Study of the rainfall-runoff models using regression and optimization techniques

Abstract

Various models representing the rainfall-runoff relationship have been in existence and are also being modified. Some of them incorporate various intermediate processes. The majority of the existing models require continuous records not only of the observed rainfall and runoff values, but also of various other parameters. Some models use effective rainfall and actual surface runoff without the baseflow, which, in turn, requires the use of other parameters. However, in the absence of continuous observations of other parameters due to various constraints, as is the case in many developing countries, a model based on observed rainfall values and streamflow must be developed. Given that the rainfall-runoff relationship is non-linear and time-variant, the general relationship between the ithi^{th}ith period/day discharge and basin rainfall values Pi,Pi?1,…,Pi?nP_{i}, P_{i-1}, \dots, P_{i-n}Pi,Pi?1,…,Pi?n, with SMqSM_qSMq as a coefficient representing the influence of soil moisture and other properties, can be expressed as: Qi=SM0[C0(Pi)x0+C1(Pi?1)x1+?+Cn(Pi?n)xn]Q_i = SM_0 [ C_0 (P_i)^{x_0} + C_1 (P_{i-1})^{x_1} + \dots + C_n (P_{i-n})^{x_n} ]Qi=SM0[C0(Pi)x0+C1(Pi?1)x1+?+Cn(Pi?n)xn] where C0,C1,…,CnC_0, C_1, \dots, C_nC0,C1,…,Cn are the empirical constants and x0,x1,…,xnx_0, x_1, \dots, x_nx0,x1,…,xn are the exponents to be determined so that collectively they represent the catchment response in the best way. Since the coefficient SMqSM_qSMq may not be constant and may depend upon the antecedent rainfall, the modified form of equation (1) will be: Qi=G+SMq(Cq(Pi)xq+SMi(Pi?1)xi+…?)Q_i = G + SM_q \left( C_q (P_i)^{x_q} + SM_i (P_{i-1})^{x_i} + \dots \right)Qi=G+SMq(Cq(Pi)xq+SMi(Pi?1)xi+…) Each of the SMq,SMi,…SM_q, SM_i, \dotsSMq,SMi,… is a function of antecedent rainfall, and they correspond to the antecedent precipitation indices described by Kohler and Linsley, varying from catchment to catchment. Assuming each of them as a product function of previous rainfall values, e.g., SMq=SM0×SM1×?×SMnSM_q = SM_0 \times SM_1 \times \dots \times SM_nSMq=SM0×SM1×?×SMn After substitution and simplification, equation (2) can be written as: Qi=C+?j=0nCj×(Pi?j)xjQ_i = C + \sum_{j=0}^{n} C_j \times (P_{i-j})^{x_j}Qi=C+j=0?nCj×(Pi?j)xj Since the various terms are in the form of posynomials, the model developed is named the Posynomial Summation Type Model (PSTModel). Assuming influence history up to nnn periods, thus neglecting terms involving rainfall prior to the (i?n)th(i-n)^{th}(i?n)th period, and combining various exponents and constants, the equation (4) can be written as: Qi=C+?j=0nCj×(Pi?j)xjQ_i = C + \sum_{j=0}^{n} C_j \times (P_{i-j})^{x_j}Qi=C+j=0?nCj×(Pi?j)xj For calibration purposes, various steps for a simplified form and the complete development of the final governing discharge equation, representing the catchment response, have been presented along with the parameters evaluated during regression analysis. Path coefficient analysis has been used at every stage and has been illustrated with an example. Averaged/grouped rainfall values for different combinations of previous days have been used as independent variables. A grouping (1,2,4,8)(1, 2, 4, 8)(1,2,4,8) means: Pi,avg=(Pi?1+Pi?2)2+(Pi?3+…?)2P_{i, avg} = \frac{(P_{i-1} + P_{i-2})}{2} + \frac{(P_{i-3} + \dots)}{2}Pi,avg=2(Pi?1+Pi?2)+2(Pi?3+…) Data from seven catchments of sizes ranging from 126 km² to 1450 km², having different characteristics all lying in the State of Karnataka, India, have been used in the study. Various models have been developed for the Lakshmanateertha basin, which has the largest area and largest number of rain gauge stations. The first multi-linear model showed that the correlation coefficient becomes almost constant after a certain period, and the inclusion of rainfall values from additional previous days did not improve the model. In the next step, the variables were raised to different powers to find out if they could model the catchment better. It was observed that uniform powers yielded better results. The Nash-Barsi model was attempted with individual daily rainfall values up to 10 previous days, but the results were not encouraging. The general polynomial form was then tested on the grouped data, which involved many combinations. The general polynomial model, with various values of nnn (from 0 to 19), and uniform power varying from 0.1 to 2.0, showed that the best uniform power is 1.79 for this catchment, with correlation values from 0.85 to 0.869 for nnn ranging from 5 to 19. The PSTModel, using (1,2,4,8)(1, 2, 4, 8)(1,2,4,8) grouping, yielded a correlation value of 0.85. To improve further and explore the effect of adding other variables representing a longer history, models M2(1,2,4,8,16)M_2(1, 2, 4, 8, 16)M2(1,2,4,8,16) and M3(1,2,4,8,16,32)M_3(1, 2, 4, 8, 16, 32)M3(1,2,4,8,16,32) were developed, which yielded correlation values of 0.894 and 0.907, respectively. Model M3M_3M3 was compared with model M4(1,3,5,7,9,11)M_4(1, 3, 5, 7, 9, 11)M4(1,3,5,7,9,11), both having six variables, thus involving the same number of steps but covering different histories. Another model M5M_5M5, using a history of up to 120 days, did not show any improvement in results. Using model M2M_2M2, it was found that the