., 1998; Canfield et al., 2001; Li et al., 2007; Zhu
and Shao, 2008; Adelsberger and Smith, 2009]. For example, Simanton et al. [1994] reported a
logarithmic relationship between surface rock fragment cover and hillslope gradient in semiarid
Arizona and Nevada. A hyperbolic equation was developed for surface rock fragment cover
from a combination of slope gradient and rock content in soil [Simanton and Toy, 1994].
Overland flow was also suggested to be the dominant factor in transporting rocks by Abrahams
et al. [1985], in the study relating mean particle size to slope gradient on the hillslopes underlain
by weak to moderately resistant rocks in the Mojave Desert. The rate of sediment transport G by
overland flow was proposed empirically to relate to horizontal distance to the slope divide x,
slope gradient S, and mean particle size D in the form pnm DSxG / , where m, n and p are all
constants of the equation with values larger than 0 [Abrahams et al., 1985]. Poesen et al. [1998]
observed a positive relationship between hillslope gradient and rock fragment cover as well as
size along semiarid hillslopes in southeast Spain, and furthermore found that rock fragment
cover was largely controlled by lithology and hillslope aspect. Dury [1966] found that particle
size decreases in orderly fashion with both slope and distance downslope. However, in some
undisturbed desert pavements on the Eastern Libyan Plateau, Egypt, Adelsberger and Smith
[2009] found no spatial relationship between pavement characteristics such as clast size, density,
lithology and orientation and local geomorphic features being typically slope gradient, and
aspect. They suggested that instead of transport mechanisms such as runoff the pavement
surface had developed by mechanical breakdown of surface clasts and in-situ pedogenesis
[Adelsberger and Smith, 2009]. Generally though, rock fragment cover and size are found to be
related to local geomorphic and geological features. Some of the results of spatial distributions
of rock fragments in arid and semi-arid regions are summarized in Table 2.1.
10
Table 2.1. A summary of previous studies of spatial distribution of rock fragments in arid and semi-arid regions.
Geomorphic location Slope shape Sample method Surface
coverage (%)
Size range(mm)
Slope gradient
(%) Rock type Result Source
RF cover RF size Gradient
A small catchment in wind-water erosion crisscross
region, Loess Plateau, China
convex-straight-concave
Digital photographing Compass 0 - 7 2-60 0 - 275 - cbSaSAc 2 [Zhu and
Shao, 2008]
Dolines in a semiarid Mediterranean mountain-
range, south Spain.
convex-straight-concave
Line-intercept method of Mueller-
Dombois and Ellenberg
Tape measure
DEM 0-100 Median
diameter:>50-32.5 Limestone
bSaAc )log(bSeaD
[Li et al., 2007]
A catchment on a highland, northern Ethiopia
convex and concave
Photography and mode-count method.
Clinometer and tape measure
57-85 Median
diameter:>56-42
Tertiary basalt and
Antalo limestone
bSaAc )log(
bSaAc
[Nyssen et al., 2002]
Semiarid hillslopes, southeast Spain.
convex-straight-concave
Photography and point-count method
Clinometer and tape measure
0-80 Median
diameter: 5-80
4-66 Micaschist, Andesite,
Conglomerate
bSaAc )log(
bc SaA
bSc eaA
[Poesen et al., 1998]
Catenas in semiarid Arizona and Nevada, USA
5 uniform, 4 convex and 3 concave
Line-point measurement
(Bonham, 1989)-
Abney level
0-75 - 2-61 - bSaAc )log( [Simanton
et al., 1994]
Debris slopes in Mojave Desert, USA
3 convex and 2
concave -
Sampling particles in a grid
Abney level
- Mean
diameter: 0-100
0-65
Gneiss, Latitic
Porphyry, Fanglomerate
baSD [Abraham
s et al., 1985]
* Ac = rock fragment coverage; D = rock fragment size; S = slope gradient. Symbols a, b and c are relationship function coefficients.
11
Particle size distributions may provide us with another means to infer landforming processes
and have been studied in soil science, fluvial geomorphology and processing engineering
[Turcotte, 1986; Bittelli et al., 1999; Cohen et al., 2009]. As fragmentation is caused by various
processes we expect different fragment size distributions depending on the type of
fragmentation such as physical weathering by salt, frost or temperature, blasting or saltation
during transport [Friedman, 1962; Grady and Kipp, 1985; Martin et al., 2009]. Scientists have
attempted to model fragmentation with different assumptions and algorithms leading to different
particle size distributions [Epstein, 1948; Turcotte, 1986; Perfect, 1997]. Among these studies,
Turcotte’s [1986] equation is widely applied in fragmentation models, which is a power-law
distribution of particle size derived from an instantaneous scale-invariant cascade of
fragmentation. In contrast, Kolmogorov [1941] (as cited by Dacey and Krumbein, 1979) argues
in an earlier paper that lognormal size distribution is caused by temporal fragmentation and sub-
particles obtained from larger particles are independent of size. In studies of surface rocks on
desert hillslopes, Dury [1966] and Dunkerley [1995] observed normal distribution of log particle
size. However, characterization of statistical distribution of particle size has been surprisingly
absent.
In addition to particle size, rock fragment shape, is considered to be an indicator of abrasion and
breakage during transport processes [Krumbein, 1941a]. On hillslopes, gravels are commonly
rounded when they are fluvial in origin, yet particle rounding can also be accomplished by
surface weathering [Dixon, 1994]. With respect to landscape origin, Al-Farraj et al. [2000]
reported changes in roundness of clasts on differently developed desert pavement surfaces of
alluvial fan. Again, to our knowledge, few studies assessed particle shape and probability
distribution on rock armoured hillslopes.
This study will identify the characteristics of the size and shape of rock fragments and the
surface cover on three natural mesa hillslopes in the Great Sandy Desert, Western Australia. In
addition, probability distribution of particle size and shape will be measured. The objective of
this study is to shed light in understanding landform stability through spatial organizations of
rock fragments on arid mesa slopes.
2.2 Materials and Methods
2.2.1 Study area
The study area was located near Newcrest Mining Ltd’s Telfer Gold Mine, 21.71ºS, 122.23ºE in
the Great Sandy Desert, Australia. The climate is arid, with typically 250 – 450 mm annual
rainfall with the majority of rainfall associated with 20 to 30 mm convective storms which
typically occur each year, while the annual potential evaporation is 3200 mm. The average
12
maximum daily temperature at Telfer varies between 25°C – 42°C [Bureau of Meteorology,
2011].
The terrain ranges from flat desert surfaces to steep mesa slopes. Interspersed with occasional
linear dune, rock fragments dominate the surface cover of mesa slopes. A duricrust capping
which is the apparent source of surface rock fragments is found at the top of the mesa (Figure
2.1). Telfer Gold Mine and the nearby mesas are part of the Paterson Range in Western Australia,
specifically the Mesoproterozoic to Neoproterozoic Yeneena Basin [Maxlow, 2005]. The mesas
are located in the Upper Yenenna Sub-Group of the lower Throssell Range Group in the west
and centre of Telfer [Maxlow 2005]. The Throssell Range Group consists of shallow marine
sediments. Mottled zone and lateritic duricrust have been eroded to reveal the underlying
saprolite and saprock [Henderson, 1996]. Samples of rock fragments were collected on three
selected mesas – simply called Mesa1, Mesa 2, and Mesa 3 (Table 2.2).
Figure 2.1. Photograph of Mesa 1 with the hard cap formed by secondary crust formation and rock
fragments with patchy vegetation covering the slope.
13
Table 2.2. General information of three mesas.
Longitude/
Latitude
Transect
aspects
Slope type Rock material Vegetation
Mesa 1 21°45’46.31’’S/
122.9’56.56’’E
NW Concave silcrete,siltstone acacia shrubs and
tussock grass
Mesa 2 21°56’0.96’’S/
122°12’43.00’’E
S Concave silcrete
conglomerate
acacia shrubs and
tussock grass
Mesa 3 21°52’3.23’’S/
122°7’29.21’’E
E Rectilinear Siltstone tussock grass
2.2.2 Field sampling
Four transects were placed along each mesa, beginning from the lower edge of the hard cap
(hard cap not included) downhill in a straight line along the direction of the steepest descent.
For Mesa 1 and 2 transects were 60 m in length, while for Mesa 3, transect lengths varied
between 24 and 51m as the hill was shorter. Sample points were located at 3 m intervals.
Positions of sample points were recorded with a differential GPS (Magellan ProMarkTM 3). At
each point a 1 m2 metal square grid was placed on the ground and digital photographs were
taken of the surface. This was repeated twice on each side of the transect line, resulting in a total
of four images per interval. This number was reduced where obstructions, such as larger plants,
prevented photographs being taken. Digital photographs were taken in JPEG/TIFF format with
the following cameras: Canon IXUS750 (3x optical zoon, 7.1 megapixels), Canon Powershot
A610 AiAF (4x optical zoom, 5 megapixels), Olympus u 1720SW (3x optical zoom, 7.1
megapixels), Olympus u850SW (5x optical zoom, 8 megapixels).
2.2.3 Image analysis
The Java based program ImageJ was used to create a binary image [Rasband, 2008]. Rock
fragments were hand traced with an Intuos3 Wacom graphics tablet system. The Turbo-Reg
plug-in for ImageJ was then used to rectify and register the image via a bilinear transformation,
altering the image to produce a standard resolution in IrfanView such that 1 pixel corresponded
to 0.28 mm2 [Thévenaz, 2008; Skiljan, 2011] (Figure 2.2). Due to the limitations of this
methodology, rock fragments less than 10 mm were excluded from the analysis.
For each individual rock fragment, the area (A) (mm2), Feret’s diameter (F) (mm), perimeter (P)
(mm) and circularity (C) (-) were determined. Feret’s diameter is defined as the longest distance
between any two points on the perimeter [Rasband, 2008]. Circularity, a measure of the shape of
particles, is defined by: 24 PAC and ranges between a small number near 0 for an
elongated thin plate and 1 for a perfect circle. For each image, surface coverage of rock
fragments (%) was calculated as:
14
%100
VT
Rc AA
AA (2.1)
where Ac, AR, AV, and AT represent rock fragment coverage percentage, the total rock fragment
area, total vegetation area and total area within the grid, respectively.
Figure 2.2. Examples of image processing procedure: original image (left), rectified image using Turbo-
Reg within ImageJ, binary image with hand-traced rock fragments and an example of fragment
characteristics as determined by the particle size analyser in ImageJ (after [Byrd, 2008]).
2.2.4 Statistical analysis
Mesa slope smoothing
The elevation at each sample location was obtained from the information recorded by the
differential GPS. To obtain a smooth transect profile, non-linear regression models of the form
bxxay )( max were fitted to the relative elevation data for each transect, where y is the
estimated relative elevation from the regression model, xmax is the horizontal length of the
transect, and x is the horizontal distance from the duricrust cap (see Figure 2.3). This enabled
the gradient at each location to be approximated by the first of the fitted equation. The following
notation is used to identify transects measured: M1 to M3 stands for Mesa 1 to Mesa 3 and T1
to T4 denotes Transect 1 to Transect 4.
Figure 2.3. Example profile from transect M1T1 showing the original elevation data obtained from the
GPS records, together with the fitted power regression line.
15
Data analysis
All statistical analyses were carried out using the software environment R [Ihaka and
Gentleman, 1996]. At each sample location, descriptive statistics such as mean, median,
standard deviation were calculated for the rock fragment size and shape measurements. The
relationship between rock fragment coverage and gradient was then analysed, in addition to the
relationship between coverage and distance along the transect. Probability densities of rock
fragment size and shape were estimated with kernel density approximation, which is a density
estimation insensitive to bin size [Sheather and Jones, 1991], to show the distribution of
particles at each sample point along the transect. The changes in mean rock fragment size
expressed either as Feret’s diameter, area or perimeter with distance along the transects were
examined by three regression models in the form of linear (Eq. 2.2), exponential (Eq. 2.3) and
power-law (Eq. 2.4).
bXLaD )(~ (2.2)
)](exp[~ XLnmD (2.3)
cXLpD q )(~ (2.4)
where D is either mean Feret’s diameter (mm), mean area (mm2) or mean perimeter (mm); L
denotes the total length of the transect (m), and X = distance from capping along each transect
(m). Model coefficients a, b, m, n, p, q, c were determined by linear and nonlinear least squares
methods [Bates and Watts, 1988; Chambers, 1992]. Regressions are deemed to be significant
with a p-value less than 0.01. In this paper, if not otherwise specified, distance refers to the
distance along the transect surface from the duricrust cap.
Hartigan’s Diptest was applied to test for unimodality of rock fragment size distribution
[Hartigan, 1985]. At each sample location, gamma, Weibull and lognormal probability
distributions were then fitted to the Feret’s diameter, while beta, Weibull and logit-normal
probability distributions were fitted to circularity. The chi-square test was applied to assess
whether the data are from the fitted distributions, and a relative measure of goodness of fit was
calculated [Ricci, 2005] (Eq. 2.5; see Appendix A).
n
ii
n
iii
y
yy
1
2
1
2* )(
(2.5)
where is the relative measure of goodness of fit; yi is expected frequency from estimated
distribution parameters; yi* is the observed frequency; n is the total number of bins, and i
denotes the ith bin. Distribution parameters were then estimated using maximum likelihood
method [Venables and Ripley, 2002].
16
As the chi-square test is sensitive to bin size, the test was carried out twice; firstly using Sturges
formula to determine bin size [Sturges, 1926] and secondly with a set number (n = 20) of bins,
each of equal width. The fit of distribution was deemed satisfactory at a 5% significance level.
Furthermore, geomorphic dependencies of the parameters of the fitted probability distributions
were assessed. The spatial dependency of the estimated distribution parameters was quantified
by linear regression as a function of distance.
2.3 Results and discussion
A summary of descriptive statistics of Feret’s diameter, area, perimeter, circularity and
percentage coverage for all transects of all three mesas comprising of a total of 222 sample
points and a total of 93415 individual rock fragments is presented in Tale 2.3.
Table 2.3. Descriptive statistics of rock fragments for all samples points on all transects from all three
Mesas.
Min Median Mean Max Standard Deviation
Coverage (%) 10.73 36.83 37.95 67.17 12.67
Feret’s Diameter (mm) 10 51 65 1008 51
Area (mm2) 53 1075 2709 484091 7456
Perimeter (mm) 24 138 184 3732 159
Circularity 0.02 0.72 0.70 0.92 0.11
2.3.1 Spatial patterns in rock fragment coverage
On most desert hillslopes, size of rock fragments has been found to decrease downslope [Cooke
et al., 1993]. We determined a significant negative linear relationship between rock fragment
coverage and transect distance from all three mesas (Figure 2.4). Although there is significant
variation in surface rock coverage at different sample points, in general higher coverage of rock
armour exists on the top part of the hillslopes.
17
Figure 2.4. Trends in rock fragment coverage (%) with respect to distance from capping on (a) Mesa 1, (b)
Mesa 2 and (c) Mesa 3.
Following Simanton and Toy [1994], Figure 2.5 shows the logarithm relationship between rock
fragment coverage and hillslope gradient on Mesa 1 and 2. Mesa 3 is not included in this
analysis because all four transects on Mesa 3 are approximately rectilinear and have nearly
constant gradients. For some individual transects, log-linear regression is able to describe the
relation between rock cover and slope gradient well, however, for most transects a linear model
better describes the spatial changes of data (see Figure 2.5).
Figure 2.5. Trends in rock fragment coverage (%) with respect to gradient on (a) Mesa 1 and (b) Mesa 2.
Above results indicate surface coverage of rock fragments is greater on the steeper parts of the
hills, where is the upper parts of concave shaped Mesa 1 and 2. However, distance provides
significant relationships with surface cover on all mesas, including Mesa 3, a rectilinearly
shaped slope with a constant slope gradient. Distance also appears to be a dependency that can
describe the changes of mean particle size and shape, and probability distribution parameters as
presented in the following sections. Although Nyssen et al. [2002] point out that distance is
0 10 20 30 40 50 60
1030
5070
(a)
R2=0.59
0 10 20 30 40 50 60
1030
5070
(b)
Distance (m)
R2=0.50
0 10 20 30 40 50 60
1030
5070
(c)
R2=0.19
Co
vera
ge
(%
)
0 10 20 30 40 50 60 70
1020
3040
5060
70
(a)
LinearR2=0.57
Log-linearR2=0.54
0 10 20 30 40 50 60 70
1020
3040
5060
70
(b)
LinearR2=0.42
Log-linearR2=0.34
Gradient (%)
Co
vera
ge
(%
)
18
important in the spatial distribution of rock fragments, there are few direct evidence for surface
cover or mean particle size – distance relationship in the literatures [Dury, 1966], as selective
particle transport process has often been hypothesized and slope gradient is closely related to
particle transport. However, the spatial changes of rock coverage and mean particle size with
respect to distance in this study are consistent with previous work on arid concave slopes
[Abrahams et al., 1985; Simanton et al., 1994; Poesen et al., 1998].
2.3.2 Spatial patterns in rock fragment size and shape
There are clear spatial patterns in the changes of probability densities of rock fragment size and
shape from top of the slope to the bottom (see Figure 2.6). The mode of size shifts from larger
to smaller sizes as one progresses downslope. Similarly the size distribution narrows downslope.
Circularity on the other hand shows the opposite trend with the mode, shifting from low to high
circularity. These distributions clearly show larger and more angular rocks with greater
variability can be expected in the upper part of the mesas, while smaller, rounder and less
variable particles are observed toward the bottom of each hill.
Figure 2.6. Changes in probability density of rock (a) area, (b) Feret’s diameter, (c) Perimeter and (d)
Circularity from the top 0m, the middle 30 m (Mesa 1), 33m (Mesa 2), 27 m (Mesa 3) to the bottom 60 m
(Mesa 1 and 2) and 51 m (Mesa 3) from cap.
