Introduction

In Mexico, there is information of daily rains measured in more than 5000 stations. The analysis of these records shows that in many stations the available data are scarce, so that the statistical analyses performed individually are not reliable (for example, CONAGUA uses only 1788 stations for the calculation of climatological normals) (CONAGUA, 2015). In the case of pluviographic records, the situation worsens, both because of the limited number of stations and the quality and accessibility of that information.

Numerous attempts have been made to estimate rainfall associated to different durations and return periods (^{Conde, Vita, Castro & López, 2014}, ^{SCT, 1990}), but an individual analysis is carried out, station by station, which often results in inconsistencies in the results.

In the last years and in different countries, there have been extensive studies on the regional analysis of rainfall frequencies, some of which are supported by the L moments technique (^{Escalante & Reyes, 2005}), as well as by principal components analysis (^{Gellens, 2002}; ^{St-Hilaire et al., 2003}; ^{Wotling, Bouvier, Danloux & Fritish, 2000}), the use of clustering techniques (cluster analysis) for regional grouping, technic of year-seasons, the ^{Dalrymple method and quantile analysis and computation of fractiles (Dalrymple, 1960}; Conleth,1988, ^{Buishand, 1991}; ^{Cunnane, 1988}; ^{Gellens, 2002}; ^{Yang et al., 2010}).Some authors propose to use the general distribution of extreme values (GEV) for the analysis of extreme rainfall for different durations (^{Rossi, Fiorentino & Versace, 1984}; ^{Gellens, 2002}; ^{St-Hilaire et al., 2003}); ^{Koutsoyiannis (2009}a and ^b) indicates in his case study that a type II extreme value function is better suited than Gumbel functions to precipitation records even with few data.

^{Berndtsson and Niemczynowicz (1988)} conducted a state-of-the-art review of the various factors involved in precipitation analysis; they emphasize the time and space scales to be used depending on the scope of the hydrological problem; the two types of precipitation analysis stand out: at a single point level to obtain the period of return of rains with different characteristics, and the analysis of the simultaneity of the precipitations considering different sites to make regional estimates.

In Mexico, storm regionalization studies have been carried out in the Valley of Mexico basin (^{Cortés, 2003}). ^{Guichard & Domínguez (1998)} in their study of regionalization of basins of the Upper Grijalva consider several studies on the factors of reduction by area, by period of return and by duration (^{Bell, 1969}; ^{Chen, 1983}).

Rainfall regionalization studies using the General Distribution of Extreme Values carried out in San Luis Potosí, Sinaloa, Mexico (^{Campos-Aranda, 2008} and ²⁰¹⁴) are highlighted. In addition to the regional estimation of convectivity factors (^{Baeza, 2007}), precipitation maps for different periods of return and durations have been made in the Mexican Republic (^{Domínguez et al.}; ^{Secretaría de Comunicaciones y Transportes, 1990}); also, the influence of regionalization on the estimation of daily maximum precipitation (^{Escalante and Amores, 2014}), the regionalization of precipitation with bivariate functions and maximum entropy (^{Escalante and Domínguez, 2001}), climate change at regional scale in precipitation and temperature (^{Magaña & Galván, 2010}).

In this work, regional analyses are carried out based on information from 2293 rain gauges, which allows the estimation of design rainfall (associated with different return periods) for different durations; a procedure using synthetic series for the validation of homogeneous zones is also presented that compares historical and synthetic coefficients of variation. In making the regional analysis, spatially consistent results are achieved, avoiding the inconsistencies that are obtained when analyzing each station individually.

In relation to rains with duration less than one day, results from ^{Baeza and Mendoza (Baeza, 2007}, ^{Mendoza, 2001}) are taken; they used the original information compiled by the Ministry of Communications and Transportation (^{Secretaría de Comunicaciones y Transportes, 1990}), and also from the former Ministry of Hydraulic Resources (mainly the Papaloapan Commission bulletins), but they managed it regionally using the concept of "convectivity" based on the theory developed by ^{Chen (1983)} and considering the topographical conformation and climatological environment of each season. The results obtained allow to estimate design logs for different time intervals and different total storm durations, using the alternating block method (^{Chow, Maidment & Mays, 1988}), for any basin of the Republic.

Methodology

Data from a total of 2293 stations in the Climate Computing Project database (CLICOM) (Figure 1) were analyzed. Table 1 shows the number of stations considered in each state as well as the corresponding average number of record years.

Figure 1 Processed stations 

For the selection of the 2293 stations it was verified that they fulfilled the following requirements

For each of the stations the annual maximum daily values were obtained and the statistics calculated were: mean, standard deviation, coefficient of variation (CV), besides the maximum and minimum values. The analysis of these statistics allowed the debugging of values that at first seemed illogical (on the one hand, very small values or even zero for the maximum precipitations in a whole year and on the other very large values, on the order of four times the mean). In the case of very large values, the original log sheets were consulted with the help of the Superficial Water Management and River Engineering (GASIR) and, according to the results of that query, the incorrect ones were eliminated.

