Simulating the spatial distribution of clay layer occurrence depth in alluvial soils with a Markov chain geostatistical approach

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1 ENVIRONMETRICS Environmetrics 2010; 21: Published online 27 March 2009 in Wiley InterScience ( Simulating the spatial distribution of clay layer occurrence depth in alluvial soils with a Markov chain geostatistical approach Weidong Li 1,2 *,y and Chuanrong Zhang 2 1 Key Laboratory of Subtropical Agriculture Resource and Environment, Ministry of Agriculture, College of Resources and Environment, Huazhong Agricultural University, Wuhan , China 2 Department of Geography and Center for Environmental Sciences and Engineering, University of Connecticut, Storrs, CT 06269, USA SUMMARY The spatial distribution information of clay layer occurrence depth (CLOD), particularly the spatial distribution maps of occurrence of clay layers at depths less than a certain threshold, in alluvial soils is crucial to designing appropriate plans and measures for precision agriculture and environmental management in alluvial plains. Markov chain geostatistics (MCG), which was proposed recently for simulating categorical spatial variables, can objectively decrease spatial uncertainty and consequently increase prediction accuracy in simulated results by using nonlinear estimators and incorporating various interclass relationships. In this paper, a MCG method was suggested to simulate the CLOD in a meso-scale alluvial soil area by encoding the continuous variable with several threshold values into binary variables (for single thresholds) or a multi-class variable (for all thresholds being considered together). Related optimal prediction maps, realization maps, and occurrence probability maps for all of these indicator-coded variables were generated. The simulated results displayed the spatial distribution characteristics of CLOD within different soil depths in the study area, which are not only helpful to understanding the spatial heterogeneity of clay layers in alluvial soils but also providing valuable quantitative information for precision agricultural management and environmental study. The study indicated that MCG could be a powerful method for simulating discretized continuous spatial variables. Copyright # 2009 John Wiley & Sons, Ltd. key words: stochastic simulation; Markov chain random field; transiogram; clay layer; alluvial soils; environmental management 1. INTRODUCTION Alluvial soils are typically composed of multiple textural layers deposited during different flooding periods in the past. They widely occur in alluvial plains which are usually major agricultural production bases for local people or even their nations. Studies indicated that the spatial arrangement of textural layers influenced field water and solute transport and crop growth (Yuan, 1980; Ye, 1985; Shi et al., 1986; Brusseau and Rao, 1990). An important natural phenomenon that occurs in alluvial soils and is *Correspondence to: W. Li, College of Resources and Environment, Huazhong Agricultural University, Wuhan , China, and Department of Geography, University of Connecticut, Storrs, CT 06269, USA. y weidong6616@yahoo.com Copyright # 2009 John Wiley & Sons, Ltd. Received 19 July 2007 Accepted 19 October 2008

2 22 W. LI AND C. ZHANG closely related with soil layering is soil salinization, which not only impacts crop production but even caused the decline of some agricultural civilizations (e.g., the ancient Egyptian Nile civilization) in human history. In the North China Plain, a soil profile of 2 m depth may contain two to ten different textural layers. On a regional scale, soil profiles may differ greatly among different sites in terms of textural classes, sequences, numbers of textural layers, and thicknesses of given textural layers (Li et al., 1999). Such a soil characteristic poses a challenge to the practice of precision farm management. Among different types of soil textural layers, clay layers play the major role in influencing soil quality and farm management, because clay has the strongest absorbability to solutes (e.g., nutrients, pesticides, and pollutants) and the strongest retarding ability to water movement (Yuan, 1980; Li and Hu, 2004; Scorza et al., 2004). Therefore, knowing whether a clay layer occurs or not in a soil body (usually in the 2-m depth), its specific occurrence depth at a location and the spatial variation of the occurrence depth at a regional scale is