0 50 100 200
0.00
00.
020
Mesa 1
Den
sitym
m1
TopMiddle
Bottom
0 50 100 200
Feret's Diameter (mm)
Mesa 2
0 50 100 200
(a)
Mesa 3
0 2000 6000 10000
0e+
006e
-04
Den
sitym
m2
0 2000 6000 10000
Areamm2
0 2000 6000 10000
(b)
0 200 400 600 800
0.00
00.
006
Den
sitym
m1
0 200 400 600 800
Perimeter (mm)
0 200 400 600 800
(c)
0.0 0.2 0.4 0.6 0.8 1.0
04
8
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
Circularity
0.0 0.2 0.4 0.6 0.8 1.0
(d)
19
The spatial trends of mean particle size along distance are described by power-law and/or
exponential regression models on all transects on Mesa 1 and 2 (Figure 2.7a shows a typical
transect). In contrast, a negative linear trend is observed between mean circularity and transect
distance (Figure 2.7b). Increases of standard deviation of size and shape with distance are
described by power-law and linear regressions, respectively (Figure 2.7c and 2.7d). Rock
fragments on Mesa 3 tend to be distributed more randomly in space (Figure 2.6). We found
smaller rock fragments in general on Mesa 3 that can result from a different lithology as we
observed more outcrops along the mesa slope. As lithology and particle size are responsible for
the spatial patterns of rock fragments [Abrahams et al., 1985; Poesen et al., 1998; Nyssen et al.,
2002], the relatively weaker spatial trends of rock particles on Mesa 3 could be controlled by a
different lithology and perhaps by a different source of surface armoured rock fragments.
Figure 2.7. Changes in (a) mean Feret’s diameter ( F ), (b) mean circularity (C ), (c) standard deviation
of Feret’s diameter (SdF) and (d) standard deviation of circularity (SdC) with respect to distance along a
typical mesa transect (Mesa 1 Transect 4).
In the above figures (Figure 2.7), power-law regressions were fitted to the changes of mean and
standard deviation of Feret’s diameter as a function of distance. By comparing r2 of regression
models, power-law regressions provided slightly better fits than exponential regressions. Using
Shapiro-Wilk normality test [Royston, 1982], residuals of power-law regression can be rejected
0 10 20 30 40 50 60
60
80
100
12
0
(a)
Me
an
Fe
ret's
Dia
me
ter
(mm
)
F = 0.011L X2.136 43.892
R2 = 0.901
0 10 20 30 40 50 60
0.6
60
.70
0.7
4
(b)
Me
an
Cir
cula
rity
C = 0.001X 0.675
R2 = 0.791
0 10 20 30 40 50 60
20
406
08
01
00
(c)
Distance (m)
Std
De
v o
f Fe
ret's
D (
mm
)
SdF = 0.003L X2.56 18.754
R2 = 0.923
0 10 20 30 40 50 60
0.0
70
.09
0.1
1
(d)
Distance (m)
Std
De
v o
f Cir
cula
rity
SdC = 0.001X 0.124
R2 = 0.783
20
as normally distributed at a p 0.1 significance level on some of the transect (Figure 2.8a and
2.8b) while a null hypothesis of normal distribution of residuals from exponential distribution is
accepted (Figure 2.8c and 2.8d), indicating that exponential regressions may be the more
appropriate description of the spatial dependence of mean rock fragment size. In fact, negative
exponential functions between particle size and distance have been observed in many sediment
studies [Krumbein, 1937; Sternberg, 1875 as cited by Krumbein, 1941a; Wentworth, 1931 as
cited by Krumbein, 1941a]. In this study, we cannot rule out unequivocally that the power-law
regression is inappropriate to describe the relationships between particle size and distance.
However, the possible consistency of particle size-distance relations in fluvial systems and
hillslope armoured rock fragments suggests the exponential relation may be applied more
generally, regardless of different geomorphic processes.
Figure 2.8. Comparison of (a) exponential and (c) power-law regression models of mean Feret’s diameter
along distance; and normal Q-Q plots of residuals from (b) exponential and (d) power-law regression
models on M2T1 as an example. W is the test statistic of Shapiro-Wilk test, denoting the ratio of the best
estimator of the variance.
The shape of rock particles is considered to be a significant indicator for rock evolution
0 10 20 30 40 50 60
60
80
10
01
20
(a)
Distance(m)
Fe
ret's
Dia
me
ter
(mm
)
F=50.441e0.014LDist
R2=0.799
-2 -1 0 1 2
-20
-10
01
02
03
0(b)
Theoretical Quantiles
Sa
mp
le Q
ua
ntil
es
Shapiro-Wilk normality test
W=0.98p-value=0.95
0 10 20 30 40 50 60
60
80
10
01
20
(c)
Distance(m)
Fe
ret's
Dia
me
ter(
mm
)
F =0.031L Dist1.878 53.845
R2=0.818
-2 -1 0 1 2
-20
-10
01
02
03
0
(d)
Theoretical Quantiles
Sa
mp
le Q
ua
ntil
es
Shapiro-Wilk normality test
W=0.92p-value=0.09
21
processes in sedimentary environments, which is usually measured by sphericity, a ratio of rock
surface area to its volume, or roundness that is based on the measurements of curvatures [Wadell,
1932; Krumbein, 1941b; Barrett, 1980]. In sedimentology, it is well known that the reduction in
particle size with transport distance down a river is usually accompanied by increased roundness,
as irregularly shaped rock particles wear. In an abrasion experiment using a tumbling barrel,
Krumbein [1941a] confirmed that both sphericity and roundness of rock fragments increased
with transport distance downstream. The concept of circularity measured in this study is similar
to sphericity, only in a 2-Demension. The increase in circularity downslope suggests that
transport is potentially a significant process contributing the spatial organisation of rock
fragments on hillslopes (Figure 2.6d and 2.7b). However, as abrasion studies show that the
increase in circularity slows down and reach a limit as rocks get more spherical during transport
[Krumbein, 1941a], the observed linear relationship between circularity and distance does not
show any obvious acceleration or deceleration in the change of shape (Figure 2.7b).
2.3.3 Relationship between rock fragment size and shape
Three size measurements – Feret’s diameter, perimeter and area are highly correlated as
expected. Circularity is an independent measurement that negatively correlated with size
variables with a comparatively weaker correlation (Table 2.4).
Table 2.4. Correlation matrix between rock fragment size and shape parameters from all samples.
Feret's Diameter Perimeter Area
Feret's Diameter - - -
Perimeter 0.989 - -
Area 0.967 0.988 -
Circularity -0.414 -0.346 -0.229
Measures of size could sometimes be influenced by particle shape [Dacey and Krumbein, 1979].
For example, Feret’s diameter, the longest distance between any two points on a certain particle,
could be exactly the same while rocks are diverse in shape – one spherical and one rectangle.
Relations between particle size and shape have also been studied in sedimentology. For example,
Krumbein [1941a] suggested a power function relating roundness and size in river rocks.
However, Russel and Taylor [1937] found no obvious correlation between the size and shape of
sand particles. In this study, we found a negative linear relationship between mean circularity
and mean Feret’s diameter (Figure 2.9).
22
Figure 2.9. Relationship between mean circularity and mean Feret’s diameter on (a) Mesa 1, (b) Mesa 2
and (c) Mesa 3.
Mean particle size on Mesa 1 and 2 are similar, however the mean circularity varies. On Mesa 3
mean particle size is smaller, but still significantly correlated with the shape. These slight
differences in size-shape correlation could be attributed to different rock textures and original
evolution processes. Among these three mesas, the cap rock of Mesa 2 consists of a
conglomerate with well-rounded pebbles embedded in the matrix. During weathering some of
those pebbles have been released from the matrix and are now part of the rock armour which
contributes to the larger circularity values.
2.3.4 Spatial trends in probability distributions
Results from Hartigan’s Diptest show that less than 2% of the locations have provided evidence
against unimodality at the 5% significance level (see Appendix B). The lognormal distribution
performed much better than gamma and Weibull distributions for describing Feret’s diameter. A
total of 85.14% sample points passed chi-square test for lognormal distribution, with 73.42%
pass for gamma distribution and 52.70% for Weibull distribution, while only 1% shows
evidence of consistency with power-law distribution. However, none of the tested probability
distributions were found to describe circularity.
Probability density function may provide us with an indication on the formation of the rock
fragment soil cover. For example, the Weibull distribution of particle size was predicted by a
cascade fragmentation model [Turcotte, 1992]. For small sized particles, Weibull distribution
can be reduced to a power-law relation by Taylor series [Turcotte, 1992]. Instead of a Weibull or
power-law distribution of particle size resulting from a single instantaneous fragmentation event,
we found lognormal distribution in particle size. As a natural form of particle size distribution,
lognormal distribution is derived from diffusion breakdown considering fragmentation as
temporal processes [Kolmogorov, 1941 as cited by Dacey and Krumbein, 1979; Epstein, 1948].
The fitted lognormal distributions are characterized by distribution parameters μ and σ from:
40 60 80 100 140
0.60
0.65
0.70
0.75
(a)
R2=0.621
RSE=0.026
Mea
n C
ircu
lari
ty
40 60 80 100 140
0.60
0.65
0.70
0.75
(b)
Mean Feret's Diameter (mm)
R2=0.751
RSE=0.013
40 60 80 100 140
0.60
0.65
0.70
0.75
(c)
R2=0.313
RSE=0.022
23
0],2
)(lnexp[
2
1),;(
2
2
xx
xxf
(2.6)
where x is the variable; μ and σ denote parameters of lognormal distribution that are the mean
and standard deviation of the variable’s natural logarithm , respectively, also known as location
parameter and the scale parameter.
Positive linear trends with distance are found for both parameters μ and σ on all three mesas,
including Mesa 3 (Figure 2.10). The negative relationship between distribution parameters and
distance is relatively weaker on Mesa 3 with lower r squared, but they are still significant. This
relationship is even stronger on each individual transect. Although the trends in the distribution
parameters seem minor, the equivalent changes in rock fragment size are immediately apparent.
For example, the change in the estimated μ on Mesa 1 decreases from μ=4.483 to μ=2.977
along the hillslope (μ = -0.014X + 4.093). This represents a threefold difference in the mean
particle size from 128 mm to 44 mm.
Figure 2.10. Trends in the parameters of the fitted distributions for all Feret’s diameter data. They are (a)
μ and (d) σ on Mesa 1; (b) μ and (e) σ on Mesa 2; and (c) μ and (f) σ on Mesa 3.
While mean or median size is often chosen to represent the spatial patterns of rock fragments on
hillslopes, few studies quantified the probability distribution of particle size and shape.
Probability distribution measured in this study overcomes the limitation that distributions with
0 20 40 60
3.0
3.5
4.0
4.5
(a)
R2=0.512
=0.014X 4.093
0 20 40 60
3.0
3.5
4.0
4.5
(b)
R2=0.498
=0.011X 3.927
0 10 30 50
3.0
3.5
4.0
4.5
(c)
R2=0.129
=0.006X3.523
0 20 40 60
0.6
0.8
1.0
1.2
1.4
(d)
R2=0.421
=0.004X 1.033
0 20 40 60
0.6
0.8
1.0
1.2
1.4
(e)
Distance (m)
R2=0.548
=0.006X 1.103
0 10 30 50
0.6
0.8
1.0
1.2
1.4
(f)
R2=0.191
=0.004X0.852
24
identical values for the mean or the median may have very different variance and skewness.
Parameters that describe shape and position of the distribution may provide us with another
window into assessing the evolution of rock armour. The linear change of the two distribution
parameters in space indicates that a simple modelling approach may be used to describe the
complex processes of rock armouring.
2.3.5 Implications for processes contributing to the formation of rock armour
Hillslopes are controlled by various processes depending upon elevation, slope angle, and
natural weathering properties of the bedrocks [Melton, 1965]. Spatial patterns of rock fragments
on hillslopes have been used to postulate the formation of rock armour [Abrahams et al., 1985;
Nyssen et al., 2002]. The downslope fining patterns of rock particles has been explained by
selective transport of fine materials by runoff events accumulating rock fragments on the
surface on both undisturbed and management affected hillslopes [Abrahams et al., 1985; Poesen
et al., 1998]. Surface runoff is also hypothesized as the dominant process in the studies of rock
fragments focusing on erosion and rock-soil interaction [Simanton et al., 1994; Li et al., 2007;
Zhu and Shao, 2008]. A few other processes were assessed by Nyssen et al. [2002], including
tillage, trampling, vertical sorting and rock fall on tillaged land. A “runoff creep” process, which
consists of larger rocks tilting and moving as the supporting fines are removed, was
hypothesized to be responsible for the spatial trends of particles too large to be transported by
overland flow [de Ploey and Moeyersons, 1975; Abrahams et al., 1990]. Dry ravel, which is
another mechanism for the movement of rock fragments by rolling, bouncing and sliding under
gravity is likely to result in downslope coarsening [Gabet, 2003].
The land surface of the Telfer region in the Great Sandy Desert is very old, probably older than
any of the landforms in the studies mentioned above [Henderson, 1996]. Rock fragments on the
mesa surface have a long history, with little human impact. Tillage can be ruled out in this study.
Trampling by animals tends to move large particles more rapidly downslope and is therefore
considered an unlikely mechanism for the observed sorting. Similarly, rock fall or dry ravel
which leads to a reverse sorting pattern, with larger particles found at the bottom of the hills,
contradict our findings and are therefore unlikely to be the sorting mechanism. It is evident that
the surface rocks are sourced from the cap on Mesa 1 and 2 as the rock texture of surface rocks
and the ones in soil profiles are very different, and there are insufficient rock particles in soil
profile to replenish the surface cover. Therefore, vertical sorting is unlikely to be significant.
While the surface coverage of rock fragments is slightly smaller on our mesa slopes in
comparison to other studies [Simanton et al., 1994; Poesen et al., 1998; Nyssen et al., 2002], the
particle size is however larger, indicating surface runoff alone is unlikely to generate the spatial
organization of rock armour. At a late stage of the landform evolution, surface weathering is
likely to be important for the formation of rock armour [Cooke, 1970] and is proposed to be a
25
highly significant process at our study sites.
Despite the importance of physical weathering processes for the formation of rock armour, few
studies have considered this process to explain the spatial distribution of rock fragments during
landform evolution. In the context of land use changes and engineered landforms, the formation
of rock armour as part of the physical weathering process and sediment redistribution is relevant
for designing stable landforms. Sediment transport and physical weathering as part of the
formation of stable land surfaces has been incorporated into landform evolution models
[Willgoose and Sharmeen, 2006; Wells et al., 2008; Cohen et al., 2009], However, few of these
models were able to describe the spatial patterns of rock fragments on mesa slopes as found in
this study. For example, instead of sourcing rock fragments from the cap, Willgoose and
Sharmeen [2006] hypothesized that surface rocks are being replenished from underneath after
finer soil particles have being washed out. They did not assess particle size distribution as
means of validating their modelling results against data. Contrary to our observations, Wells et
al. [2008] studied rock fragmentation and obtained multi-modal probability distributions during
the early stage of weathering. However, it appeared that a negative skewed unimodal
distribution was obtained as weathering proceeded, indicating that the later stages of weathering
may lead to unimodality that is consistent with the probability distribution of our dataset [Wells
et al., 2008]. The mARM model described by Cohen et al. [2009] yielded a downslope
coarsening in rock particles due to the combined effect of erosion, deposition and physical
weathering. There is a need to assess different processes such as fragmentation and creep that
could be responsible for rock armour evolution, and search for a proper model to describe our
observations.
2.4 Conclusion
With the investigation of geomorphic dependencies of rock fragment characteristics, we found
consistency with previous studies that surface coverage and mean particle size of rock
fragments increased with slope gradient on two concave shaped mesas. However, distinct spatial
patterns of both size and shape were also observed when surface distance from capping was
taken as the explanatory variable, including Mesa 3 which was approximately rectilinear with
constant gradient. Mean particle size decreased in a power-law and/or exponential function of
distance along the transects, while mean circularity increased linearly. Particle size was found to
be lognormally distributed on all the mesas. The lognormal distribution changed systematically
in space, with distribution parameters μ and σ decreasing linearly with distance down all mesa
slopes.
While wash-out certainly contributes to surface evolution, based on the age of the landform and
26
the very large particles, fragmentation is very likely to be a significant process on the old
landform. Further studies are required for deriving principles to aid the design of engineered
landforms.
27
References
Abrahams, A. D., A. J. Parsons, R. U. Cooke, and R. W. Reeves (1984), Stone movement on
hillslopes in the Mojave Desert, California: A 16 year record, Earth Surf. Process. Landforms,
9(4), 365-370.
Abrahams, A. D., A. J. Parsons, and P. J. Hirshi (1985), Hillslope gradient-particle size relations:
evidence for the formation of debris slopes by hydraulic processes in the Mojave Desert, J.
Geol., 93(3), 347-357.
Abrahams, A. D., N. Soltyka, and A. J. Parsons (1990), Fabric analysis of a desert debris slope:
Bell Mountain, California, J. Geol., 98(2), 264-272.
Adelsberger, K. A., and J. R. Smith (2009), Desert pavement development and landscape
stability on the Eastern Libyan Plateau, Egypt, Geomorph., 107(3-4), 178-194.
Al-Farraj, A. (2000), Desert pavement characteristics on wadi terrace and alluvial fan surfaces:
Wadi Al-Bih, UAE and Oman, Geomorph., 35(3-4), 279-297.
Barrett, P. J. (1980), The shape of rock particles, a critical review, Sedim., 27(3), 291-303.
Bates, D. M., and D. G. Watts (2008), Appendix 1. Data Sets Used in Examples, in Nonlinear
Regression Analysis and Its Applications, pp. 267-285, John Wiley & Sons, Inc., New York.
