Investigating Geometric Complexity of Precipitation in California … · Maskey ML, CE Puente & B...
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Remarks and Further Steps Cherry Valley Cherry Valley Cherry Valley Merced Merced Shasta Dam Merced Sacramento Shasta Dam Sacramento Shasta Dam Sacramento This work emphasizes the “deterministic complexity” of the rainfall process, as the individual sets may be represented by the FM method. Although the FM approach allows visualizing the dynamics of rainfall, there exists clear variability and no obvious trends on parameters that may be attributed to changes in climate. All sites studied may be termed “equally complex” as their FM parameters yield sparse representations, as found with total rainfall volumes. Therefore, further investigating within the space of parameters could allow discriminating rainfall complexity between sites. Investigating Geometric Complexity of Precipitation in California via the Fractal-Multifractal Method Mahesh L. Maskey, 1* Carlos E. Puente, 1 and Bellie Sivakumar 2,1 1 University of California, Davis, CA; Department of Land, Air, and Water Resources 2 University of New South Wales, Sydney, Australia; Department of Civil and Environmental Engineering * Corresponding Author: E-mail: [email protected] This work employs a deterministic geometric approach, named the fractal–multifractal (FM) method, to encode highly intermittent daily rainfall records, in order to investigate the intrinsic complexity of rainfall in various stations within the State of California. To this effect, 60+ years of daily precipitation records gathered, from South to North, at Cherry Valley, Merced, Sacramento and Shasta Dam are studied. The analysis reveals that: (a) the FM approach results in several faithful equifinal encodings of all records, by years, with mean square errors in accumulated rain that do not exceed 3%, (b) the evolution of the corresponding FM parameters while reflecting the implicit variability in the records do not exhibit discernible trends in time at every station that may be attributed to global change, and (c) a comparison of FM parameters for the four sites confirms the expected notion that all stations are equally complex.. California Extreme Precipitation Symposium Introduction Quantifying temporal and spatial complexity of rainfall process is paramount for the proper planning and design of water resources systems. Certainly, it is desirable to develop suitable techniques that may allow further understanding of the underlying complex non-linear behavior and high-intermittency behind the rainfall process, especially when the planet is undergoing substantial climate change impact. Given the limited capability of both physical and stochastic approaches in capturing intricate details of geophysical records, Puente (1996) developed a fractal geometric methodology, the fractal-multifractal (FM) approach. Such a notion models natural records as a fractal transformation of a multifractal measure rather than using a realization of an stochastic process. To date, the FM approach has already been used to encode: (a) rainfall events (Obregón et al. 2002; Huang et al. 2012), (b) daily rainfall sets gathered over a year (Maskey et al. 2015; Puente et al. 2016), (c) daily streamflow records over a year (Puente et al. 2016; Maskey et al. 2016), (d) daily temperature measurements (Puente et al. 2016), and also (e) downscaling daily rainfall and runoff from weekly, biweekly and monthly sets (Maskey et al. 2017). This work represents an effort to employ the FM method as a tool to quantify rainfall complexity of data sets collected within California. Herein, the FM notions are tested using daily rainfall sets gathered over water years (October 1 st to September 30 th ) in four sites in California, from south to north: Cheery Valley, Merced, Sacramento and Shasta Dam, and collected by NOAA’s National Climate Data Center (NCDC) References The Fractal-Multifractal Approach Overall Encodings Barnsley MF (1988) Fractals Everywhere, Academic Press, San Diego, California. Huang H, A Cortis & CE Puente (2012) Geometric harnessing of precipitation records, SERRA DOI 10.1007/s00477-012-0617-6. Obregón N, CE Puente & B Sivakumar (2002) Modeling high resolution rain rates via a deterministic approach, Fractals, 10(3), 387–394. Maskey ML, CE Puente, B Sivakumar & A Cortis (2015) Encoding daily rainfall records via adaptations of the fractal multifractal method. SERRA, 30(7), 1917-1931. Maskey ML, CE Puente & B Sivakumar (2016a) A comparison of fractal-multifractal techniques for encoding streamflow records. Journal of Hydrology, 542, 564-580. Maskey ML, CE Puente & B Sivakumar (2019) Temporal