Geo479/579: Geostatistics Ch14. Search Strategies

1. Geo479/579: GeostatisticsCh14. 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. Composite Samples • The 12 closest samples are treated as individuals, and the 38 distant ones are combined into four composite samples (similar to block kriging)

8. Composite Samples.. • 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

9. Composite Samples.. • 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

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