Global sensitivity analysis of a three-dimensional nutrients-algae dynamic model for a large shallow lake

Ecological Modelling 327 (2016) 74–84

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Global sensitivity analysis of a three-dimensional nutrients-algae dynamic model for a large shallow lake Xuan Yi a,b , Rui Zou c,d,∗ , Huaicheng Guo a,∗∗ a College of Environmental Sciences and Engineering, The Key Laboratory of Water and Sediment Sciences Ministry of Education, Peking University, Beijing 100871, China b School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14850, USA c Tetra Tech, Inc, 10306 Eaton Place, Ste 340, Fairfax, VA 22030, USA d Yunnan Key Laboratory of Pollution Process and Management of Plateau Lake-Watershed, Kunming 650034, China

a r t i c l e

i n f o

Article history: Received 21 September 2015 Received in revised form 11 January 2016 Accepted 12 January 2016 Keywords: Sensitivity analysis Morris screening EFDC model Water quality model Spatiotemporal sensitivity indices

a b s t r a c t Sensitivity analysis is a primary approach used in mathematical modeling to identify important factors that control the response dynamics in a model. In this paper, we applied the Morris sensitivity analysis method to identify the important factors governing the dynamics in a complex 3-dimensional water quality model. The water quality model was developed using the Environmental fluid dynamics code (EFDC) to simulate the fate and transport of nutrients and algal dynamics in Lake Dianchi, one of the most polluted large lakes in China. The analysis focused on the response of four water quality constituents, including chlorophyll-a, dissolved oxygen, total nitrogen, and total phosphorus, to 47 parameters and 7 external driving forces. We used Morris sensitivity analysis with different sample sizes and factor perturbation ranges to study the sensitivity with regard to different output metrics of the water quality model, and we analyzed the consistency between different sensitivity scenarios. In addition to the analysis with aggregate outputs, a spatiotemporal variability analysis was performed to understand the spatial heterogeneity and temporal distribution of sensitivities. Our results indicated that it is important to consider multiple characteristics in a sensitivity analysis, and we have identified a robust set of sensitive factors in the water quality model that will be useful for systematic model parameter identification and uncertainty analysis. © 2016 Published by Elsevier B.V.

1. Introduction Water quality models (WQMs) have been developed and applied as valuable tools for quantitative analysis of the cause-andeffect relation between management scenarios and water quality responses, and WQMs have been used widely to support decisionmaking about management of water quality (Vieira and Lijklema, 1989; Zou et al., 2006, 2007; Liu et al., 2014). The recent advances in computing power and data collection have accelerated the development of sophisticated water quality modeling, which can reproduce accurately the hydrodynamic and biochemical conditions of a water body (Castelletti et al., 2010). During the past

Abbreviations: EFDC, environmental fluid dynamics code; WQMs, water quality models; OAT, one-factor-at-a-time; Chla, chlorophyll-a; TN, total nitrogen; TP, total phosphorus; DO, dissolved oxygen; SOD, sediment oxygen demand. ∗ Corresponding author at: Tetra Tech Inc., Fairfax, VA, USA. Tel.: +1 571 830 7008; fax: +1 703 385 6007. ∗∗ Corresponding author. Tel.: +86 10 62751921; fax: +86 10 62751921. E-mail addresses: [email protected] (R. Zou), [email protected] (H. Guo). http://dx.doi.org/10.1016/j.ecolmodel.2016.01.005 0304-3800/© 2016 Published by Elsevier B.V.

decades, many complex dynamic WQMs have been developed, such as WASP, QUAL2K, CAEDYM, CE-QUAL-W2, Delft3D-ECO, PCLake (Mooij et al., 2010), and EFDC (Hamrick, 1992; Park et al., 1995). In general, a complicated water quality model represents the dynamics of water quality processes using a large number of parameters, and the majority of these parameters cannot be measured accurately. Therefore, the only way to develop a model that approximates reality is through model calibration processes that identify proper parameter values (Chapra, 1997; Zou and Lung, 2004). In practice, the complexity in model due to an increased number of parameters would cause a leap in computational requirements and increase the difficulty of calibration because of the highly interactive parameter spaces and the nonlinear, non-monotonous objective spaces (Gupta et al., 1998; Herman et al., 2013a). Therefore, it is desirable to reduce the difficulty of calibration by focusing on a subset of parameters, because in many cases, a small number of model parameters are often responsible for most of the variability in the model’s outputs (Morris et al., 2014). Sensitivity analysis has long been used to identify the subset of important input factors that control model outputs (Saltelli et al.,

