Published online 26 April 2005
Published in Vadose Zone J 4:264-270 (2005)
DOI: 10.2136/vzj2004.0113
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Defining Geometric Similarity in Soils
Bhabani S. Dasa,*,
Nathan W. Hawsb and
P. Suresh C. Raoc
a Agricultural and Food Engineering Dep., Indian Institute of Technology, Kharagpur, India 721302
b Dep. of Geography and Environmental Engineering, Johns Hopkins Univ., Baltimore, MD 21218-2686
c School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907-2051
* Corresponding author (bsdas{at}agfe.iitkgp.ernet.in)
Received 4 August 2004.
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ABSTRACT
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Scaling of soil hydraulic properties is a convenient way to characterize soil variability in a unified manner. Scaling techniques implicitly assume geometric similitude for the soils being scaled; however, the meaning of similar media is somewhat ambiguous. This study focuses on the question of how to define geometric similarity. This question is addressed using the physically based scaling (PBS) technique proposed by Kosugi and Hopmans to coalesce 247 soil water retention curves (WRCs) measured from soil cores collected across eight different counties in Indiana, USA. These soil cores represented 29 different soil series and included seven broad textural classes. Although all the 247 WRCs could be scaled together, the RMSEs of the scaling results were improved when WRCs within each textural group were scaled separately. Thus, soil texture may be used as a preliminary guide to group similar soils. This study shows that the sample standard deviations of the pore-size distribution should be used to quantify geometric similarity among soils. Specifically, the coefficient of variation among standard deviations (CV
) for selected WRCs may be used to demarcate the limit of similarity among soils. Our results show that the CV 
10% may be used as a working definition for similarity in soils.
Abbreviations: K-S, Kolmogorov-Smirnov • LNR, lognormal retention • PBS, physically based scaling • pdf, probability distribution function • S, sand • SiC, silty clay • SiCL, silty clay loam • SiL, silt loam • SL, sandy loam • WRC, water retention curve
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INTRODUCTION
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THE VALUES OF SOIL hydraulic properties can vary by orders of magnitude (Hills et al., 1989) and often depend on the scale of measurement (van Es et al., 1999). Miller and Miller (1956) introduced the similar-media concept to conveniently describe soil variability in a unified manner. They assumed that the microscopic structures of two "geometrically similar" soils differ only by a characteristic length (Warrick et al., 1977). For instance, if soil pore radius is considered as the characteristic length for two geometrically similar soils, then the pore radius of one soil would differ from that of the other only by a linear scaling factor (Tuli et al., 2001). Consequently, the characteristic length (
), pore radii (r), and the capillary pressure heads (h) for different WRCs are related as
 | [1] |
where the numbered subscripts denote the ith soil data set, and the * subscript refers to the reference soil data set, respectively (Miller and Miller, 1956; Kutilek and Nielsen, 1994). Because the radius of a water-filled pore is inversely proportional to the capillary pressure head (or matric suction head), both r and h are used interchangeably in this discussion as an indication of relative pore size. A positive sign for h indicates that it represents suction. Using Eq. [1], the scaling factor for the ith soil (
i) is defined as
 | [2] |
The scaling factor approach provides a convenient way to coalesce multiple WRCs into a single reference WRC. Alternatively, when scaling factors are known, Eq. [1] and [2] provide a convenient way to estimate unscaled capillary pressure heads from the reference capillary pressure head values. Since the introduction of similar media concepts by Miller and Miller (1956), various scaling techniques have been proposed to scale soil hydraulic properties (Warrick et al., 1977; Simmons et al., 1979; Rao et al., 1983; Ahuja and Williams, 1991; Claustnizer et al., 1992; Pachepsky et al., 1995; Kosugi and Hopmans, 1998). Most approaches compute scaling factors by minimizing the mean sum-of-squared-deviations between the observed and reference soil water retention data (Warrick et al., 1977). Recently, Kosugi and Hopmans (1998) presented an elegant PBS technique in which the median pore radius (rm) of a soil sample is taken as the characteristic length-scale. The utility of the PBS approach lies in the ability to directly estimate retention parameters for the reference soil as statistical means of parameters determined for each sample WRC. This method has also been shown to effectively scale the unsaturated soil hydraulic conductivity versus water content data (Tuli et al., 2001).
