Multifractal and Levy-stable statistics of soil surface moisture distribution derived from 2D image analysis

Soil moisture distribution usually presents extreme variation at multiple spatial scales. A Levy stable model can assist multifractal analysis for a complete statistical characterization of such extreme variability. The objectives of the present work were (i) to characterize superficial soil moisture distribution patterns using a multifractal approach and (ii) to propose a 2D/3D model for estimating the Levy stable index from the multifractal distribution. Photographs (KodakTM EasyShare C182, 12 Mpx resolutions) were taken at three different times of soil wetting process after soil irrigation (e.g., 5, 10 and 20 min.). ImageJ Software was used for image processing and multifractal analysis. Levy stable index, μ, was estimated from the f(α) curve using a 2D extension of a previous model. All gray scale images were approximately bimodal. Multifractal analysis showed that D0 was consistently <2, which gives evidence on the fractal structure of the support. All the investigated soil moisture distributions fitted well the multifractal log-Levy stable model with stability index within the range 1 < μ < 2 and Hurst exponent (H) 0.5 < H < 1. This implies that soil moisture pattern can be described by sub-Gaussian statistics (e.g., far from Central Limit Theorem) and it can arise as the outcome of persistence due to long memory effects. The results could be useful for improving the statistical characterization and the predictive capability of future models related to spatially variable soil processes.