Applied Geostatistics Miles Logsdon Mimi D’Iorio

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

Introduction to Geostatistics Z(s)Z(s) D 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

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

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 Summary of a histogram

Bivariate Scatterplots Correlation Linear Regression slope constant

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

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 * X j,Y j t j h ij =t j -t i * X i,Y i * t i (0,0) 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

Lags Variograms: How do we estimate them?

Binning Lags Variograms: How do we estimate them?

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

Definitions Variograms: What are they?

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 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

Isotropy Variograms: What are their features?

Anisotropy Variograms: What are their features?

Anisotropy Variograms: What are their features?

Anisotropy Variograms: What are their features?

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

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 **

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

Shape Metrics perimeter-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)

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)

Changing patterns MonthNPLPILSIMPFDIJI January February March April May June July August September October November December

Flying