With the advancement of scaling theory, such as the PBS technique of Kosugi and Hopmans (1998), a fundamental question remains of what constitutes a set of "geometrically similar" soils. Tillotson and Nielsen (1984) observed that different WRCs may be forced to coalesce into a reference WRC without implying a Miller–Miller scaling relationship. Thus, whether it is necessary to invoke the assumptions of Miller–Miller scaling remains an unanswered question. We examine the criteria for defining similar media and whether similar media can be identified based on easily measurable soil properties such as soil texture.
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PHYSICALLY BASED SCALING THEORY
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The PBS approach of Kosugi and Hopmans (1998) assumes a lognormal probability distribution function (pdf) for pore radii, f:
 | [3] |
where r is the pore radius, rm is the mean pore radius, and is the standard deviation of the frequency distribution. Based on this assumption, the soil water retention function, hereafter referred to as lognormal retention (LNR) model, can be expressed as (Kosugi, 1994):
 | [4] |
where is the standard deviation of the pore size distribution, Se is the relative saturation, 
is the volumetric water content at a given h, r is the residual water content, and s is at saturation (h = 0). The parameter hm is the suction large enough to drain all pores with radius larger than rm. Equation [4] shows that when all soils are saturated up to their median pore size (r = rm), the relative saturation equals 0.5; that is, Se(rm) = 0.5. Therefore, rm may be conveniently used as the characteristic length (Kosugi and Hopmans, 1998) and can be directly used to compute the scaling factor:
 | [5] |
where rm* is the median pore radius of the "reference" soil (soil to which all other soils are scaled). Equations [1] and [5] show that the PBS approach essentially shifts the quantity (lnr – lnrm) by lnrm,* (Tuli et al., 2001), and thus, all the scaled pore radii are distributed around lnrm,*. With this approach, when the natural log of scaled matric head (= ln[hii]) is plotted as a function of Se, the scaled matric heads are influenced by a single parameter, rm. Although Tuli et al. (2001) recognized the PBS approach as single parameter scaling technique, the median pore radius rm is a function of the pore size at the inflection point (r0) of the WRC and the variance of pore radii pdf (Kosugi, 1994):
 | [6] |
Thus, for each sample pore radii pdf is implicitly included in the evaluation of Eq. [5]; that is, a linear shifting in ln rm,* entails a linear shifting in 2.
Kosugi and Hopmans (1998) showed that the mean pore size distribution representing a population of WRCs may be written as
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with the parameters rm,* and * given by
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 | [9] |
where n is the number of WRCs, and i and * in the subscript corresponds to the ith sample and the population soil, respectively. Kosugi and Hopmans (1998) showed that the population attributes 
are identical to the attributes for the reference soil. Thus, the scaling factor and characteristic length scales may be directly estimated from the retention parameters for respective soil WRCs in the PBS approach.
The conditions for the application of PBS scaling technique through the use of Eq. [7] to Eq. [9] are:
- The pore radii distribution can be described by the two-parameter lognormal distribution proposed by Kosugi (1994).
- Median matric heads and, therefore, the scaling factors must follow lognormal distribution.
- The standard deviations for the pore radii distribution functions of the soils to be scaled must be identical (Fig. 2 of Kosugi and Hopmans, 1998).
The first two conditions must be satisfied before the PBS technique can be applied and, therefore, are necessary conditions. The third condition stated above is a requirement inherent to the PBS approach. The PBS theory requires that each of the scaled WRCs be obtained from "similar" media, which means that two similar WRCs differ only by their characteristic length (same as the median pore radii in the PBS approach). In reality, WRCs will differ both with respect to rm and . In this study, we examine what should be the limit of the similarity of values before the WRCs can no longer be considered as drawn from similar media.
