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Agronomy Journal 92:1279-1290 (2000)
© 2000 American Society of Agronomy

SPATIAL VARIABILITY

Joint Multifractal Analysis of Crop Yield and Terrain Slope

Alexandra N. Kravchenko, Donald G. Bullock and Charles W. Boast

Dep. of Nat. Resources and Environ. Sci., Univ. of Illinois, 1102 S. Goodwin Ave., Urbana, IL 61801-4798 USA

dbullock{at}uiuc.edu


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Materials and methods
 Results and discussion
 Summary and conclusions
 REFERENCES
 
Quantifying the spatial variability of crop yields and yield-affecting factors are important issues in precision agriculture. Topography is frequently one of the most important factors affecting yields, and topographical data are much easier to obtain than time and labor-consuming measurements of soil properties. In this study, yield variability and the relationships between yields and terrain slopes were analyzed using theories of multifractal and joint multifractal measures. Corn (Zea mays L.) and soybean [Glycine max (L.) Merr.] yield data from 1994 to 1998 were collected via yield monitors from a central 6.6 ha section of an agricultural field in eastern Indiana. Slopes were derived from a field terrain map using a GIS. Multifractal analysis of yield and slope maps revealed that both yield and slope distributions were multifractal measures. Hence, joint multifractal analysis was applied to evaluate the effect of slope on crop-yield spatial variability. Joint multifractal analysis facilitated (i) the ability to differentiate between yield distributions corresponding to field locations with high and low slopes, and (ii) the ability to make inferences about slope distributions that affect grain yield the most. Multifractal analysis revealed that during four growing seasons with moderate and dry weather conditions, larger yields were observed at low slope locations while a wide range of yield values was observed at sites with moderate and high slopes. During the wet growing season, lower yields prevailed at locations with low slopes. Joint multifractal theory was useful for the study of yield/topography relationships and was an applicable tool for the analysis of spatially distributed data.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Materials and methods
 Results and discussion
 Summary and conclusions
 REFERENCES
 
SITE-SPECIFIC MANAGEMENT BENEFITS from a thorough quantitative description of spatial and temporal within-field variability of crop yields and the factors that affect them. Topography is regarded as one of the most important factors affecting yields (Changere and Lal, 1997; McConkey et al., 1997) and as a source of easily obtained information that is useful for soil and field characterization (de Bruin and Stein, 1998; Moore et al., 1993; Odeh et al., 1994). The complexity of yield–topography relationships often cannot be appropriately characterized by traditional statistical methods. More exhaustive characterization can be achieved by using advanced statistical and mathematical procedures such as geostatistics, time series analysis, or fractal analysis. For example, Miller et al. (1988) used geostatistics to study the spatial variability and yield–landscape relationships for wheat (Triticum aestivum L.) yield on a 400- by 200-m study site and found geostatistics to be superior to either correlation or multiple regression for a yield–soil–topography analysis. Spectral analysis was utilized by Timlin et al. (1998) to study the influence of topographic location and surface curvature on corn grain yield based on data from 140 plots and five transects. Eghball et al. (1997) used fractal theory to characterize the spatial and temporal variability in corn grain yield as affected by N treatments. Fractal analysis was found to be useful for describing the temporal yield variability for 10 crops in the USA, including corn and soybean (Eghball and Power, 1995). It has also been found to be useful for depicting the effects of manure and fertilizer applications on corn yield variability (Eghball et al., 1995).

