Multifractal and joint multifractal analysis of soil invertebrate fauna, altitude, and organic carbon

ABSTRACT The objectives of this study were to evaluate the degree of multifractality of the spatial distribution of altitude, organic carbon concentration, and invertebrate fauna diversity, and to characterize the degree of joint multifractal association among these variables. Soil sampling was performed every 20 m across a 2,540 m transect, with a total of 128 sampling points in a sugarcane area in Goiana municipality, Pernambuco State. For each sampling point, the altitude, organic carbon concentration, and macrofauna diversity (diversity indices and functional groups) were evaluated. Spatial distributions of altitude, organic carbon concentration, and macrofauna diversity were characterized by the generalized dimension spectrum (Dq) and singularity spectrums [f(α) versus α], which presented multifractal behavior with different degrees of heterogeneity in scales. Joint multifractal analysis was useful for revealing the relationships at multiple scales between the studied variables, as demonstrated by the non-detected associations using traditional statistical methods. To quantify the spatial variability of edaphic fauna based on the multiple scales and association sets in the joint dimension, the impact of agricultural production systems on biological diversity can be described. All of the studied variables displayed a multifractal behavior with greater or lower heterogeneity degree depending on the variable, with altitude and organic carbon being the most homogeneous attributes.

A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.

Multifractal and joint multifractal analysis of the spatial variability of CO2 emission and other soil properties

<p>Soil is a major source and also a sink of CO<sub>2</sub>. Agricultural management practices influence soil  carbon sequestration. Identification of CO<sub>2</sub> emission hotspots may be instrumental in implemented strategies for managing carbon cycling in agricultural soils. We used multifractal analysis to assess the spatial variability of both, soil CO<sub>2</sub> emissions and associated soil physico-chemical attributes. The objectives of this study were: i) to characterize patterns of spatial variability of CO<sub>2</sub> emissions and related soil properties using single multifractal spectra, and ii) to compare the scale‐dependent relationship between soil CO<sub>2</sub> emissions and selected soil attributes by joint multifractal analysis. The study site was an experimental field managed as a sylvopastoral system, located in Selviria, South Mato Grosso state, Brazil. The soil was an Oxisol developed over basalt. Soil CO<sub>2 </sub>emission, soil water content and soil temperature were measured at 128 points every meter. In addition<strong>, </strong>soil was sampled at the marked points to analyze clay content, macro and microporosity, air free porosity, magnetic susceptibility, bulk density, and humification index of soil organic matter in absolute values and relative to organic carbon content. The generalized dimension, D<sub>q</sub> versus q, and singularity spectra, f(α) versus α, of the spatial distributions of the 11 variables studied showed various degrees of multifractality. In general, the amplitude of the generalized dimension and singularity spectra was much higher for negative than for positive q order statistical moments. Joint multifractal spectra show a positive relationship between the scaling indices of the spatial distributions of CO<sub>2</sub> and all of the other soil variables studied. However, contour plots were diagonally oriented for higher values of scaling indices and showed no distinct trend for the lower ones. Joint multifractal analysis corroborates different degrees of association between the scaling indices of CO<sub>2</sub> and all of the remaining variables studied. It also showed that CO<sub>2</sub> was stronger correlated at multiple scales than at the observation scale. Therefore, single scale analysis may not be sufficient to fully describe relationships between soil testing methods.Our study suggests that soil factors and processes driven the spatial variability of CO<sub>2</sub> and the associated variables studied may be not very different.</p><p> </p>

Joint Multifractal Analysis and Source Testing of River Level Records Based on Multifractal Detrended Cross-Correlation Analysis

The joint multifractal analysis is usually conducted in two different variables for their cross-correlations but rarely used for two records of one variable collected at two different places. It is important for the detection of change in multifractality in space. Besides, the cross-correlations in two analyzed series make the analysis of sources of joint multifractality difficult. There are few studies on the source of joint multifractality. We focus on the two issues for two level records at pairs of adjacent sites along one river and carry out an extension of our previous work which is about the single multifractality of one record with the same data set. The data set is collected from 10 observation stations of a northern China river and contains about two million high-frequency river level records. Results of joint multifractal analysis via multifractal detrended cross-correlation analysis show that the change in joint multifractality at pairs of adjacent sites caused by weak cross-correlations can be detected by comparing the single generalized Hurst exponent with the joint scaling exponent function and reveal the effects of human activities on joint multifractality. This analysis provides an approach for detecting the change in multifractality. Following the idea of our previous work, two robust hypothesis tests via a set of pairs of surrogate series are proposed for the source testing of joint multifractality. The analysis of the effects of cross-correlations is carried out via a proposed simultaneously half-shifting technique which can both minimize the cross-correlations between original series and make full use of records. Results of source analysis show not only the effects of autocorrelations in series and probability distribution of river levels but also the effects of cross-correlations between series.

Approximate multifractal correlation and products of universal multifractal fields, with application to rainfall data

Universal multifractals (UMs) have been widely used to simulate and characterize, with the help of only two physically meaningful parameters, geophysical fields that are extremely variable across a wide range of scales. Such a framework relies on the assumption that the underlying field is generated through a multiplicative cascade process. Derived analysis techniques have been extended to study correlations between two fields not only at a single scale and for a single statistical moment as with the covariance, but across scales and for all moments. Such a framework of joint multifractal analysis is used here as a starting point to develop and test an approach enabling correlations between UM fields to be analysed and approximately simulated. First, the behaviour of two fields consisting of renormalized multiplicative power law combinations of two UM fields is studied. It appears that in the general case the resulting fields can be well approximated by UM fields with known parameters. Limits of this approximation will be quantified and discussed. Techniques to retrieve the UM parameters of the underlying fields as well as the exponents of the combination have been developed and successfully tested on numerical simulations. In a second step tentative correlation indicators are suggested. Finally the suggested approach is implemented to study correlation across scales of detailed rainfall data collected with the help of disdrometers of the Fresnel platform of Ecole des Ponts ParisTech (see available data at https://hmco.enpc.fr/portfolio-archive/taranis-observatory/, last access: 12 March 2020). More precisely, four quantities are used: the rain rate (R), the liquid water content (LWC) and the total drop concentration (Nt) along with the mass weighed diameter (Dm), which are commonly used to characterize the drop size distribution. Correlations across scales are quantified. Their relative strength (very strong between R and LWC, strong between DSD features and R or LWC, almost null between Nt and Dm) is discussed.

Multifractal formalisms for multivariate analysis

Multifractal analysis, that quantifies the fluctuations of regularities in time series or textures, has become a standard signal/image processing tool. It has been successfully used in a large variety of applicative contexts. Yet, successes are confined to the analysis of one signal or image at a time (univariate analysis). This is because multivariate (or joint) multifractal analysis remains so far rarely used in practice and has barely been studied theoretically. In view of the myriad of modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, univariate analysis constitutes a major limitation. The goal of the present work is to theoretically ground multivariate multifractal analysis by studying the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation. It is notably shown that while performing well for a class of model processes, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we propose alternative formulations and examine their mathematical properties.