Practical modelling of porous media from second‐order statistics: the phase‐retrieval problem and the interpretation of the covariance

Abstract

SummaryIn many disciplines, binary numerical porous media are commonly generated as realizations of a random process characterized by a limited set of morphological measurements, e.g. the conditioning and truncated Gaussian random field (GRF) model or the Monte Carlo simulated annealing (SA) reconstruction. Whether the output microstructures of a model are quantitative or qualitative depends largely on the information carried by the constraints that are implemented. We focus on the standard two‐point correlation function, also referred to as the covariance. The isotropic situation has been previously extensively studied. Here, we concentrate on a vector distance‐dependent covariance. Relying on practical solutions to the phase‐retrieval problem (retrieving an object from its Fourier modulus) encountered in general imaging, we show empirically that a finite binary two‐ or three‐dimensional actual sample can be recovered systematically to within a pixel from its correlation integral (the volume average covariance to within a scale factor). In random modelling, this characteristic, describing uniquely a single sample of finite size, must be simplified, i.e. reduced to its statistical content, prior to be fitted in a random model (GRF, SA) as the characteristic of a series of realizations of a porous medium. We consider a common simplification, i.e. truncation, in which that part of the covariance relating to the small‐scale features of the observed sample is preserved only. This arbitrary simplification is empirically supported by numerous published numerical experiments showing that the small‐scale content of a volume‐averaged covariance can be fairly well reproduced in realizations of various random models: its statistical nature is thus illustrated with the GRF model. The unique solution to the phase‐retrieval problem confers an important sense to this numerical agreement: the correlation integral being the most complete descriptor, the actual sample and the model outputs are considered as true analogues over scales smaller or equal to the cutoff length attached to the truncation. Thus, awareness of the phase‐retrieval problem supports on an empirical basis the generation of synthetic binary objects from a direct measurement on an available sample.