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An Efficient Markov Chain Model for the Simulation of Heterogeneous Soil Structure
Keijan Wu, Naoise Nunan, John W. Crawford,* Iain M. Young, and Karl Ritz
ABSTRACT 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 impor- the lineal path function to characterize the pore geome-
tance 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 exam-
spatial statistical properties of both the physical and biological compo- ined 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 scan- these 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 dem-
based 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 proper-
feasible to extend the approach to three dimensions and to simultane- ties of the resultant biological process (Arah et al., 1997;
ously study the spatial distribution of a greater number of soil compo- Rappoldt 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 signifi-
Models of soil physical structure have been devel- cant of these is the limitation of existing imaging tech-
oped since the 1950s and used to interpret the nology.
impact of structure on function. Childs and Collis- The 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 devel- is not a trivial matter to differentiate between pore and
oped since then, most relate the structure to physical solid matrix in these visualizations. Although resolu-
processes, generally ignoring heterogeneity (for exam- tions 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 ap- the 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 parameter-
one-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 struc- The requirements of a useful model capable of de-
ture, 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 individ-
idealized but can be used to predict physical properties ual pores and microbes. Second, the model must also
from topological parameters determined from thin sec- accommodate any spatial anisotropy inherent in soil.
tions. The geometry of the pore network is explicitly Finally, the model should be capable of using parame-
described using a fractal-based approach in Crawford ters determined from two-dimensional sections. Cur-
rently, no method exists that has been demonstrated to
K. Wu and N. Nunan, Soil-Plant Dynamics Unit, Scottish Crop Re- simultaneously 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 devel-
Building, 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-
(j.crawford@simbios.ac.uk).
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
346
WU ET AL.: MARKOV CHAIN MODEL 347
dependency built into the conditional probabilities, but only
phase anisotropic media is far from a trivial extension after a number of iterations of the algorithm. Indeed a major
of the existing methodology. limitation of the application of the conventional MRF method
In this paper, we introduce an efficient method for is the tendency to a convergence slowly and an associated
modeling the architecture of soil, based on two-dimen- high computational demand. This limits the application to
sional images obtained from soil thin sections. The algo- systems where the number of states of each point is low, and
rithm is based on a Markov chain process that permits where there are only relatively short-range spatial correlations
the macroscopic (centimeter scale) properties to be de- in the structure. Furthermore, the standard implementation
termined from local (pore scale) conditions. A Monte of the method cannot reproduce nonisotropic structures, nor
Carlo implementation is used to reproduce the stochas- can it recognize concavity or convexity of shapes (Tjelmeland
tic properties of the architecture and to estimate fea- and Besag, 1998). Thus, this renders the method inappropriate
for modeling most soil structures.
tures of the posterior or predictive distribution of inter-
Elfeki and Dekking (2001) employed a raster scheme within
est by using samples drawn from images derived from a MCMC approach to simulate geological strata. Their model
real soil. was parameterized from data collected from a set of wells,
Markov chain Monte Carlo methods usually employ and they assumed that the state of a particular point in the
an iterative scheme to obtain the final spatial description strata was dependent on the points immediately to one side
(Besag and Green, 1993), and where correlated struc- and to the top. Therefore if the state of the points are known
tures of the order of the image size exist (such as mac- along a transect at the surface and in a vertical direction at a
ropores in soil thin sections) the iterative schemes are single well location, it is possible to use the model to predict
slow to converge or may even fail. The development of the state of the remaining points in the transect-well plane.
MCMC presented in this paper avoids these problems The model calculates the state of a particular point on the
by a novel choice of neighborhood for the Markov chain basis of the state of the point to the left and above, and the
and the use of a scanning scheme, based on Elfeki and conditional probabilities associated with the Markov process.
