An Efficient Markov Chain Model for the Simulation of Heterogeneous Soil Structure

et al. (1995). Finally, Yeong and Torquato (1998) use a combination of the two-point correlation function and The characterization of the soil habitat is of fundamental importhe lineal path function to characterize the pore geometance to an understanding of processes associated with sustainable management such as environmental flows, bioavailability, and soil try of a broad range of isotropic structures. ecology. We describe a method for quantifying and explicitly modeling The most useful of these models has been used to the heterogeneity of soil using a stochastic approach. The overall aim interpret the impact of structure on physical properties is to develop a model capable of simultaneously reproducing the and processes; but comparatively little work has examspatial statistical properties of both the physical and biological compoined the impact on biology. Some attempt has been nents of soil architecture. A Markov chain Monte Carlo (MCMC) made to link biological processes with soil structure; methodology is developed that uses a novel neighborhood and scanthese have generally been limited to N transformations ning scheme to model the two-dimensional spatial structure of soil, (Young and Ritz, 2000). These studies have clearly dembased on direct measurements made from soil thin sections. The model onstrated the importance of understanding the relative is considerably more efficient and faster to implement than previous spatial distribution of the physical and biotic elements approaches, and allows accurate modeling of larger structures than has previously been possible. This increased efficiency also makes it of soil structure in determining the larger-scale properfeasible to extend the approach to three dimensions and to simultaneties of the resultant biological process (Arah et al., 1997; ously study the spatial distribution of a greater number of soil compoRappoldt and Crawford, 1999). Therefore there is a nents. Examples of two-dimensional structures created by the models need to further develop models of soil structure that are presented and their statistical properties are shown not to differ are capable of integrating the physical and biological significantly from those of the original visualizations. heterogeneities that occur in most soils. However there are a number of challenges that place constraints on appropriate methodologies, and one of the most signifiM of soil physical structure have been develcant of these is the limitation of existing imaging techoped since the 1950s and used to interpret the nology. impact of structure on function. Childs and CollisThe only reliable methods for visualizing soil in three George (1950) introduced the cut-and-rejoin models of dimensions are and X-ray tomography (Rogasik et soil capillaries, which were modified by Marshall (1958). al., 1999). While the technology is rapidly improving, it While many models of soil structure have been develis not a trivial matter to differentiate between pore and oped since then, most relate the structure to physical solid matrix in these visualizations. Although resoluprocesses, generally ignoring heterogeneity (for examtions of 5 m or higher are now possible, it is still not ples see review by Young et al. [2001]), or assume simple possible to directly image soil microbes in situ and in pore-size distribution models in an attempt to take some three dimensions over comparable scales. The only qualitative account of spatial heterogeneity (Young and method for simultaneously imaging soil structure and Ritz, 2000). A number of the more sophisticated apthe distribution of microbes is by using biological thin proaches exploit the observation that the structure is sections (e.g., Nunan et al., 2001, 2003). Therefore any spatially correlated. For example, Dexter (1976) used a model of structure must be capable of being parameterone-dimensional Markov chain model for horizontal soil ized from two-dimensional data and extrapolated to structure. Moran and McBratney (1997) proposed a three dimensions. two-dimensional fuzzy random model of soil pore strucThe requirements of a useful model capable of deture, which treats the pores as a fuzzy porous set rather scribing the heterogeneity of physical and biological than explicitly dealing with geometry. In Vogel (2000) elements in soil are three-fold. First, the models must a network model for water retention and permeability be able to describe the spatial structure of multiphase is developed where the pore network is geometrically media (matrix, pore, microbe etc.) at the scale of individidealized but can be used to predict physical properties ual pores and microbes. Second, the model must also from topological parameters determined from thin secaccommodate any spatial anisotropy inherent in soil. tions. The geometry of the pore network is explicitly Finally, the model should be capable of using paramedescribed using a fractal-based approach in Crawford ters determined from two-dimensional sections. Currently, no method exists that has been demonstrated to K. Wu and N. Nunan, Soil-Plant Dynamics Unit, Scottish Crop Resimultaneously satisfy these requirements. Yeong and search Institute, Invergowrie, Dundee, DD2 5DA, UK; J.W. Crawford and I.M. Young, SIMBIOS, University of Abertay Dundee, Kydd Torquato (1998) state that their method can be develBuilding, Bell Street, Dundee DD1 1HG, Scotland; K. Ritz, National oped to satisfy these constraints, although to date no Soil Resources Institute, Cranfield University, Silsoe, Bedfordshire such modification has appeared. The development of MK45 4DT, UK. Received 4 June 2002 *Corresponding author their method based on correlation functions, to multi([email protected]). Published in Soil Sci. Soc. Am. J. 68:346–351 (2004).  Soil Science Society of America Abbreviations: MCMC, Markov chain Monte Carlo; MRF, Markov random fields. 677 S. Segoe Rd., Madison, WI 53711 USA