A spatially balanced design with probability function proportional to the within sample distance

The units observed in a biological, agricultural, and environmental survey are often randomly selected from a finite population whose main feature is to be geo-referenced thus its spatial distribution should be used as essential information in designing the sample. In particular our interest is focused on probability samples that are well spread over the population in every dimension which in recent literature are defined as spatially balanced samples. To approach the problem we used the within sample distance as the summary index of the spatial distribution of a random selection criterion. Moreover numerical comparisons are made between the relative efficiency, measured with respect to the simple random sampling, of the suggested design and some other classical solutions as the Generalized Random Tessellation Stratified (GRTS) design used by the US Environmental Protection Agency (EPA) and other balanced or spatially balanced selection procedures as the Spatially Correlated Poisson Sampling (SCPS), the balanced sampling (CUBE), and the Local Pivotal method (LPM). These experiments on real and simulated data show that the design based on the within sample distance selects samples with a better spatial balance thus gives estimates with a lower sampling error than those obtained by using the other methods. The suggested method is very flexible to the introduction of stratification and coordination of samples and, even if in its nature it is computationally intensive, it is shown to be a suitable solution even when dealing with high sampling rates and large population frames where the main problem arises from the size of the distance matrix.