especially valuable for understanding the behaviors of the soil and applying appropriate site-specific agricultural management measures. Studies on the spatial distribution of clay layers in alluvial soils are rare in literature. Li et al. (1998) estimated the spatial distribution of the thickness of first clay layers in alluvial soils in a meso-scale area in the North China Plain using kriging. The study found that the thickness of the first clay layers was lognormally distributed and their occurrence depth had a correlation range of about 1500 m. To characterize the spatial variation of a geospatial variable, geostatistical techniques are usually used. Geostatistical interpolation techniques such as kriging have been given special attention because they incorporate the spatial autocorrelation concept explicitly in the modeling process. However, kriging interpolation techniques have their limitations: They provide only one optimal prediction map, which actually smoothes out local spatial variation and ignores global statistics (Deutsch and Journel, 1998). Hence, kriging interpolation blurs variability and spatial patterns that are potentially important when assessing environmental impacts. Spatial stochastic simulations provide an alternative concept focusing on the uncertainty of predictions of geospatial variables at unsampled locations (Goovaerts, 1997, 2000; Chiles and Delfiner, 1999). They reproduce statistics such as the sample histogram and the semivariogram model in addition to honoring data values. The set of multiple realizations generated by stochastic simulation algorithms are useful to assess the uncertainty of predictions and the propagation of errors through GIS-based or ecological simulation models. Thus, spatial stochastic simulations are increasingly preferred to interpolation for applications where the spatial variation of the measured field must be preserved (Srivastava, 1996; Juang et al., 2004; Zhao et al., 2005; Grunwald et al., 2007). The major methods used for simulating continuous environmental variables include parametric sequential Gaussian simulation (SGS) or non-parametric sequential indicator simulation (SIS), both of which are based on kriging. SIS takes the form of mapping the probabilities of exceeding threshold values or discretizing a continuous variable into several intervals (or classes) (Deutsch and Journel, 1998; Goovaerts, 2001). This means that in simulation a continuous variable may be treated as a categorical variable. A recently proposed method called Markov chain geostatistics (MCG) (Li, 2007a) provided a new approach to simulate environmental variables. This method demonstrated its advantages over SIS in simulating categorical soil variables (Li and Zhang, 2007; Zhang and Li, 2007): It objectively reduced the spatial uncertainty in predicted results given the same sample datasets and strictly obeyed the interclass relationships defined by transiograms a transition probability-based spatial relationship measure. One reason may be that cross transiograms are asymmetric and therefore more powerful in representing various interclass relationships such as cross-correlation, juxtaposition, and directional asymmetry; the second reason should be that MCG is effective in incorporating those interclass relationships into simulation so that simulated realizations obtain higher accuracy through strictly obeying those relationships conveyed by input transiogram models.

3 SIMULATION OF SOIL CLAY LAYER SPATIAL DISTRIBUTION 23 The objectives of this study are (1) to gain an understanding in the spatial distribution of clay layer occurrence depth (CLOD) in alluvial soils at a watershed scale and (2) to introduce a new method for stochastic simulation of continuous spatial variables treated as categorical variables. The Markov chain sequential simulation (MCSS) algorithm was, therefore, applied to the simulation of CLOD. Four thresholds were chosen to encode the CLOD sample data into four binary indicator data sets and one five-class indicator data set. The simulations generated realization maps, occurrence probability maps, and optimal prediction maps in CLOD. These results might provide useful information for precision agricultural and environmental management as well as risk assessment of agricultural measures in the study area. 2. METHODOLOGY 2.1. Study area and data The study site is an approximately 15 km 2 rectangular area around Quzhou Experiment Station of China Agricultural University, located