Many spatial attributes are classified into mutually exclusive

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1 Published online May 16, 2007 Transiograms for Characterizing Satial Variability of Soil Classes Weidong Li* De. of Geograhy Kent State Univ. Kent, OH The characterization of comlex autocorrelations and interclass relationshis among soil classes call for effective satial measures. This study develoed a transition robability-based satial measure the transiogram for characterizing satial heterogeneity of discrete soil variables. The study delineated the theoretical foundations and fundamental roerties, and exlored the major features of transiograms as estimated using different methods and data tyes, as well as challenges in modeling exerimental transiograms. The secific objectives were to: (i) rovide a suitable satial measure for characterizing soil classes; (ii) introduce related knowledge for understanding satial variability of soil tyes described by transiograms; and (iii) suggest methods for estimating and modeling transiograms from sarse samle data. Case studies show that (i) cross-transiograms are normally asymmetric and unidirectionally irreversible, which make them more caable of heterogeneity characterization, (ii) idealized transiograms are smooth curves, of which most are exonential and some have a eak in the low-lag section close to the origin, (iii) real-data transiograms are comlex and usually have multile ranges and irregular eriodicities, which may be regarded as non-markovian roerties of the data that cannot be catured by idealized transiograms, and (iv) exerimental transiograms can be aroximately fitted using tyical mathematical models, but sohisticated models are needed to effectively fit comlex features. Transiograms may rovide a owerful tool for measuring and analyzing the satial heterogeneity of soil classes. SOIL PHYSICS Many satial attributes are classified into mutually exclusive multinomial classes (soil tye is one examle of multinomial categorical variables). For examle, in a mesoscale study area such as a watershed that stretches across dozens of square kilometers, there may exist dozens of different soil series, which may be further groued into several soil associations (Soil Conservation Service, 1962). Such soil series, articularly those within the same association, normally exhibit comlex interclass relationshis, such as cross-correlations, juxtaositions (i.e., sideby-side), and directional asymmetries in occurrence sequences. To describe the self-deendence (i.e., autocorrelation) of each soil class (e.g., soil series) and the interrelationshis between different classes, theoretically sound satial measures are needed. Markov transition robability matrices (TPMs) (or onedimensional first-order stationary Markov chain models) have a long history of use as satial measures to describe and simulate satial sequences of lithofacies in geology (e.g., Potter and Blakely, 1967; Krumbein, 1968; Carle and Fogg, 1997) and, more recently, soil layering in soil science (e.g., Li et al., 1999; Weigand et al., 2001). However, one-dimensional transition robability diagrams were rarely used or discussed before Carle and Fogg (1996), who Soil Sci. Soc. Am. J. 71: doi: /sssaj Received 26 Ar *Corresonding author (weidong6616@yahoo.com). Soil Science Society of America 677 S. Segoe Rd. Madison WI USA All rights reserved. No art of this eriodical may be reroduced or transmitted in any form or by any means, electronic or mechanical, including hotocoying, recording, or any information storage and retrieval system, without ermission in writing from the ublisher. Permission for rinting and for rerinting the material contained herein has been obtained by the ublisher. Abbreviations: TPM, transition robability matrix. suggested using transition robability diagrams in indicator kriging simulations of geological facies so that directional asymmetry and juxtaosition tendencies may be incororated. Most of the studies to date have rimarily used idealized transition robability diagrams that is, they calculated transition robability diagrams from one-ste transition robabilities based on a first-order stationary Markovian assumtion. For examle, Schwarzacher (1969) used Markov transition robability diagrams calculated from TPMs to describe the vertical change of lithofacies. Similarly, Weissmann and Fogg (1999) used such diagrams, derived from one-ste transition robabilities obtained with the transition rate method in the framework of transition-robability-based indicator geostatistics, for hydrofacies modeling. Some discussions about the roerties of idealized transition robability diagrams can be found in Carle and Fogg (1996, 1997) and Ritzi (2000). In contrast to traditional variograms, however, transition robability diagrams have rarely been estimated from samled oint data and also have not been effectively develoed as indeendent satial measures for heterogeneity characterization. The major reason may be that Markov chains have not been develoed into widely alicable multidimensional geostatistical tools for conditional simulations on sarse oint samles. Note that although Markov random field methods (including Markov mesh models) have been used in the geosciences for multidimensional simulations (e.g., Norberg et al., 2002; Wu et al., 2004), they did not involve interactions of nonadjacent neighbors (i.e., ixels), which are measured by multiste transition robabilities. Li (2006) recently roosed the transiogram concet and the use of this term for transition robability diagrams, suggesting that the transiogram might be used as an indeendent satial relationshi measure for categorical data. There are two reasons for roosing such a satial measure. First, there is a need for a hysically meaningful, readily managed, and visual measure of the comlex satial intra-class and interclass relationshis among multinomial SSSAJ: Volume 71: Number 3 May June

