Fig. 1. Pluviometer.
Maximum Daily Precipitation for Aguascalientes, México
S. I. Martínez-Martínez
Table of Contents
Abstract: The objective of the study was to create maps of maximum precipitation in Aguascalientes for several small return periods. Precipitation data were statistically processed with a standard method. The maps were drawn with ArcGIS 8.2. The resulting maps can be used to design hydraulic works. Future work can be done to smooth the isohyets shown in the maps.
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Fig. 1. Pluviometer. |
Precipitation is measured every day in climatologic stations all over the world (see Fig. 1). The measured precipitation is recorded for posterior hydroclimatologic synthesis and analysis. The recorded information permits the computation of the maximum daily precipitation for the required frequencies, known as frequency analysis, which mainly consists in the assigning of a return period to a given precipitation. The maximum daily precipitation is of interest where it is important to study the adequate disposition of excessive floods, as in the case of the design of hydraulic works, i.e. small dams, culverts, detention ponds, and urban storm water works. Some regions in Mexico, like the State of Aguascalientes (see Fig. 2), lack sufficient processed information on maximum daily precipitation.
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| Fig. 2. Map of México. |
Frequency analysis of hydrological data is believed to be first used at the end of the nineteenth century (Chow 1964). In Mexico there has been systematic recording of climatologic information since 1930 (Quintas 2000). The Mexican climatologic stations record nine variables, one of which is daily precipitation (Quintas 2000). In the State of Aguascalientes, 68 stations have been in operation in different periods since the early 1940s (Quintas 2000). Maps and tables of different climatologic variables are produced by several Mexican governmental offices, such as the Communications and Transports Ministry SCT, the National Water Commission CNA, and the National Institute of Statistics, Geography, and Informatics INEGI. Several maps of intensity of precipitation are available for the state of Aguascalientes; each map corresponds to one specific duration ranging from 5 to 120 minutes and one of a number of return periods. The maps were produced for the design of culverts and internally used by the SCT. After several hydrologists have concluded that precipitation of diverse durations and return periods have consistent quotients with some referential precipitations, Campos (1984) proposed a method to calculate precipitation of any duration from 5 minutes to one or serveral days and with return periods of 2 to 100 years, based on the maximum daily precipitation. Later, Campos et al (1990) determined for 33 Mexican stations a quotient that can be applied to calculate precipitation of any duration or return period for each of these stations.
However, there is no geographic information about the maximum daily precipitation in the State of Aguascalientes, México. The purpose of this study was to produce a series of maps showing the maximum daily precipitation in Aguascalientes for several return periods, i.e. 2, 5, 10, 20, and 25 years. The maps, apart from their usefulness as descriptors of the spatial variation of the given height of rain, can be used for designing purposes. Each map can be utilized for the estimation of the maximum precipitation of its respective return period and for durations of 5 min to several days.
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In this study, the daily precipitation of 68 climatologic stations in the Mexican State of Aguascalientes was processed to obtain several maps. Data were obtained from ERIC II© (Fast Extractor of Climatologic Information II), a CD-ROM published by the Mexican Institute of Water Technology IMTA (Quintas 2000). The combined record period of the stations is from 1940 to 1998. The stations with a record length, n, of 10 or less years were not considered for further analysis because those record lengths do not permit good estimations of the maximum daily precipitation of the required return periods. The monthly maximum daily precipitation was determined for the recorded period of each station. The annual maximum daily precipitation series were obtained from the monthly maximum daily precipitation records of each station. The series of annual exceedance of each station was determined in three steps. First, all values of the monthly maximum daily precipitation record greater than the minimum value of the maximum annual series were taken. Second, the chosen values arranged in a descending order. Third, the first n ordered values were kept. The annual maximum and annual exceedance series were subjected to a standard statistical fitting procedure. This procedure fits a given series to a theoretical probability distribution or to a regression equation depending on the comparison between the length of the series (n) and the maximum required return period (Tmax). If n <= Tmax, the series is fit to a probability distribution otherwise; the series is fit to a regression equation. Then, the fit values were calculated for each station and the return periods of 2, 5, 10, 20, and 25 years. Fig. 3 shows an example of the fitting process.
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| Fig. 3. Data fitting for the station Calvillo. |
In general, the fit values of small return periods (T <= 10 years) calculated from a series of annual exceedance were greater that those calculated from a series of annual maximum. The contrary was the case for the fit values of big return periods (T >10 years). For a safe design of hydraulic works it was decided to adopt the greater value for each station and return period.
