Scaling Relationships between Saturated Hydraulic ...

Reproduced from Soil Science Society of America Journal. Published by Soil Science Society of America. All copyrights reserved.

Scaling Relationships between Saturated Hydraulic Conductivity and Soil Physical Properties Takele B. Zeleke and Bing Cheng Si* ABSTRACT

Ks and establish rigorous relationships with soil physical properties such as particle-size distribution, OC content, bulk density (Db), etc. that usually available from soil survey data bases. The spatial variability and scaling of Ks has been studied by several authors using geostatistical techniques (Wilson et al., 1989; Bosch and West, 1998; Sobieraj et al., 2004) and the Miller and Miller (1956) Similar Media Theory (Wilson et al., 1989; Zavattaro et al., 1999). Geostatistical methods have been useful in characterizing the spatial dependency of separate measurements of Ks and in determining the degree of cross-correlation with other soil properties. However, because it is only a second-order statistical method, this technique provides poor characterization of the variability that occurs in the presence of intermittent high and low data values (Kravchenko et al., 1999; Seuront et al., 1999). The necessary conditions of stationarity and ‘quasi-gaussian’ distribution also cast certain doubts on direct applications of geostatistical and spectral techniques in the analysis of non-Gaussian and extremely variable properties such as the Ks. The use of simple scaling techniques such as the Miller and Miller (1956) scaling theory and monofractal analysis is generally limited to specific cases where abrupt changes (i.e., small scale and erratic variability) in the spatial pattern of Ks are negligible and where exact self-similarity is a rule rather than an exception. The spatial variability in Ks has been described as a result of several independent processes operating at different spatial and temporal scales with a nested hierarchy (McBratney, 1998; Sobieraj et al., 2004). Processes that are dominant at one scale may not have a significant effect at other scales. For instance, the significance of local scale processes such as biological activity and tillage-induced variations can be masked by larger scale processes and factors such as topography and soil morphological differences. Conversely, expected differences in Ks due to large-scale process variations can be defused by small-scale processes. Such superimposition of processes of different natures that act simultaneously over a range of scales (Burrough, 1983a, 1983b; McBratney, 1998) often give rise to a chaotic and nonlinear type of distributions (Phillips, 1993). A multi-scale variability in Ks of the subsoil has been reported by several authors (Liu and Molz, 1997a; Boufadel et al., 2000; Tennekoon et al., 2003). These studies reported a Le´vy type of distribution of Ks at smaller scales (which is non-Gaussian and scale-variant) and a quasi-Gaussian distribution at larger scales.

Saturated hydraulic conductivity (Ks) is an important soil hydraulic property that affects water flow and the transport of dissolved solutes. Obtaining sufficient and reliable Ks data for large-scale process modeling is always a challenge due to the extremely high spatial variability. The objectives of this study were (i) to determine if a monofractal or multifractal approach is needed to describe the variability in Ks and its soil surrogates, and (ii) to identify which soil property best reflects the spatial distribution of Ks across a wider range of scales. Saturated hydraulic conductivity and soil physical property data were collected from a 384-m transect, located at Smeaton, SK, Canada. Observation scale variability and relationships were examined using statistical and geostatistical methods. Statistical scale-invariance was evaluated through the Hurst scaling parameter (H ). Multiple scale variability and relationships were studied using multifractal and joint multifractal techniques. Results indicate that for all the studied variables 0.80 ⬍ H ⬍ 0.90, suggesting a certain degree of statistical scaleinvariance and long-range dependency. At the observation scale, the variability in Ks was significantly related to sand (SA) and silt (SI) distribution (R ⫽ 0.40 for SA and ⫺0.39 for SI, P ⬍ 0.01; n ⫽ 128), whereas, across a wider range of scales, the variability in Ks was related only to clay (CL) and organic C (OC). The result indicates scale dependent relationships between Ks and soil physical properties, which implies that the success of predictive models such as pedotransfer functions (PTFs) and Ks aggregation techniques depends largely on the correspondence between observation and implementation scales.

K

nowledge about the maximum water conducting capacity of soils is crucial in understanding and modeling several surface and subsurface processes. The partitioning between infiltration and runoff, temporary water logging in the root zone, rate of solute transport, and several other agricultural and environmental processes are dependent on the soil’s Ks. However, obtaining sufficient and reliable Ks data for large-scale process modeling remains a challenge. Inherent soil heterogeneity and extrinsic factors cause orders of magnitude variability in spatial distribution of Ks (Sobieraj et al., 2004). The existing methods for direct in situ determination of Ks such as the double ring infiltrometer, tension disk infiltrometer, and well permeameter are time-consuming and hence impractical for largescale applications that require high resolution Ks data (Jacquier and McKenzie, 1997; Zeleke and Si, 2005a). Hence, it is essential to develop a better understanding of the nature of spatial variability and scaling property of Dep. of Soil Science, Univ. of Saskatchewan, Saskatoon, SK S7N 5A8, Canada. Received 8 Mar. 2005. *Corresponding author (bing.si@ usask.ca). Published in Soil Sci. Soc. Am. J. 69:1691–1702 (2005). Soil Physics doi:10.2136/sssaj2005.0072 © Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA

Abbreviations: CL, clay; Db, bulk density; H, Hurst scaling parameter; Ks, saturated hydraulic conductivity; OC, organic carbon; PTF, pedotransfer function; SA, sand; SI, silt; UM, universal multifractal model.

