By introducing heterogeneity indices, an empirical equation is proposed for characterizing the heterogeneity of non-isotropic fields. The formula is an extension of Fairfield Smith's (1938) empirical law describing heterogeneity in isotropic fields. Based on these indices, criteria are provided for choosing optimum plot shapes in terms of minimizing the sample variance and cost. Sample plots having their largest dimension in the direction with the largest index will give more accurate results (less variable) than plots with other shapes. Relations between the optimum plot size and relative cost versus variogram parameters are given for several variogram models. These relations indicate that variograms with small effective ranges have a very profound effect on the optimum plot sizes.

A simple and efficient method was developed to simulate one-, two- and three-dimensional random fields with one particular kind of variogram for each dimension. The proposed method used an iterative numerical scheme to solve a stochastic differential equation. Since the procedure required minimal computer memory and computation time, it was especially attractive for simulating a large number of points in studying spatial distributions of soil properties. Furthermore, the procedure produced accurate realizations of random fields, in terms of simulation mean and covariance. The mean values and the covariances or variogram structures calculated from the simulated data by the proposed procedure matched closely to the theoretical values and functions. The procedure was applied to simulate spatial variability of soil water content, soil chemicals, and soil reflectance in the real field. Based on available measured data of these soil properties, conditional simulations using the proposed technique were compared with kriged results. Both methods produced very similar spatial distributions of the soil properties.

Estimation of soil particle size fractions (percentage of sand, clay and silt) with sparse data can be improved by taking into account the spatial correlation with other variables. Reflectance of the near infrared band (0.76-0.90μm) is used as an auxiliary variable in the prediction of soil particle size fractions on a 100m×100m grid in a 2200m×1300m field. The results of kriging using the textural information alone are compared with those of cokriging with the auxiliary variable. Cokriging gives a higher correlation coefficient and a lower mean squared error between estimates and measurements than kriging. The most significant improvement of the estimation is in areas of the field with sparse data for the estimated variables. The relative improvement is up to 90% in terms of reduced kriging variance and up to 33% for the mean squared error between actual measurements and estimated values. Only 17-25% of the original observations of texture are needed to obtain relatively accurate estimation when cokriging with reflectance data.

In using cokriging to study soil spatial variability, a key step is to determine cross-variograms. A recently developed approach was utilized to compute pseudo-cross-variograms, from which cross-variograms can be formulated. The approach does not require a large number of common locations where data are available for all variables used in the cokriging modeling and estimation processes. In this study, with only one-thirteenth of the original data for NO3 and Ca, valid cross-variograms, each with the electric conductivity (EC) were obtained by using pseudo-cross-variograms. Based on the cross-variograms, cokriging with EC improved the estimation of NO3 and Ca significantly. Cokriging yielded a smaller mean squared error (MSE) and kriging variance, and a higher correlation between estimates and measurements. Using 20 points of NO3 and 130 points of EC, cokriging provided a similar distribution pattern for NO3 as that generated with 100 points of NO3. Cokriging with EC reduced MSE and the mean kriging variance of the estimated Ca up to 78% and 85%, respectively, as compared with kriging.

The convection-dispersion (CDE) equation and stochasticconvective models are the most commonly used process representa-soltions for predicting solute transport in the field. The convection dispersion equation assumes that the solute is perfectly mixing in the lateral direction, whereas the stochastic-convective model assumes that the solute moves at different velocities in isolated stream tubes without lateral mixing. However, solute transport in heterogeneous preporous media cannot always be conceptualized as being either a convective-dispersive or a stochastic-convective process. In this study, a generalized transfer function model (GTF) was proposed to describe various solute transport processes in heterogenous soils. The model is similar to the convective lognormal transfer function model, but two parameters, lm and ls, are introduced to characterize the depthdependency of the mean (m) and standard deviation (s) of the loga rithm of travel time, respectively. The GTF can describe well the two extremes of solute dispersion, the convective-dispersive and stochastic-convective processes, and transport processes between the two extremes. In addition, the GTF can be used to characterize other solute transport processes in heterogeneous soils, such as those in which the mean of travel time increases with depth nonlinearly, and those in which the dispersivity is a scale-dependent function of the travel distance with any power values.