MULTIFRACTALS AND PHYSICALLY BASED ESTIMATES OF EXTREME FLOODS Phase 4A Prepared by Physics Department, McGill University Montreal, Quebec   Principal Investigator Shaun Lovejoy

Overall goal of the project: 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 To determine the relationship between flood magnitude and return period for a wide range of aggregation periods. Previous Phases: Phases 1A, 1B, 1C have focused on developing this theory for river series with weak annual cycles and demonstrating it on data series.

Goal of the subphase 4A: Create MATLAB versions of the software that performs the key analyses of Phases 1A, 1B, 1C Gives examples of how to use them for the purpose of flood frequency analysis.

The functions Codim: calculates the theoretical bare codimension function. CodimPD: calculates the theoretical dressed codimension function. DTM: applies the Double Trace Moment analysis technique for a, C1 with user defined limits. DTMauto: same but with standard limits. DTMspec: finds values of H, C1, and alpha. Ecodim: calculates the empirical codimension function. GammaDprac: an approximate method of estimating gD. GammaS: calculates gS theoretically. Hspec: Empirically estimates the exponent H (user defined frequency limits). HspecAuto: Empirically estimates the exponent H (standard frequency limits). MFFA: An all-in-one Multifractal Flood Frequency Analysis function. MFSS: MultiFractal Simulation Software. PD: Calculates the probability distribution of data. qDprac.: Calculates the theoretical probability distribution including the algebraic tails. QT: From a series, it creates a projection of stream flow as a function of the return period. Qtcompare: Same as QT but gives three projections based on the first half, second half and full series. Singularity: Outputs a singularity series. Spectrum1D: Calculates the spectrum. Theta: Calculates scaling exponent θ of the daily versus annual rank statistics. TraceMoment: Calculates the empirical trace moments.

Other deliverables MATLAB Runtimes 8 river series used as examples (from the public USGS site)

An example based on the Uchee Creek (Alabama) series Extremes (red), from power law tail A plot of the streamflow series vs. time for Uchee Creek with red areas indicating values used in fitting probability distribution tail (units: days and m3/s) The log-log probability distribution with red dots indicating fitting range and blue line showing the linear fit and indicating power law behaviour, the line indicates an exponent qD =3.03. Outputs of the function PD

Power spectrum 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) absolute logarithmic slopes high frequency (red) b = 1.86, low frequencies b = 0.47. The function HspecAuto

Double Trace Moment (a, C1) Logarithmic plot showing scaling behaviour of double trace moments for UcheeCreek. Black stars mark fitting range. l =213 corresponds to one day. Logarithmic plot of slopes of Figure 8 as functions of η. Blue stars mark fitting range for linear fit, black star marks η0 Outputs from DTMspec

Return Period projections Log-linear graph showing projected extreme values Q (in m3/s) as a function of their return period T (in years, a logarithmic plot) for Uchee Creek (dotted line) along with the actual data (circles). The theory and data are very close giving confidence in the projection.

Summary and recommendations The principle software used in phases 1A, 1B, 1C now exist in MATLAB code. They have been documented and tested on real streamflow series. Users can now use daily streamflow data to make their own projections for 1000 year return period streamflows. Recommendation: complete the project as planned Phase 2: Analysis of precipitation data with extremes and comparison with streamflow data. Phase 3: Study of streamflows with strong annual cycles and development of a stochastic model. Phase 4b: The development and documentation of MATLAB software needed in the remaining phases. The development of maps showing the distribution of exponents, parameters.