Basic geostatistics

Austin Troy

How does interpolation work•In ArcGIS, to interpolate:

•Create or add a point shapefile with some attribute that will be used as a Z value

•Click Spatial Analyst>>Interpolate to Raster and then choose the method

Three methods in Arc GIS

•IDW

•SPLINE

•Kriging

Inverse Distance Weighting•IDW weights the value of each point by its distance to the cell being analyzed and averages the values.

•IDW assumes that unknown value is influenced more by nearby than far away points, but we can control how rapid that decay is. Influence diminishes with distance.

•IDW has no method of testing for the quality of predictions, so validity testing requires taking additional observations.

•IDW is sensitive to sampling, with circular patterns often around solitary data points

•IDW: assumes value of an attribute z at any unsampled point is a distance-weighted average of sampled points lying within a defined neighborhood around that unsampled point. Essentially it is a weighted moving avg

Where λi are given by some weighting fn and

•Common form of weighting function is d-p

yielding:

Inverse Distance Weighting

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IDW-How it works

•Z value at location ij is f of Z value at known point xy times the inverse distance raised to a power P.

•Z value field: numeric attribute to be interpolated

•Power: determines relationship of weighting and distance; where p= 0, no decrease in influence with distance; as p increases distant points becoming less influential in interpolating Z value at a given pixel

IDW-How it works•There are two IDW method options Variable and fixed radius:

•1. Variable (or nearest neighbor): User defines how many neighbor points are going to be used to define value for each cell

•2. Fixed Radius: User defines a radius within which every point will be used to define the value for each cell

IDW-How it works•Can also define “Barriers”: User chooses whether to limit certain points from being used in the calculation of a new value for a cell, even if the point is near. E.g. wouldn't use an elevation point on one side of a ridge to create an elevation value on the other side of the ridge. User chooses a line theme to represent the barrier

IDW-How it works•What is the best P to use?

•It is the P where the Root Mean Squared Prediction Error (RMSPE) is lowest, as in the graph on right

•To determine this, we would need a test, or validation data set, showing Z values in x,y locations that are not included in prediction data and then look for discrepancies between actual and predicted values. We keep changing the P value until we get the minimum level of error. Without this, we just guess.

IDW-How it works•This can be done in ArcGIS using the Geostatistical Wizard

•You can look for an optimal P by testing your sample point data against a validation data set

•This validation set can be another point layer or a raster layer

•Example: we have elevation data points and we generate a DTM. We then validate our newly created DTM against an existing DTM, or against another existing elevation points data set. The computer determine what the optimum P is to minimize our error

IDW-How it works

Optimizing P value

Plot of model fitsThe blue line indicates degree of spatial autocorrelation (required for interpolation). The closer to the dashed (1:1) line, the more perfectly autocorrelated.

Where horizontal, indicates data independence Mean pred. Error near zero means unbiased

Plot of model errors

Spline Method•Another option for interpolation method

•This fits a curve through the sample data assign values to other locations based on their location on the curve

•Thin plate splines create a surface that passes through sample points with the least possible change in slope at all points, that is with a minimum curvature surface.

•Uses piece-wise functions fitted to a small number of data points, but joins are continuous, hence can modify one part of curve without having to recompute whole

•Overall function is continuous with continuous first and second derivatives.

Spline Method•SPLINE has two types: regularized and tension

•Tension results in a rougher surface that more closely adheres to abrupt changes in sample points

•Regularized results in a smoother surface that smoothes out abruptly changing values somewhat

Spline Method•Weight: this controls the tautness of the curves. High weight value with the Regularized Type, will result in an increasingly smooth output surface. Under the Tension Type, increases in the Weight will cause the surface to become stiffer, eventually conforming closely to the input points.

•Number of points around a cell that will be used to fit a polynomial function to a curve

Pros and Cons of Spline Method

•Splines retain smaller features, in contrast to IDW

•Produce clear overview of data

•Continuous, so easy to calculate derivates for topology

•Results are sensitive to locations of break points

•No estimate of errors, like with IDW

•Can often result in over-smooth surfaces

Kriging Method•Like IDW interpolation, Kriging forms weights from surrounding measured values to predict values at unmeasured locations. As with IDW interpolation, the closest measured values usually have the most influence. However, the kriging weights for the surrounding measured points are more sophisticated than those of IDW. IDW uses a simple algorithm based on distance, but kriging weights come from a semivariogram that was developed by looking at the spatial structure of the data. To create a continuous surface or map of the phenomenon, predictions are made for locations in the study area based on the semivariogram and the spatial arrangement of measured values that are nearby.