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There is limited information, however, on scale-based variability of Ks in the surface soil, and to date, there is no report on whether observed (single-scale) relationships between Ks and basic soil physical properties are valid across a wider range of spatial scales. A recent study by Sobieraj et al. (2004) reported a variation in the pattern and variability of the surface soil Ks with the scale of observation and suggested the need for a multiple scale analysis. Zeleke and Si (2005b) also reported a scale dependent variability and relationships between soil water storage at selected matric potentials and soil physical properties. The objectives of this study were (i) to determine which scaling approach (monofractal or multifractal) best describes the variability in the surface soil Ks and its soil surrogates, and (ii) to identify which basic soil physical property best reflects the spatial distribution of Ks across a wider range of spatial scales. The paper also provides close connections between variance structure (variograms) and the associated fractal and multifractal characteristics for spatially distributed soil variables.

series Z(x ) into blocks of order m and averaging over each block.

Z(x)m,k ⫽

1 m

km

兺

i⫽(k⫺1)m⫹1

Z(xi)k ⫽ 1, 2,...n/m

[2]

The index k labels the blocks and n is the total number of observations. The sample variance of the aggregated series, ␭(m ), is then calculated as

␭(m) ⫽

冤

冥

1 n/m 1 n/m [Z(x)m,k]2 ⫺ 兺兺 Z(x)m,k n/m k⫽1 n/m k⫽1

2

[3]

The same procedures are repeated for a number of m values, and then the logarithm of the ␭(m ) is plotted against log m. If the series is self-similar, the resulting points should form a straight line with a slope ␨, and furthermore, if the series is both self similar and spatially persistent (i.e., has a long range memory of the pattern), the slope ␨ is bounded between ⫺1 and 0. In practice, the slope is estimated by fitting a line to the points obtained from the plot and related to H as follows (Teverovsky and Taqqu, 1997).

H ⫽ 0.5␨ ⫹ 1

[4]

THEORY

Multifractal Analysis

Variance Structure and Statistical Scale-Invariance

The semivariogram separates and measures the amount of variability occurring in different lag distances. When all or part of the variogram follows a power law equation of the form ␥(h ) ≈ h-␨, the data are scaling in that range—that is, there is certain degree of statistical scale-invariance. It remains, however, to verify the type of scaling (simple or multiscaling), which dictates the type of fractal analysis—that is, monofractal or multifractal. For one-dimensional spatial series of property Z, the qth order structure function is defined as (Liu and Molz, 1997a)

Under the assumptions of stationarity among similar lag increments (i.e., intrinsic stationarity), spatial dependency in Ks can be calculated using the variogram (Journel and Huijbregts, 1978; Goovaerts, 1997). The traditional semivariogram estimator, ␥(h ), is calculated as

␥(h) ⫽

1 2N(h)

冦兺 [Z(x ⫹ h) ⫺ Z(x )] 冧 N(h)

i

i

2

[1]

i⫽1

where x is the distance, h is the lag, N(h ) is the number of pairs separated by h, and Z(xi) is Ks measured at spatial location xi. After fitting the experimental semivariograms to theoretical models, three model parameters, that is, nugget, range, and sill, are obtained and used to characterize the nature of spatial variability. For lag distances outside the range parameter, the variogram provides only qualitative description of the variability. For a scale-invariant and long range dependent series, quantitative characterization of the variability, which is valid across all scales, can be obtained using scaling parameters. To this end, the degree of scale-invariance and persistency in the series needs to be investigated. In the special case of exact scale-invariant distribution the scaling parameter can be obtained from the log–log plots of ␥(h ) vs. h and used as a measure of self-similarity. However, this scaling coefficient describes the scale factor of the second moment and hence does not represent the average absolute fluctuations in the series (Braun et al., 1997; Seuront, 1999). A more general and well-established method to evaluate scale invariance and persistency in spatial series is through the use of the scaling parameter (i.e., the Hurst parameter, H) (Braun et al., 1997; Pardini, 2003; Zhou et al., 2004). For a self-similar as well as spatially persistent series, the value of H is bounded between 0.5 and 1.0, and similar ‘spike’ within ‘spike’ patterns proliferate in the scatter plots. Thus, the fundamental test for self-similarity and persistency of the series reduces to verifying whether H significantly deviates from the interval (0.5, 1). The standard method to determine the H parameter is as follows (Lee, 2002; Teverovsky and Taqqu, 1997). An aggregated series of order m is first obtained by dividing the original

具[⌬Z(h)]q典 ⫽ 具|Z(x ⫹ h) ⫺ Z(h)|q典

[5]

where angle brackets indicate a statistical average. This equation generalizes the use of semivariograms (q ⫽ 2) and includes lower and higher order moments. The generalization is needed in case the process is multiscaling; it allows better determination of the probability distribution (Tennekoon et al., 2003). For a scaling series, the scale invariant structure function exponent, ␶(q ), is defined by

具[⌬Z(h)]q典 ⬀ h␶(q)