With the values cleared and considering both the coefficients of variation and the topographic conformation, homogeneous regions were defined from the point of view of the annual maximum daily precipitation, considering that, if in a region, the coefficients of variation are similar and the annual maximum values of each station are modulated by dividing them by their mean, the modulated samples will have a mean of 1 and a similar standard deviation. To test the hypothesis of homogeneity among the samples thus obtained, Fisher's test is traditionally used (^{Domínguez et al., 2014}), but considering that the maximum values do not correspond to a normal probability distribution, a new procedure is proposed in this work to verify the hypothesis that the historical samples come from a population characterized by the distribution function adjusted to the enlarged sample. For this, synthetic series are generated and the CV of a set of samples with the same size as the historical ones, obtained from the adjusted distribution function calculated. The region is considered homogeneous if the range of simulated CV includes all the historical CV (Domínguez, Bouvier, Neppel & Niel, 2005).

The homogeneous samples obtained for each region will then be made up of much more values than those corresponding to each station considered individually, so that the results obtained by adjusting them to a probability distribution function will be much more reliable and robust.

Results

Conformation of homogeneous regions

Considering on the one hand the coefficients of variation of the annual maximum values, but on the other hand the proximity between the measurement sites, the characteristics of the topography in the surroundings of those sites and the type of meteorological phenomenon that provokes the extreme precipitations, was looked for A compromise between defining regions with many stations, so that the sample obtained by assuming that they come from the same population results very large and therefore the extrapolations to larger return periods is more reliable, and limit the size of each region, which would give more confidence in the acceptance of the homogeneity hypothesis. The 59 regions shown in Figure 2 were thus defined. Also, Table 3 shows the maximum and minimum values of the coefficients variation, as well as the number of stations included in each region. It is noted that in only five cases, coefficients of variation are greater than 1 and in six, smaller than 0.2.

Figure 2 59 regions defined considering coefficients of variation and topographical and meteorological conditions. 

Table 3 Number of stations and extreme coefficients of variation in each region 

State	Region	Num. of stations	Maximum Coefficient of Variation	Minimum Coefficient of Variation
Aguascalientes	Aguascalientes	50	0.46	0.24
Baja California Norte	Baja California Norte	37	1.31	0.41
Baja California Sur	Baja California Sur	72	1.36	0.46
Campeche	Campeche	42	0.93	0.26
Chiapas	Angostura	36	0.88	0.22
	Malpaso	17	0.66	0.24
	Chicoasén	21	0.79	0.22
	Peñitas	4	0.51	0.26
	Almandro	11	0.82	0.24
	Pichucalco	3	0.35	0.29
	Teapa	3	0.50	0.23
	Costa	28	0.57	0.23
Chihuahua	Bajos	18	0.43	0.22
	Restantes	40	0.96	0.24
Coahuila	Coahuila	28	1.04	0.31
Colima	Colima	17	0.68	0.39
Ciudad de México	Ciudad de México	30	0.50	0.21
Durango	Durango	83	1.07	0.23
Estado de México	Estado de México	114	0.50	0.20
Guanajuato	Guanajuato	108	0.49	0.22
Guerrero	Norte	36	0.48	0.18
	Centro	56	0.68	0.32
	Costa	37	0.61	0.21
Hidalgo	Menores	57	0.70	0.22
	Mayores	5	0.52	0.43
Jalisco	Interior	153	0.53	0.20
	Costa	23	0.58	0.30
Michoacán	Michoacán	93	0.65	0.18
Morelos	Morelos	44	0.64	0.20
Nayarit	Costa	14	0.40	0.25
	Sierra	11	0.33	0.19
Nuevo León	I	10	0.93	0.29
	II	7	0.63	0.39
	III	38	1.01	0.38
Oaxaca	Golfo	29	0.53	0.26
	Altiplano	82	0.74	0.18
	Istmo	14	0.56	0.37
	Pacifico	4	0.59	0.43
Puebla	Norte	49	0.79	0.26
	Centro	34	0.67	0.21
	Sur	14	0.52	0.19
Querétaro	Zona Alta	7	0.49	0.26
	Zona Baja	27	0.64	0.24
Quintana Roo	Quintana Roo	20	0.64	0.28
San Luis Potosí	San Luis Potosí	103	0.94	0.27
Sinaloa	Zona I	21	0.77	0.39
	Zona II	30	0.62	0.20
Sonora	Zona I	57	0.85	0.17
	Zona II	21	0.77	0.33
	Zona III	2	0.77	0.77
Tabasco	Tabasco	32	0.53	0.24
Tamaulipas	Zona I	81	0.87	0.27
	Zona II	20	0.91	0.35
	Zona III	8	0.51	0.31
Tlaxcala	Tlaxcala	22	0.44	0.22
Veracruz	Veracruz	190	0.79	0.20
Yucatán	Yucatán	30	0.68	0.30
Zacatecas	Zona I	11	0.54	0.28
	Zona II	39	0.45	0.22
	Total	2293

Functions of distribution of the regional samples

The annual maximum precipitation values of each station were modulated by dividing them into their mean, so that, when grouped together, samples were obtained with considerable number of elements for each of the 59 regions. For each of the 59 extended samples, Gumbel (G) and Double Gumbel (DG) probability distribution functions were fitted, which gave the lowest standard error of fit; the values associated with different return periods were estimated with them as shown as example in Figure 3. Table 4 shows the results obtained for the 59 regions with the probability distributions best fitted. In addition, the average annual precipitation value can be obtained for any site in the Republic, with a map of the distribution of annual maximum precipitation averages (as in Figure 4) and multiplying by the factors by return period corresponding to the region in which the study site is located, to estimate the daily precipitation associated with any return period.

