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An Introduction to Geostatistics

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  1. 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

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

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

  4. 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

  5. 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

  6. I found this article to be relevant to what we are studying in this class. • Strongly Agree • Agree • Disagree • Strongly Disagree

  7. Based on the reading, I can see using geostatistcal tools in the future • Strongly Agree • Agree • Disagree • Strongly Disagree

  8. 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?”

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

  10. 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

  11. 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

  12. 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

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

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

  15. 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 “

  16. 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

  17. 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)

  18. 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

  19. Study site Moree, NSW

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

  21. 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?

  22. 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

  23. 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

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

  25. 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)

  26. 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)

  27. 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

  28. Direct characterization • Identify important units • Measure hydrologic properties • Assign reference Ksat values for geologic facies discontinuous predictions, assumed homogeneity within structure

  29. 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

  30. 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)

  31. 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

  32. 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?

  33. 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