in the northeast Quzhou County, Hebei Province, China (N , E ). The topography in the area is smooth and mostly occupied by a loamy depression in the Zhang River s alluvial fan. An old course of the Zhang River crosses the west part and a branch of the river cuts the middle of the study area. The height of the ground surface ranges from 35.0 to 37.3 m. The whole study area is occupied by alluvial soils, embedded with some spots of salinized soils. Surface soil texture is primarily sandy loam, clay, and light loam. One hundred fourty-two observation points were arranged on a grid with a sampling interval of 350 m (see Figure 1), but a total of 139 soil profile records were obtained due to the influence of villages. Data collection was performed by excavating pits to a depth of 2 m. Soil texture differences were determined by field texturing supported by laboratory analysis of a small number of soil samples from previous soil survey data. The occurrence depth of clay layers at every soil profile location was recorded with an observation precision of 5 cm. Related data have been used in a vertical one-dimensional simulation of soil textural profiles by Li et al. (1999) Indicator coding To simulate a continuous variable using MCG, the continuous variable needs to be first coded as indicators, of which each represents an interval of the continuous variable. Such an indicator coding is similar to that used for indicator kriging. The difference is that in MCG it is not required to use only 1 Figure 1. Study area and locations of observed soil profiles

4 24 W. LI AND C. ZHANG and 0. Given a threshold value z k, soil CLOD is actually cut into two intervals; CLOD data z(u a ) are encoded as indicator values for each threshold z k according to the following rule: iðu a ; z k Þ¼ 1 if zðu aþz k (1) 2 otherwise Thus, for each threshold we actually obtained a sample indicator data set of a binary variable, and for each binary variable we use four auto- and cross-transiograms to describe the correlations within and between the two classes of the variable (i.e., less than and not less than the threshold). If four thresholds are considered together, they actually divide the CLOD into five intervals and each interval may be treated as a class. Then the CLOD becomes a five-class categorical variable, which is encoded as 8 1 if zðu a Þz 1 >< 2 if z 1 < zðu a Þz 2 iðu a ; z k Þ¼ 3 if z 2 < zðu a Þz 3 (2) 4 if z 3 < zðu a Þz >: 4 5 if z 4 < zðu a Þ where z 1,..., z 4 represent the four thresholds from the smallest to the largest. To describe the auto- and cross-correlations among the five classes, 25 transiograms are needed Markov chain geostatistics MCG refers to the Markov chain models and simulation algorithms based on the Markov chain random field (MCRF) theory (Li, 2007a). It theoretically extends a single Markov chain into any dimensions for spatial modeling. The spatial measure for MCG is the transiogram, which is a transition probabilitybased spatial measure for characterizing auto- and cross-correlations of classes. For spatial simulation, if only the nearest known data in four cardinal directions are considered, the MCRF model can be written as PrðzðuÞ ¼kjzðu 1 Þ¼l; zðu 2 Þ¼m; zðu 3 Þ¼q; zðu 4 Þ¼oÞ ¼ P n f ¼1 p4 ko ðh 4Þp 3 kq ðh 3Þp 2 km ðh 2Þp 1 lk ðh 1Þ ½p 4 fo ðh 4Þp 3 fq ðh 3Þp 2 fm ðh 2Þp 1 lf ðh 1ÞŠ (3) where, 1, 2, 3, and 4 represent the four cardinal directions considered, h 1, h 2, h 3, and h 4 represent the distances from the uninformed location u to its nearest known neighbors u 1, u 2, u 3, and u 4 in the four cardinal directions, respectively; and k, l, m, p, and o represent the states of the Markov chain at the five locations u, u 1, u 2, u 3, and u 4, respectively, all defined in a state space S ¼ (1, 2,..., n). In directions 2, 3, and 4, transitions occur from the current uninformed location u to its nearest known neighbors, but in direction 1 (i.e., the coming direction of the Markov chain), the transition occurs from the nearest known neighbor u 1 to the current location u. With increasing lag h, any p kl (h) represents a transiogram. From Equation (3), one can see that transiograms are necessary to provide transition probability values at required lags to the MCRF model for estimating the conditional probability distribution of a random variable at a location. In this study, Equation (3) and its further simplified forms (for less than four nearest known neighbors) were used in the simulation algorithm.

