Download Explicit Spatial Markov Chain

Accounting for Interclass Dependences in Stochastic Simulation of
Categorical Soil Variables Using Markov Chain Geostatistics
Weidong Li 1 Chuanrong Zhang 2
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Department of Geography, University of Wisconsin, Madison, WI, USA
Department of Geography and Geology, University of Wisconsin, Whitewater, WI, USA
Interclass dependence is the normal characteristic of multinomial classes of categorical variables.
Strong interclass dependence exists among classes of complex categorical soil variables such as
soil types. Obviously, geostatistical simulations ignoring interclass dependence do not make full
use of the heterogeneity information conveyed by observed data of this kind of categorical
variables. Generally, interclass dependence may include cross-correlations, juxtaposition, and
directional asymmetry of spatial distribution of classes. Conventional kriging geostatistical
methods seem difficult in accounting for interclass dependence of a number of classes.
While the recent developing trend in kriging geostatistics is focusing on incorporating multiplepoint statistics from various data sources such as training images (i.e., multiple-point
geostatistics) (Journel, 2005), the recent progress in geostatistical development is seeing the
emergence of Markov chain geostatistics (MCG), which has been a long-time research topic of
the authors. Both approaches have one common purpose – to better imitate the complex spatial
structure of discrete variables. MCG uses non-linear Markov chain-based estimators. It deals
with many classes simultaneously and has no computational limitation on the number of classes.
MCG is free of some difficult issues bothering indicator kriging, such as order relation problems
and coregionalization of input parameters. Interclass dependence is naturally incorporated into
simulation through cross-transition probabilities. A new spatial heterogeneity measure transiogram was proposed recently to provide continuous transition probabilities for simulations.
Because of these advantages, realizations generated by MCG are highly imitative to the real
patterns given a number of observed data. Potentially, MCG can work with any kinds of data,
such as points, lines, small areas, and mixtures. Although MCG is still at the early stage of
theoretical and methodological development, it has shown exciting features in simulating
multinomial classes.
MCG is a new non-kriging geostatistics. The basic idea of this geostatistics is to use Markov
chains to perform multidimensional interpolation and simulation. Compared with the covariancebased (or variogram-based) geostatsitics, MCG is transition probability-based. Compared with
the kriging-based geostatistics, MCG is Markov chain-based. MCG directly uses Markov chains
to accomplish conditional simulation. Some pioneer studies that are contributive to MCG include
Carle and Fogg (1997), Elfeki and Dekking (2001), and more significantly, Pickard (1980). Our
recent publications can be seen in Li et al. (2004) and Zhang and Li (2005), which show some
preliminary results. The basic idea of MCG is that an unknown location is related on its nearest
known neighbors in different directions. With a Markov chain moving around in a space, its
conditional probability distribution at any unknown point is entirely dependent on its nearest
known neighbors in different directions. The interaction between each nearest known neighbor
and the unknown location is expressed by a transition probability at the corresponding distance.
Therefore, transiograms are the explicit components of the conditional probability function.
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Since MCG as applicable techniques (software system) is still in developing, currently we
emphasize its special capabilities – conditional simulation of multinomial classes, such as soil
types, with interclass dependence. This presentation will demonstrate how this new geostatisitcs
works and some simulated results of soil types. The fully developed MCG will work with any
discrete variables and data types.
References
Carle, S.F., and Fogg, G.E., 1997, Modelling spatial variability with one and multidimensional
continuous-lag Markov chains. Math. Geol. 29:891-918.
Elfeki, A.M., and Dekking, F.M., 2001, A Markov chain model for subsurface characterization:
theory and applications. Math. Geol. 33: 569-589.
Li, W., Zhang, C., Burt, J.E., Zhu, A.X. and Feyen, J., 2004, Two-dimensional Markov chain
simulation of soil type spatial distribution. Soil Science Society of America Journal 68, 14791490.
Journel, A.G., 2005, Beyond covariance: The advent of multiple-point geostatistics. In:
Geostatistics Banff 2004, Series: Quantitative Geology and Geostatistics, vol. 14, O.
Leuangthong and C. V. Deutsch (eds.) 2005, XXVIII, p.1167
Pickard, D.K., 1980, Unilateral Markov fields. Adv. Appl. Probab. 12: 655-671.
Zhang, C. and Li., W., 2005, Markov chain modeling of multinomial land-cover classes.
GIScience and Remote Sensing 42: 1-18.
Contact information: Weidong Li, 1563 W. Wildwood Rd., #12, Whitewater, WI 53190, USA.
Email: [email protected]
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