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Multifractal analysis can transform irregular data into a compact form and amplify small differences among the variables [ 43 , 44 ]. It uses a wider range of statistical moments, providing a much deeper insight into the data variability structure compared to the methods that use only first two statistical moments. Multifractal analysis can thereforebe used to characterize the variability and heterogeneity in soil properties over a range of spatial scales [ 44 - 47 ].

While the multifractal analysis characterizes the spatial variability in a variable, joint multifractal analysis characterizes the joint distribution of two variables along a common spatial support. It can provide information on the relationships between two variables across different spatial scales [ 47 , 48 ]. The objective of this chapter is to demonstrate what multifractal and joint multifractal analysis can do in dealing with spatial data series from the published soil science literature. In the next section we briefly describe multifractal and joint multifractal analysis methodology including the working steps and then proceed to describe some applications of these methods in soil science.

Finally we close the chapter with a discussion on future prospects for multifractal and joint multifractal methods in soil sciencewithin a brief Conclusions Section. Multifractal analysis can be used to characterize the scaling property of a soil variable measured in a direction such asa transect as the mass or value distribution of a statistical measure on a geometric or spatial support.

The geometric or spatial support indicates the extent of the sampling. This can be achieved by dividing the length of the transect into smaller and smaller segments based on a rule that generates a self similar segmentation. One such rule is the binomial multiplicative cascade [ 36 ] that can divide a unit interval associated with a unit mass, M a normalized probability distribution of a variable or a measured distribution as used in a generalized case into two segments of equal length. The parameter h is a random variable values between 0 and 1.

The new subsets and its associated mass are successively divided into two parts following the same rule. The differences among the subsets are identified using a wide range of statistical moments, which can then be used to determine the multifractal spectra of the measures [ 36 , 48 ].

Therefore, the multifractal analysis focuses on how the measure of mass varies with box size or scale and provides physical insights at all scales without any ad hoc parameterization or homogeneity assumptions [ 49 ]. A detailed description of the multifractal theory can be found in reference[36, 50, and 51]. In this section, for brevity, we will only summarize the computational techniques and concepts commonly used in examining the soil spatial variation. Past research has indicated that certain properties of a spatial series decrease with increase in scale, following a power law relationship.

However, the semivariogram only measures the scaling properties of the second moment. Similarly, we can do the same thing for the higher moments such as third, fourth, and so on. Will the scaling properties be the same at higher moments or change with the order of the moments, q? If the scaling properties do not change with q , then we say the spatial series is monofractal, i. If the scaling coefficient changes with q , then the spatial series is multifractal, i.

The type of scaling can be examined from the degree of fractality i. One such reference curve similar to the monofractal type of scaling or the theoretical model was proposed by Schertzer and Lovejoy [ 49 ], which is known as the universal multifractal model or UM model. The UM model simulates an empirical moment scaling function of a cascade process assuming the conservation of mean value.

The UM model can be used to compare and characterize the observed scaling properties as reference to the monofractal behavior or scaling. Statistically significant difference between the curves indicates multifractal scaling and non-significant difference indicates monofractal scaling. Large sum of squared difference between the curves indicates multifractal behavior and small sum of squared difference indicates monofractal behavior.

Significant difference in the slope indicates multifractal behavior. The multifractal spectrum, f q Fig. The multifractal spectrum is a powerful tool in portraying the variability in the scaling properties of the measures e. The spectrum also enables us to examine the local scaling property of soil variable.

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The wider the spectrum, higher the heterogeneity in the distribution of the soil variable. Similarly, the height of the spectrum corresponds to the dimension of the scaling indices. For many practical applications indicator parameters are selected and used, in addition to the multifractal spectrum f q vs. Generalized dimensions, D q , Fig. The D q of a multifractal measure is calculated as.

