Isohyetal Maps


Technical Details


Return Period Calculation using the Beta-P Distribution


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 – 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)




Computing Return Period Amounts for Shorter Accumulation Periods


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), PDAY is the calendar day (fixed observation time) extreme precipitation amount (inches), and P1, P2, and P3 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).



Spatial Interpolation and Contouring


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 ‘smth9’ 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.