1. Basic geostatistics
Austin Troy

2. Interpolation

3. Three methods in Arc GIS
IDW
SPLINE
Kriging

4. Inverse Distance Weighting
Weighted moving average
Weights by distance
Assumes unknown value influenced more by nearby than far away points, but we can control how rate of decay
Validity testing requires taking additional observations.
Sensitive to sampling, with circular patterns

5. Inverse Distance Weighting
Where λi are given by some weighting fn and
Common form of weighting function is d-p yielding:

6. IDW-How it works 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
Neighborhood type: standard or smooth

7. Neighborhood types: Standard
Specifications
# neighbors to include
Ellipse height/width
Sector type/ angles: to ensure inclusion of obs. from all directions

8. Neighborhood types: Smooth
3 Ellipses
Smoothing factor: size of gaps between ellipses
Inner : same weighting as “standard”
In between: weights multiplied by sigmoidal value from 1 (inner edge) to 0 (outer edge).

9. IDW Weights

10. IDW- P parameter What is the best P to use?
Minimize Root Mean Squared Prediction Error (RMSPE)
Use cross validation to check.
You can look for an optimal P by testing your sample point data against a validation data set

11. Plot of model fits
The 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

12. Plot of model errors

13. 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.

14. 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
Like IDW in that taking weighted moving average, but the weights (λ) are based on statistically derived autocorrelation measures.
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

15. Semivariance
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 which shows correlation over space

16. Kriging Method
Hence, foundation of Kriging is notion of spatial autocorrelation, or tendency of values of entities closer in space to be related.
Autocorrelation can be assessed using a semivariogram.
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

17. 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

18. 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?

19. Variogram Plots semi-variance against distance between points
Where autocorrelation exists, the semivariance should have slope
Is binned to simplify
Binned values generated by grouping empirical points using square cells one lag wide.
To show local variation around mean
Binning with average only
Binning with average and grouping

21. Functional Forms
From Fortin and Dale Spatial Analysis

22. 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

23. 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

24. Kriging: Ordinary vs. Universal
Z(s) = µ(s) + ε(s),
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