Multi-parameter identification of a two-dimensional water-quality model based on the Nelder–Mead Simplex algorithm

Demands on watercourses for waste disposal, for water supply, as a leisure resource, and as an environmental habitat are increasing. It is therefore, imperative to understand the impact and fate of pollutants introduced into these watercourses. Research has been undertaken to improve the understanding of mixing processes in open-channel flows, so that the fate of pollutants can be predicted and hence the impact of contaminants effectively controlled and appropriate consent limits set. Most rivers have a high-width/depth ratio, and pollutants become mixed vertically within a short distance from the source. Vertical mixing is important only in the so-called near field and is often neglected when considering subsequent transverse and longitudinal mixing. When dealing with practical engineering problems, it is not computationally efficient to use three-dimensional (3D) models. Instead, researchers have used two-dimensional (2D) water-quality models to simulate pollutant transport in channels. There is, however, one troublesome point in the solution of these water-quality models. Whichever method is chosen, certain parameters in the model equations must be estimated. The importance of parameter estimation in model application has been widely recognized in engineering practice. Most previous studies of parameter estimation in river water-quality models have focused on the longitudinal dispersion coefficient (Fischer et al. 1979; Deng et al. 2002; Ho et al. 2002; Seyed & Roger 2002; Fan et al. 2003; Seo & Baek 2004; Gokmen 2009; Liu et al. 2011; Ahmad 2013). However, not only the longitudinal dispersion coefficient, but also the transverse mixing coefficient need to be determined when using 2D water-quality models. Only a small body of published research exists on the transverse mixing coefficient. The methods for calculating the transverse mixing coefficient include mainly theoretical methods (Fischer et al. 1979; Boxall & Guymer 2003; Duan 2004; Baek et al. 2006; Albers & Steffler 2007), empirical formulas (Yotsukura & Sayre 1976; Baek & Seo 2011), and tracer-test methods (Holley & Abraham 1973; Zeng & Huai 2008; Zhang & Zhu 2011). The tracer-test method is thought to be more accurate and reliable than the others, but it requires a tracer-test, which is very expensive. When analyzing pollutant mixing in streams, besides the longitudinal dispersion coefficient and the transverse mixing coefficient, certain other parameters, such as the degradation coefficient, must also be properly evaluated. For practical engineering problems, the most popular parameter selection method is the trial-and-error method, which adjusts parameters continuously until an optimal agreement between predicted values and measured concentration data is achieved. This method constitutes a subjective and laborious step in the water-quality model calibration process because water-quality state variables and model parameters are interrelated. Successful use requires high computational cost and expert practical ability. To overcome these difficulties and obtain reliable water-quality model performance, researchers have used many auto-calibration models to estimate unknown parameters. The optimization method has become the main way to determine parameters because of its advantages, such as low-sampling requirements and simple and convenient calculation. However, its optimization process largely ignores field measurement errors. The Gauss–Newton method is considered the most popular method because it does not require calculation of the Hessian matrix and its rate of convergence is faster than that of other methods (Bard 1970; Yeh 1986). However, because the Gauss–Newton method cannot converge under certain conditions, it requires an algorithm. To solve the convergence problem, the Nelder–Mead Simplex (NMS) method is another efficient direct search method that determines its search direction only by comparing the function values. It is insensitive to small inaccuracies or stochastic perturbations in function values (Chang 2012) and has become one of the most widely used methods for multidimensional non-linear unconstrained optimization (Lagarias et al. 1998). It is very easy to implement in practice because it does not require gradient computation (Peter & Terry 2011). Of the methods described above for solving parameter optimization problems, the NMS method was used in this study to determine the optimal parameters of the water-quality model because it is efficient and robust for estimating parameters in non-linear models and has been successfully used to estimate parameters of non-linear Muskingum models (Reza 2011).

In this study, a NMS algorithm combined with the alternating direction implicit (ADI) method was developed to predict parameters in natural streams. Moreover, the influences of noise, observation location, and sampling frequency on parameter estimation using this method were systematically analyzed. This work could offer guidance on parameter determination for model calibration.

Only under conditions of steady uniform flow can an analytic solution of this equation be obtained. Numerical methods must be used to solve the equation under other conditions. The ADI scheme, a well-known finite-difference method, has been used here to solve the vertically averaged flow equation. ADI is a two time-level method in which solutions of the x- and y-equations are staggered one-half time step apart and each is solved using the continuity equation. The computations are carried out using finite-difference approximations on a space-staggered grid in which water levels and velocities are described at different grid points (Leendertse & Gritton 1971; Adérito et al. 2014). The concentrations are described at the same locations as the water levels. Details of the flow and advective dispersive transport computations are presented in Leendertse & Gritton (1971). The method has been used in a water-quality study of Jamaica Bay (Leendertse 1970).

Three cases of parameter identification for both steady state and dynamic flow were used to verify the method. Besides the computation method, other factors can influence the predictive value of the parameters in the water-quality model. The influence of noise, the distribution of observation points, and the richness of the data are discussed below.

This case was a purely numerical experiment, not a real test. The river width B was 100 m, and the flow depth h was 2 m. The longitudinal velocity u was 0.5 m s⁻¹, and the transverse velocity v was 0 m s⁻¹. The tracer (10 kg) was released in the upper reaches of the river. Assuming that the longitudinal dispersion coefficient E_x was 50 m²s⁻¹, E_y was 0.1 m²s⁻¹, and the degradation coefficient K was 0.3 d⁻¹. Tracer concentrations at five different locations 500 m downstream were calculated (to three decimal places) by the water-quality model and are shown in Table 1 and Figure 1. By treating the data in Table 1 as observed data and u, E_x, E_y, and K as unknown parameters to be identified, this case can be used to verify the reliability of the parameter estimation method.

