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## An Introduction to Geostatistics

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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**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. . .**• Inappropriately simple • Easier to understand • On par with the level of difficulty • More difficult to understand • Inappropriately complex**Did you feel that this article was appropriately**informative? • The article did not contain enough detail to be interesting • The article captured my attention, but was not sufficiently detailed for my level of understanding • The article was well matched to the course requirements and my level of understanding • 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. • Strongly Agree • Agree • Disagree • Strongly Disagree**Based on the reading, I can see using geostatistcal tools in**the future • Strongly Agree • Agree • Disagree • 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 approximation From: 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 distance From: 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)**Chris Vanags**Geostatistics in practice:“Predicting the field-scale hydrological impacts of shallow palæochannels in the semi-arid landscape of Northern New South Wales, Australia “**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**Background**“The areas exhibiting the largest probability of excessive DD correspond to permeable soil types associated with a prior stream channel.” • Triantafilis et al, 2003 From: Stannard and Kelly (1977)**Layers 1-5**Layers 6-10 2 1 1 4 3 1 1 2 Background “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” Water Budget Flow (m3/day) from: palaeochannel (2) to water table (4) Channel Ksat multiplier -Vanags and Vervoort, 2004**Study site**Moree, NSW**Methods**Direct Observation Ancillary Data Conceptual Model Hydrological Properties Groundwater Flow Prediction**21,22**11,12 4 5,6,7 9 Carroll Creek 8 1 2,3 Irrigation canal Groundwater response to irrigation • Perched watertable during irrigation events • Immediate response to irrigation events Source of perched water?**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**inside paddock**inside paddock Variance Southing outside paddock outside paddock combined combined Distance (m) Easting Quad-bike EM survey • Clear delineation of channel inside paddock • No delineation outside paddock • Strongly related to soil wetness 3 people hours = 2,700 data points**EM Survey: Depth sounding**480 people hours = 1000 data points**Inverted conductivity profiles**8 1 Tikh 0 McNeill • 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 Tikh 2 Tikh 1 Depth (m below surface) Distance along transect (m)**EM results: significant relation with clay content**• McNeill • poor correlation with clay content • Tikhonov 0th order • bestcorrelation, high RMSE • Tikhonov 1st order • significant correlation • Tikhonov 2nd order • significant correlation, lowest RMSE McNeill r2=0.06 RMSE = 77 Tikh 0 r2=0.40 RMSE = 42 EC (mS/m) Tikh 2 r2=0.19 RMSE = 19 Tikh 1 r2=0.20 RMSE = 23 Clay (g/g)**Groundwater flow through palæochannel**deep drainage palæochannel lateral flow heavy clays reduced clays coarse sand and gravel Depth (m below surface) coarse gravel (Narrabri Formation) permanent water table (1-2 m annual variation) fine sand**Direct characterization**• Identify important units • Measure hydrologic properties • Assign reference Ksat values for geologic facies discontinuous predictions, assumed homogeneity within structure**2.5m 1.5m 0.5m**6.5m 5.5m 4.5m 3.5m 10.5m 9.5m 8.5m 7.5m 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**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 McNeill 2m depth Tikhonov 2m depth**Conclusions**• High variability of Ksatwithin 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