Bittelli, M., G. S. Campbell and M. Flury (1999), Characterization of particle-size distribution
in soils with a fragmentation model, Soil Sci. Soc. Am. J., 63(4), 782-788.
Bureau of Meteorology (2011), Climate statistics for Australian locations - TELFER AERO,
Commonwealth of Australia, http://www.bom.gov.au/climate/averages.
Byrd, T. (2008), Investigation of size and distribution of rock fragments and vegetation on
mesas in the Great Sandy Desert, Western Australia, University of Western Australia, Perth.
Canfield, H. E., V. L. Lopes, and D. C. Goodrich (2001), Hillslope characteristics and particle
size composition of surficial armoring on a semiarid watershed in the southwestern United
States, Catena, 44(1), 1-11.
Chambers, J. M. (1992), Linear models, in Statistical Models in S, edited by J. M. Chambers
and T. J. Hastie, pp. 99-116, Wadswoths & Brooks/Cole, Belmont, California, USA.
Cohen, S., G. Willgoose, and G. Hancock (2009), The mARM spatially distributed soil
evolution model: A computationally efficient modeling framework and analysis of hillslope
soil surface organization, J. Geophys. Res., 114(F3), F03001.
Cooke, R. U. (1970), Stone pavements in deserts, Ann. Assoc. Am. Geog., 60(3), 560-577.
Cooke, R. U., A. Warren, and A. Goude (1993), Surface particle concentrations: stone
pavements, pp 68-76, in Desert Geomorphology, ULC Press, London, United Kingdom.
Dacey, M. F., and W. C. Krumbein (1979), Models of breakage and selection for particle size
distributions, J. Int. Assoc. Math. Geol., 11(2), 193-222.
de Figueiredo, T., and J. Poesen (1998), Effects of surface rock fragment characteristics on
interrill runoff and erosion of a silty loam soil, Soil Tillage Res., 46(1-2), 81-95.
de Ploey, J., and J. Moeyersons (1975), Runoff creep of coarse debris: Experimental data and
28
some field observations, Catena, 2, 275-288.
Dixon, J. C. (1994), Aridic soils, patterned ground, and desert pavements, in Geomorphology of
Desert Environments, edited by A. D. Abrahams and A. J. Parsons, pp. 64-81, Chapman&Hall,
London, United Kingdom.
Dunkerley, D. L. (1995), Surface stone cover on desert hillslopes; parameterizing characteristics
relevant to infiltration and surface runoff, Earth Surf. Process. Landforms, 20(3), 207-218.
Dury, G. H. (1966), Pediment slope and particle size at Middle Pinnacle, near Broken Hill, New
South Wales, Aust. Geogr. Stud., 4(1), 1-17.
Epstein, B. (1948), Logarithmico-normal distribution in breakage of solids, Ind. Eng. Chem.,
40(12), 2289-2291.
Friedman, G. M. (1962), On sorting, sorting coefficients, and the lognormality of the grain-size
distribution of sandstones, J. Geol., 70(6), 737-753.
Gabet, E. J. (2003), Sediment transport by dry ravel, J. Geophys. Res., 108(B1), 2049.
Grady, D. E., and M. E. Kipp (1985), Geometric statistics and dynamic fragmentation, J. Appl.
Phys., 58(3), 1210-1222.
Hartigan, P. M. (1985), Algorithm AS 217: computation of the dip statistic to test for
unimodality, J. Roy. Stat. Soc. C-App, 34(3), 320-325.
Henderson, I. (1996), A Study of Regolith Landforms in the Telfer District, Western Australia,
The University of Western Australia.
Ihaka, R., and R. Gentleman (1996), R: A language for data analysis and graphics, J. Comput.
Graph. Stat., 5(3), 299-314.
Krumbein, W. C. (1941a), The effects of abrasion on the size, shape and roundness of rock
fragments, J. Geol., 49(5), 482-520.
Krumbein, W. C. (1941b), Measurement and geological significance of shape and roundness of
sedimentary particles, J. Sediment Res., 11(2), 64-72.
Li, X.-Y., S. Contreras, and A. Sole-Benet (2007), Spatial distribution of rock fragments in
dolines: A case study in a semiarid Mediterranean mountain-range (Sierra de Gádor, SE
Spain), Catena, 70(3), 366-374.
Martin, M. A., C. Garcia-Gutierrez, and M. Reyes (2009), Modeling multifractal features of soil
particle size distributions with Kolmogorov fragmentation algorithms, Vadose Zone J., 8(1),
202-208.
Maxlow, J. (2005), Technical services group Telfer district review, Technical Report, Newcrest
Mining, Perth, Australia.
Melton, M. A. (1965), Debris-covered hillslopes of the Southern Arizona Desert: Consideration
of their stability and sediment contribution, J. Geol., 73(5), 715-729.
Nyssen, J., J. Poesen, J. Moeyersons, E. Lavrysen, M. Haile, and J. Deckers (2002), Spatial
distribution of rock fragments in cultivated soils in northern Ethiopia as affected by lateral
and vertical displacement processes, Geomorph., 43(1-2), 1-16.
29
Parker, R. M. (2006), Understanding Controls of Landform Stability of Mesa Systems to Design
Slopes of Rock Waste Dumps at Telfer Goldmine, Pilbara WA, The University of Western
Australia.
Perfect, E. (1997), Fractal models for the fragmentation of rocks and soils: a review, Eng. Geol.,
48(3-4), 185-198.
Poesen, J., and H. Lavee (1994), Rock fragments in top soils - significance and processes,
Catena, 23(1-2), 1-28.
Poesen, J. W., B. van Wesemael, K. Bunte, and A. S. Benet (1998), Variation of rock fragment
cover and size along semiarid hillslopes: a case-study from southeast Spain, Geomorph.,
23(2-4), 323-335.
Rasband, W. S. (2008), ImageJ, U.S. National Institutes of Health, Bethesda, Maryland, USA,
http://rsb.info.nih.gov/ij/.
Ricci, V. (2005), Fitting distribution with R, Vienna, Austria.
Royston, J. P. (1982), An extension of Shapiro and Wilk's W Test for normality to large samples,
J. Roy. Stat. Soc. C-App, 31(2), 115-124.
Russell, R. D., and R. E. Taylor (1937), Roundness and shape of Mississippi River sands, J.
Geol., 45(3), 225-267.
Sheather, S. J. and M. C. Jones (1991), A reliable data-based bandwidth selection method for
kernel density estimation, J. Roy. Statist. Soc. B, 53: 683–690.
Simanton, J. R., K. G. Renard, C. M. Christiansen, and L. J. Lane (1994), Spatial-distribution of
surface rock fragments along catenas in semiarid Arizona and Nevada, USA, Catena, 23(1-2),
29-42.
Simanton, J. R., and T. J. Toy (1994), The relation between surface rock-fragment cover and
semiarid hillslope profile morphology, Catena, 23(3-4), 213-225.
Skiljan, I. (2011), Download IrfanView, http://www.irfanview.com/main_download_engl.htm.
Sturges, H. A. (1926), The choice of a class interval, J. Am. Stat. Assoc., 21(153), 65.
Thévenaz, P. (2008), Download algorithms, in TurboReg, Biomedical Imaging Group, Swiss
Federal Institute of Technology Lausanne, http://bigwww.epfl.ch/thevenaz/turboreg/.
Turcotte, D. L. (1986), Fractals and fragmentation, J. Geophys. Res., 91(B2), 1921-1926.
Turcotte, D. L. (1992), Fragmentation, pp 20-34, in Fractals and Chaos in Geology and
Geophysics, Cambridge University Press, UK.
Valentin, C. (1994), Surface sealing as affected by various rock fragment covers in West Africa,
Catena, 23(1-2), 87-97.
van Wesemael, B., M. Mulligan, and J. Poesen (2000), Spatial patterns of soil water balance on
intensively cultivated hillslopes in a semi-arid environment: the impact of rock fragments and
soil thickness, Hydrol. Process., 14(10), 1811-1828.
Venables, W. N., and B. D. Ripley (2002), Univariate Statistics, in Modern Applied Statistics
with S, pp. 107-138, Springer, New York, USA.
30
Wadell, H. (1932), Volume, shape, and roundness of rock particles, J. Geol., 40(5), 443-451.
Wells, T., G. R. Willgoose, and G. R. Hancock (2008), Modeling weathering pathways and
processes of the fragmentation of salt weathered quartz-chlorite schist, J. Geophys. Res.,
113(F1), F01014.
Willgoose, G. R., and S. Sharmeen (2006), A one-dimensional model for simulating armouring
and erosion on hillslopes: 1. model development and event-scale dynamics, Earth Surf.
Process. Landforms, 31(8), 970-991.
Zhu, Y. J., and M. A. Shao (2008), Spatial distribution of surface rock fragment on hillslopes in
a small catchment in wind-water erosion crisscross region of the Loess Plateau, Sci. China
Ser. D, 51(6), 862-870.
31
Chapter 3: Does fragmentation weathering explain rock
particle sorting on arid hills?
Zhengyao Nie1, Gavan S. McGrath1, Christoph Hinz1 and Tia Byrd1
1. School of Earth and Environment, The University of Western Australia, Western Australia,
Australia.
Abstract
Transport processes are often suggested as the underlying mechanisms explaining the sorting of
rock particles on arid hillslopes, whereby mean particle sizes typically decrease in the
downslope direction. Here we show that fragmentation of particles can also reproduce similar
emergent patterns. A total of 93,415 rock fragments were digitized from 222 photographs on
three mesa hills in the Great Sandy Desert, Australia. Rock fragment size was found to be
distributed lognormally, with both the location and scale parameters decreasing approximately
linearly with distance down each transect. As particles were often much larger than those which
could be expected to be transported downslope by fluvial processes on these short hills, we
assessed whether a fragmentation process could instead reproduce the observed patterns. A
dynamic fragmentation model, with just two parameters and using the particle size distribution
at the top of the hill as the initial condition, predicted the remaining particle size distributions
downslope. The results have analogies with Sternberg's Law of abrasion in rivers, which
suggests the potential for a more general principal underpinning physical weathering of rock
particles.
Keywords: Rock fragments; particle sorting; fragmentation; alternative to fluvial transport;
mesa hillslope
3.1 Introduction
Rock fragments play a significant role in the regulation of ecological and geomorphic processes
such as infiltration, evaporation and erosion. For example, surface rock size, fractional cover
and their position in the soil all affect infiltration rates in nonlinear ways [Poesen and Lavee,
1991]. Rock fragment size and cover are also key factors affecting erosion rates [de Figueiredo
and Poesen, 1998]. Due to their impacts on these processes, observational studies have
attempted to relate rock particle size to geomorphic features of the hill, such as its slope [Poesen
et al., 1998]. Both within - and between - hillslope differences in mean particle size have been
proposed to relate to slope [Abrahams et al., 1985; Parsons et al., 2009]. A feature of these
32
studies is the use of characteristic measures of rock size such as the mean or median [Abrahams
et al., 1985; Poesen et al., 1998; Zhu et al., 2008] rather than consideration of the size
distribution of particles. Dunkerley [1995] acknowledged this limitation, pointing out that the
impact on runoff, infiltration and erosion may be very different where a narrow range the rock
size distribution exists as compared to the case of a broader distribution even the means of the
two are the same. To the best of our knowledge, and rather surprisingly, spatial patterns of
surface rock size distributions on arid hills have not yet been quantified.
Preferential erosion of fine soils often explains the emergence of rock armoured surfaces
[Poesen and Lavee, 1994]. Implicit in this has been the assumption that a similar selective
transport of small rock fragments is the mechanism explaining within hill sorting that rock
fragments on the slope surfaces are often organized and typically display a decrease in particle
size downslope [Abrahams et al., 1985; Cooke et al., 1993; Parsons et al., 2009]. However,
selective transport is often observable up to certain thresholds in particle size [Kirkby and
Kirkby, 1974; Abrahams et al., 1984]. In the case of rock armoured slopes mechanisms such as
runoff creep for a much wider ranged size have been hypothesised yet remain to be confirmed
as a particle sorting process [de Ploey and Moeyersons, 1975; Abrahams et al., 1990].
This sorting phenomenon by particle size has been observed in a variety of geomorphic systems
including desert dunes, rivers and beaches [Friedman, 1962; Jerolmack and Brzinski, 2010], yet
particularly common for rock fragments on debris mantled hillslopes in arid and semiarid
environments [Cooke, 1972; Poesen et al., 1998; Parsons et al., 2009]. In some systems, rivers
for example, physical weathering that includes abrasion and/or impact breakage, as well as
selective transport is considered necessary to explain features of downstream sorting [Sklar and
Deitrich, 2006; Jerolmack and Brzinski, 2010; Chatanantavet et al., 2010]. In a fragmentation
and abrasion model that predicted sediment fining during transportation, both mechanisms were
considered for changes in particle size distribution during an experimental study [Le Bouteiller
et al., 2011].
Untangling selective transport and physical weathering in barrel tumbling experiments found
that the mean particle size of rocks decreased exponentially with distance travelled downstream
[Sternberg, 1985 as cited by Krumbein, 1941]. This is known as Sternberg’s Law of abrasion.
Krumbein [1941] and references therein not only confirmed these results but also noted rapid
and preferential initial abrasion and breakage of the largest particles. The consideration that
physical weathering of particles may play a role in size sorting has largely been neglected in
hillslope studies, with a few exceptions studying artificial rock dump geomorphology [Wells et
al., 2008; Cohen et al., 2009].
33
In deserts, where weathering is pervasive, rock exposed to the elements at the surface can
fragment by thermal fracture, salt crystal fracture, and other physical, chemical and biological
weathering processes [Cooke et al., 1993; Eppes et al., 2010] In various contexts, fragmentation
has been modelled in a number of disciplines including soil science, engineering and geology
[Turcotte, 1986; Perfect, 1997; Bittelli et al., 1999; Bird et al., 2009; Cohen et al., 2009]. In
stochastic models of instantaneous fragmentation, such that occurs in an explosion, a cascade of
fragments from a single large particle has been shown to result in power-law size distributions
when a scale invariant cascade is assumed [Turcotte, 1986; Perfect, 1997; Bittelli et al., 1999].
In earlier studies, where fragmentation was instead considered as a temporal phenomenon,
asymptotic lognormal particle size distributions were predicted to occur [Kolmogorov, 1941 as
cited by Bird et al., 2009; Epstein, 1948; Bird et al., 2009]. More recently, Bird et al. [2009]
proposed a dynamic model combining temporal and instantaneous cascade fragmentation
processes.
Motivated by the debate about the relative importance of physical weathering and selective
transport to explain downstream fining in rivers [Sklar and Deitrich, 2006; Jerolmack and
Brzinski, 2010], here we hypothesize that observed sorting of rock particle size distributions on
hillslopes can arise from a fragmentation process. In this study, we adapt a dynamic
fragmentation model to simulate the evolution of coarse rock fragments size distributions on
three arid mesa slopes from the Great Sandy Desert, Australia. Many of these rocks are expected
to be too large to be moved by overland flow. We therefore model fragmentation without any
selective transport being considered explicitly. A rock covered surface is assumed implicitly to
be maintained by selective transport of fine soil, finer than the smallest rock fragments
considered. Unlike previous studies reporting only mean or median particle sizes, this study
characterizes, quantifies and then models changes in the statistical distributions of rock particles
on arid hillslopes. Such an approach allows physical interpretation of not only of rock
weathering but also of the geomorphology of the hills studied.
3.2 Methods
3.2.1 Site description and sampling
Three mesa slopes in the Great Sandy Desert, Australia were selected for particle size analysis,
simply named as Mesa 1, Mesa 2 and Mesa 3. They are located at 21°45’46’’S, 122.9’57’’E;
21°56’01’’S, 122°12’43’’E; and 21°52’3’’S, 122°7’29’’E. These mesa slopes have a surface
coverage of up to 67.2% siltstone or silcrete conglomerate rock armour sourced from a duricrust
cap which had undergone secondary silicification, giving rise to relatively hard rocks in
comparison to the underlying siltstone [Henderson, 1996]. The typical lithology revealed by
excavation on several hills showed that the surface rock armour overlay a highly weathered
34
zone between 10 – 200 cm thick, including a clear zone without any rock fragments, before a
siltstone bedrock was intercepted. Vegetation on these mesas are usually acacia shrubs and
tussock grass, but only the latter were found on Mesa 3. The climate is arid, with a mean annual
rainfall of around 300 mm, and an average daily temperature varying seasonally between 25°C
– 42°C [Bureau of Meterology, 2011]. These mesa hills are analogous to those described in
Ollier and Tuddenham [1962] in which a duricrust capped hills, covered in a thin armour layer
sourced from the cap material retreated laterally over time. In addition, retreat of the cap was
hypothesised to control the rate of hillslope retreat.
Four transects were placed along each mesa, beginning from the lower edge of the hard
duricrust cap downhill in a straight line in the direction of the steepest descent, where duricrust
cap was not included in sampling. Transects on the three mesas were facing different aspects
respectively as northwest, south and east. Sample points were located at three meter intervals
along each transect to a distance of 60 m on Mesa 1 and 2, whereas varying from 24 m to 51 m
on Mesa 3 as the hillslope intercepted aeolian sandy deposits at the base of the hill made it
shorter. Slopes on Mesa 1 and 2 are concave shaped, with slope gradient ranging from 1% to
68%, and 2% to 61%; yet Mesa 3 slopes are approximately rectilinear, with gradient of 36%.