Several tailor-made computer programs were utilized for processing the precipitation. The programs ran in PCs and were written in Pascal for DOS. The statistical procedure followed for processing the precipitation can be consulted, for example, in Chow et al. (1988) or Martínez (2000).
Afterwards, a map of México was downloaded from the site of the Federal Highway Administration. A outline of Aguascalientes was extracted from the map. A digital elevation model (DEM) of Aguascalientes was obtained from the National Imagery and Mapping Agency. The DEM was used to construct a raster of elevations. Then, contour curves were calculated from the raster. The fit precipitation and corresponding geographical data were used to create the feature class stations. Other geographic data that were used to create the feature class towns were found in the web site of the INEGI. A base map was drawn with the outline, stations, towns, raster of elevations, and topographic contour curves. For each of the five return periods two maps were constructed in three steps from the base map. First, the attribute of the adopted precipitation of the feature class stations was interpolated to a raster to create a raster of precipitation. Second, from the raster a feature class of isohyets (curves of equal precipitation) was interpolated (first map). The isohyets showed a complex geometry dificult to read. For the maps to be useful it was necessary to provisionally correct the isohyets. Third, the correction consisted in to apply a focalmean operation to the raster of precipitation before obtaining the isohyets (second map). Complementary, the maps of the original isohyets show the raster of elevations, outline, and towns. The maps of smoothed isohyets show the raster of smoothed precipitation, outline, and towns. The maps were produced using the ESRI® ArcGIS™ 8.2 software.
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The base map is presented in Fig. 4. The map shows the stations, main towns, outline of Aguascalientes, contour curves, and a scale of colors to represent elevation.
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| Fig. 4. Base map. |
The maximum daily precipitation (MDP) maps obtained for return periods of 2, 5, 10, 20, and 25 years are shown in Fig. 5 through Fig. 9 respectively.
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| Fig. 5a. MDP isohyets (mm) return period of 2 years. | Fig. 5b. Smoothed MDP isohyets (mm) return period of 2 years. |
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| Fig. 6a. MDP isohyets (mm) return period of 5 years. | Fig. 6b. Smoothed MDP isohyets (mm) return period of 5 years. |
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| Fig. 7a. MDP isohyets (mm) return period of 10 years. | Fig. 7b. Smoothed MDP isohyets (mm) return period of 10 years. |
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| Fig. 8a. MDP isohyets (mm) return period of 20 years. | Fig. 8b. Smoothed MDP isohyets (mm) return period of 20 years. |
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| Fig. 9a. MDP isohyets (mm) return period of 25 years. | Fig. 9b. Smoothed MDP isohyets (mm) return period of 25 years. |
Initially, only 68 stations were considered for drawing the maps; but in order to get a better delineation of the isohyets near the state limit, it soon proved necessary to include another 11 stations located in the neighboring states. In virtually all the 79 stations it was necessary to discard one or more years of doubtful or incomplete record. Twenty-three stations were eliminated due to a corrected record length of less than ten years (Table 1). The 112 (2 x 56) series had, on average, 28.9 elements.
| Stations | Number | % Total |
|---|---|---|
| Used | 56 | 70.9 |
| Discarded | 23 | 29.1 |
| Total | 79 | 100.0 |
Several probability distributions were fit to the series. The Gumbel, log Pearson, Pearson, exponential, Weibull, and lognormal probability distributions were the better fit. It was also considered to fit a regression equation to the data of stations with records longer than 25 years. For a graphic comparison, see Fig. 10. The average and maximum mean quadratic error were 3.2 and 9.3 mm respectively. Detailed information about stations and maximum precipitation is shown in the Table 2.
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| Fig. 10. Distributions and regression vs number of series. |
Contrary to expectations, it was found that the topographic elevation was not correlated with the maximum daily precipitation. This can be seen in the plot of the maximum daily precipitation of return period of 2 years vs. the elevation, as shown in Fig. 11.
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| Fig. 11. Elevation vs MDP of return period of 2 years. |
The proposed maps (Fig. 5b to Fig. 9b) can be useful in designing hydraulic works. Let us assume that the precipitation of duration t minutes and return period of T = 2, 5, 10, 20 or 25 years in a place located in Aguascalientes is needed for durations from 5 minutes to several days. First, the maximum daily precipitation of return period of 2 years, P2d, can be obtained from Fig. 2. Second, P2d must be converted to maximum precipitation in 24 hours, P224, applying the equation (Campos 1984)
P224 = 1.13 P2d (1)
Third, once P224 is known, the maximum hourly precipitation of return period of 2 years, P21, must be calculated. For Aguascalientes, P21/P224 = 0.38 (Campos et al. 1990). Fourth, any precipitation P of duration t minutes and return period T years can be calculated with a formula, for example, the Bell’s formula (Campos 1984)
P = (0.35 Ln T + 0.76) (0.54 t0.25 – 0.50) P21 (2)
Valid for 5 <= t <= 120 minutes and 2 <= T <= 100 years. Fifth, the maximum daily precipitation can be read from the map of the return period T required and then converted to maximum precipitation in 24 hours,
PT24 = 1.13 PTd (3)
Sixth, a number of precipitations of return period of T years and durations from 5 to 120 minutes calculated with the equation 2 can be plotted in a logarithmic paper. The precipitation PT24 can also be plotted in the same paper. Then, a smooth curve can be drawn through the points. It is valid to draw a straight line from the point representing the precipitation for 120 minutes (2 hours) and PT24. It is possible to continue the straight line to 2 or 3 days. The resulting graph can be similar to Fig. 12.
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| Fig. 12. Precipitation-duration-return period curves. |
This procedure is applicable to a runoff area A equal to 25 km2 or less. If the watershed of interest has an area greater than 25 km2, it is necessary before obtaining P224 and PT24 to apply a reduction due to magnitude of watershed, Fr, adimensional. For example, applying the equation, simplified from Campos (1987),
Fr = 1 – 0.0935 (1 – e-0.00579A) (4)
In the equations 1 and 3,
P224 = 1.13 Fr P2d (1’)
PT24 = 1.13 Fr PTd (3’)
Finally, the smoothed isohyets in the maps show a rather complex geometry. In future studies, particularly when more and better data are available, attempts should be made to smooth, even more, the isohyets.
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The purpose of this study was to obtain a set of maps of maximum daily precipitation in Aguascalientes. The isohyets in the resulting maps show a complex behavior of the precipitation. This is opposed to an expected simpler and gradually varying behavior. A reason for the suspected distortion of the isohyets can be linked to the processed data. Although these data were carefully examined, there still existed the possibility that several maximum values were not taken into account for generating the statistical series. Specifically, a number of incomplete data years were not eliminated because it was assumed that they already had a maximum value. Another reason can be related to the estimation of parameters of the diverse probability distributions considered. The parameters were estimated only with one method, the method of moments. It is desirable to estimate the parameters with more than one method. Certainly, statistical tools more powerful than the ones used here can be applied to build better maps; in fact, it should be the case when more and better data are available. In conclusion, future work may be necessary to verify the geometry of the isohyets. Nevertheless, this study has generated new material that enriches the knowledge of the maximum daily precipitation and proposes a new tool for designing hydraulic works in Aguascalientes.
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The author would like to thank to Dr. David Maidment for his advice and support; Dr Susan Murphy, and Vaughan Mak for their careful editing of an earlier version of this paper.
Selected Bibliography and References
Campos, D. F. (1984). Processes of the hydrologic cycle. Autonomous University of San Luis Potosí, San Luis Potosí, México. (In Spanish).
Campos, D. F. (1987). "Model of precipitation-runoff events." PhD thesis, Faculty of Engineering, National Autonomous University of México, México. (In Spanish).
Campos, D. F., Gómez, R. (1990). “Procedure to obtain I-D-Tr curves from pluviometric records.” Hydr. Engrg. in Mexico, IMTA, II epoch, 5(2), 39-52. (In Spanish).
Chow, V. T. (1964). “Statistical and probability analysis of hydrologic data.“ Handbook of applied hydrology, V. T. Chow, ed., Section 8-I, McGraw-Hill Inc., New York, 8-1 – 8-42.
Chow, V. T., Maidment, D. R, Mays, L. W. (1988). Applied hydrology. McGraw-Hill Inc., New York.
Martínez, S. I. (2000). Introduction to surface hydrology. Autonomous University of Aguascalientes, Aguascalientes, México. (In Spanish).
Quintas, E. (2000). Fast extractor of climatologic information II. IMTA, México. (In Spanish).
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Last actualization: December 6, 2002.