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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

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A brief experiment... Is it hot it here?

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Follow up from assigned reading “Analyzing the Consequences of Chernobyl Using GIS and Spatial Statistics”

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Relative to your other course readings was this article Inappropriately simple 2. Easier to understand 3. On par with the level of difficulty 4. More difficult to understand 5. Inappropriately complex

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● 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

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I found this article to be relevant to what we are studying in this class. 1. Strongly Agree 2. Agree 3. Disagree 4. Strongly Disagree

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Based on the reading, I can see using geostatistcal tools in the future 1. Strongly Agree 2. Agree 3. Disagree 4. Strongly Disagree

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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?”

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A brief history of geostatistics ● $962Billion Global mining industry Georges Matheron (1930 – 2000) Gold deposits in Witwaterstand, SA

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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 approximation From: Wikipedia Commons

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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 distance From: USGS

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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

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Fundamental concepts: covariability ● “Using information that is easy to obtain to predict information that is difficult to obtain” ● Trend Kriging ● Regression Kriging ● Co-Kriging

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Where to go from here... ? ● Indicator kriging (right) ● Stochastic modeling (below)

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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

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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

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“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

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Water Budget Flow (m 3 /day) from: palaeochannel (2) to water table (4) Background Layers 1-5Layers Channel K sat 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

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Study site Moree, NSW

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Methods Direct Observation Hydrological Properties Conceptual Model Ancillary Data Groundwater Flow Prediction

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Groundwater response to irrigation Perched watertable during irrigation events Immediate response to irrigation events Source of perched water? 1 2,3 8 5,6, ,12 21,22 Irrigation canal Carroll Creek

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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

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Easting Southing Quad-bike EM survey Clear delineation of channel inside paddock No delineation outside paddock Strongly related to soil wetness Distance (m) Variance inside paddock outside paddock combined outside paddock inside paddock combined 3 people hours = 2,700 data points

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EM Survey: Depth sounding 480 people hours = 1000 data points

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1 8 Inverted conductivity profiles McNeill McNeill Discontinuous profiles Discontinuous profiles Channel not delineated Channel not delineated Tikhonov 0 th order Tikhonov 0 th order Laterally smooth profiles Laterally smooth profiles Large range in predictions Large range in predictions Tikhonov 1 st order Tikhonov 1 st order Smooth profiles Smooth profiles Channel delineated Channel delineated Tikhonov 2nd order Tikhonov 2nd order Smooth profiles Smooth profiles Channel delineated Channel delineated McNeill Tikh 0 Tikh 1 Tikh 2 Distance along transect (m) Depth (m below surface)

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EM results: significant relation with clay content McNeill poor correlation with clay content Tikhonov 0 th order best correlation, high RMSE Tikhonov 1 st order significant correlation Tikhonov 2 nd order significant correlation, lowest RMSE Tikh 2 r 2 =0.19 RMSE = 19 Tikh 0 r 2 =0.40 RMSE = 42 Clay (g/g) McNeill r 2 =0.06 RMSE = 77 EC (mS/m) Tikh 1 r 2 =0.20 RMSE = 23

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Groundwater flow through palæochannel heavyclays coarse sand and gravel finesand coarse gravel (Narrabri Formation) palæochannel reduced clays permanent water table (1-2 m annual variation) deep drainage lateral flow Depth (m below surface)

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Direct characterization Identify important units Measure hydrologic properties Assign reference K sat values for geologic facies discontinuous predictions, assumed homogeneity within structure

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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

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Improved groundwater model 2D laterally-continuous K sat from EM data X 20 layers Simulated input from irrigation channel and deep drainage Limited temporal prediction (single event)

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MODFLOW Simulations K sat predictions from 2D EM surveys Continuous K sat in slice (projected in 3 dimensions) K sat = Kref x λ Top boundary input Inversion method accounted for 33% of predicted deep drainage Tikhonov 2m depth McNeill 2m depth

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Conclusions High variability of K sat 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?

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Future research Continue groundwater monitoring Calibrate groundwater model Improve K sat 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

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