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## Tecnología y ciencias del agua

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*versión On-line* ISSN 2007-2422

### Tecnol. cienc. agua vol.5 no.3 Jiutepec may./jun. 2014

Notas técnicas

**Assessment of obtaining IDF curve methods for Mexico**

**Evaluación de métodos de obtención de curvas IDF para México**

**Francisco Manzano-Agugliaro*, Antonio Zapata-Sierra, Juan Francisco Rubí-Maldonado**

*Universidad de Almería, Spain*

* Corresponding author

** Quetzalcoatl Hernández-Escobedo**

*Universidad Veracruzana, Mexico*

**Institutional address of authors**

*Dr. Francisco Manzano Agugliaro*

*Dr. Antonio Zapata Sierra*

*Dr. Juan Francisco Rubí Maldonado*

Dpt. Engineering

Universidad de Almería

La Cañada de San Urbano

04120 Almeria (Spain)

Teléfono: +34 (950) 015 693

fmanzano@ual.es

ajzapata@ual.es

rubimal@ual.es

*Dr. Quetzalcóatl Hernández Escobedo*

Facultad de Ingeniería

Universidad Veracruzana

Campus Coatzacoalcos

Av. Universidad Veracruzana km 7.5,

96535 Col. Santa Isabel

Coatzacoalcos, Veracruz, México

Teléfono: +52 (921) 2115 700, extensión 59223

qhernandez@uv.mx

Received: 20/06/12

Accepted: 25/10/13

**Abstract**

This paper assesses the methods of obtaining IDF curves for the country of Mexico: modified Wencel, Chen, modified Chen, Témez and modified Témez. The data came from a total of 63 automated weather stations distributed throughout the country, recording data every 10 minutes for a minimum of 7 years. For the analysis, stations 50 km or less from the coast were identified as coastal and the remaining as inland. For each station, all of the parameters for the methods mentioned to calculate the IDF curves were evaluated for durations of 10 minutes to 24 hours, and return periods of 2 to 500 years. It was shown that when rainfall records for 10 minutes or less are used the Wencel method is recommended, and when the records are hourly the Chen method is recommended. When rainfall data are daily for durations under 2 h, the modified Temez method is required, and for durations of more than 2 h the best method is the modified Chen for inland areas and modified Temez for coastal areas.

**Keywords**: Mexico, IDF, extreme rainfall, coastal, inland, Wencel, Chen, Témez.

**Resumen**

En este trabajo se evalúan los métodos de obtención de curvas IDF para México: Wencel modificado, Chen, Chen modificado, Témez y Témez modificado. Los datos proceden de 63 estaciones automáticas (EMAS), distribuidas por todo el país, con registros cada 10 minutos y durante siete años como mínimo. Para el análisis se han diferenciado estaciones de costa cuando están a 50 km o menos de esa zona, y las demás como de interior. Se han valorado para cada una de las estaciones, todos los parámetros de los métodos de cálculo de curvas IDF mencionados, para duraciones entre 10 minutos y 24 horas, y para periodos de retorno de 2 a 500 años. Se ha comprobado que cuando se tienen registros de lluvia cada 10 minutos o menos, se recomienda el método de Wencel; cuando se tienen registros de lluvia horarios, se aconseja el método de Chen; cuando los datos de lluvia son diarios, para duraciones menores de 2 h, se necesita el método de Témez modificado; para duraciones de más de 2 h, el mejor es el de Chen modificado para las zonas del interior y Témez modificados para las zonas costeras.

**Palabras clave**: México, IDF, lluvia extrema, costa, interior, Wencel, Chen, Témez.

**Introduction**

The dimensioning of hydraulic structures is based on the design flood (Singh and Hao, 2011). The level of desired performance is often determined by the potential damage and severity of weather hazards that could cause failure, malfunction or overflow structure in question (Soro *et al*., 2010). Thus, in the case of stormwater management, the dimension of various components of the infrastructure system (case of pipes and canals sanitation) is based on the return period of heavy rainfall events (Monhymont and Demarée, 2007; Segond *et al*., 2007). This information is often expressed as Intensity-Duration-Frequency (IDF) curves obtained from a statistical study of extreme events.

For the country of Mexico there is a map of IDF curves developed by the Secretaría de Comunicaciones y Transportes (SCT, 1990); in the literature are studies such as Campos (1990) who obtained IDF curves Cazadero, Zacatecas, applying the equations of Bell and Chen, widespread heavy rains; also Pereyra-Díaz *et al*. (2004) in a preliminary study, adjusted equations Sherman (1931), Wenzel (modified by Chow *et al*., 1988) and Koutsoyiannis *et al*. (1998) to the intensities of 11 severe storms recorded during the period 1999-2002. All of these studies show the need for continuous records of precipitation for major cities, to use extreme rainfall in urban design.

