Probable Maximum Precipitation Estimation in a Humid Climate

Due to the importance of probable maximum precipitation (PMP) for designing and planning hydraulic structures, 5 the aim of this study is the estimation of 24-hour PMP (PMP24) by using the statistical and physical methods in a humid climate of Qareh-Su Basin which is located in the northern part of Iran. For statistical estimate of PMP, the equations of empirical curves of Hershfield method were extracted. Then the standard and revised approaches of Hershfield method were written in JAVA programming language, as a user friendly and multi-platform application called the PMP Calculator. Convergence model was considered to calculate PMP by physical method. The depthareaduration (DAD) curves were 10 extracted to estimate PMP24 using physical method and then PMP24 was estimated for each storm. The results showed that for the standard and revised approaches, Km was found to be varied the range of 17-18.0 and 2.2-5.3, respectively. The maximum values of PMP24 for the first approach was obtained 447.7 mm and for second approach was 200.7 mm. Using the physical method, PMP24 was 143.1 mm. The results of this study will be helpful for planning, designing, and management of hydraulic structures and water resources projects in the study area. 15


Statistical Hershfield's method 20
The procedure as developed by Hershfield is based on the general frequency equation (WMO, 2009;Chow, 1951). The equation of this method as follows: Nat. Hazards Earth Syst. Sci. Discuss., https://doi.org/10.5194/nhess-2018-38 Manuscript under review for journal Nat. Hazards Earth Syst. Sci where X PMP is the PMP estimate for a certain station at the particular duration, ̅ and are the average and standard deviation of the annual extreme series for a given duration, respectively. K m is frequency factor (K m ). In this approach, K m is calculated by K m chart given in the manual for estimation of probable maximum precipitation (WMO, 2009, Fig. 4.1, Page 65) based on duration and average of annual maximum rainfall that these curves were extracted based on records of 24-hour rainfall for some 2700 stations in the climatological observation of America. The standard approach of Hershfied method is 5 modified by Desa et al., (2001) in Malaysia. In modified approach, K m is calculated by the following Eq. (2): where is maximum observed rainfall data, ̅ and are the average and standard deviation of the annual extreme series without the largest value, respectively. First, the parameters in Eq. (1) are estimated. Next, the K m values for all the stations are mapped against the ̅ values respectively and a smooth envelope curve is drawn. The K envelope value is picked up from the curve for each station's ̅ . The value of PMP for each station is then estimated using Eq. (1) by 10 replacing K m with K envelope value (Alias and Takara, 2013).

Physical Method
There are two synoptic models namely the mountainous and convergence models to calculate PMP (Joos et al., 2005). The convergence model is based on physical characteristic of storm, i.e. dew point temperature, wind speed, wind direction and etc. The main steps to calculate PMP, using the convergence model are the selection of severe storms, producing the depth-15 area-duration (DAD) curves, moisture maximization and wind maximization. Severe and widespread storm is a weather condition that leads to produce precipitation in all stations in the basin and even around the basin. The most severe and widespread storms are selected based on maximum discharge and maximum 24 hours rainfall data. Producing isohyets maps are one of the main steps in preparation of DAD curves. Using an analysis of the storms, DAD curves can be obtained. DAD curves are also applied to generalized relations for other areas or other basins with similar climate and topographic 20 characteristics. The first step to develop DAD curve is collecting the precipitation data for all areas in the storm.

Moisture maximization
The moisture maximization method is one of the acceptable procedures to calculate PMP for specific areas and consists of moisture maximization adjustment to observed areal rainfall values associated with severe storms (Rakhecha and singh, 2009). This method assumes that the atmospheric moisture would hypothetically rise up to a high value that is regarded as 25 the upper limit of moisture and the mentioned limit is estimated from historical records of dew point temperature. After selection of severe and widespread storms and calculation of average rainfall depth for study area, it is necessary to calculate maximum humidity source in order to maximize selected storms. By converting mean monthly pressure data at each station to 1000 mb pressure level, the effect of topography could be ignored. Dew point temperature and maximum 12-hour where W m is the maximum precipitable water in the 1000 to 200 mb levels, which can be obtained on the basis of the maximum 12-hour duration dew point with 50-year return period and W s is the maximum precipitable water at 1000 to 200 mb levels which can be obtained on the basis of maximum 12-hour duration dew point in a simultaneous period with storm 5 (WMO, 2009).