2 THEORY Transiogram The Transiogram Concet A brief introduction of the transiogram concet has been rovided in Li (2006), and is summarized here. A transiogram is defined as a onedimensional transition robability (i.e., two-oint conditional robability) function across the distance lag h: ( h) = Pr [ z( x+ h) = j z( x) = i] [1] Fig. 1. Illustration of different tyes of transiograms: (a) idealized auto-transiogram; (b) idealized cross-transiogram; (c) exhaustive auto-transiogram; (d) exhaustive cross-transiogram; (e) exerimental auto-transiogram; (f) exerimental crosstransiogram. Exerimental transiograms are estimated from sarse samles and therefore they are comosed of discontinuous oints; (i,j) reresents a transition robability from class i to class j where h is the satial lag searation vector. classes. Although TPMs may reresent interclass relationshis, they only measure one-ste satial relationshis and fail to quantify the comlex satial relationshis among classes across a number of satial stes. Thus, transiograms may reveal more information than a TPM does. Transiograms directly estimated from data are not limited by the first-order stationary Markovian assumtion and manifest the effect of satial-lag-deendent interactions among observed data, thus they contain a wealth of features that reveal satial heterogeneity of multinomial classes. Second, Markov chain conditional simulation models (Li, 2007b) need a more owerful satial measure such that they are more flexible for working with various tyes of data (e.g., sarse samle oint data) and incororating comlex satial heterogeneity into simulations of multinomial classes. A comrehensive assessment of transiograms is resented with resect to (i) basic roerties and tyes; (ii) theoretical background of idealized transiograms; (iii) comlex features of exerimental transiograms estimated from multinomial satial data (i.e., a mesoscale soil series ma); and (iv) modeling of exerimental transiograms estimated from sarse oint samles. The rimary objective was to introduce a suitable satial measure for characterizing the comlex satial variability of discrete soil variables, articularly the interclass relationshis among multinomial soil classes. Note that the word class is used as a general term for reresenting a category and the word state refers to the status of a Markov chain at a location. where (h) is the transition robability of random variable z from class i to class j. As lag h increases from the origin, (h) generates a diagram, referred to as the transiogram. Therefore, (h) is used to denote a transiogram, with the understanding that ii (h) denotes the auto-transiogram of class i and (h) (i j) notates the cross-transiogram from class i to class j. Here h may be unidirectional (e.g., west to east) or non-unidirectional (e.g., multidirectional or omnidirectional). In a discrete sace, h can be reresented by the number of satial stes (i.e., ixels). A transition satial ste means that the satial rocess (e.g., Markov chain) moves from one ixel to the next adjacent ixel, that is, the distance of a ixel size. In a continuous sace, h is reresented by a continuous distance measure (e.g., meters). Because cross-transition robabilities are normally asymmetric, i and j are not interchangeable in (h); for convenience, class i is called the head class, and class j is called the tail class. In Eq. [1], Z is assumed to be a second-order stationary satial random variable, that is, (h) deends only on the lag h and not on the secific location x. This assumtion, and the ergodic hyothesis (i.e., that satial statistics are equivalent to ensemble statistics), allows estimation of transiograms directly from satial data airs in a satial data set. Note that this framework and assumtion are similar to that underlying variogram estimation, and as such refers to the model, since there is no hysical basis on which the hyothesis can be tested with resect to the data er se (e.g., see Deutsch and Journel, 1998, ; Chilès and Delfiner, 1999, ). Therefore, a transiogram also reresents a two-oint satial continuity/discontinuity measure, similar to an indicator variogram. Auto-transiograms reresent self-deendence (i.e., autocorrelations) of single classes; however, cross-transiograms reresent interclass relationshis (including cross-correlations, juxtaositions, and directional asymmetries). Tyes of Transiograms Transiograms may be divided into two tyes based on how they are estimated: idealized transiograms and real-data (or observed) transiograms. The former refers to transiograms estimated using one-ste transition robabilities based on the first-order stationary Markovian assumtion. The latter refers to transiograms directly estimated from real data sets. Real-data transiograms may be further slit into two subtyes based on the tye of data sets used: exhaustive transiograms and exerimental transiograms. Exhaustive transiograms refer to those transiograms directly estimated from mas or images where data are exhaustive. Exerimental transiograms refer to those directly estimated from sarsely samled data. Figure 1 shows examles of different tyes of auto- and cross-transiograms. The first row (i.e., Fig. 1a and 1b) is a air of idealized autoand cross-transiograms. They are idealized because they are calculated from a one-ste TPM, which imlicitly imoses the first-order stationary Markovian assumtion (i.e., that data are satially stationary, aeriodic, and deendent on only immediate neighbors). Idealized transiograms are tyified by smooth curves with stable sills and aarent correlation ranges. The second row of Fig. 1 shows a air of exhaustive auto- and cross-transiograms (see Fig. 1c and 1d), which obviously have more features and seldom show stable sills or clear correlation ranges. Here the term features refers to the aarent eaks and troughs as well as multile ranges shown 882 SSSAJ: Volume 71: Number 3 May June 2007