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2004; Janse et al., 2010; Makler-Pick et al., 2011; Nossent et al., 2011; Ciric et al., 2012; Neumann, 2012; Sun et al., 2012). There are two main branches of methods for sensitivity analysis: local and global methods. Local sensitivity analysis involves the one-factorat-a-time (OAT) experiment, where a single factor is perturbed while all other factors are fixed to assess the variation in output. This method can generate sensitivity information that is generally local to the parameter values that are taken, except for models that are linear or weakly nonlinear (i.e., can be adequately represented by a first-order polynomial approximation) (Ahmadi et al., 2014). On the contrary, global sensitivity analysis explores the influence of a factor throughout the full multi-dimensional space by varying all factors simultaneously, therefore it is well suited for factor interactions and non-linear relationships between factors and model outputs (Saltelli et al., 2008). However, global sensitivity analysis is often computationally expensive (Campolongo et al., 2007) and has relatively fewer applications in complex modeling of water quality. Modern WQMs are usually large-scaled, sophisticated (nonlinear and factors interaction), and computationally expensive. Thus, various criteria should be considered in selecting an appropriate sensitivity analysis method. Makler-Pick et al. (2011) suggested that a method selection process should include the following key criteria: (i) the computational cost, (ii) the ability to account for the interactions between factors, (iii) the ability to account for the nonlinearities and non-monotonicity in models, (iv) the input data required for the analysis, and (v) the ability to use the output of sensitivity analysis. Considering these criteria, only a small number of approaches are suitable for complex modeling of water quality. In this study, the method of Morris (1991) was selected and applied to a complex water quality model of a shallow lake. This method is a global sensitivity analysis that requires less computational demand than other global methods, such as the variance-based sensitivity analysis (e.g., Sobol’s (Saltelli, 2002), EFAST (Saltelli et al., 1999), etc.). The method of Morris is a screening method proposed by Morris (1991) and modified by Campolongo et al. (2007). Unlike the variance-based or regression-based methods, which can be prohibitive computationally for water quality models with large number of parameters and long simulation times (Saltelli et al., 2008) or require assumptions regarding the types of functions underlying the model, screening methods are more appropriate for complex WQMs due to their ability to capture general sensitivity structures with a significantly lower computational requirement. Although the method of Morris does not implement individual computation of sensitivity indices for interactions, this is not a critical limitation preventing us from achieving the goal of this study. This can be justified by many previous studies using the method of Morris to successfully identify the influential and non-influential factors in models and derive information regarding parameter interactions and model non-linearity (Morris, 1991; Gamerith et al., 2013; King and Perera, 2013). A few studies (Campolongo and Saltelli, 1997; Saltelli et al., 2006) have proven the robustness of this method, and Herman et al. (2013a) demonstrated that this method performs well in comparison to Sobol’s analysis in identifying influential parameters at a greatly reduced computational expense. Campolongo and Saltelli (1997) and DeJonge et al. (2012) also found a strong correlation between the total sensitivity indices of Morris and FAST/Sobol’s methods, and they suggested that the Morris sensitivity indices could be used quantitatively. Previous applications of global sensitivity analysis focused generally on watershed models (Sun et al., 2012; Ahmadi et al., 2014) and spatially aggregated aquatic ecosystem models (Makler-Pick et al., 2011; Ciric et al., 2012; Zheng et al., 2012; Morris et al., 2014), but only a few studies involved complex water quality models of multi-dimensional lakes/reservoirs because of the computational limitation. A relevant study (Salacinska et al., 2010) applied Morris to a two-dimensional ecological model (GCM) to find sensitivity

