Second-phase Spatial Sampling: Local and Global Objectives to Optimize Sampling Patterns

Eric Delmelle
Department of Geography and Earth Sciences University of North Carolina - Charlotte Charlotte, U.S.A.

Abstract: In geographic sampling, once initial samples of the primary variable have been collected, it is possible to take additional measurements, an approach known as second-phase sampling. It is generally desirable to collect such additional samples in areas far away from existing observations to reduce redundancy, which coincide with regions where the kriging variance is maximum. However, the kriging variance is independent of data values and computed under the assumption of stationary spatial process, which is often violated in practice. Weighting the kriging variance with another criterion, giving greater sampling importance to locations exhibiting significant spatial roughness, can serve as an alternative objective (Delmelle & Goovaerts 2009). This roughness is computed by a spatial moving average window. Another objective function consists of locally determined variogram models to obtain local kriging variances, reflecting non-stationarity (Haas 1990). The benefits and drawbacks of these three approaches are illustrated in a case study using an exhaustive remote sensing image. Combinations of first-phase systematic and nested sampling designs (or patterns) are generated, while the location of additional observations is guided in a way which optimizes each objective function. Augmented sampling sets minimizing the weighted kriging variance or minimizing the kriging variance computed by local variograms lead to better reconstruction of the true image, while patterns minimizing the kriging variance computed by a global variogram lead to reconstruction similar to a random addition. This indicates that accounting for spatial roughness in second-phase sampling improves the overall accuracy of the prediction.

Keywords: sampling pattern, local variance, spatial roughness, weighted kriging variance

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