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

possibility or good guess of soil occurrence in the survey area. An interpolated map using standard interpolation Due to our imperfect knowledge of soil distributions acquired from techniques may represent an optimal guess based on field surveys, spatial uncertainties inevitably arise in mapping soils at unobserved locations. Providing spatial uncertainty information along the dataset and the interpolation method used, but does with survey maps is crucial for risk assessment and decision-making. not reflect the real spatial variation characteristics beThis paper introduces a novel probability vector approach for spatial cause of uneven smoothing effects (Goovaerts, 1997, uncertainty modeling of soil classes based on an existing two-dimenp. 369–370). As Journel (1997, p. viii) pointed out: “The sional Markov chain model for conditional simulation. The objective very reason for geostatistics and the future of the disis to find an accurate and efficient way to represent spatial uncertaintcipline lie in the modeling of uncertainty, at each node ies that arise in mapping soil classes. Joint conditional probability through conditional distribution and globally through distribution (JCPD) represented by a set of occurrence probability stochastic images (conditional simulations).” Therefore, vectors (PVs) of soil classes is directly calculated from conditional soil survey maps should be accompanied by related spaMarkov transition probabilities, rather than the conventional approxitial uncertainty information. Data reflecting spatial unmate estimation from a limited number of simulated realizations. By visualizing the calculated PVs, information reflecting spatial uncercertainty usually include occurrence probability maps tainty of soil distribution can be quickly assessed. We hypothesize and realizations provided by random field models (Zhang that these directly calculated PVs are equivalent to the PVs estimated and Goodchild, 2002; Zhang and Li, 2005). This is particfrom an infinite number of realizations and thus realizations visualized ularly useful for risk assessment and decision-making. from the calculated PVs represent the spatial variation of soil distribuIn addition to informing users about the existence and tion. This hypothesis is supported by simulation results showing that: degree of the spatial uncertainty in delineated maps, a (i) with increasing the number of realizations generated by the Markov significant utility of conditionally simulated data using chain model from 10 to 100 and to 1000, PVs estimated from these random field models is that they can be introduced into realizations gradually approach the calculated PVs; (ii) similar to application models (e.g., process-based ecological modsimulated realizations, realizations visualized from calculated PVs els or hydrological models) to further infer response disalso can reflect the spatial patterns of soil classes and approximately reproduce the complex indicator variograms of soil classes of the tributions of variables of interest (e.g., water budgets) original soil map. (Goovaerts, 1996; Kyriakidis and Dungan, 2001; Li et al., 2001). There are several problems hindering spatial uncertainty modeling: (1) It is difficult to mathematically calS mapping is crucial for natural resource evaluaculate the JCPD of a random variable at all unknown tion and environmental protection. However, the locations in a study area of even moderate size. So far knowledge of soil distribution acquired through field we have not yet found any existing geostatistical method survey (and other ways) is always imperfect. Thus spatial that realizes this goal. The normal way for spatial unceruncertainties inevitably arise in soil mapping; for examtainty modeling is through generating a set of alternative ple, a prominent problem is the difficulty in accurately realizations and then approximately estimating the determining the boundaries of multinomial soil classes JCPD (represented as a series of probability maps) from in making choropleth maps on unsurveyed locations. a number of realizations (Zhang and Goodchild, 2002; Given the same observed dataset for a survey area, Zhang and Li, 2005). Thus, the accuracy of probability different persons normally delineate similar but differmaps is largely dependent on the number of realizations ent area-class maps of soil distribution because of their used. (2) Many random field models have difficulties different interpretations over the unobserved portion generating a sufficiently large number of realizations of the landscape. A human-delineated soil map based within acceptable computation time and computer storon a set of observed sparse data only represents one age (Dubrule and Damsleth, 2001), particularly when the number of classes or thresholds is large. With the ongoing W. Li and J.E. Burt, Dep. of Geography, Univ. of Wisconsin, Madison, development of computer techniques, this problem has WI 53706; C. Zhang, Dep. of Geography and Geology, Univ. of been relaxed in recent years. For example, the sequenWisconsin, Whitewater, WI 53190; A-X. Zhu, State Key Lab. of Retial indicator simulation is an efficient variogram-based sources and Environmental Information System, Inst. of Geographical Sciences and Natural Resources Research, Chinese Academy of Scisimulation method; in recent years it has been used for ences, Beijing, China and Dep. of Geography, Univ. of Wisconsin, Madison, WI 53706. Received 29 July 2004. *Corresponding author Abbreviations: CCDF, conditional cumulative distribution function; ([email protected]). CMC, coupled Markov chain; JCPD, joint conditional probability distribution; PV, occurrence probability vector; PV-realizations, visuPublished in Soil Sci. Soc. Am. J. 69:1931–1942 (2005). Soil Physics alized realizations from the calculated PVs; TMC, triplex Markov chain; TPM, transition probability matrix; TP-realizations, simulated doi:10.2136/sssaj2004.0258 © Soil Science Society of America realizations using the TMC model through conditional transition probabilities. 677 S. Segoe Rd., Madison, WI 53711 USA 1931 Published online October 27, 2005