Academic journal article Geographical Analysis

Area-to-Point Prediction under Boundary Conditions

Academic journal article Geographical Analysis

Area-to-Point Prediction under Boundary Conditions

Article excerpt

This article proposes a geostatistical solution for area-to-point spatial prediction (downscaling) taking into account boundary effects. Such effects are often poorly considered in downscaling, even though they often have significant impact on the results. The geostatistical approach proposed in this article considers two types of boundary conditions (BC), that is, a Dirichlet-type condition and a Neumann-type condition, while satisfying several critical issues in downscaling: the coherence of predictions, the explicit consideration of support differences, and the assessment of uncertainty regarding the point predictions. An updating algorithm is used to reduce the computational cost of area-to-point prediction under a given BC. In a case study, area-to-point prediction under a Dirichlet-type BC and a Neumann-type BC is illustrated using simulated data, and the resulting predictions and error variances are compared with those obtained without considering such conditions.

Introduction

In environmental research, downscaling of model outputs to the typically different observation level is required if, for instance, a groundwater model predicts an average phreatic surface for an area, while the effects of changing groundwater depth on vegetation are measured by observing plants in small plots within that area (Bierkens, Finke, and Willigen 2000). In a similar context, spatial data used in socioeconomic research are collected and reported in diverse zonal systems, say, census tracts or school districts, while modeling often requires individual-level data. The downscaling problem essentially consists of reconstructing a spatial attribute surface at a finer scale, given that only attribute values are known at a coarser scale. Downscaling is a particular case of change of support, for which Gotway and Young (2002) provide a comprehensive review of existing approaches.

The existence of a boundary in the study area, however, whether natural or anthropogenic, might have a significant impact on downscaling results. When the region of interest has a natural boundary, it may be reasonable to expect attribute values at sites on the geographic boundary to have different properties from attribute values at sites in the geographic interior (Martin 1989).

Consider the task of modeling a spatial process in a specific subregion where potential interaction with the external neighborhood of the study area is expected. The most common procedure would be to neglect the influence of the boundary and assume the true process is defined on the internal finite study area. Obviously, this will lead to a model misspecification problem in any estimation endeavor (Upton and Fingleton 1985). In a series of articles, Griffith (1980, 1983) and Griffith and Amrhein (1983) discussed the geographical boundary value problem, also known as "boundary effects" or "edge effects" in spatial statistics (Ripley 1981). In general, the following two different types of boundaries are considered in geographic research: a region surrounded by a set of fixed natural boundaries, for example, coastlines or fault lines in geology, so that no external neighbors exist (Henley 1981), or a subregion of a larger region defined by artificial boundaries, for example, an air pollution boundary in environmental modeling or an urban and rural boundary in population migration (Haining 1990). In a closed region delineated by natural boundaries, the spatial process controlling the spatial distribution of attribute values halts at the boundaries and generates abrupt discontinuities in attribute values at the borders. Such a process continues beyond the edge of a study area demarcated by artificial boundaries. Given the significance of boundary effects, Griffith (1983) proposed nine correction techniques in removing boundary-induced bias from inference. Some correction techniques, however, are based on a very simple or unrealistic assumption, for example, that of an infinite and homogeneous surface (Griffith and Amrhein 1983). …

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