Multifractal and multiscale entropy scaling of in-situ soil moisture time series: Study of SMOSMANIA network data, southwestern France

Abstract

Soil moisture processes exhibit a strong variability in space and time due to the variability of the meteorological forcing and the spatial heterogeneity of soil properties. This study aims at providing a statistical description of soil moisture variability by analyzing data from nine in situ stations located over an West-East gradient in southwestern France (SMOSMANIA stations, distributed over an area of about 300 × 200 km). For each station, four time series of soil moisture observed at four different depths ranging from 5 cm to 30 cm are analyzed. First, possible scaling properties are investigated within the Fourier domain with the help of spectral analysis tools. Red noise-like (1/f2) scaling properties could be noticed over a fairly wide scale range (1000 h–1 h) with relatively homogeneous scaling parameters for 5 cm depth soil moistures regardless of the station. These properties are confirmed at other depths with slightly steeper spectra and more heterogeneity across stations. In a second step, multifractal analysis has been carried out on the same data. Multifractal scaling is observed over a narrower scaling range (128 h–1 h). Moment scaling functions can be parameterized within the framework of Universal Multifractals: typical parameters are C1≈0.25 and α ≈ 1.6–1.8 for surface data while both parameters are subject to strong changes (C1 increases and α decreases) as the depth increases. In a third step, Multiscale Entropy (MSE) analysis has been applied in order to analyze the dataset from an information theory point of view and to infer whether multifractal properties could have a signature on MSE estimates. The MSE function has been found to follow a power law of the aggregation time with a scaling exponent close to 0.3–0.4 for surface data. These exponents were generally close to the Hurst exponent H estimated by first-order structure functions. While it is already known that MSE should follow scaling properties in the case of monofractal signals, the results suggest that the latter property holds for natural multifractal processes. Finally, complementary numerical tests based on synthetic multifractal time series are done in order to assess the relationship between the MSE scaling exponent H′ and multifractal parameters C1, α and H. A dependency on the three multifractal parameters has been observed, yet in practice the approximation H′ ≈ H seems somewhat acceptable for processes with relatively large Hurst exponents and/or low multifractal intermittency (C1). On a more broad perspective, the existence of relationships between multifractal scaling and MSE properties could help to refine the physical interpretation of observed scaling properties in many geophysical processes.