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Extrapolation for the 500-Year Flood Only recently has the USGS begun to publish at- site estimates of the 500-year flood or to publish regional regression equations for estimating the 500- year flood at un gaged sites, Therefore, most of the USGS statewide reports do not contain regression equations or at-site estimates for the 500-year flood, A procedure is given in the NFF program for extrapolat- ing the regional regression equations in any State to the 500-year flood. The extrapolation procedure basically consists of fitting a log-Pearson Type III curve to the 2- to 100-year flood discharges given by NFF and extrap- olating this curve to the 500-year flood discharge, The procedure consists of the following steps for a given watershed: I. Determine the flood-peak discharges for selected return periods from the appropriate regional regression equations given in NFF. At least three points are needed to define the skew coef- ficient required in a subsequent step. Use of additional points improves the definition of the frequency curve that is defined by the regional equations and helps to average out any minor irregularities that may exist in the relations among the regional equations. The NFF pro- gram uses all available regional equations for selected return periods to define the frequency curve, I 2, Fit a quadratic curve to the selected points on log- probability paper using least squares regression computations, The variables used in the regres- sion computations are the logarithms of the selected discharges and the standard normal deviates associated with the corresponding probabilities, The purpose of this quadratic curve is to obtain a smooth curve through the selected flood-peak discharges from step I above. The quadratic curve is an approxima- tion of the log-Pearson Type III curve that will be computed. I I I 3. Determine the skew coefficient of the log-Pearson Type III frequency curve that passes through the 2-, 10-, and 100-year floods defined by the quadratic curve. The skew coefficient is defined approximately by the formula (Inter- agency Advisory Committee on Water Data, 1982) G = -2,50 + 3,12 log (QlOoIQ10) flog (QloIQzj, 4. Replot (conceptually) the selected discharges and return periods using a Pearson Type III proba- bility scale defined such that a frequency curve with the computed skew plots as a straight line, This scale is defined by plotting probability values p at positions x on the probability axis, where x is defined by the standardized Pearson Type III deviate (K values) for the given skew and probability. A Wilson-Hilferty approxima- tion (Kirby, 1972) is used to compute the K value. 5. Fit a straight line by least-squares regression to the points plotted in step 4, and extrapolate this line to the 500- year flood-peak discharge. The variables used in the least squares computation are the logarithms of the selected discharges and the Pearson Type III K values associated with the corresponding probabilities. Figure 2 is an example of a flood-frequency curve computed by this procedure for the Fenholloway River near Foley, Florida. The solid triangles (fig. 2) are the regional flood-frequency values as estimated by the equations given by Bridges (1982), which are incorporated in the NFF program. The 500-year value shown as a solid circle (fig. 2) (\ 2,800 cubic feet per second) is estimated using the extrapolation procedure described above. Note that the extrapolated 500-year value is a reasonable extension (see dotted line) of the regional frequency curve. The solid triangle (fig. 2) (11,500 cubic feet per second) for the 500-year value is the regional value as obtained directly from the 500-year equation given in Bridges (1982). The 500-year flood for the Fenholloway River can be estimated without extrapo- lation since Florida is one of the few States for which 500-year regression equations have been published. The difference between the two 500-year values is 11.3 percent. This is typical of several comparisons of extrapolated 500-year floods to published regional equations that has indicated most results agree within plus or minus 15 percent. Details of these comparisons are given in a later section, For comparison and evaluation, the NFF pro- gram compares each extrapolated 500-year flood-peak discharge with the maximum flood-envelope curves given by Crippen and Bue (1977) and Crippen (1982), Because there is no frequency of occurrence associated with the envelope-curve estimates, the comparison of these values to the extrapolated SOO-year flood is 14 Nationwide Summary of U.S. Geological Survey Regional Regre.slon Equations for estlmetlng Magnitude and Frequency of Flood. lor Ung.ged 511..,1993 i I i I 1 , , I " I I . i I , ! , I i , 1