Overview

The original method M5 to estimate extreme precipitations and PMP values was proposed by the ^{NERC (1975)} of England, it uses: (1) the precipitation of 24-hour duration and 5 year-return period, designated M5 as index variable; (2) a regional estimate of the coefficient of variation (Cv) and (3) the value of the reduced variable (y) of the Gumbel or General Extreme Value Distribution (GVE) type I, whose shape parameter k is equal to zero. It is a regional method that allows estimations in sites without data, based on the map of isovalue curves of the M5 in the British Isles.

^{Eliasson (1994)} indicates that both the method of David M. Hershfield (^{Hershfield, 1961}; ^{Hershfield, 1965}) and M5, which statistically estimate the PMP, are incorrect when using a probability distribution not bounded at its right end, since by definition the PMP has a physical upper limit. Eliasson (1994) also points out that the use of regional envelope curves of extreme precipitation values, to which the Gumbel distribution is fitted, tends in the high return periods, to a limit value of the reduced variable, which can be used to estimate the PMP.

^{Elíasson (1997)} points out that when the Gumbel distribution, which is a straight line in the paper of extreme probability, is fitted to series of annual maximum daily precipitation (PMD), usually the data are close in the middle part of such model, but they show deviations in the low and large values. These deviations and the lack of an upper limit in the Gumbel model, detract from the reliability of the original M5 method to estimate the PMP.

Estimation of Extreme Predictions

In the generalization of the M5 method, due to ^{Elíasson (1997)}, its anomalies cited, are eliminated them by transforming the random variable of the Gumbel distribution and delimiting the reduced variable (y _lim), to obtain a truncated Gumbel model. Now the M5 method depends on another local parameter called slope factor (C _i ) that is a function of the Cv; its equations are (^{Elíasson, 2000}):

being:

when 25 < M5_d < 200 mm/day, then:

In the above expressions, the undefined variables are: X _Tr is an extreme precipitation in 24 hours or random variable with return period Tr, M5 is the maximum precipitation of 24-hour duration and Tr = 5 years, that is with y≅ 1.50 and a probability of 0.80 of non-exceedance; on the other hand, M5_d is a daily value. The C _i tends to a value of 0.19 with a standard deviation of 0.035, according to the curve of C _i against M5 for the processed values (^{Elíasson, 1997}).

Estimation of the PMP

^{Elíasson (1997)} found that applying Equation 4 extreme predictions values of the ^{NERC (1975)} and of the PMP of the US National Weather Service for the state of Washington in USA, in a 25.9 km² area are reproduced. This is the other generalization of the M5 method. y _lim is a regional parameter and M5 and C _i are local parameters that have the same probabilistic performance as the mean and Cv in the frequency analysis. Equation 4 defines extreme values of y _lim of 10.5225 with a Tr of 37142.5 years and of 9.280 with a Tr of 10721.9 years. To estimate the punctual PMP of 24-hour duration, y _lim is evaluated with expression 4, then such value is substituted in Equation 3 and the respective Tr is cleared, which is finally replaced in the following expression by ^{Alfnes and Förland (2006)}:

Available Extreme Predictions

During a study conducted in 2013, ^{Campos-Aranda (2014)} processed 98 records of annual maximum daily precipitation (PMD) of the state of Zacatecas, Mexico, whose minimum amplitude was 25 years and the maximum of 68 data. The lapses of such records varied from 1943 to 2012 and the statistical quality tests detected that 17 series showed deterministic components, reason why they were eliminated, leaving 81 records to be processed. In columns 2 and 3 of Table 1, the names of the rain-gauge stations and the amplitudes of each series of annual PMD are indicated. In columns 5 and 6 of Table 1, their respective values of the M5_d and Cv are shown.

The probabilistic processing of the 81 series of annual PMD, based on the General Extreme Value (GVE) and Log-Pearson type III (LP3) distributions, adopting (column 4) the one that reported a lower standard error of fitting (^{Kite, 1977}), led to the extreme daily predictions of Tr = 100, 1000 and 10000 years, shown in columns 7, 10 and 13 of Table 1, from ^{Campos-Aranda (2014)}. By multiplying the predictions of PMD by 1.13, those of 24-hour duration are obtained (^{Weiss, 1964}; ^{WMO, 2009}).

Probable Maximum Precipitation (PMP) in 24 hours

The 24-hour punctual PMP of the state of Zacatecas, Mexico, was estimated based on the statistical method of David M. Hershfield (^{Hershfield, 1961}; ^{Hershfield, 1965}; ^{Campos-Aranda, 1998a}; ^{Campos-Aranda, 1998b}; ^{WMO, 2009}), whose results are in column 16 of Table 1, for the 81 series of annual PMD processed; such values come from ^{Campos-Aranda (2014)}.

