12. Spatial Analysis Using a Proportional Effect Semivariogram Model

The assumption of zero trend seems unlikely for many soil properties that change systematically across the landscape. Removal of a trend may result in a bias in estimating the semivariogram and in many studies has not resulted in apparent improvements in kriging estimates. This study was conducted with the objective to use a proportional effect semivirogram model approach for handling a region with a pronounced trend aside from trend removal. To demonstrate this approach, soil samples (0–30-cm depth) were collected along four transects and analyzed for the electrical conductivity (EC) of the soil/water (1:5) extracts. The 50-m long transects centered on the points (x _j ) were from four subregions with maximum contrast in salinity means [m(x _j )]. Fractile diagrams and goodness-of-fit analysis at the 0.05 significance level indicated that the normal distribution function fitted the EC values of the four transects. Except for transect T(x ₁), a lognormal distribution function might also be accepted. A proportional effect stationary semivariogram model: \( {\gamma }_{{e}}(h,{x}_{j}){ /}\Phi \left[m({x}_{j})\right]={\gamma }_{s}(h)\)was fitted to the local experimental semivariograms γ _e(h, x _j ) by the steepest-descent optimization method. The predicted local dispersion variances were reasonable when compared with the experimental variances thereby supporting validity of the proposed model. Statistical analysis of cross validation (kriging) results confirmed the adequacy of the stationary semivariogram model and the validity of the estimated parameters. The assumption of quasi-stationary instead of stationary along the transect improved predictions of true kriging variance. The proportional effect quasi-stationary semivariogram models may offer a possible approach for krig handling a regionalized variable having a pronounced trend without the need for trend removal.