Probabilistic Assessment of Contemporary Soil Evolution in the South of Western Siberia Based on Analysis of Soil Monitoring Data

Abstract

The aim of this article was to introduce a new general and theoretical approach to the probabilistic assessment of contemporary soil evolution (CSE) by analyzing soil monitoring data. The CSE is considered a continuous block process of change in soil conditions over a period extending from 10 to 100 years. The soil condition is considered as its position in the range of n soil properties in k soil horizons. As previous investigations have shown the essential intrinsic changeability of soil properties in this range, even in homogeneous objects, my proposed idea to assess CSE was to evaluate changes in the probability distribution functions (pdfs) of soil properties at different moments in time. Taking into account soil variability at different scales, I have introduced three categories for the spatial changeability of soil properties. Assessments of the variability of soil properties at the field level are of most importance for the evaluation of CSE. This variance is presented as the sum of variances induced by elementary soil processes, the micro- or meso-heterogeneity of factors of soil formation and elementary landscape processes, distinctions in the anthropogenous factor, and how the soil reacts to them. I developed a method that consists of (1) identifying the pdfs of soil properties, which means a quantitative evaluation of the kind and parameters of pdfs according to data samples resulting from soil investigations; (2) calculating probabilistic indicators such as the statistical entropy of pdfs as probabilistic characteristics of soil status and informational divergence that is a measure of pdf difference. A case study has been done on the large territory in the south of Western Siberia. New findings were the changes in the probability structure of Kastanozem soil properties during CSE under natural processes and anthropogenous influences. Distinctions in pdfs were evaluated from the values of informational divergence and increment in statistical entropy, which were quantitatively different for soils of different granulometric composition, that is, useful to point out the most vulnerable soils in the territory under investigation. It may be concluded that it is necessary to use probabilistic indicators to assess CSE from pdf alterations. They characterize a degree of influence of soil-forming factors and anthropogenous influences on the probability structure of the properties of a soil and its stability. Thus, they could be reliable indicators of environmental transformation, that is, important for land resources research, land use policy planning, and basic research.

Appendix: Identification of Probability Distribution Functions

The statistical method of constructing mathematical models of probability distribution involves selecting the type and parameters of distribution functions and fitting them for use as experimental data. Thus, the basic task was to determine pdfs. The standard procedures for verifying hypotheses on how close the statistical distribution studied is to the theoretical one often fail to give a satisfactory, unambiguous answer. This is explained by methodological problems and the ambiguity of hypothesis testing. Thus, we used the approach and software developed by the Department of Applied Mathematics and Informatics at Novosibirsk State Technical University (Lemeshko 2005; Mikheeva and Kuzmina 2000). This has overcome problems in the definition of mathematical functions pdfs using a principle, whereby many hypotheses about coincidence were checked with a large set of theoretical functions. In each case, the only function and corresponding parameters chosen is that which is the best approximation coordinated with the set of parametric and nonparametric criteria. The identification of distribution involved selecting of the best theoretical probability distribution from among the thirty known functions to describe the empirical distribution. An incomplete list of such functions is shown in Table 3.

Table 3 Density functions of statistical probability distributions of soil properties

For this purpose, the parameters of each of these functions were estimated from the factual data of the samples studied using the method of maximum likelihoods. Then, statistical hypotheses about the agreement between the empirical data and the tested distribution were checked.

Most researchers select the best distribution using a single-fitting criterion for a specified significance level (usually 0.01, 0.05, or 0.1) and accept the hypothesis of agreement when the corresponding statistics do not suggest agreement with other criteria. Our experience shows that several criteria with different measures of agreement should be used, and the decision should be based on their integrity. Our program system used six criteria for testing the hypotheses; the Pearson χ² criterion (two modifications), the likelihood ratio test (two modifications), the Kolmogorov criterion, the Smirnov criterion, and two Mises criteria (ω ² and Ω²).

The conventional procedure of hypothesis testing involves the nonrejection of the null hypothesis. However, frequently there is no reason to reject several hypotheses; that is, a number of alternatives remain. At the same time, the distribution with the best fit should be selected. Therefore, the probability is calculated for each ith criterion and each jth theoretical distribution (where i is the index of one of the К criteria used, and j is the index of one of the R tested theoretical distributions):

$$ P\{ S_{i} > S_{{ij}}^{*} \} = \int\limits_{{S_{{ij}}^{*} }}^{\infty } {g_{i} (s){\text{d}}s = \alpha _{{ij}} ,} $$

where S* is the corresponding statistics of the criterion used for the ith distribution, and g _i (s) the known distribution density function of statistic S _i provided that the H ₀ hypothesis is true.

Thus, in testing the hypothesis about the agreement with the jth distribution by the ith criterion, if α _ij  > α (where α is the preset significance level), there is no reason to reject the hypothesis of agreement with the jth distribution according to the ith criterion. There is also no reason to reject the hypothesis about the agreement with many laws noted by indices R1 (of R indices). Thus, the distribution law is selected for which $\forall i\alpha_{ij} = \mathop {\hbox{max} }\limits_{j \subset Rl} \alpha_{ij} .$

An unambiguous conclusion may usually be drawn. However, if any uncertainty remains (e.g., in the case of similar distribution laws), the multicriterion problem of decision making is solved. In this case, a simple compromise criterion is composed in the form $\mathop {\hbox{max} }\limits_{j \subset Rl} \sum\nolimits_{i} {\omega_{i} \alpha_{ij} ,}$ where ω _i is the weighting coefficient of the ith criterion, $\sum\nolimits_{i} {\omega_{i} = 1.}$

We have applied this statistical procedure, which has allowed pdfs to be identified with very high probability values. The usage of large samples (n = 50…600) gives us confidence that the form of probabilistic distribution received by this statistical procedure is close to its real form. We have identified pdfs of soil properties for each chestnut soil series; and as a result, this has given us a database of pdfs of soil properties on the Kulunda steppe at different stages of soil usage (Mikheeva 2001, 2005a, b).

Substituting the specific values of the parameters in a distribution equation in Table 3, we obtain the specific form of the pdfs. For example, the pdf of the clay content in the A horizon of the loamy sandy chestnut soil is the Su-Johnson distribution, with parameters θ ₀ = −0.43, θ ₁ = 1.78, θ ₂ = 2.21 and, θ ₃ = −7.17. Substituting the parameter values in the corresponding equation in Table 3 gives us the specific function of pdfs for this property:

$$ W(x) = \frac{1.78}{{\sqrt {2\pi } \sqrt {(x - 7.27)^{2} + 2.21^{2} } }}\; \times \exp \left\{ { - \frac{1}{2}\left[ { - 0.43 + 1.78\ln \left\{ {\frac{x - 7.27}{2.21} + \sqrt {\left( {\frac{x - 7.27}{2.21}} \right)^{2} + 1} } \right\}} \right]^{2} } \right\} $$

The seeming awkwardness of the equation poses no problem, as all the calculations are performed by a computer.