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Published in Agron. J. 96:1082-1090 (2004).
© American Society of Agronomy
677 S. Segoe Rd., Madison, WI 53711 USA

SPATIAL VARIABILITY

Scaling Properties of Topographic Indices and Crop Yield

Multifractal and Joint Multifractal Approaches

Takele B. Zeleke and Bing Cheng Si*

Dep. of Soil Sci., Univ. of Saskatchewan, Saskatoon, SK, Canada, S7N 5A8

* Corresponding author (Bing.Si{at}usask.ca).

Received for publication September 11, 2003.

   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Topography controls soil water distribution in semiarid environments where water is the major growth-limiting factor. Identification of the topographic index that best represents the spatial variability and scaling properties of crop yield is important for precision farming. Our objective was to characterize the scaling properties of four topographic indices [relative elevation (RE), wetness index (WI), upslope length (USL), and curvature (CR)] and their relationships to wheat (Triticum aestivum L.) grain yield and biomass using multifractal and joint multifractal approaches. Wheat grain yield and terrain data were collected at 6-m intervals along a 576-m-long transect on a nonlevel landscape with dominant soil type of Aridic Ustoll, under the semiarid environment of Saskatchewan, Canada. Results indicated that CR and RE had a fractal type of scaling only for a narrow range of moment orders. Wetness index showed a monofractal scaling with fractal dimension of 0.98; whereas yield, biomass, and USL showed a multifractal scaling. Joint multifractal analyses showed a high correlation coefficient between the scaling indices of grain yield and USL (r = 0.93). Wetness index appeared to be effective as a yield covariate only at low slope areas and depressions where it has similar scaling to that of USL. Results from this study suggested that USL was the best indicator of grain yield and biomass at any scale. The implication for precision farming is that USL can be used as a guideline for varying production inputs such as fertilizer as well as for yield prediction.

Abbreviations: BM, aboveground biomass • CR, curvature • CV, coefficient of variation • RE, relative elevation • USL, upslope length • WI, wetness index • YL, crop yield


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
A GROWING INTEREST in site-specific management in precision farming demands a wealth of data on the spatial and temporal distribution of yield and growth conditions. Direct and repeated measurement of these variables, however, is resource intensive, and indirect estimation of these variables based on data from easily measured and relatively stable parameters (i.e., covariates) is desirable. Topographic indices are such covariates that can readily be measured and used for simulation and interpolation purposes (Bakhsh et al., 2000). Topographic indices became more appealing as readily available spatial data with the development and increased access to digital elevation models that provide important topographic information with increased accuracy (Moore et al., 1993; Wilson et al., 1998). In a nonlevel landscape, topographic features are the dominant factors determining soil physical properties and fertility status as well as controlling water distribution after snowmelt and rainfall. Consequently, crop yield (YL) and growth parameters such as total biomass output can be mapped based on topographic indices (slope, RE, aspect, CR, USL, etc.) to provide input data for varying production inputs such as fertilizer and for yield prediction before harvest.

Numerous studies indicate that the relationship between YL and topographic indices is scale dependent (Miller et al., 1988; Si and Farrell, 2004). Unfortunately, measurement and/or management may be done at different scales. Therefore, transfer of information from one scale to another (scaling) is essential. Inherent differences in scaling properties and the complex nature of spatial variability across scales requires the use of statistical tools that can reveal scaling properties at both local and global levels. Application of geostatistics (Miller et al., 1988), spectral analyses (Timlin et al., 1998), multifractal as well as joint multifractal analyses (Kravchenko et al., 1999, 2000), state-space analyses (Li et al., 2002), and wavelets (Si and Farrell, 2004) have been used as techniques to reveal certain levels of complexities that were overlooked by traditional statistical tools and monofractal analyses.