3 on a transiogram. These features usually should not be regarded as data noise. The third row of Fig. 1 rovides a air of exerimental auto- and cross-transiograms (see Fig. 1e and 1f). Each is comosed of a grou of discontinuous oints. Because real world data are usually strongly nonstationary and non-first-order Markovian, real-data transiograms have far more comlex features than idealized transiograms have. Basic Proerties Both real-data and idealized transiograms, as transition robability diagrams, have the following basic roerties: 1. Non-negative. As robabilities, transiograms are nonnegative, that is, ( h ) 0 [2] 2. Sum to 1. As transition robabilities, values of transiograms involving the same head class sum to 1 at any estimated lag h, that is, n j= 1 ( h) = 1 [3] where n is the number of classes, i is the head class, and j is the tail class. 3. No nugget effect. For mutually exclusive classes, theoretically and hysically, transiograms should not have a nugget effect, that is, we have (0) = 0, i j [4] for cross-transiograms and ii (0) = 1 [5] for auto-transiograms, which must always hold true. Therefore, auto- and cross-transiograms should include the oints (0,1) and (0,0), resectively, in their curves (see Fig. 1). 4. Asymmetry. Transiograms are tyically asymmetric, that is ( h) ( h) ji [6] The asymmetry roerty of transiograms rovides more information about satial juxtaositional relationshis between classes. But if transiograms are estimated bidirectionally or omnidirectionally, the following relationshi holds: j ( h) = ji ( h) i [7] where i and j are the roortions of class i and class j, resectively. This means that if we know ji (h) and the roortions of the two related classes in a study area (or a data set), we can directly obtain (h). 5. Irreversibility. If transiograms are estimated unidirectionally, they are tyically irreversible; that is, ( h) ( h) [8] The irreversibility roerty of unidirectional transiograms reflects the directional asymmetry of satial atterns of classes. The first three roerties also constitute the set of constraint conditions of transiogram modeling for Markov chain simulation. The last two roerties are secial features of transiograms that other widely used satial measures (e.g., variograms) may not have. Idealized Transiograms Although idealized transiograms are not a true reflection of realworld data or henomena, they have theoretical and alication value. For examle, they can cature the basic correlation characteristics of classes and their roerties are articularly helful to interreting realdata transiograms and modeling exerimental transiograms. Idealized transiograms have been used exlicitly (Weissmann and Fogg, 1999) or imlicitly (through multiste transition robabilities calculated from one-ste TPMs; Li et al., 2005) in geostatistical conditional simulations, with one-ste transition robabilities estimated from borehole data or survey line data. For the sake of simlicity and efficiency, idealized transiograms may be used as simlified models when one-ste transition robabilities are available. Therefore, understanding idealized transiograms is crucial, articularly for understanding real-data transiograms. There are two simle methods to obtain idealized transiograms from one-ste transition robabilities. The first uses one-ste TPMs, while the second uses the transition rate matrix method, to derive transiograms (Carle and Fogg, 1997). Both methods are based on the first-order stationary Markovian assumtion. The First-Order Markovian Assumtion The first-order Markovian assumtion states that the conditional distribution of a future state, given ast states and the resent state, is indeendent of ast history and deends only on the resent state. The first-order Markovian assumtion (or roerty) can be exressed as Pr [ zm ( + 1) = kzm ( ) = lzm, ( 1) = r,..., z(0) = s] = Pr [ zm ( + 1) = kzm ( ) = l] = tlk [9] where z(m + 1),, z(0) reresent a satial state sequence of the random variable z; k, l, r, and s reresent states of the random variable in a state sace (1, 2,, n); and t lk reresents the transition robability from state l to state k. A first-order stationary Markov chain can be described by a TPM. For examle, a three-state first-order Markov chain can be fully described by a TPM P with diagonal entries: P t t t = t21 t22 t 23 t t t [10] where each entry of P reresents a one-ste transition robability between different states or within the same state with a fixed ste length. In a discrete sace, Eq. [10] is commonly used. Sometimes, a first-order stationary Markov chain is exressed by a TPM without diagonal entries. Thus, the three-state first-order Markov chain given in Eq. [10] can be rewritten as π12 π 13 Q = π 21 π 23 π31 π32 [11] where each entry of Q reresents a transition robability between different states. The Markov chain described by Eq. [11] is called an embedded Markov chain, because the satial continuity of a state has to SSSAJ: Volume 71: Number 3 May June

4 be decided by a secific robability distribution and embedded into a Markov chain simulation (Potter and Blakely, 1967; Li et al., 1999). Idealized Transiograms Derived from Transition Probability Matrices Under the first-order stationary Markovian assumtion, if n and m reresent two distances in a satial sequence and m < n, the firstorder Markovian roerty can be formulated as tlk( n) = tlj ( m) t jk( n m) j [12] where t lk (n) denotes the transition robability from state l to state k across a distance n. This is the Chaman Kolmogorov equation for homogeneous (i.e., stationary) rocesses (Agterberg, 1974,. 420). If transition robabilities are arranged in the matrix format, Eq. [12] reduces to P( n) = P( m) P ( n m) [13] where P(n) reresents a TPM with the distance n. In a discrete sace, m and n can be numbers of satial stes. Letting m = 1, we have P( n) = P(1) P ( n 1) [14] Successively alying Eq. [14], we get P( n ) = P (1) n [15] where P(n) reresents an n-ste TPM and P(1) a one-ste TPM. As n increases, the entries in P(n) gradually become stable and reach the stationary robabilities that is, different rows in P(n) will have the same values, with each entry t lk (n) being equal to the roortion of the tail class k. Equation [15] reresents the method to derive multiste transition robabilities from a one-ste TPM. As n increases, the calculated multiste transition robabilities t lk (n) form a continuous diagram an idealized transiogram lk (h). If the one-ste TPM is estimated from sufficient data and the study area is sufficiently large comared with the normal olygon size, the derived transiogram will have a sill equal to the roortion of the tail class k. Therefore, Eq. [15] is one method for deriving idealized transiograms. The rerequisite for this method is that we must have reliable oneste TPMs. The merits of this method are its simlicity and efficiency. Idealized Transiograms Derived from Transition Rates A general solution satisfying the Chaman Kolmogorov equation (Agterberg, 1974,. 