Multifractal analysis (Mandelbrot, 1974) has been utilized successfully to characterize several factors that affect yields, including rainfall (Olsson and Niemczynowicz, 1996), soil strength (Folorunso et al., 1994), soil particle size distribution (Grout et al., 1998), soil P and K concentrations, and organic matter content (Kravchenko et al., 1999). It has also been shown to provide additional detailed information about soil spatial variability compared with traditional fractal approaches (Folorunso et al., 1994). Multifractal analysis is applicable to variables that can be regarded as multifractal measures, i.e., variables self-similarly distributed on a geometric support that is represented by a plane, volume, or fractal set (Feder, 1988). As an example of a multifractal measure, let us consider the distribution of groundwater within a certain geographical area (Evertsz and Mandelbrot, 1992). If this area is divided into two equally sized parts, the groundwater contents of each part will be different even though the areas for both parts are equal. If one of the parts is further subdivided into two equally sized pieces, their corresponding groundwater contents will again be different. This subdivision can be continued until the amount of water contained within one rock pore will be different from that of another. That is, the distribution of groundwater is irregular at all scales. If the irregularity in the variable's distribution remains statistically similar at all studied scales (Evertsz and Mandelbrot, 1992), then the variable is assumed to be self similar or multifractal.

An extension of multifractal theory for the analysis of more than one variable was developed by Meneveau et al. (1990) and is called joint multifractal theory. Joint multifractal theory can be used for the simultaneous analysis of several multifractal measures existing on the same geometric support, and hence for quantifying the relationships between the measures studied. If crop yields and soil properties or topographical features of the field are shown to be multifractal measures, then joint multifractal analysis can be applied to study the influence of soil properties or field topography on crop yields.

The first objective of this research was to examine the applicability of multifractal analysis for describing and quantifying crop yield spatial variability. The second objective was to apply joint multifractal theory for analyzing crop yield–topography relationships.


   Materials and methods
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
Results and discussion
 Summary and conclusions
 REFERENCES
 
Data
The studied area was a square-shaped 6.6 ha central portion of a 50 ha agricultural field located in eastern Indiana. The field was in a corn–soybean rotation during the study period and for at least several decades before it. The dominant soil of the field was a fine-loamy, mixed, mesic Aquic Hapludalf.

Corn and soybean grain yield data were collected with yield monitors (Ag Leader Technol., Ames, IA) in 1994 through 1998. Geo-referenced yield measurements were recorded on a 1-s interval integrating across an area of about 10 m2 (2 m is the avg. forward distance traveled by the combine during 1 s, and 5 m is the width of the combine header). Due to terrain irregularities and combine speed variations and dynamics, yield monitors are prone to produce errors in the yield measurements and coordinate values of individual data points (Birrell et al., 1994). To smooth out the effect of possible errors in individual data points, the yield measurements were interpolated, and interpolated data were used in further analysis instead of the actual data. The inverse-distance interpolation method provided by ArcView Spatial Analyst (ESRI, 1996), with a power to distance of 2 and a no. of the closest neighboring points equal to 12, was used to convert yield point data into cell-based maps where interpolated yield values were obtained for the centers of each cell. Each yield map consisted of a 32 by 32 array of 8-m square map cells. The choice of the inverse-distance parameters and map cell size was dictated by necessity to balance two counteracting goals: (i) averaging out yield measurement errors without oversmoothing and (ii) obtaining a sufficient number of map cells for further multifractal and joint multifractal analyses.

Elevation measurements were made with SOKKIA SET 5 total station (SOKKIA, Overland Park, KS). The distance between the measurements varied from 2 to 50 m, depending on the complexity of the terrain. Measurements on the level parts of the field were made at larger distances while marked depressions and hills were measured more intensely. The elevation measurements were also converted into cell-based terrain maps using ArcView Spatial Analyst because the number of elevation data points in the studied area was not sufficient for the multifractal data analysis. As a result of the elevation measurement strategy that was employed in the study, areas with diverse topography were represented by a large number of measurement points while homogeneous areas had less elevation measurements. Therefore, a relatively high accuracy in the interpolated terrain map was reached using a minimum number of measurement points. Inverse-distance weighting, with a power to distance of 2 and six closest neighboring points, was used as an interpolation method for creating the terrain maps. The number of closest neighboring points was selected so that it would be sufficient for estimation at sparsely sampled level areas, and at the same time, would not produce oversmoothing in densely sampled depressions/hills. The terrain slope was derived from the terrain map on the same cell basis as the yield maps (ESRI, 1996). The tangent of the slope was calculated as a ratio of the difference in elevation between the centers of adjacent cells to the horizontal distance between them. The slope for each cell was obtained based on a set of 3 by 3 neighboring cells using the average maximum technique (Burrough, 1986, p. 50) and was measured in degrees. Using the map cell values instead of the actual data in further analysis provided an easy and reliable way to study the relationships between the variables because neither the yield nor the elevation data were measured at exactly the same locations. Figure 1 shows the elevation measurement scheme in the studied area and the terrain map.