Multifractals have been found to represent many natural phenomena, including soil particle size distributions (Grout et al., 1998; Posadas et al., 2001), fluid flow in porous media (Liu and Molz, 1997; Boufadel et al., 2000), and spatial distribution of soil properties (Kravchenko et al., 1999). Multifractal behavior is associated with systems where the underlying physics are governed by a random multiplicative process (i.e., successive division of a measure and its geometric support based on a given rule), and systematic analyses of this behavior enable characterization of complex phenomena in a fully quantitative fashion (Stanley and Meakin, 1988). Lee (2002) described multifractal analyses as a technique that transfers irregular data into a more compact form and amplifies slight differences among the variables' distribution. He also remarked that the method can be used to investigate the relationship between a variable and the underlying physical processes responsible for its spatial distribution. Kravchenko et al. (1999) emphasized the advantages of multifractal analyses over geostatistical methods by stating "... variograms use only the first two statistical moments of the variable, while multifractal approach uses a wider range of statistical moments, providing a much deeper insight into data variability structure." These and other researchers stressed the strength of multifractal approach in analyzing and describing the distribution of random variables in spatial and temporal domains (Meneveau et al., 1990; Olsson, 1995).

Kravchenko et al. (2000) used multifractal and joint multifractal approaches to analyze YL variability as well as the relationship between YL and terrain slopes. They reported that both slope and YL distributions were multifractal measures. It was also found that joint multifractal analyses allowed differentiation between YL distributions corresponding to field locations with high and low slopes as well as making inferences about slope distributions that most affect YL. These authors, however, used only a single parameter of topography in their analyses (terrain slope), assuming slope at a given point was the dominant control of water and soil distribution. In semiarid zones, soil water distribution is dominated by snowmelt and rainfall runoff. Slope alone does not dictate the differences within knolls, midslope area, and depressions. Thus, the spatial and temporal variability of soil water in a semiarid area is a complex process determined by the interrelated effects of factors such as contributing area, CR, aspect, terrain slope, and vegetation cover.

In periods when precipitation continually exceeds evapotranspiration, the upslope area of a given point is the dominant control (Gomez-Plaza et al., 2001). Conversely, Sinai et al. (1981) reported that the degree of concavity or convexity largely influences water accumulation and runoff processes. Furthermore, factors controlling soil water distribution in humid regions are different from those in semiarid regions (Gomez-Plaza et al., 2001), rationalizing a need for independent studies of the two contrasting environments. Independent evaluation of the different topographic indices and their composites and identifying the ones that show similarity in scaling and spatial variability with YL and growth parameters are important in precision farming.

The objectives of this study were to: (i) characterize the scaling properties of YL, biomass, and four topographic indices (WI, RE, CR, and USL) using the theory of multifractal analyses and (ii) characterize the joint scaling properties of YL and topographic indices such as the WI and USL using the joint multifractal theory. We hypothesize that YL and different topographic indices have different scaling properties.


   THEORY
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Multifractal Analyses
The scaling property of a variable (such as YL and topographic indices) measured along a transect can be characterized using multifractal analyses as a mass distribution of a measure on a geometric support. This is achieved by dividing the transect into smaller and smaller segments based on a rule that generates a self-similar segmentation. The binomial multiplicative cascade is one such rule and can easily be described (Evertsz and Mandelbrot, 1992). Consider a unit interval associated with a unit mass, M (measure as used in a generalized case), and divide the unit interval into two segments of equal length. Also partition the associated mass into two fractions, h x M and (1 – h) x M, and assign them to the left and right segments, respectively. The parameter h is a random variable, governed by a probability density function, 0 ≤ h ≤ 1, and each new subinterval and its associated weight are successively divided into two parts following the same rule. A method of moments (Evertsz and Mandelbrot, 1992; Kravchenko et al., 2000) can then be used to determine the multifractal spectra of the measures.