At each of these points a one meter square reference frame was placed on the ground and digital
photographs were taken of the surface with the following cameras: Canon IXUS750 (3x optical
zoon, 7.1 megapixels), Olympus u 1720SW (3x optical zoom, 7.1 megapixels), and Olympus
u850SW (5x optical zoom, 8 megapixels). Four reduplicate photographs were taken at each
sample point, but this number was reduced if taking photographs was prevented by obstructions,
such as shrubs. All rock fragments included in those four photographs were combined as rock
samples at each sample point for further analysis. Rock fragments larger than 1 m2 surface area
thus are not able to be included. The software ImageJ was used to rectify and standardize
images to a resolution of 0.28 mm2 pixel-1 [Rasband, 2008]. Individual rock fragments were
hand-traced with a graphics tablet system (Intuos 3 Wacom) in order to create binary images.
Rock fragment size was quantified by the Feret’s diameter, the longest distance between any
two points on the perimeter of a particle. Due to the limitations of this methodology, rock
fragments less than 10 mm in diameter, were excluded from the analysis.
The chi-square test was applied to assess whether it is reasonable to assume that particles at
each location came from truncated lognormal distributions. A significance level of 5% was
adopted for this test. Parameters of upper truncated lognormal distributions were estimated by
maximum likelihood methods using the R software environment [Ihaka and Gentleman, 1996].
35
3.2.2 Modelling fragmentation
The model, adapted from Bird et al. [2009], describes the temporal evolution of the cumulative
mass fraction of particles equal to or less than a size [L]. This mass fraction is denoted iM [-],
where i denotes to the ith size class, in which the largest size class being i = 0 and the smallest i
= n. Fragmentation causes a proportion p of this fraction to break down into smaller size, i.e.
ii pMM 1 . With this definition, the fraction of mass at time t in a given class i is:
)()()( 1 tMtMtm iii (3.1)
A size dependent proportion iQ of this mass is assumed to break down to the next size class 1ir
at time t through a time-dependent fragmentation. In addition, at time t the mass fraction
passing via a cascade from larger particles to a number of smaller sizes equal to and less than ir
through instantaneous fragmentation is given by:
)1()()( tMtMtc iii (3.2)
A size dependent proportion iP of this mass fraction is smaller than ir , then the total mass
fraction balance equation is given by:
)1()()1()( 11 tmQtcPtMtM iiiiii (3.3)
Substituting Eq.3.3 with Eq.3.1 and Eq.3.2 results in the fragmentation model:
)]1()1([)]1()([)1()( 111 tMtMQtMtMPtMtM iiiiiiii (3.4)
In order to relate the mass fraction for model to the measured Feret’s diameter, we assume mass
m can be calculated from size r via barm where [M L-3] denotes the rock mass density,
and for spherical particles 6/a and 3b . As we will demonstrate later, observed rock size
distributions were consistent with lognormal distributions. Given the above mass-size
relationship then rock mass is also lognormally distributed, with distribution parameters * and
* , that can be derived from size distribution parameters and by the form 2* b
and * [Sklar and Dietrich, 2006].
The boundary conditions necessary to solve the time dependent fragmentation model are given
by 1)(0 tM and the initial cumulative mass fraction size distribution )0(0 FM i which we
obtained from parameterised regression estimates of the and at the top of the hillslope. As
the model specifies a largest particle class, we used a truncated lognormal distribution function
with a upper bound 0r , which gives the mass fraction of class i as:
*2
*)log(1
*2
*)log(1
0
rerf
rerf
M
i
i (3.5)
36
Bird et al. [2009] gave examples of model behaviour with various forms of iP and iQ . One
particular case they presented iii pQP where p < 1 and < 1 that produced similar
preferential weathering of large particles, however, was found to be unable to reproduce
observed changes in particle size distributions. This failure initialized a search for a different
type of model, in which the probability of fragmentation was a function of particle size
explicitly. The simplest and most predictive model found was the following:
)/exp( iii rQP (3.6)
where β [T-1 or equivalently in space m-1] is a size-invariant rate of fragmentation, controlling
how quickly the distribution shifts to smaller sizes and α [L-1] is a rate coefficient controlling the
size dependence of the fragmentation rate. Best fit parameters were identified using a systematic
examination of the parameter space using the root mean square (RMS) error between modeled
size distributions and distributions fitted to the data (see Appendix C; Figure 3.1 is an
example). In each case the error surface was smooth giving confidence that the estimated
parameters were close to the global optimum. We used 50 size classes, binned exponentially,
between 10 mm and 3000 mm for modeling. The largest size class r0 = 3000 mm was
determined from estimates of the thickness of the duricrust capping on the mesas.
Figure 3.1. Results of best fit parameters (α = -87, β = 0.31) on Mesa 2 as an example of systematic
examination of model parameters by minimizing root mean square (RMS) error between modeled
distributions and observed ones.
In summary we assume rocks armoring on slope surface are sourced from the duricrust cap at
the top of the hill, and weather in place as the hill retreats laterally the surface rock armor
weathers in place, relative to surrounding rock [Ollier and Tuddenham, 1962; Ollier, 1963]. One
model time step is assumed equivalent to one additional meter from the hard cap material, a
-90 -89 -88 -87 -86 -85 -84 -83 -82 -81
0.26
0.27
0.28
0.29
0.3
0.31
0.32
0.33
0.34
0.35
0.0
99
10
.10
18
0.1
04
50
.10
73
>=
0.1
1
RMS Error
37
time for space substitution. Selective transport of fine soil, finer than the smallest rock particle
considered, 10 mm, is assumed to maintain the surface rock armor but is not considered
explicitly in the modeling. Non-selective transport such as creep, moving the surface rock
irrespective of size, may also be occurring, but is also not considered explicitly.
3.3 Results
A total of 93415 rocks were digitized from the image. Size measurements of rock fragments in
85% of 222 photographs could not be rejected as coming from truncated lognormal distributions
(p < 0.05). There are obvious systematic changes of the distributions on Mesa 1 and 2, with
modes shifting from larger to smaller particle size as well as the distributions narrowing as one
progresses downslope (Figure 3.1). Although changes in distributions are not as obvious on
Mesa 3, all hills display a significant negative linear relationship between the fitted distribution
parameters (μ and σ) and slope surface distance from the cap (Figure 3.2). All slope coefficients
were found to be significantly different from zero (p < 0.01).
Figure 3.2. Changes in the size distribution of rock fragments with position on each hill, with the
frequency histogram of observed data (bars), truncated lognormal probability density functions (pdf)
fitted to the data (line), and modeled initial (solid circle) and predicted (open circles) size distributions.
The top is 0 m, the middle 30 m (Mesa 1 and 2), 27 m (Mesa 3) and the bottom 60 m (Mesa 1 and 2) and
51 m (Mesa 3) from the duricrust cap.
0.0
00
0.0
15
Fitted lognormal pdf
Model predictions
Initial condition
Top
Me
sa 1
Middle
Bottom
Pro
babi
lity
Den
sity
Me
sa 2
0 100 200 300 400
0.0
00
0.0
10
0.0
20
Me
sa 3
100 200 300 400
100 200 300 400Feret's Diameter (mm)
38
Table 3.1. Linear regression coefficients; best fit model parameters and root mean square (RMS) errors.
For the linear regressions x denotes distance in meters from the top of each transect.
Mesa 1 Mesa 2 Mesa 3
Best-fit Parameters α -81 -87 -75
β 0.31 0.31 0.21
Linear Regressions μ 4.385-0.001x 4.329-0.008x 4.131-0.007x
σ 0.753-0.005x 0.708-0.005x 0.595-0.003x
RMS Error Model vs. data 0.10 0.10 0.14
Regression vs. data 0.57 0.48 0.36
Values of model parameters α and β are summarized in Table 3.1. All the values are in a
narrow range and both α and β are smaller on Mesa 3. The modeled μ and σ were derived from a
nonlinear least squares fit of Eq. 3.5 to the modeled particle distribution using the conversion of
distribution parameters between mass fraction and particle size described earlier. The modelled
µ and σ both decrease with distance downslope, corresponding well with observed distribution
parameters (Figure 3.2). While σ decreases in an approximately linear way, μ appears a
nonlinearly decrease on Mesa 1 and 2. Comparing predictions of observed parameters with
linear regression and fragmentation model, root mean square errors in Table 3.1 are showing
advantages of the model.
Figure 3.3. Changes in particle size lognormal distribution location μ and scale σ parameters (Eq. 3.5) as
a function of distance from duricrust cap. Shown are the empirical fits to observed data (crosses), fitted
linear regressions and 95% confidence intervals (lines) and predicted distribution parameters by the
fragmentation model (circles) given the initial particle size distribution, emphasized by the large point.
3.8
4.0
4.2
4.4
Fitted lognormal
Modelled
Linear reg.
95% Cl
Mesa 1
Mesa 2 Mesa 3
0 10 20 30 40 50 60
0.3
0.5
0.7
0 10 20 30 40 50 60 0 10 20 30 40 50 60
Distance (m)
39
3.4 Discussion
In this study, the consideration of the whole particle size distribution has elucidated systematic
variation in two parameters which quantify the whole size distribution. The lognormal
distribution was found to provide reasonable fits to the data, and its two parameters were found
to decrease approximately linearly with distance in a surprisingly simply way. A lognormal
particle size distribution is consistent with the sizes emanating from a temporal fragmentation
process [Epstein, 1948; Bird et al., 2009] rather than a pure instantaneous cascade which tend to
produce power-law distributions [Turcotte, 1986; Perfect, 1997]. Moreover, beyond general
measures of rock characteristics, the lognormal distribution allows the fragmentation model
reproducing observation well without considering real physical fracture of individual rock
fragments, and give clues not only to the rock fracture but also to the geomorphology evolution.
For the first time, a fragmentation model other than a regression to observations has described
changes in surface rock particle size distributions along a hillslope. Consistent with the results
from investigations of relationship between a characteristic particle size (e.g., mean and/or
median size) and geomorphic features (e.g., slope gradient), we found mean rock size to be
larger where the slope is steeper, as two of the three hills were concave in shape. However an
approximately rectilinear hill, Mesa 3, also had rock particle sizes decreasing downslope with
respect to distance, while size and slope obviously obeyed on relationship. Few studies report
distance downslope but similar trends are implied from some studies based upon descriptions of
hillslope shape (e.g., Abrahams et al., 1985). It remains unclear how the distribution in rock
fragment size within a hillslope relate to its form. The relationship between rock size and slope,
in the case of the concave hills, may just be coincidental and not causal.
The apparent behaviour of these rock fragments has a number of similarities with observations
of abrasion studies. Changes in the size distribution downslope suggest a preferential reduction
in the numbers of larger particles. In the condition that α < 0 the fragmentation rates of larger
rocks are faster than smaller ones; when α > 0 smaller sized rock particles preferentially
fragment (Figure 3.4). This more frequently breaking down of larger rocks had also been
observed by Cooke [1970]. In the context of rivers, Krumbein [1941] noted preferential abrasion
and some breakage of the largest rocks in a tumbler experiment, particularly during the initial
phase of the experiment. Based on similar experiments Sternberg [1875] and Schoklitsch [1914]
(as cited by Krumbein, 1941) found mean rock size, r , decreases downstream exponentially as a
function of travelled distance, x, via )exp()0()( xrxr , known as Sternberg’s Law of
abrasion. For lognormally distributed rock sizes, mean particle size )2/exp()( 2 xr . In
this study as described above, 2 and is approximately a linear function of distance
40
cx , we obtain )exp()0()( xrxr for rock particle size on mesa hills. This suggests
the potential that Sternberg’s Law is potentially a more general result applicable to physical
weathering of rock fragments.
Figure 3.4. Changes in cumulative mass fractions of (a) empirical mass data converted from observed
particle size distribution; (b) best fit from model prediction (α < 0); (c) model prediction while α = 0; and
(d) model prediction while α > 0.
Selective transport of fine materials by runoff events is the dominant explanation for the sorting.
However, more than 30% of rock fragments in this study above the threshold previously
reported for transport by overland flow of 65 mm, and with sorting already evident at very large
rock sizes, we find this explanation unsatisfactory [Abrahams et al., 1984]. A mechanism
observed by de Ploey and Moeyersons [1975] which induces a process of slow creep as a result
of selective removal of supporting fines by runoff, has been hypothesised as a sorting process
[Abrahams et al., 1990]. However, there is currently no evidence for sorting of large rock
fragments by this process. Another mechanism for the movement of dry debris and sorting as a
result of rolling, bouncing and sliding is dry ravel [Gabet, 2003]. However, this results in
downslope coarsening, opposite to the observations in our field sites. Deflation processes
described by Cooke [1970] illuminated how coarse rock fragments remained and concentrated
on surface during removal of underlying fines, which corresponded with hillslope retreating
0.0
0.2
0.4
0.6
0.8
1.0
(a) (b)
10 50 200 500 2000
0.0
0.2
0.4
0.6
0.8
1.0
(c)
10 50 200 500 2000
(d)
Diameter (mm)
Cu
mu
lativ
e M
ass
Fra
ctio
n
41
process in this study. Yet this hypothesis failed to explain particle sorting still. Nevertheless,
irrespective the mechanism for the sorting of rock fragments, we consider that selective
transport of fines which maintains the rock armour layer, removing fine physically weathered
fragments is still an essential process on these hills.
Thermal expansion and contraction, which is due to extremes of diurnal variability, significant
seasonal variations and sometimes short term fluctuations in temperature, is believed to be a
likely mechanisms, leading to rock fragmentation in the form of parallel breakdown and
irregular crazing crack [Ollier, 1963; Smith, 1988; Cooke et al., 1993; Eppes et al., 2010]. With
large fluctuations in temperature occurring in the study site, those rock fragments partially
buried into fine matrix are said to be prone to insolation weathering, because of constraint
volume change and reduced heating and cooling in the buried part leading to higher stress
gradient for fracture [Ollier, 1963; Smith, 1988]. Yet, it is unlikely that pure temperature change
having sufficient energy to break down rocks. Salt crystallization, chemical and biological
activities, abrasion and collision within subsequent fluvial, and freeze-thaw could also
contribute to particle breakdown [Cooke, 1970; Smith, 1988; Cooke et al., 1993].
In a pediment survey on Broken Hill in New South Wales of Australia, Lanford-Smith and Dury
[1964] claimed that the standard slope elment are independent of the presence of caprock.
However, rock material of the fragments on mesas in this study is consistent with the one of the
caprock. Ollier and Tuddenham [1962] suggested that the rate of hillslope retreat is controlled
by the rate of retreat of the duricrust cap. Therefore, it is the weathering of mesa cap being the
rate-limiting factor of hillslope retreat and the supply of surface rock armour. Preferential
fragmentation of larger particles will remain a rock mantled surface, but in contrast, if smaller
particles more easily breakdown, the rate of rock supply from the cap would be unable to catch
up with the rate of rocks weathering further downslope, resulting in the failure of maintaining
surface armour along the entire slope. A definitive assessment of the hypothesized
fragmentation process as an explanation for particle sorting appears to require a study to date
rock fragment surfaces. A physical mechanism explaining the estimated probability of fracture
)/exp( ii rP , (where 0 ) remains to be determined.
3.5 Conclusion
We investigated the spatial distribution of rock fragments on three debris mantled mesa in the
Great Sandy Desert, Australia. This study demonstrated the size distributions were consistent
with lognormal distributions and that the distributions changes systematically as a function of
distance from the hard duricrust cap, thought to be the source of the rock armour. A dynamic
42
model of fragmentation reproduced the observed changes in rock size distributions along the
hills. In addition some similarities with another physical weather process, abrasion, were found
with similarly preferential weathering of large particles initially and an approximate change in
mean particle size that may be consistent with Sternberg’s Law of abrasion, indicating an
exponential decline in mean particle size with distance (time). We believe further studies to
assess whole particle size distributions will be valuable for interpreting the geomorphologic and
ecological processes altering arid hillslopes.
43
References
Abrahams, A. D. (1984), Stone movement on hillslopes in the Mojave Desert, California: A 16
year record, Earth Surf. Process. Landforms, 9(4), 365-370.
Abrahams, A. D., A. J. Parsons, and P. J. Hirshi (1985), Hillslope gradient-particle size relations:
evidence for the formation of debris slopes by hydraulic processes in the Mojave Desert, J.
Geol., 93(3), 347-357.
Abrahams, A. D., N. Soltyka, and A. J. Parsons (1990), Fabric analysis of a desert debris slope:
Bell Mountain, California, J. Geol., 98(2), 264-272.
Bird, N. R. A., C. W. Watts, A. M. Tarquis, and A. P. Whitmore (2009), Modeling dynamic
fragmentation of soil, Vadose Zone J, 8(1), 197-201.
Bittelli, M., G. S. Campbell and M. Flury (1999), Characterization of particle-size distribution
in soils with a fragmentation model, Soil Sci. Soc. Am. J., 63(4), 782-788.
Bureau of Meteorology (2011), Climate statistics for Australian locations - TELFER AERO,
Commonwealth of Australia, http://www.bom.gov.au/climate/averages.
Chatanantavet, P., E. Lajeunesse, G. Parker, L. Malverti, and P. Meunier (2010), Physically
based model of downstream fining in bedrock streams with lateral input, Water Resour. Res.,
46(2), W02518.
Cohen, S., G. Willgoose and G. Hancock (2009), The mARM spatially distributed soil evolution
model: A computationally efficient modeling framework and analysis of hillslope soil
surface organization, J. Geophys. Res., 114(F3), F03001.
Cooke, R. U. (1970), Stone pavements in deserts, Ann. Assoc. Am. Geog., 60(3), 560-577.
Cooke, R. U. (1972), Relations between debris size and the slope of mountain fronts and
pediments in the Mojave Desert, California, Zeitschrift fr Geomorphologie, 0372-8854, 76-
82.
Cooke, R. U., A. Warren, and A. Goude (1993), Weathering forms and processes, pp 23-44, in
Desert Geomorphology, ULC Press, London, United Kingdom
de Figueiredo, T., and J. Poesen (1998), Effects of surface rock fragment characteristics on
interrill runoff and erosion of a silty loam soil, Soil Tillage Res., 46(1-2), 81-95.
de Ploey, J., and J. Moeyersons (1975), Runoff creep of coarse debris: Experimental data and
some field observations, Catena, 2, 275-288.
