Lecture 5 Geostatistics Lecture Outline Spatial Estimation Spatial Interpolation Spatial Prediction Sampling Spatial Interpolation Methods Spatial Prediction Methods I Interpolating l i Raster R Surfaces S f with i h ArcGIS A GIS Spatial Estimation/Prediction Spatial Prediction: Estimate values at unsampled locations. Why do we need to do this? – Resource limitations (time and money). – You can can’tt measure every single location. – Access or safetyy constraints. – Missing or unsuitable samples. Spatial Interpolation The p prediction of variables at unmeasured locations based on the sampling of the same variables at known locations. Usually used to estimate air and water temperature, soil moisture, elevation, population density, etc. Spatial Prediction The estimation of variables at unsampled locations, based partially on other variables. variables Ex. Use elevation to measure temperature. – Combine C bi elevation l ti and d temperature t t layers l to t better predict temperature at unknown locations. locations The MAUP MAUP: The Modifiable Areal Unit Problem –Ap potential source of error that can affect spatial studies which utilize aggregate data sources. sources – Commonly applied to demographic analysis. – Can also be applied to physical geography and GIS. Sampling Two characteristics of sampling: – Location of samples (How they are spread around) – Number of samples (How many can you afford?) Sometimes we can’t control sampling. i.e. You may be limited to occurrences of an event. Common Sampling Patterns a) Syste Systematic at c Sa Sampling: p g Simple, uniform intervals Randomly pick first observation then select observation, observations at intervals from sampling frame. b) Random Sampling: Randomly placed samples. samples Each observation has the same opportunity to be selected. l d Common Spatial Sampling Patterns c) Cluster Sampling: Groups samples d) Adaptive Sampling: Higher h sampling l densities where feature of interest is more variable. Common Spatial Sampling Patterns Transect Sampling: Select the transects first Select sampling points along the transects Contour Sampling: Select sampling sites along contour lines Spatial Interpolation Methods No one interpolation p method is superior p for all datasets. Method choice depends on: – – – – Characteristics of the variable to be measured Cost of sampling Available resources Accuracy requirements of the users Methods differ in the mathematical functions used to weight each observation, and the number of observations used. Methods: h d – – – – Nearest Neighbor IDW Fixed Distance Spline Spatial Interpolation Methods Types of Interpolators: Exact E Interpolators I l – An interpolation method where the estimated value is identical to the observed value at sampling locations Inexact Interpolators – An interpolation method where estimated value is predicted/estimated from a measured value. Methods: Global Methods – Use information at all sampled location to estimate the value for each unknown location – Trend surface interpolation Local Methods – Use information at nearby locations to estimate the value at locations of interest Geostatistical Methods – Spatial autocorrelation Spatial Interpolation Methods Nearest Neighbor i hb Nearest Neighbor* Neighbor (Thiessen Polygon) Assigns a value to an unsampled l d location that is equal q to the value found at the nearest sample location. Exact interpolator: Value at each sample point is preserved. *Referred to as Natural Neighbor/Voronoi Polygons in lab exercise. Spatial Interpolation Methods IDW (Inverse ( Distance i Weighted) i h d) Uses distance and values to nearby known points. Reduces the contribution of distant points. Weight g of each sample p point p is an inverse proportion to the distance. Further points = less weight Closer points = more weight Exact Interpolator Value l at each h known k point (50,52,30) ( ) are averaged, with the weights based on the distances (d1, d2, d3) from the interpolated point. Spatial Interpolation Methods Fixed i d Radius di and d Spline S li Fixed Radius: Cell values estimated based on average of nearby b samples. l Depends on search radius. Spline: Used to interpolate along a smooth curve. Force a smooth line to pass through a set of points. Spatial Prediction Methods Often generated via a statistical process process. Type of predictive modeling. Primary Pi Methods: M th d – Spatial Autocorrelation – Spatial Regression – Kriging Spatial Prediction Methods Autocorrelation l i Tendency of nearby objects to vary together. “Everything in the universe is related to everything else, but closer things are more related.” – Tobler’s First Law of Geography Spatial Prediction Methods S i l Regression Spatial i Establishes stab s es relationships e at o s ps bet between ee numerous u e ous input variables and presents the relationships in a succinct manner. A regression i analysis l i has h two parts: – The dependent variable, which is the phenomenon whose level or presence you are trying to predict or explain for each location in a study site. – The independent variables variables, which are the known attributes of the locations that influence the level or presence of the dependent variable. Spatial Prediction Methods Kriging i i Weights g the surrounding g measured values to derive a prediction p for each location. However, the weights are based not only on the distance between the measured points and the prediction location but also on the overall spatial arrangement among the measured points. 3 Components: 1. Spatial Trend: increase or d decrease in i variable i bl depending d di on direction. Ex. Temperature to NW 2. Spatial Autocorrelation: Tendency for points near each other to have similar values. 3. Random variation of measured points. Spatial Prediction Methods Kriging i i Lag Distance: – For paired points, the distance between two points. – Symbolized by h – Defines the neighbors. Semivariance: Semivariance: – Based on values at nearby sample points. – For a given h, h there is a value for semivariance semivariance.. – If values are similar to each other for locations at n lags apart, you will see a smaller value of semivariance. semivariance. – So, S smaller ll semivariance i i i di t nearby indicates b locations l ti are similar to each other or stronger spatial autocorrelation. Spatial Prediction Methods Kriging i i Variogram/Semivariogram – A graph showing the relationship between lag distance and semivariance. Nugget – Initial semivariance. – Where autocorrelation is typically highest. Sill – Points where the variogram levels off – Where little spatial autocorrelation occurs. Range – Lag at which sill is reached Spatial Estimation in ArcGIS Analysis performed using Spatial Analyst Tools ÆInterpolation Toolbox. Toolbox OR Geostatistical Analyst Toolbar: