k = 1, 2, 3, where k = 1 refers to the variance at h = 0, i.e. the nugget variance; the

components PC were computed. The data were then rotated to obtain scores that were kriged separately avoiding the necessity of modeling the cross-variograms required by cokriging. There- fore, variography was used to study the spatial correlation of both the most relevant measured variables and the principal components. Omnidi- rectional sample variograms were calculated using the version 2.0 of the software package GSLIB Deutsch and Journel, 1998. To represent the variogram of each variable studied and of the principal components, a double exponential model with a nugget was fitted, by using a semi-empirical least-squares method im- plemented in the NLIN procedure in SASSTAT SAS, 1998. The model in its isotropic form is: g h = C + C 1 {1 − e − ha 1 } + C 2 {1 − e − ha 2 } for h \ 0 g 0 = 0 1 where, h is the lag distance, C is the nugget variance, C 1 is the sill of the short-range variance, C 2 is that of the long-range variance, a 1 and a 2 are the distance parameters of the short- and long- range structures, respectively. The distances 3a 1 and 3a 2 are the effective ranges, that is the h-val- ues, where the variogram is approximately 95 of its sill. However, as the statistically independent PC scores are rarely spatially orthogonal, we pre- ferred to incorporate PC technique in geostatistics using factorial kriging Vargas-Guzman et al., 1999a,b. Multivariate geostatistical analysis was performed using only six soil properties SAND, SILT, P01, P1, P15, OC; the variables CLAY, P05, P2 and P10 were eliminated from coregional- ization analysis because they were redundant. All the variables were standardized to zero mean and unit variance, so that the covariance matrix equals the correlation matrix between the attributes. In this case, principal component analysis computed from the covariance matrix produces the same results as that from the correlation matrix. The classical approach to the principal compo- nent analysis PCA neglects the spatial relation- ships among variables, whereas factorial kriging analysis FKA recognizes the correlation struc- tures of measured soil properties. The theory un- derlying FKA has been described in several publications Matheron, 1982; Goovaerts, 1992, 1997; Goovaerts and Webster, 1994; Wacker- nagel, 1994; Vargas-Guzman et al., 1999a,b. Here we describe the major steps in coregionalization analysis as applied to our data. According to the linear model of coregionaliza- tion LMC, we assumed that all the auto and cross-variograms are modeled as linear combina- tions of the same set of three basic variogram functions g k

h, k = 1, 2, 3, where k = 1 refers to the variance at h = 0, i.e. the nugget variance;

k = 2 and k = 3 refer short and long-range vari- ance components, respectively. So, for any couple of variables i and j, the variogram g ij has the following form: g ij h = 3 k = 1 b ij k g k h 2 where, b ij k is a coefficient to be determined by the data. The LMC can be also expressed in matrix terms as: G h = 3 k = 1 B k g k h 3 where Gh is the p × p variogram matrix p is the number of variables, in our case 6 and B k is a positive semi-definite matrix of the coefficients b ij k , called the coregionalization matrix Goovaerts, 1992. To satisfy the condition of positive semi- definiteness of the coregionalization matrices B k , one must check that, for each structure g k

h, the

matrix of b-coefficients is positive semi-definite. A symmetric matrix is positive definite if its determi- nant and all its principal minor determinants are non-negative Goovaerts, 1997. Following the experience with the principal components Eq. 1, we modeled the whole set of auto and cross variograms by two exponential models plus nugget: g ij h = b ij 1 + b ij 2 {1 − e − ha 1 } + b ij 3 {1 − e − ha 2 } 4 For the parameters a 1 and a 2 , we chose the same values determined from the model of the principal components. The LMC was then fitted under the constraint of positive semi-definiteness of B k using the iterative procedure developed by Goulard and Voltz 1992, which utilizes as a criterion of fitting the sum of weighted least squares. Each coregionalization matrix B k describes the relations between the six chosen variables at the particular spatial scale k, defined by the basic variogram function g k h. However, a unit-free measure of correlation between any two variables i and j is to be preferred because the effect of differences in variance is compensated. Such a measure is the structural correlation coefficient r ij k , defined as: r ij k = b ij k b ii k b jj k 5 which focuses on a specific spatial scale k, filtering the effect of other scales of variation. A PCA of each coregionalization matrix B k was then used to summarize further the relations among the vari- ables at the different spatial scales Wackernagel, 1989. The PCA of the coregionalization matrices yielded sets of spatial principal components, called regionalized factors, separately for each spatial scale k. Whereas, the principal compo- nents, estimated from the classical variance – co- variance matrix, combine the contributions from the spatially dependent correlations at different scales, they are separated from one another by FKA in order to understand the spatial patterns Webster et al., 1994. Finally, maps of some variables and regional- ized factors were obtained using ordinary cokriging.

3. Results and discussion