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  		Fig. 1 Locations of the elevation measurements taken with a SOKKIA SET 5 total station (SOKKIA, Overland Park, KS) and the elevation map used for deriving the terrain slope. The elevation map was obtained by inverse-distance weighting of the measured data with power of 2 and the no. of the closest neighbors equal to 6. The map cell size is equal to 8 m

 
Multifractal Theory
Multifractal Spectrum
A method of moments is used to compute multifractal spectra of the crop yield data and terrain slope (Evertsz and Mandelbrot, 1992). The interpolated yield and slope maps represent distributions of the studied variables, and these distributions are hypothesized to be multifractal measures on a geometric support of the studied field. The measure contained in each map cell i of the size {epsilon} is characterized by a probability mass function

(1)

Five cell sizes ranging from 8 to 128 m were used in the study. The yield and slope values for the cells of the smallest size , µi, were obtained directly from the maps, and µsum was calculated as the sum of all µi values from the studied field. For the following cell sizes, the variable values for each cell were calculated as the sum of the µi values of the cells included in the cell of that size (Kravchenko et al., 1999). The larger the number of cell sizes considered in the study, the more accurate is the multifractal spectrum. However, calculations for a wide range of cell sizes require a large number of the cells of the smallest size. For example, in this study, the number of the smallest cells was equal to 1024 (corresponds to 32 by 32 map cells of the study area), with the numbers of cells of the following four sizes equal to 256, 64, 16, and 4, respectively. Hence, the data for multifractal calculations with the method of moments should be either very large grid-sampled data sets or outputs of reliable interpolations of data that are not grid sampled or sparse such as those used in this study.

For multifractal measures, the probability mass function of the cell, µi({epsilon}), scales with the cell size as

(2)
where {alpha} is called a coarse-grained Hölder exponent or a singularity strength. The number of cells of size {epsilon} with {alpha} values falling within an {alpha} to {alpha} + d{alpha} interval, N{alpha}({epsilon}), scales with the cell size as

(3)
where the exponent f({alpha}) characterizes the abundance of cells with a certain {alpha}. The maximum value of f({alpha}) is equal to the box-counting dimension of the geometrical support of the studied measure. In this study, the maximum value of f({alpha}) is equal to 2 (box-counting dimension of a plane). The parameters {alpha} and f({alpha}) characterize the spatial variability of the measure by describing its local scaling properties ({alpha}) and numbers of locations where certain scaling properties are observed [f({alpha})].

The method of moments estimates {alpha} and f({alpha}) values based on a partition function, {chi}q({epsilon}), calculated from the µi({epsilon}) values as

(4)
where n is the total number of cells of the size {epsilon}, and q is a real number ranging from -{infty} to {infty}. Following the derivation provided by Evertsz and Mandelbrot (1992), the partition function can be expressed as

(5)
where µi({epsilon}) is replaced with its equivalent from Eq. [2], and the sum is represented as an integral evaluating the contribution of cells with {alpha} values within a range of {alpha} to {alpha} + d{alpha} (Eq. [3]). A mass exponent of order q, {tau}(q), defining the scaling properties of the partition function

(6)
is related to the {alpha} and f({alpha}) values of the studied measure as

(7)

The singularity strength, {alpha}, and the parameter f({alpha}) are determined by a Legendre transformation of the {tau}(q) curve (Evertsz and Mandelbrot, 1992) as

