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Deterministic simulation of highly intermittent hydrologic time series

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Abstract

Application of a deterministic geometric approach for the simulation of highly intermittent hydrologic data is presented. Specifically, adaptations of the fractal-multifractal (FM) method and a Cantorian extension are advanced in order to simulate rainfall records measured at the daily scale and encompassing a water year. It is shown, using as case studies 2 years of rainfall sets gathered in Laikakota, Bolivia and Tinkham, Washington, USA, that the FM approach, relying on only at most 8 parameters, is capable of closely preserving either the whole record’s histogram (therefore including moments), the whole data’s Rényi entropy function and/or the maximum number of consecutive zero values present in the sets, resulting in suitable rainfall simulations, whose overall features and textures are similar to those of the observed sets. The study hence establishes the possibility of simulating highly intermittent sets in time in a deterministic and holistic way as a novel parsimonious methodology to supplement available stochastic frameworks.

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Abbreviations

NSH:

Nash–Sutcliffe statistic for histograms

PS90:

Percent histogram mass in simulated sets corresponding to 90% in observed data

NZR:

Number of zeroes in observed and (simulated) rainfall sets

EZR:

Percent error of zeros preserved by FM simulation

NSE:

Nash–Sutcliffe statistic for entropies

MCZ:

Maximum consecutive zeros in records and (simulated) sets

NSA:

Nash–Sutcliffe statistic for autocorrelations

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Acknowledgements

The research leading to this article was supported by a JASTRO Award provided to the first author by the University of California, Davis. We are thankful to Ministerio de Medio Ambiente y Agua, Bolivia for providing rainfall records gathered at Laikakota and also to the team of National Resource Conservation Service for the availability of rainfall records in its web portal. Bellie Sivakumar acknowledges the financial support from the Australian Research Council (ARC) through the Future Fellowship Grant awarded to him (FT110100328). Comments and suggestions by anonymous reviewers helped improve the manuscript and are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Land, Air & Water Resources, University of California, Davis, CA, 95616, USA

    Mahesh L. Maskey, Carlos E. Puente & Bellie Sivakumar

  2. School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia

    Bellie Sivakumar

Authors
  1. Mahesh L. Maskey
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  2. Carlos E. Puente
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  3. Bellie Sivakumar
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Corresponding author

Correspondence to Carlos E. Puente.

Appendix 1: Parameter values for FM representation of Figs. 3–8

Appendix 1: Parameter values for FM representation of Figs. 3–8

See Tables 3 and 4.

Fig. 3
figure 3

A measured set of daily rainfall at Laikakota, Bolivia gathered from September 1965 to May 1966 (top), followed by its histogram, Rényi entropy and autocorrelation functions; and four suitable rainfall simulations (bottom), based on fits of the records’ histogram. While simulations A and B emanate from fractal wires, C and D come from Cantorian attractors. While sets A and C use a horizontal threshold, B and D employ a vertical one

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Fig. 4
figure 4

A measured set of daily rainfall at Laikakota, Bolivia gathered from September 1965 to May 1966 (top), followed by its histogram, Rényi entropy and autocorrelation functions; and four suitable rainfall simulations (bottom), based on fits of the records’ entropy function. While simulations A and B emanate from fractal wires, C and D come from Cantorian attractors. While sets A and C use a horizontal threshold, B and D employ a vertical one

Full size image
Fig. 5
figure 5

A measured set of daily rainfall at Laikakota, Bolivia gathered from September 1965 to May 1966 (top), followed by its histogram, Rényi entropy and autocorrelation functions; and four suitable rainfall simulations (bottom), based on fits of the records’ distribution of zeros. While simulations A and B emanate from fractal wires, C and D come from Cantorian attractors. While sets A and C use a horizontal threshold, B and D employ a vertical one

Full size image
Fig. 6
figure 6

A measured set of daily rainfall at Tinkham Creek, Washington gathered from October 2000 to September 2001 (top), followed by its histogram, Rényi entropy and autocorrelation functions; and four suitable rainfall simulations (bottom), based on fits of the records’ histogram. While simulations A and B emanate from fractal wires, C and D come from Cantorian attractors. While sets A and C use a horizontal threshold, B and D employ a vertical one

Full size image
Fig. 7
figure 7

A measured set of daily rainfall at Tinkham Creek, Washington gathered from October 2000 to September 2001 (top), followed by its histogram, Rényi entropy and autocorrelation functions; and four suitable rainfall simulations (bottom), based on fits of the records’ entropy function. While simulations A and B emanate from fractal wires, C and D come from Cantorian attractors. While sets A and C use a horizontal threshold, B and D employ a vertical one

Full size image
Fig. 8
figure 8

A measured set of daily rainfall at Tinkham Creek, Washington gathered from October 2000 to September 2001 (top), followed by its histogram, Rényi entropy and autocorrelation functions; and four suitable rainfall simulations (bottom), based on fits of the records’ distribution of zeros. While simulations A and B emanate from fractal wires, C and D come from Cantorian attractors. While sets A and C use a horizontal threshold, B and D employ a vertical one

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Table 3 “Wire” representations
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Table 4 Cantorian representations
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Maskey, M.L., Puente, C.E. & Sivakumar, B. Deterministic simulation of highly intermittent hydrologic time series. Stoch Environ Res Risk Assess 31, 2719–2732 (2017). https://doi.org/10.1007/s00477-016-1343-2

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