Precipitation amounts corresponding to various recurrence intervals (return periods) were computed by fitting a Beta-P distribution (Mielke and Johnson, 1974) to the daily extreme precipitation partial-duration series at each station (Table 1). It has been shown in a previous study that the Beta-P distribution provides the best representation of extreme precipitation in the Northeast U.S., compared to other commonly used theoretical distributions (Wilks 1993). Fitting a theoretical distribution to the observed data serves two purposes: 1) when computing precipitation amounts for shorter return periods, sampling irregularities are smoothed out of the observed data, and 2) due to the limited length of observation records at most stations, computation of precipitation amounts corresponding to longer (50-yr or 100-yr) return periods requires extrapolation beyond the observed data.

The following details about the Beta-P distribution and its use in return period computation follow those described by Wilks and Cember (1993), and were used identically in the present analyses. The probability density function for the Beta-P distribution is

, (1)

where x is the random variable (here, daily extreme precipitation partial-duration amounts), which must be nonnegative. The distribution has three parameters: a and q are dimensionless shape parameters, and b is a scale parameter having the same physical units as the random variable. The distributions were fit to data for each station by maximum likelihood, using the Levenberg-Marquardt method (Press et al. 1986), as described in Wilks (1993). One convenient feature of the Beta-P distribution is that it is analytically integrable, so that its cumulative distribution function can be written in closed form. That is, Beta-P probabilities can be obtained using

. (2)

Average return periods, R, relate to cumulative probabilities, F, of the distributions of partial-duration data according to

, (3)

where w is
the average frequency with which the partial-duration data samples the full
record of daily observations, in years^{-1}. The average sampling frequency was chosen to be close to 1
yr^{-1}, but because individual data records may start and stop on different
dates and may contain different numbers of missing data, w
varies slightly from station to station.
Let N represent the number of daily observations passing the quality
control screening that are available for a particular station. The partial-duration data were then
constructed to consist of the largest n precipitation accumulations, where n is
the greatest integer not exceeding N/365.25. This convention results in the average sampling frequency being

. (4)

Precipitation amounts, x, corresponding to specified return periods are obtained by inverting Equation 2 (i.e., solving it for x), and substituting the expression F(x)=1 Ð 1/wR obtained by rearrangement of Equation 3. These operations yield the expression for precipitation amounts as a function of return period, and of the parameters of the fitted Beta-P distribution,

. (5)

New York State was divided into a set of subregions using a Smirnov test-based clustering algorithm (DeGaetano 1998). Using this procedure, stations were grouped based on geographic proximity and similarities among empirical extreme precipitation half-partial-duration series. Statistical comparisons between stations were performed on half-partial-duration series (e.g. the greatest 25 components for a 50 year data sample) rather than on full-partial-duration series (e.g. 50 components for a 50 year data sample) for this purpose to better reflect the spatial pattern of the most extreme precipitation events. Each subregion formed from this procedure (15 subregions in New York) consisted of stations that have no statistical differences between their empirical extreme precipitation half-partial-duration series.

The procedure for computing regional extreme precipitation return period amounts was similar to that used for computing these amounts for individual stations. The only difference involved the construction of the extreme precipitation partial-duration series, for which extreme precipitation events at all stations within a subregion were used to construct a ÔregionalÕ partial-duration series. If multiple stations within a subregion observed precipitation over the same time period (±3 days), only the largest daily precipitation total is retained for consideration in the regional partial-duration series. This eliminates the dependence of regional results on the density of stations within a subregion, since the same precipitation event may influence a large area that encompasses many of the stations. Each daily precipitation amount retained for consideration in the regional partial-duration series can be considered an independent subregional precipitation event. Once the construction of the regional partial-duration series was complete, the computation of return period amounts using the Beta-P distribution was used.

Surface water runoff includes not only precipitation reaching the EarthÕs surface in liquid form, but also the melting of a snowpack that has accumulated over time. Some applications may be more interested in the return period amounts associated with surface runoff rather than those associated with precipitation alone. Across most of New York State, less than 10% of the precipitation events that comprise the extreme precipitation partial duration series are potentially enhanced by the melting of a snowpack. The precipitation events were determined to be potentially enhanced by snowmelt if these extreme events were accompanied by decreases in snow depth.

In order
to assess the importance of snowmelt consideration when computing extreme
precipitation return period amounts, observations of snowpack water equivalent
(SWE) were obtained. Measurements
of SWE are taken once-daily at first-order stations, 11 of which are utilized
in New York and surrounding states (Table 5). Data at most of these stations were
available from 1953-2003. Since SWE is measured each day at 18 UTC (1PM EST),
daily changes in SWE reflect the change in snowpack between this hour on the
current day, and the same hour on the previous day. Return periods were
computed for daily extreme snowmelt events, daily extreme precipitation events,
and the combination of these events.
If snowmelt events influenced the extreme precipitation partial duration
series at a station, its magnitude was often less than the 50^{th}
percentile of the series. As a
result, the influence of snowmelt on extreme precipitation return period
amounts was minimal (< 2-3%) for recurrence intervals of 2 and 5 years, and
often non-existent for lower-frequency return periods.