(8)

(9)

From the definition of the partitioning function (Eq. [4]), {chi}q({epsilon}) is determined by a large µi({epsilon}) for high positive q values while small µi({epsilon}) values contribute to {chi}q({epsilon}) the most for high negative q values. Hence, the parameters {alpha}(q) and f[{alpha}(q)] at a low q are mainly influenced by cells with low variable values, and at high q, they are defined primarily by the properties and abundance of cells with high variable values. A plot of f[{alpha}(q)] vs. {alpha}(q) for a range of q values is called a multifractal spectrum.

The mass exponent for each of the q values was obtained by plotting log {chi}q({epsilon}) vs. log {epsilon}. If the plot is a straight line, then the method of moments is applicable for analyzing the measure on the selected scale range. In this study, the plots were fitted with linear equations, and the least-square fitting procedure was used to find the slope of the log {chi}q({epsilon})/log {epsilon} line. The decision about the linearity of the plots was made in each particular case by visual inspection (Evertsz and Mandelbrot, 1992) assisted by an analysis of the linear regression coefficients. After the {tau}(q) values were calculated, the multifractal spectra for the studied variable were obtained using Eq. [8] and [9]. Because very high and very low data values are relatively scarce in most data sets, the linearity of the log {chi}q({epsilon})/log {epsilon} line at very low or high q values is more likely to be distorted. In this study, we considered q values ranging from -3 to 3 in 0.2 increments. For this range of q values, the log {chi}q({epsilon})/log {epsilon} plots of all of the variables used in the study were linear, and hence valid multifractal spectra could be constructed using the method of moments.

The steps of the multifractal calculations using the method of moments can be summarized as follows: (i) the probability mass functions are calculated for the cells of smallest size based on the actual data (Eq. [1]); (ii) the probability mass functions are calculated for the cells of the larger sizes based on the probability mass functions of the smallest size (Eq. [1]); (iii) the partition functions are calculated for each cell size {epsilon} and for each q value from the selected q range (Eq. [4]); (iv) for each q, the partition functions are plotted vs. the corresponding cell sizes in a log-log format; (v) log-transformed plots are checked for linearity, and mass exponent {tau}(q) values are determined as slopes of the linear plots (Eq. [6]); (vi) the multifractal parameters {alpha}(q) and f[{alpha}(q)] are calculated based on the {tau}(q) values for each q (Eq. [8] and [9]). Further details on the method theory and calculation procedure are available in Multifractal Measures by Evertsz and Mandelbrot (1992).

Joint Multifractal Spectrum
Joint multifractal spectra for corn and soybean grain yields vs. terrain slope were obtained based on the procedure proposed by Meneveau et al. (1990). The probability mass functions for two measures coexisting on the same geometrical support (e.g., an agric. field) are defined as

(10)

(11)
where superscripts 1 and 2 refer to the first and second studied variables, respectively. Because both of the studied variables are hypothesized to be multifractal measures, their probability mass functions scale with cell size as

(12)

(13)
where {alpha}1 and {alpha}2 are the corresponding singularity strengths of the two measures. The number of cells with a singularity strength for the first variable within a range from {alpha}1 to {alpha}1 + d{alpha}1 and with a singularity strength for the second variable ranging from {alpha}2 to {alpha}2 + d{alpha}2, N{epsilon} ({alpha}1, {alpha}2), scales with cell size as

(14)
where the parameter f({alpha}1,{alpha}2) is a characteristic describing the abundance of cells with common {alpha}1 and {alpha}2 values.

