Hydrologic models are useful tools for simulating water resources and flux(contain Runoff and evaporation) variations. Determining hydrological model parameters is primary condition for good simulating of hydrological process. With the estimated values of this parameters via the calibration process, the model can well simulate the natural system. The success of the existing models to simulate reality does not depend on the complexity and number of parameters. The structure of the model, identify influential parameters and calibration method can have a significant impact on improving the model outputs.

Optimization methods are from the useful automatic calibration methods. Practical experiences of hydrological models calibration have shown that single objective optimization methods often not enough to measure all important structures of observational data. In this research four different evolutionary multi-objective optimization algorithms are used for calibration and estimating main 4 parameters of GR4J (in French, mod`ele du G´enie Rural `a 4 param`etres au pas de temps Journalier) hydrological model over Danube basin. GR4J is a simple rainfall-runoff model and belongs to the family of hydrological models that focus on the soil moisture compartment.

All the four used evolutionary optimization techniques, including the Non-dominated Sorting Genetic Algorithm II (NSGA-II), the Multi-objective Particle Swarm Optimization (MPSO), the Pareto Envelope-Based Selection Algorithm II (PESA-II) and the Strength Pareto Evolutionary Algorithm II (SPEA-II) applied in the mode of two objective function. Both objective functions of these algorithms are regarded base on Nash–Sutcliffe model efficiency coefficient which can be calculated as follow:

where  and  are observed and simulated values, respectively, and  stands for the temporal mean of observations (, runoff or GRACE dTWS anomalies).

First objective is base on daily observed and simulated runoff and the other is according to daily Total Water Storage (dTWS) changes derived from the Gravity Recovery And Climate Experiment (GRACE) and its corresponding simulated daily total water storage using GR4J model. For the study basin, GR4J-derived Total Water Storage (TWS) at time t was calculated as the sum of amount of four water comportments which extracted from different simulation steps of GR4J as:

TWS_t = S_t + R_t + V_t + W_t

where S and R are the amount of water stored in the production store and routing store at the end of each day respectively. V and W are the remaining water in unite hydrographs (UH1 and UH2) during the simulation process at the end of each day, respectively. Finally, basin averaged TWS anomalies (dTWS) are computed as a difference between daily TWS (TWSt) and the temporal mean of TWS () during the study period as:

GR4J parameters are estimated by maximizing the objective functions through each algorithm process. Some quality measures including Number of Pareto Solution(NPS), Generation Distance(GD), Spacing(SP) and Maximum Spread(MS) are used to compare the calibration results. The results indicate that NPS of all methods at the last iteration are the same. NSGA-II has better SP and MS criteria rather than other techniques for calibrating GR4J using GRACE dTWS and in-situ runoff data. MPSO show better response in all criteria respect to PESA-II and SPEA-II method and its GD is also better than NSGA-II. The calculated Nash-Sutcliffe model efficiency coefficient in calibration and evaluation periods show that GR4J simulation results can be judged as satisfactory results.