Second-phase sampling designs for non-stationary spatial variables

In spatial 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. Additional samples are usually collected away from observation locations, or where the kriging variance is maximum. However, the kriging variance (also known as prediction-error variance) is independent of data values and computed under the assumption of stationary spatial process, which is often violated in practice. In this paper, we weight the kriging variance with another criterion, giving greater sampling importance to locations exhibiting significant spatial roughness that is computed by a spatial moving average window. Additional samples are allocated using a simulated annealing procedure since the weighted objective function is non-linear. A case study using an exhaustive remote sensing image illustrates the procedure. Combinations of first-phase systematic and nested sampling designs (or patterns) of varying densities are generated, while the location of additional observations is guided in a way which optimizes the proposed objective function. The true pixel value at the new points is extracted, the semivariogram model updated, and the image reconstructed. Second-phase sampling patterns optimizing the proposed criterion lead to predictions closer to the true image than when using the kriging variance as the main criterion. This improvement is stronger when there is a low density of first-phase samples, and decreases however as the initial density increases.