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SOIL WATER RETENTION DATA AND DATA ANALYSIS
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The soil water retention measurement data used in this study were compiled in the Purdue University Station Bulletin 452 (Wiersma, 1984) as a part of intensive soil survey in Indiana, USA from 1957 to 1965. Soil physical properties and WRCs were measured for a total of 136 soil profiles containing multiple soil horizons. The quality of the water content and pressure measurements used to construct WRCs was maintained by including duplicate samples of a standard soil during each measurement. A composite silt loam soil was used as the standard soil. Several runs of this composite soil sample were made at pressures ranging from 0.1 to 0.15 MPa to establish the standard WRC for this composite silt loam sample. Measured retention data for any given run were discarded when there was a difference >0.5% water content between the average water content of the two standard soil samples and their water contents predetermined during the standardization step (Wiersma, 1984). Thus, any variation in estimated parameters among different soils may be attributed to inherent soil properties and not measurement error.
A set of 247 WRCs was selected from the database of more than 300 WRCs measured during this survey. Selection was based on the criterion of a WRC measured across the range of suctions from 0 to 1500 kPa. The set of 247 WRCs came from eight counties, represented 29 soil series, and included seven dominant soil textures. Wiersma (1984) also reported the dry bulk density values for each soil, which were used in this study to convert the measured gravimetric water contents to volumetric water contents. Mean values for soil physical properties according to soil texture are listed in Table 1. The profile description was conducted on site (Wiersma, 1984, p. 111); therefore, the textural classes were defined without a quantitative determination of sand, silt, and clay fractions. As a result, a quantitative application of textural classification in terms of particle size was not possible for this study.
An Excel (Microsoft Office Spreadsheet Application) macro was developed to estimate water retention parameters for subset of Wiersma (1984) soils according to the LNR model. The retention parameters hm and were estimated using the least-squares optimization routine in Excel's solver function (Wraith and Or, 1998). The parameters saturated water content (s) and residual water content (r) were assumed to be equal to the measured at h = 0 and h = 1500 kPa, respectively, and were kept constant during parameter estimation. Initially, all 247 WRCs were scaled to a single reference soil. Later, WRCs were grouped into seven subpopulations based on the soil texture defined in Wiersma (1984): sand (S), sandy loam (SL), loam (L), silt loam (SiL), clay loam (CL), silty clay loam (SiCL), and silty clay (SiC) (Table 1). Each textural class subpopulation was then scaled to its respective subpopulation mean. Textural grouping was assumed to be a sufficient determination of self-similarity. The validity of such an assumption was then evaluated in the light of scaling results.
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RESULTS AND DISCUSSION
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Retention Parameters for Different Soil Textures
All the 247 WRCs (Fig. 1)
were successfully described by the LNR model (fitted data not shown) with more than 92% of WRCs having R2 > 0.95 and only two WRCs with R2 < 0.9. Such high values of R2 indicate the effectiveness of the LNR model in describing measured retention data. To test the condition that the ln(hm) values are normally distributed, the Kolmogorov-Smirnov (K-S) test statistic was estimated through the KSTEST function of MatLab program for all the 247 ln(hm) values. The K-S test statistic of 0.087 against the cutoff value of 0.103 at a significance level of 0.01 and a p value of 0.044 suggested that the null hypothesis that ln(hm) is normally distributed could not be rejected. Because the scaling factor is directly proportional to its hm value for a given sample, the scaling factors must also be lognormally distributed.
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Fig. 1. Unscaled soil water retention curves for sand (S), sandy loam (SL), loam (L), silt loam (SiL), clay loam (CL), silty clay loam (SiCL), silty clay (SiC), and all 247 soils combined together.
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All of the estimated log-transformed median matric heads (lnhm,i) and the standard deviations of pore size distribution (i) are compared in Fig. 2
. Each data point (symbols) in this figure represents retention parameters for a particular soil, and a given color identifies data points within any textural group. A linear trend line is also presented to clearly show textural clusters. The R2 values for these trend lines ranged from 0.27 (n = 69) for silt loams to 0.73 (n = 24) for silty clays. Although such R2 values are small, the negative slopes for these trend lines suggests that soils with smaller median pore radius (or, larger ln[hm]) may be associated with a smaller standard deviation for pore radii pdf within a given textural class. Figure 2 also shows that coarser soils tend to have smaller values for lnhm,i and the opposite is true for the finer soils. Similar results were also observed by Tuli et al. (2001).
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Fig. 2. Retention parameters for seven soil textural classes (S, sand; SL, sandy loam; L, loam; SiL, silt loam; CL, clay loam; SiCL, silty clay loam; SiC, silty clay). Symbols represent estimated parameters and solid lines represent the trend lines through each textural cluster.