5 SIMULATION OF SOIL CLAY LAYER SPATIAL DISTRIBUTION 25 A transiogram p ij (h) refers to a diagram of a transition probability from class i to class j over the distance lag h. An auto-transiogram p ii (h) represents the self-dependence (i.e., auto-correlation) of a single class i and a cross-transiogram p ij (h)(i 6¼ j) represents the cross-dependence of class j on class i. Here class i is called a head class and class j is called a tail class. Transiograms are always positive. One typical property of transiograms is that the sill of a transiogram is theoretically equal to the proportion of the tail class in the study area, that is, its stable transition probability. The second typical property of transiograms is that the values of all transiograms headed by the same class at any specific lag sum to one (Li, 2007b). Practically, an experimental transiogram _ p ij ðhþis directly estimated from sample data by counting the transition frequency from a class to itself or another class with different lags (e.g., numbers of pixels for raster data) by the following equation: ^p ij ðhþ ¼F ij ðhþ=n i ðhþ (4) where N i ðhþ ¼ P n j¼1 F ijðhþ is the total of elements in the ith row of a transition frequency matrix at the lag h, F ij (h) represents the frequency of transitions from class i to class j at the lag h, and n is the total number of classes. To acquire reliable experimental transiograms from sparse samples, one has to consider a lag tolerance Dh around each lag value, which may be decided by users according to the density of samples. If anisotropies are considered, experimental transiograms have to be estimated directionally with or without a tolerance angle, similar to estimation of variograms. Experimental transiograms estimated from limited samples are scattered points and therefore cannot be used directly in simulations. Two methods were suggested to acquire continuous transiogram models. The first one uses mathematical models to jointly fit experimental transiograms (Li, 2007b). This method is relatively time consuming when the number of classes is large, but it permits incorporation of expert knowledge in estimation of transiogram models. Therefore, this method is more flexible and widely applicable, particularly when samples are sparse and cannot provide reliable experimental transiograms. The second one interpolates experimental transiograms into continuous models (Li and Zhang, 2005). This method is efficient but eliminates the chance of incorporating expert knowledge. Therefore, this method is suitable only when samples are sufficient and experimental transiograms are reliable. In this study, we use the interpolation method for transiogram modeling because the grid sample data set can provide quite reliable experimental transiograms. The linear interpolation method is given as the following equation: p ij ðhþ ¼^p ijðh k Þðh kþ1 hþþ^p ij ðh kþ1 Þðh h k Þ h kþ1 h k (5) where h k þ 1 and h k are the corresponding lags of two estimated neighboring values in an experimental transiogram, and p ij (h) is the value to be interpolated (or estimated) at the lag h between h k þ 1 and h k. In simulation, the four cardinal directions are replaced by four quadrants of a search circle, that is, in each quadrant the simulation algorithm looks for a nearest known neighbor. The finally found nearest known neighbors in the quadrants constitute the MCRF model for estimating the uninformed location. In case there is no data (i.e., informed locations) occurring in some quadrants (e.g., at the beginning of simulation or on boundaries), the nearest known neighbors found will be less than four; thus MCRF models further simplified from Equation (3) based on the number of the found nearest known neighbors will be used. A detailed introduction of the MCSS algorithm was introduced in Li and Zhang (2007). As to how to choose the search radius, it is the user s decision based on the density of their samples, and it

6 26 W. LI AND C. ZHANG should avoid using such a small search radius that the search circle covers no nearest known neighbor frequently at the initial stage of a simulation. Note that when we simulate the continuous variable discretized by four threshold values, we actually simulate it as a five-class categorical variable. Thus the simulated realizations will be polygonal maps, not maps with continuous values. It is feasible to simulate it as a continuous variable with continuous values by interpolating/extrapolating the discrete cumulative conditional distribution function into a continuous curve, similar to that performed in indicator kriging (Goovaerts, 1997, pp ) Simulations Whether clay layers occur at a depth less than a certain threshold is of great concern in precision agriculture and environmental management. In this study we chose four thresholds (i.e., 50 cm, 75 cm, 100 cm, and 150 cm) to simulate the spatial distribution of the CLOD. For simulation, the study area was discretized into a raster with a pixel size of m. Experimental transiograms were estimated from each of the five sets of indicator data coded for the four thresholds, which made four binary variables and one five-class variable. It seems huge work might be needed to