Dunkerley, D. L. (1995), Surface stone cover on desert hillslopes: parameterizing characteristics
relevant to infiltration and surface runoff, Earth Surf. Process. Landforms, 20(3), 207-218.
Eppes, M. C., et al. (2010), Cracks in desert pavement rocks: Further insights into mechanical
weathering by directional insolation, Geomorph., 123(1-2), 97-108.
Epstein, B. (1948), Logarithmico-normal distribution in breakage of solids, Ind. Eng. Chem.,
40(12), 2289–2291.
Friedman, G. M. (1962), On sorting, sorting coefficients, and the lognormality of the grain-size
distribution of sandstones, J. Geol., 70(6), 737-753.
44
Gabet, E. J. (2003), Sediment transport by dry ravel, J. Geophys. Res., 108(B1), 2049.
Henderson, I. (1996), A study of the regolith landforms in the Telfer district, Western Australia.
MSc thesis, The University of Western Australia
Ihaka, R. and Gentleman, R. (1996), R: A language for data analysis and graphics, J. Computat.
Graphic. Stat., 5, 299-314.
Jerolmack, D. J. and T. A. Brzinski (2010), Equivalence of abrupt grain-size transitions in
alluvial rivers and eolian sand seas: A hypothesis, Geology, 38(8), 719-722.
Kirkby, A., and M. J. Kirkby (1974), Surface wash at the semi-arid break in slope., Zeitschrift
fur Geomorphologie, (21), 151-176.
Krumbein, W. C. (1941), The effects of abrasion on the size, shape and roundness of rock
fragments, J. Geol., 49(5), 482-520.
Langford-Smith, L., and G. H. Dury (1964), A pediment survey at middle pinnacle, near Broken
Hill, New South Wales, J. Geol. Soc. Aust., 11(1), 79-88.
Le Bouteiller, C., F. Naaim-Bouvet, N. Mathys, and J. Lave (2011), A new framework for
modeling sediment fining during transport with fragmentation and abrasion, J. Geophys. Res.,
116(F3), F03002.
Ollier, C. D. (1963), Insolation weathering; examples from central Australia, Am. J. Sci., 261(4),
376-381.
Ollier, C. D. and W. G. Tuddenham (1962), Slope development at Coober Pedy, South Australia,
J. Geol. Soc. Aust., 9(1), 91-105.
Parsons, A. J., A. D. Abrahams and A. D. Howard (2009), Rock-Mantled Slopes, pp 233-263, in
Geomorphology of Desert Environments 2nd ed., edited by A. J. Parsons and A. D.
Abrahams, Springer.
Perfect, E. (1997), Fractal models for the fragmentation of rocks and soils: a review, Engineer.
Geol., 48(3-4), 185-198.
Poesen, J. W. A., and H. Lavee (1991), Effects of size and incorporation of synthetic mulch on
runoff and sediment yield from interrils in a laboratory study with simulated rainfall, Soil
Tillage Res., 21(3-4), 209-223.
Poesen, J., and H. Lavee (1994), Rock fragments in top soils significance and processes, Catena,
23(1-2), 1-28.
Poesen, J. W., B. van Wesemael, K. Bunte and A. S. Benet (1998), Variation of rock fragment
cover and size along semiarid hillslopes: a case-study from southeast Spain, Geomorph.,
23(2-4), 323-335.
Rasband, W.S. (2008), ImageJ, U. S. National Institutes of Health, Bethesda, Maryland, USA,
http://rsb.info.nih.gov/ij/
Sklar, L. S., and W. E. Dietrich (2006), The role of sediment in controlling steady-state bedrock
channel slope: Implications of the saltation-abrasion incision model, Geomorph., 82(1-2),
58-83.
45
Smith, B. J. (1988), Weathering of superficial limestone debris in a hot desert environment,
Geomorph., 1(4),355-367.
Turcotte, D. L. (1986), Fractals and fragmentation, J. Geophys. Res., 91(B2), 1921-1926.
Wells, T., G. R. Willgoose and G. R. Hancock (2008), Modeling weathering pathways and
processes of the fragmentation of salt weathered quartz-chlorite schist, J. Geophys. Res.,
113(F1), F01014.
Zhu, Y. J., and M. A. Shao (2008), Spatial distribution of surface rock fragment on hillslopes in
a small catchment in wind-water erosion crisscross region of the Loess Plateau, Sci. China D,
51(6), 862-870.
46
Chapter 4: Self-organisation of rock fragment cover on
engineered and natural mesa slopes
Zhengyao Nie1, Christoph Hinz1, Gavan S. McGrath1 and Erin Poultney1
1. School of Earth and Environment, The University of Western Australia, Western Australia,
Australia.
Abstract:
Erosion control and slope stabilization is essential for mine waste rock dumps (WRD). Rock
fragment armour on hillslopes protects subsoils from erosion. With the interest of the initial
surface development and self-organisation on engineered slopes in post–mining landscapes, the
study compared rock fragment characteristics on a mine waste rock dump and natural mesas,
and further hypothesized a wash-out process responsible for early evolution of rock armour. The
waste rock dump was designed to mimic concave shaped natural mesa to prevent erosion and to
blend in with the natural surrounding; two treatments were applied on the slope surface in the
construction, one with topsoils only, and the other with soil and rock fragment mixtures. Rock
fragments were characterized by surface cover, size, shape, and their probability distributions.
Rock fragments on the artificial slope were generally smaller, and more angular. Particle size
followed lognormal distribution, consistent with its natural analogues; circularity were from
beta distribution that has not been found on natural mesas. However, rock fragments on the
artificial slope did not exhibit a spatial pattern as natural mesa did. The differences in artificial
and natural mesas could be attributed to different lithology and evolution process. Erosion rills
measured on the artificial slope suggested surface soils without the presence of rock fragments
were less resistant to erosion. Fines were more easily to be washed out with less rock content
until the formation of rock armour which converged to values of about 32% irrespective of
initial rock content which was slightly lower than natural mesa slopes but fairly close. Rainfall
simulation experiments using different mixtures of top soil and rock fragments experiment
confirmed the convergence to a constant surface coverage after only 50 minutes of applied
rainfall. This study indicates that the formation of surface rock armour is a self-organized
phenomenon caused by different processes acting at different time scales for engineered and
natural hillslopes.
Keywords: rock fragments; artificial slope; rill erosion; rainfall simulation; water wash process;
initial surface development; self-organized pattern
47
4.1 Introduction
Reconstructing disturbed lands requires a design of stable landforms that will allow us to re-
establish post-disturbance land use and prevents rapid erosion. Post-mining landscapes are the
typical example for which stable landforms need to be engineered. A common problem with
current practice of engineered post-mining landforms is severe erosion risk in particular shortly
after slopes have been established [Lin and Herbert Jr, 1997; Lefebvre et al., 2001; Walker and
Powell, 2001; Johnson and Hallberg, 2005; Hancock et al., 2008]. The placement of rock
material and disturbed soil on the surface of engineered slopes makes the erodibility much
higher on mine waste rock dump than natural surfaces [Riley, 1995]. As a result rill and gully
erosion are common on such post-mining lands in particular in arid and semi-arid climates
[Nearing et al., 1997, Poesen et al., 2003; Hancock et al., 2008].
Hancock et al. [2003] proposed to use natural landforms as an analogue for engineered slopes of
rock waste dumps. Based on a case study in the Pilbara, Western Australia, they carried out a
geomorphic analysis of the surrounding landscape to derive the shape of the slopes that
engineered mine waste rock dumps should be designed with and suggested that concave slopes
are more likely to prevent erosion. While such concave slopes are common in arid lands,
engineered rock dumps consist usually of linear slopes interrupted by berms that accumulate
surface runoff and are major cause of tunnel and gully erosion. Concave slopes along with rock
mulch on the surface are the more desirable design to stabilize engineered landforms, but no
experimental evidence was provided to support this [Hancock et al., 2003]. Waste rock dumps at
Telfer goldmine followed the traditional design of a series of terraced benches with linear slopes
of approximately 20° with reverse graded berms. Problems of surface and tunnel erosion as well
as weed invasion occurred on those linear slopes in contrast to the natural concave shaped
slopes. Therefore a field trial was established to assess how engineered concave slopes with
different covers improve the resilience of the rehabilitation effort. Those were designed based
on natural mesas, which are stable landforms with flat tops often capped with hard debris, and
concave slopes armoured by rock fragments. These rock fragments placed in topsoils act as
mulch and play an overall positive role in infiltration and erosion control, to stabilize hillslopes
against interrill, rill and gully erosion [Brakensiek and Rawls, 1994; Poesen and Lavee, 1994;
Valentin, 1994; van Wesemael et al., 1996; Cerdà, 2001]. Accordingly, this experimental study
will focus on evaluating the role of rock fragments on stabilizing an engineered rock dump
slope and assess the self-organizing processes leading to rock armour.
Cooke [1970] proposed a wide range of different processes responsible for surface armour
development. In the early age of a hill, wash by water dominates the formation of the stone
pavements, before surface weathering gains significance. Vertical sorting with large stones
48
tending to concentrate in the soil surface due to saturation and desiccation cycles becomes
subsequently more important. In the long term all processes slow down and surface weathering
becomes the most important process in stone pavement development [Cooke, 1970]. Based on
this we hypothesize that rock fragment accumulation on the young engineered slope is caused
by wash-out. In contrast, rock fragment distribution of the old mesa slopes is most likely caused
by surface weathering and fragmentation as outlined in Chapter 3.
Accordingly, this paper reports on the characterisation of rock fragment distribution of two
contrasting site both located in the Great Sandy Desert in Western Australia, one being a
concave engineered slope and the other being natural mesa slopes with established rock armour.
The engineered slope was covered by rock fragments extracted from an open cut mined mixed
with topsoils. As this field trial was carried out as part of the mine operation to support closure
planning, the initial state of the trial was not assessed and documented. We therefore treat the
site the same way as the natural mesa site: We observe one point in time during the evolution of
the land surface and compare both sites based on the same measurements. The objective was to
perform a comparative field assessment of the coverage and size and shape distribution of
surface rock fragments on both landforms and measure the erosion indicators with a clear view
to assess how effective engineered rock pavements protect the surface in comparison to
naturally evolved surfaces. Furthermore, we carried out controlled rainfall simulation with
various mixes of soil and rock fragment to determine if our wash-out hypothesis contributes to
the self-organisation of rock armour during very early stage of hillslope evolution
4.2 Method
4.2.1 Study site
Telfer goldmine is located on the edge of Great Sandy Desert, Western Australia, at 21.71ºS,
122.23ºE. Telfer experiences hot summers and warm winters, with daily average maximum
temperature varying from 25ºC to 42ºC. The study site is in an arid environment, and the annual
rainfall is 250 – 400 mm mostly delivered as convective rain [Bureau of Meteorology, 2011].
A 1.5 year old waste rock dump in the study region was constructed to mimic local mesas, with
a plateau of large rock fragments on top, and decreasing steepness downhill: top slope is 37°,
middle 20° and bottom 15°. This mesa shaped design was to minimize erosion and to assimilate
the rock dump with the natural surroundings. The waste rocks from blasting and crushing are
siltstones, sand stones and quartzites. A 20 cm depth layer of sandstone or quartzite was placed
on top of the waste rock dump, and two treatments were applied during slope construction.
Treatment 1 was spreading topsoil to 5 cm depth on the slope surface; Treatment 2 was
spreading a mixture layer of 5 cm depth of topsoil and 20 cm depth of sandstone that was
49
premixed. Topsoils in the treatments were designed for the purpose of revegetation. They are
not from natural mesa slope, but from swales, which is essential as native plants will only return
with topsoil as the initial growth medium. Soil properties were very similar in both treatments.
Since the completion of the construction in 2006, the waste rock dump had experienced three
tropical cyclones – cyclone George, Jacob and Kara on March 2007, which brought 466 mm of
rainfall in one-month time, contributed 81.2% of annual total rainfall, and the highest rainfall
daily reached 93 mm [Bureau of Meteorology, 2011].
Five transects were placed along the waste rock dump slope from top to bottom for rock
fragment sampling, named as Transect A, B, M, C and D (Figure 1). Transect A and B were 70
m in length whereas the other three were longer at 90 m. Transect A and D intersected
Treatment 1, with bare soils on top; Transect B and C intersected Treatment 2, that surface soils
were mixed with rock fragments, forming a mixture layer on slope surface; Transect M was
located in between of the two treatments.
Figure 4.1. The digial elevation model (DEM) of designed waste rock dump and sample points on five
selected transects.
4.2.2 Rock fragments
Sample points were located at 10 m interval on each transect, resulting in a total of 41 locations
50
(Figure 4.1). A 1 m2 metal frame was placed on the surface at each sample point, where two
replicated photographs were taken for further image analysis, using a Canon Digital IXUS 70
(7.1 mega pixels) and an Olympus μ850SW (8.0 mega pixels) in JPEG format. Locations of the
sampling points were recorded by a Magellan ProMark3 RTK Differential Global Positioning
System (GPS).
All the images taken at sample points in field site and in rainfall simulation were rectified to
square and transformed to bilinear format using TurboReg plugin in Image J [Rasband, 2008].
Individual rock fragments in these bilinear images were then traced using a Wacom tablet and
Analyse Particles plug-in in ImageJ. Pixels in images were then converted to physical length
and area of rock fragments, where 1 pixel corresponded to 0.28 mm2pixel-1. Rock fragments
were characterized by size and shape, represented by Feret’s diameter and circularity
respectively, where Feret’s diameter (F) (mm) was defined as the longest distance between any
two points in the rock boundary; circularity (C) (-) was calculated based on area (A) (mm2) and
perimeter (P) (mm) via 2/4 PAC [Rasband, 2008]. Surface coverage (Ac) (%) was
calculated as:
%100T
Rc A
AA (1)
where AR denotes the sum of rock fragment area and AT is the total image size of a sample point,
which is 106 mm2.
Software environment R was used for statistical analysis of rock fragments [Ihaka and
Gentleman, 1996]. In addition to the descriptive statistics of rock fragments fragment such as
mean and median of size and shape, probability distributions were assessed. Four probability
distributions (lognormal, gamma Weibull distributions) were fitted to Feret’s diameter, and two
(beta and logit-normal distributions) to circularity. Null hypotheses of fitted distributions were
accepted at a significance level of 5% through the chi-square goodness of fit test [Ricci, 2005].
Parameters of the fitted lognormal (Eq. 4.2) and beta distribution (Eq. 4.3) were estimated by
maximum likelihood method [Venables and Ripley, 2002]. Spatial changes of descriptive
statistics and probability distributions of rock fragments were assessed, as well as inter-
relationships among surface cover, particle size and shape. These characteristics of rock
fragments were than compared to the ones on the natural analogue of this artificial slope.
0],2
)(lnexp[
2
1),;(
2
2
xx
xxf
(4.2)
where x is the variable; μ and σ denote parameters of lognormal distribution that are the mean
and standard deviation of the variable’s natural logarithm , respectively, also known as location
parameter and the scale parameter.
51
10,)1(),(
1),;( 11
2121
21 xxxB
xf
(4.3)
where x is the variable; β1 and β2 denote two positive shape parameters of beta distribution; B is
a normalization constant to ensure that the total probability integrates to unity.
4.2.3 Field evidence of erosion
In the field, erosion rills were measured in addition to rock fragment sampling. The position of
the rills were mapped using a 100 m measuring tape and a 3 m tape was used to measure rill
geometry [Nearing et al., 1997]. The data were interpolated to estimate the total length of rills,
and the rill volume. Rill shape was highly variable on the slope, however for the purpose of
estimating the rill volume, the shape was assumed to be rectangular.
4.2.4 Rainfall simulation
Rainfall simulations were conducted on the topsoil collected from the field and rock fragments
for the rock dump at Telfer, using an oscillating boom rainfall simulator [Loch et al., 2001].
Rainfall simulator SMI smart motor (Smart Motor SM2315D, Animatics Corporation, USA)
was used in this experiment to increase and decrease rainfall intensity by stepwise changing the
rate of sweeps across the plot.
Rainfall simulation trays were 75 cm x 75 cm, and were 20 cm deep with the bottom 15 cm
filled with coarse yellow sand. Shade cloth was placed on the top of yellow sand before rock
fragment and topsoil mixtures were applied, to allow movement of water while preventing
movement of soil. Rock fragments (> 2 mm) were separated from silt (< 2 mm) using sieves.
Rock fragments larger than 10 cm were not included in the simulations because of their large
size in relation to the total tray area influencing runoff [de Figueiredo and Poesen, 1998]. A
single slope angle at 15° was set up for all rainfall simulation experiments [Evans, 1980; Fox
and Bryan, 1999].
Four different compositions comprised of 50%, 60%, 70% and 80% of rock fragments by
volume with the remaining volume being top soil. Rainfall was applied at an intensity of 100
mm/hr for a short period of time until the soil surface was homogeneously wetted. Simulations
ran with rainfall intensities from lowest to highest (20, 40, 60, 80 and 100 mm/hr) for each tray,
and then from highest to lowest (100, 80, 60, 40 and 20 mm/hr). For every intensity in one run,
simulated rainfall lasted for 5 minutes, resulting in 25 minutes for one run and 50 minutes for
the complete simulation. Three trays with same rock fragment volume were used for each
simulation. Photos were taken before, between and after simulation runs to determine change in
surface rock fragment cover (Ac) using the image analysis method described in Section 4.2.2
52
4.3 Result and discussion
4.3.1Descriptive data of rock fragments
A comparison of rock fragments on artificial and natural mesas by descriptive statistics is
summarized in Table 4.1. Average surface rock cover is slightly higher on natural mesas, but
there is no major difference, consistent with our field observation that appearances of rock cover
are very similar on natural and artificial slopes. Rock fragments on natural mesas have larger
Feret’s diameter and smaller circularity meanwhile, indicating larger and rounder particles.
