New scaling model for variables and increments with heavy‐tailed distributions

Many hydrological (as well as diverse earth, environmental, ecological, biological, physical, social, financial and other) variables, Y , exhibit frequency distributions that are difficult to reconcile with those of their spatial or temporal increments, Δ Y . Whereas distributions of Y (or its logarithm) are at times slightly asymmetric with relatively mild peaks and tails, those of Δ Y tend to be symmetric with peaks that grow sharper, and tails that become heavier, as the separation distance (lag) between pairs of Y values decreases. No statistical model known to us captures these behaviors of Y and Δ Y in a unified and consistent manner. We propose a new, generalized sub‐Gaussian model that does so. We derive analytical expressions for probability distribution functions (pdfs) of Y and Δ Y as well as corresponding lead statistical moments. In our model the peak and tails of the Δ Y pdf scale with lag in line with observed behavior. The model allows one to estimate, accurately and efficiently, all relevant parameters by analyzing jointly sample moments of Y and Δ Y . We illustrate key features of our new model and method of inference on synthetically generated samples and neutron porosity data from a deep borehole.