Post on 25-Feb-2016
description
transcript
An Introduction to Geostatistics
Presented to Math 216, Spring, 2012
Chris Vanags, Ph.D.Associate Director, Vanderbilt Center for Science Outreach
Instructor, School for Science and Math at Vanderbilt
A brief experiment . . .
Is it hot it here?
Follow up from assigned reading
“Analyzing the Consequences of Chernobyl Using GIS and Spatial Statistics”
Relative to your other course readings was this article. . .
1. Inappropriately simple2. Easier to understand3. On par with the level of difficulty4. More difficult to understand5. Inappropriately complex
●Did you feel that this article was appropriately informative?
1. The article did not contain enough detail to be interesting
2. The article captured my attention, but was not sufficiently detailed for my level of understanding
3. The article was well matched to the course requirements and my level of understanding
4. The article captured my attention, but was overly detailed for my level of understanding
I found this article to be relevant to what we are studying in this class.
1. Strongly Agree2. Agree3. Disagree4. Strongly Disagree
Based on the reading, I can see using geostatistcal tools in the future
Strongly
Agree Agree
Disagre
e
Strongly
Disagre
e
25% 25%25%25%1. Strongly Agree2. Agree3. Disagree4. Strongly Disagree
Why am I here?
“How do geostatistics differ from "normal" statistics in terms of determining the probability of given events assuming they have these large, vaguely defined sample sizes?”
A brief history of geostatistics
● $962Billion Global mining industry
Georges Matheron (1930 – 2000)Gold deposits in Witwaterstand, SA
Fundamental concepts: interpolation models
● Nearest neighbor (right)– Exact values
● Inverse-distance weighting– Interpolation based on
distance from known values
● Trend analysis– Interpolation based on
distance and variation
Nearest neighbor approximationFrom: Wikipedia Commons
Fundamental concepts: the (semi)variogram
● Change in distance vs. change in property
● Used to weight estimates of variation between known points
● Key terms:– Nugget– Range– Sill
Semivariogram of topsoil clay content vs. lag distanceFrom: USGS
Fundamental concepts: Kriging
● Interpolation based on the modeled semivariogram
● Provides estimates of properties AND estimates of uncertainty of the prediction (right)
● Multi-dimensional● Computationally
expensive
Fundamental concepts: covariability● “Using information that
is easy to obtain to predict information that is difficult to obtain”
● Trend Kriging● Regression Kriging● Co-Kriging
Where to go from here. . . ?
● Indicator kriging (right)● Stochastic modeling
(below)
Geostatistics in practice:“Predicting the field-scale hydrological impacts of shallow palæochannels in the semi-arid landscape of Northern New South Wales, Australia “
Chris Vanags
Background“Expansion of flood irrigation in the Lower Macquarie Valley of New South Wales, Australia has been suggested as a major cause of increased groundwater recharge”
- Willis et al, 1997
www.boreline.co.uk
“The areas exhibiting the largest probability of excessive DD correspond to permeable soil types associated with a prior stream channel.”
Background
From: Stannard and Kelly (1977)
- Triantafilis et al, 2003
Water BudgetFlow (m3/day) from:
palaeochannel (2) to water table (4)
Background
Layers 1-5 Layers 6-1011
1 12
2
3 4
y = 8.9302x + 1.1831R2 = 0.9896
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5
Channel Ksat multiplier
“a two fold increase in the contrast in saturated conductivity between channel sediments and those surrounding the channel increases the predicted deep drainage by 64% in our Modflow simulation”-Vanags and Vervoort, 2004
Study siteMoree, NSW
MethodsDirect
Observation
HydrologicalProperties
ConceptualModel
AncillaryData
GroundwaterFlow
Prediction
Groundwater response to irrigation Perched watertable
during irrigation events
Immediate response to irrigation events
Source of perched water?