A joint partition function is calculated from the probability mass functions of the two variables weighted by the real numbers q1 and q2 as

(15)

When q1 or q2 is equal to 0, the joint partition function in Eq. [15] becomes equal to the partition function of a single variable (Eq. [4]), and the joint spectrum reduces to a single multifractal spectrum described above. As for a single multifractal spectrum, at low q1 or q2, joint partition function values are defined mostly by the low values of the first or second variable while at high q1 or q2, the partition function depends mostly on high data values. The maximum f({alpha}1, {alpha}2) value of the joint multifractal spectrum is equal to the box-counting dimension of the geometrical support, and it is reached when both q1 and q2 are equal to zero. The mass exponent of order q1/q2, {tau}(q1, q2), is obtained by analyzing the scaling properties of the joint partition function

(16)

The {tau}(q1, q2) values were obtained as slopes of the log-transformed joint probability mass functions plotted vs. the log-transformed cell sizes. The linearity of the plots was determined as described above for a single multifractal spectrum and the only values used in further analysis were {alpha}1, {alpha}2 and f({alpha}1, {alpha}2), which were obtained based on the {tau}(q1, q2) from linear plots. Double Legendre transformations of the {tau}(q1, q2) curve result in expressions for calculating the singularity strengths for the two measures and the joint f({alpha}1, {alpha}2) value as

(17)

(18)

(19)

To demonstrate how various cases of the relationships between variables can be described by joint multifractal spectra, we simulated three two-variable data sets using the simulated annealing procedure from the software package GSLIB (Deutsch and Journel, 1998). Each of the data sets consisted of 1024 data points located in a regular grid pattern. The distance between the points corresponded to the minimum cell size that was used for the multifractal calculations, i.e., 8 m. In the first data set, the variables were positively correlated and in the second data set, they were uncorrelated . For the third data set, the variables were uncorrelated for the bulk of the data while most of the highest values of the first variable were collocated with the lowest values of the second variable. This data set was obtained by manually modifying the simulated uncorrelated data. It represents the case where a positive or negative influence of one variable on another exists only at a certain range of variable values. Scatter plots of the three data sets are shown in Fig. 2a through 2c along with their joint multifractal spectra. Figure 2 also shows single multifractal spectra for the second variable (Variable 2) that were obtained at the lowest and highest available q1 values, -3 and 3, respectively. These single multifractal spectra allow for the studying of the distribution of the values of Variable 2 that correspond to the lowest and highest values of Variable 1. This is because the lowest values of Variable 1 contribute most to the multifractal spectrum at q1 = -3 while the highest values contribute most at q1 = 3.



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  		Fig. 2 Scatter plots of the three simulated data sets with correlation coef. between the two variables equal to (a) 0.91, (b) 0.00, and (c) -0.12. Joint multifractal spectra for the two variables and single multifractal spectra for the second variable at the lowest and highest q values of the first variable demonstrate the effect of data correlation on the shape of the joint and single multifractal spectra

 
The high correlation between {alpha}1 and {alpha}2 that was observed in the joint multifractal spectrum of the first data set is consistent with the high correlation between the two variables (Fig. 2a). The single multifractal spectrum of Variable 2 at q1 = -3 is clearly asymmetrical with a longer right part (higher {alpha}2 values), which indicates that the lower values of Variable 1 corresponded to the lower values of Variable 2. At q1 = 3, the multifractal spectrum indicates that the higher values of the two variables were also located together.

For the uncorrelated data set (Fig. 2b), the single multifractal spectra of Variable 2 were symmetrical and similar in width for both , indicating that the whole range of Variable 2 values can be found at locations with both low and high Variable 1 values. The partial nature of the correlation between the variables in the third data set is reflected in the joint multifractal spectrum (Fig. 2c). The single spectra for q1 = -3 and q1 = 3 are noticeably different. The spectrum at q1 = -3 is asymmetrical with a longer left part, implying that the higher values of Variable 2 dominated the data distribution at the low values of Variable 1. The spectrum at q1 = 3 was relatively symmetrical, and indeed, the whole range of Variable 2 data was present at the high values of Variable 1, except for a few very high values. These three data sets are simple examples with apparent relationships between the variables that can be easily detected from the scatter plots and correlations. For real data, the relationships between the variables are rarely that simple, and hence the more valuable the information is that can be obtained from joint multifractal spectra. This information can aid in depicting those aspects in the relationships between variables that might otherwise be missed by traditional methods of data analysis.