Figure 3 Distribution function adjusted to the annual normalized maxima of the Oaxaca region. 

Table 4 Factors per return period for the 59 regions 

State	Homogeneous region	Region	Regionalization factors
			Return Period YEARS
			2	5	10	20	50	100	200	500	1 000	2 000	5 000	10 000
Aguascalientes	1	Aguascalientes	0.94	1.25	1.45	1.64	1.90	2.08	2.27	2.52	2.71	2.89	3.14	3.33
	2	Baja California Norte	0.84	1.36	1.81	2.23	2.73	3.09	3.45	3.91	4.27	4.62	5.09	5.43
Baja California S	3	Baja California Sur	0.77	1.36	2.03	2.65	3.38	3.90	4.41	5.08	5.57	6.07	6.72	7.21
Campeche	4	Campeche	0.86	1.24	1.64	2.04	2.51	2.85	3.19	3.62	3.95	4.27	4.69	5.01
Chiapas	5	Angostura	0.90	1.19	1.46	1.81	2.29	2.65	3.00	3.46	3.81	4.15	4.61	4.95
	6	Malpaso	0.90	1.22	1.50	1.80	2.20	2.49	2.77	3.15	3.44	3.72	4.10	4.36
	7	Chicoasén	0.93	1.21	1.44	1.68	1.99	2.22	2.46	2.76	2.99	3.23	3.53	3.76
	8	Peñitas	0.91	1.22	1.50	1.79	2.18	2.47	2.76	3.13	3.41	3.69	4.07	4.35
	9	Almandro	0.89	1.22	1.53	1.89	2.36	2.70	3.03	3.47	3.80	4.13	4.55	4.90
	10	Pichucalco	0.95	1.24	1.44	1.63	1.87	2.05	2.23	2.47	2.66	2.84	3.08	3.26
	11	Teapa	0.94	1.25	1.46	1.66	1.91	2.10	2.29	2.54	2.73	2.92	3.17	3.36
	12	Costa	0.91	1.20	1.48	1.81	2.19	2.46	2.72	3.05	3.30	3.55	3.88	4.13
Chihuahua	13	Bajos	0.90	1.43	1.78	2.11	2.54	2.87	3.19	3.62	3.94	4.26	4.69	5.01
	14	Restantes	0.89	1.25	1.56	1.89	2.34	2.68	3.02	3.46	3.79	4.12	4.55	4.87
Coahuila	15	Coahuila	0.85	1.28	1.68	2.09	2.60	2.98	3.34	3.82	4.19	4.55	5.03	5.41
Colima	16	Colima	0.85	1.26	1.69	2.10	2.58	2.93	3.27	3.71	4.05	4.38	4.82	5.15
Ciudad de México	17	Ciudad de México	0.95	1.21	1.39	1.55	1.77	1.93	2.09	2.30	2.46	2.62	2.83	3.00
Durango	18	Durango	0.91	1.25	1.52	1.80	2.16	2.43	2.70	3.05	3.32	3.58	3.93	4.21
Estado de México	19	Estado de México	0.95	1.21	1.38	1.55	1.76	1.93	2.09	2.30	2.46	2.62	2.83	2.99
Guanajuato	20	Guanajuato	0.95	1.23	1.42	1.60	1.83	2.00	2.17	2.40	2.57	2.75	2.98	3.15
Guerrero	21	Norte	0.95	1.21	1.38	1.54	1.75	1.91	2.07	2.27	2.43	2.59	2.79	2.95
	22	Centro	0.92	1.34	1.62	1.89	2.24	2.50	2.76	3.10	3.36	3.62	3.96	4.21
	23	Costa	0.90	1.24	1.52	1.80	2.15	2.40	2.65	2.99	3.24	3.49	3.82	4.07
Hidalgo	24	Menores	0.92	1.35	1.63	1.90	2.25	2.51	2.78	3.12	3.38	3.64	3.99	4.25
	25	Mayores	0.87	1.35	1.71	1.98	2.30	2.53	2.76	3.06	3.28	3.50	3.80	4.01
Jalisco	26	Interior	0.92	1.24	1.45	1.64	1.86	2.03	2.19	2.41	2.57	2.74	2.95	3.12
	27	Costa	0.93	1.30	1.55	1.79	2.09	2.32	2.55	2.85	3.08	3.31	3.61	3.84
Michoacán	28	Michoacán	0.92	1.21	1.45	1.69	1.98	2.20	2.42	2.70	2.92	3.14	3.42	3.64
Morelos	29	Morelos	0.95	1.21	1.39	1.56	1.77	1.93	2.10	2.31	2.47	2.63	2.84	3.01
Nayarit	30	Costa	0.95	1.22	1.40	1.58	1.80	1.97	2.13	2.36	2.52	2.69	2.91	3.08
	31	Sierra	0.95	1.20	1.37	1.53	1.73	1.88	2.04	2.24	2.39	2.54	2.74	2.90