[6]

where the symbol “⬀” means proportionality. To differentiate between simple scaling (i.e., monofractal) and multiscaling (i.e., multifractal), one uses many values of q and determines the slope of log 具[⌬Z(h )]q典 versus log h. If the plot of ␶(q ) vs. q has a single slope (i.e., a linear line), then the series is a simple scaling (monofractal) type. On other hand, if ␶(q ) vs. q is nonlinear and convex (facing downward), then the series is a multiscaling (multifractal) type. For a scaling series, it turns out that the qth order normalized probability measures of a variable (also known as the partition function), ␮(q, ε), vary with the scale size, ε, in a manner similar to Eq. [6] (Meneveau et al., 1990; Evertsz and Mandelbrot, 1992), that is,

␮i(q,ε) ⫽

[Pi(ε)]q ⬀ (ε/L)␶(q) 兺 [Pi(ε)]q

[7]

i

where Pi(ε) is the probability of a measure in the i th segment of size ε units and calculated by dividing the value of the variable in the segment to the whole support length; for exam-

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ple, to the whole transect for one dimensional spatial series data from a transect of length L units. In simpler terms, Pi(ε) measures the concentration of a variable of interest (say CL content, OC, etc.) in a given segment relative to the whole support length. The ␶(q ) function in Eq. [7] is given a new name, ‘the mass exponent,’ because it relates the probability of mass distribution in a given segment to the size of the segment (scale) and is used widely in multifractal analysis. After calculating ␮(q,ε), the multifractal spectrum, f(q ), which is the fractal dimension of the subsets of segments of size ε units having coarse Ho¨lder exponents (local scaling indices) of ␣ in the limit as ε→0, can be calculated as (Chhabra et al.,1989; Evertsz and Mandelbrot, 1992),

ε→0

⫺1

冤冢Lε 冣冥

f(q) ⫽ lim log

兺i ␮i(q,ε)log␮i(q,ε)

[8]

and the local scaling indices, ␣, are given by ⫺1

冤冢冣冥

ε ␣(q) ⫽ lim log ε→0 L

兺i ␮i(q,ε)logPi(ε)

[9]

In the preceding presentation, multifractal measures were characterized by their spectrum of dimensions, f(q ) vs. ␣(q ). Often, for many practical applications few indicator parameters are selected and used to describe the scaling property and variability of a process. The two widely used models that provide such parameters are the generalized dimensions, Dq, and the universal multifractal model (UM) of Schertzer and Lovejoy (1987). The Dq of a multifractal measure is calculated as

Dq ⫽

1 lim q ⫺ 1 ε→0

log兺 Pi(ε) i

log(ε)

[10]

The value at q ⫽ 1, D1, is referred to as the information dimension and provides information about the degree of heterogeneity in the distribution of the measure (Voss, 1988). The Dq value at q ⫽ 2, D2, is known as the correlation dimension; it is mathematically associated to the correlation function and measures the average distribution density of the measure (Grassberger and Procaccia, 1983). The UM model, using a small number of parameters, simulates the ␶(q ) function of a cascade process (Schertzer and Lovejoy, 1987). Under the assumption of conservation of the mean value of the variable, the UM model describes the ␶(q ) function as

C1 (q␣⬘ ⫺ q) ␣⬘ ⫺ 1 ␶(q) ⫽ C1log(q) ␶(q) ⫽

␣⬘ ⬆ 1

[11a]

␣⬘ ⫽ 1

[11b]

where ␣⬘ (commonly known as the Le´vy index) indicates the degree of multifractality and is bounded between 0 and 2, which correspond to the monofractal and log-normal cases, respectively (Seuront et al., 1999). The Parameter C1 expresses the codimension (i.e., C1 ⫽ d ⫺ D, where d is the dimension of the observation space and D is the fractal dimension) of the set of values lower than the mean of the variable and thus characterizes the sparseness of the values.

Joint Multifractal Analysis Extending the single multifractal analyses theory to the joint distributions of two variables, the partition function (i.e., the normalized ␮-measures) for the joint distributions of two variables with probability distribution of Pi(ε) and Ri(ε), weighted by the real numbers q and t, can be calculated by

(Chhabra et al., 1989; Meneveau et al., 1990; Zeleke and Si, 2004).

␮i(q,t,ε) ⫽

Pi(ε)qRi(ε)t N(ε)

兺

j⫽1

[12]

[Pj(ε)qRj(ε)t]

The local scaling indices (coarse Ho¨lder exponents) with respect to the two probability measures Pi(ε) and Ri(ε), which are represented, respectively, by ␣(q,t ) and ␤(q,t ), are calculated as follows:

␣(q,t) ⫽ ⫺{log[N(ε)]}⫺1 ␤(q,t) ⫽ ⫺{log[N(ε)]}⫺1

N(ε)

兺 {␮i (q,t,ε)log[Pi(ε)]}

[13]

i⫽1

N(ε)

兺 {␮i (q,t,ε)log[Ri(ε)]}

[14]

i⫽1

The dimension [i.e., f(␣, ␤)] of the set on which ␣(q, t ) and ␤(q, t ) are the mean local exponents of both measures is given by

f(␣,␤) ⫽ ⫺ {log[N(ε)]}⫺1

N(ε)

兺 {␮i (q,t,ε)log[␮i (q,t,ε)]}

[15]

i⫽1

When q or t is set to zero, the joint partition function shown in Eq. [12] reduces to the partition function of a single measure, and hence the joint multifractal spectrum defined by Eq. [15] becomes the spectrum of a single measure. When both q and t are set to zero, the maximum [f(␣, ␤)]is attained, which is the dimension that would be obtained if all the segments contain a similar concentration of mass. Therefore, different pairs of ␣ and ␤ are scanned by varying the parameters q and t. More importantly, by using selected values of q or t, it is possible to examine the distribution of different intensity levels of one variable with respect to different intensity levels of the other variable.

