## 1. Introduction

Numerical models have become useful behavioral representatives of complex environmental systems. In water-related domains, the growth of knowledge about the underlying processes and recent developments regarding numerical solvers and computational techniques have increased the performance and speed of simulations. However, a significant challenge remains for modelers to minimize the misfit between simulation outputs and corresponding physical observations, which is necessary to obtain a reliable predictive model [

1,

2]. Hydro-morphodynamic models are characterized by a large number of input parameters dealing with a considerable amount of uncertainty. This uncertainty arises from the sophisticated behavior of environmental fluid systems, the simplified structure of models, implemented empirical equations, unknown boundary conditions, and imprecise input data. Some of the physical input parameters, which are subject to calibration, can only be quantified at specific locations over a limited period, and for some it is not physically feasible to measure them.

Modeling practice can be defined as a four-step task, involving model setup, calibration, validation, and application [

3]. Consequently, the model accuracy depends not only on the model structure and the quality of input data but also on the model calibration and validation. Calibration of numerical models is the foremost step for quantifying and accrediting computational simulations, involving two major stages: parameter specification (i.e., selection of sensitive parameters that are subject to adjustment) and parameter estimation (i.e., determination of optimal or quasi-optimal values of specified parameters). Calibration is an inverse, multistep problem, with the aim of uncertainty diminution by updating the model through subsequent comparisons between observations and computational results, in order to achieve a good agreement with a reasonable tolerance [

4]. In other words, calibration is the process of model adaption by adjusting independent variables to gain accordance between computed and measured distributions of dependent variables, so that the model output deviates marginally from the real data specified in the performance criteria [

5,

6]. Nevertheless, even calibrated models potentially involve a certain amount of uncertainty and seldom address the underlying deficiencies because of both model errors (modeling assumptions, simplifications, and approximations) and data errors (the lack of error-free measurements) [

7,

8]. Hence, a calibrated model needs to be validated before applying it to practical problems to ensure reliable predictions [

9,

10,

11]. On the other hand, according to Refsgaard and Henriksen [

12], a validated model operates in a precise manner with regards to site-specific applications and predefined accuracy criteria, and thus the validity of a model is always limited in terms of time, space, boundary conditions, and the type of application.

Model calibration can be accomplished manually, automatically, or by using a multistep method combining the two approaches [

13,

14,

15]. The most widely used trial-and-error method is not only highly dependent on users’ knowledge of the model structure and their level of expertise, but also on their understanding of the environmental system characteristics and the properties of measured data [

16]. The trial-and-error method follows the simple approach of manual adjustment of uncertain or unknown parameters and the comparison of predictions with measured values, involving human judgment to attain the best fit for model parameters [

17]. As the model structure becomes complicated, the manual calibration procedure becomes laborious, time- and cost-consuming, and for models with a large number of uncertain or unknown variables (e.g., morphological models), sometimes impractical. The demanding practice related to the manual method has been persuading modelers to improve the complex inverse calibration technique based on optimization algorithms (e.g., deterministic, metaheuristic, stochastic, or uncertainty-based). This can be done by coupling the model with an optimization engine, with the aim of speeding up the calibration process and establishing an objective scheme to address the “user-subjectivity” issue [

18,

19,

20]. As an example of the user influence on model calibration results, Botterweg [

21] compared the outcomes of a hydrological model that was calibrated independently by two users with an identical set of measured data and reported how different sets of calibrated parameter values could yield reasonable results.

Principally, automated calibration consists of three elements: an objective function to assess the differences between model outputs and observations, an optimization algorithm for sequential adjustment of preselected model parameters with regard to the reduction of the objective function’s value, and a convergence criterion [

22,

23,

24]. Optimization algorithms can be categorized into two classes: global methods based on sampling the proposed values of parameters over the entire space, and local methods based on point estimation by finding the optimum point where no further progress can be achieved in the adjacent space of the parameter. Regarding the local methods, which are computationally efficient and need far fewer model runs, the initial values should be chosen carefully, as model calibration proceeds from this point towards the gradient descent of the objective function. If nonlinearity affects the model, there is no assurance that the inverse problem is unimodal, which means local methods can be simply trapped in local minima points instead of finding the global minimum. Therefore, since the calibration process is strongly dependent on the initial starting point, it is required that the user continuously assesses results, adjusts starting values, and restarts the model [