Sometimes, D 1 is also known as the entropy dimension. A value of D 1 close to unity indicates the evenness of measures over the sets of a given cell size, whereas a value close to 0 indicates a subset of scale in which the irregularities are concentrated. The D 2 , known as the correlation dimension, is associated with the correlation function and measures the average distribution density of the measure [ 55 ]. For a distribution, with simple scaling monofractal , the D 1 and D 2 become similar to the D 0 , the capacity dimension.

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Calculate the probability measure p from a linear distance for a transect, or from a rectangle for an area. Regarding the minimum number of samples required to carry on the analysis, the multifractal analysis method has the flexibility over other methods such as Fourier transform or wavelet transform, which generally require a larger dataset with regular interval between samples.

For certain q values or statistical moments e. If they are not, then they are not multifractal and no further multifractal analysis is needed. If they are, continue the following. While the multifractal analysis characterizes the distribution of a single variable along its spatial support, the joint multifractal analysis can be used to characterize the joint distribution of two or more variables along a common spatial support. The dimension i.

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When q or t is set to zero, the joint partition function shown in Eq. Because, high q or t values magnify large values in the data and negative q or t values magnify small values in the data, by varying q or t , we can examine the distribution of high or low values different intensity levels of one variable with respect to that of the other variable.

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Pearson correlation analysis can be used to quantitatively illustrate the variation of the scaling exponents of one variable with respect to another variable across similar moment orders. By perturbing q and t , we can examine the association of similar values highs vs. The bottom left part of the contour graph shows the joint dimension of the high data values of the two variables, while the top right part represents the low data values [ 57 ].

The diagonal contour with low stretch indicates strong correlation between values corresponding to the variables in the vertical and horizontal axis Fig. A typical of multifractal spectra for joint distribution of two variables with a strong correlation and b weak correlation. Calculate the probability measure p for variable Y from a linear distance for a transect or a rectangle from an area. For certain q values e.

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Multifractal analysis for two-dimensional or three dimensional fields is the same as for a transect. Multifractal analysis, like spectral analysis, is based on the global statistical properties of spatial series. Therefore, localized information is lost, which is different from wavelet analysis. However, multifractal analysis provides information regarding the higher moments and how the higher moments change with scale.

Multifractal analysis does not require regularly spaced samples. Any sampling scheme can be analyzed by multifractal analysis. For highly spatially variable fields, the probability p for some locations may be very small or even zero. As such, the negative power of p can be very large and the partition function will be dominated by this single value.

In this case, the multifractal method may diverge and the process of division needs to be stopped. Joint multifractal analysis 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 measurements studied.

Joint multifractal analysis is based on the assumption that the individual variable is multifractal. Simulation of a synthetic field according to the measured multifractal distribution may enhance our understanding of the effects, spatial scaling, and spatial variability of soil properties on various soil processes.

Fractal theory [ 37 ] has been used to investigate and quantitatively characterize spatial variability over a large range of measurement scales in different fields of geosciences including soil science [ 4 , 58 ]. A detailed review of the applications of fractal theory in soil science can be found in reference[ 38 ]. The fractal theory applications in soil science used monofractal approach, which assumes that the soil spatial distribution can be uniquely characterized by a single fractal dimension.

However, a single fractal dimension might not be sufficient to represent complex and heterogeneous behavior of soil spatial variations. Multifractal analysis, which uses a set of fractal dimensions instead of one, is useful in characterizing complex soil spatial variability. Among the earlier works, Folorunso et al. Since then, multifractal analysis has been used in soil science to study various issues including spatial variability of soil properties e. While multifractal analysis is used to characterize the spatial variability of soil properties over a range of scales, the joint multifractal analysis can be used in soil science to characterize the joint distributions between soil properties over a range of scales.

In this section we review previous applications of multifractal and joint multifractal analysis in soil science. Various soil properties have shown multifractal behavior and the variability in those soil properties has been characterized using multifractal analysis. For example, Kravchenko et al. The study used a ha agricultural field in central Illinois, USA. One set of grid size was used and the authors reported only one unique set of multifractal spectra for each soil property.