estimated runoff is not very sensitive to the selection of the data period, although the final governing equation may have different coefficients and exponents. Work on six other catchments was undertaken mainly to test the applicability of the PSTModel. For Hemavathi, additional studies were done to find out whether the PSTModel with grouping behaves better than various other polynomial forms using the same grouping. The best uniform power developed by the general polynomial form is found to be 1.29 for this catchment, and the PSTModel with grouping yields better results than all other polynomial forms. Comparison of the proposed method for developing the PSTModel with Tyskin's method showed that the proposed method involves far fewer steps and results in similar correlation values. It has also been shown that PSTModels can help indicate various inconsistencies and discrepancies in the data used for the analysis. Though PSTModels generally produce good results, they do not perform better if the last variable has zero values. This fact is clear from the general governing equation of PSTModels, as Pi?nP_{i-n}Pi?n appears in every term. Another model, Path Coefficient Type Model (PCTModel), has also been developed. During its development, coefficients m0,m1,…m_0, m_1, \dotsm0,m1,… (corresponding to SMqSM_qSMq, etc., of the PSTModel) were assumed in a general polynomial form involving nnn antecedent rainfall values, e.g., m0=Cq+?k=1n(Pi?k)xkm_0 = C_q + \sum_{k=1}^{n} (P_{i-k})^{x_k}m0=Cq+k=1?n(Pi?k)xk The governing equation of the PCTModel with rainfall values restricted to nnn previous periods is: Qi=C+?j=0nCj×(Pi?j)xj+?k=j+1nCjk×(Pi?k)xkQ_i = C + \sum_{j=0}^{n} C_j \times (P_{i-j})^{x_j} + \sum_{k=j+1}^{n} C_{jk} \times (P_{i-k})^{x_k}Qi=C+j=0?nCj×(Pi?j)xj+k=j+1?nCjk×(Pi?k)xk As seen in equation (6), each period's direct contribution and the indirect contribution via all other periods have been separately taken into account. For this reason, the model performs better, as shown for all seven catchments using (1,2,4,8,16)(1, 2, 4, 8, 16)(1,2,4,8,16) grouping. Since CjC_jCj and CjkC_{jk}Cjk are similar to the direct and indirect path coefficients from path coefficient analysis, the model has been named the Path Coefficient Type Model. Since the regression analysis for the PCTModel involved too many combinations, a non-linear optimization technique has also been used for the exact evaluation of various parameters. For unconstrained optimization, the Gauss-Newton method has been used, and its specific application to the PCTModel has been presented, along with the formulation of various matrices and the algorithm to be used. The constrained optimization has been formulated using the Penalty Function approach to make the PCTModel physically realizable. For this purpose, the constraint placed on the various CjC_jCj, Cjj?C_{jj'}Cjj? coefficients is that they remain positive. Studies conducted to examine the effect of different sets of initial powers chosen for unconstrained optimization showed that the convergence of the objective function (?[Q?Qobs]2)\left( \sum [ Q - Q_{obs} ]^2 \right)(?[Q?Qobs]2) is faster for those input power sets with higher correlation values. Various input sets were initially developed through step-wise regression analysis. Regression analysis performed to develop a physically realizable PCTModel involved keeping a watch on the correlation values as well as on the regression coefficients. Only those terms were added to the equation that yielded a higher correlation value than the previous step and did not make any of the coefficients negative, although attempts were made to include as many terms as possible. The correlation values were found to be low for all the catchments. Similar trends (higher objective function values) were also observed during constrained non-linear optimization. The effects of various parameters like the relaxation coefficient, penalty, etc., have also been studied during constrained optimization, along with the lower bound values (of the computing system). A very high penalty imposed if constraints are violated results in a singular matrix. From various exponents and coefficients developed up to 40,000 cycles, it was found that very few terms remain significant. The objective function values obtained by constrained optimization are much higher compared to those obtained by unconstrained optimization. Different types of comparative studies have been undertaken. The comparison of different terms of a PCTModel has been done to identify which term contributes more than others, and under what conditions. Similarly, comparisons have been made for the corresponding terms' effects for different catchments. A comparison between the Curve Number method, PSTModel, and PCTModel for the Malathi catchment showed that the PCTModel produces better results (and even better with unconstrained non-linear optimization). For comparison of different catchments' behavior, a new term, "apparent saturation value," has been introduced. This is defined as the value of uniform rainfall, which, when applied to a catchment for a specific duration, results in the same runoff at the end of the period. The apparent saturation values have been found to correspond to the various characteristics of the general soil types existing in different catchments.