Trend Surface Analysis
Trend surface analysis fits a mathematical surface (typically a polynomial) to spatially distributed data points, separating the broad regional trend from local variation. It is used to identify large-scale spatial patterns in elevation, pollution, temperature, and other continuous phenomena.
Trend surface analysis is a global interpolation method that fits a polynomial regression surface to a set of georeferenced data points. The fitted surface represents the large-scale, systematic spatial trend in the data, while the residuals (differences between observed and predicted values) represent localized variation or noise. By decomposing spatial data into trend and residual components, the technique helps analysts understand broad geographic patterns.
Polynomial Surfaces
The simplest trend surface is a first-order (linear) plane, which represents a constant gradient across the study area. Second-order (quadratic) surfaces can model a single dome, basin, or saddle shape. Third-order (cubic) and higher-order surfaces capture increasingly complex regional patterns. The polynomial coefficients are estimated using ordinary least squares regression with x and y coordinates (and their products and powers) as independent variables. The order of the polynomial is selected to balance goodness of fit with parsimony.
Applications
Geologists use trend surface analysis to model regional geological structures, separating broad tectonic trends from local anomalies that may indicate mineral deposits. Environmental scientists fit trend surfaces to pollution measurements to identify regional gradients and local hotspots. Meteorologists model large-scale temperature or pressure patterns. Archaeologists use trend surfaces to model regional artifact density patterns, with residuals revealing local concentrations of interest.
Limitations
Trend surface analysis uses global polynomial fitting, so the surface is influenced by all data points simultaneously and cannot adapt to local variations. High-order polynomials may exhibit edge effects and oscillation. The method is best suited for identifying broad spatial trends rather than for precise local prediction, for which krigingKrigingKriging is an advanced geostatistical interpolation method that uses the spatial covariance structure of sample data ... or local interpolation methods are preferred.
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