Nuevo León	32	I	0.88	1.26	1.61	1.99	2.51	2.89	3.26	3.75	4.12	4.49	4.97	5.34
	33	II	0.91	1.38	1.69	1.99	2.38	2.67	2.96	3.34	3.63	3.92	4.30	4.59
	34	III	0.82	1.28	1.77	2.30	2.94	3.40	3.85	4.44	4.89	5.33	5.91	6.37
Oaxaca	35	Golfo	0.94	1.24	1.44	1.63	1.87	2.06	2.24	2.48	2.66	2.84	3.08	3.27
	36	Altiplano	0.91	1.22	1.46	1.76	2.2	2.53	2.86	3.28	3.6	3.92	4.33	4.64
	37	Istmo	0.87	1.36	1.72	1.97	2.27	2.48	2.69	2.96	3.17	3.38	3.65	3.85
	38	Pacífico	0.86	1.25	1.67	2.10	2.61	2.98	3.33	3.80	4.15	4.50	4.96	5.33
Puebla	39	Norte	0.93	1.32	1.57	1.82	2.14	2.38	2.61	2.93	3.16	3.40	3.72	3.95
	40	Centro	0.94	1.27	1.49	1.71	1.98	2.19	2.40	2.67	2.87	3.08	3.35	3.55
	41	Sur	0.94	1.26	1.47	1.67	1.92	2.12	2.31	2.57	2.76	2.95	3.21	3.40
Querétaro	42	Zona Alta	0.93	1.29	1.53	1.75	2.04	2.26	2.48	2.77	2.99	3.21	3.49	3.71
	43	Zona Baja	0.94	1.27	1.49	1.70	1.98	2.18	2.38	2.65	2.86	3.06	3.33	3.53
Quintana Roo	44	Quintana Roo	0.88	1.26	1.60	1.91	2.28	2.55	2.82	3.17	3.44	3.70	4.05	4.32
San Luis	45	San Luis	0.88	1.29	1.63	1.94	2.33	2.61	2.89	3.25	3.52	3.80	4.16	4.44
Sinaloa	46	Zona I	0.84	1.28	1.76	2.15	2.61	2.93	3.25	3.67	3.98	4.29	4.70	5.01
	47	Zona II	0.94	1.27	1.49	1.70	1.97	2.18	2.38	2.65	2.85	3.06	3.33	3.53
Sonora	48	Zona I	0.93	1.29	1.53	1.75	2.05	2.27	2.49	2.78	3.00	3.22	3.50	3.72
	49	Zona II	0.88	1.26	1.61	1.96	2.38	2.69	3.00	3.40	3.70	4.00	4.41	4.71
	50	Zona III	0.77	1.50	2.23	2.77	3.38	3.82	4.25	4.82	5.24	5.67	6.22	6.64
Tabasco	51	Tabasco	0.94	1.26	1.47	1.67	1.94	2.13	2.33	2.59	2.78	2.98	3.23	3.43
Tamaulipas	52	Zona I	0.89	1.31	1.61	1.88	2.20	2.43	2.66	2.96	3.19	3.42	3.72	3.95
	53	Zona II	0.85	1.26	1.69	2.18	2.81	3.26	3.70	4.27	4.71	5.14	5.72	6.12
	54	Zona III	0.91	1.32	1.59	1.82	2.09	2.29	2.49	2.75	2.94	3.14	3.39	3.59
Tlaxcala	55	Tlaxcala	0.95	1.22	1.40	1.57	1.79	1.96	2.13	2.35	2.51	2.68	2.90	3.06
Veracruz	56	Veracruz	0.90	1.27	1.56	1.83	2.15	2.39	2.62	2.93	3.16	3.39	3.69	3.93
Yucatán	57	Yucatán	0.93	1.32	1.59	1.84	2.17	2.41	2.66	2.98	3.22	3.47	3.79	4.03
Zacatecas	58	Zona I	0.92	1.34	1.62	1.88	2.23	2.48	2.74	3.08	3.33	3.59	3.92	4.18
	59	Zona II	0.95	1.23	1.42	1.60	1.84	2.02	2.19	2.42	2.60	2.77	3.00	3.18

Figure 4 Distribution of means of annual maximum daily rainfall in the Mexican Republic 

Estimates for storms with durations less than 1h

Figure 5 shows the map of convectivity factors estimated by ^{Baeza (2007)}. To obtain the precipitation associated with a duration of 1h, only multiply that obtained for one day by the corresponding factor. In addition, and according to the results obtained by ^{Chen (1983)}, to estimate the values associated with other durations of less than 24 h, those of 1 h are multiplied by the factors indicated in Table 5.