This is repeated in a step-wise way as the chain moves from
Dekking (2001) but without the requirement for precon- left to right across the domain in a raster fashion. Crucially,
ditioning, in which the model image is constructed from however, to get an accurate representation the chain must be
a single-pass raster. The resulting structure models are conditioned on several wells across the transect. Therefore
quantitatively compared with the original images to test the probabilities are adjusted to guarantee agreement with
the validity of the approach. A brief discussion of the the well data (Elfeki and Dekking, 2001). This approach is
applicability of the method to extrapolate to three- appropriate for data on geological strata since the number of
dimensions is presented, although a full treatment is wells that can be drilled always limits one, and the aim is
deferred to a forthcoming paper. Finally, the trivial ex- to reproduce the actual structure as closely as possible. The
tension to simultaneously modeling the biological and purposes are quite different to those of the algorithm pre-
physical components of soil architecture in three-dimen- sented here. In our case, we have complete information about
sions is outlined. the state of the points in a two-dimensional domain (i.e., soil
thin section), and we aim to reproduce the functionally impor-
tant statistical properties of the structure rather than a literal
MATERIALS AND METHODS copy of the structure itself. The method should also be ex-
tendable to three dimensions, and for the reasons outlined
Methodology above, we are restricted to methods that can extrapolate from
A detailed description of Markov random fields (MRF) can two-dimensional data. The method of Elfeki and Dekking
be found in Besag (1974), Geman and Geman (1984) and (2001) is then inappropriate.
Cressie (1993). Markov theory lends itself to modeling soil
precisely because the structure of soil is spatially correlated. Multidimensional Markov Chain Model
This means that the structural state at any particular point in
space is conditionally dependent on the state in the vicinity. We consider an image made up of pixels arranged in a
Formally, the neighborhood where such dependency prevails rectangular array. The standard implementation of the MRF
is predefined and these dependencies are expressed in the method assumes that the state of a particular point in an image
form of conditional probabilities. depends on the state of an isotropic neighborhood centered
In using the MRF method to model visualizations such as about the point. As this is unsuitable for modeling anisotropic
those in the current application, a central assumption is that soil structures, we proceed by removing this constraint.
the state of the structure at some point conditionally depends
on only a relatively small number of points in a predefined The Potential Function
neighborhood. Implementation of the algorithm commences
with the derivation of the conditional probabilities from direct The standard MRF method calculates the required condi-
measurements of the probabilities of different neighborhood tional probability from a potential function defined in an iso-
structures in an image of soil structure. The generation of tropic neighborhood. The fundamental framework contains
model structures starts with an initial estimate of the spatial the following components:
distribution of states (e.g., a spatially random distribution). (i) a set S⫽{x|x僆S} of (pixel) sites;
The structure at each point is then updated in accordance (ii) a set N⫽{n
x
|n僆N} attached to the site;
with the conditional probabilities derived from the image. (iii) a probability model, for the joint distribution of the S.
This update is then repeated by successive applications of
the conditional probabilities at each point, until the statistical p(x)⬀exp
冤
兺
xεS
G
x
(n
x
)
冥
[1]
properties of the resulting spatial distribution converge (i.e.,
do not change significantly between successive iterations).
Larger-scale correlations emerge as a consequence of the local Here we redefine the neighborhood as one based on five pixels
348 SOIL SCI. SOC. AM. J., VOL. 68, MARCH–APRIL 2004
However, the apparent compactness is a consequence of the
assumption that the potential functions can be expressed as
a linear function of the parameters. Because of this, the param-
eters can be estimated using maximum-likelihood methods
(Qian and Titterington, 1991). This appears to work for many
kinds of images, but its applicability to images of soil, where
relatively long-range correlations exist, has not been verified.
We attempted to model the structure of our soil samples by
calculating the potential function. However, the resulting
model structures were inadequate. For moderate-sized images,
the modeled structure showed substantial departures from the
Fig. 1. Notation relating to neighborhood system adopted in this original (parent) structure, and as the size of the modeled
study. (a) Two-pixel neighborhood; (b) five-pixel neighborhood. domain increased, the agreement deteriorated further (Fig. 2).