   Results and discussion
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
 Results and discussion
Summary and conclusions
 REFERENCES
 
A statistical summary of the corn and soybean grain yield data is shown in Table 1 . The corn yields were less variable than the soybean yields with lower coefficients of variation and negatively skewed distributions with high kurtosis. The distribution of the soybean yield was positively skewed in all 3 yr with low kurtosis values. The average terrain slope for the study area was equal to 1.17°, with a minimum slope of 0.00° and a maximum of 4.66°. The slope was negatively correlated with the yield for 4 out of 5 yr, with correlation coefficients equal to -0.11, -0.18, -0.07, and -0.07 for the yield data from 1994, 1995, 1996, and 1998, respectively . There was no significant correlation between corn yield and slope in 1997.


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  		Table 1 Statistical summary of the crop yield data

 
Figures 3a, 4a, 5a, 6a, and 7a show the multifractal spectra for grain yields in 1994, 1995, 1996, 1997, and 1998. The joint multifractal spectra for the studied yields are presented in Fig. 3b, 4b, 5b, 6b, and 7b. For clarity, the joint multifractal {alpha} values for the two studied variables, slope and yield, are denoted as {alpha}slope and {alpha}yield in further discussion and in the figures, corresponding to {alpha}1(q1, q2) and {alpha}2(q1, q2) from Eq. [17] and [18]. The values of q1 and q2 are denoted as qslope and qyield.



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  		Fig. 3 (a) Multifractal spectrum, (b) joint multifractal spectrum, and yield multifractal spectra at (c) low and (d) high slope values for the 1994 corn yield. The notations {alpha}yield and {alpha}slope correspond to {alpha}1(q1, q2) and {alpha}2(q1, q2) defined by Eq. [17] and [18], respectively. Intersections of the horizontal and vertical lines mark the points with the max. f({alpha}1, {alpha}2) values [f({alpha}1, {alpha}2) = 2] that are reached at q1 = q2 = 0

 


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  		Fig. 4 (a) Multifractal spectrum, (b) joint multifractal spectrum, and yield multifractal spectra at (c) low and (d) high slope values for the 1995 corn yield. The notations {alpha}yield and {alpha}slope correspond to {alpha}1(q1, q2) and {alpha}2(q1, q2) defined by Eq. [17] and [18], respectively. Intersections of the horizontal and vertical lines mark the points with max. f({alpha}1, {alpha}2) values [f({alpha}1, {alpha}2) = 2] that are reached at q1 = q2 = 0

 


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  		Fig. 5 (a) Multifractal spectrum, (b) joint multifractal spectrum, and yield multifractal spectra at (c) low and (d) high slope values for the 1996 corn yield. The notations {alpha}yield and {alpha}slope correspond to {alpha}1(q1, q2) and {alpha}2(q1, q2) defined by Eq. [17] and [18], respectively. Intersections of the horizontal and vertical lines mark the points with max. f({alpha}1, {alpha}2) values [f({alpha}1, {alpha}2) = 2] that are reached at q1 = q2 = 0

 


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  		Fig. 6 (a) Multifractal spectrum, (b) joint multifractal spectrum, and yield multifractal spectra at (c) low and (d) high slope values for the 1997 corn yield. The notations {alpha}yield and {alpha}slope correspond to {alpha}1(q1, q2) and {alpha}2(q1, q2) defined by Eq. [17] and [18], respectively. Intersections of the horizontal and vertical lines mark the points with max. f({alpha}1, {alpha}2) values [f({alpha}1, {alpha}2) = 2] that are reached at q1 = q2 = 0

 