Figure 5 Map of convectivity factors (^{Baeza, 2007}) 

Table 5 Factors K _d ⁽¹⁾ depending on R to and duration d. Source: Adapted from ^{Luna, 2013}  

d(min)	d(hrs)	R=P1T/P24T K in function of d and the convectivity factor
		R=0.10	R=0.20	R=0.30	R=0.40	R=0.45	R=0.50	R=0.60	R=0.65
10	0.17	0.293	0.39	0.432	0.454	0.462	0.469	0.481	0.487
15	0.25	0.38	0.485	0.536	0.565	0.575	0.584	0.6	0.608
30	0.5	0.612	0.699	0.745	0.773	0.79	0.793	0.809	0.816
60	1	1	1	1	1	1	1	1	1
90	1.5	1.378	1.248	1.185	1.146	1.13	1.119	1.097	1.088
120	2	1.646	1.424	1.317	1.25	1.22	1.203	1.166	1.151
150	2.5	1.934	1.595	1.435	1.337	1.30	1.268	1.215	1.193
180	3	2.207	1.75	1.538	1.41	1.35	1.322	1.254	1.226
210	3.5	2.468	1.892	1.631	1.475	1.41	1.367	1.286	1.253
240	4	2.719	2.024	1.715	1.532	1.45	1.407	1.314	1.275
270	4.5	2.961	2.148	1.793	1.584	1.508	1.443	1.337	1.294
300	5	3.196	2.266	1.865	1.631	1.547	1.475	1.358	1.311
360	6	3.649	2.485	1.997	1.716	1.616	1.531	1.395	1.339
420	7	4.081	2.686	2.115	1.791	1.676	1.579	1.425	1.362
480	8	4.497	2.874	2.223	1.858	1.73	1.621	1.451	1.382
540	9	4.899	3.05	2.322	1.919	1.778	1.659	1.474	1.399
600	10	5.289	3.216	2.414	1.975	1.822	1.694	1.494	1.415
660	11	5.669	3.375	2.501	2.026	1.862	1.725	1.513	1.429
720	12	6.039	3.527	2.582	2.074	1.9	1.754	1.53	1.441
840	14	6.756	3.812	2.734	2.162	1.968	1.807	1.56	1.463
960	16	7.445	4.078	2.872	2.241	2.029	1.853	1.586	1.482
1080	18	8.112	4.328	2.999	2.313	2.084	1.895	1.609	1.499
1200	20	8.758	4.564	3.117	2.379	2.134	1.933	1.63	1.513
1320	22	9.388	4.789	3.228	2.441	2.18	1.968	1.649	1.527
1440	24	10.001	5.004	3.333	2.498	2.223	2	1.667	1.539

Example for interpretation of results

To facilitate the interpretation of the results obtained, the following example is proposed:

Suppose that the centroid of a basin in the Oaxaca Istmo region has coordinates -94,559 Longitude West and 16.733 Latitude North, in such a way that the average of the maximum annual daily precipitation is 140 mm (see Figure 4, or for more detail Figure 6). According to Table 4, this value is multiplied by 2.48 to estimate the precipitation associated with 100 years of return period, resulting in a value of 347.2 mm for 1-day rainfall with 100 years of return period. If the concentration-time of the watershed was 1.5 h, a hyetograph with half-hour intervals and a total duration of 4 h could be estimated as follows. According to Figure 7, the convectivity factor would be 0.45, so that the precipitation in 1 h results in 156.2 mm. For this value of the convectivity factor, Table 5 indicates the factors to be considered for the different durations. These factors are reproduced in Table 7. The third column of Table 7 was obtained by multiplying the factors by 156.2 mm; in the fourth one the increments are indicated every half hour and in the fifth those same values ordered by alternate blocks, to obtain the hyetograph of Figure 7.

Figure 6 Istmo Region Oaxaca (94,559 West Length and 16,733 North Latitude) 

Figure 7 Hyetograph for 30-minute intervals and total duration of 4h. 

Table 7 Estimation of a hyetograph for 30-minute intervals and total duration of 4 h 

Duration minutes	Factor respect to 1h	Accumulated precipitation (mm)	Increment (mm)	Sorted values (mm)
30	0.79	148.52	148.52	10.68
60	1.00	188.00	39.48	14.80
90	1.13	213.31	25.31	25.31
120	1.22	229.36	16.05	148.52
150	1.30	244.16	14.80	39.48
180	1.35	253.80	9.64	16.05
210	1.41	264.48	10.68	9.64
240	1.45	272.60	8.12	8.12

Discussion

The results obtained allow to estimate in a robust and spatially consistent manner, rainfall for any part of the republic, any period of return and any duration. With these results, design hyetographs can also be built.

To verify the statistically homogeneous behavior of the maximum rainfall sample data in a region, the Fisher test is used traditionally, comparing the ratios of the variances corresponding to the different sites to the limits of the Fisher distribution function for a probability of being exceeded by, for example, 5%.

The values obtained from the squared ratios of the coefficients of maximum variation between the minimum of each region are shown in Table 8, which also indicates the type of distribution function that best fit the extended sample corresponding to each region.

