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)

Return
period amounts for accumulation periods of 1 day were obtained directly from
the fit of the Beta-P distribution to the observed daily extreme precipitation
partial-duration series. Precipitation
events are not bound by the observation times of the observers however, and if
an extreme precipitation event spans the time of observation, return period
amounts for a 24-hr accumulation period would be underestimated. Additionally, the actual duration of
events is unknown when observations are taken once-daily.

Empirical relationships were used to convert precipitation
amounts associated with calendar day accumulation periods to amounts associated
with 6-, 12-, 18- and 24-hr accumulation periods, independent of observation
time (Table 6).
These conversion factors were determined empirically for the Northeast
U.S. by McKay and Wilks (1995), and were found to
be slightly lower than factors given by Huff and Angel
(1992) for smaller accumulation periods (£12-hr).

Regression-based
estimates of return period amounts associated with shorter accumulation periods
(1-, 2- and 3-hr) from extreme calendar-day precipitation amounts were obtained
using the equations developed by McKay and Wilks
(1995) for the Northeast U.S.
This set of equations is given by

;

;
(6)

,

where f is
the station latitude (degrees North), l is the station longitude
(degrees West), z is the station elevation (feet), P_{DAY} is the
calendar day (fixed observation time) extreme precipitation amount (inches),
and P_{1}, P_{2}, and P_{3} are the resulting return
period estimates (inches) for 1-, 2- and 3-hr accumulation periods,
respectively. Regression-based
estimates were found to have a lower root mean squared error for these shorter
accumulation periods than estimates using empirical factors (McKay and Wilks 1995).

The Grid Analysis and Display System
(GrADS) is used for spatial interpolation and creation of maps. GrADS software is used to place station
values onto a 0.2¡ grid before contouring the gridded values by automated
means. GrADS uses an objective
analysis scheme developed by Cressman (1959).

Maps
produced for comparison with those of Wilks and Cember
(1993) also place station values onto a 0.2¡ grid, and then this grid
of values is subject to a weighted 9-point smoothing (using Ôsmth9Õ in GrADS). Maps are generated in this fashion
using data only through 1993 to simulate the results and methods used by Wilks and Cember (1993), and then compared to similar
maps generated using an additional decade of data (1994-2003). Since these maps
are subjected to the weighted 9-point smoothing, they differ from the maps presented
as products.