A physically motivated index of subgrid‐scale pattern

Abstract

Earth Observation data on land surface properties such as albedo are typically collected at a pixel resolution of 1 km or less. Global climate models, on the other hand, are constrained by current limits on computing power to run at a grid box resolution of 10 km or more. This mismatch in spatial scales means that large amounts of pixel‐scale information are condensed into a small number of grid‐box‐scale summary statistics before use in global climate models. Subgrid‐scale patterns in land surface properties may have a significant effect but the summary statistics currently in use, such as the grid box mean and grid box variance, are insensitive to the spatial arrangement of pixels. To address this gap in the information available to global climate models we define here a new grid‐box‐scale summary statistic, the Laplacian pattern index, that is sensitive to the spatial arrangement of pixels. This dimensionless index is based on the mean‐squared value of the Laplacian filter ∇2 within the grid box and is motivated by the physics of diffusive heat transport. We investigate the value of the index in some simple cases and show that it is a measure of the local correlation structure within a grid box. This allows us to generate random grid boxes in which the index takes (on average) a prescribed value. The Laplacian pattern index is designed to be a useful measure of subgrid‐scale pattern in numerical climate models, but can be used as a measure of pattern in any two‐dimensional array of real‐valued data.