Event-based analysis of the dry spell phenomenon
One form of drought is the interruption of the rainy season by a sa called dry spell. Dry spell can be defined as a sequence of dry days including days with less than a threshold value of rainfall. A dry spell, defined a on daily scale, becomes untraceable by statistics using longer than one-day-long equidistant time intervals. If the daily discretization of the rainy season is te be avoided, an intermediate technique is needed. Event-based analysis of the rainfall and dry spell provides This approach. The method is demonstrated with data from the Dodoma Region, situated in the semiarid highlands of Tanzania. The climate is characterized by one rainy tesson from the and of November until the end of April. The occurrence of rainfall is erratic. The average seasonal precipitation is about 600 mm with variations between 450 and over 800 mm. Rainy seasons are separated by an almosl 7 month long dry season. During the rainy season convective type storms prevail. Single storms lasi a few hours, but their occurrence is clearly grouped within the timespan of a few days, separated by the dry spells which are usually much longer. Conventional statistics of dry spells are summarized in tables 1 and 2 using 1.0 mm daily precipitation as the threshold. It is shown that dry spells occur randomly during the rainy season. For the event-based analysis dry spell is detined as a dry event. Dry events are considered as a sequence of dry days separated by rainfall events from each other. Thus the rainy season is detined as a series of rainfall and subsequent dry events. Rainfall events are defined as the uninterupted sequence of rainy days, when at least on one day more than a threshold amount of rainfall has been observed- Rainy days with less Man the threshold depth of precipitation are accounted for the rainfall event if they occur in an uninterrupted sequence. Only isolated subthreshold rainfall will be discarded, and considered as part of a dry event (fig. 2). ln this analysis the threshold value of 5.0 mm/day was seiected. The comparison of tables 1 and 4 shows that the length of the mean maximum dry spell doubles by replacing the 1.0 mm/day threshold by 5.0 mm/day. The sequence of rainfall and dry events is characterized by Dn, m, duration of the mth rainfall of the nth rainy season, and by the inter-event time Zn, m (duration of the dry event) between the end at the preceding and the start of the succeeding rainfall event. In case of convective type storms the series of the subsequent events (either dry or rainfall) could be considered independent, thus their number/season should follow the Poisson distribution. In case of independence of subsequent events, the waiting lime for a new event must follow the exponential pdf. By measuring the waiting time in days the discrets equivalent of the exponential pdf can be used. Since the sequences of convective type storms de not contain purely independent events, the waiting time t follows instead the discrete counterpart of the 2 parameter gamma pdf, the negative binomial pdf. This modified Poisson-modal, Poisson pdf for the number of rainfall events and the negative binomial pdf for the length of the inter-event lime has been applied to describe the rainy season. Table 3 summanzes the parameters r and p for different rein gauges. By focusing on the dry spell event, the duration of the rainfall events Dn, m will in fact be identified as inter-event time. This change of rotes fils the original Poisson model better. Since rainfall events are shorter, their duration follows the geometrical pdf, as theoretically required. The Poisson pdf seems to fit slightly better the number of dry events than the rainfall ones (fig. 5). If has to be pointed out that the event-based definition of the rainy season dues not exactly fit the theoretical precondition, i. e. to have a certain fixed period. Rainy seasons have variable longths, as they are a stochastic fonction of the events themselves. For "modal fitting" the consideration of the core of the rainy season, from January to April would be a better choice. However it would truncate the physical phenomenon wilh the potentiel omission of extrema long dry events. Therefore, in spite of mediocre fitting, the Poisson modal will be used ln analysis. Tables 4 and 5 summarize the statistical characteristics of the dry events for the selected rain gouges for bath the whole, and for the core of the rainy season. Dry events accurring in the core of the rainy season were identified as those ending within the timespan of January -April. The mean lengths of the longest dry spells in the core are less than the corresponding value for the whole season. However, at two stations, Kondoa and Gwandi, in more han 70 % of the season the longest dry spell did occur during the core. This coincidence was only 40 % for the Farkwa rain gauge. For planning purposes, the longest dry spells associated with the varions statistical recurrence periods are derived on the basis of the fitted Pearson III type probability distribution functions (fig. 6, table 6). The event-based analysis, relying on the expected number ot events/season and the negative binomial pdf for the length of the dry events, can also be used to approximate the distribution of the extreme long dry spells. Contrary to the Pearson III distribution fitted to the seasonal extreme values, the negative binomial pdf f (n) determines the probability that a random dry event would last n days. Consequently, the exceedence probability Pe (N), that an extreme long dry event would occur at least once within a given statistical recurrence period of T years must be equal to the reciprocal value of the product λ•T, where λ denotes the expected number of dry eventslyear (season). λ•T specifies the expected number of "trials" needed to observe at least once the extreme duration of N days associated with the return period of T years. The length of this extreme dry spell can then be obtained from the cumulative negative binomial pdf (table 6). The deviations observed for low number of "trials" between the event-based and the extreme seasonal value approach are due to the conceptual difference. Table 7 displays the simultaneous occurrence of dry events at several rain gauges. By using Farkwa as the reference station, table 7 does not account for dry events that might have occurred simultaneously at Gwandi and Kandoa without having been recorded at Farkwa. Except for the very short (1 or 2 days) and the very long dry events (over 30 days), the overwhelming majority of the dry events occurred at least at two stations simultaneously. Furthermore, excluding the 1, 2 or more than 30 day - long events, more than 63 % of the dry events have been observed at all three stations. Two (or three) dry events were only classified as simultaneous if more than hall of the duration of the reference event at Fartwa was overlapped by an uninterrupted dry event at the other station (s). Event-based analysis, even if it is carried out on the bases of a few years of observation, can rely on large number of data points (table 3). While the expected number of events/season is still derived from very few data, this estimate is more reliable than the approximative expected length of the longest seasonal dry spell, since the variability of the former is usually less than that of the latter, for the same data sets (table 8).