Geostatistics Mike Goodchild

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

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

Linear interpolation n Along a line –geocoding with address ranges x 2,y 2 address 2 x 1,y 1 address 1 x,y address

In a triangle

In a rectangle n Bilinear interpolation (24) (34) (29)

Characteristics of linear interpolation n Exact n 0-order continuity n Contours are straight –but not parallel in bilinear case

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

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

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

The variogram n Relationship between variance and distance n Formalization of Tobler's First Law n 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

Estimation n Data points z i (x i ) n Interpolate at x –stochastic process –multiple realizations variance obtained from variogram n A set of weights i unique to x –chosen such that the estimate is unbiased minimum variance

Kriging prediction

Results of Kriging n A mean surface n A variance surface –minimum at observation points n 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

Kriging standard error

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

z = f (elevation)

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

Co-Kriging prediction

Co-Kriging standard error

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

Indicator Kriging n Assign Class 1, notClass 1 n Among notClass 1, assign Class 2, notClass 2 n Continue to Class n-1 –notClass n-1 = Class n

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

Advantages and disadvantages n Theoretically based n Not a black box n Statistical –variance estimates n Sensitivity to sample design