The detailed procedure for calculating the multifractal spectrum of a measure is given in Chhabra et al. (1989), Meneveau et al. (1990), and Evertsz and Mandelbrot (1992). The singularity strength (the local scaling index), {alpha}(q), and the multifractal spectrum, f(q), were calculated as

[1]

[2]
where Pi({epsilon}) is the probability of a measure in the ith segment of size {epsilon} unit, µ(q,{epsilon}) is the partition function of moment order q, and L is the total length of the support. For multifractally distributed measures, the partition function scales with the segment size {epsilon} as (Evertsz and Mandelbrot, 1992; Meneveau et al., 1990)

[3]
where {tau}(q) is the mass or correlation exponent of order q. The connection between the scaling exponent f[{alpha}(q)] and {tau}(q) is made through a Legendre transformation as (Callen, 1985; Chhabra and Jensen, 1989)

[4]

The generalized (Rényi) dimension, Dq, was determined as

[5]

The Dq value at q = 0, D0, is called the capacity dimension or the box-counting dimension of the geometric support of the measure. The Dq value at q = 1, D1, is referred to as the information dimension and provides information about the degree of disorder in the distribution of the measure in analogy to the entropy of an open system in thermodynamics (Voss, 1988). Hence, the name entropy dimension is sometimes used for D1. From Eq. [5], D1 is undetermined because the denominator becomes 0. Evertsz and Mandelbrot (1992) recommended a graphical tool for determining D1. Here, we derived an analytical method by solving the first derivatives of the numerator and denominator of the right-hand side of Eq. [5] with respect to q and taking the limit of the resulting expression as q approaches 1 (i.e., l'Hôpitals rule), which provides,

[6]

The generalized dimension at q = 2, D2, is known as the correlation dimension and is mathematically associated to the correlation function and measures the average distribution density of the measure (Grassberger and Procaccia, 1983).

Joint Multifractal Analyses
The multifractal theory discussed above analyzes the distribution of a single variable (such as YL or RE) within or along its geometric support. It is more interesting to know the joint distribution of two or more multifractally distributed variables (such as YL vs. RE) on a common geometric support (for instance along a transect). This can be achieved by extending the single multifractal theory discussed above to the joint distributions of two or more interacting variables.

As in the multifractal analysis for a single variable, the geometric support is divided into segments of size {epsilon} and define the probability of the measure in the ith segment of the first variable as Pi({epsilon}) and the second variable as Ri({epsilon}). The local singularity strengths corresponding to these variables can then be defined as (Meneveau et al., 1990)

[7]
and

[8]
where {alpha} and ß are the local singularity strengths or Hölder exponents corresponding to Pi({epsilon}) and Ri({epsilon}), respectively.

Then the joint distributions of {alpha} and ß and the dimensions of the set resulting from the intersection of segments with iso-{alpha} and iso-ß values are needed for identifying the scaling property of one variable with respect to the other. If we let N{epsilon}({alpha}, ß)d{alpha}dß denote the number of segments of size {epsilon} with {alpha} values in the range {alpha} ± d{alpha} and ß values in the range ß ± dß, then it is possible to define the dimension f({alpha}, ß) of the set resulting from the intersection of segments with iso-{alpha} and iso-ß values as (Meneveau et al., 1990)

[9]

The dimension f({alpha}, ß) is thus the multifractal spectra of the joint distributions of the two variables considered. We will again use the direct method of obtaining f({alpha}, ß) using the method of µ-weighted averaging (Chhabra et al., 1989; Meneveau et al., 1990).

Extending the single multifractal analyses theory to the joint distributions of two variables, the partition function (the normalized µ measures) for the joint distributions of Pi({epsilon}) and Ri({epsilon}), weighted by the real numbers q and t, can be calculated by

[10]

The average value of {alpha} = ln[Pi({epsilon})]/ln({epsilon}/L) with respect to the µ measures is given by

[11]
and the average value of ß = ln[Ri({epsilon})]/ln({epsilon}/L) with respect to this µ measures is given by

[12]

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

[13]