downscaling rainfall and streamflow records through a deterministic fractal geometric approach. Journal of Hydrology, 568, 447-461. Puente CE (1996) A new approach to hydrologic modeling: derived distributions revisited, Journal of Hydrology, 187, 65-80. Puente CE, ML Maskey & B. Sivakumar (2016) Combining fractals and multifractals to model geoscience records. In: B. Ghanbarian, A. Hunt (eds.) Fractals: concepts and applications in geoscience, CRC Press, Boca Raton, FL Abstract Geometric Classification of rainfall patterns As seen in Figure 5, both evolutions of rainfall classes based on successive FM parameters (red) and the deciles of the records (blue) are equally complex and lead to similar and broad transition matrices, on the right. Encodings of Rainfall Sets Figure 1 illustrates how a disperse attractor is constructed using two affine maps (Barnsley 1988) whose end-points are 0,0 , . , . and . , −. , 1, −. , namely: 1 0.41 0 0.97 −0.32 2 0.19 0 −4.53 −0.43 0.80 −4.02 and when such are iterated following independent outcomes of a 33- 67% biased coin. As seen, such a process defines an attractor, from x to y, a Cantorian measure over x (Mandelbrot, 1982), and an interesting “rainfall” sets over y (Maskey et al. 2015). Figure 2 demonstrates the ability of the FM approach, based on 8 geometric parameters (the ones in bold above) to encode daily rainfall normalized records (adding up to one) at the four California sites. Such were found solving an inverse problem aimed at fitting the accumulated records. Despite having geometrically disparities, the close approximation of accumulated sets explains that the FM method gives reasonable fits as implied by small values of encoding errors as tabulated in Table 1. It ought to be noted that close “solutions” exist but having distinct parameters (equifinality). Figure 3 includes best FM representations over the whole records available: 59 years for Cherry Valley, 116 for Merced, 115 for Sacramento and 72 for Shasta Dam. This figure includes the observed rainfall set (top), and the corresponding FM fit (bottom) obtained by upgrading annual volumes (depths). Figure 2. A measured set of daily rainfall sets at four sites of California gathered from October 1 st to 30 th September of the water years shown, followed by relevant mass function (accumulated set), histogram and Rényi entropy, and scatter plot of accumulated records. Table 2. Summary of Overall encodings at all places shown in Figure 3. Figure 3. Rainfall records in four sites of California for the shown water years (top-blue) and best FM representations (year by red). The scales of the rain sets are in inches/day. Geometric Dynamics of Rainfall Figure 1. A generalized FM approach: from a Cantorian texture , to a projection , via a disperse attractor from to . The set is found pruning below a threshold . Figure 4. Evolution of the best FM parameters for all sites (blue) and averages over 5 years (red). Figure 5. Evolution of best FM rainfall classes obtained via k-means clustering of FM parameters for all sites in blue and evolution of classes based on deciles of data in red (left), followed their corresponding Markov matrices, right. � Cherry Valley (1957-2015) Merced (1900-2015) Sacramento (1901-2015) Shasta Dam (1944-2015) Best Data Cherry Valley Sacramento Merced Shasta Dam Spatial Comparison of geometric pattern of rainfall set Qualifiers Cherry Valley Merced Sacramento Shasta Dam RMSEAR 1.6±0.2 1.6±0.2 1.6±0.2 1.8±0.2 MAXEAR 7.1±1.5 6.9±1.3 7.1±1.2 7.9±2.2 Sites (water years) RMSEAR MAXEAR NSH NSE Cherry Valley (1998-1999) 1.40 4.04 99.9 99.0 Merced (1931-1932) 1.65 5.42 99.9 99.1 Sacramento (1939-1940) 1.43 4.87 100 95.0 Shasta Dam (2006-2007) 2.28 5.50 99.9 97.2 Table 1. Performance of FM model for the best representations for all sites and years shown in Figure 2 0 1 365 365 362 1 10 0.1 5.1 1 1 1 0 1 1 1.2 1.2 1 0 0 0 0 0.5 1 Cherry Valley (1998-1999) Merced (1931-1932) Sacramento (1939-1940) Shasta Dam (2006-2007) 0 0 1 366 366 366 1 10 0.1 5.1 1 1 1 0 1 1 1.2 1.2 1 0 0 0 0.5 1 0 1 366 366 366 1 10 0.1 5.1 1 1 1 1 0 1 1 1.2 1.2 1 0 0 0 0 0.5 � ∑ � ∑ FM Set Scatter ∑ Data Histogram Entropy 1 0 1 1 1.2 1.2 1 0 1 365 365 365 0 0 1 10 0.1 5.1 1 0 0 1 1 0.5 Accumulated The goodness of the FM encodings is further corroborated by small range of RMSEAR (optimized) and maximum error in accumulated records MAXEAR over the years (Table 2). Figure 4 includes the time evolution of the best FM parameters for all four sites. The FM geometric parameters vary wildly and often swing from high to low values and vice-versa. There are no noticeable trends in the best FM parameters (not even when averaged every five years, in red) and