457) is Ph ( ) = ex( Rh ) [16] where R is a transition rate matrix (Krumbein, 1968), which is indeendent of the lag h. For an m-state sace, r11 r 1m R = rm1 r mm [17] where entry r lk of R reresents the rate of change er unit length (ste) from class l to class k. Off-diagonal entries of R can be calculated from transition robabilities π lk of an embedded Markov chain and mean length L (for soil classes, a mean length refers to the mean olygon l size [i.e., length or width] of a class, and is generally called the mean boundary sacing) of a class as r = π L l k [18] lk lk l Analogously, the diagonal entries of R can be calculated by r kk = 1 L k [19] For more details on this method, see Carle and Fogg (1997) and Weissmann and Fogg (1999). If R is known, with increasing lag h, each entry of the TPM P(h) makes a continuous curve an idealized transiogram. Similarly, in a discrete sace, h can be the number of satial stes. If h = 1, Eq. [16] reduces to P(1) = ex( R ) [20] Thus, for n discrete stes, we have P( n) = ex( Rn) = P (1) n [21] Therefore, in a discrete sace, the transition rate method and the one-ste TPM method for transiogram estimation are theoretically identical. Comaring the two methods, it can be seen that the transition rate method does not require estimation of the diagonal transition robabilities, but the mean boundary sacing of each class must be known. When data are few and exert knowledge has to be used, it may be easier to estimate aroximate mean boundary sacings than to estimate one-ste self-transition robabilities. When detailed line data or exhaustive data (e.g., a training image) are available, both one-ste self-transition robabilities and mean boundary sacings can be easily estimated. Major benefits of the transition rate method, according to Carle and Fogg (1997), are continuity in functional reresentation of the Markov chain model and flexibility afforded in choosing samling intervals. Proerties of Idealized Transiograms Besides the basic roerties shared by all transiograms, idealized transiograms have the following roerties, which may not hold for real-data transiograms. An idealized auto-transiogram ii (h) starts from oint (0,1) and with increasing h gradually decreases to a stable value, the sill C i (Fig. 1a). This sill, in a sufficiently large area, is equal to the roortion i of the class i in that area. Therefore, we have lim ( h) = C = h ii i i [22] The lag distance at which the auto-transiogram aroaches its sill is called the autocorrelation range, denoted by a i. The autocorrelation range refers to the range within which observations of the same class are correlated. If the transiogram aroaches its sill asymtotically, the effective range is set equal to the distance where the transiogram reaches 95% of its sill, similar to the effective range definition for asymtotic variogram models (Deutsch and Journel, 1998,. 25). While the autocorrelation range reresents the distance of selfdeendence of the class i, it does not directly tell the mean size of olygons of the class. If we draw a tangent of the auto-transiogram from 884 SSSAJ: Volume 71: Number 3 May June 2007

5 oint (0,1) to the x axis, the lag h where the tangent crosses the x axis is equal to the mean of the olygon size of the class, denoted by L, i.e., i ii ( h) h h = 0 = 1 Li [23] (see Fig. 1a) (Carle and Fogg, 1996). An idealized cross-transiogram (h) starts from oint (0,0) and with increasing h gradually increases to a stable value, the sill C (Fig. 1b). The sill, for a sufficiently large area, is equal to the roortion j of the tail class j. That is, we have lim ( h) = C = h j [24] Similarly, we define the cross-correlation range a as the lag distance at which the sill is reached (or aroached asymtotically). The cross-correlation range reresents the distance within which the two involved classes are interrelated. It should be noted that if one-ste transition robabilities are estimated from a small area, sills of idealized transiograms, articularly idealized unidirectional cross-transiograms, may not be equal to roortions of corresonding tail classes in the area because boundary effects become substantial in small areas. The term boundary effects in this context refers to the fact that a class may have smaller transition frequencies in relation to other classes than those corresonding to the whole field if the class is more frequently adjacent to the boundaries of the study area, because boundary olygons are incomlete and transitions to other classes beyond boundaries are unknown (Li, 2006). In addition, because of the roerties of idealized transiograms given in Eq. [22] and [24], aarently if class i and class j have different roortions, idealized cross-transiograms (h) and ji (h) have different sills. Actually, even if j = i, (h) and ji (h) will generally not be equal at any arbitrary lag h less than their resective correlation ranges, because they may not have the same curve shae due to the different juxtaosition characteristics of classes. These exlain why idealized cross-transiograms are asymmetric. Real-Data Transiograms The Non-Markovian Effect The term Markovian roerty usually refers to the first-order Markovian roerty and a Markov chain tyically means a firstorder Markov chain. So-called high-order Markov chains are actually Markovian reresentations of discrete non-markovian rocesses. We will use the term non-markovian, rather than the term high-order Markovian, to reresent the non-first-order Markovian roerty of a discrete rocess. The non-markovian roerty of data has seldom been considered in revious Markov chain models in the geosciences. The first-order Markovian roerty is an assumtion for the convenience of creating first-order Markov chain models, not a roerty of the data. Real-world data sets tyically have many features that are not consistent with a Markovian assumtion. These non-markovian roerties imly that the current satial state not only deends on its adjacent state but also deends on some nonadjacent states in a satial state sequence. If a real-world data set is strongly non-markovian, it may not be roer to use a first-order Markov chain model to deal with the data set. Real-data transiograms have