The aim of this paper is to assess the different procedures for obtaining the IDF relationships for Mexico, based on two approaches: the reference method and empirical method, in order to first determine whether there are differences in behaviour between coastal and interior geographical areas, and if so, which model is best suited to each zone, depending on rainfall data available.

**Data and methods**

*Data*

To assess the different methods for obtaining IDF ratios in coastal and inland areas, records were used from the network of automatic weather stations (EMAs) administered by the General Coordination of National Weather Service (CGSMN) with satellite transmission. This network has 133 automatic weather stations installed throughout the country. The age of the series of records of this network of stations is variable depending on the station, so 63 stations have been selected with record set which are limited between 1999 and 2008, see figure 1. The choice of these 63 stations have been allowed for this work with data sets a minimum period of 8 years for those 86%, increasing to 90% when the minimum age of the series is 7 years. Works realized in other countries also use short lengths of series to analyze these phenomena if they do not arrange of longer series, for example Lam and Leung (1994) in Hong Kong (China) or Zapata-Sierra *et al*. (2009) in Spain, that a similar length of the series gave similar results to longer series. For Mexico, Escalante y Reyes (2004), have observed that for records longer than 20 years, the *R* (ratio of rain for 1 h to 24 h) becomes stable, and Mendoza-Resendiz *et al*. (2013) use series of data of 7 years length for the calculation of synthetic rains.

Precipitation records used in this work are made by the height of precipitation (in mm) in 10 minutes (GMT) for each station, for each month and year of the study period. Thus, we have had a total of 105, 120 records per station and year. Table 1 lists the stations included in the study, and table 2 offers their classification in the coastal (C) or inland (I) zone and the period of data used. Figure 1 shows the spatial distribution of the 63 automatic weather stations on the country of Mexico.

*Intensity-duration-frequency analysis*

Some authors propose the use of double Gumbel distribution in areas where there is possibility of rain with two different generation mechanisms (Guichard-Romero *et al*., 2009). But since in the central regions of Mexico has found a better fit for the Gumbel distribution (Domínguez-Mora *et al*., 2013), and that in the case of short data series, Gumbel distribution gives good results (Tung and Wong, 2013), this one has been chosen. Figure 2 shows an adjustment to the Gumbel distribution for observed data (annual maximum) at Acapulco station. At each station, frequency analysis was carried out using the maximum annual rainfall for each of the rainfall durations selected, by fitting each series to a Gumbel distribution using the maximum-likelihood method (Zapata-Sierra *et al*., 2009).

For the return periods *T* = 2, 5, 10, 25, 50 and 100 years, the rainfall-height values, *R*_{t}^{T} , were obtained for each rainfall duration considered, *t*, and the corresponding intensities, *r*_{t}^{T}.

*Intensity-duration-frequency relationships*

IDF relationship can be described mathematically by means of various expressions (Wenzel, 1982). The most common one, also called reference method, which groups the various intensity-duration curves for the various return periods in a single formula, is equation (1), which is applicable to locations with observatories keeping records for rainfall durations between 10 min and 24 h:

where *r*_{t}^{T} is the mean intensity (mm h^{-1}) for the duration *t* (min) and the return period *T* (years), and *a, b, c* and *d* are parameters to be determined by fitting.

In cases where only 24 h rainfall data is available, regional rainfall characterization studies are carried out analyzing the ratios between short-lasting rainfall and rainfall over 1 h and/or 24 h (Bell, 1969; Chen, 1983; Froehlich, 1993 and 1995). Using isohyetal rainfall maps for large regions of the USA, Chen (1983) obtained a ratio between the rainfall height for 1 h and 24 h, regardless of the return period, (*R*_{1}^{T}*/R*_{24}^{T}), that varies very little according to the geographical location, ranging between values of 0.1 and 0.6, with an average value of 0.4.

The equations used in this work are, Chen (1983):

where *a = a*_{1}*r*_{1}^{T}, *b = b*_{1} and *c = c*_{1}. The fitting parameters *a*_{1}, *b*_{1} and *c*_{1} can be obtained from the known rainfall data from a given station by using optimization techniques and the least squares method.

Chen modified equation:

where *x* = *R*_{24}^{100} / *R*_{24}^{10}. These equations allow us to obtain the IDF ratios from 24 h rainfall data, Témez, 1987:

where *t* is rainfall duration in h.

Témez modified (Zapata-Sierra *et al*., 2009):

where the coefficients α_{1} and β_{1} can be determined by using optimization techniques based on the observed intensity data.

**Results**

Each equation parameters studied was obtained by optimization techniques, being minimum square error for data from each station using the frequency analysis, here in after observed data.