such a fact clearly precludes the possibility of readily finding rainfall forecasts from such geometric information or discerning effects due to climate change. Year Cherry Valley Merced Sacramento Shasta Dam 1 57 76 96 15 00 19 39 59 79 99 15 01 20 40 60 80 00 15 43 63 83 03 15 1 0 1 0 1 0 1 0 0.7 0 0.7 0 0.8 0 0.7 0 1 2 2 1 2 3 57 67 77 87 97 07 15 0 13 57 67 77 87 97 07 15 0 13 01 01 11 21 51 71 91 10 31 41 61 81 0 4.6 01 11 21 51 71 91 10 31 41 61 81 0 4.6 01 00 10 20 50 70 90 10 30 40 60 80 0 2.7 00 00 10 20 50 70 90 10 30 40 60 80 0 2.7 00 44 54 74 84 14 64 94 0 12 04 44 54 74 84 14 64 94 0 12 04 Figure 6 portrays pairwise comparisons of sites based on best FM solutions, as in Figures 3-5. Such graphs also include a visual comparison of the class evolutions for concurrent years and the corresponding scatter plots. As seen, the rather erratic class evolutions result in fairly uncorrelated scatter plots yielding no clear correlations among sites, hence emphasizing the complexity of the rainfall records in space. Figure 6. Pairwise site-comparison of best FM rainfall class evolutions for different sites and their scatter plots. The evolutions use blue for the set in the x-axis and red for the one in the y-axis. Sizes of circles are proportional to class repetitions. Year Year
Transcript of Investigating Geometric Complexity of Precipitation in California … · Maskey ML, CE Puente & B...
Remarks and Further Steps
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This work emphasizes the “deterministic complexity” of the rainfall process, as the individual sets may be represented by the FM method. Although the FM approach allows visualizing the dynamics of rainfall, there exists clear variability and no obvious trends on parameters that may be attributed to changes in climate. All sites studied may be termed “equally complex” as their FM parameters yield sparse representations, as found with total rainfall volumes. Therefore, further investigating within the space of parameters could allow discriminating rainfall complexity between sites.
Investigating Geometric Complexity of Precipitation in California via the Fractal-Multifractal Method
Mahesh L. Maskey,1* Carlos E. Puente,1 and Bellie Sivakumar2,1 1University of California, Davis, CA; Department of Land, Air, and Water Resources
2 University of New South Wales, Sydney, Australia; Department of Civil and Environmental Engineering
* Corresponding Author: E-mail: [email protected]
This work employs a deterministic geometric approach, named the fractal–multifractal (FM) method, to encode highly intermittent daily rainfall records, in order to investigate the intrinsic complexity of rainfall in various stations within the State of California. To this effect, 60+ years of daily precipitation records gathered, from South to North, at Cherry Valley, Merced, Sacramento and Shasta Dam are studied. The analysis reveals that: (a) the FM approach results in several faithful equifinal encodings of all records, by years, with mean square errors in accumulated rain that do not exceed 3%, (b) the evolution of the corresponding FM parameters while reflecting the implicit variability in the records do not exhibit discernible trends in time at every station that may be attributed to global change, and (c) a comparison of FM parameters for the four sites confirms the expected notion that all stations are equally complex..
California Extreme Precipitation Symposium
Introduction Quantifying temporal and spatial complexity of rainfall process is paramount for the proper planning and design of water resources systems. Certainly, it is desirable to develop suitable techniques that may allow further understanding of the underlying complex non-linear behavior and high-intermittency behind the rainfall process, especially when the planet is undergoing substantial climate change impact.
Given the limited capability of both physical and stochastic approaches in capturing intricate details of geophysical records, Puente (1996) developed a fractal geometric methodology, the fractal-multifractal (FM) approach. Such a notion models natural records as a fractal transformation of a multifractal measure rather than using a realization of an stochastic process. To date, the FM approach has already been used to encode: (a) rainfall events (Obregón et al. 2002; Huang et al. 2012), (b) daily rainfall sets gathered over a year (Maskey et al. 2015; Puente et al. 2016), (c) daily streamflow records over a year (Puente et al. 2016; Maskey et al. 2016), (d) daily temperature measurements (Puente et al. 2016), and also (e) downscaling daily rainfall and runoff from weekly, biweekly and monthly sets (Maskey et al. 2017).