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parameters for algae blooms, but it only identified sensitive parameters for a single state variable. Practically, to implement a sensitivity analysis, the model output at a specific time (Morris et al., 2014; Li et al., 2015) or in an aggregate form (Salacinska et al., 2010) is used to measure the response of the simulated state variables. Water quality in lakes and reservoirs has inherent temporal and spatial variability due to climate, hydrodynamics, inputs of pollutants, and bathymetry (Missaghi et al., 2013). In a model with a multi-dimensional representation of physical, chemical, and biological processes, sensitive factors that control the model’s behavior might also vary across the spatial domain. Furthermore, time-dependent sensitivity should be considered also in a dynamic model, because time-varying sensitivity may occur (Wang et al., 2013; Herman et al., 2013b, 2013c). The objective of this study was to identify the influential and non-influential factors (including parameters and external drivers) for a three-dimensional water quality model for Lake Dianchi, China. The result of this study will be used to facilitate the future enhancement of the existing Lake Dianchi model into an uncertainty based watershed management decision support platform for Lake Dianchi. In order to understand the robustness of the sensitivity analysis result, we designed a variety of analysis to explore the variability in sensitivity results with regard to different settings in the Morris sensitivity analysis, including the perturbation range, sample size, constituents, and aggregation metrics. In addition, we also analyzed the spatial and temporal variability of the sensitivity distribution to gain further insights regarding the system behavior. The paper is organized as follows: The theoretical background of the EFDC model and the sensitivity analysis method, and the computational experiments are introduced briefly in Section 2. The results are described in detail and discussed in Section 3. Finally, we present a summary and a discussion of future work. 2. Materials and methods 2.1. Study area Lake Dianchi, the sixth largest lake in China, is located on the Yunnan-Guizhou Plateau of southwestern China (Fig. 1), at an altitude of 1887 m. The surface area of the lake is approximately 300 km2 and the watershed area is approximately 2920 km2 (latitude 24◦ 28 –25◦ 28 N, longitude 102◦ 30 –103◦ 00 E). The lake has a total storage capacity of 1.5 × 109 m3 , and an average depth of 5.2 m. There are 29 rivers flow into the lake and 1 outlet drains the lake. The climate in this area is subtropical, moist monsoon with an annual precipitation of 932 mm (Zhou et al., 2014). Lake Dianchi was historically a clean lake. However, rapid urbanization and industrial development that began in the 1980s produced tremendous nutrient loads in the lake, causing deterioration of the lake’s water quality (Wang et al., 2014). Among all the inflows, those from the north of the basin contribute most of the nutrient loading. During the past decades, Lake Dianchi has lost its function as a source of drinking and irrigation water, and it has become one of the three most polluted large lakes in China. Monthly water quality data were collected from eight regular monitoring sites (Fig. 1), and these data have been used to calibrate and validate the modeling of water quality of Lake Dianchi (Wang et al., 2014). 2.2. Lake Dianchi water quality model The water quality model of Lake Dianchi was developed based on a sophisticated computational platform titled

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The governing equations of the water quality module of EFDC can be represented mathematically as follows:

∂(mx my HC) ∂ ∂ ∂ + (my HuC) + (mxHvC) + (mx my wC) ∂t ∂x ∂y ∂z ∂ = ∂x



my HAx ∂C mx ∂x



∂ + ∂y

+ mx my HSc

Fig. 1. Location and monitoring sites in Lake Dianchi.

Environmental Fluid Dynamics Code (EFDC) (Hamrick, 1992), which has been calibrated and validated previously (Wang et al., 2014). EFDC is a comprehensive multi-dimensional model capable of simulating hydrodynamics, salinity, temperature, eutrophication dynamics, and fate and transport of toxicants (Hamrick, 1992).



mx HAy ∂C my ∂y



∂ + ∂z



Az ∂C mx my H ∂z



(1)

where C is the concentration of a water quality state variable; u, v, w are velocity components in the curvilinear, sigma x-, y-, and z-directions, respectively; A is turbulent diffusivities; Sc is internal and external sources and sinks per unit volume; H represents water column depth; and m is the horizontal curvilinear coordinate scale factor. The EFDC model is capable of simulating 21 water column state variables, including algae, dissolved oxygen, carbon, nitrogen, phosphorus, silicon, total metals, and bacteria. EFDC also includes a sediment diagenesis module capable of simulating kinetic processes that occur in the sediment bed and its interactions with the water column (Park et al., 1995). The structure of the water quality model and the interactions between state variables are illustrated in Fig. 2. Previously, Wang et al. (2014) developed a 3-dimensional hydrodynamic and water quality model based on EFDC to simulate the eutrophication dynamics in Lake Dianchi, including state variables such as chlorophyll-a (Chla), nitrogen, phosphorus, and dissolved oxygen (DO). The model used a boundary-fit curvilinear grid of 664 cells to represent the lake horizontally, and six layers to resolve the vertical variability. The model was calibrated and validated with data and applied to evaluate a series of watershed management scenarios (Wang et al., 2014). Global sensitivity analysis was conducted using this model as the working platform with four water quality constituents that were analyzed and compared: Chla, total nitrogen (TN), total phosphorus (TP), and DO. The reason of choosing these four constituents is that they are the most widely used indicators for evaluating eutrophication and water quality impairments, and many watershed management efforts

Fig. 2. EFDC water quality sub-model schematic (Hamrick, 1992).

