Geostatistic Analysis Topic 5 Starting 2 27 2007 Purposes Estimating or interpolating values at unsampled sites within the area covered by existing observations Visiting every location is usually difficult or expensive Assumption spatially distributed objects are spatially correlated in other words things are close together tend to have similar characteristics first law of geography also called spatial autocorrelation example Few sample points to fill all cells Example Point elevation to surface Global interpolation global interpolators determine a single function which is mapped across the whole region a change in one input value affects the entire map global algorithms tend to produce smoother surfaces with less abrupt changes are used when there is an hypothesis about the form of the surface e g a trend Local interpolation Local interpolators apply an algorithm repeatedly to a small portion of the total set of points On average values at points closer in space are more likely to be similar than point further apart spatial autocorrelation A change in an input value only affects the result within the window Two important steps for local interpolation Define sampling neighborhood Find points samples in the neighborhood If no directional influence If there are no directional influences in the data you want to give equal weight to sample points regardless of their direction from the prediction location This means that you probably want your neighborhood to be a circle If directional influence if there is directional influence in your data such as might be caused by water draining downhill then you may want to make an ellipse with the major axis running uphill downhill Sample points Once a shape is specified you can restrict which sample points within the neighborhood are used You do this by specifying the maximum and minimum numbers of points to use and by dividing the neighborhood into sectors If the neighborhood is sectored then the maximum and minimum constraints are applied to each part 1 Explore data before interpolation Before creating a surface explore data ED tool enables you to gain a deeper understanding of the phenomena you are investigating so that you can make better decisions on issues relating to your data Exploring the distribution of the data looking for global and local outliers looking for global trends examining spatial autocorrelation and understanding the covariation among multiple datasets Tools includes Histogram Voronoi Map Normal QQPlot Trend Analysis Semivariogram Covariance Cloud General QQPlot and Crosscovariance Cloud Only works on point and polygon layers 1 1 Histogram provides a univariate onevariable description of your data displays the frequency distribution for the dataset of interest and calculates summary statistics measures of location mean median and quartiles measures of spread standard deviation variance measures of shape skewness kurtosis 1 2 QQPlot The quantile quantile q q plot is a graphical technique for determining if two data sets come from populations with a common distribution A q q plot is a plot of the quantiles of the first data set against the quantiles of the second data set By a quantile we mean the fraction or percent of points below the given value That is the 0 3 or 30 quantile is the point at which 30 of the data fall below and 70 fall above that value The q q plot is used to answer the following questions Do two data sets come from populations with a common distribution Do two data sets have common location and scale Do two data sets have similar distributional shapes Do two data sets have similar tail behavior standard Normal QQPlot General QQPlot QQPlots are graphs on which quantiles from two distributions are plotted relative to each other a cumulative distribution is produced by ordering the data and producing a graph of the ordered values versus cumulative distribution values 1 3 Voronoi map Voronoi maps are constructed from a series of polygons thiessen polygon formed around the location of a sample point The value for each polygon can be calculated using any of methods simple mean mode cluster entropy median standard deviation IQR interquartile range 1 4 Trend analysis You may be interested in mapping a trend or you may wish to remove a trend from the dataset before using kriging The Trend Analysis tool can help identify global trends in the input dataset globe trend and anisotropy globe trend is an overriding process that affects all measurements can be represented by a mathematical formula polynomial anisotropy is a random process that shows higher autocorrelation in one direction than in another directional autocorrelation or directional influence the reason for directional influence may not be known but they can be statistically quantified 1 5 Semivariogram covariance cloud The semivariogram covariance cloud shows the empirical semivariogram half of the difference squared and covariance for all pairs of locations within a dataset and plots them as a function of the distance between the two locations the empirical semivariogram for the i j th pair is simply 0 5 z si z sj 2 and the empirical covariance is the cross product where is the average of the data The semivariogram covariance cloud can be used to examine the local characteristics of spatial autocorrelation within a dataset and look for outliers Creating Variography Semivariogram depicts the spatial autocorrelation Understanding a semivariogram range sill and nugget The distance where the model first flattens out is known as the range The value at which the model attains the range is called the sill The value at which the model intercepts the y axis is called the nugget Fitting the semivariogram Circular spherical exponential Gaussian and linear Making a prediction prediction based on the semivarioogram model and the measured values that are nearby using search radius fixed search radius requires a distance and minimum number of points variable search radius number of points needs to be specified You can also specify a maximum distance radius that the search radius cannot exceed To explore a directional influence in the semivariogram cloud we use the Search Direction tools the direction the pointer is facing determiners which pairs of data locations are plotted on the semivariogram lag size is the size of a lag distance used to reduce the larger number of possible combinations it is the size of the cells in the semivariogram surface number of cells is