The initial values of u, E_x, E_y, and K were set to 1 ms⁻¹, 100 m²s⁻¹, 1 m²s⁻¹, and 0.5 d⁻¹, respectively. By entering the data in Table 1 directly into the model, the results shown in Table 2 can be obtained. The identified values are satisfactory except for K. The error in determining K is large compared to that for the other parameters. It is very difficult to estimate K because its assumed value was 0.3 d⁻¹, but the total length of time over which measurements were taken was only 40 minutes, which was a much shorter time scale.

This numerical test was performed to test the parameter identification approach for steady-state flow and continuous pollutant release. The river width B was 100 m, and the depth h was 2 m. The tracer was released continuously at a rate of 100 gs⁻¹ in the upper reaches of the river. The longitudinal velocity u was 0.5 ms⁻¹, and the transverse velocity v was 0 ms⁻¹. The effect of the longitudinal dispersion coefficient E_x can be neglected in this case. With E_y set to 0.1 m²s⁻¹ and the degradation coefficient K to 0.3 d⁻¹, the tracer concentration downstream can be calculated by the water-quality model. Treating the calculated data as observed data and u, E_y, and K as unknown parameters to be identified, this case can be used to verify the reliability of the parameter estimation method and to investigate the effect of observation location on the parameter estimation results.

Assuming five sampling points located on the transverse distribution in the 500 m downstream section shown in Figure 1, data for (500,40), (500,20), (500,0), (500, −20), and (500, −40) were calculated by the water-quality model. Treating these data as observed data, the parameter estimation results using the parameter identification model are shown in Table 4. As in Case 1, the results show that u and E_y have high accuracy, but that the identified value of K is unsatisfactory.

Assuming five sampling points located on the longitudinal distribution, three different conditions were designed and are shown in Figure 2 and Table 5. The parameter estimation results obtained from the parameter identification model are shown in Table 6. As in Case 1, the results show that u and E_y have higher accuracy than K. Of the three conditions, parameter identification under condition 3 was more accurate than under conditions 1 and 2, which shows that increasing the value of y can lead to better results. However, increasing the value of y also leads to lower concentrations, which may be more difficult to observe. When working with real-world observation distributions, this balance needs to be considered.

A comparison of the parameter results under the longitudinal and transverse distributions shows that the longitudinal distribution is well suited for identification of K and the transverse distribution for subsequent identification of E_y.

When the flow is dynamic, the concentration will vary with time. If the sampling frequency is inadequate, the accuracy of parameter identification will be affected by the insufficient richness of the observed data. However, the increased number of sampling times will also lead to a heavy workload. Therefore, the influence of sampling frequency on parameter identification was studied here. Using the target function J ≤ 0.000001 as a termination condition, the parameter estimation results using the parameter identification model under different sampling time scales are shown in Table 8. Using a maximum of 100 iterations as a termination condition, the parameter estimation results are shown in Table 9. The parameter evolution process is illustrated in Figure 5.

These results show that the sampling frequency, which is directly related to the sampling time scale, has a great influence on the precision of the identified parameters. Generally, the more frequent the sampling, the higher will be the precision obtained, but the convergence rate may be slow and the computational time lengthy. Therefore, when dealing with practical problems, a reasonable balance should be determined between the amount of calculation and the parameter estimation accuracy. In this case, 12 is a reasonable number of sampling times which can reflect the processes of change within one hydraulic cycle.

Field observation data (Fischer 1966) were used to determine the dispersion coefficient in the Green River, Washington. Two gallons of Rhodamine B dye were injected as a tracer from the Orillia Bridge. After the tracer was released, water was sampled downstream at Renton Junction. Lateral stations were located starting from the right end of the bridge. Concentration samples were taken at 10-minute intervals at stations 65, 75, 85, 95, 105, 115, 125, and 135. The concentrations measured at each lateral position are shown in Figure 6. The measured average velocity was approximately 0.35 ms⁻¹. For the investigated reach of the Green River, the calculated E_x was 8.3 m²s⁻¹ and the calculated E_y was 0.02 m²s⁻¹ by the proposed method. This value of E_x is very close to the result of 7.8 m²s⁻¹ obtained by integration of the theoretical profile with the velocity distribution.

To avoid the shortcomings of traditional optimization algorithms, a new multi-parameter estimation model based on the NMS algorithm coupled with the ADI method was proposed in this paper. Computational results for three numerical cases indicated that the proposed parameter estimation model has high accuracy and good anti-noise properties. It can give precise results under both steady state and dynamic flow and can be used for both single-parameter and multi-parameter identification, including hydrodynamic parameters and water-quality parameters. The proposed method has a simple and easily understood theoretical basis, lower dependence on field data, and higher accuracy compared to previous methods.

This research was supported by the National Key Technology R&D Program (Grant No. 2012BAB03B04), the National Nature Science Foundation of China (Grant Nos 51479064, 51379060, 51379058, and 11202037), the National Science and Technology Pillar Program (Grant No. 2012BAK10B04), the National Basic Research Program of China (973 Program) (Grant No. 2008CB418202), the Ministry of Water Resources Special Fund for Scientific Research on Public Causes (Grant No. 201001028), the National Public Research Institutes for Basic R & D Operating Expenses Special Project (Grant No. CKSF2013024/SL), the ‘Six Talent Peak’ Project of Jiangsu Province (08-C), and the China Scholarship Council (File No. 201206715006).