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Soil Physics

A Markov Chain-Based Probability Vector Approach for Modeling Spatial Uncertainties of Soil Classes

^a Dep. of Geography, Univ. of Wisconsin, Madison, WI 53706
^b Dep. of Geography and Geology, Univ. of Wisconsin, Whitewater, WI 53190
^c State Key Lab. of Resources and Environmental Information System, Inst. of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China and Dep. of Geography, Univ. of Wisconsin, Madison, WI 53706

^* Corresponding author (weidong6616{at}yahoo.com)

Due to our imperfect knowledge of soil distributions acquired from field surveys, spatial uncertainties inevitably arise in mapping soils at unobserved locations. Providing spatial uncertainty information along with survey maps is crucial for risk assessment and decision-making. This paper introduces a novel probability vector approach for spatial uncertainty modeling of soil classes based on an existing two-dimensional Markov chain model for conditional simulation. The objective is to find an accurate and efficient way to represent spatial uncertainties that arise in mapping soil classes. Joint conditional probability distribution (JCPD) represented by a set of occurrence probability vectors (PVs) of soil classes is directly calculated from conditional Markov transition probabilities, rather than the conventional approximate estimation from a limited number of simulated realizations. By visualizing the calculated PVs, information reflecting spatial uncertainty of soil distribution can be quickly assessed. We hypothesize that these directly calculated PVs are equivalent to the PVs estimated from an infinite number of realizations and thus realizations visualized from the calculated PVs represent the spatial variation of soil distribution. This hypothesis is supported by simulation results showing that: (i) with increasing the number of realizations generated by the Markov chain model from 10 to 100 and to 1000, PVs estimated from these realizations gradually approach the calculated PVs; (ii) similar to simulated realizations, realizations visualized from calculated PVs also can reflect the spatial patterns of soil classes and approximately reproduce the complex indicator variograms of soil classes of the original soil map.

Abbreviations: CCDF, conditional cumulative distribution function • CMC, coupled Markov chain • JCPD, joint conditional probability distribution • PV, occurrence probability vector • PV-realizations, visualized realizations from the calculated PVs • TMC, triplex Markov chain • TPM, transition probability matrix • TP-realizations, simulated realizations using the TMC model through conditional transition probabilities

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