## Problems with trend surfaces

There are a number of disadvantages to a global fit procedure. The most obvious of these is the extreme simplicity in form of a polynomial surface as compared to most natural surfaces. A first-degree polynomial trend surface is a dipping flat plane. A second-degree surface may have only one maximum or minimum. In general, the number of possible inflections in a polynomial surface is one less than the number of coefficients in the trend surface equation. As a consequence, a trend surface generally cannot pass through the data points, but rather has the characteristics of an average.
Polynomial trend surfaces also have an unfortunate tendency to accelerate without limit to higher or lower values in areas where there are no control points, such as along the edges of maps. All surface estimation procedures have difficulty extrapolating beyond the area of data control, but trend surfaces seem especially prone to the generation of seriously exaggerated estimates.

Computational difficulties may be encountered if a very high degree polynomial trend surface is fitted. This requires the solution of a large number of simultaneous equations whose elements may consist of extremely large numbers. The matrix solution may become unstable, or rounding errors may result in erroneous trend surface coefficients.

Nevertheless, trend surfaces are appropriate for estimating a grid matrix in certain circumstances. If the raw data are statistical in nature, with perhaps more than one value at an observation point, trend surfaces provide statistically optimal estimates of a linear model that describes their spatial distribution.

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