model all needed experimental transiograms. However, in MCG the joint modeling of experimental transiograms is quite easy because it has no strict constraint conditions as required for joint variogram modeling in kriging. For simplicity, in this study we simply interpolated the experimental transiograms into continuous models for use in simulations. The search radius used for all simulations is 50 pixel lengths (i.e., 1250 m), which is sufficiently large for the sample density. For each of the four binary variables and one five-class categorical variable, one hundred realizations were generated. From those realizations for each variable, maximum occurrence probability maps and single-class occurrence probability maps were estimated; and based on the maximum occurrence probabilities a corresponding optimal prediction map was further acquired. Simulations were done on an ordinary personal computer. The computation time used for generating 100 realizations and other related maps for each sample data set is similarly about 1 h. In fact, the computation time needed mainly depends on the total number of pixels of the simulation domain, rather than the number of classes. 3. RESULTS AND ANALYSIS The probability distribution of CLOD estimated from sample data apparently has two raised tails (Figure 2), the left tail representing the probability of COLD ¼ 0 cm and the right one representing the probability of COLD > 200 cm. With these two large extreme values, it is inappropriate to use SGS to simulate COLD because SGS requires a normality of the probability distribution and cannot effectively deal with extreme values. Similar to indicator auto- and cross-variograms, auto-transiograms represent spatial autocorrelations of single classes and cross-transiograms represent spatial cross-correlations between classes. However, cross-transiograms are normally asymmetric, that is, p ij (h) 6¼ p ji (h). Figure 3 shows the four experimental auto- and cross-transiograms estimated from the sample indicator data set for the 75 cm threshold and their interpolated continuous models. It can be seen that the two classes are obviously auto-correlated and also cross-correlated of each other. Class 1 (i.e., CLOD 75 cm) has an auto-correlation range of about 40 pixels (i.e., 1000 m) and class 2 (i.e., CLOD > 75 cm) has a far longer auto-correlation range. The cross-correlation range for transiogram p 12 (h) is about 45 pixels

7 SIMULATION OF SOIL CLAY LAYER SPATIAL DISTRIBUTION 27 Figure 2. Probability distribution of clay layer occurrence depth estimated from samples Figure 3. Experimental transiograms estimated from the indicator data set of samples for the 75 cm threshold (i.e., 1125 m) and that for transiogram p 21 (h) is much larger (about 90 pixels). In addition, from the approximate sills of these experimental transiograms, one can find that the proportion of class 1 is about 38% and that of class 2 is about 62%. To simulate the CLOD as a five-class categorical variable, we need totally 25 auto- and cross-transiorgam models. Figure 4 displays a subset of experimental transiograms estimated from the five-class sample indicator data set. This subset is headed by class 5 (i.e., CLOD > 150 cm). Similarly, the auto- and cross-correlation ranges of related classes can be approximately discerned, and the proportions of all the five classes can also be approximately estimated from the sills of corresponding experimental transiograms. For example, class 5 has an auto-correlation

8 28 W. LI AND C. ZHANG Figure 4. A subset (headed by class 5) of experimental transiograms estimated from the five-class indicator data set of samples range of about 50 pixels (i.e., 1250 m) and a proportion of about 18%. The cross-correlation range for class 5 and class 2 is about 40 pixels (i.e., 1000 m) and the sill of p 52 (h) indicates that class 2 has a proportion of about 17% among the five classes. Conditional simulations were conducted for each data set. Figure 5A shows the optimal prediction maps for the four thresholds. It can be seen that within 50 cm depth clay layers occur mainly at the south-west part and between 75 and 100 cm depths clay layers occur at most places in the middle and east parts of the study area. In the middle-west part, there is a clear south-north stripe where no clay layers occur in less than 150 cm depth. This stripe is where the old course of the Zhang River is located and soils along this stripe tend to be sandy. In general, within 150 cm depth clay layers tend to occur almost in the whole area except for one south-north stripe, and through mapping the clay layer occurrence within this depth the old course of the Zhang River and its approximate width are clearly displayed. The optimal prediction maps only show where clay layers tend to occur within a depth (i.e., have an occurrence probability of more than 0.5). The corresponding maximum occurrence probability maps, as shown in Figure 5B, provide the exact credibility of clay layer occurrence at every location. Because there are only two classes, the lowest maximum probability is 0.5, normally occurring on the boundaries of polygons of the two classes (see the shallow-white stripes in Figure 5B).

