Trend Surface Models

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A scalar field is one in which the height is defined as some function of position. A trend surface is a specified form for such a function that is fitted to observed data using the least squares criterion of goodness of fit. Typically, as is illustrated by the example of a simple inclined plane, trend surfaces are specified by simple, linear polynomials. In this sense ‘trend’ is defined as any large-scale systematic change that extends smoothly and predictably from one map edge to the other and a variety of functional forms are possible. Subtraction of the predicted values at each of the input observed data locations from those observed gives a set of residuals, which are regarded as due to local effects. A statistical interpretation is that the coefficients of any such trend model are estimates of some unknown population parameters and appropriate significance tests are outlined. Finally, it is noted that these models find most application in physical geography where continuous scalar fields are more often of concern and where point-valued height data are more commonly obtained.

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