Two-dimensional Markov Chain Simulation of Soil Type Spatial Distribution

At present, such research is still very rare in soil science literature. Soils typically exhibit complex spatial variation of multi-categorical For characterizing the spatial correlation of categorivariables such as soil types and soil textural classes. Quantifying and cal variables in geosciences, the main descriptive tools assessing soil spatial variation is necessary for land management and currently used are indicator variograms provided by inenvironmental research, especially for accurately assessing the water and solute transport processes in watershed scales. This study dedicator geostatistics and transition probability matrices scribes an efficient Markov chain model for two-dimensional modeling (TPMs) provided by Markov chains. Currently, indicaand simulation of spatial distribution of soil types (or classes). The tor geostatistics (Journel, 1983), especially the sequenmodel is tested through simulations of a simplified soil map. The tial indicator simulation (Deutsch and Journel, 1997), application of the model for predictive soil mapping with parameters are more widely used. Indicator methods usually deal estimated from survey lines is explored. Analyses of both simulated with multiple classes by considering each class binomimaps and associated semi-variograms show that the model can effecally and using indicator variogams class-by-class to reptively reproduce observed spatial patterns of soil types and their spatial resent spatial correlation. This approach has been proven autocorrelation given an adequate number of survey lines. This indisuitable for modeling cutoffs (i.e., thresholds) of contincates that the model is a feasible method for modeling spatial distribuuous variables (Goovaerts, 1997, 1999; Brus et al., 2002). tions of soil types (or classes) and the transition probability matrices But for categorical variables that are normally classified of soil types in different directions can adequately capture the spatial interdependency relationship of soil types. The model is highly effiinto multinomial classes with complex spatial depencient in terms of computer time and storage. The model also provides dences, indicator geostatistics seem insufficient to capture an approach for assessing the uncertainty of soil type spatial distributhe complex spatial patterns of multinomial classes with tion in areas where detailed survey data are lacking. The major conlimited measured data (Bierkens and Weerts, 1994; straint on applications of the model at this stage is that the minor soil Ehlschlaeger, 2000; Weissmann and Fogg, 1999; McGwire types are relatively underestimated when survey lines are too sparse. and Fisher, 2001). For example, indicator geostatistics have difficulties in dealing with sharp boundaries and autocorrelation of nominal classes simultaneously (MowC spatial variation of multi-categorical soil rer and Congalton, 2000; McBratney et al., 2000), coping variables, such as soil types and soil textural with anisotropies in multinomial classes (Wingle and classes, is a typical feature of soils in the real world. On Poeter, 1993; Ehlschlaeger, 1998), respecting the juxtathe one hand, traditionally the information on the spatial position relationships between classes (Weissmann and distribution of soil types can only be obtained by deFogg, 1999), and integrating of expert knowledge (Carle tailed field surveys, and soil maps are drawn according and Fogg, 1996; Scull et al., 2003; Weissmann and Fogg, to experts’ empirical judgment based on visual field 1999). They are also highly demanding in computation observations and visual interpretation of air photos and when the number of classes is large (Zhang and Goodtopographic maps. For some regions with limited physichild, 2002), which hinders application over large areas cal access, or without enough survey data, the soil distriand in high-resolution simulation. bution is difficult to assess. On the other hand, we are The Markov chain theory is a stochastic process thestill short of suitable mathematical methods to quantitaory, which describes how likely one state is to change tively characterize the spatial distribution of categorical to another state through one or more time or space soil variables such as soil types and various soil classes. steps. The one-dimensional Markov-chain method has Because an understanding of the spatial distribution of been widely used in geology to simulate stratigraphic categorical soil variables is crucial to soil management sequences since 1960s (Harbaugh and Bonham-Carter, and environmental research (Kite and Kauwen, 1992; 1980; Krumbein, 1968). It also has been used in soil Zhu and MacKay, 2001; Bouma et al., 2002), it is essenscience to describe the spatial order of parcels of differtial to develop suitable mathematical models for characent soil classes (Burgess and Webster, 1984a, 1984b) and the vertical spatial change of textural layers in alluterization of the spatial distribution of such variables. vial soils (Li et al., 1997, 1999) in one-dimension. Although one-dimensional Markov chain is simple and W. Li and J.E. Burt, Dep. of Geography, Univ. of Wisconsin, Madison, easy to use, extending it into multidimensions for condiWI 53706; C. Zhang, Dep. of Geography and Geology, Univ. of Wiscontional simulation is difficult because of the difficulties of sin, Whitewater, WI 53190; A.-Xing Zhu, State Key Lab. of Resources conditioning on measured data and choosing a suitable and Environmental Information Systems, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Scisimulation ordering. ence, Beijing 100101, China; J. Feyen, Institute for Land and Water “Unlike one-dimensional application of Markov chains, twoand Management, Catholic Univ. of Leuven, B-3000 Leuven, Belgium. Received 7 Apr. 2003. *Corresponding author ([email protected]). three-dimensional applications are difficult because there is not Published in Soil Sci. Soc. Am. J. 68:1479–1490 (2004).  Soil Science Society of America Abbreviations: CMC, coupled Markov chain; TMC, triplex Markov chain; TPM, transition probability matrix. 677 S. Segoe Rd., Madison, WI 53711 USA