Encoding daily rainfall records via adaptations of the fractal multifractal method

Abstract

A deterministic geometric approach, the fractal-multifractal (FM) method, already found useful in modeling storm events, is adapted here in order to encode, for the first time, highly intermittent daily rainfall records gathered over a water year and containing many days of zero rain. Through application to data sets gathered at Laikakota in Bolivia and Tinkham in Washington, USA, it is demonstrated that the modified FM approach can represent erratic rainfall records faithfully, while using only a few FM parameters. It is shown that the modified FM approach, by capturing the rain accumulated over the season, ends up preserving other statistical attributes as well as the overall ''texture'' of the records, leading to FM sets that are indistinguishable from observed sets and certainly within the limits of accuracy of measured rainfall. This fact is further corroborated comparing 20 consecutive years at Laikakota and a modified FM representation, via common statistical qualifiers, such as histogram, entropy function, and inter-arrival times.

A comparison of fractal-multifractal techniques for encoding streamflow records

“…The study clearly demonstrates how the notions that encompass the FM approach (i.e., projections of measures defined over fractal attractors) could be employed as a suitable tool to describe (encode) streamflow and other geophysical records that are complex and 'seemingly random' in nature. As illustrated herein, and elsewhere for rainfall sets (Maskey et al, 2015), such a geometric procedure could supplement and complement many of the existing stochastic procedures that aim at the modeling of complex sets.…”

“…"wires" and "leaves") but also the goodness of "closing the loop" notions that combine various attractors while sharing a common (multifractal) input measure. Given that the FM methods have been found useful in the modeling of rather complex rainfall patterns (Puente and Obregón, 1996;Cortis et al, 2009Cortis et al, , 2013Huang et al, 2012Huang et al, , 2013Maskey et al, 2015), it is not surprising that the same notions would do very well with the generally lesscomplex streamflow sets. This work, however, shows, for the first time, that streamflow records may indeed be thought of as transformations of multifractal measures via fractal attractors, in a manner that possesses a physical interpretation as a special outcome of a non-trivial multiplicative cascade (Cortis et al, 2013).…”

“…An examination of the FM parameter trends for long periods of streamflow records, with possible linkages to streamflow dynamics and climatic changes via plausible correlations of FM parameters and climatic attributes, is certainly a natural extension to the present study. We also envision that joint FM codings of rainfall (Maskey et al, 2015) and streamflow records (as done here), and within the same catchment, may help define, via the intrinsic complexity of FM evolutions of rainfall and runoff, distinct climatic/geomorphic regimes. interpolating function : → .…”

“…Indeed, the FM approach and its variants have been shown to work reasonably well in defining a vast class of patterns (Puente, 2004), over one and higher dimensions, preserving not only the key statistical indicators (e.g., moments, autocorrelation, power spectrum, multifractal spectrum) but also the overall geometric features and textures present in the data sets. While stochastic approaches are typically not capable of preserving finer details beyond the main statistical features, the FM methodology is found to yield indeed fruitful results in encoding a host of geophysical records, such as precipitation sets (e.g., Puente and Obregón, 1996;Cortis et al, 2009Cortis et al, , 2013Huang et al, 2012Huang et al, , 2013Maskey et al, 2015), fluid turbulence (Puente and Obregón, 1999), river width function (Puente and Sivakumar, 2003), and groundwater contamination patterns (Puente et al, 2001a, b).…”

A probabilistic gridded product for daily precipitation extremes over the United States

“…In traditional analysis of precipitation extremes using gridded products (see, e.g., Wehner 2013;Sylla et al 2013, and many others), the extreme climatology is estimated separately for each grid cell using a univariate extreme value analysis. However, we assert that gridded daily precipitation products are problematic data sources for constructing these extreme climatologies because daily precipitation is well-known to exhibit fractal scaling (e.g., Lovejoy et al 2008;Maskey et al 2016, and numerous references therein), and therefore any spatial smoothing or averaging during the gridding process will diminish variability and extreme values. Additionally, a recent thread of research (King et al 2013;Gervais et al 2014;Timmermans et al 2019) explicitly questions the appropriateness of using gridded products as a substitute for observed extremes.…”