Thus, the simplifying assumptions underlying the linear formu-
forming an anisotropic neighborhood as illustrated in Fig. 1.
lation of the potential functions are incompatible with the
Thus, the state of any given pixel is assumed to depend on the
image structures associated with soil. We therefore developed
two pixels immediately to its left, and on the three immediately
an alternative method based on direct measurement of the 64
above it.
conditional probabilities associated with all possible configura-
The conditional probabilities are then determined from
tions of the five-pixel neighborhood (Fig. 1).
the relation:
P(X
i
⫽x
i
|x
i
⫺
1
,...,x
1
)⫽p(x
i
|x
i
⫺
1
)Markov Chain Monte Carlo Method
⬀exp
冦
兺
N
j
⫽
1
g
ij
(x
ij
)⫹
兺
N
⫺
1
j
⫽
1
[G
ij
(x
ij
,x
i,j
⫺
1
)⫹G
ij
(x
ij
,
i,j
⫹
1
)]
冧
While the definition of the neighborhood as defined in Fig. 1
provides the potential to treat anisotropy, this change alone
[2] is clearly insufficient, and an alternative to the determination
of conditional probabilities from the potential functions mustwhere Nis the number of neighbors, g
ij
and G
ij
are the potential
functions, and x
ij
represents the state of the pixels at positions be found. To this end, we replace the potential functions with
the full set of conditional probabilities that define all possible(i,j). Because of the difficulty in parameterizing the potential
function, these models are usually generalized to involve an combinations of states for the neighborhood. In the case stud-
ied here, we aim to model the relative position of pore spacearbitrary structure of pairwise pixel interactions (Gimel’farb,
1999). There have only been a few attempts to broaden the and solid, and so each pixel can be in one of two possible states.
The standard methods for implementing MCMC are itera-class of these models by introducing potential functions with
more complex neighborhoods, including the Gibbs model tive, for example, the Gibbs sampler version of the Metropolis
Hastings algorithm (Geman and Geman, 1984). However,(Moussouris, 1974) and region maps (Derin and Elliot, 1987).
The associated potentials for the formulae have only two val- these suffer from long convergence times as discussed above,
and are not suitable for modeling large correlated structuresues: G
x
(n
x
)⫽
x
, if all states xin the neighborhood n
x
are
equal, and ⫺
x
otherwise. For higher-order interactions, the such as are found in soil. Here, we use a more efficient method
based on the scanning scheme algorithm proposed by Qianpotential functions are assumed to be a linear function of
parameters, which can be derived from an image only under and Titterington (1991), modified to cope with long-range
correlations in the structural heterogeneity found in soil im-such simplified assumptions.
The advantage of determining the probabilities from the po- ages. The modification replaces the potential function with a
more explicit determination of the transition probabilities astential function is that the neighborhood interactions of an
entire image can be represented by relatively few parameters. detailed below.
Fig. 2. Comparison of simulated soil images with real images. The size of each image is 1.6 by 1.2 cm. (a) Original soil image with anisotropic,
linear pore structure. (b) Simulated image using the potential function; (c) simulated image using the scan scheme. (d) Original soil image
containing pores of the order of the image size; (e) simulated image using the potential function; (f) simulated image using the scan scheme.
WU ET AL.: MARKOV CHAIN MODEL 349
The Scanning Scheme Algorithm Sampling
After considering several different forms for the neighbor- Soil cores were collected from an arable field and thin
hood, we determined that the neighborhood given in Fig. 1 sections were produced using the method described in Nunan
was the smallest capable of reproducing the observed soil et al. (2001). Soil pore maps were obtained by subtracting
properties. The modeling proceeds in two steps. First, the images obtained with cross-polarized light from images cap-
state of the pixel located at the point (i,j ) is determined from tured using transmitted bright-field light. The resultant images
knowledge of those at (i,j ⫺1), (i⫺1, j⫺1), (i⫺1,j), and were then segmented into solid and void. The images em-
(i⫺1,j⫹1) using the associated four-neighborhood condi- ployed in this study were binary pore maps of dimension 760
tional probability. Second, the state of the pixel at (i,j ⫹1) by 570 pixels, representing an area of 1.6 by 1.2 cm. Images
is obtained from knowledge of the new state at (i, j ) together were selected to represent a range of characteristic soil struc-
with the state of the those at (i,j ⫺1), (i⫺1,j⫺1), (i⫺1, tural properties, as shown below.
