Lognormal and Moving Window Methods of Estimating Acid Deposition

Abstract

The deposition of heightened levels of sulfuric and nitric acid through rainfall in the United States may adversely affect the environment. For example, soils may become toxic to native tree species because of soil acidification. Ecological effects models being built to study these potential problems have a need for regional deposition estimates with associated measures of uncertainty. However, statistical estimation of the deposition process is complicated by a strong spatial trend (mean nonstationarity) and apparently a spatial covariance structure dependent on location (covariance nonstationarity). The available data for calculating deposition estimates consist of several hundred point observations at irregularly spaced sampling locations across the United States. The spatial estimation technique of kriging is the foundation of four deposition estimation methods evaluated in this study. These are lognormal kriging with a single model of the spatial covariance structure (the variogram); single-estimate lognormal kriging within a moving window using a local model of the variogram; ordinary kriging within a moving window, also with a local variogram model; and planar regression with spatially correlated errors within a moving window used to estimate residuals that are then input to the moving window ordinary kriging algorithm to calculate a (local) trend-adjusted estimate. Each of these four methods has some capacity for accommodating mean and covariance nonstationarity. The moving window methods are new. It is shown that the log transform may not be the correct transform for the deposition process. However, confidence intervals found from a lognormal kriging method do not include a negative interval, as can frequently occur with intervals found from ordinary kriging. Assuming that a partially negative confidence interval for deposition is interpretable, the method of local planar regressions followed by residual kriging gives confidence interval widths that are the most resistant to inflation due to trend effects on the variogram and trend-dependent variance.

Key Words:

• Spatial models
• Geostatistics
• Variogram
• Nonlinear regression
• Two-stage regression