MATERIALS AND METHODS Site Description and Field Sampling The research site is located at Smeaton, SK, Canada (53⬚40⬘ N lat. and 104⬚58⬘ W long.). The soil at the site is classified as Gleyic Luvisol with texture dominated by sandy loam developed from glacio-fluvial and fluvial-lacustrine sands and gravels. The topography of the site is gently undulating and the climate is classified as cold and subhumid. The long-term annual temperature, rainfall, and potential evapotranspiration are 0.1⬚C, 393 mm, and 530 mm, respectively (Anderson and Ellis, 1976). A north–south transect of 384-m length was established on a gently sloping land with a variable texture and OC content. After preliminary observations, a 3-m sampling interval was established along the transect and core samples were collected in September 2003 using 54-mm-diam. by 60mm-long aluminum rings. The undisturbed core samples were used to determine Ks and basic physical property variables of the surface soil at 128 points along the transect.

Ks and Soil Physical Property Determination Saturated hydraulic conductivity of the undisturbed core samples was determined using the constant head method (Klute and Dirksen, 1986). To this end, the samples were saturated for 3 d using 0.005 M CaSO4 solution. The saturated samples were transferred to a constant head water application system. After applying about 1.5-cm head of water on each core samples, the system was left to run for about 30 min to assure good saturation conditions and also equilibration in

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Table 1. Descriptive statistical test results for variability, data distribution, and observation scale correlations with saturated hydraulic conductivity.†

Db SA SI CL OC Ks

CV

Skewness‡

% 6.41 9.47 17.79 47.42 68.24 45.55

⫺0.33 ⫺0.21 ⫺0.30 0.43 0.67 (S)§ 0.59 (S)

K-S max(D )¶ 0.90 0.89 0.91 0.90 0.99 0.96

(S) (S) (S) (S) (S) (S)

R with Ks ⫺0.19 0.40 (S) ⫺0.39 (S) ⫺0.06 ⫺0.09 N/A

† Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity § (S) ⫽ significant at P ⬍ 0.01. ‡ The critical skewness value is given by 2 ⫻ (6/n )0.5 where n is the number of observations (Tabachnick and Fidell, 1996). ¶ The critical departure value from Gaussian distribution in K-S test is given by (1.36/n )0.5 (Massey, 1951).

the new setup. Then the outflow water was collected for 1 to 2 h depending on the infiltration rate, and Ks values were determined using Darcy’s equation. Soil bulk density was determined from the volume–mass relationship for each core sample. Particle-size distributions were determined based on the hydrometer method (Gee and Bauder, 1986). Organic C content was determined using LECO-12 carbon determinator (LECO Corp., St. Joseph, MI).

Preliminary Data Analysis Distribution of the raw data was characterized using classical statistical tools (Table 1). The Kolmogorov-Smirnov normality test and skewness were used in deciding whether log transformation of the data set was necessary during semivariogram analysis. The relationships between Ks and the soil physical property data at the observation scale were determined using the Pearson correlation analysis. Percentage of nugget, range, and best-fit theoretical model were determined from the semivariogram analysis (Eq. [1]). Spatial persistency and type scaling were determined from the Hurst coefficient and the structure function exponent analysis.

Multifractal and Joint Multifractal Analysis To apply the multifractal techniques, the particle-size distribution and OC data need to be converted into the distribution of mass along a geometric support or mass content per segment of a given size. To this end, the percentage of SA, SI, CL, and OC from the soil cores were converted into their mass equivalents from 3-m long, 0.06-m thick, and 0.05-m wide segments and used as mass of SA, mass of SI, mass of CL, and mass of OC. Saturated hydraulic conductivity and the Db from the cores were used as representative values for the 3-m segments. The probability of the measure in each segment of size ε, that is, Pi(ε), was determined by dividing the values of the measure in the segment by the sum of the measure in the whole transect. These Pi(ε) values were considered as a density of the measure in the segment of size ε and a dyadic (Lk ⫽ 2⫺k; k ⫽ 0, 1, 2,...) multiplicative cascade procedure was applied to divide the transect into subsequently smaller segments. The multiplicative procedure, which was applied to the 384-m transect carrying the 128 data points resulted in seven segment sizes: 192, 96, 48, 24, 12, 6, and 3 m; carrying, respectively, 2, 4, 8, 16, 32, 64, and 128 data points. In calculating the multifractal parameters, we used the method suggested by Chhabra and Jensen (1989). To estimate the ␣⬘ parameter in the UM model, the double trace moment method (DTM) by Lavalleé (1991) was employed. In calculat-