When q or t is set to zero, the joint partition function shown in Eq. [10] reduces to the partition function of a single measure, and hence the joint multifractal spectrum defined by Eq. [13] becomes the spectrum of a single measure. When both q and t are set to zero, the maximum f({alpha}, ß) is attained, which is equal to the box dimension of the geometric support of the measures. Therefore, different pairs of {alpha} and ß 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
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Site Description
Detailed description of the site as well as the sampling procedures is reported in Si and Farrell (2004). Briefly, the study site is located in a semiarid environment at Alvena, SK, Canada. The geographical location of Alvena is 49°44' W and 107°35' N. This site has a hummocky landscape (Fig. 1) with a dominant soil type of an Aridic Ustoll. The texture of the soil at the site is loam to clay loam. Average annual air temperature at the site is 2.2°C, and the long-term average annual precipitation is 350 mm. The potential evapotranspiration is as high as 624 mm yr–1, resulting in a water deficit of 274 mm yr–1.



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  		Fig. 1. The spatial distribution of the six variables along the transect: (a) measured variables (RE = relative elevation, YL = yield, and BM = biomass) and (b) calculated variables (USL = upslope length, WI = wetness index, and CR = curvature). The distance was measured from south to north.

 
Crop Yield and Topographic Indices Measurement
A north-south 576-m-long transect was established on a non-level land with a sinusoidal type of landscape. On 26 Apr. 2001, spring wheat was seeded, and all the necessary management practices were implemented. Fertilizer was applied at a rate of 50 kg N ha–1. Growing season precipitation was 85 mm, which was only one-quarter of the long-term average. On 30 Aug. 2001, a 1.0-m2 area along the transect was selected at 6-m intervals and hand-harvested for determination of total aboveground biomass (BM) and grain yield.

Four topographic indices—RE, CR, USL, and WI—were determined using a Laser theodolite (Sokkisha Electronic Total Station, Set 5, Sokkisha, Tokyo, Japan). The CR was calculated using the approximation (Sinai et al., 1981; Pennock et al., 1994)

[14]

The derivatives in Eq. [14] were approximated using the empirical form

[15]
where Z is elevation at a point and i and j represent the indices for the x and y coordinates, respectively. Upslope length was calculated as the distance from the point of flow origin (drainage divide) to the measurement point. The WI of Beven and Kirkby (1979) was calculated as

[16]
where {gamma} is the contributing area per unit contour length and tan({phi}) is the local slope at a point.

Although the field measurements provided a total of 103 data points, only 96 points were used in the multifractal and joint multifractal analyses. Seven data points from either ends of the transect were removed for two reasons: (i) to obtain the number of data points suitable for dyadic segmentation and (ii) to avoid unnecessary border effects on the grain yield and biomass data.

Determination of Multifractal and Joint Multifractal Parameters
Grain yield, biomass, and four topographic indices were analyzed as a one-dimensional spatial data series. The data was converted into a probability distribution, Pi({epsilon}), by dividing the values of the measure in a given segment by the sum of the measures in the whole transect. These probability values, Pi({epsilon}), were viewed as a relative concentration of the measure in the segment of size {epsilon}. A dyadic (Lk = 2–kL) multiplicative cascade procedure was then used to divide the transect into subsequently smaller segments. The multiplicative procedure, which was applied to the 576-m transect (carrying the selected 96 data points), resulted in five segment sizes: 228, 144, 72, 36, and 18 m. These segments were used in the analyses of the scaling property of the measures along the transect. For those measures with multifractal scaling property, the multifractal and joint multifractal parameters were analyzed using 6-m segments, the minimum size defined by the sampling interval. The range of moment orders used in the analyses was –15 to 15 for multifractal analyses and –20 to 20 for joint multifractal analyses. Although the use of a wide range of moment orders enables one to obtain a wide range of the joint dimension, 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. The data values for the larger segment sizes were determined as the normalized sum of data values for the smallest segment sizes contained within the larger segment.