The analysis in Table 8 shows that, in some cases, the square of the ratio between the maximum and minimum coefficients of variation is very large, so that in applying the Fisher test, it would be concluded that some stations should be removed from the corresponding region, although care was taken to ensure that climates and exposure to similar meteorological phenomena were present for each region and that the separation between regions corresponded to a well-defined watershed. It is also noted in Table 8 that most of these cases correspond to regions in which the best fit for the enlarged sample was obtained with the Double Gumbel Function. Two very notable cases are Sonora zone 1 and Veracruz, so these two cases are commented next: in Sonora, the largest CV (0.85) was in station 26034, in which, in 1972, a maximum of 407 mm was recorded whereas, of all the other stations the maximum measured was of 176 mm, and in all the period of record of all the stations the maximum measured is of 240 mm; if it is eliminated, the CV of the maximums in that station would be of 0.525.At the other extreme, the minimum CV value is 0.174 and corresponds to station 26101 in which the maximum recorded is very low (106 mm) even though the average is one of the highest in the region, possibly because it did not record in the years in which the major storms occurred (1978, 1994 and 2007). If these two values are eliminated, the extreme CV are left at 0.69 (at station 26010 which did measure in critical years) and 0.211, so that the squared ratio is 10.6 instead of 23.9; if the two extreme values are still eliminated, the squared ratio is 3.89.

Something similar occurs in Veracruz, where there is a record of 690 mm in the station 30031, in 1978, a year that was not characterized by major storms and in which the maximum recorded in the other 189 stations was 364 mm. If that value is eliminated, the maximum CV results from 0.67 and the square ratio between extreme values is reduced from 15.60 to 11.2. Also, if the two smaller values of the CV are eliminated from the 190 stations in this region, the ratio is reduced to 7.2, a value which is logical if it is considered to correspond to 187 stations.

Generation of synthetic series

Considering that these are extreme events whose probability distribution is not Normal, tests were performed by comparing the coefficients of variation of the historical samples to those obtained by generating random numbers with the adjusted probability distribution to the regional samples. For each of the 59 regions, 10 synthetic samples were generated, with the same number of values as the corresponding historical sample. Figures 8 and 9 compare the cumulative frequency distributions of the historical and synthetic CV for the extreme cases of the regions in Morelos and La Angostura in Chiapas. In each case 10 synthetic samples were generated, each of which consists of the same number of values as the one of years recorded in each station of the corresponding region.

Figure 8 Comparison between the empirical frequencies of the coefficients of variation of the historical sample and the sample generated. Morelos Region 

Figure 9 Comparison between the empirical frequencies of the coefficients of variation of the historical sample and the sample generated. La Angostura Region, in Chiapas 

In the first case (Morelos region) a good behavior is observed except for a station with a coefficient of variation of 0.64, (which was the maximum in that area in which the next value was 0.45). The station is number 17045 and in its data of maximum annual precipitations appears a value of 200 mm in 1977, year in which the maximum registered in all the other stations was of 87 mm, reason why it was resorted to verify the data, with the help of GASIR, in the original registry, and it was found that the maximum in that year was 57.4 and when changing it the coefficient of variation fell to 0.332.

In the case of the Angostura region, in Chiapas, although the behavior of the relative frequencies of the historical CV and the CV generated is similar (Figure 9) the value of the historical maximum CV (0.88, corresponding to station 7119, with 66 years ) was higher than the maximum generated with the 10 synthetic samples (0.81), so it was analyzed in more detail and was found to correspond to a record of 320 mm on October 5, 2005, when several high values were presented in the near stations, reason why it was decided to conserve that value.

Table 9 shows the coefficients of variation of each of the other 12 regions (of the 59 considered) in which the square ratio of the coefficients of variation is greater than 10.

Table 9 Variation coefficients of each station for the 12 regions (out of the 59 considered) in which the square ratio of the coefficients of variation is greater than 10. 