9 SIMULATION OF SOIL CLAY LAYER SPATIAL DISTRIBUTION 29 Figure 5. Optimal prediction maps of clay layer occurrence depth for different thresholds (column A) and their corresponding maximum occurrence probability maps (column B) (a): 50 cm; (b): 75 cm; (c): 100 cm; and (d): 150 cm From one set of conditioning data and related transiogram models, one can obtain only one optimal prediction map, which cannot demonstrate the uncertainty in spatial distribution of a random variable caused by incomplete sampling (or observation). Realizations obey the same spatial statistics (i.e., transiogram models) with different specific patterns, each representing a possible occurrence (i.e., a configuration) of the variable. Thus, a number of realizations may demonstrate the spatial uncertainty of distribution of a variable. Figure 6A shows three realization maps for CLOD to be no greater than 100 cm depth. It is clear that the realizations display different patterns, but also have some similarity to some extent. The difference in patterns indicates the spatial uncertainty, and the similarity is contributed by the common conditioning data and spatial correlation information represented by transiogram models as simulation inputs. While a series of realizations may demonstrate spatial uncertainty, it is more efficient and also clearer to use occurrence probability maps of single classes to represent spatial uncertainty. Figure 6B provides the single-class occurrence probability maps of the CLOD to be no greater than 50 cm, 100 cm, and 15 cm, respectively. It can be seen that with increasing depth clay layers tend to appear at more and more locations. A jump in probability values occurs at the depth interval from 75 to 100 cm, which means that at many places clay layers occur within that depth range. The Zhang River s

10 30 W. LI AND C. ZHANG Figure 6. Simulated realizations for the threshold value of 100 cm depth (column A) and the occurrence probability maps of the clay layer occurrence depth to be no greater than 50 cm, 100 cm, and 150 cm, respectively (column B) old course becomes clear with increasing observation depth, because clay layers have little chance to occur within the old river course. Compared with the corresponding optimal prediction maps shown in Figure 5, occurrence probability maps convey much more information, that is, they provide the exact probability values of CLOD at every location (i.e., pixel). Thus, these probability maps (especially high resolution raster maps with pixel probability values) are much more useful than optimal prediction maps in soil management, particularly in risk assessment. For example, the applications of chemical fertilizers and pesticides should consider the possibility of occurrence of clay layers in soils at a specific place because clay can absorb many chemicals and thus reduce or delay the leakage of chemicals to ground water (Scorza et al., 2004). The simulations aforementioned considered only a single threshold value. When four threshold values were considered together, the CLOD became a categorical variable with five classes representing the five intervals. The merit of such an approach to simulate the CLOD is that the spatial variation of CLOD at different depth ranges can be clearly displayed together like a contour map. Figure 7 shows some of simulated results of CLOD as a five-class variable. The optimal prediction map here is similar to an optimal contour map with four contours to some extent. However, the simulated realization maps and occurrence probability maps indicate the complexity and uncertainty in spatial distribution of CLOD at different depth ranges and deliver far more information than the single optimal prediction map. It is normal that the spatial distribution of different CLOD classes in simulated realizations does not follow the order from shallow to deep because the first clay layers at different places may be formed in different times at different depths and are consequently discontinuous in the space.

11 SIMULATION OF SOIL CLAY LAYER SPATIAL DISTRIBUTION 31 Figure 7. Simulated results of clay layer occurrence depth as a five-class variable. The five classes 1, 2, 3, 4, and 5 refer to the depth intervals 0 50, 50 75, , , and >150 cm, respectively. Left column the optimal prediction map based on maximum occurrence probabilities and two simulated realizations. Right column the maximum occurrence probability map and two occurrence maps of single classes 4. CONCLUSIONS The MCG simulation approach for categorical variables was used to simulate a continuous environmental variable the clay layer occurrence depth in alluvial soils in a meso-scale area. Although the CLOD is a continuous variable, simulation can be conducted by transforming it into categorical variables through indicator coding with one or several threshold values. Because of the efficiency of the algorithm and the incorporation of interclass relationships, the simulated results, especially the occurrence probability maps estimated from a large number of realizations, effectively captured the spatial distribution of CLOD and related spatial uncertainty. Simulated results showed that different intervals of CLOD, which were encoded as indicators, are spatially both auto-correlated and cross-correlated. Within 150 cm, clay layers occurred at most of the study area, except for a north-south stripe at the middle-west part and a piece in the center of the middle-east part. The north-south stripe in fact clearly indicated an old river course, where no clay layers were found in the observation depth and the soils actually tended to be sandy. The simulated results sufficiently mapped the various spatial distributional characteristics of CLOD in the study area, and might provide useful information for effective agricultural and environmental management.