Fig. 1. Plots of the spatial distribution of saturated hydraulic conductivity and soil physical properties measured at a 3-m intervals along the 384-m transect.

ing the joint multifractal parameters, we used the method discussed in Meneveau et al. (1990), which is also a multivariate extension of the method described by Chhabra and Jensen (1989) (Eq. [12]–[15]). The range of moment orders (i.e., q values) used in the analyses was ⫺6 to 6 for multifractal analyses, and ⫺15 to 15 for joint multifractal analyses. Positive values amplify the distribution of high data values whereas negative values amplify the distribution of low data values. Hence the use of both positive and negative values serves as a mathematical microscope to look at the distribution of different data ranges. The use of a wide range of moment orders enables one to obtain a wide range of the joint dimensions. However, we limited our computation to the above values so as to avoid instability of the multifractal parameters because higher moment orders may magnify the influence of outliers in the measurements. All analyses were done using programs written in Mathcad 2000 Professional (Mathsoft Inc., Cambridge, MA) and Statistical Analyses Software-SAS Version 8 (SAS Institute Inc., Cary, NC).

RESULTS AND DISCUSSION Variability, Distribution, and Observation Scale Relationships between the Variables Descriptive statistical results on the overall variability, distribution, and symmetry, and the degree of linear association between Ks and soil physical properties are presented in Table 1. The Db and SA data were relatively uniform, with a coefficient of variation (CV) of 6.41 and 9.47%, respectively. The highest variability was observed for OC data with CV ⬎ 68%. The variability in



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Fig. 2. Histograms of soil physical property variables and saturated hydraulic conductivity data. Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity.

CL and Ks data was comparable (CV ⫽ 45 and 47%). The high variability in OC was caused mainly by largescale processes (Fig. 1). An increase in OC content of about 165% was observed toward the right end of the transect (280–370 m). The variability in Ks and CL was, however, dominated by small-scale fluctuations in the data. The distributional symmetry of the data was evaluated by the skewness function and tested for significance using the method suggested in Tabachnick and Fidell (1996). The critical skewness value for the 128 observations was 0.433. The skewness in Db, SA, SI, and OC (Table 1 and Fig. 2) were not significant and hence do not necessitate data transformation. The distribution of OC and Ks were positively skewed with values of 0.67 and 0.59, respectively (significant at P ⫽ 0.01). The test for normality of the data was evaluated by the Kolmogorov-Smirnov (K-S) goodness-of-fit test and the level of significance was calculated using the method discussed in Massey (1951) (Table 1). The K-S test showed that the distribution of the data for all the variables was

significantly different from the normal (Gaussian) type of distribution. The observed high positive skewness of OC data was the effect of larger scale variations in the distribution of this variable. Positively skewed and lognormal distribution of Ks is expected due to preferential flow phenomenon and has been reported by several authors (Logsdon, 2002; Romano, 1993). The degree of linear association between Ks and the soil physical properties at the observation scale were evaluated using Pearson’s correlation analysis. The distribution of Ks was correlated to SA and SI with a correlation coefficient (R) of 0.40 and ⫺0.39, respectively (significant at P ⫽ 0.01; n ⫽ 128). Saturated hydraulic conductivity was also slightly correlated to Db (R ⫽ ⫺0.19, not significant at P ⫽ 0.01). Organic C and CL, however, did not show any relationships to Ks at this scale. It appears that at this scale (i.e., the 3-m observation interval) the variability in Ks distribution was explained mainly by the trade-off between SA and SI content and other factors that determine the distribution of macropores. The significant correlation between


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Fig. 3. Semivariograms of soil physical property variables and the natural log transformed saturated hydraulic conductivity data. Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity.

Ks and SA at this scale is in agreement with the observation by Sobieraj et al. (2004) who reported increase in relationships between the two variables with decrease in observation scale (i.e., increase in resolution) from 25 to 1 m. The slight variations in CL and OC content (Fig. 1) seem to be masked by the more important local scale processes such as biological activities and tillage induced perturbations as suggested in Tsegaye and Hill (1998). Table 2. Geostatistical analysis results (percentage of observation scale variability, range of spatial dependency, and semivariance structure) and the Hurst scaling parameter values.†

Db SA SI CL OC Ks ln(Ks)

Nugget

Range

Best fit model (SSE)‡

% 42 43 36 8 16 34 46

m 98 116 173 43 43 50 78

Exponential (71) Exponential (24) Exponential (25) Spherical (94) Spherical (78) Spherical (128) Spherical (157)

H 0.83 0.82 0.83 0.86 0.88 0.87

(0.028)§ (0.021) (0.023) (0.033) (0.011) (0.006) N/A

† Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity. ‡ The model with the least sum square of errors (SSE) was selected as the best fit one. § Values in the parentheses are 95% confidence intervals.

Figure 3 shows empirical semivariograms and the fitted models. The contribution of measurement scale variability to the total variance was estimated using percentage of nugget values (Table 2). The local scale variability in OC and CL was low compared with that of Ks, which is in agreement with the poor relationships between Ks and these variables at the 3-m observation scale. The similarity in local scale variability between Ks and SA is also in agreement with the significant relationships between these variables as observed from the linear correlation analysis. The range of spatial dependency was relatively smaller for CL, OC, and Ks (with log transformed Ks having the largest range); their semivariance has a distinct upper bound and best fits the spherical model. The largest range of spatial dependency was observed for SI, followed by SA and Db. The semivariance of these variables increases continuously (i.e., unbounded type). Beyond the range of autocorrelation, the semivariogram of CL, OC, and log(Ks) has a cyclic pattern where short cycles were followed by longer ones, which is indicative of the presence of nested scales of variability.