BCN	Campeche	Chiapas	Chiapas	Chihuahua	Coahuila	Durango	Hidalgo	Michoacán	Oaxaca	Puebla	Nuevo León
		Chicoasén	Almandro	Restantes			Menores		Altiplano	Centro	Reg. 1
Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.	Coef. var.
1.315	0.934	0.792	0.822	0.958	1.044	1.070	0.700	0.647	0.738	0.669	0.933
0.969	0.658	0.461	0.465	0.714	0.875	0.879	0.620	0.618	0.680	0.483	0.573
0.950	0.655	0.409	0.410	0.662	0.744	0.634	0.602	0.593	0.662	0.481	0.522
0.826	0.643	0.389	0.406	0.655	0.632	0.583	0.569	0.586	0.648	0.438	0.499
0.777	0.635	0.386	0.383	0.601	0.628	0.582	0.535	0.555	0.598	0.393	0.475
0.776	0.634	0.367	0.369	0.599	0.624	0.561	0.516	0.501	0.588	0.386	0.433
0.738	0.627	0.365	0.312	0.492	0.609	0.523	0.515	0.494	0.565	0.376	0.418
0.663	0.596	0.350	0.308	0.489	0.597	0.521	0.513	0.490	0.564	0.375	0.355
0.616	0.594	0.330	0.298	0.484	0.596	0.513	0.512	0.449	0.557	0.371	0.322
0.602	0.579	0.326	0.264	0.482	0.591	0.505	0.502	0.440	0.557	0.355	0.289
0.596	0.576	0.296	0.245	0.479	0.588	0.502	0.497	0.435	0.557	0.355
0.581	0.554	0.286		0.469	0.553	0.492	0.485	0.435	0.554	0.354
0.581	0.544	0.272		0.467	0.548	0.485	0.472	0.427	0.537	0.352
0.572	0.540	0.272		0.457	0.529	0.484	0.464	0.425	0.532	0.332
0.571	0.512	0.269		0.455	0.513	0.481	0.448	0.417	0.526	0.331
0.570	0.511	0.266		0.442	0.510	0.480	0.447	0.411	0.513	0.331
0.570	0.511	0.246		0.437	0.497	0.477	0.444	0.409	0.505	0.324
0.556	0.503	0.237		0.435	0.497	0.454	0.433	0.404	0.500	0.323
0.550	0.478	0.234		0.432	0.495	0.453	0.430	0.395	0.491	0.318
0.550	0.473	0.219		0.429	0.491	0.452	0.428	0.394	0.477	0.308
0.545	0.465			0.427	0.491	0.450	0.420	0.390	0.466	0.300
0.528	0.461			0.417	0.490	0.447	0.420	0.381	0.458	0.289
0.528	0.461			0.412	0.490	0.440	0.408	0.376	0.432	0.287
0.520	0.459			0.411	0.482	0.438	0.404	0.374	0.428	0.285
0.517	0.458			0.404	0.466	0.433	0.402	0.373	0.427	0.279
0.516	0.454			0.399	0.462	0.432	0.400	0.368	0.418	0.272
0.510	0.445			0.398	0.444	0.431	0.389	0.365	0.410	0.266
0.507	0.436			0.395	0.436	0.431	0.387	0.364	0.408	0.265
0.482	0.436			0.364	0.435	0.429	0.387	0.361	0.406	0.263
0.477	0.432			0.352	0.429	0.423	0.382	0.360	0.404	0.255
0.476	0.424			0.340	0.428	0.419	0.378	0.358	0.402	0.252
0.469	0.419			0.335	0.404	0.415	0.362	0.357	0.401	0.244
0.459	0.412			0.328	0.369	0.414	0.354	0.349	0.399	0.244
0.450	0.412			0.319	0.321	0.408	0.354	0.349	0.398	0.210
0.440	0.394			0.317	0.313	0.406	0.351	0.348	0.390
0.421	0.381			0.315		0.405	0.342	0.339	0.387
0.420	0.374			0.303		0.401	0.339	0.339	0.381
0.407	0.367			0.279		0.399	0.334	0.338	0.380
	0.366			0.263		0.393	0.330	0.333	0.373
	0.363			0.240		0.389	0.326	0.333	0.369
	0.360					0.385	0.324	0.333	0.368
	0.344					0.384	0.323	0.330	0.367
	0.335					0.379	0.319	0.329	0.366
	0.303					0.379	0.315	0.328	0.363
	0.288					0.373	0.309	0.324	0.363
	0.263					0.370	0.307	0.320	0.358
						0.367	0.306	0.317	0.354
						0.366	0.297	0.315	0.354
						0.365	0.293	0.313	0.353
						0.365	0.288	0.311	0.350
						0.362	0.285	0.310	0.348
						0.362	0.272	0.308	0.348
						0.361	0.271	0.307	0.348
						0.357	0.265	0.305	0.344
						0.357	0.260	0.303	0.340
						0.354	0.249	0.303	0.340
						0.354	0.229	0.301	0.340
						0.354	0.218	0.295	0.339
						0.350		0.293	0.338
						0.349		0.292	0.315
						0.346		0.290	0.312
						0.341		0.289	0.311
						0.339		0.288	0.309
						0.337		0.288	0.307
						0.334		0.284	0.304
						0.329		0.283	0.297
						0.325		0.281	0.295
						0.315		0.281	0.294
						0.299		0.280	0.287
						0.298		0.274	0.282
						0.298		0.272	0.282
						0.282		0.272	0.276
						0.279		0.267	0.275
						0.276		0.260	0.273
						0.275		0.257	0.272
						0.273		0.255	0.258
						0.266		0.247	0.253
						0.266		0.246	0.251
						0.260		0.244	0.241
						0.253		0.243	0.241
						0.251		0.241	0.202
						0.238		0.231	0.175
						0.228		0.231
								0.230
								0.229
								0.221
								0.220
								0.208
								0.207
								0.199
								0.198
								0.186
								0.180

