Glenn E. Moglen Department of Civil & Environmental Engineering Virginia Tech Flood Frequency: Peak Flow Regionalization CEE 5324 –Advanced Hydrology – Lecture 20

Questions? Announcement: Exam 2 on April 9 (take home) Flood Frequency PeakFQ Flowchart comment Begin Regression Equation Regionalization US Wide Documents Virginia Documents Regression equation Fundamental Terms Regression equation Goodness of Fit Today’s Agenda

Flowchart in PeakFQ document Suggested Bulletin 17B analysis sequence.

What does “regionalization” mean? Ans. Regionalization is the extension of the results from flood frequency analyses at a set of gaged locations in a region to all ungaged locations within a region. How is regionalization typically done? Ans. Typically, flood frequency analysis results are related to watershed characteristics through multiple regression. What are some examples of this regression approach? Begin Regionalization

National Streamflow Statistics (NSS) Program

These equations are from Miller (1978). The Bisese (1995) equations supersede the ones shown here. “Old” Virginia equations

Current rural peak flow report for Virginia

Northern Valley and Ridge indicated Peak discharge regions identified by Bisese (1995)

Regional peak flow equations in Virginia

Regional peak flow equations in Virginia – “NV” region only

See pages 8-10 of Jennings (1993) report for summary Urban Equations

Shameless promotion… Moglen & Shivers (2006) Urban Equations – 2 nd Try

Classic linear model structure: Basic power law structure: Log-Transform of power law structure: Regression Equation Fundamentals: Definitions and Structure Criterion Variable Predictor Variables

In log-transform, note that if x 2 (or any criterion variable) is zero then logarithm is undefined. For this reason, an arbitrary constant is often added: So power law structure changes mildly: Regression Equation Fundamentals: Model Structure

Calibration: using a known set of observations of both predictor and criterion variables and determining (through regression or other methods) the coefficients and exponents in a given model structure. Prediction: using a calibrated equation where predictor variables have been determined to estimate the criterion variable. Regression Equation Fundamentals: Calibration vs. Prediction

Consider the model: Does the sign of the exponents on drainage area ( A, in mi 2 ) and forest cover ( F, in % ) make sense? If so, then these are “rational”. L is main channel length (in miles). What should the sign of c 3 be? Regression Equation Fundamentals: Rationality

The quality of regression equation in predicting a criterion variable is quantified through measures of GOF. S e /S y Bias USGS measures: ( S e,percent, N r ) Note: The correlation coefficient ( R or R 2 ) is probably what you are most familiar with. But is not truly appropriate here because we are NOT dealing with linear relationships. Goodness-of-Fit (GOF)

Standard error ( S e ) is appropriate: where = n – p – 1, and p is the number of predictor variables. Standard error is analogous to the standard deviation ( S y ). Both measure variability about a central tendency. Goodness-of-Fit (GOF)

Relative standard error is simply the ratio of standard error to the standard deviation. S e /S y less than 1 is good. The closer to zero the better. In the context of this course, please note that in the above equation, the “ y ” values are discharges ( Q ’s). Important: NOT log( Q )’s (unless otherwise indicated). Goodness-of-Fit (GOF)

Standard Error of Prediction (note: S e,USGS is in “log units”) USGS Goodness-of-fit values

Equivalent Years of Record USGS Goodness-of-fit values

Comments here are limited to power model functional form favored by the USGS Classic (and quick) method is log-log transform of data, followed by linear regression Another, slightly more complicated method, is non-linear numerical optimization. Methods give different results. Let’s explore… Methods of Regression Equation Calibration

We’ve seen this before in the context of the Horton Ratio calculations. Recipe: 1. Take log of predictor and criterion variables. 2. Use linear regression on log transformed data 3. Note that: Methods of Regression Equation Calibration: Log-log transform / linear regression