Spatial covariance modelling in a complex coastal domain by multidimensional scaling

In aquatic studies, spatial interactions may be both easier to interpret and to quantify by using water distance than by using geographic distance. The water distance is the shortest path between those two sites that may be traversed entirely over water. One problem is that water distances may be non-Euclidean, and thus covariance and variogram functions are not necessarily valid when using the water distance as a distance metric. Another problem is that the computation of water distances for a large set of spatial locations is computationally expensive. Our alternative is a computationally efficient method for calculation of a Euclidean approximation to water distances. The first step of the method is to define a triangular grid covering the complex domain of interest. Using this triangular grid, we pre-compute approximate water distances using a graph search algorithm. These water distances are then approximated by multidimensional scaling, giving a Euclidean space. Finally, we use linear interpolation to move the data locations into the new Euclidean space. By using this method, subsequent computations of water distances between any locations can be done very fast and the method leads to a theoretically valid spatial covariance model. We apply our method to herring data from the Vestfjord system in Northern Norway.