Statistical Surfaces Any geographic entity that can be thought of as containing a Z value for each X,Y location –topographic elevation being the most obvious example –but can be any numerically measureble attribute that varies continuously over space, such as temperature and population density (interval/ratio data)

Surfaces Statistical surface Continuous Discrete

Statistical Surfaces Two types of surfaces: –data are not countable (i.e. temperature) and geographic entity is conceptualized as a field –punctiform: data are composed of individuals whose distribution can be modeled as a field (population density)

Statistical Surfaces Surface from punctiform data Distribution of trees Find # of trees w/in the neighborhood of each grid cell Point dataDensity surface

Statistical Surfaces Storage of surface data in GIS –raster grid –TIN –isarithms (e.g. contours for topographic elevation) –lattice

Statistical Surfaces Isarithm

Statistical Surfaces Lattice: a set of points with associated Z values RegularIrregular

Statistical Surfaces Interpolation –estimating the values of locations for which there is no data using the known data values of nearby locations Extrapolation –estimating the values of locations outside the range of available data using the values of known data We will be talking about point interpolation

Statistical Surfaces Estimating a point here: interpolation Sample data

Statistical Surfaces Estimating a point here: interpolation Estimating a point here: extrapolation

Statistical Surfaces Interpolation: Linear interpolation Elevation profile Sample elevation data A B If A = 8 feet and B = 4 feet then C = (8 + 4) / 2 = 6 feet C

Statistical Surfaces Interpolation: Nonlinear interpolation Elevation profile Sample elevation data A B C Often results in a more realistic interpolation but estimating missing data values is more complex

Statistical Surfaces Interpolation: Global –use all known sample points to estimate a value at an unsampled location Use entire data set to estimate value

Statistical Surfaces Interpolation: Local –use a neighborhood of sample points to estimate a value at an unsampled location Use local neighborhood data to estimate value, i.e. closest n number of points, or within a given search radius

Statistical Surfaces Interpolation: Distance Weighted (Inverse Distance Weighted - IDW) –the weight (influence) of a neighboring data value is inversely proportional to the square of its distance from the location of the estimated value

Statistical Surfaces Interpolation: IDW x 1 = x 1.8 = x 4 = / (4 2 ) = / (3 2 ) = / (2 2 ) = /.0625 = /.0625 = /.0625 = 4 WeightsAdjusted Weights = / 6.8 = 175

Statistical Surfaces Interpolation: 1st degree Trend Surface –global method –multiple regression (predicting z elevation with x and y location –conceptually a plane of best fit passing through a cloud of sample data points –does not necessarily pass through each original sample data point

Statistical Surfaces Interpolation: 1st degree Trend Surface x y z x y In two dimensionsIn three dimensions

Statistical Surfaces Interpolation: Spline and higher degree trend surface –local –fits a mathematical function to a neighborhood of sample data points –a ‘curved’ surface –surface passes through all original sample data points

Statistical Surfaces Interpolation: Spline and higher degree trend surface x y z x y In two dimensionsIn three dimensions

Statistical Surfaces Interpolation: kriging –common for geologic applications –addresses both global variation (i.e. the drift or trend present in the entire sample data set) and local variation (over what distance do sample data points ‘influence’ one another) –provides a measure of error