First, Wenzel's equation (1) was fitted to the rainfall-intensity data obtained from each station by means of frequency analysis ("observed data"), obtaining values for the parameters *a*, *b*, *c*, and *d* expressed in table 1. We then proceeded to estimate the rainfall-intensity values for the different durations and return periods by applying Chen's equation (2) with the coefficients *a*_{1}, *b*_{1} and *c*_{1}, determined by using optimization techniques; Chen's modified equation (3), applying the coefficients *a*_{24}, *b*_{24} and *c*_{24}; and Témez's equation (4) and its modified equation (5), optimizing the parameters a_{1} and b_{1}. The values for the parameters in equations (1), (2), (3) and (5) determined by optimization are shown in table 1.

In order to compare the estimates made by each procedure, we defined a coefficient of variation (CV) as the ratio between the square root of the mean squared error and the mean of the rainfall values observed:

where *x*_{i}_{0} are the values obtained for the rainfall heights of the different rainfall durations and return periods, *x*_{ic} are the rainfall heights calculated for the different durations (10 min to 24 h) and return periods (2 to 100 years), and *n* is the number of rainfall data employed for each equation.

The equations of Témez modified (5) and Chen modified (3) require the use of parameters calculated for a nearby area. This may be done using the parameters obtained in this work (table 1).

Table 2 shows the values of the coefficients of variation (CV) (in bold indicate the lowest CV obtained in each EMA) obtained with the different expressions to generate the complete set of 10 minutes to 24 h the heights of rain for different return periods. We observe that equations (1), (2) and (5) are those with a greater number of stations with minimum CV value. Témez equation (4) is the worst result offers, surpassing in some cases the CV values of 0.1.

The figures 3 and 4 show the average values of CV obtained at coastal and inland stations, obtained for the two periods studied, less than 2 hours and less than 24 hours. In all cases, seen as the Témez equation gets higher CV, and then it is not recommended for use without particularization proposed in equation (4).

The results obtained for durations between 10 minutes to 24 hours and for a return period between 2 to 500 years, due the length of data was 7 years, the supported validity for these estimates is limited to the duration of one series of data. But where there is no other information, here is provided some guidance for hydrological design.

**Conclusions**

The conclusions obtained in this work for data lengths of at least seven years were as follows:

Wencel´s method generally shows the best results as expected, which justifies their use when available short-term rainfall data such as every 10 minutes, which is not always possible. For these cases, this work shows that equations are more appropriate depending on the country's geographical area of Mexico, coast or inland, where rainfall data are available for longer.

Chen's equation gives very good results for rainfall durations between 2 h and 24 h, but requires data of maximum rainfall in one hour. This data can be more accessible but not widespread except for relatively modern stations. When only rainfall data available 24 hours, this is the most common situation, the estimation of rain of short duration (< 2 h) necessary to obtain IDF curves, then the best equations are Témez modified.

For durations longer than 2 h and for the coastal zone, the best equation is always Témez modified. While for the inland area depending on the duration of the rainfall to be estimated should be used: the equation Témez modified for durations less than 2 h and Chen modified for longer durations.

**References**

BELL, F.C. Generalized Rainfall-Duration-Frequency Relationships. *J. Hydraulic Division*. Vol. 95, No. HY1, 1969, pp. 311-327. [ Links ]

CAMPOS, D.F. Procedimiento para obtener curvas I-D-T a partir de registros pluviométricos. *Ingeniería hidráulica en* *México*. Vol. 5, núm. 2, 1990, pp. 39-52. [ Links ]

CHEN, C.L. Rainfall Intensity-Duration-Frequency Formulas. *Journal of Hydraulic Engineering*. Vol. 109, No. 12, 1983, 1603-1621. [ Links ]

CHOW, V.T., MAIDMENT, D.R., and MAYS, L.W. *Applied* *Hydrology*. New York: McGraw-Hill, 1988, 572 pp. [ Links ]

DOMÍNGUEZ-MORA, R., ARGANIS-JUÁREZ, M.L., MENDOZA-RESÉNDIZ, A., CARRIZOSA-ELIZONDO, E., ECHAVARRÍA-SOTO, B. Storms Generator Method that Preserves their Historical Statistical Characteristics. Application to Mexico City Basin Daily Rainfall Fields. *Atmosfera*. Vol. 26, No. 1, 2013, pp. 27-43. [ Links ]

ESCALANTE, C.A. y REYES, L. Influencia del tamaño de muestra en la estimación del factor de lluvia. *R.* *Información Tecnológica*. Vol. 15, No. 4, 2004, pp. 105-110, doi: 10.4067/S0718-07642004000400015. [ Links ]

FROEHLICH, D.C., Short-Duration-Rainfall Intensity Equations for Drainage Design. *J. Irrigation and Drainage* *Engineering*. Vol. 119, No. 5, 1993, pp. 814-828. [ Links ]