Quantitative measure of contrast

The relative error (ER) in percentage, calculated with following equation, was adopted as a basic indicator of the contrasts:

in which, X _Tr is the prediction of 24-hour duration and return period (Tr) in years, estimated with Equation 1 of the generalized M5 method by Jónas Elíasson and PMD _Tr is the daily extreme prediction of equal Tr. Both predictions are expressed in millimeters. When PMP is contrasted, the previous equation is transformed into the following:

being, PMP _M5 the punctual probable maximum precipitation in 24 hours estimated with Equation 5 and PMP _MH that calculated with the Hershfield method, of 24-hour duration; both in millimeters. In expressions 6 and 7, when the estimates of the M5 method exceed the predictions previously calculated in the state of Zacatecas, Mexico (^{Campos-Aranda, 2014}), positive relative errors are obtained. When predictions of M5 method are lower the ER are negative.

Regarding predictions of Tr = 100 years

All the contrasts performed were analyzed based on the ER, considering that when such an indicator is less than 10.0%, a fairly approximate estimate was obtained. For the case of predictions of Tr = 100 years, as observed in column 9 of the Table 1, only in 15 rain-gauge stations or series of annual PMD, ERs greater than 10.0% were observed, whereby, in 81.5% of the processed records a fairly approximate prediction was achieved with the generalized M5 method. In the 15 records that have the highest dispersions with the predictions of the GVE or LP3 models, nine were by default and six by excess, with maximum values of -20.6% and 23.7%, shown in the last two lines of the last column of Table 1 (first part).

Taking into account that the prediction of Tr = 100 years, extrapolates most of the processed records, initially it was sought to relate the 15 ER greater than 10.0% with the short records, but such dependence does not occur in a generalized manner. What can be observed is that positive ERs occur when the Cv of the annual PMD series is less or close to 0.260 and negative ERs are presented in Cv greater than 0.320; the anomalies of the previous one are in the Felipe Ángeles and Mezquital del Oro stations, both with short records of 26 years.

Regarding the predictions of Tr < 100 years

Based on the ER of the predictions of Tr = 100 years (column 9 of Table 1), the three stations with ERs negative maximums were selected, with ER minimums and ER positive maximums. In such stations a contrast was made of their predictions of Tr of 2, 5, 10, 25, 50 and 100 years, which is shown in Table 2. It is observed that all ERs are smaller in such predictions and close to zero, or less than 10%, in the Tr of 5, 10 and 25 years; except in Trancoso (Station No. 76). Therefore, it is concluded that the generalized M5 method reproduces in a fairly approximate manner the predictions of Tr <100 years.

Regarding extreme predictions

In the Tr predictions of 1000 and 10000 years of the generalized M5 method, ERs lower than 10.0% are significantly reduced, with only 34.6% of the records in the greater Tr, according to column 15 of Table 1. In addition, the maximum values of such ERs increase considerably, being -54.1% and 74.7%, in the extreme predictions of Tr = 10000 years.

In a punctual study of each record with negative ER, it was detected that they occur in annual PMD series having maximum scattered values (outliers) and as a result, they are fitted to the GVE distribution with negative shape parameter (k) or Fréchet model. On the contrary, positive ERs occur when the records follow the Weibull model or GVE distribution with positive k, since there is now an upper limit. In summary, the generalized M5 method cannot reproduce the extreme elevated predictions of the distribution Fréchet, neither mark out or delimit those of the Weibull model.

Due to space limitations, rain-gauge stations with ERs negative maximums are not mentioned, nor are their respective k values cited, but they occur in the numbers: 11, 42, 47, 43, 65, 58 and 69, with k varying from -0.24 to -0.11. In contrast, the ERs positive maximums are in stations: 76, 22, 24, 67, 77 and 28, with k fluctuating from 0.35 to 0.17.

Regarding the PMP

According to the ERs shown in the final column of Table 1, 36 records have ERs lower than 10.0%, which corresponds to 44.4% of the 81 series of annual PMD, processed. In addition, in another 40 records the PMP estimated with the generalized M5 method, led to ER positive, that is, it overestimates the value of the PMP obtained with the Hershfield method. Finally, negative ERs were obtained in the five missing records, with a maximum value of -16.4%. The above is generally considered an excellent approximation of a method that allows PMP estimates in 24 hours in sites without data, based only on two regional parameters, the M5_d and the Cv.

With respect to the five underestimations of the PMP, they occur when the Cv of the annual PMD series is less than 0.251; On the other hand, significant overestimates, considered those of an ER greater than 30.0%, are generally associated with values greater than 0.386 of the Cv.

Regarding the regional analysis

In his study, ^{Campos-Aranda (2014)} carried out two regional analyses, one in the Hydrological Region No. 12 Partial (Rio Santiago) in the Juchipila River basin, which included 13 stations and another in Hydrological Region No. 37 (El Salado), which included 19 records. Selecting from Table 1 the lines of the stations belonging to each hydrological region, two tabulations were integrated. It was observed in both tabulations, that now their values tend to be similar; that is, they present less dispersion. The previous one was verified in its final lines of minimum and maximum values, which showed less amplitude or range. This generates confidence in the results of the application of the generalized M5 method, in areas or geographical zones of a hydrological region. Due to space limitations, just the final portion of the stations of the Juchipila river basin is shown in Table 3, which are: Excamé, Huanusco, Huitzila, Jalpa, Juchipila, La Villita, Los Campos, Mezquital del Oro, Moyahua de Estrada, Nochistlán, Teúl de González Ortega, Tlachichila and Tlaltenango.