In calculating the multifractal parameters, we used the method suggested by Chhabra and Jensen (1989). This method is known to be superior over other methods for experimental data with low dimension or limited scaling range (Chhabra et al., 1989; Posadas et al., 2001). The {tau}(q) vs. q plots were used to identify if the scaling property of the variables can be categorized under single or multiple scaling. The distinction was made by fitting two linear lines (i.e., one for q < 0 and the other for q > 0) to the {tau}(q) vs. q data and testing if the slopes of the two lines were different. To this end, the Student's t test for significantly different means (Press et al., 1992, p. 617) was used.

In calculating 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). The correlation between the scaling exponents was used to quantify the degree of association between the scaling exponents of two variables at different intensity levels. This was obtained as follows: The joint partition function of the two variables for the selected moment orders, q and t values, was calculated using Eq. [10]. This partition function is defined using q values for the first variable and t values for the second one. Average values of the scaling indices of each variable with respect to the joint partition function were then determined using Eq. [11] and [12]. After calculating the scaling indices for selected pairs of q and t values (i.e., to obtain different intensity levels of the variables), the Spearman correlation coefficient between them was determined. Analyses were done using programs written in Mathcad 2000 Professional (Mathsoft Inc., Cambridge, MA) and statistical analyses software SAS Version 8 (SAS Inst., Cary, NC).


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
The data for RE, YL, and BM are shown in Fig. 1a (measured variables). High YL and BM values were observed for lower RE values, i.e., depression areas, with correlation coefficients (r) of –0.79 for YL and –0.80 for BM. Lower YL and BM values corresponded to the higher RE values (knoll areas), which is expected for semiarid environments where soil water is the major limiting factor. Figure 1b shows the relationship between the spatial distribution of the remaining three variables—USL, WI, and CR.

The major statistical and multifractal properties of YL, BM, RE, USL, WI, and CR are presented in Table 1. The highest coefficients of variation (CV) in the data was observed for CR and RE, 270 and 125%, respectively. The lowest variability was observed in WI, with a CV of 24%. The RE has a symmetrical distribution, all other variables having a slightly positively skewed distribution. The range of spatial dependence, as determined from sample variograms, was the highest for RE (16.2 m) and the lowest for CR (6.5 m) while the other four variables had similar values between these two.


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  		Table 1. Summary of the statistical and multifractal properties of crop yield (YL, kg ha–1), upslope length (USL, m), wetness index (WI), aboveground biomass (BM, kg ha–1), relative elevation (RE, m), and surface curvature (CR, m–1) data at the Alvena site.

 
Logarithmic plots of the partition functions, µq({epsilon}), against the measurement scales, {epsilon}, for YL, RE, USL,WI, BM, and CR are shown in Fig. 2 for moment orders of –15, –10, –5, –2, 0, 2, 5, 10, and 15. For YL, WI, USL, and BM (Fig. 2a, 2b, 2c, and 2e, respectively), the logarithm of µq({epsilon}) varied linearly with the logarithm of {epsilon} for all moment orders, resembling self-similar random measures. Their coefficient of determination (r2) was >0.99. However, for RE and CR (Fig. 2d and 2f, respectively), the r2 relationship between log µq({epsilon}) and log {epsilon}, especially for higher negative moments (q << –1), was <0.96. The distribution of a measure is considered as a fractal, mono- or multifractal, only if all the moments obey power laws; in other words, the double log plots of all the partition functions against the measurement scales should fit (i.e., r2 {approx} 1) a linear line (Evertsz and Mandelbrot, 1992; Lee, 2002). According to this rule, YL, USL, WI, and BM have a fractal distribution, whereas the distribution of the RE and CR did not show this relationship for higher negative moments. This suggests that monofractal and multifractal models may not adequately describe the spatial scaling structure of RE and CR for this and similar landscape morphology.



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  		Fig. 2. Natural logarithms of the normalized µ measures (partition functions) plotted against natural logarithms of the measurement scales for q values of 15 (filled triangle), 10 (open diamond), 5 (filled diamond), 2 (open square), 0 (filled square), –2 (open and inverted triangle), –5 (filled and inverted triangle), –10 (open circle), and –15 (filled circle). YL = yield, WI = wetness index, USL = upslope length, RE = relative elevation, BM = biomass, and CR = curvature.

