An integrative hierarchical stepwise sampling strategy for spatial sampling and its application in digital soil mapping

Sampling design plays an important role in spatial modeling. Existing methods often require a large amount of samples to achieve desired mapping accuracy, but imply considerable cost. When there are not enough resources for collecting a large set of samples at once, stepwise sampling approach is often the only option for collecting the needed large sample set, especially in the case of field surveying over large areas. This article proposes an integrative hierarchical stepwise sampling strategy which makes the samples collected at different stages an integrative one. The strategy is based on samples' representativeness of the geographic feature at different scales. The basic idea is to sample at locations that are representative of large-scale spatial patterns first and then add samples that represent more local patterns in a stepwise fashion. Based on the relationships between a geographic feature and its environmental covariates, the proposed sampling method approximates a hierarchy of spatial variations of the geographic feature under concern by delineating natural aggregates (clusters) of its relevant environmental covariates at different scales. The natural occurrence of such aggregates is modeled using a fuzzy c-means clustering method. We iterate through different numbers of clusters from only a few to many more to be able to reveal clusters at different spatial scales. At a particular iteration, locations that bear high similarity to the cluster prototypes are identified. If a location is consistently identified at multiple iterations, it is then considered to be more representative of the general or large-scale spatial patterns. Locations that are identified less during the iterations are representative of local patterns. The integrative stepwise sampling design then gives higher sampling priority to the locations that are more representative of the large-scale patterns than local ones. We applied this sampling design in a digital soil mapping case study. Different representative samples were obtained and used for soil inference. We started with samples that are the most representative of the large-scale patterns and then gradually included the samples representative of local patterns. Field evaluation indicated that the additions of more samples with lower representativeness lead to improvements of accuracy with a decreasing marginal gain. When cost-effectiveness is considered, the representative grade could provide essential information on the number and order of samples to be sampled for an effective sampling design.