This work represents an effort to employ the FM method as a tool to quantify rainfall complexity of data sets collected within California. Herein, the FM notions are tested using daily rainfall sets gathered over water years (October 1st to September 30th) in four sites in California, from south to north: Cheery Valley, Merced, Sacramento and Shasta Dam, and collected by NOAA’s National Climate Data Center (NCDC)
References
The Fractal-Multifractal Approach
Overall Encodings
Barnsley MF (1988) Fractals Everywhere, Academic Press, San Diego, California. Huang H, A Cortis & CE Puente (2012) Geometric harnessing of precipitation records, SERRA DOI 10.1007/s00477-012-0617-6. Obregón N, CE Puente & B Sivakumar (2002) Modeling high resolution rain rates via a deterministic approach, Fractals, 10(3), 387–394. Maskey ML, CE Puente, B Sivakumar & A Cortis (2015) Encoding daily rainfall records via adaptations of the fractal multifractal method. SERRA, 30(7), 1917-1931. Maskey ML, CE Puente & B Sivakumar (2016a) A comparison of fractal-multifractal techniques for encoding streamflow records. Journal of Hydrology, 542, 564-580. Maskey ML, CE Puente & B Sivakumar (2019) Temporal downscaling rainfall and streamflow records through a deterministic fractal geometric approach. Journal of Hydrology, 568, 447-461. Puente CE (1996) A new approach to hydrologic modeling: derived distributions revisited, Journal of Hydrology, 187, 65-80. Puente CE, ML Maskey & B. Sivakumar (2016) Combining fractals and multifractals to model geoscience records. In: B. Ghanbarian, A. Hunt (eds.) Fractals: concepts and applications in geoscience, CRC Press, Boca Raton, FL
Abstract
Geometric Classification of rainfall patterns As seen in Figure 5, both evolutions of rainfall classes based on successive FM parameters (red) and the deciles of the records (blue) are equally complex and lead to similar and broad transition matrices, on the right.
Encodings of Rainfall Sets
Figure 1 illustrates how a disperse attractor is constructed using two affine maps (Barnsley 1988) whose end-points are {(0,0), (𝟎𝟎.𝟒𝟒𝟒𝟒,𝟒𝟒.𝟎𝟎𝟎𝟎)} and {(𝟎𝟎.𝟎𝟎𝟎𝟎,−𝟒𝟒.𝟎𝟎𝟎𝟎), (1,−𝟎𝟎.𝟑𝟑𝟑𝟑)}, namely:
𝑤𝑤1𝑥𝑥𝑦𝑦 = 0.41 0
0.97 −0.32𝑥𝑥𝑦𝑦
𝑤𝑤2𝑥𝑥𝑦𝑦 = 0.19 0
−4.53 −0.43𝑥𝑥𝑦𝑦 + 0.80
−4.02
and when such are iterated following independent outcomes of a 33-67% biased coin. As seen, such a process defines an attractor, from x to y, a Cantorian measure over x (Mandelbrot, 1982), and an interesting “rainfall” sets over y (Maskey et al. 2015).
Figure 2 demonstrates the ability of the FM approach, based on 8 geometric parameters (the ones in bold above) to encode daily rainfall normalized records (adding up to one) at the four California sites. Such were found solving an inverse problem aimed at fitting the accumulated records. Despite having geometrically disparities, the close approximation of accumulated sets explains that the FM method gives reasonable fits as implied by small values of encoding errors as tabulated in Table 1. It ought to be noted that close “solutions” exist but having distinct parameters (equifinality).
Figure 3 includes best FM representations over the whole records available: 59 years for Cherry Valley, 116 for Merced, 115 for Sacramento and 72 for Shasta Dam. This figure includes the observed rainfall set (top), and the corresponding FM fit (bottom) obtained by upgrading annual volumes (depths).
Figure 2. A measured set of daily rainfall sets at four sites of California gathered from October 1st to 30th September of the water years shown, followed by relevant mass function (accumulated set), histogram and Rényi entropy, and scatter plot of accumulated records.
Table 2. Summary of Overall encodings at all places shown in Figure 3.
Figure 3. Rainfall records in four sites of California for the shown water years (top-blue) and best FM representations (year by red). The scales of the rain sets are in inches/day.
Geometric Dynamics of Rainfall Figure 1. A generalized FM approach: from a Cantorian texture 𝑑𝑑𝑥𝑥, to a projection 𝑑𝑑𝑦𝑦, via a disperse attractor from 𝑥𝑥 to 𝑦𝑦. The set 𝑑𝑑𝑦𝑦𝑣𝑣 is found pruning 𝑑𝑑𝑦𝑦 below a threshold 𝜙𝜙.
Figure 4. Evolution of the best FM parameters for all sites (blue) and averages over 5 years (red).
Figure 5. Evolution of best FM rainfall classes obtained via k-means clustering of FM parameters for all sites in blue and evolution of classes based on deciles of data in red (left), followed their corresponding Markov matrices, right.
𝑅𝑅
𝑅𝑅�
𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌
Cherry Valley (1957-2015) Merced (1900-2015)
Sacramento (1901-2015) Shasta Dam (1944-2015)
Best Data
Cherry Valley
Sacramento
Merced
Shasta Dam
Spatial Comparison of geometric pattern of rainfall set
Qualifiers Cherry Valley Merced Sacramento Shasta
Dam RMSEAR 1.6±0.2 1.6±0.2 1.6±0.2 1.8±0.2 MAXEAR 7.1±1.5 6.9±1.3 7.1±1.2 7.9±2.2
Sites (water years) RMSEAR MAXEAR NSH NSE Cherry Valley (1998-1999) 1.40 4.04 99.9 99.0
Merced (1931-1932) 1.65 5.42 99.9 99.1 Sacramento (1939-1940) 1.43 4.87 100 95.0 Shasta Dam (2006-2007) 2.28 5.50 99.9 97.2
Table 1. Performance of FM model for the best representations for all sites and years shown in Figure 2
0 1 365 365 362 1 10 0.1 5.1 1 1
1
0
1 1 1.2 1.2 1
0 0 0 0 0.5 1
Cherry Valley (1998-1999)
Merced (1931-1932)
Sacramento (1939-1940)
Shasta Dam (2006-2007)
0 0 1 366 366 366 1 10 0.1 5.1 1 1
1
0
1 1 1.2 1.2 1
0 0 0 0.5 1
0 1 366 366 366 1 10 0.1 5.1 1 1 1
1
0
1 1 1.2 1.2 1
0 0 0 0 0.5
𝑅𝑅� 𝑝𝑝 𝐻𝐻 ∑𝑅𝑅�
𝑡𝑡 𝑞𝑞 ∑𝑅𝑅
FM Set Scatter
𝑅𝑅
𝑡𝑡 𝑌𝑌
∑𝑅𝑅
𝑡𝑡
Data Histogram Entropy 1
0
1 1 1.2
0
1.2 1
0 1 365 365 365 0 0
1 10 0.1 5.1 1 0 0
1 1 0.5
Accumulated
𝑥𝑥
𝑑𝑑𝑥𝑥
𝑑𝑑𝑦𝑦 𝑑𝑑𝑦𝑦𝑣𝑣
The goodness of the FM encodings is further corroborated by small range of RMSEAR (optimized) and maximum error in accumulated records MAXEAR over the years (Table 2).
Figure 4 includes the time evolution of the best FM parameters for all four sites. The FM geometric parameters vary wildly and often swing from high to low values and vice-versa. There are no noticeable trends in the best FM parameters (not even when averaged every five years, in red) and such a fact clearly precludes the possibility of readily finding rainfall forecasts from such geometric information or discerning effects due to climate change.
Year
Cherry Valley Merced Sacramento Shasta Dam
𝑥𝑥1
𝜙𝜙
57 76 96 15 00 19 39 59 79 99 15 01 20 40 60 80 00 15 43 63 83 03 15
1
0
1
0
1
0
1
0
0.7
0
0.7
0
0.8
0
0.7
0
𝑑𝑑1
𝑑𝑑2
𝑝𝑝
𝑥𝑥2
𝑦𝑦1
𝑦𝑦2
𝑦𝑦3
57 67 77 87 97 07 15 0
13
57 67 77 87 97 07 15 0
13
01 01 11 21 51 71 91 10 31 41 61 81
0
4.6
01 11 21 51 71 91 10 31 41 61 81 0
4.6
01
00 10 20 50 70 90 10 30 40 60 80 0
2.7
00
00 10 20 50 70 90 10 30 40 60 80 0
2.7
00
44 54 74 84 14 64 94 0
12
04
44 54 74 84 14 64 94 0
12
04
Figure 6 portrays pairwise comparisons of sites based on best FM solutions, as in Figures 3-5. Such graphs also include a visual comparison of the class evolutions for concurrent years and the corresponding scatter plots. As seen, the rather erratic class evolutions result in fairly uncorrelated scatter plots yielding no clear correlations among sites, hence emphasizing the complexity of the rainfall records in space.
Figure 6. Pairwise site-comparison of best FM rainfall class evolutions for different sites and their scatter plots. The evolutions use blue for the set in the x-axis and red for the one in the y-axis. Sizes of circles are proportional to class repetitions.
Year
Year