j), and (i⫺1, j⫹1) using the associated five-neighborhood
conditional probability. These probabilities are obtained from Comparison of Real and Simulated Images
the original image by sampling the four- and five-neighbor-
hoods, and enumerating the different realizations of the state To compare the simulated and real images we selected a
of the point (i,j) and (i,j ⫹1) respectively for each configura- range of quantitative metrics that characterize the heterogene-
tion of the neighborhood. ity and connectivity of the structures under investigation. The
To initiate the model we need to assign states to all the most obvious of these is the porosity and this is readily deter-
cells in the first row, and the first cell of the next row. The mined from both the real and simulated structures. The corre-
cells in the first row are obtained using a two-neighborhood sponding values are listed in Table 1 and range from 7 to
Markov chain where the state of a cell is conditionally depen- 24% in the real structures. There was no significant difference
dent on the state of the cell to its left. The parameters for this between the porosities in the simulated and real structures
Markov chain are obtained as above, but using the smaller (P⬎0.05, paired ttest).
two-cell neighborhood. The state of the first cell of the next The second metric adopted was the mass fractal dimension,
row is determined using the same two-neighborhood condi- which essentially characterizes the degree of aggregation of
tional probability. Using these boundary values, the chain runs the solid matrix. The mass (solid) fractal dimension was deter-
from the left-hand corner of the image and progresses in raster mined by the box counting method (Hastings and Sugihara,
fashion across the image to the right-hand side. First, the state 1993). The calculated values are listed in Table 1, and again
of cell (i,j) in the neighborhood is evaluated, followed by the there was no significant difference between the simulated and
state of cell (i,j ⫹1). The neighborhood is then advanced two real structures (P⬎0.05, paired ttest).
cells to the left and the process is repeated. This continues The third and fourth metrics chosen characterize the pore
until the last cell on the right-hand side is reached and its space, where visually obvious differences between the samples
state is evaluated. At this point, the cells in the first two rows were present. The third is the variance of the porosity as
have been evaluated. Next, the neighborhood remains at the measured in a 0.4 by 1.6 mm sampling window placed at 50
right-hand side but moves one row down. The chain now random locations in each image. For a given porosity, this is
reverses direction, and instead of deriving the states of the a measure of the connectivity of the pore space (Mandelbrot,
(i,j) and (i,j ⫹1) cells in terms of the state of the others, it 1985). We used the Chi-Square goodness-of-fit test, and tested
is the states of cells (i,j ) and (i,j⫺1) that are determined. the null hypothesis that the variances in each pair of simulated
However, before the chain can proceed leftwards, the state and original images were different. This could be rejected at
of the first cell on the right-hand side of the third row must the 95% confidence level indicating that this aspect of the
be evaluated. This is done in the same way as when the neigh- structure of the pore space was not significantly different in
borhood was at the left-hand side of the domain, using the the real and simulated images. The fourth metric measured
two-neighborhood conditional probability. The chain can now the spatial correlation of the pore space by determining the
advance leftwards until the left-hand side of the domain is
semivariogram. Figure 3 shows the variograms for the different
reached. The neighborhood then moves down one row, and
soil samples used in this study, and there are clear differences
the chain reverses as before. Thus, the whole domain is
scanned in a raster-like fashion on this basis, until the states in these between the different soil sections. The figure shows
of all the required cells are obtained. the comparison between the variograms for the real and simu-
The scanning scheme algorithm converges rapidly. In the lated structures. There is no formal statistical way of compar-
examples reported here, we observed the transition kernel ing the properties of semivariograms, however the high degree
(i.e., the matrix of conditional probabilities for all possible of correspondence between the curves for the measured and
neighborhood configurations), calculated from the recon- simulated structures is good, adding further support for the
structed image as the chain progressed. Almost all the proba- modeling methodology.
bilities had become stable after the chain had completed 200
rows, which is equivalent to a depth of 4.0 mm in the original
Table 1. Comparison of soil properties between real image and
soil section and takes about 10 s of computing time on a 1.7-
simulated image
GHz Pentium IV computer. In other words, the minimum size
Real soil Simulated
of a simulated representative soil image should be 0.6 by 0.4
image image
cm
2
, and it takes only a few minutes to generate an image
covering several squared.
Sample 1 Porosity, % 7 7
Mass fractal dimension, D
m
1.9642 1.9654
Sample 2 Porosity, % 17 16
Mass fractal dimension, D
m
1.9382 1.9322
Validation
Sample 3 Porosity, % 24 22
The method was validated using images obtained from soil
Mass fractal dimension, D
m
1.9204 1.9366
Sample 4 Porosity, % 12 11
thin sections and selected to represent a broad contrast in
Mass fractal dimension, D
m
1.9599 1.9503
structural properties.