nothing to do with the first-order Markovian assumtion. They are a direct reflection of the satial variation characteristics of the original data. If transiograms from a real data set are basically identical with the corresonding idealized transiograms, we may say the data set is consistent with a first-order Markovian assumtion. Otherwise, the extra features manifested by the real-data transiograms are a reflection of the non-markovian roerties of the data. Therefore, we may generally call such features the non-markovian effect because they cannot be catured with a first-order Markov chain model. Real-data transiograms aarently rovide a way to quantitatively and visually reresent the non-markovian roerty of data, which reveal secial features not reresented by idealized transiograms. When real-data transiograms or well-fitted transiogram models are used in simulations, the non-markovian effect of a data set is incororated into the simulation. Thus, constraints imosed by firstorder Markov chain models may be relaxed to some extent. Exhaustive Transiograms An exhaustive soil area-class ma is essentially a simlified reresentation of the real soil landscae in a survey area based on field survey data, soil taxonomy, and exert interretation. It is imossible to obtain a 100% accurate soil ma, as an exhaustive survey is not feasible. The information contained in reliable soil survey mas such as those rovided by the USDA are the result of ractical exerience gained through long-term soil survey efforts and reresent the collective knowledge of exerienced survey teams under suort of the government. These soil mas contain considerable knowledge on satial variability of soil tyes and their satial relationshis as discerned by exerienced soil surveyors. It is thus aroriate to regard a reliable human-delineated soil ma as a good reresentation of soil class satial distributions (i.e., to assume the soil ma bears the satial statistics [e.g., transiograms] of the real soil distribution in the maed area). Thus, transiograms estimated from a reliable soil ma may reveal many characteristics of soil satial distributions that are not readily aarent from a visual examination of the soil ma. Exhaustive transiograms may be used (i) as a data-mining tool to extract knowledge on satial correlations and interrelationshis of soil classes from the existing digital soil mas, as the extracted knowledge is helful for users to understand and interret the maed soils; and (ii) to rovide direct transiogram models or exert knowledge for estimating transiogram models for soils in other areas that bear similarities to the maed region. Exerimental Transiograms Samles usually account for only a very small ortion of the whole study area and yet often reresent the major source of information on which a soil ma is delineated. Therefore, the information conveyed by the samle data is most recious. Because of the costs associated with collection, and the subsequent sarsity of data, there is no doubt that exerimental transiograms estimated from such samling will be a series of discontinuous oints and may not reveal nearly as abundant information as is rovided in exhaustive transiograms. Even so, exerimental transiograms can be used to characterize soil satial variability and develo continuous transiogram models for stochastic simulation, similar to the way traditional exerimental variograms have been used. Modeling of Exerimental Transiograms Accurate modeling of transiograms is crucial to fully exloit the utility of transiograms for both soil heterogeneity characterization and Markov chain simulation. For heterogeneity characterization, following the convention in traditional geostatistical alications, the goal is to choose a suitable tye of function model and to secify (statistically fit) model arameters to the samle data, thus roviding a characterization of satial heterogeneity in terms of a few simle arameters (e.g., such as the range, sill, wavelength, and model tye, for instance, Gaussian, exonential, or sherical). For Markov chain simulation of soil classes, a crucial ste is to arameterize a continuous transiogram model that catures as many major features of SSSAJ: Volume 71: Number 3 May June

6 Table 1. Some basic mathematical models for transiograms. Model tye Function For auto-transiograms Linear ii (h) = 1 (1 i )h/a i ; h < a i ii (h) = i ; h a i Sherical ii (h) = 1 (1 i )[1.5(h/a i ) 0.5(h/a i ) 3 ]; h < a i ii (h) = i ; h a i Exonential ii (h) = 1 (1 i )[ 1 ex( 3h/a i )] Gaussian ii (h) = 1 (1 i ){1 ex[ (3h/a i ) 2 ]} Cosine exonential ii (h) = 1 (1 i )[1 ex( 3h/a i )cos(bh)] Cosine Gaussian ii (h) = 1 (1 i )[1 ex( 3h 2 /a 2 i )cos(bh)] For cross-transiograms (i j) Linear (h) = j h/a ; h < a (h) = j ; h a Sherical (h) = j [1.5(h/a ) 0.5(h/a ) 3 ]; h < a (h) = j ; h a Exonential (h) = j [1 ex( 3h/a )] Gaussian (h) = j {1 ex[ (3h/a ) 2 ]} Cosine exonential (h) = j [1 ex( 3h/a )cos(bh)] Cosine Gaussian (h) = j [1 ex( 3h 2 /a 2 )cos(bh)] a i = auto-correlation range; a = cross-correlation range; i = roortion of class i; b = 2π/λ, where λ is the wavelength of the cosine function. The linear, sherical, exonential, and Gaussian models for auto-transiograms were also rovided in Ritzi (2000). the exerimental transiograms as ossible. This allows the soil satial heterogeneity conveyed by the samle data to be effectively incororated into the stochastic simulation, thus ensuring that such simulations more accurately reflect the samle data. When samle data are insufficient, exerimental transiograms may not convey reliable information (e.g., they may exhibit surious fluctuations). In this situation, exert knowledge regarding lausible values for correlation ranges, sills, and transiogram model tye are required to build a suitable model. Basic Transiogram Models One may use any suitable mathematical model to reresent an exerimental transiogram, such as some of the classic mathematical models that are widely used in modeling auto-variograms (Deutsch and Journel, 1998,. 