Geostatistics

1. Geostatistics Mike Goodchild

2. Spatial interpolation • A field • variable is interval/ratio • z = f(x,y) • sampled at a set of points • How to estimate/guess the value of the field at other points?

3. Characteristics of interpolated surfaces • Representation • raster, isolines, TIN • Form • rugged or smooth • exact or approximate • continuity • 0-order • 1-order • 2-order • Uncertainty • variance estimators?

4. Linear interpolation • Along a line • geocoding with address ranges x2,y2 address2 x,y address x1,y1 address1

5. In a triangle 30 40 20

6. (24) 30 20 40 30 (34) In a rectangle • Bilinear interpolation (29)

7. Characteristics of linear interpolation • Exact • 0-order continuity • Contours are straight • but not parallel in bilinear case

8. IDW • Advantages • quick, universal, theory-free • Disadvantages • theory-free • directional effects • non-spatial • characteristics of a weighted average • when all weights are non-negative

9. 4.5 4 3.5 3 2.5 2 1.5 1 0.5 1 4 7 0 10 13 16 19 22 25 28 37 40 43 46 49 52 55 58 61 64 76 79 82 85 88 91 94 97 31 34 67 70 73 100

11. Characteristics of IDW surfaces • Pass through each data point (exact) • if negative power distance function • 1/0b =  • 0-, 1-, 2-order continuous • except at data points • Underestimate peaks • volcanoes • unless peak is observation point • Extrapolate to the global mean • Noisy extrapolations

12. Kriging • Geostatistics as theoretical framework • Estimation of parameters from data • Use of estimated model to control interpolation • Many versions • not a simple black box • highlights • demonstration

13. The variogram • Relationship between variance and distance • Formalization of Tobler's First Law • Estimated from data • how well can a given data set estimate variogram? • distribution of sample points is critical • at peaks and pits • samples the range of possible distances • uniform spacing not desirable • but often out of the user's control

15. Estimation • Data points zi(xi) • Interpolate at x • stochastic process • multiple realizations • variance obtained from variogram • A set of weights i unique to x • chosen such that the estimate is • unbiased • minimum variance

16. Kriging prediction

17. Results of Kriging • A mean surface • A variance surface • minimum at observation points • Mean surface is smoother than any realization • is not a possible realization • a mean map is not a possible map • compare a univariate process • average rainfall versus rainfall from a single storm • conditional simulation

18. Kriging standard error

21. Kriging variants • Co-Kriging • interpolation process guided by another variable (field) • hard and soft data • observations of interpolated data are hard • guiding variable is soft

22. 70 55 83 68 z = f (elevation)

23. Co-Kriging • Linear relationship f • Point observations are hard • accurate, sparse • Elevation observations are soft • inaccurate (errors in measurement or prediction) • dense

24. Co-Kriging prediction

25. Co-Kriging standard error

26. Indicator Kriging • Binary field • c {0,1} • Obtained by thresholding an interval/ratio field • c=1 if z>t else c=0 • estimate variogram from observations of c • z is hidden • The multivariate case • sequential assignment

28. Indicator Kriging • Assign Class 1, notClass 1 • Among notClass 1, assign Class 2, notClass 2 • Continue to Class n-1 • notClass n-1 = Class n

30. Universal Kriging • Simple Kriging is all second order • trend results from random walk • Stochastic process plus trend • trend is first order • remove trend before analysis • restore trend after analysis

31. Advantages and disadvantages • Theoretically based • Not a black box • Statistical • variance estimates • Sensitivity to sample design