# A Potential Problem with Mean Dimensionless Hydrographs at Ungaged Sites

*by*Kenneth L. Wahl, U.S. Geological Survey, Lakewood, United States,

James G. Rankl, U.S. Geological Survey, Lakewood, United States,

**Abstract:**A flood hydrograph for an ungaged stream site can be estimated from a mean dimensionless hydrograph and estimates of instantaneous peak discharge (Q) and total storm runoff volume (V). However, the time base of the resulting flood hydrograph can be inversely related to the magnitude of the peak discharge if estimates of runoff volume were defined using ordinary least-squares regression relations of the form V=f(Q). Such an inverse relation is not hydrologically consistent. The problem can be solved in several ways. Redefining the relation between V and Q using an alternative model that preserves the variance in V generally will produce exponents for Q that are near 1. The resulting estimated flood-hydrograph volumes will nearly match the original volumes near the mean of the logarithms of V and Q, but will differ as volumes and peak discharges depart from the mean values. The difference will depend on how much the original exponent of Q differed from 1. Another solution is to simply hold T', the multiplier needed to expand the time base of a mean dimensionless hydrograph into the time base of an estimated flood hydrograph, constant. That solution is a questionable approach if basins vary either in size, shape, or slope. A third solution is to define T' as a function of time to peak, Tp. Flood volume then depends only on Q, Tp, and the dimensionless hydrograph, thereby removing the need to define a relation for estimating volume.

**Subject Headings:**Hydrographs | Floods | Water discharge | Runoff | Rivers and streams | Storms | Least squares method

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