25), which may be readily adated to model auto- and cross-transiograms. Such suitable models include the exonential, sherical, Gaussian, and linear mathematical (theoretical) models. The observed characteristics of exerimental auto- and crosstransiograms can be used to develo anisotroic models that satisfy the theoretical constraints imlicit in a transiogram. Recall that sills of idealized transiograms are theoretically equal to the roortions of their resective tail classes. Although exerimental transiograms are normally more comlex in shae than idealized transiograms and may thus not manifest a stable sill, the magnitudes are still a reflection of the tail class roortions. Therefore, setting the sill of a transiogram model to the roortion of the tail class is aroriate. Ritzi (2000) rovided some basic mathematical models for auto-transition robabilities (i.e., auto-transiograms) with the model sills being set equal to the class roortions. The shae of a cross-transiogram with increasing lag is similar to that of an auto-variogram. Thus, we can simly set sills of mathematical models to the roortions of the corresonding tail classes for modeling cross-transiograms (Li, 2007a). The exonential model for cross-transiograms can thus be given as ( h) = j 1 ex( 3 h a ) [25] Since the maximum value of an auto-transiogram occurs at h = 0, the model sills are rescribed differently. The corresonding exonential model for auto-transiograms can be given as [ ] ( h) = 1 (1 ) 1 ex( 3 h a ) ii i i [26] (Ritzi, 2000). Equation [26] ensures that the sill of the auto-transiogram model is equal to the roortion of the modeled class. Sills of other models can be set similarly. Basic mathematical models for auto- and cross-transiograms are given in Table 1. Note that the nugget model used for modeling variograms is discarded here because transiograms for multinomial classes should not have a nugget effect. In many cases, these basic models are sufficient to fit the general trend of an exerimental transiogram. They are not caable of modeling finer details of exerimental transiograms, however, articularly a crosstransiogram that exhibits a eak or significant non-markovian effects. When a study area is small and the sill of an exerimental transiogram obviously deviates from the theoretically exected sill (i.e., the roortion of the tail class) because of the boundary effect, setting the sill of the transiogram model to the roortion of the tail class may result in a fit to the exerimental transiogram that aears inaccurate; however, the transiogram model thus derived should be a better reflection of the satial heterogeneity. Ma and Jones (2001) suggested using multilicative-comosite models to fit hole-effect auto-variograms, or at least to fit the first eak if a holeeffect auto-variogram is irregular. The cosine exonential model and the cosine Gaussian model used by Ma and Jones (2001) are adated here in the same way for modeling eriodicities of transiograms (see Table 1). These two models attenuate the sinusoidal amlitude with increasing lag; however, the cosine Gaussian model has higher eak heights than the cosine exonential model. In many situations, one may find these hole-effect models still do not adequately fit exerimental transiograms of multinomial classes, because the latter are too comlex and irregular. Nested models constructed from these basic models may be useful for some comlex transiograms. Transiogram Model Constraints for Stochastic Simulation If the urose in constructing transiogram models is solely to characterize soil satial variability (e.g., with no attemt to create a stochastic simulation), there is no need to strictly adhere to the theoretical constraints. When creating transiogram models for use in Markov chain simulations, however, the aforementioned three constraint conditions (Eq. [2 4]) must be satisfied to correctly erform simulations. The requirement that the transiogram model is void of a nugget effect (i.e., Eq. [4]) can be satisfied by including the oints at 0 lag [i.e., oint (0,1) for auto-transiograms and oint (0,0) for cross-transiograms] into exerimental transiograms and honoring these values in fitted models. One otion for the transiogram models fitting a subset of exerimental transiograms headed by the same class to satisfy the summing-to-1 constraint (i.e., Eq. [3]) is to infer one of them from others by the following equation: ( h) 1 ( h) ik n = j= 1 j k [27] where n is the number of classes, i is the head class, and ik (h) is the inferred model (Li, 2007a). 886 SSSAJ: Volume 71: Number 3 May June 2007

7 To guarantee that ik (h) is non-negative and consistent with the exerimental data, the model-fitting rocess of other transiograms headed by the same class i may need reetitive tuning. Normally, if the sill of every transiogram model is set to the roortion of the corresonding tail class and the section close to the origin is aroriately fitted by choosing a suitable model tye and range, ik (h) will be consistent with the exerimental data. The non-negative condition is not a concern for the other exerimental transiograms fitted by mathematical models because exerimental transiograms are never negative, which means models fitting them will not be negative. CASE STUDIES Data Sets Case studies here aim to demonstrate the features of different tyes of soil class transiograms and the corresonding modeling of those exerimental transiograms. An exhaustive data set was used for estimating idealized transiograms and exhaustive real-data transiograms. This data set is a large digital soil series ma, which covers a art of Iowa County, Wisconsin. The county soil ma was downloaded from the NRCS Soil Survey Geograhic Database (SoilDataMart.nrcs.usda.gov/). The entire county soil ma is too large to dislay. Therefore, the county was divided into several subareas and only one subarea was chosen for the uroses of this study. The maing unit in the original ma is the soil hase, which is a classification level for the urose of soil management. The soil hases were merged into corresonding soil series, a higher classification level that is based on edogenic roerties. The clied soil series ma of the chosen subarea is given in the left side of Fig. 2, which has an extent of about 9.6 by 9.6 km 2. For the convenience of data analysis, the ma was discretized into a raster image of about 500 by 500 ixels with a ixel size of 20 by 20 m 2. The soil ma contains 48 soil series, thus 48 by 48 transiograms may be calculated for the study