Applied Geostatistics Miles Logsdon mlog@u.washington.edu Mimi D’Iorio mimid@u.washington.edu

1. Applied GeostatisticsMiles Logsdonmlog@u.washington.eduMimi D’Ioriomimid@u.washington.edu

2. "An Introduction to Applied Geostatistics" by Edward H. Isaaks and R. Mohan Srivastava, Oxford University Press, 1989. • "Spatial Data Analysis: Theroy and Practice" by Robert Haining, Cambridge University Press, 1993. • "Statistics for Spatial data" by Noel a. c. Cressie, Wiley & Sons, Inc. 1991. 

3. D Z(s) Introduction to Geostatistics • D is the spatial domain or area of interest • s contains the spatial coordinates • Z is a value located at the spatial coordinates {Z(s): s D} • Geostatistics: Z random; D fixed, infinite, continuous • Lattice Models: Z random; D fixed, finite, (ir)regular grid • Point Patterns: Z  1; D random, finite

4. GeoStatistics • A way of describing the spatial continuity as an essential feature of natural phenomena. • The science of uncertainty which attempts to model order in disorder. • Recognized to have emerged in the early 1980’s as a hybrid of mathematics, statistics, and mining engineering. • - Now extended to spatial pattern description • Univariate • Bivariate • Spatial Description

5. Univariate • One Variable • Frequency (table) • Histogram (graph) • Do the same thing (i.e count of observations in intervals or classes • Cumulative Frequency (total “below” cutoffs)

6. Summary of a histogram • Measurements of location (center of distribution • mean (m µ x ) • median • mode • Measurements of spread (variability) • variance • standard deviation • interquartile range • Measurements of shape (symmetry & length • coefficient of skewness • coefficient of variation

7. Bivariate Scatterplots Correlation Linear Regression slope constant

8. Autocorrelation • Values at locations that are near to each other are more similar than values at locations that are farther apart.

9. * Xj,Yj tj hij=tj-ti * Xi,Yi * ti (0,0) Spatial Description - Data Postings = symbol maps (if only 2 classes = indicator map - Contour Maps - Moving Windows => “heteroscedasticity” (values in some region are more variable than in others) - Spatial Continuity (h-scatterplots Spatial lag = h = (0,1) = same x, y+1 h=(0,0) h=(0,3) h=(0,5) correlation coefficient (i.e the correlogram, relationship of p with h

10. Lags • Variograms: How do we estimate them?

11. Binning Lags • Variograms: How do we estimate them? 1 1 2 2 3 3 4 4

12. 1 1 3 2 2 2 2 3 15 3 10 4 1 4 12 5 11 Geostatistics Let’s review: Univariate - Bivariate - Spatial Description - VECTOR OR RASTER • Data Postings => symbol maps • Contour Maps • Moving Windows => “heteroscedasticity” • Spatial Continuity h-scatterplots Lag bins Spatial Lag = h = distance Values at locations that are near to each other are more similar than values at locations that are farther apart. = Autocorrelation

13. Definitions • Variograms: What are they?

14. moment of inertia = • Correlogram = p(h) = the relationship of the correlation coefficient of an h-scatterplot and h (the spatial lag) • Covariance = C(h) = the relationship of the coefficient of variation of an h-scatterplot and h • Semivariogram = variogram = = moment of inertia OR: half the average sum difference between the x and y pair of the h-scatterplot OR: for a h(0,0) all points fall on a line x=y OR: as |h| points drift away from x=y

15. Isotropy • Variograms: What are their features?

16. Anisotropy • Variograms: What are their features?

17. Anisotropy • Variograms: What are their features?

18. Anisotropy • Variograms: What are their features?

19. Represent the Data Represent the Data Explore the Data Explore the Data Fit a Model Fit a Model Perform Diagnostics Perform Diagnostics Compare the Models Compare the Models Structured Process in Geostatistics

22. Physiognomy / Pattern / structure • Composition = The presence and amount of each element type without spatially explicit measures. • Proportion, richness, evenness, diversity • Configuration = The physical distribution in space and spatial character of elements. • Isolation, placement, adjacency • ** some metrics do both **

23. Types of Metics • Area Metrics • Patch Density, Size and Variability • Edge Metrics • Shape Metrics • Core Area Metrics • Nearest-Neighbor Metrics • Diversity Metrics • Contagion and Interspersion Metrics

24. Shape Metricsperimeter-area relationships • Shape Index (SHAPE) -- complexity of patch compared to standard shape • vector uses circular; raster uses square • Mean Shape Index (MSI) = perimeter-to-area ratio • Area-Weighted Mean Shape Index (AWMSI) • Landscape Shape Index (LSI) • Fractal Dimension (D), or (FRACT) • log P = 1/2D*log A; P = perimeter, A = area • P = sq.rt. A raised to D, and D = 1 (a line) • as polygons move to complexity P = A, and D -> 2 • A few fractal metrics • Double log fractal dimension (DLFD) • Mean patch fractal (MPFD) • Area-weighted mean patch fractal dimension (AWMPFD)

25. Contagion, Interspersion and Juxtaposition • When first proposed (O’Neill 1988) proved incorrect, Li & Reynolds (1993) alternative • Based upon the product of two (2) probabilities • Randomly chosen cell belongs to patch “i” • Conditional probability of given type “i” neighboring cells belongs to “j” • Interspersion (the intermixing of units of different patch types) and Juxtaposition (the mix of different types being adjacent) index (IJI)

26. Changing patterns

27. Flying