Deterministic Interpolation

Fo Polynomial Trend

A polynomial trend surface, an inexact global basis model based on spatial coordinates, is used as a fitting regression procedure of a global surface for smoothing, filtering and data interpolation by separating the variable into two components: Large-scale variation (trend or regional features) and random error (non-systematic white noise fluctuation due to local features).

Developed by Whitten in 1957, the unknown b_i coefficients of the polynomial regression coordinates are found by solving a set of simultaneous non-linear equations of the cross-products sums of the (x,y) coordinates and their Z_i values. If the n terms presented under the A matrix (Figure 1) correspond to the coordinates’ data points and the n terms of vector Z equal the correspondent variable values, b0 of vector B represents the constant term while the remaining b1 and b2 parameters are the regression coefficients of the linear polynomial (algebraically, this least square solution equals B=(A’A)^-1(A’Z), where A’ represents the transpose of A matrix). Thus, the estimation of a particular (x_i,y_i) location equals b0+b1xA_xi+b2xA_yi, if a linear polynomial is considered, or b0+b1xA_Xi+ b2xA_yi+b3xA²_Xi+b4xA_xixA_yi +b5xA_yi², if a quadratic one is contemplated.

It is this simplicity that makes this approach worth using and unlike Kriging, it avoids the estimation of the variogram weights. Yet, like any OLS approach, this method is highly susceptible to edge effects and rarely passes exactly through the original data points.

Figure 1.

The linear (a constant dip in a single direction) spatial polynomial matrices used by myGeoffice^© to compute the B vector parameter

Analogous to the traditional regression analysis, the F global test can be used to check for a significant linear regression relationship between the present variable (Z vector) and the space coordinates (A matrix). According to a pre-defined level of confidence (95%, usually), if the computed F ratio (2.42 in this particular case) is smaller than the chosen F-table critical value (close to 3.03, with df1=3-1=2, df2=140-3=137), the present spatial regression does not account for the present dataset behavior (statistically speaking, the null hypothesis should be accepted, i.e., the estimator b1 and b2 of the linear regression cannot be considered different from zero). In other words, this linear trend surface does not explain the variation of the sample data.

Figure 2.

Green and pink points in the right map represent positive and negative residuals, respectively (the difference between the sample values and the linear trend estimation)

So Polynomial Trend

Generically, polynomial surface analysis is useful in two situations: 1) To detect physical or biological global trends attached to the spatial variable under study; 2) To overcome the lack of the intrinsic assumption of Kriging. In this case, the polynomial fits a plane on the data on a global scale, where Kriging is applied to the residuals to analyze the local scale structure (see section “Kriging With Trend”).

In particular, second order polynomials model spatial drifts with a reasonable amount of hills and holes across the sampling area. Since this approach respect OLS, again, this surface analysis is highly susceptible to edge effects and rarely passes exactly through the original data points (inexact interpolator). Another limitation is the lack of uncertainty assessment.

Figure 3.

The quadratic (a bowl or dome shape) spatial polynomial matrices used to assess the B vector estimators under myGeoffice^©