area in each direction. In this study, only some of transiograms along axis directions were estimated and analyzed. Among these soil tyes, the names, roortions, and olygon information for 14 of the 48 are rovided in Table 2. These data indicate that some soil tyes (e.g., 6, 11, and 13) have smaller olygons and some tyes (e.g., 4 and 5) have larger olygons and that each soil tye accounts for a different area roortion. From Fig. 2 it can be seen that Soil Tye 4 dominates the study area and the right boundary. The olygon data shown for these soil tyes may be indicative of the more general comlexity of olygon shaes for soils. Note that the soil tye numbers listed in Table 2 are identical with those used in the following related figures of transiograms. To estimate exerimental transiograms, two sarse data sets one relatively large and the other considerably smaller were used. The larger sarse data set was obtained by a random samling, with n = , of the large raster soil ma (Fig. 2, right). The Fig. 2. A large raster soil series ma (art of Iowa County, Wisconsin) and a corresonding random samle data set obtained by simulated samling of the ma. The ma has a total of 48 soil series; of these, the hues of the 14 most dominant soil series are given (their corresonding names are given in Table 2). small, sarse data set was obtained using a grid-based samling with 136 oints (Fig. 3, bottom), from a small soil ma with seven soil tyes (Fig. 3, to). The seven soil tyes of the small soil ma are distributed relatively uniformly within a 4.0- by 1.7-km 2 area, which was discretized into an 80 by 34 raster grid with a ixel size of 50 by 50 m 2 (Fig. 3) (also see Li et al. [2005]). The urose of using this small data set with a small number of soil tyes is to demonstrate how to jointly model exerimental transiograms for Markov chain simulation, because transiogram models for simulation have to be estimated one subset at a time so that they can meet the constraint conditions. Here a subset contains all transiograms headed by the same class. The reason for using regular data is because they are more efficient for estimating reliable exerimental transiograms; otherwise the small data set of 136 oints would be insufficient to estimate the 49 (i.e., 7 by 7) exerimental auto- and cross-transiograms. The secific Table 2. Names, area roortions, and some olygon characteristics of the first 14 of the 48 soil series (or land tyes) aearing in the soil ma shown in Fig. 2. No. Name Proortion Polygons Average area of olygons Average erimeter of olygons no. m 2 m 1 Northfield, sandy loam Dubuque, stony silt loam Fayette, silt loam, valleys Dubuque, dee Huntsville, silt loam Chaseburg, silt loam Orion, silt loam Ettrick, silt loam Tell, silt loam Lawson, silt loam Northfield, loam Water Gale, silt loam Dubuque, ordinary SSSAJ: Volume 71: Number 3 May June

8 Fig. 3. A small soil ma with seven classes and a corresonding regular samle data set (n = 136) from this ma. names for these soil tyes are not relevant here, so the numbers (i.e., 1, 2, 3,, 7) are used instead to reresent the seven different soil tyes. Estimation of Transiograms from Data Transiograms can be estimated by counting transition frequencies of soil tyes in a data set with different lags. The equation is given as follows: ( h) = F ( h) F ( h) j= 1 n [28] where F (h) reresents the frequency of transitions from class i to class j at the lag h, and n is the total number of soil tyes. If directional asymmetry and anisotroy are not considered, transition frequencies in different directions may be ooled together to get multidirectional or omnidirectional transiograms. Fig. 4. Idealized transiograms calculated from the one-ste transition robability matrix (estimated from the large soil ma (shown in Fig. 2) in the west-to-east direction): (a) auto-transiograms; (b), (c), and (d) cross-transiograms. The notation like (i,j) refers to the transiogram i,j (h), reresenting a transition robability from class i to class j where h is the satial lag searation vector. The scale along the h axis is the number of grid units. When transition frequencies are counted from exhaustive data sets, the lag h may be set exactly with a zero tolerance width. Thus, estimated transiograms are continuous functions of lag searation, which may be conveniently exressed as the corresonding number of ixels. As such, they may cature considerable fine-scale resolution of satial variability structures among and between the several soil tyes. When data are sarse, however, the number of observed transitions at certain lags will robably not be sufficient to rovide an aroriate robability estimate (e.g., estimated transition robabilities may be zero or unrealistically high). Therefore, a tolerance width should be used when estimating exerimental transiograms from sarse data. A tolerance width Δh (e.g., 30 m) means that all transitions within a lag interval from (h Δh) to (h + Δh) (e.g., 400 to 460 m) are counted into the frequency of transitions at one lag h (e.g., 430 m). Thus, only transition robabilities at a limited number of lags are estimated. Consequently, exerimental transiograms are comosed of a small number of discrete, tyically equally saced oints, similar to the convention followed for variogram estimation with sarse data. In this study, because raster data were used to estimate transiograms, the ixel number was used as a relative distance to reresent the lag h. RESULTS AND DISCUSSION Idealized Transiograms To examine the features of idealized transiograms, one-ste TPMs were estimated from the large soil ma in different directions and transiograms were derived from these TPMs using Eq. [15]. Figure 4 shows some of these idealized transiograms in the west-to-east direction, all of which are smooth curves. Among these, the auto-transiograms are exonential, each with a generally unique correlation range. The sill values for these idealized transiograms are in accordance with the roortions of the corresonding soil tyes given in Table 2. The shaes of most cross-transiograms are also reasonably consistent with an exonential model, although some of them, such as 1,13 (h) and 1,11 (h) (see Fig. 4d) are not monotonically increasing, with a eak, or maximum value, occurring