NCEP/Climate Prediction Center ATLAS No. 5

A Precipitation Climatology for Stations in the Tropical Basin; Effects of ENSO

5. Frequencies of Rainfall Deficits of Given Intensity and Duration

From the raw data used to form the histograms provided in Fig. 3a and Fig. 3b (see the abstract for access to the data), users can derive a wide variety of statistics for individual stations or sets of stations. One such statistical set might consist of counts of exceedances (or shortfalls) of given amount and/or duration thresholds. For example, one might be interested in the number of instances in which total precipitation has been less than a given percentile for at least a given number of consecutive running 3-month periods. This would help identify locations at which rainfall deficits tend to occur in protracted episodes rather than in more frequent but shorter-lived events. It is assumed that the former characteristic is more strongly associated with economically unfavorable conditions.

An example of such an analysis is shown in Table 3. Here, we tabulate the number of cases in the 1955-96 period in which rainfall was below a given percentile rating for a given number of consecutive running (overlapping) 3-month periods, such as Jan-Feb-Mar, Feb-Mar-Apr, etc. The five percentile points used are 10 percentile points apart in the dry half of the distribution (50 %ile, 40 %ile, 30 %ile, 20 %ile and 10 %ile; each represented by one column), and the number of cases of rainfall being below the given percentile for the given number of running consecutive periods (the number of periods represented by separate rows) is shown in the body of the table. These interior numbers represent the frequency of occurrence over the 42-year period. Suppose, for example, that rainfall was below the 10 %ile level at a station from Feb-Mar-Apr to May-Jun-Jul in one of the years. This would represent a case of four consecutive running 3-month periods. Note that this duration of four periods actually covers six consecutive months; the number of months is always two greater than the number of running 3-month periods. While two cases of three running 3-month seasons would be contained in that period (namely Feb-Mar-Apr to Apr-May-Jun and Mar-Apr-May to May-Jun-Jul), Table 3 would not include these; three running seasons are tabulated only when they represent the full duration of a running dry period. In other words, these counts are for individual (not cumulative) occurrences of the specific number of running seasons. In the same manner, results are non-cumulative in the percentile dimension: for a given number of consecutive seasons, while cases that qualify for below the 10 %ile also qualify for below the 20 %ile, the 30 %ile, etc., they are not tallied for the less extreme percentiles unless their duration did not become larger for the less extreme percentile. The process of deriving Table 3 can be visualized as the drawing of a straight horizontal line through Fig. 3b from 1955 to 1996 at a given percentile level, and tallying the durations of dry spells whose histogram bars are cut by that line. We choose to tally the elemental individual results as such and let users develop more highly derived tables as they see fit. Thus, the counts in Table 3 do not necessarily increase or remain equal from left to right in any row, as they do not do so from the bottom to the top of a percentile column. Cumulative versions of this table could be developed with respect to either (or both) dimensions, and the accumulation of individual results could be done in more than one way (i.e., using differing rules) depending on what is desired. Furthermore, the entries could be expressed in terms of percentages of the total number of possible cases instead of as raw totals. The potential uses of the basic data are varied and numerous.

The interpretation of the frequencies in Table 3 is illustrated by inspecting that table along with the percentile histogram (Fig. 3b ) corresponding to the same station. Consider Honolulu, for example. Table 3 shows one case of 15 consecutive seasons having rainfall below the 50 %ile. Figure 3b indicates that from Mar-Apr-May 1995 through May-Jun-Jul 1996 (spanning 15 running 3-month periods), the rainfall was at or below the median. Although the exact percentile is not indicated in Fig. 3b, we can assume that the rainfall for Jan-Feb-Mar 1996, shown as near-50 %ile in Fig. 3b, was actually in the 47.5-50.0 %ile range rather than the 50.0-52.5 %ile range. In fact, it is shown in a later figure (Fig. 7) that 49 was the actual percentile. (Note: fortunately, because of the number of years [42], no percentile is exactly 50.0, or any other exact multiple of 10.0; this reduces ambiguities.) The illustration beneath the caption of Table 3 points out this specific string of 15 running below-median periods in 1995-96 in Honolulu, taken from Fig. 3b. Table 3 claims that the second longest-lasting below-50 %ile rainfall period was 11 seasons duration. Inspection of Fig. 3b shows that this dry spell occurred mainly in 1976, but lasted into early 1977. Another possibility would have been from spring 1994 to Jan-Feb-Mar 1995; however, since the number of 11-seasons-long spells is 1 rather than 2, the Mar-Apr-May 1994 season, shown as near-50 %ile in Fig. 3b, must have been slightly higher than 50 %ile (up to 52.5 %ile). For 10 consecutive seasons, Table 3 indicates one instance for less than the 50 %ile, one instance below 40 %ile, and also one below 30 %ile. Fig. 3b shows that the 30 %ile and 40 %ile counts come from the same 1976-77 dry period, whose 50 %ile count was 1 for 11 seasons duration. The one instance of 50 %ile or less for 10 seasons comes from the 1994 dry period (Apr-May-Jun 1994 through Jan-Feb-Mar 1995). This is an appropriate example from which to illustrate the reasoning behind counting individual occurrences as opposed to cumulative occurrences. If cumulative counts were made with respect to percentile and all three of the above cases of below 30%ile and 40%ile were also counted as being below 50%ile (which is true, and initially would seem desirable to acknowledge), then the event that lasted for 11 seasons would also be counted as lasting for 10 seasons; the counts would then become cumulative in duration as well as percentile. Clearly, all counts would become much larger. While this outcome would not necessarily be undesirable, it would be governed strongly by the number of percentile thresholds used and the number of duration thresholds used, both of which are arbitrary. For example, if percentiles intervals of 2 were used instead of 10, the cumulative tallies would become far higher. Thus, the non-cumulative results shown in Table 3 appear to contain the most fundamental information of interest, and would not change appreciably with the use of somewhat different percentile or duration thresholds. Completing the demonstration using Honolulu, we turn finally to the frequency of 10 %ile or less. It is noted from Fig. 3b that a 5-season-long period of such severe dryness may have occurred in 1975, in 1976, 1977 or 1995. However, only three of these four possibilities actually qualified, depending on which three did not have any 3-month periods in the 10-12.5 %ile range. While Fig. 3b and Table 3 could be constructed using intervals that would avoid such ambiguities, the mutual near-confirmation demonstrated in the above examples helps establish their accuracy while illustrating their potential utility.

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