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Encoding daily rainfall records via adaptations of the fractal multifractal method

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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.

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Abbreviations

RMSEAR:

Root mean square error in accumulated rainfall (observed vs. FM fitted)

MAXEAR:

Maximum error in accumulated rainfall (observed vs. fitted)

NSR7:

Nash–Sutcliffe statistic between observed rainfall at the weekly scale (observed vs. fitted)

NSACR:

Nash–Sutcliffe statistic between rainfall autocorrelations (observed vs. fitted)

AL0:

Lag where autocorrelation reaches zero for the first time

NSHR:

Nash–Sutcliffe statistic between rainfall histograms (observed vs. fitted)

PZMR:

Percent of zeros matched between observed and fitted rainfalls

PF90:

Percent histogram mass in fitted records corresponding to 90 % in observed data

NSER:

Nash–Sutcliffe statistic between rainfall entropies (observed vs. fitted)

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Acknowledgments

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). The thoughtful comments from anonymous reviewers are gratefully acknowledged, as they resulted in an improved manuscript, both technically and in presentation. Those are gratefully acknowledged.

Author information

Authors and Affiliations

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

    M. L. Maskey, C. E. Puente & B. Sivakumar

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

    B. Sivakumar

  3. GE Oil & Gas, Houston, TX, 77032, USA

    A. Cortis

Authors
  1. M. L. Maskey
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  2. C. E. Puente
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  3. B. Sivakumar
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  4. A. Cortis
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Corresponding author

Correspondence to C. E. Puente.

Appendix 1: FM parameters for rainfall encoding

Appendix 1: FM parameters for rainfall encoding

Figure

Maps

X1

X2

X3

X4

Y1

Y2

Y3

Y4

Y5

d1

d2

d3

p1 (%)

p2 (%)

p3 (%)

\(\phi_{\varvec{v}}\) (%)

3A

3

0.525

0.704

0.390

0.907

−4.583

2.478

−1.907

1.185

−0.911

0.116

0.455

−0.573

24.4

25.5

50.1

30.8

3B

2

0.033

0.327

1

−

2.737

−2.758

−1.417

–

–

−0.136

−0.715

–

22.7

77.3

–

26.3

3C

3

0.367

0.630

0.443

0.958

1.578

0.575

−1.903

−2.002

1

−0.358

−0.250

0.128

43.8

17.7

38.5

16.1

5A

2

0.068

0.584

1

–

4.951

−0.249

3.709

–

–

0.536

0.288

–

42.3

57.7

–

28.7

5B

3

0.759

0.256

0.316

0.947

0.526

−2.137

1.222

−3.538

0

−0.692

0.127

−0.003

31.4

39.4

29.2

5.7

5C

3

0.000

0.613

0.271

0.374

4.992

2.287

−2.847

−1.669

5

0.143

0.173

0.010

43.0

32.4

24.6

3.3

7A

3

0.277

0.545

0.589

0.988

−2.688

−2.431

1.697

0.015

5

0.184

−0.374

−0.312

32.9

38.7

28.4

12.1

7B

3

0.117

0.463

0.149

0.385

4.643

2.531

0.798

−0.915

2.023

−0.316

−0.530

−0.255

43.2

20.5

36.3

19.9

7C

3

0.101

0.000

0.760

0.566

−0.756

−0.570

−2.026

−2.206

5

−0.590

−0.191

0.269

37.7

13.1

49.2

14.5

8A

3

0.379

0.996

0.802

0.482

4.023

−3.796

4.909

0.057

5

−0.737

−0.239

0.461

64.8

32.8

2.4

27.7

8B

3

0.428

0.434

1

–

−0.368

−3.998

3.577

–

–

−0.069

0.341

−0.325

25.2

30.7

44.1

10.3

8C

3

0.003

1.000

0.366

0.327

−5.000

4.861

0.635

−3.766

0

0.182

−0.456

0.116

25.2

44.5

30.3

16.1

  1. The domain for all cases goes from 0 to 1, i.e., X 0  = 0, X 3 or X 5  = 0. The first vertical value is always 0, i.e., Y 0  = 0
  2. All the cases come from FM Cantorian representations except 8B that uses a FM wire passing through four interpolating points

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Maskey, M.L., Puente, C.E., Sivakumar, B. et al. Encoding daily rainfall records via adaptations of the fractal multifractal method. Stoch Environ Res Risk Assess 30, 1917–1931 (2016). https://doi.org/10.1007/s00477-015-1201-7

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