1. Geo479/579: Geostatistics Ch14. Search Strategies

2. Introduction  Search Strategy controls the samples that are included in a local estimation an important consideration in any local estimation  Choice of a Search Strategy Are there enough nearby samples? Are there too many nearby samples? Are there nearby samples that are redundant? Are the nearby samples relevant? (stationarity)

3. Search Neighborhood  an area within which all available samples will be used to contribute to estimation  an ellipse centered on the point being estimated  Anisotropy The shape of ellipse is oriented with its major axis parallel to the direction of maximum continuity If anisotropy is detected, orientation and anisotropy ratio of the search neighborhood have to be determined for each point being estimated

4. Are there enough nearby samples?  The size of the search neighborhood It must be big enough to include some samples  Determined by the geometry of the data set If the sample points are on a pseudo regular grid, the size of the search ellipse must include the four closest samples In practice, one typically tries to have at least 12 samples

5. Are there enough nearby samples..  For irregularly gridded data, the search neighborhood should be slightly larger than the average spacing between the sample points  This is the minimum size of the search neighborhood

6. Are there too many nearby samples? The maximum size is determined by Computation time: it is proportional to the cube of the number of samples It can be reduced by combining several farthest samples into a single composite sample When samples come from farther and farther away, a stationary random function model becomes doubtful

7.  The 12 closest samples are treated as individuals, and the 38 distant ones are combined into four composite samples (similar to block kriging) Composite Samples

8.  Composite samples can be treated as the average covariance between any two samples  If both are individuals, the average is a point-to- point covariance  If one is individual and another is composite, the average is over n point-to-point covariances between a sample to n points that make the composite  If both are composites, the average is over nxm point-to-point covariances Composite Samples..

9.  Weight assigned to a composite sample of each block is equally distributed among the individual samples of which it is composed  If few samples are within the range, the addition of samples beyond the range often improves local estimation Composite Samples..

10. Are the nearby samples redundant?  Clustering of data points Less of a concern for ordinary kriging which accounts for possible redundancies through the C matrix Inverse distance techniques with a search strategy that takes into account clustering will show noticeable improvements  Benefits of reducing redundant samples Reduce the adverse effects of clustering Reduce the number of calculations

11. Quadrant search  Divide the search neighborhood into 4 quadrants and specify the maximum number of samples per quadrant  If a quadrant has samples less than the maximum, keep all samples. Otherwise keep the closest ones

12. Quadrant Search..  Quadrant search accomplishes some declustering  Effect more noticeable on methods that do not decluster by themselves, e.g. inverse distance squared  In fact, it is not a bad idea to screen all data using a quadrant search for not only inverse distance techniques, but also before kriging

13. Are the nearby samples relevant?  Relevance assumption in estimation: the sample values used in the estimation are somehow relevant and they belong to the same group of population as the point being estimated.  Relevance of sample point is decided based on the objective of the estimation  Deciding which samples are relevant for the estimation may be more important than the choice of an estimation method

14. Relevance of nearby samples and stationary models  Stationary random function model requirements - The mean of the probability distribution of each random variable is the same - Models are unbiased only when the weights sum to one

15. Relevance of nearby samples and stationary models..  Using an inappropriate model will result in the actual estimates being very different from their model counterparts  If one does not have the time and curiosity for good estimation, then polygonal or triangulation may limit damage done by a poor search strategy