For each of these 12 regions, 10 synthetic samples were also generated, each with the same number of values as the historical record. In Table 9, the historical coefficients of variation greater than 20% of the maximum obtained with the generated series were marked with green, which are greater or less than the extreme values obtained with the samples generated but in which the differences are less than 20%. It can be observed that of the 12 regions, in five of them the historical coefficient of variation exceeds the synthetic by more than 20% and in one there are two stations in which this occurs; there are also 13 stations (of more than 2000 studied) in which the historical coefficient of variation exceeds the synthetic one but in less than 20%, and eight in which the historical minimum is less than the synthetic one in less than 20%. For the cases that appeared to be the strangest, a request for verification was made to CONAGUA's Superficial Water Management and Engineering (GASIR), with the results shown in Table 10. According to the information provided by GASIR, the values of the coefficients of variation of table 8 are modified as follows:

Table 10 Results of doubtful data verification  

DOUBTFUL DATA ON CLIMATOLOGICAL STATIONS
State	Station	Value (mm)	Year	Month	Day
Baja California Norte	2046	240	1972	10	4	GASIR recommends elimination
	2046	170	1973	9	25	GASIR does not find information, it was eliminated
Campeche	4006	380	1950	10	8	GASIR says it is 38
	4006	200	1995	8	20	Correct
Chiapas	7033	400.9	1985	7	6	GASIR says it is 40.9
	7067	201.1	1970	7	5	correct
Puebla	21114	250	1963	7	3	correct
	21080	179.1	2000			is 17.9
		171	1999	9	13	is 17.1
Sonora	26034	407	1972	8	9	correct
Veracruz	30031	690	1978	7	10	GASIR does not find information, it was eliminated
Morelos	17045	200	1977	8	24	is 57.4, day 12th
Chihuahua	8005	320	1929			Eliminated, monthly maximum is 8 mm
Coahuila	5003	280	1976	9	19	Is 28
	5003	200	1978	9	27	GASIR does not find information, it was eliminated
	5003	280	1977	7	14	Is 20
Durango	10016	280	1997	7	27	GASIR does not find information, it was eliminated
	10033	250	1974	9	2	Correct
		240	1987	8	30	GASIR does not find information, it was eliminated
	10045	220	1978	8	30	correct
Nuevo León	19009	324	2010	6	31	Correct, Alex Hurricane
		243	2005	7	20	Correct, Emily Hurricane

The value of 1.315 in BCN changes to 0.817; the value 0.934 of Campeche passed to 0.74; the value 0.792 of Chicoasén is correct, the 0.822 of Almandro passed to 0.51; for Chihuahua disappears the 0.958, for Coahuila disappears the 1.044, for Durango disappears the 1.070 and the 0.879 passes to 0.755 and for Puebla center, disappears the 0.669. That is to say that with these corrections there are only two stations in which the historical CV exceeds the synthetic in more than 20% and 12 in which it exceeds but in less than 20%.

Conclusions and recommendations

In this work, the record for 2193 stations with more than 20 years of annual maximum rainfall was analyzed. The importance of an effort to do control quality to the annual maximum daily precipitation data obtained from the CLICOM database is shown. With the data cleared 59 regions were defined as homogeneous (about annual maximum daily precipitation). To define the regions, the coefficients of variation of the annual maximum precipitation recorded in each station were considered, but also their geographic location and the degree of exposure to extreme meteorological phenomena.

The hypothesis of homogeneity was verified with a method proposed in this work that is based on generating synthetic samples obtained from the distribution function of annual maximum rains modulated by dividing each value between the average maximums corresponding to each station.

The analysis of the records of the 2193 stations that fulfilled the stated conditions, allowed to arrive at results with which the values of precipitation associated to any period of return can be obtained in a reliable way for durations shorter than one day and for durations between one and thirty days, for any place of the Mexican Republic.

The method is based on obtaining a map with the values of the means of the annual maximum precipitation, the values of mean are considered very reliable because in all cases they were calculated with at least 20 years of registration. To obtain the values corresponding to different return periods, the values of the mean of each station are multiplied by regional factors obtained for 59 homogeneous regions.

A broad discussion is presented regarding the definition of the 59 regions considered homogeneous. This discussion shows, on the one hand, that the Fisher test is not enough to compare the coefficients of variation obtained in each region (mainly because it is designed for samples that follow a Normal distribution) and that a more complete analysis can be done by generating samples which correspond to the actual distribution function, so that the cumulative frequency curve of the CV obtained from these samples can be compared with that corresponding to the historical values.

To go from the values corresponding to durations shorter than one day Chen's theory was used, but adapted by Baeza to the conditions of Mexico. The results allow the estimation of design hyetographs for basins with concentration time less than one day. In the case of rains associated with durations greater than one day (required for the analysis of large basins), it was possible to obtain, for all the regions, factors that relate them to those of a day, in a way that can be obtainedhyetographs of design with daily values.

It is recommended to carry out studies throughout the republic (there are already some for specific regions) that allow to estimate factors of reduction by area, with which it is possible to pass from the point hyetographs to those referred to in the previous paragraphs to average hyetographs for each basin.

Detailed information on the different results of this study (maps such as Figure 6, distribution functions adjusted to the annual normalized maxima for each of the 59 regions, etc.) can be requested from the authors of this work.