FROEHLICH, D.C. Intermediate-Duration-Rainfall Intensity Equations. *Journal of Hydraulic Engineering*. Vol. 121, No. 10, 1995, pp. 751-756. [ Links ]

GUICHARD-ROMERO, D., DOMÍNGUEZ-MORA, R., FRANCÉS-GARCÍA, F. y GARCÍA-BARTUAL, R. Análisis de la densidad de estaciones en zonas de lluvias convectivas. Caso del Mediterráneo español. *Ingeniería* *Hidráulica en Mexico*. Vol. 24, núm. 3, 2009, pp. 35-49. [ Links ]

KOUTSOYIANNIS, D., KOZONIS, D., and MANETAS, A. A Mathematical Framework for Studying Rainfall Intensity-Duration-Frequency Relationships. *Journal of Hydrology*. Vol. 206, No. 1-2, 1998, pp. 118-135. [ Links ]

LAM, C.C., and LEUNG, Y.K. *Extreme rainfall Statistics and* *Design Rainstorm Profiles at Selected Locations in Hong Kong*. Hong Kong: Royal Observatory, 1994. [ Links ]

MENDOZA-RESENDIZ, A., ARGANIS-JUAREZ, M., DOMINGUEZ-MORA, R., and ECHAVARRIA, B. Method for Generating Spatial and Temporal Synthetic Hourly Rainfall in the Valley of Mexico. *Atmospheric* *Research*. No. 132-133, 2013, pp. 411-422 [ Links ]

MONHYMONT, B., and DEMARÉE, G.R. Intensity-Duration-Frequency Curves for Precipitation at Yangambi, Congo, Derived by Means of Various Models Of Montana Type. *Hydrol. Sci. J*. Vol. 51, 2007, pp. 239-253. [ Links ]

PEREYRA-DÍAZ, D., PÉREZ-SESMA, J.A. y GÓMEZ-ROMERO, L. Ecuaciones que estiman las curvas intensidad-duración-período de retorno de la lluvia. *GEOS*. Vol. 24, núm. 1, 2004, pp. 46-56. [ Links ]

SCT. *Isoyetas de intensidad-duración-frecuencia. República* *mexicana*. México, DF: Subsecretaría de Infraestructura de la Secretaría de Comunicaciones y Transportes, 1990, 495 pp. [ Links ]

SEGOND, M.L., NEOKLEOUS, N., MAKROPOULOS, C., ONOF, C., and MAKSIMOVIC, C. Simulation and Spatio-Temporal Disaggregation of Multi-Site Rainfall Data for Urban Drainage Applications. *Hydrol. Sci. J*. Vol. 52, 2007, pp. 917-935. [ Links ]

SHERMAN, C.W. Frequency and Intensity of Excessive Rainfall at Boston, Mass. *Trans. Am. S.C.E.* Vol. 95, 1931, pp. 951-960. [ Links ]

SINGH, V.P. and HAO, Z. Entropy-Based Probability Distribution for IDF Curves. World Environmental and Water Resources Congress 2011: Bearing Knowledge for Sustainability. *Proceedings of the 2011 World Environmental* *and Water Resources Congress*, Palm Springs, California, USA, American Society of Civil Engineers, 2011, pp. 1265-1272. [ Links ]

SORO, G.E., GOULA, BI T.A., KOUASSI, F.W., and SROHOUROU, B. Update of Intensity-Duration-Frequency Curves for Precipitation of Short Durations in Tropical Area of West Africa (Cote D'ivoire). *Journal of* *Applied Sciences*. Vol. 10, No. 9, 2010, pp. 704-715. [ Links ]

TÉMEZ, J.R. *Cálculo hidrometeorológico de caudales máximos en* *pequeñas cuencas naturales*. Madrid: Dirección General de Carreteras, MOPU, 1987. [ Links ]

TUNG, Y.K., and WONG, C.L. Assessment of Design Rainfall Uncertainty for Hydrologic Engineering Applications in Hong Kong. *Stochastic Environmental Research and Risk* *Assessment*. Vol. 27, No. 6, 2013, pp. 1-10. [ Links ]

WENZEL, H.G. Rainfall for Urban Stormwater Design. In *Urban Storm Water Hydrology*. Kibler, D.F. (editor). Washington, DC: Water Resources Monograph, 7, AGU, 1982. [ Links ]

ZAPATA-SIERRA, A.J., MANZANO-AGLUGIARO, F., and AYUSO-MUÑOZ, J.L. Assessment of Methods for Obtaining Rainfall Intensity-Duration-Frequency Ratios for Various Geographical Areas. *Spanish Journal of* *Agricultural Research*. Vol. 7, No. 3, 2009, pp. 699-705. [ Links ]