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Fig. 4. Linear fits of the log-log plots of the aggregated variance versus level of aggregation. Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity.

Persistency in Spatial Series and Statistical Scale-invariance The semivariogram analysis has provided useful information on the significance of small scale variability and the range of spatial dependency. However, for lag distances outside the range of spatial dependency, the variogram provides only a qualitative description of the variability. For statistically scale-invariant and long range dependent series quantitative characterization of the variability can be obtained using scaling parameters. To this end, the degree of statistical scale-invariance and persistency in the series were investigated using the loglog plots of the aggregated variance (␭m) vs. the level of aggregation (m) and the Hurst scaling parameter, H. The log(␭m) of all the variables varied linearly with log(m) indicating the presence of statistical scale-invariance (Fig. 4). The coefficient of determination (R2) for a linear fit (n ⫽ 7) for SA, Db, SI, OC, CL, and Ks were, respectively, 0.95, 0.94, 0.94, 0.90, 0.77, and 0.86 (all significant at P ⫽ 0.01). Hence, the degree of statistical self-similarity in the variance structure was the highest for SA, followed by Db and SI. The relatively lower R2 values for OC, CL, and Ks indicate a tendency toward more nonlinearity in the log variance vs. log scale relationships than SA, Db, and SI.

The Hurst scaling parameter, H, was determined from the log-log plots of ␭m vs. m and presented in Table 2. The H values of all the variables lie within the theoretical range for a self-similar and long range positively correlated spatial series; that is, 0.5 ⬍ H ⬍ 1.0. There were, however, slight differences in H values. Relative to Db, SA, and SI, the H values for CL, OC, and Ks were slightly higher (but not statistically significant at P ⫽ 0.01). The slightly higher H values of CL, OC, and Ks are likely the consequence of uncertainties involved (the low R2 values) in extracting the slope of the loglog plots or the consequence of higher long-range correlations. The following section presents results based on multifractal and joint multifractal analysis. Note that monofractals are a special case of multifractals.

Multifractal Analysis Results Plots of the empirical ␶(q) functions of all the variable are shown in Fig. 5. The sum of square of deviations from linearity [SSD␶(q)] and their statistical significance based on the Chi-square goodness-of-fit analysis are presented in Table 3. The ␶(q) plots of Db and SA were straight lines and not significantly different from the fitted linear lines. The ␶(q) plots of SI also looks straight line (Fig. 5) and has a relatively small SSD␶(q) value (only


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Fig. 5. The mass exponents of the variables (q ⫽ ⫺6 to 6 at 0.5 increments). The solid line is a superimposed straight line passing through ␶(0). Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity.

Fig. 6. The multifractal spectra of all the variables studied (q ⫽ ⫺6 to 6 at 0.5 increments). Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity.

3.7, relative to CL ⫽ 91.6), but fails the significance test for linearity at P ⫽ 0.01. The ␶(q) plots of CL, OC, and Ks were convex (facing downward) and significantly different from a linear line. Therefore, Db and SA have a monofractal type of distribution, whereas CL, OC, and Ks have a multifractal type. The distribution of SI appears to be represented by either model. Selected indicator parameters from the Dq function and the ␣⬘ values of the UM model of Schertzer and Lovejoy (1987) are presented in Table 3. Values of D1 and D2 of a variable with a monofractal type of scaling become similar to the capacity dimension, D0, whereas D0 ⬎ D1 ⬎ D2 if the distribution has a tendency of multifractal type of scaling (Turcotte, 1997). For Db, SA, and SI, values of D1 and D2 are similar to the capacity dimension, whereas for CL, OC, and Ks, D0 ⬎ D1 ⬎ D2. The ␣⬘ parameter, also called the levy index, indicates the degree of multifractality. Based on their ␣⬘ values, Db and SA have similarities in terms of their scaling properties, which is in good agreement with the observations from the Dq analysis. The ␣⬘ values for CL, OC, and Ks showed a multifractal tendency.

To discern the local scaling patterns that are imbedded within the larger ones, the multifractal spectrum [i.e., f(q) vs. ␣(q)] of the variables were calculated and presented in Fig. 6. The wider the spectrum (i.e., the higher the ␣max⫺ ␣min value), the higher is the heterogeneity in the local scaling indices of the variable and vice versa. The height of the spectrum, f(q), corresponds to the dimension of these scaling indices. The small f(q) values correspond to rare events (extreme values in the distribution), whereas the highest value of f(q) is the capacity dimension, which is obtained by assuming similar values in all the segments. The spectra of Db and SA have a narrow width (range of ␣ values) and an average dimension similar to the capacity dimension of the support (i.e., 1.0), which implies scaling uniformity or the possibility of representing the scaling property by a single value. Clay and Ks had the most heterogeneous scaling indices with (␣max⫺ ␣min) ⬎ 0.5, followed by OC with (␣max⫺ ␣min) ≈ 0.35. The scaling dimensions for SI, CL, OC, and Ks vary from 0.37 to 1.0, meaning that representation of the scaling property of these variables requires numerous dimensions whose values are bounded between 0.37 and 1.0. The spectra of these variables (i.e., SI, CL, OC, and K) have a longer tail to the right of the maximum f(q) value, which is a typical characteristic of multifractal measures. Note that the right side of the spectrum corresponds to lower data values that are amplified by ⫺q values, and hence the right skewed feature is the result of more heterogeneity in the distribution of lower data values. The results presented above clearly elucidate the similarity and differences between the variables in terms of statistical distribution as a function of spatial scales. At smaller scales the variability in Ks was only significantly related to SA and SI distributions. This relationship appears to be the reflection of direct influence of higher