Coefficient of variation of both particle size and shape tends to be greater on the waste rock
dump, suggesting rock fragments on the artificial slope are more variable. The statistics of rocks
on the waste rock dump is closer to Mesa 3, an approximately rectilinear shaped mesa. As
described in Chapter 2, we did not find the same sorting of rock fragments on Mesa 3 as on
Mesa 1 and 2 when assessing mean particle size and surface cover, indicating rock fragments on
Mesa 3 were more randomly distributed in space. With outcrops consisting of quartzitic
siltstone found along the hillslope, the source for rock fragments on Mesa 3 consists of
materials from the cap as well as weathering resistant rock from within the slope. However,
lognormal distribution of particle size followed the same spatial patterns on Mesa 3 as on the
other two mesas.
Table 4.1. Descriptive statistics of rock fragment characteristics on the waste rock dump and natural mesa
slopes.
Mesa 1 Mesa 2 Mesa 3 WRD
No. of sample points 83 83 56 41
No. of rocks 37632 26693 29089 18606
Surf. Cover (%) 39.83 39.09 34.98 32.02
Feret's Diameter
Mean
(mm) 69 69 57 47
CV (%) 82 78 65 82
Min (mm) 11 14 10 11
Max (mm) 911 1008 710 762
Circularity
Mean 0.68 0.71 0.7 0.66
CV (%) 16.38 14.8 17.22 17.98
Min 0.1 0.11 0.02 0.2
Max 0.91 0.89 0.92 0.93
The differences between artificial and natural mesas are significant at p<0.05 level in the t-test,
Further assessments in probability distributions and spatial patterns of rock characteristics will
provide further insights into the characteristics of both systems.
53
4.3.2 Probability distributions
Among lognormal, gamma, Weibull and power-law probability density functions describing the
Feret’s diameter distribution of rock fragments at each sample point, lower-end truncated
lognormal distribution described the observations best. A total of 92.68% sample points passed
the chi-square test of lognormal distribution at 0.05 significance level, very close to the number
of 92.77% on Mesa 1. Maximum likelihood estimated parameters μ and σ are significantly
correlated with each other on natural mesa slopes, but not on the artificial one (Figure 4.2).
There is a marked difference in the distribution of the parameters for the mesas and the waste
rock dump. A significant positive relationship exists between μ and σ for the mesa sites, in
contrast to the artificial slope. This relationship suggests that a group of particles with smaller
mean size are less variable compared to a group of larger particles on natural mesas. This
relationship does not exist on the artificial slope. With most points having lower μ values than
the natural site, which is indicative of a smaller rock fragments, the variation in particle size on
the artificial slope is relatively greater. The μ-σ relation reflects the spatial pattern of probability
distributions on the natural site as both μ and σ decrease downslope, which does not occur on
the rock dump (see Figure 4.4). According to our fragmentation model described in Chapter 3,
μ will decrease in a weakly nonlinear pattern and σ decrease more linearly once a fragmentation
process occurs. Considering the distribution of the rock dump as the initial condition of the
model of Chapter 3, it is unlikely the probability distribution of particle size on the artificial
slope will evolve in a similar way as the natural site.
Figure 4.2. Relationship between lognormal distribution parameter ì and σ from all sites on (a) natural
mesa slopes and (b) the waste rock dump. The dark grey, grey and light grey solid circles are the
combination of distribution parameters on the top, middle and bottom of a typical transect on each system
– natural mesas and the waste rock dump, respectively as shown in Figure 4.4.
While most of the blasting studies use Weibull or power-law particle size distribution to
quantify rock fragment size distribution [Kuznetsov, 1973; Turcotte, 1986], we found that rock
0.6 0.7 0.8 0.9 1.0 1.1 1.2
2.5
3.0
3.5
4.0
4.5
(a)
0.6 0.7 0.8 0.9 1.0 1.1 1.2
(b)
54
fragments on the artificial slope are lognormally distributed, consistent with its natural
analogues. The lognormal distribution can be the result of further grinding and crushing of mine
rocks after blasting. Selective transport of fines and movement of rock fragments during
placement may have further affected the distribution. Fragmentation and creep mechanisms that
are considered to be important occurring on natural mesas for ages are less likely to dominate on
the young artificial slope.
Beta, logit-normal and Weibull distributions were assessed for the shape measurement of rock
fragments. Circularity from 70.73% locations on the artificial slope are consistent with beta
distribution. Two beta distribution parameters β1 and β2 are positively correlated and the
correlation is 0.71. Beta distribution of circularity was not found on the natural mesa slopes.
Lithology and preferred tectonic directions are likely to be the controlling factors in rock
fracture and lead to different shape distribution.
4.3.3 Spatial patterns of rock fragments
On natural Mesa 1 and 2, distinct spatial patterns of surface rock coverage, particle size and
shape were found along transect distance. The spatial trends on Mesa 3 are less obvious, but still
significant. However, these patterns hardly exist on the artificial slope. For example, although
the surface cover of rock fragments on the two contrasting systems are similar, the significantly
decreasing trends in surface cover with distance downslope are absent on the artificial slope
(Figure 4.3).
Figure 4.3. Spatial changes of rock fragment surface coverage on (a) Mesa 1; (b) Mesa 2; (c) Mesa 3 and
(d) the waste rock dump.
Furthermore, we observed linearly decreasing distribution parameters μ and σ on all three
natural mesas (Figure 4.4a and 4.4b), with distribution shape narrowing down and distribution
mode shifting to the smaller size down the mesa slopes (Figure 4.4c, 4.4d and 4.4e). No
obvious spatial trends in probability distributions are found on the waste rock dump (Figure
4.4f to 4.4j).
0 20 40 60
1030
5070
(a)
Su
rfa
ce C
ove
rag
e (
%)
0 20 40 60
(b)
0 20 40 60
(c)
0 20 40 60 80
(d)
Distance (m)
55
Figure 4.4. Lognormal distribution parameters of particle size change with respect of distance are shown
as changes of (a) μ and (b) σ down a typical mesa transect and (f) μ and (g) σ down a typical waste rock
dump transect; probability density of Feret’s diameter with fitted lognormal distribution line changes on
the mesa transect from (c) the top at 0 m from cap, (d) middle at 30 m, to (e) bottom at 60 m, and on the
waste rock dump from (h) the top at 0m, (i) middle at 40m, to (j) bottom at 80m.
Spatial trends of rock fragment characteristics (e.g., coverage, mean or median size) have been
found widely found on arid and semiarid hillslopes [Simanton et al., 1994; Abrahams et al.,
1985; Poesen et al., 1998]. For example, Poesen et al. [1998] found surface coverage of rock
fragments up to 80% and it increased with slope gradient; with the mean particle size up to 100
mm, Abrahams et al. [1985] found it increased with respect to slope gradient. As the particle
size on the natural mesas is in a similar range with Abrahams’ [1985] study, the surface cover is
much lower than Poesen’s [1998] observations. However, as concave shaped slopes were
2.8
3.4
4.0
Natural Mesa
(a)
0 20 40 60
0.7
0.9
1.1
(b)
Distance (m)
00.
03
(c)
00.
03
Pro
b. D
ensi
ty
(d)
0 100 300
00.
03
Feret's Diameter (mm)
(e)
WRD
(f)
0 20 40 60 80
(g)
Distance (m)
(h)
(i)
0 100 300
(j)
Feret's Diameter (mm)
56
investigated in these two studies, surface rock cover and mean particle size can be expected to
decrease downslope, which is consistent with our observations on natural mesas. Rock
fragments on the waste rock dump are more randomly distributed possibly due to the initial
placement of rock fragments and topsoil using earthmoving equipment (see Appendix D).
However, the unexpected similar surface coverage of rock fragments and the same probability
distribution found on the two contrasting systems indicate some degree of self-organized surface
development on the artificial slope. Based on Cooke’s [1970] surface evolution and the major
cyclone events accompanied with a significant amount of rainfall the engineered site
experienced, it is very likely that wash-out by surface runoff dominates the early stage of rock
armour development on the waste rock dump. The role of sediment transport by water is well
documented by the field observation of rill erosion on the waste rock dump.
4.3.4 Rill erosion in the field
In field observation, rills are affecting a much higher percentage of transect length where
Treatment 1 was applied (topsoil only); the total lengths of rills measured on Transect A and D
are 23.02 m and 60.84 m, much higher than the number on Transect B and C, which intersected
with Treatment 2 mixing topsoil with rock fragments (Table 4.2). Total rill volume is 20.33 m3
on two transects in Treatment 1, but only 3.2 m3 in Treatment 2. The longest rills with the
highest starting point are observed on Transect D with the bare soil treatment, which also has
the lowest mean rock fragment surface coverage (21.74%) (see Appendix E). On all three
natural mesas however, no rills were found.
Table 4.2. Results of rill erosion and the corresponding surface cover of rock fragments on each transect
on the waste rock dump.
Treatment 1
(bare soil)
Treatment 2
(soil and rock
fragment mixture)
Transect A D B C M
Total length of all rills (m) 23.02 60.84 4.07 6.79 -
Rill volume (m3) 7.02 13.31 1.85 1.35 -
Surface rock cover (%) 33.25 21.74 34.32 32.38 39.21
The presence of rills clearly proves the importance of sediment transport by water during the
initial stage of development. After 716.6 mm total rainfall from 69 rain days during the 18
months since the completion of construction, the treatments showed marked differences. Daily
rainfall was less than 1 mm in 33 days out of 69, 1 – 20 mm in 27 days, 20 – 50 mm in 6 days,
and 50 – 100 mm in 5 days. Daily rainfall more than 50 mm were all brought from the tropical
cyclones in one month, with the highest number of 93 mm per day. As the field trial was
designed and executed within an operational context of the mine site, we were able to collect
57
evidence only at one point in time, without a thorough characterisation of the initial conditions.
The increased stability on Transect B and C can be attributed to the presence of rock fragments
in surface soils, forming a layer of rock armour as fine sediments are removed, and protecting
the soils beneath from further erosion [Barrett, 1980; Poesen et al., 1994, Cerdà, 2001; Knapen
et al., 2007]. Accordingly greater volume of soil has been eroded from Treatment 1 without rock
fragments in surface soils on Transect A and D. Although Transect D has a lower surface cover
comparing to the convergent coverage on the other four transects, it is likely the rock cover will
increase as more soil will be eroded with less surface rock protection, and finally evolve to a
similar coverage. This convergence is also indicated by Poesen et al. [1998], that surface
coverage seemed to be in a similar range on different hillslopes when lithology was the same.
To further investigate the accumulation of rock fragment on the soil surface as a self-organizing
process, we will evaluate rainfall simulation experiments.
4.3.5 Rainfall simulation
The changes in surface coverage of rock fragments suggest surface cover appears to converge to
a full coverage through the simulated rainfall events irrespective to the initial rock content
(Table 4.3). Surface cover increases rapidly in the first turn of rainfall, and the increase slows
down in the second turn. For the mixture with 80% of rock fragments initially, surface cover
reaches 100% during the first turn of simulated rainfall, and does not change anymore. The
surface with lowest rock fragment volume changed the most in total; the surface with the
highest rock fragment volume changed the least.
With rock fragments mixed in surface soils, rock armours develop quickly as fine materials are
washed away. Surface rock cover tends to converge to 95% coverage and more independent of
the initial rock volume ≥ 50% in the mixture. Results from rainfall simulation support our wash-
out hypothesis in addition to the field observation, suggesting selective transports of fine
materials by surface runoff will cause self-organization of surface rocks. The rock content in the
mixtures is higher than our observations in the field, but is in a similar range with the
investigations by Simanton et al. [1994] (up to 75%), Poesen et al. [1998] (up to 80%) and
Nyssen et al. [2002] (up to 85%). While the experimental conditions are different from field
situations in which rainfall is highly variable even during a single event and the spatial scale is
much larger, the mixing of topsoil and rock fragments in the trays reflect the disturbance of the
material during placement in the field. The advantage of using rainfall simulation is that the
material variability and the rainfall intensity and duration can be controlled and reproducibly
repeated, and our results should be viewed as a proof of concept rather than an experimental
mimicking of the field situation.
58
Table 4.3. Changes in rock fragment surface coverage for each replicate of each volume ratio during
rainfall simulation.
Rock
Fragment
Volume
(%)
Replicate
Surface cover (%) Difference (%)
Before Between After Before -
Between
Between -
After
Before -
After
1 66.26 90.43 94.14 24.17 3.71 27.88
50 2 66.49 92.78 95.54 26.29 2.76 29.05
3 68.25 94.92 96.89 26.67 1.97 28.63
1 73.43 93.58 94.44 20.15 0.86 21.01
60 2 74.08 92.78 100 18.7 7.22 25.92
3 72.25 92.61 95.54 20.36 2.93 23.29
1 73.63 89.71 93.53 16.08 3.82 19.9
70 2 71.74 90.06 94.23 18.32 4.17 22.49
3 74.22 91.53 96.82 17.31 5.29 22.61
1 97.35 100 100 2.65 0 2.65
80 2 95.33 100 100 4.67 0 4.67
3 94.87 100 100 5.13 0 5.13
In summary, sorting of rock fragments in size and surface cover that are widely observed on arid
and semiarid hillslopes including the natural mesas in our study site are however absent on the
artificial slope. The lack of spatial patterns can be attributed to the engineered construction
instead of natural hillslope evolution processes, as well as a different lithology. However, the
similar surface rock coverage on the two very contrasting systems initialized the hypothesis of
wash-out process leading to a potential self-organized surface development on the 18 months
old waste rock dump. Rills found on the artificial slope as field evidence of erosion suggest
surface soil with rock content are more resistant to erosion. However, all transects seem to
evolve to the same surface coverage irrespective to the initial rock content as a result of
selective transport as fines being washed away. The rainfall simulation in a controlled laboratory
environment has proved the convergence in rock cover with different rock content initially, and
provides evidence that early stage rock fragment accumulation at the soil surface will occur very
quickly [Brakensiek and Rawls, 1994; Poesen et al., 2003; Hancock et al., 2008].
4.4 Conclusion
A wash-out process by water was hypothesized to be responsible for the initial surface
development on the artificial slope. Evidence were observed as: (1) a lognormal distribution in
particle size of rock fragments on the artificial slope was found rather than Weibull or power-
law distribution usually resulting from engineered processes, indicating a likely transport
59
process on slope leading to a self-organization. (2) Rill erosion was heavier on Treatment 1 with
topsoil coverage without rock fragments; in contrast, rock armour developed as fines were
removed, leading to a convergence in surface rock cover. (3) A rainfall simulation with mixes of
top soil and rock fragments clearly showed that irrespective of the initial conditions, the surface
coverage converged generally to a value of ≥95%. The independent evidences support the
formation of self-organizing and self- stabilizing surface rock armour can occur very quickly on
the young engineered slope. It furthermore indicates that placement of rock fragments on
engineered slopes will indeed stabilize the surface and may become a leading practise in
rehabilitation of engineered landforms in particular in semi-arid and arid climates in which
vegetation alone is insufficient to prevent erosion. While wash-out is the most significant
process in the initial phase of rock armour formation, results from the natural site indicates that
other much slower processes such as fragmentation and creep will continue to replenish rock
armour and provide a means of long term stabilisation of arid hillslopes. Further research is
required to determine if there is an optimum of rock fragment content in relation to the type of
fine earth matrix providing the best protection against erosion.
60
References
Abrahams, A. D., A. J. Parsons, and P. J. Hirshi (1985), Hillslope gradient-particle size relations:
evidence for the formation of debris slopes by hydraulic processes in the Mojave Desert, J.
Geol., 93(3), 347-357.
Brakensiek, D. L., and W. J. Rawls (1994), Soil containing rock fragments: effects on
infiltration, Catena, 23(1-2), 99-110.
Barrett, P. J. (1980), The shape of rock particles, a critical review, Sedimentology, 27(3), 291-
303.
Bureau of Meteorology (2011), Climate statistics for Australian locations - TELFER AERO,
Commonwealth of Australia, http://www.bom.gov.au/climate/averages.
Cerdà, A. (2001), Effects of rock fragment cover on soil infiltration, interrill runoff and erosion,
European Journal of Soil Science, 52(1), 59-68.
Cooke, R. U. (1970), Stone pavements in deserts, Annals of the Association of American
Geographers, 60(3), 560-577.
de Figueiredo, T., and J. Poesen (1998), Effects of surface rock fragment characteristics on
interrill runoff and erosion of a silty loam soil, Soil and Tillage Research, 46(1-2), 81-95.
Dunkerley, D. (2008), Rain event properties in nature and in rainfall simulation experiments: a
comparative review with recommendations for increasingly systematic study and reporting,
Hydrol. Process., 22(22), 4415-4435.
Evans, R. (1980), Mechanics of water erosion and their spatial and temporal controls: an
empirical viewpoint, in Soil erosion, edited by M. J. Kirkby and R. P. C. Morgan, pp. 109-
128, Chichester, Wiley.
Fox, D. M., and R. B. Byran (1999), The relationship of soil loss by interill erosion to slope
gradient, Catena, 38, 211-222.
Hancock, G. R., R. J. Loch, and G. R. Willgoose (2003), The design of post-mining landscapes
using geomorphic principles, Earth Surf. Process. Landforms, 28(10), 1097-1110.
Hancock, G. R., D. Crawter, S. G. Fityus, J. Chandler, and T. Wells (2008), The measurement
and modelling of rill erosion at angle of repose slopes in mine spoil, Earth Surf. Process.
Landforms, 33(7), 1006-1020.
Ihaka, R., and R. Gentleman (1996), R: A language for data analysis and graphics, J. Comput.