before reaching the sill value. This eak is a tyical indication of juxtaosition relationshis, which results when one class frequently occurs in association with another class. Examining the transiogram sills, it is aarent that these cross-transiograms also have sills that are identical to the roortions of the corresonding tail classes. For examle, the sill of 1,14 (h) is , which is equal to the roortion of Soil Tye 14 (see Table 2). Idealized cross-transiograms are asymmetric and, if estimated unidirectionally (i.e., on the basis of unidirectional TPMs), are also irreversible. These characteristics can be seen from Fig. 5, which shows idealized cross-transiograms between Class 1 and Class 6 in the west-to-east direction and the east-to-west direction. Asymmetry indicated by the discreancy between 1,6 (h) and 6,1 (h) is clear because of their different sills. But the discreancy between cross-transiograms 1,6 (h) and 1,6 ( h) is evident only at short lag searations, because these two cross-transiograms have equal sills corresonding to the roortion of the same Tail Class 6. Notice that the slight difference in the sills of 6,1 (h) and 6,1 ( h) shown here (see Fig. 5) results from the boundary effect of the limited 888 SSSAJ: Volume 71: Number 3 May June 2007

9 study area; theoretically, this effect would not occur if the study area were infinitely large. It also should be noted that the idealized transiograms shown here are similar to the continuous transition robability models derived using the transition rate matrix method as shown in Carle and Fogg (1997) and Weissmann and Fogg (1999), since they are all based on the first-order Markovian assumtion and are essentially the equivalent reresentations in a discrete sace. Although not able to cature non-markovian effects of satial data, transiograms derived using one-ste TPMs rovide a simle and efficient way for transiogram estimation and modeling. This aroach may save considerable time and effort if one-ste TPMs are available (e.g., from survey line data or training images). A requirement for using this aroach is that one-ste TPMs estimated from data must be reliable; that is, they should reflect the transition relationshis among different classes. If survey line data or large training mas are not available, or the study area is too small comared with the arcel sizes of the soil tyes, it may be difficult to obtain reliable one-ste TPMs. Exhaustive Transiograms To demonstrate tyical features of exhaustive transiograms, the latter were estimated from the large soil ma using a tolerance width of zero. Both unidirectional and four-directional transiograms (i.e., comuted as the average of the transiograms calculated in the four cardinal directions east, west, south, and north) were estimated. These transiograms rovide an oortunity to gain general understanding and insight into the satial structure of the variability in soil tyes, and more imortantly, it demonstrates tyical features of real transiograms and the satial features that cannot be catured with a first-order Markovian characterization. Shae and Periodicity Figure 6 shows exhaustive real-data auto- and cross-transiograms estimated from the large soil ma in an easterly direction. It can be seen that all auto-transiograms shown here can be reasonably described using exonential models. Note that the auto-transiogram 5,5 (h) has a longer auto-correlation range than the other soil tyes, which is a consequence of its large average olygon size (see Table 2). Note also that the fluctuations in auto-transiograms are relatively weak comared with those in cross-transiograms. Aarent fluctuations can be seen in crosstransiograms, esecially with an exanded h axis (see Fig. 6d). These cross-transiograms are aroximately consistent with exonential or sherical models for lags less than the first eak, although the satial atterns and eriodicities are tyically distinct for each cross-transiogram. A eriodic attern in transition frequencies indicates characteristic scales in satial atterns of soil tyes, such as occurs in swell swale landscae sequences (Haws et al., 2004) (i.e., arcels of soil tyes aear in a sace with a satial regularity or rhythm). Four-directional (not truly omnidirectional) transiograms are dislayed in Fig. 7. As a consequence of the ooling across directions, these average curves are considerably smoother than Fig. 5. Illustration of the asymmetry and irreversibility of idealized transiograms. The transiograms (1,6) and (6,1) [referring to 1,6 (h) and 6,1 (h), resectively] are estimated along the west-to-east direction, whereas the transiograms (1,6) and (6,1) [referring to 1,6 ( h) and 6,1 ( h), resectively] are estimated along the east-to-west direction. their unidirectional counterarts. As a consequence of the averaging rocess, the eriodicities dislayed in the four-directional crosstransiograms generally differ from the corresonding unidirectional cross-transiograms. Four-directional or omnidirectional transiograms may be used in Markov chain simulations when it is deemed that anisotroies and directional asymmetries are not of interest. Correlation Range and Non-Markovian Effect Examination of the correlation ranges associated with the exhaustive transiograms in Fig. 6 and 7 show that some of the autotransiograms have a rincial autocorrelation range of about 15 to 30 ixels (i.e., m). That is, they first decrease quickly to a rincial range, and then, rather than remaining stable at a fixed level, gradually decrease and [see 4,4 (h) and 1,1 (h) in Fig. 6a and 7a] aroach their theoretical sills (i.e., the roortions of the corresonding classes). This rincial range characteristic and subsequent transition to the theoretical value differs from that of an idealized auto-transiogram, which reaches its theoretical sill quickly and stabilizes at this level with increasing lags. Figure 8 contrasts these two tyes of transiograms, demonstrating the differences between each. Fig. 6. Exhaustive transiograms estimated from the large soil ma (shown in Fig. 2) in the west-to-east direction: (a) and (b) auto-transiograms; (c) and (d) cross-transiograms. The right column [i.e., (b) and (d)] have longer lags so as to demonstrate irregular fluctuations. SSSAJ: Volume 71: Number 3 May June