25,

26,

27]. “Equifinality”, as described by Beven [

28,

29], should also be considered regarding the automated calibration, which may propose the same model prediction by using different parameter sets (also see Straten and Keesman [

30] for the term “equally possible” parameter sets). In other words, uncertainty-based calibration methods such as Generalized Likelihood Uncertainty Estimation (GLUE) [

31,

32,

33] or the Metropolis–Hastings algorithm of Markov Chain Monte Carlo (MCMC) method [

34], which sample different parameter combinations to produce a calibrated probability distribution, detect several feasible parameter sets, unlike the other methods, which focus on pinpointing a single optimal solution [

35].

The performance of autocalibration techniques has been extensively studied for environmental models in the fields of hydrology and groundwater over several decades. However, in the field of fluvial hydraulics, there are relatively few studies available in the literature regarding the application of automated inverse methods for calibrating hydro-morphodynamic models [

36,

37,

38,

39,

40,

41,

42]. In the present study, a 3D hydro-morphodynamic model of a 180° curved channel is developed according to a physical model and calibrated against the measured bed elevations of 54 points along different longitudinal and cross-sections. The model calibration is carried out with a local optimization method (Gauss–Marquardt–Levenberg method), using the Parameter ESTimation (PEST) package [

43] and three population-based global optimization algorithms: Particle Swarm Optimization (PSO) [

44], Shuffled Complex Evolution (SCE-UA) [

45], and Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [

46]. The used methodology is further explained in

Section 2. This work aims to evaluate the efficiency of the automatic calibration procedure based on the local optimization method as compared to the global methods, with the main objective of predicting a single set of optimal parameter values for the model, which is presented in

Section 3.1. Further, in order to test the performance of the most efficient method, different experimental runs with various discharge rates are used to set up and calibrate additional models, presented in

Section 3.2. Finally, the results are summarized and conclusions are presented in

Section 4.

## 4. Summary and Conclusions

In the present study, four different optimization methods (GML, SCE-UA, CMA-ES, and PSO) are employed for automatic calibration of a 3D morphodynamic numerical model of a 180° bend. Three input parameters, namely roughness height, active layer thickness, and the sediment volume compared to water at the bed, are selected for calibration, using their initial values and a reasonable range based on the literature. The results of the model calibration using all optimization methods (a local method and three global methods) show very similar calibrated values. All of the methods facilitate the calibration process by reducing the user intervention. However, considering the convergence speed, using the GML algorithm of PEST software is considerably more efficient compared to the other methods, as it needs far fewer model runs. Accordingly, PEST is selected for further investigation (In total, nine models are calibrated by using three different discharge rates and the sediment transport formulae of van Rijn, Wu, and Engelund-Hansen). Nonetheless, as PEST uses a gradient-based algorithm, it potentially involves a high risk of being trapped in a local minimum point over the search space rather than finding the global minimum. Therefore, the initial parameter values should be taken with care. It is also worthwhile reassessing the calibration procedure with different starting values to ensure that PEST finds the global minimum of the objective function. Moreover, characters and discontinuous values, which are defined in models (such as the selection of the sediment transport formula), cannot be processed by PEST and have to be adjusted manually.

The final bed levels predicted by the calibrated numerical models are compared with the measurements at various longitudinal and cross-sections. The overall capabilities of the numerical models are evaluated using the coefficient of determination and the root mean squared error statistics. It is concluded that the calibrated models using Wu’s formula can predict the general characteristics of bed deformations (regions of deposition, erosion, and their magnitudes) better than the other two formulae by having the highest correlation and the lowest error between simulations and experimental data. The formula of van Rijn also gives reasonably acceptable results. However, the models which use the formula of Engelund-Hansen have the highest disagreement with the observations. It can also be mentioned that the bedforms, developed at the outer bend by increasing the discharge rate, cannot be accurately simulated by the numerical model. This effect is independent of the selection of the sediment transport formula and the calibration routine.