769300 769400 769500 769600 769700 7698006751700
6751800
6751900
6752000
6752100
6752200
1 2,38
5,6,749
11,1221,22
Irrigation canalCarroll Creek
-9
-8
-7
-6
-5
-4
-3
25/10/2005 19/11/2005 14/12/2005 8/01/2006 2/02/2006 27/02/2006Date
wat
erle
vel i
n m
bel
ow th
e su
rfac
e
A: well 2 (9m) below PC B: well 3 (5m) inside PC C: well 4 (9m) outside PC
A
B
C
Ancillary information“… the high costs and intrinsic features of invasive sampling techniques such as drilling and cone penetrometer technologies limit their use to a finite number of sampling locations and do not allow complete coverage of the area under consideration”
-Borchers et al 1997
EM 31
DGPS height-adjustable stabiliser
Easting
Sou
thin
g
769400 769600 769800
6751500
6751700
6751900
6752100
6752300
Quad-bike EM survey
Clear delineation of channel inside paddock
No delineation outside paddock Strongly related
to soil wetness
Distance (m)
Varia
nce
inside paddock
outside paddock
combined
outside paddock
inside paddock
combined
3 people hours = 2,700 data points
EM Survey: Depth sounding
Tx Rxseparation
½ separation
TxTx RxRxseparation
½ separation
Bi var i at e Fi t of 2 By ydi st Xval ue=2
Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_0_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_0_V"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_v "Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1_V"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 34_10"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 34_20"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 38_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 38_V"
405060708090
100110120130140
2
0 10 20 30 40 50 60 70 80 90 100
y dis t
Bi var i at e Fi t of 1 By ydi st Xval ue=2
Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_0_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_0_V"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_v "Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1_V"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 34_10"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 34_20"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 38_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 38_V"
405060708090
100110120130140
1
0 10 20 30 40 50 60 70 80 90 100
ydis t
1m Rx Tx
Bi var i at e Fi t of 2 By ydi st Xval ue=2
Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 31_0_H"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 31_0_V"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_H"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_v "Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 31_1_H"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 31_1_V"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 34_10"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 34_20"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 38_H"Sm oot hin g Spline Fit , lam bda=0. 1 I D==" EM 38_V"
405060708090
100110120130140
2
0 10 20 30 40 50 60 70 80 90 10 0
ydis t
Bi var i at e Fi t of 1 By ydi st Xval ue=2
Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_0_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_0_V"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_v "Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 31_1_V"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 34_10"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 34_20"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 38_H"Sm oot hing Spline Fit , lam bda=0. 1 I D==" EM 38_V"
405060708090
100110120130140
1
0 10 20 30 40 50 60 70 80 90 100
ydist
Bi va r i ate F i t of 2 By ydi st Xval ue=2
Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 31_0_H"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 31_0_V"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 31_1. 5_H"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 31_1. 5_v "Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 31_1_H"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 31_1_V"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 34_10"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 34_20"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 38_H"Sm oot hing Spline Fit , lam bd a= 0. 1 I D==" EM 38_V"
405060708090
100110120130140
2
0 10 20 30 40 50 60 70 80 90 100
y dis t
Bi va r ia te Fi t of 1 By ydi st Xval ue=2
Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 31_0_H"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 31_0_V"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_H"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 31_1. 5_v "Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 31_1_H"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 31_1_V"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 34_10"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 34_20"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 38_H"Sm oo t h ing Spline Fit , lam bda=0. 1 I D==" EM 38_V"
405060708090
100110120130140
1
0 10 20 30 40 50 60 70 80 90 100
y dis t
480 people hours = 1000 data points
1
8
Inverted conductivity profiles
McNeill Discontinuous profiles Channel not delineated
Tikhonov 0th order Laterally smooth profiles Large range in predictions
Tikhonov 1st order Smooth profiles Channel delineated
Tikhonov 2nd order Smooth profiles Channel delineated
McNeill Tikh 0
Tikh 1 Tikh 2
Distance along transect (m)
Dep
th (m
bel
ow s
urfa
ce)
EM results: significant relation with clay content
McNeill poor correlation with clay
content
Tikhonov 0th order best correlation, high
RMSE
Tikhonov 1st order
significant correlation
Tikhonov 2nd order
significant correlation, lowest RMSE
Tikh 2r2=0.19RMSE = 19
Tikh 0r2=0.40RMSE = 42
Clay (g/g)
McNeillr2=0.06RMSE = 77
EC
(mS
/m)
Tikh 1r2=0.20RMSE = 23
Groundwater flow through palæochannel
heavyclays
coarsesand andgravel
finesand
partic le size c lusters a long channel
S outhing
dept
h be
low
sur
face
(m)
-15
-10
-5
6751800 6751900 6752000 6752100
1
2
3
coarse gravel (Narrabri Formation)
palæochannel
reduced clays
permanent water table (1-2 m annual variation)
deep drainage
lateralflow
Dep
th (m
bel
ow s
urfa
ce)
Direct characterization
Identify important units
Measure hydrologic properties
Assign reference Ksat values for geologic facies
discontinuous predictions, assumed homogeneity within structure
Testing the scaling methods with 3D regression kriging
Use trend from ancillary data to weight direct observations
Assumptions: Direct observations
are related to ancillary data
Weighting is based on regression analysis
2.5m 1.5m 0.5m
6.5m 5.5m 4.5m 3.5m
10.5m 9.5m 8.5m 7.5m
Improved groundwater model
2D laterally-continuous Ksat from EM data X 20 layers
Simulated input from irrigation channel and deep drainage
Limited temporal prediction (single event)
MODFLOW Simulations
Ksat predictions from 2D EM surveys Continuous Ksat in slice
(projected in 3 dimensions)
Ksat = Kref x λ Top boundary input
Inversion method accounted for 33% of predicted deep drainage
Tikhonov2m depth
McNeill2m depth
Conclusions
High variability of Ksat within palæochannel 2 orders of magnitude within channel
3 orders between palæochannel and surrounding sediments
Is direct characterization possible for a landscape scale effort?
Palæochannel associated with deep drainage AND lateral flow In presence of irrigation channel: lateral flow >> deep drainage
31 Ml water lost during a single year of irrigation on one site. where is this water headed? what is the water quality?
Future research
Continue groundwater monitoring Calibrate groundwater model
Improve Ksat predictions Incorporate measured data i.e. Kriging with trend
Incorporate uncertainty from ptf and scaling Generate stochastic groundwater model
Translate to landscape scale Use prediction method for larger data set