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An important observation in Fig. 2 is that the boundary of textural clusters between coarse soils is relatively more distinct than for fine soils, especially with regard to the median pore radii values. This aspect is more clearly shown in Fig. 3
, where the mean values for ln(hm) and the standard deviations () for the pore radii pdf are plotted against soil texture on the primary and secondary y axes, respectively. The error bars for these two curves indicate the standard error (SE). The mean ln(hm) values slightly increase as the texture becomes finer, and are almost identical for soil textures finer than loam. This result suggests that it is difficult to classify soil textures on the basis of the characteristic scaling length alone, especially for finer soils. Failing to group retention parameters on the basis of soil texture and bulk density, Tuli et al. (2001) arbitrarily chose ln(hm) = 6 to classify their soils; soils with ln(hm) <6 were classified as coarser-textured soils and those with ln(hm) > 6 were classified as finer-textured soils. The textural class loam (Fig. 3) in this analysis shows a mean ln(hm) = 5.6 (Table 2), with a range between 5 and 7. Loam is generally considered a medium-textured soil, and soils finer than loam are classified as fine-textured soils. Thus, it is difficult to classify soil textures on the basis of the characteristic scaling length alone. Other parameters such as percentage composition of sand, silt, and clay fractions; the orientation of soil particles; and connectivity of soil pores also influence water retention behavior in soil.
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Fig. 3. Mean ln(hm) and mean standard deviations of the pore size distribution () for seven soil textural classes. The error bars indicate the standard error for each textural class. The numbers in parentheses shown in abscissa are the number of water retention curves used in each textural class.
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The mean values are not as distinct as the ln(hm) values for different soil textures (Fig. 3), with the exception of the silt loam and clay loam soils. High SE values for the mean indicate a wide range for this parameter within each textural class. As recognized earlier, greater variation in decreases the effectiveness of the single parameter PBS approach. Because the variation in is inevitable in measured WRCs, it is useful to define an acceptable limit of variation for this parameter, such that those WRCs that are within this limit may be classified as "geometrically similar" soils and can be efficiently scaled together.
Soil Texture as a Criterion to Classify Similar Media
As a preliminary step to group similar soils, we applied the PBS approach to soils with similar texture. Figure 4
shows scaled WRCs (filled circles) for seven soil textures and the reference soil WRC (open circles) for each textural class. Scaled WRCs for the pooled data of 247 soils are also shown in this figure. Resulting reference retention parameters are shown in Table 2. The effectiveness of scaling within respective textural group is evaluated by estimating the squared deviations between scaled (filled circles) and the reference (open circles) WRCs (Fig. 4) and are represented with the RMSE in Table 2. The RMSE value is highest when all the 247 WRCs are scaled together (Table 2). Although it is recognized that the soil texture is not a sufficient criterion to group WRCs collected from different soils, Table 2 and Fig. 4 show that the RMSE values are reduced when soils are scaled after grouping them by soil textures. Thus, as expected, soils that are separated by textural group are more similar than when all soils are considered together. Consequently, soil texture may serve as a preliminary guide for distinguishing similar media.
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Fig. 4. Scaled water retention curves along with their reference retention curve for sand (S), sandy loam (SL), loam (L), silt loam (SiL), clay loam (CL), silty clay loam (SiCL), silty clay (SiC), and all soils combined together.
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To show how the variation in i values is responsible for the scatter around the reference WRCs, the coefficients of variation for i (CV), defined as the ratio of the arithmetic mean of i to the standard deviation of i, were plotted on the primary y axis and the RMSE on the secondary y axis as a function of soil texture in Fig. 5
. Minimum RMSE may be observed for the soil texture for which CV is the minimum, suggesting that the effectiveness of scaling (or scalability) is directly linked with the variation of and the CV may be used as a criterion to choose WRCs for scaling.
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Fig. 5. Coefficient of variation CV (%) of sample standard deviations for the pore radii distribution function and the RMSE for each soil texture. The numbers in parentheses shown in abscissa are the number of water retention curves used in each textural class.