 
The mass exponents, {tau}(q), of the variables were calculated with Eq. [3] and plotted for q values ranging from –15 to 15, in 0.5 increments (Fig. 3) . The {tau}(q) vs. q plots for YL, USL, and BM (Fig. 3a, 3b, and 3d, respectively) were obviously nonlinear with a convex downward shape. The slope of the fitted linear lines to the {tau}(q) vs. q plots of YL, USL, and BM were significantly different at a significance level of 0.001. However, for WI, the two slopes were not significantly different at a significance level of 0.001.



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  		Fig. 3. The mass exponent (of the qth moment) evaluated at 0.5 increments in q values for (a) yield, (b) upslope length, (c) wetness index, and (d) biomass.

 
The shape of {tau}(q) curve describes the type of scaling involved. A curve with a single slope implies a monofractal type scaling, whereas a convex downward curve with different slopes for negative and positive moment orders implies a multiple scaling (i.e., multifractal) (Evertsz and Mandelbrot, 1992; Olsson and Niemczynowicz, 1996). The {tau}(q) vs. q plots discussed above clearly shows the presence of multiple scaling in the spatial distribution of YL, USL, and BM and a single scaling (monofractal type) in that of WI. For variables with a single scaling property, the fractal dimension can be determined from the slope of the {tau}(q) vs. q plots for a range of q values. Accordingly, the fractal dimension of the distribution of WI was found to be 0.985 using a linear regression of the {tau}(q) vs. q values of WI (Fig. 3c). The implication was that the statistical properties of the distribution of WI across different scales can be related using a single value, i.e., the fractal dimension, unlike that of YL, USL, and BM. The closeness of this value to the capacity dimension of the measure (i.e., 1.0, which would have been obtained if WI had the same value through out the transect) also shows that WI had a relatively uniform distribution along the transect.

The marked differences in the range of Dq values between the variables (Fig. 4) for all analyzed moment orders clearly show differences in the scaling properties of these variables. This is substantiated by the variations in the absolute differences among the magnitudes of selected Dq values, such as the capacity, information, and correlation dimensions (D0, D1, and D2, respectively) within a given variable (Table 1). The Dq values for YL, USL, and BM showed a strong dependence of Dq on values of q, which is a typical characteristic of multifractals. For WI, however, Dq showed only a slight dependence on values of q. For higher data values (positive q side), YL, BM, and USL have similar scaling properties (Fig. 4). For lower data values (negative q side), there were slight differences between these variables. Nonetheless, the differences between these three variables were much smaller than the difference between WI and any of the three. The difference between the capacity dimension (D0) and the information dimension (D1), which can be used as a heterogeneity index (Caniego et al., 2003), was 0.01 for WI, 0.04 for BM, 0.07 for USL, and 0.04 for YL (Table 1), again indicating the least heterogeneity in the overall distribution of WI. The closeness among D0, D1, and D2 values of WI is another important observation depicting the monofractal tendency (Turcotte, 1997) of this variable. This monofractal scaling property observed in the spatial distribution of WI appears to be introduced by the slope angle, which has been known to have a monofractal scaling (Huang and Turcotte, 1989; Moore et al., 1993).



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  		Fig. 4. The generalized dimensions (Dq) at –10 to 10 ranges of q evaluated at 0.5 increments. YL = yield, USL = upslope length, WI = wetness index, and BM = aboveground biomass.

 
The f({alpha}) spectrum (Fig. 5) and selected multifractal parameters such as the difference between the maximum and minimum values of {alpha} and f({alpha}) (Table 1) provided the characteristics of the dimension function and the range of the local scaling exponents. The width of the f({alpha}) spectrum (i.e., {alpha}max – {alpha}min) showed the degree of heterogeneity in scaling property of the variables. The YL, USL, and BM variables exhibited a wide range of {alpha} values, reinforcing the claim that these variables have a multifractal scaling nature. Conversely, WI showed a narrow range of {alpha}, which was more than four times smaller than the scaling range of the other three variables. The range of {alpha} values decreased in the order of USL > YL > BM > WI, implying that USL had the most heterogeneity and WI had the least.



