Finite Element Method Application in Areal Rainfall Estimation Case Study; Mashhad Plain Basin

Introduction: The hydrological models are very important tools for planning and management of water resources. These models can be used for identifying basin and nature problems and choosing various managements. Precipitation is based on these models. Calculations of rainfall would be affected by displacement and region factor such as topography, etc. Estimating areal rainfall is one of the basic needs in meteorological, water resources and others studies. There are various methods for the estimation of rainfall, which can be evaluated by using statistical data and mathematical terms. In hydrological analysis, areal rainfall is so important because of displacement of precipitation. Estimating areal rainfall is divided to three methods: 1- graphical. 2-topographical. 3-numerical. This paper represented calculating mean precipitation (daily, monthly and annual) using Galerkin’s method (numerical method) and it was compared with other methods such as kriging, IDW, Thiessen and arithmetic mean. In this study, there were 42 actual gauges and thirteen dummies in Mashhad plain basin which is calculated by Galerkin’s method. The method included the use of interpolation functions, allowing an accurate representation of shape and relief of catchment with numerical integration performed by Gaussian quadrature and represented the allocation of weights to stations. Materials and Methods:The estimation of areal rainfall (daily, monthly,…) is the basic need for meteorological project. In this field ,there are various methods that one of them is finite element method. Present study aimed to estimate areal rainfall with a 16-year period (1997-2012) by using Galerkin method ( finite element) in Mashhad plain basin for 42 station. Therefore, it was compared with other usual methods such as arithmetic mean, Thiessen, Kriging and IDW. The analysis of Thiessen, Kriging and IDW were in ArcGIS10.0 software environment and finite element analysis did by using of Matlab7.08 software environment. The finite element method is a numerical procedure for obtaining solutions to many of the problems encountered in engineering analysis. First, it utilizes discrete elements to obtain the joint displacements and member forces of a structural framework and estimate areal precipitation. Second, it uses the continuum elements to obtain approximate solutions to heat transfer, fluid mechanics, and solid mechanics problems. Galerkin’s method is used to develop the finite element equations for the field problems. It uses the same functions for Ni(x) that was used in the approximating equations. This approach is the basis of finite element method for problems involving first-derivative terms. This method yields the same result as the variational method when applied to differential equations that are self-adjoints. Galerkin’s method is almost simple and eliminates bias by representing the relief by suitable mathematical model and incorporating this into the integration. In this paper, two powerful techniques were introduced which was applied in Galerkin’s method: The use of interpolation functions to transform the shape of the element to a perfect square. The use of Gaussian quadrature to calculate rainfall depth numerically . In this study, Mashhad plain is divided to 40 elements which are quadrilateral. In each element, the rain gauge was situated on the node of the stations. The coordinates are given according to UTM, where x and y are the horizontal and z, the vertical (altitude) coordinate. It was necessary at the outset to number the corner nodes in a set manner and for the purpose of this paper, an anticlockwise convention was adopted. Results and Discussion: This paper represented the estimation of mean precipitation (daily, monthly and annual) in Mashhad plain by Galerkin’s method which was compared with arithmetic mean, Thiessen, kriging and IDW. The values of Galerkin’s method by Matlab7.08 software and Thiessen, kriging and IDW by ArcGIS10.0 were calculated. The base of the comparison was isohyetal method, because it showed the relief and took into account the effect of rain gauges, therefore it could represent rainfall data and region condition completely. The most accurate method was isohyetal method in estimating mean precipitation. Cross-validation was usually used to compare the accuracy of interpolation method. In this study, root mean square error (RMSE) was used as validation criteria. Meanwhile, in the present study, the effects of altitude were neglected for two reasons. First, partial correlation coefficient of rainfall/altitude gradients was weak and second, the storms data were not accessible. Conclusions: In this study, the estimation of areal rainfall by Galerkin’s method was an innovative step. The case study was Mashhad basin (9909 km2) which included 42 rain gauges. Comparing other methods indicated that: Galerkin’s method was more efficient in comparison with arithmetic mean and it had more accurate results. Result of Galerkin’s method was similar to Kriging, IDW and Thiessen method. Unlike other methods, mesh of finite element could be used for calculating runoff, sediment and temperature and it did not need station weights. Even within one network the number of interpolation points can be varied, so that in a rugged region the number can be increased with little increase in effort, while in a more uniform region fewer are necessary.