Recurrence Intervals and Plotting Positions

The recurrence interval is one of the primary methods in hydrology to express risk.  Engineers all know it as a return interval, like the "2 year return", then inverse representing the probability that the event will be equaled or exceeded in any one year.  Hence a 50 year return interval has a probability of 1/50 or 0.02 or 2% of being equaled or exceeded in any single year. 

The return interval is found by analyzing a series of data for maximum annual events  For example, one could analyze forty years worth of precipitation and condense it to the peak hourly runoff rate that occurred in each of the forty years.  That data could then be subjected to a number of probabilistic models.  There are numerous models, the most common that are used in hydrology are normal, gamma (also known as Pearson Type 3), lognormal, log gamma (also known as Log Pearson Type 3), and Gumbel (also known as Extreme Value Type 1). 

HSPF Toolkit™ currently supports the log gamma probabilistic model.  The methodology is taked directly from "Guidelines For Determining Flood Flow Frequency (Revised) 1982 - Bulletin 17B".  Much has been written about it so we skip the technical explanation and equations.  The log gamma or Log Pearson Type 3 Distribution is probably so well known because it is in Bulletin 17B.  The log Pearson Type 3 computation requires two constants in addition to the data series.  They are the MSEG (Mean-Square Error of Generalized Skew) and the Station Skew.  The documentation (17B) says that when the Generalized Skew is read from "Plate 1", the MSEG should be 0.302.  Hence if you use Plate 1, then MSEG should be 0.302.  Plate 1 is the standard graphic that is shown in most hydrology books and is included here for so that you know what we are talking about.  Incidentally, the chart below is what is in our version of 17B.  The chart shown in the link above is slightly different.

 

Note that the title on the graphic states log of annual maximum streamflow as the basis of the map.  We point this out because there are numerous probabilistic methods and sophisticated arguments as to why one method might be better than another. We are not weighing in on the discussion.  We are only pointing out why some may feel that log pearson type 3 is not really appropriate for computing return frequencies of precipitation data. 

From the standpoint of a typical HSPF analysis, if the UCI aggregate runoff from several basins that are interconnected by reaches or streams, then it would seem appropriate to use the Log Pearson Type 3 to estimate return frequencies for that stream.  However, if one were to apply that same analysis on the precipitation time series in the WDM, one could argue that precipitation data is not stream data, hence a Log Pearson Type 3 probabilistic approach utilizing a generalized skew based on annual maximum streamflow is a stretch.

Also consider the return frequencies of the stream.  It might be appropriate to use Log Pearson Type 3 before the stream enters a detention pond, but the effects of the ponds outlet structure really isn't natural, so too, it might be a stretch to apply the Log Pearson Type 3 probabilistic approach to the time series downstream of the pond.  For that those reasons, HSPF Toolkit current supports a number of Plotting Position formulas. 

The whole point of the plotting formulas goes back to the use probability paper to plot the points and see which type of paper comes closest to a straight line. The plotting formulas supported by the Toolkit are Cunnane, Gringorton, Gumbel, Log Normal, and Weibull.  They are all very similar and we will leave it to the engineer to decide which is most appropriate for the dataset. 

An indepth discussion of the most appropriate probabilistic methodology or suitability for one plotting formula over another is very much outside the scope and ability of Engenious Systems, Inc.  Suffice to say that the program supports the above methods.  Over time, it is our intent to support additional methods.

reference: Hydrology and Floodplain Analysis, Philip B. Bedient, Wayne C. Huber, Copyright 1998, Addison-Wesley Publishing Company.