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FrequencyAnalysis ICVEEN 4410: Engineering HydrologyTopic: Frequency AnalysisLearningObjectivesGeneral Goal: Use frequency analysis of historical data to forecast hydrologicevents, e.g., specific flood magnitudes, annual runoff values, etc.RandomVariablesSpecific goals/outcomes for this presentation: define random variables define PDFs and PMFs define Method of Moments define what is a probability distribution PDFsandPMFsMethod ofMomentsuse general (non-graphical) frequency analysis to forecast the probabilityof a given size flood in cfs(specifically: an example problem is assigned here, and you are tasked with working through thisproblem at home; we will then work through this example problem together in class)Probabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisOcean

FrequencyAnalysis IConsider: Peak intensity (in in/hr) of next rainfall event in SLC Peak streamflow (in cfs) of next year in EmigrationCreek Snowmelt runoff volume (in ac-ft) in thod ofMomentsTake 1 minute and estimate these valuesThink about these values and decidewhether we can predict these values withcertainty!Values that cannot be predicted withcertainty are called random variablesFrequency Analysis ICVEEN 4410 Engineering HydrologyProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysis2

FrequencyAnalysis ILearningObjectivesIs a coin flip outcome a random variable?RandomVariablesIs the outcome of a roll of a die a randomvariable?PDFsandPMFsMethod ofMomentsYes!Probabilitydistributions These types of random variables that canonly take on values from a specifieddomain of discrete values are calleddiscrete random variablesFrequency Analysis ICVEEN 4410 Engineering Hydrologysome criticalaspectsGeneral (nongraphical)FrequencyAnalysis3

FrequencyAnalysis ILearningObjectives Hydrologic random variables can take on valuesfrom a domain of continuous values –continuous random variables If continuous data are assigned to intervals(quantiles), then data are said to be quantizedcontinuous (e.g., histograms)RandomVariablesPDFsandPMFsMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology4

FrequencyAnalysis ILearningObjectivesSpend 1 minute (pause this video) and thinkabout how you would quantify the probabilityof streamflow exceeding 16,000 cfs in a riverany given yearRandomVariablesPDFsandPMFsMethod ofMomentsAnalysis of Historical Data!Probabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology5

FrequencyAnalysis ILearningObjectivesPDFs (and PMF, the discrete equivalent):RandomVariablesPDFsandPMFsMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology6

FrequencyAnalysis IMethod of Moments:LearningObjectives Hypothesize underlying ethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology7

FrequencyAnalysis IMethod of Moments:LearningObjectives Hypothesize underlying ethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology8

FrequencyAnalysis IMethod of Moments:LearningObjectives Hypothesize underlying probabilitydistribution Obtain sample; compute sample momentsRandomVariablesPDFsandPMFs Equate sample moments and parameters ofthe proposed density functionMethod ofMoments Construct frequency curve Check data fit to frequency curveProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology9

FrequencyAnalysis ILearningObjectivesSample Moments: the first three moments arethe mean, standard deviation, and skew:RandomVariablesPDFsandPMFsMean:Method ofMoments2ùé 1S êXi - X úåë n - 1 i 1ûnStandard Deviation:nStandardized Skew:Frequency Analysis Ig ((n å Xi - Xi 1))Probabilitydistributionssome criticalaspects3( n - 1) ( n - 2 ) S 3CVEEN 4410 Engineering Hydrology0.5General (nongraphical)FrequencyAnalysis10

FrequencyAnalysis ITwo Methods (in general) ofFrequency Fs1. Method of Moments2. Graphical MethodsMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology11

FrequencyAnalysis ILearningObjectivesDifferent Probability Distributions forFrequency AnalysisRandomVariables1. NormalPDFsandPMFs2. Log-normalMethod ofMoments3. BinomialProbabilitydistributions4. Exponentialsome criticalaspects5. Gamma6. Log-Pearson Type IIIToday -- we’ll discuss applications that assume the normal distributionFrequency Analysis ICVEEN 4410 Engineering HydrologyGeneral (nongraphical)FrequencyAnalysis12

FrequencyAnalysis IMethod of Moments forAssumed Normal DistributionLearningObjectivesRandomVariables1. Assume random variable has a normaldistribution with population parameters and 2. Equate the sample moments to the populationparametersPDFsandPMFsMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology13

FrequencyAnalysis ILearningObjectivesSome critical aspects:RandomVariablesThe cumulative distribution function, or CDF:PDFsandPMFsF(x) P(X x) 1 - 1/T; note that T 1 / (1 - F)Return Period T 11 1- (non - exceedance probability) exceedance probabilityThe CDF represents the probability that a randomvariable X takes on a value less than or equal to x. T isthe return period -- an annual maximum event has areturn period of T years if its magnitude is equaled orexceeded once, on average, every T years. F is alsothe NON-EXCEEDANCE PROBABILITY.Frequency Analysis IMethod ofMomentsCVEEN 4410 Engineering HydrologyProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysis14

FrequencyAnalysis ILearningObjectivesSome critical aspects:RandomVariablesThe frequency factor:PDFsandPMFsx-xK SxMethod ofMomentsWhere the variables in the numerator are the randomvariable and its mean, respectively, and Sx is thestandard deviationProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology15

FrequencyAnalysis ILearningObjectivesSome critical aspects:RandomVariablesThe frequency factor K also corresponds to thevariable “z”, also called the “standard normalvariate”:z x-msSee Appendix D of text NormalDistribution Table, includingtabulation of frequency factors(standard normal variate, z)PDFsandPMFsMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology16

FrequencyAnalysis ILearningObjectivesSome critical aspects:RandomVariablesThe magnitude of a random variable x for a givenreturn period is given by:PDFsandPMFsx x z iSxMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysisFrequency Analysis ICVEEN 4410 Engineering Hydrology17

FrequencyAnalysis IGeneral (non-graphical) Frequency Analysisusing a normal distribution:LearningObjectivesRandomVariables1. Assume random variable has a normaldistribution with population parameters and 2. Compute the sample momentsand SPDFsandPMFs3. For the normal distribution, the parameters andsample moments are related by and S4. Calculate the CDF: F(z) 1 - 1/T5. Obtain the standard normal variate “z” frompublished tables of values (Appendix D of text)6. Use x x z iSx to calculate magnitudeWith these steps, you can calculate how frequent an event is likelyto occur, and what the magnitude of that event might be.Frequency Analysis ICVEEN 4410 Engineering HydrologyMethod ofMomentsProbabilitydistributionssome criticalaspectsGeneral (nongraphical)FrequencyAnalysis18

In-Class Exercise: The normal distribution is fit todischarge data for the Siletz River. What is the 100 year flood magnitude?What is the probability that a flood will beless than or equal to 30,000 cfs?Please review this problem at home, and be prepared towork through it in-class (as an “in-class” exercise).Frequency Analysis ICVEEN 4410 Engineering Hydrology19

Ocean General Goal: Use frequency analysis of historical data to forecast hydrologic events, e.g., specific flood magnitudes, annual runoff values, etc. Specific goals/outcomes for this presentation: define random variables define PDFs and PMFs define Method of Moments define what is a probability distribution use general (non-graphical) frequency analysis to forecast .

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