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Measure of Similarity between Two Porous Media
To demonstrate that CV may be used as a criterion to classify geometric similarity, scaling exercises were conducted by selecting WRCs in two different approaches, a random selection approach and selection by incremental addition of WRCs. In the random selection approach, 36 sets of 20 WRCs were chosen randomly from the pool of 247 WRCs. To accomplish this, sample identifiers (numbers between 1 and 247) for each WRC were assigned to each dataset. An Excel macro was then created to generate 20 random numbers between 1 and 247. Water retention curves having these 20 random numbers as sample identifiers were then scaled together, and the resulting RMSE values were plotted as a function of the coefficients of variation for i (CV) in Fig. 6a
. The insert in this figure highlights the RMSE values up to the CV = 8.3. This figure shows that the RMSE fluctuates around low values for CV < 8.3 and then increases rapidly as CV exceeds 8.3. Inspection of mean hm and values within these 36 sets of WRCs indicated that both the maximum and minimum values for the mean hm and values occurred within the CV < 8.3, suggesting that the observed trend in the RMSE was not influenced by a biased sampling of mean hm and .
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Fig. 6. Root-mean-squared errors plotted as a function of the coefficient of variation (%) of the standard deviations for the scaled WRCs representing different sub-populations of (a) all soils and (b) silt loam and silty clay loam soil.
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In the selection by incremental addition of WRCs approach, new sets of WRCs were constructed by adding a few WRCs with larger to an initially selected set of WRCs with smaller . Silt loam and silty clay loam soils were chosen for this approach because of the large number of WRCs available. For example, 69 WRCs in the silt loam textural group were arranged in ascending order of their i values from 1 = 1.56 to 69 = 3.44. A total of 13 subpopulations of WRCs were obtained by incrementally adding five WRCs to each preceding subpopulation. Ten WRCs (those having 1 to 10) were chosen as the first subpopulation. Thus, the second subpopulation had 15 WRCs corresponding to standard deviations 1 to 15, and the third subpopulation had 20 WRCs corresponding to standard deviations 1 to 20, and so on. Each of these subpopulations was separately scaled using the PBS approach. This approach was also followed for the silty clay loam soil. In addition, for the silty clay soil, 20 WRCs were also randomly selected and scaled as in the previous analysis with the 247 soils. These results of RMSE vs. CV are summarized in Fig. 6b. This figure shows that the RMSE remained relatively constant up to a CV of 10% for both soils, after which it rapidly increased. The upward inflection point in RMSE values beyond CV = 10% provides a convenient working definition of geometric similarity as CV 10%.
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SUMMARY
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Similar-media scaling based on a PBS technique (Kosugi and Hopmans, 1998) was applied to find a practical definition of geometric similarity in soil. A total of 247 soil WRCs collected across eight different counties in Indiana were scaled for this purpose. As in Kosugi and Hopmans (1998) and Tuli et al. (2001), our results showed that the PBS approach is an effective tool for directly estimating scaling factors from estimated median pore radii (Eq. [5]). Although all the 247 soil WRCs were scaled together, the effectiveness of scaling was significantly improved when WRCs within each textural group were scaled separately. However, the sample standard deviation is more useful than soil texture to demarcate geometric similarity of soils. Specifically, the coefficient of variation among standard deviations (CV) for selected WRCs may be used to define the limit of similarity among soils. Our results show that the CV 10% may be used as a working definition for similarity in soils. This study also shows that it is possible for soils with different textures to have identical median pore radii, yet much different WRCs. Thus, soil textural information alone may not be sufficient to predict the water retention behavior of soil as is done in some pdf approaches. A slight limitation of the present analyses is the lack of quantitative data to define soil texture. The availability of sand, silt, and clay fractions of each soil would more clearly distinguish the effects of soil texture on the characteristic scaling length. However, as Fig. 6a demonstrates, the lack of quantitative textural data does not affect our major conclusion that the variation in standard deviations of the pore radii pdfs is useful to define similar soils.
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ACKNOWLEDGMENTS
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Sincere appreciation is expressed to Dr. L.S. Lee of Purdue University for the support of the senior author during his visit to Purdue University where part of this work was carried out. Acknowledgment is also given to Mrs. P. Das for her assistance with data entry.
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REFERENCES
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ABSTRACT
INTRODUCTION