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  		Fig. 5. The multifractal spectrum [f({alpha})] for yield (YL), upslope length (USL), wetness index (WI), and biomass (BM) for q values of –10 to 10, evaluated at 0.5 increments.

 
The overlap of the left-hand side of the spectra (q > 0) for YL, USL, and BM (Fig. 5) shows a strong similarity in scaling behavior of these three variables. For the lower data values (q < 0), however, the spectra for each variable were divided, especially for the USL, which showed a marked difference from the other two variables. Consequently, for lower data values (i.e., at knoll positions), USL did not have exactly the same scaling behavior as that of YL.

To further understand the physical meaning of multifractal parameters, correlation analyses between traditional statistical parameters and multifractal parameters were performed, and the results are summarized in Table 2. The variance, skewness, and kurtosis (Table 1) did not show significant correlations with any of the multifractal parameters analyzed. The CV, however, showed a positive correlation of 0.99 with {alpha}max – {alpha}min (significant at 0.001 probability level) and a negative correlations of 0.96 and 0.99 with D1 and D2, respectively. The range of autocorrelation determined from variogram showed a significant (at 0.001 probability level) negative correlation of r = –0.84 with D1. The observed correlations for CV and the variogram range were consistent with the results reported by Kravchenko et al. (1999) based on two-dimensional spatial data on soil properties.


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  		Table 2. Correlation coefficients (r) between the statistical and multifractal parameters evaluated across the parameters within the six variables studied.

 
Figure 6 shows contour plots of the joint multifractal spectra, f({alpha}, ß), as separate functions of the local fractal dimensions (coarse Hölder exponents) {alpha}(q) and ß(t). The r among the local fractal dimensions of the paired variables was also analyzed at a range of intensity levels (high, low, and intermediate) of the variables (Table 3). Inherent to all the plots (for all the pairs) was the existence of a unique maximum value that was attained when both q and t values were set to zero; there was a slight divergence (from closed and smooth contour lines) of the contours of lower dimensions depending on the range of exponents analyzed. The maximum value, referred to as the capacity dimension of the geometric support of the measures, was equal to 1.0 for all the pairs as would be expected for the one-dimensional spatial data.



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  		Fig. 6. Joint multifractal spectrum, f({alpha}, ß), for the joint distribution of (a) yield (YL) vs. upslope length (USL), (b) YL vs. wetness index (WI), (c) YL vs. aboveground biomass (BM), and (d) USL vs. WI evaluated for q and t values (order of exponents) ranging from –20 to 20.

 

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  		Table 3. Correlation coefficients (r) among the singularity exponents [{alpha}(q) and ß(t)] of the paired variables at selected moment orders (q and t combinations).

 
The r between the scaling exponents of YL [{alpha}(q)YL] vs. BM [ß(t)BM] (Table 3) was 0.99, implying a strong association between the local fractal dimensions of YL and BM. The contour plots for the joint multifractal spectrum of YL vs. BM (Fig. 6a) showed high correlation (perfectly diagonal gradient) between the scaling exponents of the two variables and were instrumental in portraying the type of contour surfaces that should be expected when the two variables are highly correlated at all scales. The diameter of the ellipse across the tilt is small compared with the one along the tilt, implying low variability in the joint scaling indices of YL and BM. Moreover, the contours are stretched equally along the horizontal and vertical axis, implying similarity in the degree of variability in the scaling indices of the two variables.

