SPATIAL VARIABILITY Joint Multifractal Analysis of Crop Yield and Terrain Slope

a yield–soil–topography analysis. Spectral analysis was utilized by Timlin et al. (1998) to study the influence of Quantifying the spatial variability of crop yields and yield-affecting topographic location and surface curvature on corn factors are important issues in precision agriculture. Topography is frequently one of the most important factors affecting yields, and grain yield based on data from 140 plots and five trantopographical data are much easier to obtain than time and laborsects. Eghball et al. (1997) used fractal theory to characconsuming measurements of soil properties. In this study, yield variterize the spatial and temporal variability in corn grain ability and the relationships between yields and terrain slopes were yield as affected by N treatments. Fractal analysis was analyzed using theories of multifractal and joint multifractal measures. found to be useful for describing the temporal yield Corn (Zea mays L.) and soybean [Glycine max (L.) Merr.] yield data variability for 10 crops in the USA, including corn and from 1994 to 1998 were collected via yield monitors from a central soybean (Eghball and Power, 1995). It has also been 6.6 ha section of an agricultural field in eastern Indiana. Slopes were found to be useful for depicting the effects of manure derived from a field terrain map using a GIS. Multifractal analysis of and fertilizer applications on corn yield variability (Eghyield and slope maps revealed that both yield and slope distributions were multifractal measures. Hence, joint multifractal analysis was ball et al., 1995). applied to evaluate the effect of slope on crop-yield spatial variability. Multifractal analysis (Mandelbrot, 1974) has been utiJoint multifractal analysis facilitated (i) the ability to differentiate lized successfully to characterize several factors that afbetween yield distributions corresponding to field locations with high fect yields, including rainfall (Olsson and Niemczynowand low slopes, and (ii) the ability to make inferences about slope icz, 1996), soil strength (Folorunso et al., 1994), soil distributions that affect grain yield the most. Multifractal analysis particle size distribution (Grout et al., 1998), soil P and revealed that during four growing seasons with moderate and dry K concentrations, and organic matter content (Kravweather conditions, larger yields were observed at low slope locations chenko et al., 1999). It has also been shown to provide while a wide range of yield values was observed at sites with moderate additional detailed information about soil spatial variand high slopes. During the wet growing season, lower yields prevailed at locations with low slopes. Joint multifractal theory was useful for ability compared with traditional fractal approaches the study of yield/topography relationships and was an applicable tool (Folorunso et al., 1994). Multifractal analysis is applicafor the analysis of spatially distributed data. ble 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, volS management benefits from a thorough ume, or fractal set (Feder, 1988). As an example of a quantitative description of spatial and temporal multifractal measure, let us consider the distribution of within-field variability of crop yields and the factors that groundwater within a certain geographical area (Evertsz affect them. Topography is regarded as one of the most and Mandelbrot, 1992). If this area is divided into two important factors affecting yields (Changere and Lal, equally sized parts, the groundwater contents of each 1997; McConkey et al., 1997) and as a source of easily part will be different even though the areas for both obtained information that is useful for soil and field parts are equal. If one of the parts is further subdivided characterization (de Bruin and Stein, 1998; Moore et into two equally sized pieces, their corresponding al., 1993; Odeh et al., 1994). The complexity of yield– groundwater contents will again be different. This subditopography relationships often cannot be appropriately vision can be continued until the amount of water concharacterized by traditional statistical methods. More tained within one rock pore will be different from that exhaustive characterization can be achieved by using of another. That is, the distribution of groundwater is advanced statistical and mathematical procedures such irregular at all scales. If the irregularity in the variable’s as geostatistics, time series analysis, or fractal analysis. distribution remains statistically similar at all studied For example, Miller et al. (1988) used geostatistics to scales (Evertsz and Mandelbrot, 1992), then the variable study the spatial variability and yield–landscape relais assumed to be self similar or multifractal. tionships for wheat (Triticum aestivum L.) yield on a An extension of multifractal theory for the analysis 400by 200-m study site and found geostatistics to be of more than one variable was developed by Meneveau superior to either correlation or multiple regression for 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 A. Kravchenko and D. Bullock, Dep. of Crop Sci. and C. Boast, Dep. of Nat. Resources and Environ. Sci., Univ. of Illinois, 1102 S. Goodwin same geometric support, and hence for quantifying the Ave., Urbana, IL 61801-4798. Received 3 Feb. 2000. *Corresponding relationships between the measures studied. If crop author ([email protected]). yields and soil properties or topographical features of the field are shown to be multifractal measures, then Published in Agron. J. 92:1279–1290 (2000).