Graph. Stat., 5(3), 299-314.
Johnson, D. B., and K. B. Hallberg (2005), Acid mine drainage remediation options: a review,
Sci. Total Environ., 338(1-2), 3-14.
Knapen, A., et al. (2007), Resistance of soils to concentrated flow erosion: A review, Earth-Sci.
Rev, 80(1-2), 75-109.
Kuznetsov, V. (1973), The mean diameter of the fragments formed by blasting rock, J. Min. Sci.,
9(2), 144-148.
Lefebvre, R., et al. (2001), Multiphase transfer processes in waste rock piles producing acid
61
mine drainage: 1: Conceptual model and system characterization, J. of Contam. Hydrol.,
52(1-4), 137-164.
Li, X. Y., S. Contreras, and A. Sole-Benet (2007), Spatial distribution of rock fragments in
dolines: A case study in a semiarid Mediterranean mountain-range (Sierra de Gádor, SE
Spain), Catena, 70(3), 366-374.
Lin, Z., and R. B. Herbert Jr (1997), Heavy metal retention in secondary precipitates from a
mine rock dump and underlying soil, Dalarna, Sweden, Environ. Geol., 33(1), 1-12.
Loch, R. J. and T. A. Donnollan (1983), Field stimulator studies on two clay soil of Darling
Downs, Queensland. I. The effect of plot length and tillage orientation on erosion processes
and runoff and erosion rate, Aust. J. Soil Res., 39: 599-610
Nearing, M. A., et al. (1997), Hydraulics and erosion in eroding rills, Water Resources Res., 33,
865-876.
Parsons, A. J. (1988), Hillslope form and climate, in Hillslope Form, pp. 47-67, Routledge, New
York.
Poesen, J., and H. Lavee (1994), Rock Fragments in Top Soils - Significance and Processes,
Catena, 23(1-2), 1-28.
Poesen, J., et al. (2003), Gully erosion and environmental change: importance and research
needs, Catena, 50(2-4), 91-133.
Rasband, W. S. (2008), ImageJ, U.S. National Institutes of Health, Bethesda, Maryland, USA,
http://rsb.info.nih.gov/ij/.
Ricci, V. (2005), Fitting distribution with R, Vienna, Austria.
Riley, S. J. (1995), Issues in assessing the long-term stability of engineered landforms at Ranger
Uranium Mine, Northern Territory, Australia, J. Proc. R. Soc. NSW, 128, 67-78.
Simanton, J. R., E. Rawitz and E. D. Shirley (1984), Effects of rock fragments on erosion of
semiarid rangeland soils, Soil Sci. Soc. Am. Spec. Publ., 13, 65-72.
Turcotte, D. L. (1986), Fractals and Fragmentation, J. Geophys. Res., 91(B2), 1921-1926.
Valentin, C. (1994), Surface Sealing as Affected by Various Rock Fragment Covers in West-
Africa, Catena, 23(1-2), 87-97.
van Wesemael, B., et al. (1996), Suface roughness evolution of soils containing rock fragments,
Earth Surf. Process. Landforms, 21(5), 399 - 411.
Venables, W. N., and B. D. Ripley (2002), Univariate Statistics, in Modern Applied Statistics
with S, pp. 107-138, Springer, New York.
Walker, L. R., and E. A. Powell (2001), Soil Water Retention on Gold Mine Surfaces in the
Mojave Desert, Restor. Ecol., 9(1), 95-103.
62
Chapter 5: Summary
Obvious sorting of rock fragments was found on three selected natural mesas in the Telfer
region. Distance along mesa slopes was found to be a better predictor of rock fragment patterns
than slope. Surface rock coverage and mean particle size were found to decrease linearly with
distance down each transect. Rock fragment shape, measured by circularity, was found to
increase (become rounder) linearly downslope. The spatial trends while significant, were
weaker on Mesa 3. Downslope decreases in cover and size have been widely observed [Cooke et
al., 1993] and have previously been associated with the change in slope (e.g., Abrahams et al.
[1985]; Poesen et al. [1998]). Rock lithology and hillslope morphology were suggested to be
both responsible for and associated with the spatial organisation of rock fragments [Abrahams et
al., 1985; Abrahams et al., 1990; Poesen et al., 1998]. As we observed minor outcropping along
the Mesa 3 slope, and slightly smaller rock fragments, it may be that the weaker spatial trends
on Mesa 3 resulted from a different lithology and perhaps a different source of surface rocks.
For example the rocks on Mesa 1 and 2 were more of a conglomerate of alluvial pebbles in a
siltstone matrix, whereas Mesa 3 was much more a homogeneous siltstone. The probability
distributions of particle size and shape were assessed in addition to descriptive statistics of rock
characteristics. A lognormal distribution of Feret’s diameter was found to fit the data well at
most of the sample points, including those on Mesa 3. The two distribution parameters μ and σ
decreased approximately linearly with distance downslope.
It is widely accepted that particle sorting on hillslopes results from selective transport processes,
as a result of overland flow, mass movement, rockfall and rolling, animal trampling etc.
[Abrahams et al., 1985; Abrahams et al., 1990; Simanton et al., 1994; Nyssen et al., 2002;
Poesen et al., 1998]. At our field sites rock fragments were much larger than could be expected
to be transported by overland flow, and yet downslope fining was observed, even near the top of
the hill [Kirkby and Kirkby, 1974; Abrahams et al., 1990]. Other than selective wash out, a wide
range of processes are suggested by Cooke [1970] that may take place during the evolution of
surface rock armour, including creep, vertical sorting of rock fragments, and surface weathering
that dominates in a later stage of the landforms. Due to the very old age of the natural mesa
slopes, surface weathering is likely to be more important than other processes. Therefore, we
hypothesized an in-situ fragmentation accompanied by hillslope retreat yielded the observed
sorting, while not ruling out the possibility that other non-sorting transport processes may also
be at work. With a fragmentation model based upon the probability distribution of rock size, we
assumed a space for time substitution, such that a 1 m displacement along the surface of the
slope was equivalent to one model time step. The modelled fragmentation was considered as a
diffusion process with a probability of fragmentation decreasing as particle size decreased. The
63
model described changes in the rock size distribution well, particularly the preferential loss of
larger particles from the size distribution. The observed changes in mean particle size and the
preferential loss of large particles have analogies with the phenomena of abrasion in rivers
[Krumbein, 1941]. These similarities indicate that the weathering of rocks, probably by salt or
thermal fracture, on hills and the physical weathering by abrasion in rivers share similar
emergent patterns.
Rock fragments were also studied on an artificial waste rock dump, designed to mimic the
natural mesas. The artificial mesa hill was concave shaped and covered by various mixtures of
topsoils and rock fragments, aiming to prevent erosion and achieve long term stability. With a
similar surface coverage, mean particle size and circularity on the artificial slope were slightly
smaller than natural mesas. However, rock fragments on the waste rock dump failed to show a
distinct spatial pattern. The lognormal distribution of Feret’s diameter was found again to
describe the artificial hills rock fragments, consistent with natural mesas. The Weibull or power-
law distributions predicted by rock blasting models however did not fit well [Kuznetsov, 1973;
Turcotte, 1986]. A wash-out process was hypothesized to dominate in the initial development of
rock armour on this engineered slope. Rills as field evidence of erosion supported out
hypothesis that fines were more easily to be eroded without the presence of rock fragments in
soil surface, and a convergence in surface cover is likely to be achieved eventually irrespective
of the initial rock content in soil. During a controlled rainfall simulation in laboratory conducted
on mixed soil and rocks, rock armour developed quickly as fines were washed away, and the
surface cover converged with different rock content initially. The results indicate that there is
likely a rapid initial phase of self-organizing of rock cover on the artificial hills. As rock armour
developed, they appeared to slow down the erosion and play a positive role in stabilization
[Brakensiek and Rawls, 1994; Poesen and Lavee, 1994; Cerdà, 2001]. From the field
observation and laboratory experiment, wash-out is likely to be the dominating mechanism on
the young artificial slope responsible for surface development rather than in-situ fragmentation
of rock fragments, which possibly occurs on natural mesas.
With a unique dataset including 112,142 rock fragments from 263 locations on natural and
artificial arid mesa slopes, size and shape of individual rocks were measured, and the particle
size and shape distributions were quantified for the first time. This opened a window to
understanding the self-organisation of rock spatial patterns in the context of geomorphic
processes and rock particle weathering. The comparison study of rock fragments on the
engineered waste dump also suggested an initial rapid self-organization of surface rock cover,
but through a wash-out process at the early stage of landform evolution. Therefore, the two
contrasting sites developed rock armoured surfaces contributing to stabilization by self-
organization, but through two very different processes.
64
Overall, the series of studies characterized, quantified and modelled surface rock fragments on
arid hillslopes, and tried to find implications to slope surface evolution of natural mesas and
artificial waste rock dumps. With these interests, the studies hopefully will contribute to a better
understanding in rock fragments and their geomorphologic impacts. Further studies of dating
the age of natural mesas will contribute to confirming rock surface processes leading to the
sorting phenomenon.
65
References
Abrahams, A. D., A. J. Parsons, R. U. Cooke, and R. W. Reeves (1984), Stone movement on
hillslopes in the Mojave Desert, California: A 16 year record, Earth Surf. Process.
Landforms, 9(4), 365-370.
Abrahams, A. D., A. J. Parsons, and P. J. Hirshi (1985), Hillslope gradient-particle size relations:
evidence for the formation of debris slopes by hydraulic processes in the Mojave Desert, J.
Geol., 93(3), 347-357.
Abrahams, A. D., N. Soltyka, and A. J. Parsons (1990), Fabric analysis of a desert debris slope:
Bell Mountain, California, J. Geol., 98(2), 264-272.
Brakensiek, D. L., and W. J. Rawls (1994), Soil containing rock fragments: effects on
infiltration, Catena, 23(1-2), 99-110.
Cerdà, A. (2001), Effects of rock fragment cover on soil infiltration, interrill runoff and erosion,
Eur. J. Soil Sci., 52(1), 59-68.
Cooke, R. U. (1970), Stone pavements in deserts, Ann. Assoc. Am. Geog., 60(3), 560-577.
Cooke, R. U., A. Warren, and A. Goude (1993), Surface particle concentrations: stone
pavements, pp 68-76, in Desert Geomorphology, ULC Press, London, United Kingdom.
Kirkby, A., and M. J. Kirkby (1974), Surface wash at the semi-arid break in slope, Zeitschrift
fur Geomorphologie Supplement Band(21), 151-176.
Krumbein, W. C. (1941), The effects of abrasion on the size, shape and roundness of rock
fragments, J. Geol., 49(5), 482-520.
Kuznetsov, V. (1973), The mean diameter of the fragments formed by blasting rock, J. Min. Sci.,
9(2), 144-148.
Nyssen, J., J. Poesen, J. Moeyersons, E. Lavrysen, M. Haile, and J. Deckers (2002), Spatial
distribution of rock fragments in cultivated soils in northern Ethiopia as affected by lateral
and vertical displacement processes, Geomorph., 43(1-2), 1-16.
Poesen, J., and H. Lavee (1994), Rock fragments in top soils significance and processes, Catena,
23(1-2), 1-28.
Poesen, J. W., B. van Wesemael, K. Bunte and A. S. Benet (1998), Variation of rock fragment
cover and size along semiarid hillslopes: a case-study from southeast Spain, Geomorph.,
23(2-4), 323-335.Turcotte, D. L. (1986), Fractals and Fragmentation, J. Geophys. Res.,
91(B2), 1921-1926.
Turcotte, D. L. (1992), Fragmentation, in Fractals and Chaos in Geology and Geophysics, pp.
20-34, Cambridge University Press, New York.
66
Appendices
Appendix A. The R script: Assessment of gamma, Weibull and lognormal distributions for
particle size
fitting<-function(data){
f.g<-fitdistr(data,"gamma") # fit the gamma distribution
g.shape.est<-f.g[[1]][[1]] # extracting the parameter estimates
g.rate.est<-f.g[[1]][[2]]
f.w<-fitdistr(data,"weibull") # fit the weibull distribution
w.shape.est<-f.w[[1]][[1]]
w.scale.est<-f.w[[1]][[2]]
f.l<-fitdistr(data,"lognormal") # fit the lognormal distribution
meanlog.est<-f.l[[1]][[1]]
sdlog.est<-f.l[[1]][[2]]
return(data.frame(g.shape.est,g.rate.est,w.shape.est,w.scale.est,
meanlog.est,sdlog.est))}
library(MASS) # required for fitdistr () function
chi.fit.shift<-function(data){
data<-data-floor(min(data))
b<-seq(min(data),max(data),length=20)
data.bin<-cut(data,breaks=b)## binning data into 19 bins of equal length with breaks
defined by 'b' (see above line)
binned.data.table<-table(data.bin) ## put the data frequencies for each bin in a table
f<-fitting(data) ## fit the three different distributions to the data and estimate the
parameters
# FREQUENCY = PROBABILITY * TOTAL NUMBER
# GAMMA
f.ex.g<-vector()
for(i in 1:(length(b)-1)) f.ex.g[i]<-(pgamma(b[i+1],shape=f[[1]],rate=f[[2]])-
pgamma(b[i],shape=f[[1]],rate=f[[2]]))*length(data)
f.ex.g<-ceiling(f.ex.g) # expected frequencies vector
# WEIBULL
f.ex.w<-vector()
for(i in 1:(length(b)-1))
f.ex.w[i]<-(pweibull(b[i+1],shape=f[[3]],scale=f[[4]])-
pweibull(b[i],shape=f[[3]],scale=f[[4]]))*length(data)
f.ex.w<-ceiling(f.ex.w) ## expected frequencies vector
# LOGNOMRAL
f.ex.l<-vector()
67
for(i in 1:(length(b)-1))
f.ex.l[i]<-(plnorm(b[i+1],meanlog=f[[5]],sdlog=f[[6]])-
plnorm(b[i],meanlog=f[[5]],sdlog=f[[6]]))*length(data)
f.ex.l<-ceiling(f.ex.l) ## expected frequencies vector
# OBSERVED FREQUENCY
f.os<-vector()
for (i in 1:length(binned.data.table)) f.os[i]<-binned.data.table[[i]]
# Calculating the chi-sq statistics
X2.g<-sum(((f.os-f.ex.g)^2)/f.ex.g); X2.w<-sum(((f.os-f.ex.w)^2)/f.ex.w); X2.l<-
sum(((f.os-f.ex.l)^2)/f.ex.l)
# degrees of freedom is equal to (# bins - # parameters estimated (2 for each
distribution) - 1)
df<-(length(b)-1)-2-1
# Calculating the p-values
p.g<-1-pchisq(X2.g,df)
p.w<-1-pchisq(X2.w,df)
p.l<-1-pchisq(X2.l,df)
best<-vector()
if(p.l>=0.05) {best<-"Lognormal"} else {
if (p.g>=0.05) {best<-“Gamma”}} else{
if(p.w>=0.05) {best<-“Weibull”}}
n<-length(f.os)
##gamma distribution##
g.rgof.ind<-sum(abs(f.os-f.ex.g))/sum(f.os)
g.rgof.ind2<-(sum((f.os-f.ex.g)^2/n))^0.5/sum(f.os/n)
g.rgof.ind22<-(sum((f.os-f.ex.g)^2)/sum((f.ex.g)^2))^0.5
##weibull distribution##
w.rgof.ind<-sum(abs(f.os-f.ex.w))/sum(f.os)
w.rgof.ind2<-(sum((f.os-f.ex.w)^2/n))^0.5/sum(f.os/n)
w.rgof.ind22<-(sum((f.os-f.ex.w)^2)/sum((f.ex.w)^2))^0.5
##lognormal distribution##
l.rgof.ind<-sum(abs(f.os-f.ex.l))/sum(f.os)
l.rgof.ind2<-(sum((f.os-f.ex.l)^2/n))^0.5/sum(f.os/n)
l.rgof.ind22<-(sum((f.os-f.ex.l)^2)/sum((f.ex.l)^2))^0.5
p<-round(c(p.g,p.w,p.l),digit=3)
d22<-round(c(g.rgof.ind22,w.rgof.ind22,l.rgof.ind22),digit=3)
return(c(f,d22,p,best)) ## the output of the function consists of a vector of: the
estimated parameters, the p-values of the tests, the relative % goodness of fit measures,
the degrees of freedom and the best fitting distribution
}
colnames<-
c("GammaShape","GammaRate","WeibShape","WeibScale","LnormMean","LnormSd","d22Gamma","d22
68
Weib","d22Lnorm","pGamma","pWeib","pLnorm","BestFit")
# Apply to M1T1 as an example
DistrFit.F.m1t1<-t(sapply(F.m1t1,chi.fit.shift))
69
Appendix B. Results of the probability distribution assessments
Hartigan's Diptest
When Hartigan’s Diptest [Hartigan, 1985] was carried out on the data, the results were positive
with very few datasets providing evidence against unimodality at the 5% level. These are noted
in Table 6.1 below.
Table 6.1. Hartigan's Diptest Results. Please note that the distance here in the “Transect and Location”
column is the distance from the bottom of the transect.
Transect and Location Data displaying evidence against unimodality
M1T1_54m Circularity
M1T2_39m Circularity
M1T3_60m Circularity
M2T2_54m Perimeter and Feret’s Diameter
M3T3_27m Perimeter, Circularity and Feret’s Diameter
M3T3_36m Perimeter, Circularity and Feret’s Diameter
M3T3_39m Feret’s Diameter
M3T4_6m Perimeter
Fitting distributions
Lognormal, gamma and Weibull probability distributions were assessed for all the sample
locations, and results are tabulated below. In Table 6.2, the “percentage passed chi-square test”
refers to the proportion of locations for which the data can be assumed to come from the
corresponding fitted distribution, i.e. p-value ≥0.05. “Best fit” refers to the best fitting
distribution, i.e. the distribution with the highest proportion of locations passed, the highest
mean p-value or the lowest mean d22 value.
