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

• Published in: Water resources research Vol. 51; no. 6; pp. 4623 - 4634
• Main Authors: Riva, Monica, Neuman, Shlomo P., Guadagnini, Alberto
• Format: Journal Article
• Language: English
• Published: Washington Blackwell Publishing Ltd 01-06-2015; John Wiley & Sons, Inc
• Subjects: Behavior; Boreholes; heavy-tailed distributions; neutron porosity; parameter estimation; Porosity; Probability distribution; scaling; Statistical models; sub-Gaussian
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1002/2015WR016998

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, ΔY. Whereas distributions of Y (or its logarithm) are at times slightly asymmetric with relatively mild peaks and tails, those of ΔY 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 ΔY 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 ΔY as well as corresponding lead statistical moments. In our model the peak and tails of the ΔY 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 ΔY. We illustrate key features of our new model and method of inference on synthetically generated samples and neutron porosity data from a deep borehole.
Key Points:
A new statistical scaling model is developed, explored and applied
Apparent inconsistency between variable and increment statistics is eliminated
Parameters are estimated using variable and increment moments jointly