On the fractal structure of soil moisture fields

We study the spatial structure of soil moisture fields within savanna ecosystems, whose persistence is vital because it is the driver of the entire ecological structure and function. These include changes in the physical and biogeochemical conditions of the landscape, affecting vegetation state, soil composition, water fluxes, and solar radiation. We focus on computations of the probabilistic structure of islands of soil moisture, known empirically to be related to that of tree clusters, defined as crossing properties of simulated soil moisture fields. Rainfall is modelled via Cox-Isham space-time fields endowed with characteristic scales. Results show that clusters of soil moisture islands are characterized by robust scale-free structures in the region of a phase transition whose order parameter depends on mean soil moisture. Signatures of this fractal structure are well-defined power laws of size distributions of soil moisture clusters; their perimeters-vs-area relations; variance-vs- area of the fields. These characteristics allow for the estimation of the fractal dimension of the field, and its Hurst coefficient. From the general covariance equation of a fractal field, spatial simulations are possible because its mean and variance are known from the probabilistic structure of soil moisture at a point. Our results identify the statistics of hotspots of microbial activity deduced from proper moisture islands, unattainable otherwise, and thus may guide the design of field and remote observations. The critical order parameter characterizing the phase transition establishes where the fractal structure of soil moisture fields exists as a function of the climatic drivers, and the thresholds reflecting where vegetation survives in the field. An example of application of the phase transition diagram presented here is carried out with reference to the Nylsvley savanna in South Africa.