Geometric harnessing of precipitation records: reexamining four storms from Iowa City

Abstract

Complex geometries often present in hydrologic data sets such as precipitation records have been difficult to model in their totality using classical stochastic methods. In recent years, we have developed extensions of a deterministic procedure, the fractal-multifractal (FM) method, whose patterns share fine details and textures of individual data sets in addition to the usual key statistical properties. This work discusses our latest efforts at encoding four geometrically distinct storms gathered in Iowa City with parameters found running a modified particle swarm optimization procedure. The results reaffirm the capabilities of the FM method as all storms are closely fitted within measurement errors. All sets may be encoded with a compression ratio exceeding 350:1, have a maximum error in cumulative distribution less than 2.5 %, and closely preserve the autocorrelation, power spectrum, and multifractal spectrum of the records.

Appendices

Appendix 1

See Tables 1, 2, 3, 4.

Table 1 Results of searching for a fit for the set Iowa A: $\ell^{2}$-norm of averaged error on data, $\ell^{2}$-norm of averaged cumulative distribution deviations, maximum deviation between cumulative distributions, Nash–Sutcliffe model efficiency coefficient for the data, Nash–Sutcliffe coefficient for the autocorrelation, fractal dimension, multifractal spectrum’s entropy dimension, power spectrum exponent, autocorrelation lag at 1/e, and autocorrelation lag at 0

Table 2 Results of searching for a fit for the set Iowa B

Table 3 Results of searching for a fit for the set Iowa C

Table 4 Results of searching for a fit for the set Iowa D

Appendix 2

See Figs. 1, 2, 3, 4, 5, 6.

Fig. 1

The FM approach and three of its extensions. a Top left the original method: from a multifractal, bottom, to a projection, right, via a fractal interpolating function, center. b Top right an extension with overlaps and/or holes: from a multifractal, bottom, to a projection, right, via a “broken” attractor. c Bottom left, an extension with nonlinear perturbations: from a multifractal, bottom, to a projection, right, via an attractor with overlaps and holes generated with maps that contain nonlinear perturbations. d Bottom right, an extension to higher dimensions: from a multifractal, bottom, to a projection, right, of the y-marginal of an attractor in y and z

Fig. 2

Iowa A renderings. From top to bottom, representations using the original FM approach (using three maps), the extension with overlaps and holes (four maps), the nonlinear variant of the extension (three maps), and the higher dimensional extension (three maps). Left column the fit, in black, imposed on the original data, in gray. Middle column the cumulative distributions of the data and the fit, with errors colored as area between the curves. Right column scatterplot of the data (horizontal) versus the fit (vertical). Diagonal lines indicate ±10 % lines. Readers interested in reconstructing the plots may navigate to the website, where all parameters may be found: http://puente.lawr.ucdavis.edu/omake/serra2012.html

Fig. 3

Iowa B renderings. The row and column setup is as in Fig. 2 in Appendix 2. From top to bottom, the representations were constructed with four, three, three, and three maps

Fig. 4

Iowa C renderings. The row and column setup is as in Fig. 2 in Appendix 2. From top to bottom, the representations were constructed with four, three, three, and two maps

Fig. 5

Iowa D renderings. The row and column setup is as in Fig. 2 in Appendix 2. From top to bottom, the representations were constructed with four, four, three, and three maps

Fig. 6

High resolution renderings. Cumulative distributions, autocorrelation functions, and power spectra (plotted in log–log) for the four best renderings in black, imposed upon that of the original storms in gray. From top to bottom: Iowa A, B, C, D, all represented using the basic extension (Sect. 2.2) with four, three, three, and four maps