70
Table 6.2. Results of fitted distributions (lognormal, gamma and weibull) to Feret’s diameter of rock
fragments on three mesas.
Mesa 1 - 83 locations Lognormal Gamma Weibull Best Fit
Percentage passed chi-square test (>= 0.05) 92.77% 71.08% 54.22% Lognormal
Mean p-value 0.341 0.280 0.220 Lognormal
Mean d22 value 0.118 0.154 0.185 Lognormal
Mesa 2 - 83 locations
Percentage passed chi-square test (>= 0.05) 83.13% 85.54% 69.88% Gamma
Mean p-value 0.312 0.349 0.260 Gamma
Mean d22 value 0.164 0.175 0.198 Lognormal
Mesa 3 - 56 locations
Percentage passed chi-square test (>= 0.05) 76.79% 58.93% 25.00% Lognormal
Mean p-value 0.330 0.231 0.067 Lognormal
Mean d22 value 0.129 0.149 0.215 Lognormal
Model validation
Further model validation was carried out using graphical residual analysis (Figure 6.1 as an
example). There are no major concerns of the fit of linear models in general.
The plots in Figure 6.1 are as follows:
1. A plot of the data points (distance v. Parameter value at that point) together with the
fitted regression line;
2. A histogram of the residuals εi = yi – ẏi where yi are the observed parameter values and
ẏi are the estimated parameter values from the fitted regression line;
3. A plot of the fitted values against the residuals εi;
4. A QQ-plot of the residuals with 95% confidence envelope for the normal distribution. If
the points lie within the envelope, there is no reason to suppose the residuals are not
normally distributed.
71
Figure 6.1. Graphical residual analysis for the fitted regression lines for the area data from Mesa 1.
72
Appendix C. The R script: Searching for best model parameters (Mesa 1 as an example)
# loading field observation data
load("E:\\Chapter 1 Spatial Distribution of Rock Fragments\\R
outputs\\data_prepare_filtered.RData")
# required library
library(stats); library(MASS)
###########################################################
############ Prepare Mesa 1 #############
###########################################################
# combine all 4 transects on mesa 1 with distance from bottom
# NB: RF samples were taken from slope bottom to top. So 1 is the bottom and 21 is the
top
F.m1<-list();
F.m1[[1]]<-c(F.m1t1[[1]],F.m1t3[[1]],F.m1t4[[1]])
for (i in 2:21) {F.m1[[i]]<-c(F.m1t1[[i]],F.m1t2[[i-1]],F.m1t3[[i]],F.m1t4[[i]])}
##smallest and largest value of Feret's D in Mesa 1 for truncating distribution later
min<-sapply(F.m1,min); max<-sapply(F.m1,max) ##max and min value of each F in each
interval
F.min<-min(min); F.max<-max(max) ##max and min of F in whole trasects
# Truncated lognormal distribution with upper bound
trnclognormal<-function(x,a,b){
pdf<-vector();
pdf<- dlnorm(x ,meanlog=a,sdlog=b)/plnorm(F.max,meanlog=a,sdlog=b,
lower.tail = TRUE)
return(pdf)
}
fit<-function(x){
fit.l<-fitdistr(x,trnclognormal,start = list(a = mean(log(x)), b =
sd(log(x))),method = "SANN")
meanlog.est<-fit.l[[1]][[1]]
sdlog.est<-fit.l[[1]][[2]]
meanlog.sd<-fit.l[[2]][[1]] ##estimated standard error of meanlog
sdlog.sd<-fit.l[[2]][[2]] ##estimated standard error of sdlog
return(data.frame(meanlog.est,sdlog.est,meanlog.sd,sdlog.sd))
}
# fitting lognormal distribution to observed Feret's D
fit.data<-sapply(F.m1,fit)
meanlog.data<-vector(); sdlog.data<-vector(); meanlog.sd<-vector();sdlog.sd<-vector()
for (i in 1:21) {meanlog.data[i]<-fit.data[,i]$meanlog.est; sdlog.data[i]<-
fit.data[,i]$sdlog.est}
meanlog.sd<-vector();sdlog.sd<-vector()
for (i in 1:21) {meanlog.sd[i]<-fit.data[,i]$meanlog.sd; sdlog.sd[i]<-
fit.data[,i]$sdlog.sd}
# linear regression of parameters
meanlm<-lm(meanlog.data~rev(m1t1.d)); am<-meanlm$coef[[2]]; bm<-meanlm$coef[[1]]; rm<-
summary(meanlm)$r.squared
73
meanlog.reg<-am*m1t1.d+bm
sdlm<-lm(sdlog.data~rev(m1t1.d)); as<-sdlm$coef[[2]]; bs<-sdlm$coef[[1]]; rs<-
summary(sdlm)$r.squared
sdlog.reg<-as*m1t1.d+bs
# transfer rock class size into Diameter represented as "D"
n<-50 ##50 size class of rocks
# let D.min be the smallest value of Feret's.
# Because the sampling method is limited to get larger particle size than 1m square
which may be in the real case. So assume a larger D.max is 3000 mm
D.min<-F.min; D.max<-3000
b<-(D.max/D.min)^(1/(n-1)) ##base
D<-vector()
for (i in 1:n) {D[i]<-F.min*b^(n-i)}
# the initial particle size distribution can be generated from the regression data
initial.cdf.freq<-plnorm(D,meanlog=meanlog.reg[1],sdlog=sdlog.reg[1])/
plnorm(D.max,meanlog=meanlog.reg[1],sdlog=sdlog.reg[1],lower.tail = TRUE)
# which means the initial mass distribution is known
meanlog.mass<-vector(); sdlog.mass<-vector()
meanlog.mass[1]<-meanlog.reg[1]+3*sdlog.reg[1]^2; sdlog.mass<-sdlog.reg[1]
initial.cdf.mass<-plnorm(D,meanlog=meanlog.mass[1],sdlog=sdlog.mass[1])/
plnorm(D.max,meanlog=meanlog.mass[1],sdlog=sdlog.mass[1],lower.tail = TRUE)
# cdf.mass: restores cumulative mass in every 3 time step, to correspond cumulative
frequency in dataset
cdf.mass<-list(); cdf.mass[[1]]<-initial.cdf.mass
###########################################################
##### Systematic Search of the Best Fit ALPHA and BETA ####
###########################################################
# the determination of these two parameters is to minimize the sum of RMS (root mean
square) of the modelled results and the observation
# beta and alpha are two model parameters. Beta is in [0,1], alpha is the size sensitive
parameter, when alpha<0 preferential fragmentation of larger particles; when alpha=0
no size dependency; when alpha>0 preferential fragmentation of small particles
opt<-function(m){
beta<-df[m,]$beta; alpha<-df[m,]$alpha
##INITIALIZATION
cdf.mass.old <-initial.cdf.mass; cdf.mass.new <-vector(); cdf.mass.new[1]<-1;
time<-60; ##total time steps for fragmentation process
time.inter<-3; ##time interval
for (t in 1:time){
for (i in 1:(n-1))
{cdf.mass.new[i+1]<-cdf.mass.old[i+1]+beta*exp(alpha/D[i])*(cdf.mass.new[i]-
cdf.mass.old[i+1])}
cdf.mass.old<-cdf.mass.new;
# restore the CDF every 3 time step (= 3 m length on slope)
74
if(t%%time.inter==0) {
j<-t/time.inter+1;
cdf.mass[[j]]<-cdf.mass.old
d<-data.frame(D,cdf.mass.old)
fit.mod<-nls(cdf.mass.old~plnorm(D,meanlog=a,sdlog=b,lower.tail = TRUE, log.p =
FALSE)/
plnorm(D.max,meanlog=a,sdlog=b,lower.tail = TRUE, log.p = FALSE),
data=d,start=list(a=3,b=0.5),
lower=c(0.1,0.01),upper=c(10,1),algorithm = "port",
control = list(maxiter = 50, printEval = FALSE, warnOnly = TRUE))
meanlog.mass[j]<-coef(fit.mod)[1];
sdlog.mass[j]<-coef(fit.mod)[2];
} ##end of if
}##end of for
# convert back from mass distr to particle size distr
meanlog.p.mod<-vector(); sdlog.p.mod<-vector();
sdlog.p.mod<-sdlog.mass;
meanlog.p.mod<-meanlog.mass-3*sdlog.p.mod^2;
# difference between modelled one and observed one
rms.meanlog.data<-(sum((meanlog.p.mod-
rev(meanlog.data))^2)/length(meanlog.data))^0.5
rms.sdlog.data<-(sum((sdlog.p.mod-rev(sdlog.data))^2)/length(sdlog.data))^0.5
rms.meanlog.reg<-(sum((meanlog.p.mod-meanlog.reg)^2)/length(meanlog.data))^0.5
rms.sdlog.reg<-(sum((sdlog.p.mod-sdlog.reg)^2)/length(sdlog.data))^0.5
rms.sum<-rms.meanlog.data+rms.sdlog.data
return(data.frame(beta,alpha,rms.sum))
}
# 10 time 10 combination of beta and alpha. We aimed preferential fragmentation of
larger particles to reproduce field observation, so alpha<0
beta<-rep(seq(0.1,1,0.1),10)
alpha<-c(rep(-80,10),rep-(85,10),rep(-90,10),rep(-95,10),rep(-100,10),
rep(-105,10),rep(-110,10),rep(-115,10),rep(-120,10),rep(-125,10))
df<-data.frame(cbind(p=p,alpha=alpha))
a1<-t(sapply(c(1:10),opt)); a2<-t(sapply(c(11:20),opt)); a3<-t(sapply(c(21:30),opt));
a4<-t(sapply(c(31:40),opt)); a5<-t(sapply(c(41:50),opt)); a6<-t(sapply(c(51:60),opt));
a7<-t(sapply(c(61:70),opt)); a8<-t(sapply(c(71:80),opt)); a9<-t(sapply(c(81:90),opt))
a10<-t(sapply(c(91:100),opt))
par.best.fit<-rbind(a1,a2,a3,a4,a5,a6,a7,a8,a9,a10)
write.table(par.best.fit,file="par.best.fit.txt")
# from the result we narrow down the range beta in 0.25:0.35; alpha in -85:-76
p<-rep(seq(0.26,0.35,0.01),10)
alpha<-c(rep(-85,10),rep(-84,10),rep(-83,10),rep(-82,10),rep(-81,10), rep(-80,10),rep(-
79,10),rep(-78,10),rep(-77,10),rep(-76,10),)
75
df<-data.frame(cbind(p=p,alpha=alpha))
b1<-t(sapply(c(1:10),opt)); b2<-t(sapply(c(11:20),opt)); b3<-t(sapply(c(21:30),opt));
b4<-t(sapply(c(31:40),opt)); b5<-t(sapply(c(41:50),opt)); b6<-t(sapply(c(51:60),opt));
b7<-t(sapply(c(61:70),opt)); b8<-t(sapply(c(71:80),opt)); b9<-t(sapply(c(81:90),opt));
b10<-t(sapply(c(91:100),opt))
par.best.fit<-rbind(b1,b2,b3,b4,b5,b6,b7,b8,b9,b10)
write.table(par.best.fit,file="par.best.fit.txt")
# The best fit result is: beta=0.31 and alpha=-81
76
Appendix D. Further results of rock fragment analysis on the artificial waste dump slope
as a comparison to its natural analogues
Characteristics of rock fragment size and shape
Welch two sample t test performed on two groups of rock size from natural site and waste show
the two means are statistically different from each other. Boxplots in Figure 6.2, drawn with
width proportional to the square-roots of the number of observations in the group, is showing
that waste rock dump and natural mesas have rock fragments sized in the similar range, and
both of them have extreme values. Rock size on waste rock dump is slightly smaller.
Figure 6.2. Boxplots of all Feret’s diameter data on three natural mesa slopes and the waste rock dump.
With the result from natural mesas, smaller rock fragments usually accompany with larger
circularity. However, even the rock size is smaller on the waste rock dump circularity is still
lower, indicating more angular rock fragments on the waste rock dump. The plot whiskers
extended range is wider on the waste rock dump comparing natural mesas, however with fewer
extremes (Figure 6.3).
Figure 6.3. Boxplots of all circularity data on three natural mesa slopes and the waste rock dump.
1020
5010
020
050
0
Fer
et's
Dia
met
er (
mm
)
M1 M2 M3 WRD
0.0
0.2
0.4
0.6
0.8
1.0
Circ
ular
ity
M1 M2 M3 WRD
77
Probability distributions
Beta, logit-normal and Weibull distributions were assessed for the shape measurement of rock
fragments. Circularity from 70.73% locations on the artificial mesa are consistent with beta
distribution. Two beta distribution parameters β1 and β2 are positively correlated (Figure 4a). As
two beta distribution parameters increase, circularity concentrates at a higher peak and the value
goes larger (Figure 6.4b, 6.4c and 6.4d). On natural mesas however, null hypotheses of beta,
logit-normal and weibull distributions of circularity have all been rejected.
Figure 6.4. (a) relationship between beta distribution parameters β1 and β2, with (b), (c) and (d) density
circularity histograms corresponding to solid circles of different combination of β1 and β2, and the fitted
beta distribution line.
Interrelationships among between rock fragment cover, size and shape
Mean particle size of rock fragments is found to be significantly related to particle shape on
Transect A, B and C (Figure 6.5a). Although relationship is missing on the other transects,
particle size is overall negatively correlated to the shape with the correlation of -0.52,
suggesting that larger rock fragments are often more angular. The size and cover relation is even
stronger appearing on four of the transects (Figure 6.5b). However, there are no strong
relationship between circularity and surface cover (Figure 6.5c).
8 10 12 14 16
45
67
89
1
2
(a)
Circ.
Den
sity
0.2 0.4 0.6 0.80
12
34
5 (b)
Circ.
Den
sity
0.2 0.4 0.6 0.8
01
23
45 (c)
Circ.
Den
sity
0.2 0.4 0.6 0.8
01
23
45 (d)
78
Figure 6.5. Inter-relationship between (a) mean circularity and mean Feret’s diameter; (b) rock surface
coverage and mean Feret’s diameter and (c) rock surface coverage and mean circularity on five transects
on the waste rock dump.
The negative relationship between rock fragment size and shape is usually recognized as an
indicator of transport processes, during which particles wear, getting smaller in size and rounder
in shape [Krumbein, 1941; Barrett, 1980]. Surface cover is higher on sample locations where
rock fragments are generally larger. This cover-size relation can be resulted from the limitations
in methodology, as individual large rocks occupy relatively large areas in a limited sample
square (1 m2), resulting in over-estimated rock coverage. However, we observed a more
frequent emergence of “stone pavement” in the field where larger rock fragments were found. It
can also be an indicator of transport process and fine materials were more easily to be washed
away at these locations where high intensity runoff was experienced.
Spatial changes in rock fragment characteristics
On natural Mesa 1 and 2, distinct spatial patterns of surface rock cover, particle size and shape
were found along transect distance, as well as the probability distribution of particle size.
However, spatial patterns are barely found on the artificial slope. No relationship was found
between surface rock coverage and distance, while Feret’s diameter shows a decreasing trend
downslope only on Transect B; circularity was significantly related with distance on Transect B
and C, but randomly distributed on other transects (Figure 6.6).
30 50 70
0.60
0.64
0.68
A(a
)Cir
cula
rity
30 50 70
B
30 50 70
C
Feret's Diameter (mm)30 50 70
D
30 50 70
M
30 50 70
1030
50(b
)Sur
face
Cov
er(%
)
30 50 70 30 50 70Feret's Diameter (mm)
30 50 70 30 50 70
0.60 0.64 0.68
1030
50(c
)Sur
face
Cov
er(%
)
0.60 0.64 0.68 0.60 0.64 0.68Circularity
0.60 0.64 0.68 0.60 0.64 0.68
79
Figure 6.6. Spatial changes of (a) surface rock cover; (b) mean Feret’s diameter and (c) mean circularity
along distance on five transects on the waste rock dump.
Size characteristics on top, middle and bottom slopes tell the spatial patterns in a clear way
(Figure 6.7). Larger rock fragments locate on the top of mesa slopes, resulting less samples in
the same sample area. Feret’s diameter decreases downslope, with smaller ranges in size.
Similar trends are not shown on waste rock dump transect.
Figure 6.7. Changes of Feret’s diameter orienting downslope on top, middle and bottom of a typical mesa
transect and a typical waste rock dump transect.
1030
50
A
(a)S
urfa
ce C
over
(%) B C D M
3050
70(b
)Fer
et's
Dia
met
er(m
m)
0 20 60
0.60
0.64
0.68
(c)C
ircul
arity
0 20 60 0 20 60Distance (m)
0 20 60 0 20 60
1020
5020
050
0
Fer
et's
Dia
met
er (
mm
)
Top Middle Bottom
(a)
1020
5020
050
0
Top Middle Bottom
(b)
80
Appendix E. Field evidence of erosion
Rill erosions on the waste rock dump are visualized in the photographs taken from the top of the
slope looking down (Figure 6.8). No distinct rill erosion can be seen on Transect B (Figure
6.8a), while rill erosion was clearly evident on Transect D (Figure 6.8b).
Figure 6.8. Photographs looking downslope from the top of (a) Transect B (in Treatment 2) and (b)
Transect D (in Treatment 1).
81
References
Barrett, P. J. (1980), The shape of rock particles, a critical review, Sedim., 27(3), 291-303.
Krumbein, W. C. (1941), The effects of abrasion on the size, shape and roundness of rock
fragments, J. Geol., 49(5), 482-520.
Hartigan, P. M. (1985), Algorithm AS 217: computation of the dip statistic to test for
unimodality, J. Roy. Stat. Soc. C-App, 34(3), 320-325.