350 SOIL SCI. SOC. AM. J., VOL. 68, MARCH–APRIL 2004
Fig. 3. Some examples of soil thin section images and associated simulated images (each image size is 1.6 by 1.2 cm) (a) original images (i.e.,
soil thin sections); (b) associated simulated image; (c) semivariograms of pore space in original and simulated images.
CONCLUSIONS AND DISCUSSION structures than has previously been possible and so link
pore-scale to core-scale. By linking with suitable models
We describe a new method for modeling the complex for physical processes, it should now be possible to
architecture of soil. The method is based on a MCMC search for scaling laws from first principals that relate
approach, but incorporates a novel neighborhood defi- the impact of microscopic detail on macroscopic be-
nition and scanning scheme to model large-scale spatial havior.
correlations with rapid convergence. In this approach The method is easily extendable to simultaneously
the state of a pixel in the model image is conditionally model the spatial distribution of a variety of components
dependent on the state of the pixels in the predefined of soil architecture. In the current paper, we verified
neighborhood. The associated conditional probabilities the approach by modeling the physical elements hence
are calculated directly from a segmented image obtained each pixel could be in one of two states—pore or solid.
from a thin section of soil. The method reproduces the In a forthcoming publication, we have extended the
mean and spatial variance of the porosity, and fractal approach to model the relative spatial distribution of
dimension of the matrix as well as the spatial variogram microbes in soil, parameterized directly from soil thin
of the pore space, as estimated from original thin section sections that have been prepared in a manner that pre-
images. Agreement was obtained with images of widely serves the microbes in situ (Nunan et al., 2003).
contrasting soil structures. As well as being capable of
While it is of interest to be able to produce two-
modeling broad-scale heterogeneity, the method is also
dimensional models of the different components of soil
more efficient and therefore faster than previous ap-
proaches. As a consequence, we can simulate larger architecture that have the same statistical properties as
WU ET AL.: MARKOV CHAIN MODEL 351
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dimensions. The efficiency of the algorithm together 267–278.
with a Markov process that is based on local neighbor- Elfeki, A., and M. Dekking. 2001. A Markov chain model for subsur-
face characterization: Theory and applications. Math. Geol. 33:
hood, without the need to condition the chain on existing 569–589.
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dimensional structures. This means it is possible to pro- tions, and the Bayesian restoration of images. IEEE T. Pattern
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Gimel’farb, G.L. 1999. Image textures and Gibbs random field. Kluwermicrobes in structured soil for the first time. This can
Academic Publishers, Dordrecht.
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linking microbes and their microhabitats directly in this Moussouris, J. 1974. Gibbs and Markov random systems with con-
straints. J. Stat. Phys. 10:11–33.
way, the potential for theoretical and experimental soil
Nunan, N., K. Wu, K. Ritz, I., Young, and J.W. Crawford. 2001. Quanti-
ecology is significantly advanced. fication of the in situ distribution of soil bacteria by large-scale
imaging of thin-sections of undisturbed soil. FEMS Microbiol. Ecol.
ACKNOWLEDGMENTS 36: 67–77
Nunan, N., K. Wu, I., Young, J.W. Crawford, and K. Ritz, 2003. Spatial
This work was part-funded by a UK Department of Trade distribution of bacterial communities and their relationships with
and Industry Award in the Biological Treatment of Soil and the micro-architecture of soil. FEMS Microbiol. Ecol. 44: 203–215
Water LINK Programme, and is carried out in association with Qian, W., and D.M. Titterington. 1991. Multidimensional Markov-
Aventis and QuantiSci. The Scottish Crop Research Institute chain models for image-textures. J. Roy. Stat. Soc. Ser. B 53:
receives grant-in-aid from the Scottish Executive Environment 661–674.
and Rural Affairs Department. Rappoldt, C., and J.W. Crawford. 1999. The distribution of anoxic
volume in a fractal model of soil. Geoderma 88:329–347.
Rogasik, H., J.W. Crawford, O. Wendroth, I.M. Young, M. Joschko,
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