Wind maximization
Wind maximization is most commonly used in orographic regions when it appears that observed storm rainfall over a mountain range might vary in proportion to the speed of the moisture-bearing wind blowing against the range. The wind maximization ratio is simply the ratio of the maximum average wind speed for some specific duration and critical direction 10 obtained from a long record of observations, e.g. 50 or 100-years, to the observed maximum average wind speed for the same duration and direction in the storm being maximized. The wind speed maximization factor (MW) is defined by Eq. (4).
where MW 1 and MW 2 are the maximum wind speed with 100-year return period and the maximum persisting 12-hour wind speed during the storm, respectively (WMO, 2009).

Statistical approaches
The focus of the study is the calculation of PMP 24 by using statistical and physical method in north of Iran. In order to calculate PMP using the statistical method, the equations of adjustment factors of Hershfield method were extracted, based on the coefficient of determination (R 2 ). Adjustment factors that are applied in statistical estimation of PMP values are K m , adjustment of average and standard deviation for maximum observed event and for sample size, adjustment for fixed 20 observational time intervals and area reduction curves. These equations permit estimation to be carried out rapidly by using a computer. The 24-hour duration K m was gained by Eq. (5).
-8 3 -5 2 m K = -5×10 x +8×10 x -0.052x +19.794 (5) where x is 24-hours mean annual maximum rainfall (mm). Thus, a user friendly and multi-platform JAVA application, which is called PMP Calculator, was developed that is shown on Fig. 2. This application was supported by all operating system such as Windows, Linux, and Macintosh OS X. Also, in order to compare PMP in all stations, this application 25 calculates the ratio of PMP to maximum depth of rainfall as criteria independent of climatic conditions. result of PMP 24 using statistical method in study area, which was calculated by PMP Calculator application. Fig. 3 shows PMP 24 isohyetal map by using standard and revised approach in study area.

Physical method
In this study based on maximum discharge and the daily rainfall data with 24-hours duration obtained from Iran water 5 resources management company and IRIMO as a reliable source, 8 storms were selected as the most severe and widespread storms during 1981 to 2013. The date of occurrence these storms have been given in Table 3. After selection of severe and widespread storms, the isohyet maps for each storm were plotted in ArcGIS 9.3. To produce the DAD curves, the area bounded by each isohyet line was calculated in ArcGIS 9.3. Based on Fig. 4 that shows the spatial distribution of precipitation during the storm of September 2008 as one of the most severe storm, the greatest amount of precipitation 10 occurred over the western parts of basin that is nearest to sea, whereas the smallest amounts of precipitation occurred over the eastern parts of basin. Also, Fig. 5 shows DAD curve for the storm of September 2008. Table 4 shows the moisture and wind speed maximizations at 1000 mb for selected storms in Gorgan station.   Table 5 shows the PMP values estimated by physical method for selected storms over the study area. Using the PMP Calculator application, PMP can be calculated by two approaches of Hershfield method for durations such as 5 minutes, 1, 6 and 24-hour durations. In this study, maximum 24 hours duration rainfall values for selected stations 5 located in north of Iran with record length of 33 years was adopted to estimate the appropriate K m values. The results showed that for the standard approach, K m was found to be varied the range of 17 and 18. The minimum and maximum values for point PMP 24 were 232.4 and 447.7 mm. In revised approach, just the maximum values were considered and caused a severe and perceptible decrease in K m values which were more rational. Therefore, the corresponding values of K m for this approach ranged from 2.2 to 5.3 and the minimum and maximum values for point PMP 24 were 111.4 to 200.7 mm. The ratio of areal 10 PMP 24 to maximum of 24 hours precipitation (P 24 ) max , as a criteria independent of climatic conditions was used to compare PMP 24 in the study area. The maximum and minimum value of the ratio of PMP 24 to (P 24 ) max for standard approach was obtained 2.8 and 4.3 whereas these values for revised approach was obtained 1.4 and 1.9. Finally, based on revised method, the maximum K m of Hershfield equation in the study area was found to be 5.3. The approximated K m is in accordance with corresponding research in the Atrak watershed (Ghahraman, 2008) and (Desa et al., 2001;Desa and Rakhecha, 2007) 15 Malaysia. Much research has been done on K m in standard approach but all of them lead to high estimation of PMP. In the revised approach, just the maximum values were considered and caused a severe and perceptible decrease in K m values Nat. Hazards Earth Syst. Sci. Discuss., https://doi.org/10.5194/nhess-2018-38 Manuscript under review for journal Nat. Hazards Earth Syst. Sci. Discussion started: 16 February 2018 c Author(s) 2018. CC BY 4.0 License.