Daily rainfall interpolation of Mashhad Drainage basin

Daily Rainfall estimation usually performed with classical interpolation methods (Dingman,2002).To have a responsible accuracy in using new geostatistical methods, and neural networks methods we need a dense distributed stations (Goovaerts,2000º Rahimi-BondarAbadi and Saghafian, 2007). However, Modified Inverse Distance Method(MIDW) can be used in mountainous areas with low density (LO,1992(.Elevation to the distance ratio (with equal power) appears in MIDW. MIDW-F is the advanced version of MIDW that considers the elevation and distance as the inverse with unequal power (m and n). It is analyzed with fuzzy mathematics and is optimized with Genetic Algorithm (GA) (Chang et al, 2005). The purpose and innovation of this paper is to provide MIDW-F with a new alignment of MIDW-F which named GMIDW-F.
2 - 2.1 Study area and data :The study area is Mashhad Drainage basin (dry and semi-dry climate) with longitude 58° ,20´ to 60°,8' Easting and Latitude 36°-0' to 37°-5' Northing(North East of Iran) with total area of ​​9909.4 km2. Number of rain gauges within and adjacent the area are 49 with over a period of 16 years combined(1993-2009). 215daily rainfall (at least 50% of the stations have rainy day at the same time) was used for modeling in this study.
2-2 Modified inverse distance method Based on Fuzzy Mathematics
MIDW method considers the ratio of elevation(h) to distance(d) with equal power (LO, 1992). Advanced version of this is MIDW-F (Eq.8) that powers are unequal (Chang et al, 2005. The weights of elevation and distance(Eqs.1 and 2) are fuzzy. and are the Fuzzy membership functions d,and. and are the membership degrees. They can be integrated with the fuzzy operators, minimum, maximum, multiplied and sum of squares (Eqs.3 to 7) (Vahidian-Kamyad and Tarqyan, 2002). The phrase is integrated weight. We can consider the role of elevation directly in these area . We applied two different alignments to MIDW-F which named GMIDW-F method (Eq.8). If weights(h and d) appear in reverse (as), it was named GMIDW-F(1). The caseis named GMIDW-F(2).(1)(2) (3) (4) (5) (6) (7) (8) GMIDW-F equation
2.3 Genetic Algorithms:The GA is useful to estimate and optimize the parameters m and n of equation 8. The error function is regional sum of absolute errors(RSAE).
2.4 Data screening and normalization :Reforming data due to wrong registration, incorrect transmission, system failure, etc. is called screening. The normalization is for unification the scales of elevation and distance (Eqs 9 and 11). If the role of elevation is assumed to be negative, normalized by Eq.(10) and in direct mode can be done with Eq.(11) (Chang et al, 2006). (9) (10) (11)
3 - The MIDW-F considers elevation and distance inversely with unequal powers (m and n) in MIDW. We added a new alignment elevation to the distance ratio (GMIDW-F). Optimization of m and n was conducted for 215 daily rainfalls. Rainfalls were classified into 5-10, 10- 20, 30-40, 40-50 etc (in mm). Screening and normalization were also performed. Integration was examined with five fuzzy functions(Eqs. 4 to 8). GA is applied to optimize the parameters.
RSAE for each equation and for each category was calculated(Eq. 10, Tables 1 and 2). This classification did not show any specific results. Contribution of minimum and multiply operators is more frequenty (Table 2). Some statistical features of RSAE increase with rainfall classification(Table 3).
Without classification the optimum function was obtained in 66% of cases with and 34% of cases with. The Best operator was minimized (57%) and then multiplied (31%) (Tables 1, 2 and 4). The multiplication operator showed that in 76% of cases the effect of elevation and distance are inversed when and in 24% of the cases the effect of distance is direct while elevation effect is inverse when (Tables 2 and 4). The zoning of a daily precipitation (11/04/2009) by GMIDW-F and IDW methods were compared in a graph with RSAE values of 213 and 252 (in mm) respectively. By using IDW method, precipitation was estimated zero when it was at least 7(in mm), so it is overestimate, while it was estimated 1.5 mm by at the same values. It could be concluded that zoning by GMIDW-F provides better results than IDW method.
4 - The results of analysis showed that the minimum and multiplication operators are the best (Table1). Type of alignment is effective. Function improved in 66% of cases by applying GMIDW-F(1) and 44% of cases by applying GMIDW-F(2). The best function and alignment is determined by h and d. The classification does not affect for choosing the Fuzzy operator (Table1). It can be concluded that there is no restriction for parameters, classification is ineffective, the minimum and multiplication operators have priority and the alignment of h and d should be considered.