Multifractal parameters were studied over a range of moment orders q from to The generalized fractal dimensions were also calculated for all positive q values. An excellent fit between the theoretical fractal dimension model and the D q curve also indicated the multifractal nature of the soil properties except OM and soil pH. A high correlation between the multifractal parameters and exploratory statistics e. Multifractal spectra of the soil properties studied carried a large amount of spatial information and allowed quantitative differentiation between the soil variability patterns.

Kravchenko et al. Interpolation methods based on the multifractal scaling relationship improves the mapping of soil properties. Multifractal spectra, f q , for a phosphorus, b potassium, c organic matter, d pH, e exchangeable Ca, f exchangeable Mg, and g cation exchange capacity adopted from Various other studies from different parts of the world also indicate the multifractal behavior of soil properties. Caniego et al. Though the authors reported the multifractal nature of the soil properties along the long transect, the variability along the short transects were more homogeneous.

The variability along the short transect was explained by simple random fractal noise close to a white noise and thus a single scaling index was consideredsufficient to transform information from one scale to another [ 61 ]. The variability in the long transect might have a deterministic component reflecting changing geological features and differences in soil forming factors with distance.

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The presence of nonlinearity together with the environmental features along the long transect resulted in the observed multifractal behavior of soil properties. Monofractal behavior of sand and silt was reported by Wang et al. Zeleke and Si [ 52 , 57 ] characterized the distribution of bulk density Db , sand content SA and to some extent silt SI content as monofractal along a gently sloping m long transect in semi-arid central Saskatchewan, Canada.

However, these authors did report multifractal behavior of clay CL , saturated hydraulic conductivity K s and organic carbon OC from the same study. For example, Fig. Wang et al. A decrease in the variability of EC was reported with an increase in the EC value itself [ 63 ]. Db is the bulk density, SA is the sand content, SI is the silt content, CL is the clay content, OC is the organic carbon content, Ks is the saturated hydraulic conductivity.

The multifractal behavior of effective saturated hydraulic conductivity in a two-dimensional spatial field was reported by Koirala et al. The authors used a normalized mass fraction of a randomized multifractal Sierpinski carpet to represent the areal distribution of saturated hydraulic conductivities in an aquifer. The effective saturated hydraulic conductivity was related to the generalized dimensions of the multifractal field.

This relationship may be helpful to predict the effective saturated hydraulic conductivity of an aquifer from an empirical Dq spectra determined by the multifractal analysis [ 64 ]. The multifractal nature of the hydraulic conductivity was also reported by Liu and Molz [ 46 ]. The multifractal behavior of soil water content [ 53 , 63 , 65 , 66 ], soil water retention at different suction [ 57 , 62 , 67 , 68 ] and theoretical water retention parameters e.

The volumetric as well as gravimetric soil water content was found to exhibit multifractal behavior [ 65 ]. The variability in soil water content decreases with the increase in soil water and increases with the increase in the representative area of the measurement [ 66 ]. Though the water retention at saturation showed monofractal behavior, water retention at kPa and kPa showed multifractal behavior [ 57 , 67 ]. Filling up the pore with water during saturation irrespective of the size of the pores might result in characteristic that is persistent over a range of scales.

However, at kPa and kPa, the relation between pore size and pore water retention might result in multifractal behavior [ 57 , 67 ]. Similar behavior of water retention parameters was reported by Liu et al. However, Guo et al. The fractal behavior of soil water estimates derived from the satellite images were reported earlier [ 73 , 74 ]. The relationship between the variance in soil water and the area of measurement or aggregation area or scale indicated self-similarity or scale invariance in soil water estimates.

Later this relationship was used to develop models for downscaling information on soil water estimates from satellite images. A number of studies have developed and used multifractal models to downscale soil water estimates from satellite images to represent small areas[ 75 - 79 ]. Often the passive remote sensing images provide estimates of soil water for a large area coarse spatial resolution;e. Downscaling models are necessary tools to characterize and reproduce soil water heterogeneity from the remotely sensed estimates. The presence of multifractal behavior helped developing downscaling models [ 76 ].