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  		Fig. 7 (a) Multifractal spectrum, (b) joint multifractal spectrum, and yield multifractal spectra at (c) low and (d) high slope values for the 1998 corn yield. The notations {alpha}yield and {alpha}slope correspond to {alpha}1(q1, q2) and {alpha}2(q1, q2) defined by Eq. [17] and [18], respectively. Intersections of the horizontal and vertical lines mark the points with max. f({alpha}1, {alpha}2) values [f({alpha}1, {alpha}2) = 2] that are reached at q1 = q2 = 0

 
The multifractal spectrum of the corn grain yield in 1994 (Fig. 3a) has an asymmetrically long right part with the {alpha} reaching 2.3 and the f({alpha}) descending to as low as 1.3. High {alpha} values indicate the presence of areas with extremely low yields while low f({alpha}) values suggest that the number of cells that cover low yield areas is relatively small. The overall distribution of corn yield over the field can be described as relatively homogeneous because only a few points on the multifractal spectrum have high {alpha} and low f({alpha}) values. An analysis of the joint multifractal spectrum (Fig. 3b) reveals certain trends in the 1994 relationship between the corn yield and terrain slope. The widest range of {alpha}yield values (from 2.37 at ) is observed at . Hence, we can infer that the areas of extremely low yields that cause a pronounced asymmetry in the yield multifractal spectrum (Fig. 3a) are associated with slopes that are lower than average (but not the lowest slopes). The yield multifractal spectrum at the lowest available qslope value, -3, is asymmetrical with the longer part corresponding to lower {alpha}yield values, i.e., higher yields (Fig. 3c). This indicates that at the lowest slopes, most of the yields were relatively high. The yield multifractal spectrum at (Fig. 3d) was more narrow than the spectrum at (Fig. 3a) and more symmetrical than the spectrum at (Fig. 3c) although its left part was lower than the right. The narrow spectrum indicated that the yield variability at locations with high slopes was less than the total field yield variability and that none of the locations with extremely low yields that caused asymmetry in the 1994 yield spectrum (Fig. 3a) had high slope values. Symmetry of the spectrum implies that, except for the extremely low yields, a wide range of yield values was observed at the locations with high slopes.

The spectra for soybean yields (1995, 1996, and 1998) are relatively symmetrical, with the right portions of the spectra just a little longer than the left portions (Fig. 4a, 5a, and 7a). The spectrum for soybeans in 1995 (Fig. 4a) is wider than the those of 1996 and 1998, with {alpha} ranging from 2.21 at , which indicates that the overall variability of the soybean yield in 1995 was higher than that in the other years. This observation is consistent with the highest coefficient of variation of the soybean yield in 1995 (Table 1). An analysis of the joint multifractal spectrum for 1995 (Fig. 4b) shows that substantially different yield distributions existed at low and high slopes. At high slopes, , the range of {alpha}yield values is relatively large (from 2.19 at ), i.e., the whole range of yield values from the minimum to maximum yields is observed at high-slope locations (Fig. 4d). At low slopes, , the yield spectrum is rather narrow (from 1.97 at ) and skewed to the left (Fig. 4c). The differences between the yield spectra at high and low slopes suggested that the locations with lower slopes in 1995 were beneficial for soybean yield while high slopes did not influence yield either negatively or positively. A similar trend was also observed for soybean yield in 1996, with low-slope locations influencing the yields positively and high slopes having no influence on the yields.

The multifractal spectrum for corn in 1997 was not as asymmetrical as it was in 1994; however, asymmetry is still present. This indicates, as in the case with corn in 1994, the presence of a few areas with relatively low yields (Fig. 6a). The spectrum is narrow, with {alpha} ranging from 2.02 at , which is consistent with the low coefficient of variation for the corn yield in 1997 (Table 1). A wide range of {alpha}yield values (from ) corresponds to low slopes, (Fig. 6c), while the range of values for high slopes, (Fig. 6d), is relatively small (from 2.01 at at ). Compared with the part of the spectrum at (Fig. 6a), the part of the multifractal spectrum corresponding to high slopes, (Fig. 6d), is relatively symmetrical. The small range of shows that the lowest yields that caused asymmetry in the multifractal spectrum of corn in 1997 (Fig. 6a) were not present in the areas with high slopes. The spectrum at low slopes is skewed toward high yields, implying that overall higher yields existed at low-slope locations, which is similar to the results observed in the previous 3 yr. However, a relatively wide range of {alpha}yield values at suggests that some of the areas with the lowest yields are located at lower slopes. An analysis of the joint multifractal spectrum indicates that low-slope locations were beneficial for the corn yield in 1997, which is similar to the crop yields in previous years. The absence of a significant correlation between the yield and slope values in 1997 is the result of a number of areas with low yields collocated with the low-slope sites.