--from ESRI Help

Kriging Method•Kriging is a geostatistical method and a probabilistic method, unlike the others, which are deterministic. That is, there is a probability associated with each prediction. Kriging has both a deterministic and probabilistic component, respectively Z(s) = μ(s) + ε(s), where both are functions of distance

•Assumes spatial variation in variable is too irregular to be modeled by simple smooth function, better with stochastic surface

•Interpolation parameters (e.g. weights) are chosen to optimize fn

•Assumes that variable in space can be modeled as sum of three components: 1) structure/deterministic part, 2) random but spatially correlated part and 3) spatially uncorrelated random part

Kriging Method

•Hence, foundation of Kriging is notion of spatial autocorrelation, or tendency of values of entities closer in space to be related.

•This is a violation of classical statistical models, since observations are assumed to be independent.

•Autocorrelation can be assessed using a semivariogram, which plots the difference in pair values (variance) against their distances.

•Where autocorrelation exists, the semivariance should increase until certain distance where SV= variance around mean, so flattens out. That value is called a “sill.” The sloped area, or “range” is where values are related to each other. Intercept is nugget

Semivariance

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•Semivariogram(distance h) = 0.5 * average [ (value at location i– value at location j)2] OR

•Based on the scatter of points, the computer (Geostatistical analyst) fits a curve through those points

•The inverse is the covariance matrix whichshows correlation over space

Steps

• Variogram cloud; can use bins to make box plot

• Empirical variogram: choose bins and lags

• Model variogram: fit function through empirical variogram– Functional forms?

Variogram• Plots semi-variance

against distance between points

• Is binned to simplify• Can be binned based on

just distance (top) or distance and direction (bottom)

• Where autocorrelation exists, the semivariance should have slope

• Look at variogram to find where slope levels

Binning based on distance only

Binning based on distance and direction

Variogram• SV value where it flattens

out is called a “sill.”

• The distance range for which there is a slope is called the “neighborhood”; this is where there is positive spatial structure

• The intercept is called the “nugget” and represents random noise that is spatially independent

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Functional Forms

From Fortin and Dale Spatial Analysis

Kriging Method

•We can then use a scatter plot of predicted versus actual values to see the extent to which our model actually predicts the values

•If the blue line and the points lie along the 1:1 line this indicates that the kriging model predicts the data well

Kriging Method

•The fitted variogram results in a series of matrices and vectors that are used in weighting and locally solving the kriging equation.

•Basically, at this point, it is similar to other interpolation methods in that we are taking a weighting moving average, but the weights (λ) are based on statistically derived autocorrelation measures.

• λs are chosen so that the estimate is unbiased and the estimated variance is less than for any other possible linear combo of the variables.

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Kriging Method

•Produces four types of prediction maps:

•Prediction Map: Predicted values

•Probability Map: Probability that value over x

•Prediction Standard Error Map: fit of model

•Quantile maps: Probability that value over certain quantile

Kriging Method•Semivariograms measure the strength of statistical correlation as a function of distance; they quantify spatial autocorrelation

•Because Kriging is based on the semivariogram, it is probabilistic, while IDW and Spline are deterministic

•Kriging associates some probability with each prediction, hence it provides not just a surface, but some measure of the accuracy of that surface

•Kriging equations are determined by fitting line through points so as to minimize weighted sum of squares between points and line

•These equations are weighted based on spatial autocorrelation, which is determined from the semivariograms

Kriging: Ordinary vs. Universal

•Known as Kriging in the presence of universal trends.

•Universal kriging is used where there is an underlying trend beyond the simple spatial autocorrelation

•Generally this trend occurs at a different scale

•Trend may be fn of some geographic feature that occurs on one part of the map

Example•Here are some sample elevation points from which surfaces were derived using the three methods

Example: IDW•Done with P =2. Notice how it is not as smooth as Spline. This is because of the weighting function introduced through P

Example: Spline•Note how smooth the curves of the terrain are; this is because Spline is fitting a simply polynomial equation through the points

Example: Kriging•This one is kind of in between—because it fits an equation through point, but weights it based on probabilities

Kriging output: prediction