Stochastic investigation of long-term persistence in two-dimensional images of rocks

Determining the geophysical properties of rocks and geological formations is of high importance in many fields such as geotechnical engineering. In this study, we investigate the second-order dependence structure of spatial (two-dimensional) processes through the statistical perspective of variance vs. scale (else known as the climacogram) instead of covariance vs. lag (e.g. autocovariance, variogram etc.) or power vs. frequency (e.g. power spectrum, scaleogram, wavelet transform etc.) which traditionally are applied. In particular, we implement a two-dimensional (visual) estimator, adjusted for bias and for unknown process mean, through the (plot of) variance of the space-averaged process vs. the spatial scale. Additionally, we attempt to link the climacogram to the type of rocks and provide evidence on stochastic similarities in certain of their characteristics, such as mineralogical composition and resolution. To this end, we investigate two-dimensional spatial images of rocks in terms of their stochastic microstructure as estimated by the climacogram. The analysis is based both on microscale and macroscale data extracted from grayscale images of rocks. Interestingly, a power-law drop of variance vs. scale (or else known as long-term persistence) is detected in all scales presenting a similar power-exponent. Furthermore, the strengths and limitations of the climacogram as a stochastic tool are discussed and compared with the traditional tool in spatial statistics, the variogram. We show that the former has considerable strengths for detecting the long-range dependence in spatial statistics.