Prediction of soil water storage as affected by soil microtopography or microrelief can also be made using the multifractal approach [ 80 - 81 ]. Soil surface roughness and microtopography showed multifractal behavior, the quantification of which help characterizing spatial features in topography and water storage in micro-depressions. The soil surface roughness created by tillage operations can determine soil strength, penetration of roots and susceptibility to wind and water erosion.

The multifractal behavior of soil surface roughness can be used to explain the structural complexity created by tillage operations and the effect of various tillage implements. Various studies have reported the multifractal behavior of soil surface roughness [ 80 - 85 ]. Generally, these studies measured the soil surface roughness in a very small area. The degree of multifractality was found to increase at higher statistical moments [ 84 ].

The variability in the distribution of soil units also showed multifractal behavior [ 86 - 87 ]. Information on the variability within and between soil pedological units pedotaxa shows promise in analyzing and characterizing the complexity of soil development at multiple scales. A large number of studies reported the multifractal behavior of soil particle size distribution [ 56 , 88 - 95 ].

Most of the studies used soil particle size distributions measured using laser diffraction. One of the studies also used a piece-wise fractal model for very fine and coarse particle sizes [ 95 ]. Multifractal behavior of soil particles sizes at different land can be used to characterize soil physical health and its quality [ 90 ].

Distribution of soil particles size determines the porosity, which in turn determines the flow and transport of water and chemicals in soil. Soil thin sections have been used to study the pore arrangements and distributions in soil [ 96 - 98 ]. The pore geometry showed multifractal behavior, which is useful for classification of soil structure and determining the fluid flow parameters [ 96 ]. The multifractal nature of soil porosity was used to identify the representative elementary area using photographs of soil and confocal microscopy [ 99 ]. The use of two dimensional binary or grayscale images[ 96 - 98 ] of soil thin sections helped in characterizing the pore structure and movement of water in soil.

With the advancement of technology, the images from Magnetic Resonance Imaging MRI [ ] or Computer Tomography CT [ , ] helped in characterizing the soil as a porous media and the fluid flow through it. Three-dimensional images of soil systems helped to identify soil pores and their connectivity in three-dimensions, which in turn helped understand the movement of water in soil and the characterization of preferential flow paths [ ].

The multifractal behavior of soil pore systems is often used to develop models for creating simulated media, which helps to study flow and transport of water and chemicals in soil or other porous media [ ]. The infiltration of water into soil often shows multifractal behavior. Use of dye is a common approach to study the preferential flow paths of water in soil. The distribution of dye along with the infiltration water helped illustrate multifractal behaviour[ - ].

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The multifractal nature of infiltration helped in identifying the input parameters for various rainfall-runoff models [ - ]. Perfect et al. The radioactive cesium Cs fallout at small spatial scales has been found to be multifractal in nature [ ]. Grubish [ ] reviewed research on the Cs fallout and reported that the multifractal nature stemmed from the erosion and deposition of soil materials, which showed a log-normal distribution.

The behavior of soil chemical processes such as nitrogen absorption isotherms was also found to be multifractal [ ]. The authors studied the isotherm characteristics from 19 soil profiles in a tropical climate. The asymmetric singularity spectra shifted to the left indicated the highly heterogeneous and anisotropic distribution of the measure Fig.

Various studies also reported the multifractal behavior of soil landscape properties. For example, Wang et al. The multifractal pattern was also found in different terrain indices. While the upslope area and wetness index were multifractal, the relative elevation was monofractal [ 47 , ].

This may be due to inclusion of multi-scale characteristics in calculating secondary terrain indices such as the wetness index. The spatial variability of crop yield also found to exhibit multifractal in nature [ 47 , - ]. A stochastic simulation study showed that the spatial variability in the production of wheat crop was multifractal, while the production of corn was monofractal [ ]. Multifractal spectra of the joint distribution of saturated hydraulic conductivity vertical axis and other soil properties horizontal.

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