Estimating precipitation intensity and its spatial distribution based on fractal theory (Case study: Tireh-Borujerd watershed)

Introduction
Estimating the amount and intensity of precipitation and its spatial distribution in various return periods is necessary for flood estimation hydrological models. This information is obtained based on traditional methods through intensity-duration-frequency curves with many assumptions, such as the choice of suitable distribution for each period, and the requirement of many parameters in different return periods. If a study area has incomplete data or a lack of data, traditional methods are limited. For this reason, the fractal method is used to transform the precipitation hyetograph in different durations and transfer the precipitation data from one place to another. The fractal method is a self-similar method; It means that every part of it is similar to the whole, like a pine tree, where every branch is like a whole tree. This method has high reliability, convenient access, and less number of parameters, which can be used to create daily precipitation data over long and short periods. It is noteworthy, that in the past valuable research has been done in the field of using the fractal theory to extract IDF curves. Nevertheless, in this research, in addition to extracting the characteristics of precipitation based on the above method in all the stations inside and outside the selected watershed, the spatial distribution of the rainfall intensity mapped based on the Co-Kriging method, and has been compared with the Ghahraman method.
 
Materials and Methods
The study area is the Tireh watershed in the Borujerd-Dorud region, which is located between east longitudes from ̍28˚48 to ̍17˚49 and north latitudes from 51˚33 to ̍35˚33. This watershed, with an area of 2127.28 km², is in the northernmost part of the large Karun River watershed and in the south of Oshtorinan town. The average rainfall in the mountains and plain areas has been estimated as 611.4 and 410.6 mm, respectively. The average annual temperature of the plain with an average elevation of 1493.3 m is 13.4 ̊C and in the highlands with an average elevation of 2025 m, it is 8.5 ̊C. In addition, the amount of evaporation in the highlands and plains is 1852.2 and 2148.8 mm per year respectively. In this research, the maximum intensity and amount of precipitation were estimated based on the fractal theory using the daily precipitation data for 12 stations with a statistical period of 31 years recorded from 1990 to 2021. The research method was conducted based on studies of Azhdary Moghaddam and Heravi, (2018) in the following steps. A) data extraction of the maximum amount of precipitation in different durations of 1, 2 and … days B) determining the maximum intensity of annual precipitation by dividing the maximum precipitation values by their durations C) calculating the weighted moment of the data ( ) in different orders (r) and durations (d) and then drawing linear graphs on a logarithmic scale, D) and then, using the related relationship, the maximum rainfall was calculated in the specified duration and return period. Since hourly precipitation data are not available in most of the stations, therefore, at this step, the IDF curves were extracted using the fractal theory and were compared with the experimental method of Ghahraman (which is based on the maximum daily precipitation) by the Pearson correlation coefficient. The Co-kriging method was used to create spatial distribution maps of precipitation intensity and amount based on fractal theory. The geostatistical Co-kriging interpolation method is similar to kriging and auxiliary variables can be used for better spatial analysis. Optimal spatial distribution maps of rainfall intensity and amount are provided by the existing point data extracted from the fractal method, and introducing different auxiliary layers such as maximum daily rainfall in the 24-hour continuity period.
 
Results and discussion
Fractal analysis of precipitation data showed that there is a linear relationship between scale power and moment order in all stations. Therefore, the maximum precipitation data in the study area have a mono-fractal nature, which means that by using the fractal theory, the precipitation data can be converted from one duration to another. The results of the density of precipitation zoning based on fractal theory using the Co-kriging method showed that the accuracy of interpolation increases with the increase of the return period. Indeed the calculated values have a suitable fitting with the observed values and are close to the fitted line. Contrary to this, the results of precipitation zoning based on the Ghahraman method using the Co-kriging method showed the most scattered points around the fitting line; which actually shows the low accuracy of this method in estimating and zoning the area precipitation. The results of the 24-hour rainfall interpolation error using the fractal method showed an increase in the RMSE value with the increase of the return period based on only the auxiliary variable of the rainfall intensity data produced by the fractal method. The RMSE was calculated based on adding auxiliary data such as the amount and annual average of precipitation and the value of the maximum one-day precipitation intensity of the original data to precipitation intensity prepared by the fractal theory. According to this, the RMSE in the return periods of 2, 5, 25, 50, 100, 200, and 300 years equals 0.09, 0.27, 0.74, 0.18, 0.25, 0.059, and 0.13, respectively, have a decreasing trend compared to the use of only auxiliary variable of precipitation magnitude. This composition has reduced the error criterion values to less than a fifth compared to the initial state (only by precipitation intensity covariate) in the return period of over 50 years.
 
Conclusion
The analysis of statistical moments showed that the precipitation maximum intensity data has a mono-fractal nature. In other words, the changes in the power of the scale are completely linear with respect to moment order, and it can be used to produce the data in different durations. The statistical analysis of the results of estimating the intensity of precipitation in the different return periods and durations with this method compared to the Ghahraman method showed that in most stations there is a significant relationship with a correlation coefficient of over 99 % at a confidence level of 99 %. Generally, the results of the spatial distribution error of precipitation intensity using the Co-kriging method based on fractal in the return periods of 2, 25, 100, and 200 years showed acceptably reduced interpolation error by adding different auxiliary data.