Table 3. Parameters describing the scaling nature of the variables studied. SSD␶(q) ⫽ sum square of deviations from a straight line. D1 ⫽ information dimension. D2 ⫽ correlation dimension. ␣ⴕ is a multifractality index in the universal multifractal model of Schertzer and Lovejoy (1987).† Db SA SI CL OC Ks

SSD␶(q)

D1

D2

0.00 0.01 3.76 (S)‡ 91.26 (S) 47.00 (S) 60.49 (S)

1.00 1.00 1.00 0.97 0.96 0.97

1.00 1.00 0.99 0.95 0.92 0.95

␣ⴕ 0.00 0.02 0.03 0.22 0.19 0.16

(0.001)§ (0.006) (0.009) (0.055) (0.002) (0.028)

† Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity. ‡ (S) ⫽ significant at P ⬍ 0.01. § Values in the parentheses are 95% confidence intervals.



SA content on Ks. Other factors such as OC content, root biomass, microbial population and activity do not seem to fluctuate in these short measurement intervals. Nonetheless, the high positive skewness and CV of Ks indicate the presence of locations and scales with extremely high values. Besides preferential flow phenomenon, the extreme values of Ks could also be the result of processes operating at different scales (Burrough, 1983a, 1983b; McBratney, 1998). The scaling property presented above shows that the distribution of CL, OC, and Ks changes with scale, whereas that of Db, SA, and to some extent SI do not. This also means that the observed linear relationships between Ks, SA, and SI may not be maintained at scales other than the observation one. Thus, a universal predictor of Ks should put more weight on soil variables that vary with scale of observation, in this case CL and OC. Nonetheless, the distribution of all the variables has good spatial continuity whereby, on average, high values are most likely followed by high values and low values are most likely followed by low ones.

Joint Multifractal Analysis Results The scaling dimensions for the joint distributions of Ks and the five soil physical properties were analyzed using the joint multifractal analysis technique. Figure 7 shows the joint multifractal spectra and Table 4 provides the correlation coefficients between scaling indices of variables at a selected range of data values. The contour lines represent the joint dimensions, f(␣,␤), of a pair of variables on each plot. The bottom left part of the contours show the joint dimension of high data values of the two variables, whereas the top right part shows that of low data values. Diagonal contours with low stretch indicate high correlation between values corresponding to the variables in the vertical and horizontal axis (Si and Kachanoski, 2000; Zeleke and Si, 2004). Figure 7A shows the contour lines for the joint scaling dimensions of Ks and Db. There appears to be some relationships between the scaling dimensions of Ks and Db for both high and low data values, which is evident from the slightly diagonal feature of the plots and the high correlation coefficients between the scaling indices of the two variables. The highest correlation coefficient (R ⫽ ⫺0.57) between scaling indices of Db and Ks was obtained for the high data values of the two variables. Figure 7B as well as Table 4 shows that the scaling indices of SA were not related to that of Ks; however. Mass of SI (Fig. 7C) has significant (but negative) relationships to Ks for all ranges of data values. The contour lines in Fig. 7D were diagonal and pulled together indicating that the high and low scaling indices of Ks were associated, respectively, with the high and low scaling indices of CL. The correlation coefficient between the scaling indices of Ks and CL indicates strong relationships (R ⫽ ⫺0.97 to ⫺0.98, significant at P ⬍ 0.01) over all the ranges of data values. The joint dimensions of Ks vs. OC (Fig. 7E) also show strong relationships between the two variables. The correlation coefficient between the scaling indices of the Ks and OC values