The yield–slope relationship for 1998, as reflected by the joint multifractal spectrum (Fig. 7a), is substantially different from that of the previous years. For low slopes , the yield spectrum is skewed toward the high {alpha}yield values, which signifies that generally lower yields prevailed at low slopes (Fig. 7c). The wide range of {alpha}yield values at suggests that the yield distribution at high slopes is not different from that of the whole field and that both high and low yields are observed at high-slope locations (Fig. 7d).

The low soybean yields at low slopes can be explained by the weather patterns during the growing season of 1998 (Fig. 8) . In June 1998, an abnormally high precipitation of 216 mm (compared with 100 mm of long-term avg. monthly precipitation) fell on the study area, with as much as 58 mm falling during a single day. This resulted in drainage problems in low-sloped areas of the field and subsequently caused lower yields. The weather patterns of the previous 4 yr were rather moderate. The relatively high amounts of precipitation that fell on the field in May 1995 and 1996 did not affect the plants; however, they provided more water for further redistribution within the field and caused more pronounced patterns of higher yields at depression sites with low slopes.



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  		Fig. 8 Monthly precipitation data for Mar. through Sept. 1994–1998 and the historical avg. from 1961–1990 for the studied field

 

   Summary and conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
 Results and discussion
 Summary and conclusions
REFERENCES
 
An analysis of the yield–slope relationships in the studied field revealed that even on a territory with a diverse terrain, the effect of topographical features on crop yields can be relatively small. The effect of the slope on soybean and corn yields during four growing seasons was manifested mainly by higher yields that were consistently observed at lower located sites with low slopes. Locations with steep slopes usually produced a wide range of yield values, indicating that the negative effect that such locations might have had on yields was masked by other factors affecting yields. For four growing seasons with moderate and dry weather conditions, these observations were remarkably consistent. For the fifth studied growing season, which was very wet during early summer, drainage problems at the sites with lower slopes inhibited plant growth and caused lower yields.

The variability of the corn and soybean yields and the relationships between the yields and the slope were examined with multifractal and joint multifractal analyses. Multifractal analysis was found to be applicable for studying spatial distributions of crop yields. Joint multifractal analysis was shown to be a useful tool to study the relationships between two spatially distributed variables. When applied to yield–slope relationships, joint multifractal analysis allowed us to make interferences regarding the variability of the crop yields corresponding to different terrain slope values. Although negative yield–slope correlation coefficients indicated the presence of certain relationships between the yields and slopes, only joint multifractal analysis allowed for a more detailed view on the slope values that contributed the most to these relationships and were of importance for explaining the yield variability across the field. Joint multifractal analysis was particularly beneficial for an analysis of the relationships in the extreme data ranges such high or low yields and their associations with high or low slopes.

Further possibilities for applying multifractal analysis in agriculture require additional investigation. Based on the study, we conclude that it can be a very helpful tool in depicting certain aspects in relationships between the studied variables that might otherwise be missed by such traditional methods as correlation analysis. However, a number of factors, including data clustering, spatial correlation, and their effect on multifractal and joint multifractal spectra need to be examined for further effective utilization of multifractal methods for spatial data analysis.

Received for publication February 3, 2000.
   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
 Results and discussion
 Summary and conclusions
 REFERENCES
 




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