The contour plots for the joint dimensions of YL vs. USL are shown in Fig. 6b. These plots are elliptical and tilted to the right, indicating that higher scaling exponents of YL ({alpha} values) were associated with higher scaling exponents of USL (ß values). Therefore, there was a positive correlation between the local scaling indices of these two variables. Furthermore, the plots are stretched equally in the vertical and horizontal direction, indicating the presence of similar degree of variability in the scaling indices of USL and YL. On the other hand, Fig. 6c shows poor correlation between the scaling exponents of WI and YL as the higher scaling indices of YL did not correspond to the higher scaling exponents of WI. The plots were more stretched in the horizontal direction than that of the vertical, implying higher variability (uncertainty) in the scaling indices of YL than that of WI.

The r between the scaling exponents of YL [{alpha}(q)YL] vs. USL [ß(t)USL] and YL [{alpha}(q)YL] vs. WI [ß(t)WI] (Table 3), for all the analyzed q and t values (–20 to 20 at the interval of 0.5), were 0.93 and 0.65, respectively, implying high association between the local scaling indices of YL and USL in general. High correlation between the scaling indices of YL and WI was observed only at large values of both variables. The scaling indices for low yield values (q = –20) did not show significant correlations with that of WI but showed a negative correlation with USL.

The similarity in scaling properties and high correlation in scaling properties between YL and USL indicates that for yield-modeling purposes in the area (semiarid environment), the USL appears to be the superior topographic index over the classic WI. Considering the relationship between WI and USL, i.e., WI = ln[USL/tan({phi})] (Beven and Kirkby, 1979), the obvious implication was that both the small-scale and large-scale statistical properties of yield are related more to the USL than to the local slope [tan({phi})]. The two topographic indices (USL and WI) had similarities only at level positions and depressions where WI attains large values. This appears to be the consequence of minimum differences in slope angle [tan({phi})], which is the only difference between the two topographic indices. The results from local wavelet spectrum analyses by Si and Farrell (2004) and a correlation study by Western et al. (1999) also support the above observations signifying the superiority of USL as topographical index over the commonly used WI under semiarid environments.

Although some of the observations made on the scaling properties of YL, BM, USL, and WI using joint multifractal analyses had certain similarities to what was obtained from the single multifractal analyses, joint multifractal analyses have provided more information about the scaling properties at various data ranges of the variables involved by formulating a joint dimension function. In practical applications, the information from the joint multifractal analyses can be utilized for site-specific management purposes.

The strong association between USL and YL in semiarid regions may be explained as follows. Water is a limiting factor in this region, and snowmelt is the main mechanism for water distribution. As the slope length increases, more snow will accumulate in the lower area (i.e., areas with higher USL), increasing soil water content. Hence, the spatial distribution of YL is expected to follow the same pattern as that of USL compared with the other topographic indices.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
Multifractal and joint multifractal analyses have been used to characterize the spatial distribution of YL, biomass, and four topographic indices [USL, WI of Beven and Kirkby (1979), RE, and CR] in a semiarid environment. Scaling property analyses showed that YL, biomass, and USL had a similar scaling property, which could be represented by a multifractal type of scaling. The spatial distribution of the WI of Beven and Kirkby (1979) showed a strong tendency toward monofractal type of scaling. Analyses of the level of heterogeneity in spatial distribution of the four variables using the generalized fractal dimensions and f({alpha}) spectrum showed that the distribution of YL and USL was the most heterogeneous one, whereas that of WI was the least. Comparative study using joint multifractal analyses also showed similar spatial distribution among YL, biomass, and USL. Wetness index showed similar scaling property to that of yield only at level positions and depressions. Results suggested that, in terms of scaling property and spatial variability, the distribution of yield and biomass was better reflected in USL than other topographic indices studied in here. The implications for precision agriculture are (i) USL can be used as a guideline for varying production inputs such as fertilizer, especially when detailed recommendations at a given scale of interest are not available, and (ii) site-specific prediction of the final YL (i.e., before harvest) can be made using USL regardless of scale.


   ACKNOWLEDGMENTS
 
The research was funded by the National Science and Engineering Research Council of Canada (NSERC). Technical help from Shawn Campbell and Lindsay Tallon is appreciated. We thank Bob Hilkewich and his family for allowing us unlimited access to their farm field.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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