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indicates strong relationships (R ⫽ 0.79, significant at P ⬍ 0.01) over all the ranges of data values. The relationship between Ks and OC was even stronger (R ⫽ 0.92) when analyzed for high vs. high data values of the two variables, which implies that locations with high OC content will always have high Ks. The joint multifractal results presented above provided a clear picture of the distribution of different intensity levels (high or low data values) of one variable that coexisted with different intensity levels of the other variable; where the ‘highs’ and ‘lows’ refer to values that are above and below the average value for the whole transect. The relatively higher negative correlation coefficient between the high data values of Db and Ks, compared with the correlation for the whole data range shows that those locations with high Db values seem to have the higher impact on the distribution of Ks than those with average and low Db values. In other words, locations with high Db will always have low Ks; regardless of scale. This observation can be explained based on the inverse relationships between Db and porosity through Poiseuille’s Law. High Db usually means little or no meso- and macropores. The strong association between the scaling indices of Ks vs. CL and Ks vs. OC indicates that the relationships between the two variables are valid across a wider range of spatial scales and that the spatial variability in one of the variable is very well reflected in the variability of the other. The negative correlations as well as the left tilt of the contours indicate the negative influence of CL on Ks, which implies that locations with high CL content usually have low Ks and vice versa, regardless of the spatial scale. At this point, it seems difficult to provide literature support as to why CL described the variability in Ks at all spatial scales better than SA and SI. One possible speculation is the possibility of swelling and shrinkage that usually determines the number and size of preferential flow paths and the multitude of other processes that are directly affected by CL content. The results also showed that locations with high OC values affect Ks the most. High OC usually implies better soil aggregates and abundance of enhanced micro- and macro-organism activities. Thus, Ks variation at scales where OC significantly changes should be expected. The poor relationship between these two variables at the observation scale is explained by the relatively uniform distribution of OC at that scale. With increase in the spatial scales, there was a significant variation in OC content. This variation, in turn, brings in variation in the size and distribution macropores that masks local scale processes that also affect Ks. In general, the above results provide significant evidence about the presence of statistical scale-invariance in spatial distribution of the variables and scale dependent relationships between them. Such relationships are likely the consequence of variations in the spatial scale of physical and biological processes that determine the spatial distribution of different soil properties (Zeleke and Si, 2005b). Variation in the maximum water conducting capacity of soils is a function of the amount and continuity of macropores (Bouma et al., 1977; Vepraskas


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Fig. 7. Multifractal spectrum of the joint distribution of saturated hydraulic conductivity (vertical axis) and the soil properties (horizontal). Contour lines show the joint scaling dimensions of the variables. Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity.

et al., 1991; Sobieraj et al., 2002). At smaller scales, slight variation in macropore distribution may result from local scale processes such as tillage induced perturbations and microbial activities (Tsegaye and Hill, 1998). However, with aggregation of scales, geological processes, mineralogy, soil erosion, topography, and cropTable 4. Correlation coefficient (R ) between the scaling indices of soil properties and that of saturated hydraulic conductivity as determined from joint multifractal analysis at different intensity levels of the variables (selected q and t values).†

All values H vs. H† L vs. L‡

Db

SA

SI

CL

OC

⫺0.43 ⫺0.57 0.39

R 0.29 0.09 0.18

⫺0.71 ⫺0.85 ⫺0.98

⫺0.98 ⫺0.97 0.98

0.79 0.92 0.48

† Db, bulk density; SA, sand content; SI, silt content; CL, clay content; OC, organic carbon content; Ks, saturated hydraulic conductivity. ‡ H ⫽ high data values. § L ⫽ low data values.

ping history become more important. This induces variation in crop productivity (increase in root biomass and OC), earthworm population, and the type and amount of clay minerals (with possibility of cracks), etc., with consequential variation in the size and density of macropores (Bodhinayake and Si, 2004). The above findings on scaling and variability analysis of measured saturated hydraulic conductivity were based on the notion of spatial continuity. Note that continuity in spatial distribution of Ks is usually expected because of continuity in such soil variables as SA and organic matter content. In this regard, this article can be considered as continuation of past works on spatial variability and scaling of measured Ks data (Liu and Molz, 1997a; Mohanty and Mousli, 2000; Olsson et al., 2002; Sobieraj et al., 2004). Another interesting approach for characterizing the spatial distribution of Ks is based on the notion of connectivity (Vogel, 2000; Neuman and Federico,



2003; Knudby and Carrera, 2004), which is a conceptual paradigm based on interconnected paths within the soil. Future work in characterization of Ks need to include thorough comparison between these approaches and identify which one is more appropriate under a given set of conditions and goals such as extrapolation and design of sampling strategies.

CONCLUSIONS Spatial variability and scaling of Ks and its soil surrogates were studied for a gently sloping 384-m transect with variable texture and OC content. The spatial variance structure and the presence of statistical scale-invariance were analyzed through geostatistical and stochastic spatial series tools such as the Hurst scaling parameter and structure functions. Variability and relationships across a wider range of spatial scales (multiple scales) were studied using multifractal and joint multifractal techniques. The distribution of Db, SA, and to some extent SI had a simple scaling type distribution that can be characterized by monofractals, whereas that of CL, OC, and Ks had a multiple scaling type of distribution that needs to be characterized using multifractals. Observation scale analysis using Pearson correlation coefficients and the proportion of nugget variance showed that the variation in Ks (at the observation scale) is significantly related to SA and SI distribution. However, multiple scale analysis using multifractal and joint multifractal techniques showed that, across a wider range of spatial scales, the distribution of Ks is more related to CL and OC than SA and SI. The implication of such scale dependent variability is that single scale analysis may not be sufficient to fully describe the relationships between the soil and soil water properties that is valid across all spatial scales. Hence, future research in improving predictive models such as PTFs as well as Ks aggregation and interpolation techniques should incorporate several spatial scales during the modeling and validation process. ACKNOWLEDGMENTS Funding for this project was provided by the National Science and Engineering Research Council of Canada (NSERC). Technical help from W. Bodhinayake and L. Tallon is greatly appreciated.

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