12 32 W. LI AND C. ZHANG The study also indicated that the MCG approach represented a practical method for simulating continuous spatial variables because it is easy to acquire a valid set of auto- and cross-transiogram models and the simulation algorithm is efficient in computation and powerful in generating polygonal spatial patterns. REFERENCES Brusseau ML, Rao PSC Modeling solute transport in structured soils: a review. Geoderma 46: Chiles JP, Delfiner P Geostatistics: Modeling Spatial Uncertainty. John Wiley & Sons: New York. Deutsch CV, Journel AG GSLIB: Geostatistics Software Library and User s Guide. Oxford University Press: New York. Goovaerts P Geostatistics for Natural Resources Evaluation. Oxford University Press: New York. Goovaerts P Geostatistical modeling of uncertainty in soil science. Geoderma 103: Goovaerts P Estimation or simulation of soil properties? An optimization problem with conflicting criteria. Geoderma 97: Grunwald S, Reddy KR, Prenger JP, Fisher MM Modeling of the spatial variability of biogeochemical soil properties in a freshwater ecosystem. Ecological Modelling 201: Juang KW, Chen YS, Lee DY Using sequential indicator simulation to assess the uncertainty of delineating heavy-metal contaminated soils. Environmental Pollution 127: Li B, Li W, Shi Y Some distribution features of textural layers of regional soils in a fluviogenic plain. Acta Pedologica Sinica 35: Li YZ, Hu KL Simulation for the effect of clay layer on the transport of soil water and solutes under evaporation. Acta Pedologica Sinica 41: Li W. 2007a. Markov chain random fields for estimation of categorical variables. Mathematical Geology 39: Li W. 2007b. Transiograms for characterizing spatial variability of soil classes. Soil Science Society of America Journal 71: Li W, Li B, Shi Y Markov-chain simulation of soil textural profiles. Geoderma 92: Li W, Zhang C Application of transiograms to Markov chain simulation and spatial uncertainty assessment of land-cover classes. GIScience & Remote Sensing 42: Li W, Zhang C A random-path Markov chain algorithm for simulating categorical soil variables from random point samples. Soil Science Society of America Journal 71: Scorza RP Jr, Smelt JH, Boesten JJTI, Hendriks RFA, van der Zee SEATM Preferential flow of bromide, bentazon, and imidacloprid in a Dutch clay soil. Journal of Environmental Quality 33: Shi YC, Li YZ, Lu JW The Water and Salt Movement in Salinized Soils. Beijing Agricultural University Press: Beijing. Srivastava MR An overview of stochastic simulation. In Spatial Accuracy Assessment in Natural Resources and Environmental Sciences: Second Int. Symposium, Mowrer HT, Czaplewski RL, Hamre RH (eds.). US Dept. of Agriculture, Forest Service, General Technical Report RM-GTR-277: Fort Collins; Yuan JF Affect of clay interlayers on ascending movement of groundwater. Acta Pedologica Sinica 17: Ye WH A study on the relationship between solum structure patterns and growth of crops in farm-land of North China Plain. Acta Geographica Sinica 40: Zhang C, Li W Comparing a fixed-path Markov chain geostatistical algorithm with sequential indicator simulation in categorical variable simulation from regular samples. GIScience & Remote Sensing 44: Zhao Y, Shi X, Yu D, Wang H, Sun W Uncertainty assessment of spatial patterns of soil organic carbon density using sequential indicator simulation, a case study of Hebei province, China. Chemosphere 59: