report for cea technologies inc. (ceati)gang/ftp.transfer/dsig.final... · 2012. 10. 12. · cea...

Report for

CEA TECHNOLOGIES Inc. (CEATI) 1155 Metcalfe Street, Suite 1120

Montreal, Quebec, Canada H3B 2V6 Website: www.ceatech.ca

Dam Safety Interest Group: DSIG

MULTIFRACTALS AND PHYSICALLY BASED ESTIMATES OF EXTREME FLOODS-

Phase 4A

Prepared by Physics Department, McGill University

Montreal, Quebec

Principal Investigator Shaun Lovejoy

October 2012

Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A

ABSTRACT The overall goal of this project is to better (statistically) predict floods using a physically based approach established on systems which respect a scale symmetry over a wide range of space-time scales and to determine the relationship between flood magnitude and return period for a wide range of aggregation periods. Previous phases of this report have focused on developing this theory and demonstrating it on data series. This phase of the project consisted of creating MATLAB versions of the software that performs the analysis and writing examples of how to use them. The main object of this phase is to explain how to use software developed in MATLAB for the purpose of flood frequency analysis.

ii Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A

TABLE OF CONTENTS

Page

ABSTRACT..............................................................................................................................................ii 1.0 INTRODUCTION........................................................................................................................5 2.0 EXAMPLES..................................................................................................................................5 2.1 Codim.............................................................................................................................................6 2.2 CodimPD........................................................................................................................................7 2.3 DTM...............................................................................................................................................8 2.4 DTMauto.....................................................................................................................................10 2.5 DTMspec......................................................................................................................................11 2.6 Ecodim.........................................................................................................................................13 2.7 GammaDprac..............................................................................................................................14 2.8 GammaS......................................................................................................................................15 2.9 Hspec............................................................................................................................................16 2.10 HspecAuto...................................................................................................................................18 2.11 MFFA...........................................................................................................................................19 2.12 MFSS............................................................................................................................................20 2.13 PD.................................................................................................................................................21 2.14 qDprac.........................................................................................................................................23 2.15 QT................................................................................................................................................24 2.16 QTcompare..................................................................................................................................26 2.17 Singularity...................................................................................................................................27 2.18 Spectrum1D.................................................................................................................................28 2.19 Theta............................................................................................................................................29 2.20 TraceMoment..............................................................................................................................30 3.0 ADDITIONAL EXAMPLES.....................................................................................................32 4.0 APPENDIX..................................................................................................................................35 4.1 Appendix A: Uchee Creek data.................................................................................................35 4.2 Appendix B: Additional streamflow data.................................................................................36

iii Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A

1.0 INTRODUCTION

In phase 1A of the project “Multifractals and Physically Based Estimates of Extreme Floods” various statistical properties of a data base of North American streamflows were investigated using statistical scaling techniques. In order to characterize the statistics over a wide range of time scales, multifractal scaling methods were used and the time series were characterized by three “universal multifractal” exponents: α, C1, H with a fourth exponent qD characterizing the extreme tails of the probability distribution. In phase 1B, various techniques were discussed for conveniently estimating them, the MATLAB software described in the following report cover all the techniques necessary for producing a Multifractal Flood Frequency Analysis (MFFA), i.e. the relationship between the magnitude of the flood event Q (in m3/s) and the return time T (in years).

2.0 EXAMPLES

The following are examples of how to use the MATLAB functions provided. They include a basic summary of each function, its inputs, outputs, and an example of its usage with streamflow data. With these examples the user should be able to successfully execute the functions and obtain values of the desired parameters needed for flood frequency analysis. It is recommended that the user read the “Inputs” section carefully so as to avoid any errors with the function. It is also important that all functions be in the chosen MATLAB directory since many functions require the use of another (this is specified in each example). Further comments about each function are provided in the code.

5 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.1 codim Basic Summary This is the theoretical “bare” codimension function c(γ). It is a function of gamma, alpha, and C1. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.1. Inputs There are three inputs. C1 and alpha can be found from a separate function (e.g TraceMoment) and different values of gamma can be substituted into the function to find different values of c(γ). Outputs There will be one output value, the value of c(γ). Example Input: gamma=1 alpha=1.9435 C1=0.3336 Output: 1.2802 Note: This function requires GammaDprac, GammaS, qDprac, and GammaD Alternatively, to display the graph of c(γ) keeping alpha and C1 constant, type: fplot(@(gamma) codim(gamma,1.9435,0.3336), [-.5 .5]) (here the range is -.5 - .5 but can be changed) To obtain:

Figure 1: The codimension function For the paramters above. Note: c(γ) = 0 for γ ≤ -C1/(α-1).

6 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.2 codimPD Basic Summary This function is similar in every point to the previous one, codim, except that instead of calculating γD and qD using practical estimate techniques with the function GammaDprac and qDprac, they are inputted directly. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.1. Inputs There are five inputs. C1 and alpha can be found from a separate function (e.g TraceMoment) and different values of gamma can be substituted into the function to find different values of c(γ). To find γD and qD, the probability distribution function, PD, can be used. Outputs There will be one output value, the value of c(γ) at the specified γ. Example Input: codimPD(gamma,alpha,C1,gammaD,qD) = codimPD(1,1.9435,0.3336,0.8162,2.5701) Output: ans =1.4704

7 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.3 DTM Basic Summary This is the Double Trace Moment function (DTM). This method allows the user to obtain an estimate for the codimension function’s fixed point, C1, and the multifractality index, α. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 3.4.2. Inputs There are five inputs for this function. The first one, “field”, should be a one-dimensional data series on which the DTM method will be performed. The second, q, determines the statistical order of the process. The third and fourth inputs, kfitmin and kfitmax, determine the range over which the log2M a function of log2λ should be fitted. Finally, the last input, range, determines how many points around η0 should be used for the fit of log10M as a function of log10η. Note that η0 is selected such that the condition q × η0 = 1 holds. Outputs The outputs of the function are two values and two graphs. The values are C1 and α. The graphs are a graph of log2M vs. log2λ and a graph of log10M vs. log10η. In the former plot the black stars mark the range for fitting; in the latter plot the cyan stars mark the range centered on the black star which marks η0. Note: This function requires Flux and K. Example Input: DTM(field,q,kfitmin,kfitmax,range)=DTM(UcheeCreek,2,8,13,3) Output: 'C1' [0.2834] 'alpha' [1.5626] Figure 2: Logarithmic plot showing scaling behaviour of double trace moments on the Uchee Creek data. Black stars mark fitting range.

8 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A

DTM (continued)

Figure 3: Logarithmic plot of slopes of Figure 2 as functions of η. Blue stars mark fitting range for linear fit, black star marks η0.

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2.4 DTMauto Basic Summary This is the Double Trace Moment function (DTM). It uses second order moments (q =2) and a fixed 32 day range for fitting, which corresponds to the scaling break between low-frequency and high-frequency weather. It outputs values of the parameters C1 and alpha. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 3.4.2. Inputs There is one input of this function: field. This is a one-dimensional data series Outputs The outputs of the function are two values and two graphs. The values are C1 and alpha. The graphs are a graph of Log(M) vs. Log(λ) and a graph of Log(M) vs. Log(η). In the former plot the black stars mark the range for fitting, in the latter plot the blue stars mark this range and the black star marks η0. Note: This function requires Flux and K. Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) Output: 'C1' [0.2834] 'alpha' [1.5626] Graphs outputted are the same as above.


2.5 DTMspec Basic Summary DTMspec finds values of H, C1, and alpha by automatically using the Double Trace Moment method to find alpha and C1 and then using Hspec to find H over low and high frequencies (see Hspec example page for more information). The fitting range used in Hspec is automatically chosen to be (32 days)-1 and lower for the low frequencies and above this for the high frequencies. This cutoff corresponds to the break in the scaling regime at ~32days. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 4.3.2. Inputs The only input for this function is “field” which is a one-dimensional array of data. Outputs The outputs are the four values and three graphs. The values are low-frequency H, high-frequency H, C1, and alpha. The graphs are the same three graphs from Hspec if the automatic DTM option is chosen, except that there are two different fits for low and high frequencies. These graphs are a spectrum of frequencies showing the fit through the range (the red stars), the Log(M) vs.Log(λ) and the Log(M) vs. Log(η) graph. (Again see Hspec example page). Note: This function requires DTMauto, HspecAuto, Flux, Spectrum1D, Window1D, Theta, and K Example Input: field=“UcheeCreek” (1x21643 streamflow series in ft3/s) Output: 'alpha' [ 1.4690] 'C1' [ 0.2974] 'Beta high freq.' [ 1.8571] 'H high freq.' [ 0.6721] 'Beta low freq' [ 0.4673] 'H low freq.' [-0.2663] Figure 4: Power spectrum with linear fits over low- and high-frequency ranges for UcheeCreek.

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DTMspec (continued) Figure 5 Logarithmic plot showing scaling behaviour of double trace moments for UcheeCreek. Black stars mark fitting range. λ =213 corresponds to one day.

Figure 6: Logarithmic plot of slopes of Figure 1 as functions of η (for UcheeCreek). Blue stars mark fitting range for linear fit, black star marks η0

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2.6 Ecodim Basic Summary Ecodim takes a field as its input and outputs a graph of the empirical codimension function, c(γ). It then degrades the function by a factor of two and plots c(γ) again (superposed on the previous plot) until the field has been degraded to approximately one hundredth of its original length. This function does not require values of C1 and alpha prior to computing c(γ). More theoretical information can be found in “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 3.3.2. Inputs There is only one input, “field”. It represents a one dimensional array of data. Outputs There are two outputs, a plot of c(γ) and the values of the codimension (y-values on the plot). These values only represent the y-values of the last line plotted. Note: This function requires Flux, Theta, Singularity, DTMspec, DTMauto, HspecAuto, Spectrum1D, Window1D, K, and PD or Degrade, Singularity Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) Output: Columns 1 through 7 0.1374 0.3737 0.0679 0.3256 0.2772 0.2921 1.0012 Columns 8 through 14 0.3660 0.0728 0.2063 0.1074 0.0324 0.1197 0.0270 (this matrix was 1x216 so only a small part is shown here)

Figure 7: Codimension functions plotted after degrading by a factor of two for UcheeCreek. 13


2.7 GammaDprac Basic Summary This function requires inputs of parameters C1 and alpha, and outputs a practical estimate of the theoretical γD. γD is a critical singularity; when γ<γD, the c(γ) becomes linear and the tail of the corresponding probability distribution becomes hyperbolic (power law) with exponent qD. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.2. Inputs C1 and alpha are found using a separate function (e.g TraceMoment) and used as input arguments (alpha as the first argument, C1 as the second). Outputs The output value is the value of γD. Note: This function requires GammaS Example Input: alpha=1.9435 C1=0.3336 Output: 0.7081

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2.8 GammaS Basic Summary This function takes the inputs alpha and C1 and outputs a value of γS. γS is the highest order singularity present in the data (here assumed to be a single series, d=1). For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.2. Inputs C1 and alpha are found using another function (e.g TraceMoment) and used as input arguments (alpha as the first argument, C1 as the second). Outputs The output is the value of γS. Example Input: alpha=1.9435 C1=0.3336 Output: 0.8173

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2.9 Hspec Basic Summary This function estimates the exponent β from the spectrum and uses it, along with alpha and C1, to compute the scaling exponent of the mean field (H). For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 4.3.2. Inputs There are five inputs in this function: field, a, b, alpha, and C1. “field” is the usual one dimensional series of data. The next two inputs, “a” and “b”, specify the range over which the fit is computed to obtain the (absolute) logarithmic slope β. These can be modified to obtain the best fit. Finally, the parameters alpha and C1 can be determined from a separate function (e.g TraceMoment) or they can be determined automatically using the double trace moment (DTM) method. In the latter case the user should input 0 for both alpha and C1. Outputs The outputs are the values of β and H or β,H,alpha, and C1 as well as either one graph or three graphs. The reason for this is whether the automatic DTM method is used (which outputs the alpha and C1 as well as two more graphs) or whether alpha and C1 are manually inputted. The first of the graphs is the one that pertains to both. It is the spectrum and shows the fit that was manually inputted (a and b are the red stars on the graph). The second graph pertains to the DTM only and is a plot of Log(M) vs. Log(λ). The final graph pertains to DTM and is a plot of Log(M) vs. Log(η). Note: This function requires DTMauto, Spectrum1D, Window1D, Flux, and K Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) a=2000 (cycles/T) (here T is the duration of the series) b=8000 (cycles/T) alpha=0 C1=0 Output: 'C1' [0.2537] 'alpha' [1.4988] 'Slope beta' [1.7366] 'H' [-1.1582]


Hspec (continued)

Figure 8: Power spectrum for Uchee Creek with user-specified fit.

Figure 9: Logarithmic plot showing scaling behaviour of double trace moments for UcheeCreek. Black stars mark fitting range. λ =213 corresponds to one day.

Figure 10: Logarithmic plot of slopes of Figure 8 as functions of η. Blue stars mark fitting range for linear fit, black star marks η0.

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.10 HspecAuto Basic Summary This function is similar to the previous function, Hspec, except that it fits the power spectrum on two predefined range of frequencies: from 1-32 days, i.e. high-frequencies, and from 64 days to the full length of the series, i.e. low-frequencies. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 4.3.2. Inputs There is one input, field. “field” is the one dimensional series of data. Outputs The outputs are the values of β and H for high and low frequencies, as well as one graph. It is the spectrum and shows the two fits: the high-frequencies fit is the red curve whereas the blue curve corresponds to the low-frequencies fit. Note: This function requires DTMauto, Spectrum1D, Window1D, Flux, and K. Example Input: HspecAuto(field,alpha,C1)=HspecAuto(UcheeCreek,1.4690,0.2974) Output: 'Slope beta high' [ 1.8571] 'H high freq.' [ 0.6721] 'Slope beta low' [ 0.4673] 'H low freq.' [-0.2663]

Figure 11: Power Spectrum with red fit performed on the high frequencies between the red stars and with the blue fit performed for low frequencies between the blue stars (for UcheeCreek).

18 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.11 MFFA Basic Summary This function is meant to be an all-in-one analysis function. Its only input is field, but it outputs 10 graphs and the values of 10 parameters. The functions it uses include TraceMoment, DTM, Theta, PD, GammaD, Ecodim, and QTcomp. Inputs The only input is the data field to be analyzed, this is assumed to be a time series of daily data at least several years in duration. Outputs As mentioned above, there are numerous outputs. The graphs are, in order or appearance, a graph of the raw data, the two graphs from TraceMoment, the three graphs from DTMspec, the graph from Theta, the two graphs from PD, and a final graph from QT. Note: This function requires Amax, ATLL, codimPD, dK, DTMauto, DTMspec, EcodimMffa, Flux, Flux2nd, Fluxall, HspecAuto, PDmffa, QTmffa, Singularity, Spectrum1D, Theta, TLI, SortTally, Tally, accumulate, and TraceMomentAuto Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) Output: 'alpha TM' [ 0.3306] 'C1 TM' [ 1.9582] 'Kq TM' [2x21 double] 'alpha DTM' [ 1.4690] 'C1 DTM' [ 0.2974] 'H high freq.' [ 0.6721] 'H low freq.' [ -0.2663] 'theta' [ 1.1892] 'qD' [ 2.5673] 'gammaD' [ 0.8155] Since all ten of the graphs are pictured in other examples, they will not be re-pictured here.

19 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.12 MFSS Basic Summary The function MFSS, which stands for MultiFractal Simulation Software, allows the user to generate a multifractal field with given parameters α and C1. Inputs The first input is the multifractality index α and the second is C1. The third input, power, determines the length of the generated field and should be an integer, the length will be 2power. Outputs The function outputs a one-dimensional series, i.e. the generated multifractal field. Note: This function requires Levy Example Input: MFSS(alpha,C1,Power)=MFSS(1.8,0.1,10) Output: Columns 869 through 875 2.1830 7.1606 9.4595 4.2709 24.4407 8.9974 9.3922 Columns 876 through 882 4.9470 10.0584 3.0150 0.8491 3.5321 2.5476 7.3115 (Only a small portion of the output is shown.)

Figure 12: Multifractal field simulated with α=1.8 and C1=0.1 and a length of 1024.

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.13 PD Basic Summary This function show a probability distribution of data calculated by sorting the data points into rank order. It also shows this on a log-log plot and finds the slope (using the extreme “n” points). More information can be found in “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 3.2.1. Inputs There are two inputs, field and n. “field” is the one dimensional series of data. Note that if it is of length 350, for example, it can be inputted as a 1x350 or 350x1 series. “n” is the number of extreme values used in the probability distribution. Outputs The outputs are the value of the slope of the log-log plot and two graphs. The first graph is simply the data as a function of time. The red areas represent the points used for the probability distribution. The second graph is a logarithmic graph of with red dots indicating the range over which the fit was estimated. Note: This function requires DTMauto, Flux, K, and dK Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) n= 40 Output: ans = -3.0575

Figure 13 A plot of the streamflow series vs. time for Uchee Creek with red areas indicating values used in fitting the probability distribution tail (see fig. 15).

21 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A PD (continued)

Figure 14: The log-log probability distribution with red dots indicating fitting range and blue line showing the linear fit and indicating power law behaviour (see red points in fig.13), the line indicates an exponent qD =3.03.

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.14 qDprac Basic Summary This function takes values of alpha and C1 and outputs a theoretical value of probability distribution exponent qD using the practical estimation technique. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.2. Inputs There are two inputs, alpha and C1. These can be found using a separate function (e.g TraceMoment). Outputs The only output is the theoretical value of qD. Note: This function requires GammaD and GammaS Example Input: alpha=1.9435 C1=0.3336 Output: ans = 1.5858

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.15 QT Basic Summary This function takes a field and creates a curve of its values as a function of their return period. It shows the values of extreme events with period of up to one thousand years. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.3. Inputs There is one input: “field”, which should be a one-dimensional series. Outputs The function outputs a two-dimensional series of the projected values of the field Q and their corresponding projected return periods T, with the actual data superposed as points. Note: This function requires DTMspec, DTMauto, HspecAuto, Flux, Spectrum1D, Window1D, Theta, K, IcodimM, Amax, ATLL, PDq, dK, and codimPD. Example Input: QT(field) = QT(UcheeCreek) Output: ans= 1.0e+005 * Columns 1 through 8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 0.0013 0.0018 0.0025 0.0035 0.0049 0.0068 0.0094 Columns 9 through 16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0113 0.0113 0.0114 0.0114 0.0115 0.0115 0.0115 0.0116 (Only a small part of the entire output is shown.)

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A QT (continued)

Figure 15: Log-linear graph showing projected extreme values Q as a function of their return period T (in years, a logarithmic plot) for Uchee Creek (dotted line) along with the actual data (circles).

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.16 QTcompare

Basic Summary This function splits the data used to plot QT in half, performs the analysis on both halves, and compares the two with the result from the entire field of data. This could be used to get an idea of the uncertainties or alternatively – if the series was broken at the point when a dam was constructed, to judge its impact. It plots them after taking the logarithm base 10 of their periods and also shows their empirical curves. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.1.3. The difference between the projections based on the first and second halves of the data can be used to judge the uncertainties. Inputs There are three inputs, field, position, and Switch. “field”, which should be a one-dimensional series. “position” and “Switch” Note: This function requires QT, Frag, DTMspec, DTMauto, HspecAuto, Flux, Spectrum1D, Window1D, Theta, K, IcodimM, Amax, ATLL, PDq, dK, and codimPD Outputs The function outputs one graph as described above. Example Input: Qtcompare(field, position, Switch)=QTcompare(UcheeCreek,1,1)

Figure 16: Graph showing projected extreme values Q as a function of their return period T (in years, a logarithmic plot). The black curve is made using the entire field, the blue and red curves represent the results of the analysis from each half (for UcheeCreek).

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.17 Singularity Basic Summary This function takes a field as its input and outputs a singularity series. More theoretical information can be found in “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 3.3.1. Inputs There is one input: “field”. This must be one-dimensional. Note that if it is of length 350, for example, it can be inputted as a 1x350 or a 350x1 series. Outputs There is one output: the singularity series. This is the mean of the logarithm (base λ) of the normalized field. Example Input: field= “UcheeCreek” (1x21643 series of streamflow data in ft3/s) Output: Columns 12062 through 12068 -0.1499 -0.1611 -0.1697 -0.1750 -0.1791 -0.1911 -0.1943 Columns 12069 through 12075 -0.1994 -0.2047 -0.2047 -0.2047 -0.2047 -0.2047 -0.2047 (output was 1x21643 so only a small part is shown here)

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.18 Spectrum1D Basic Summary This function takes a one-dimensional field and a value of β and outputs a spectrum of frequencies using Fourier transforms. It multiplies (“compensates”) the spectrum by the frequencies raised to the user-specified power β. The spectrum uses a Haar window to reduce spectral leakage. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, sections 3.1.3 and 4.3.1. Inputs There are two inputs, field and beta. “field” is the one dimensional series of data. Note that if it is of length 350, for example, it can be inputted as a 1x350 series or a 350x1 series. “beta” is the theoretical value which can be obtained from a separate function (e.g Hspec). Outputs The outputs are a plot of the spectrum and a matrix of the x and y values used in the plot. Note: This function requires Window1D Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) beta= -1.7366 Output: Columns 3676 through 3682 3.5654 3.5655 3.5656 3.5657 3.5658 3.5660 3.5661 -2.1499 -0.6352 -0.5829 -1.0658 -1.4693 -1.1581 -1.2325 (output was 2x10821 so only a small part is shown here)

Figure 17: Basic power spectrum using Fourier analysis for UcheeCreek.

28 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.19 Theta Basic Summary The Theta function takes an input field of daily data and outputs a graph of the rank of daily maxima as a function of the rank of annual maxima. The scaling exponent θ needed to transform daily statistics into annual maximum statistics is outputted. For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1B”, section 3.3 and 3.3.1. Inputs There is only one input: “field”. This field must be one dimensional and contain daily measurements. It is important to note that the daily data inputted must be at least two years long. Outputs There are two outputs, a graph and a value (“ans”). The graph is a log-log plot of the rank of daily maxima as a function of the rank of annual maxima. On the graph iD stands for rank of daily maxima and iA stands for rank of annual maxima. The value is the value of θ discussed in the basic summary above. Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) Output: ans=1.6275

Figure 18: Logarithmic plot of daily vs. annual ranks. Blue line shows linear fitting for UcheeCreek.

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 2.20 TraceMoment Basic Summary The TraceMoment function takes an input field with H=0 and finds C1 (codimension which measures mean inhomogeneity), and α (multifractality index) using trace moments. To do so it uses inputs of independent strings of data (“field” input) and a value that determines whether to use no differences(“Switch”=0), first differences(“Switch”=1), or second differences(“Switch” anything else). For more theoretical information see “Multifractals and Physically Based Estimates of Extreme Floods Phase 1A”, section 3.4.1. Inputs It should be noted that the output will be the same no matter whether “field” is transposed or not. Changing “Switch” should only change the values slightly. Finally, “low” and “high” represent the fit parameters. These can be experimented with to obtain the best fit. Outputs The outputs are K(q), C1, α and two graphs. The first is a graph of Log(M) as a function of Log(λ), the second is a graph of K as a function of q. Note: This function requires Flux, Flux2nd, and Fluxall. Example Input: field= “UcheeCreek” (1x21643 streamflow series in ft3/s) Switch=1 low=1 high=35 Output: C1=0.3336 alpha=1.9435 Kq= Columns 1 through 7 0.0001 0.1001 0.2001 0.3001 0.4001 0.5001 0.6001 -0.0045 -0.0343 -0.0569 -0.0723 -0.0805 -0.0819 -0.0767 Columns 8 through 14 0.7001 0.8001 0.9001 1.0001 1.1001 1.2001 1.3001 -0.0655 -0.0487 -0.0267 0.0000 0.0310 0.0659 0.1043 Columns 15 through 21 1.4001 1.5001 1.6001 1.7001 1.8001 1.9001 2.0001 0.1459 0.1904 0.2375 0.2870 0.3387 0.3922 0.4476

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A TraceMoment (continued) Figure 19: Scaling behaviour of statistical moments for various orders of q (for UcheeCreek).

Figure 20: The scaling moment function K(q).

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 3.0 ADDITIONAL EXAMPLES Here are some examples for comparison of the final graph of extreme values of Q as a function of their return period produced by QT. Each is performed on a separate river; these series are included with the software (they are from the public USGS site).

Blackwater River Chatta hoochee River

Choctawhatchee River Conecuh River

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Little Double Bridges Creek Murder Creek

Panther Creek Patsaliga Creek

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Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 4.0 APPENDIX 4.1 Appendix A: Uchee Creek data The series used for all of the examples is a stream flow series obtained from the (public) USGS site. This site offers many different options for types of data. The series chosen is daily data in ft3/s and is from the state of Alabama. It is of length 21,643 days beginning Oct. 1, 1946 and ending on Jan. 1, 2006. Here is a link to where “UcheeCreek” can be selected: http://waterdata.usgs.gov/nwis/dv?referred_module=sw&state_cd=al&dv_count_nu=7300&index_pmcode_00060=1&sort_key=site_no&group_key=NONE&format=scroll_list&sitefile_output_format=html_table&column_name=agency_cd&column_name=site_no&column_name=station_nm&range_selection=days&period=365&date_format=YYYY-MM-DD&rdb_compression=file&list_of_search_criteria=state_cd%2Cobs_count_nu%2Crealtime_parameter_selection<http://waterdata.usgs.gov/nwis/dv?referred_module=sw&state_cd=al&dv_count_nu=7300&index_pmcode_00060=1&sort_key=site_no&group_key=NONE&format=scroll_list&sitefile_output_format=html_table&column_name=agency_cd&column_name=site_no&column_name=station_nm&ran The selection process starts at http://waterdata.usgs.gov/nwis/sw >Daily data >State , then select a state, '7300' for number of observations (to ensure that at least 20 years of daily data is obtained), 'Streamflow', 'ft3/s', and 'Scroll list of sites'. It is important to note that the output format of data from the USGS site is not compatible with MATLAB. The data was copied and pasted into a separate .txt file.

Here is a plot of the Uchee Creek streamflow vs. time: Graphs of data can also be obtained by choosing 'Graphs of data' instead of 'Scroll list of sites'.

34 Multifractals and Physically Based Estimates of Extreme Floods, Phase 4A 4.2 Appendix B: Additional stream flow data The data used in section 3.0 Additional Examples was also obtained from the (public) USGS site. The rivers were randomly selected from the state of Alabama. Below is a chart detailing their lengths: River Length of data (days)

Blackwater River 16307 Chattahoochee River 11754 Choctawhatchee River 16800 Conecuh River 16802 Little Double Bridges Creek 9807 Murder Creek 16792 Panther Creek 7670 Patsaliga Creek 13772

These data files are included with the software.

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