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Table 1 Parameters and drivers of EFDC-WQM. Parameter

Description

Units

Initial value

Pc Pd Pg Rc Rd Rg Dc Dd Dg KEb KEc KHNc KHNd KHNg KHPc KHPd KHPg TMc1 TMc2 TMd1 TMd2 TMg1 TMg2 TMp1 TMp2 Sc Sd Sg SRP SLP KLN KDN KLP KDP KLC KDC KN DOPTc DOPTd DOPTg KTG1c KTG2c KTG1d KTG2d KTG1g KTG2g CPprm1 Wser CN CS NN NS PN PS

Max growth rate of algae group 1 Max growth rate of algae group 2 Max growth rate of algae group 3 Basal respiration rate of algae group 1 Basal respiration rate of algae group 2 Basal respiration rate of algae group 3 Predation death rate of algae group 1 Predation death rate of algae group 2 Predation death rate of algae group 3 Background light extinction coefficient Chlorophyll-a induced light extinction coefficient Nitrogen half saturation coefficient for algae group 1 Nitrogen half saturation coefficient for algae group 2 Nitrogen half saturation coefficient for algae group 3 Phosphorus half saturation coefficient for algae group 1 Phosphorus half saturation coefficient for algae group 2 Phosphorus half saturation coefficient for algae group 3 Lower bound of optimal temperature for algae group 1 Upper bound of optimal temperature for algae group 1 Lower bound of optimal temperature for algae group 2 Upper bound of optimal temperature for algae group 2 Lower bound of optimal temperature for algae group 3 Upper bound of optimal temperature for algae group 3 Lower optimal temperature for diatom predation Upper optimal temperature for diatom predation Settling rate of algae group 1 Settling rate of algae group 2 Settling rate of algae group 3 Settling rate of refractory organic matters Settling rate of labile organic matters LPON hydrolysis rate DON decay rate LPOP hydrolysis rate DOP decay rate LPOC hydrolysis rate DOC decay rate Base nitrification rate Optimal depth for algae group 1 growth Optimal depth for algae group 2 growth Optimal depth for algae group 3 growth Suboptimal temperature effect coefficient for algae group 1 growth Superoptimal temperature effect coefficient for algae group 1 growth Suboptimal temperature effect coefficient for algae group 2 growth Superoptimal temperature effect coefficient for algae group 2 growth Suboptimal temperature effect coefficient for algae group 3 growth Superoptimal temperature effect coefficient for algae group 3 growth Algae Phosphorus-to-Carbon ratio Wind velocity change rate Carbon reduction rate in the north source Carbon reduction rate in the south source Nitrogen reduction rate in the north source Nitrogen reduction rate in the south source Phosphorous reduction rate in the north source Phosphorous reduction rate in the south source

d−1 d−1 d−1 d−1 d−1 d−1 d−1 d−1 d−1 m−1 m−1 per gm−3 mg L−1 mg L−1 mg L−1 mg L−1 mg L−1 mg L−1 ◦ C ◦ C ◦ C ◦ C ◦ C ◦ C ◦ C ◦ C m d−1 m d−1 m d−1 m d−1 m d−1 d−1 d−1 d−1 d−1 d−1 d−1 d−1 m m m – – – – – – g C per g P – – – – – – –

2.95 2.8 2.5 0.14 0.15 0.14 0.04 0.15 0.04 0.3 0.012 0.02 0.02 0.02 0.001 0.001 0.001 26 30 10 15 22 25 18 25 0.1 0.25 0.15 0.2 0.2 0.04 0.05 0.04 0.05 0.05 0.07 0.05 1 1 1 0.008 0.008 0.005 0.012 0.008 0.008 42 1 1 1 1 1 1 1

adopt these constituents to measure management targets. Methodology Morris sensitivity analysis was chosen as the preferred method for sensitivity analysis in this study because it can distinguish the influential and non-influential factors in a computationally efficient manner (Herman et al., 2013a). Among the over 100 parameters in EFDC, forty-seven parameters and seven external drivers (Table 1) were selected as potentially important factors for this analysis based on previous studies on modeling Lake Dianchi. The initial values were obtained from Wang et al. (2014).

2.2.1. Morris sensitivity analysis framework Morris sensitivity analysis (Morris, 1991) provides a measure of global sensitivity from a set of local derivatives (also called elementary effects), sampled on a grid covering the entire parameter space (Herman et al., 2013a). It is used widely in factors fixing and

insensitivity analysis (King and Perera, 2013; Wang et al., 2013; Herman et al., 2013a, 2013b; Morris et al., 2014). The Morris screening is based on an OAT method, in which the elementary effect of a change i of each factor xi creates a trajectory through the factor space: EEi =

f (xi, . . .xi, . . ., xn) − f (x) i

(2)

where EEi is the elementary effect for the ith factor, f(x) represents the prior point in the trajectory, n is the number of model factors, i = p/[2(p − 1)] represents the grid size where p is the number of ‘levels’ in the input space. Because the OAT is highly dependent on the random initial x position and does not concern the interaction among factors, the procedure was performed r times to provide r trajectories. The mean i is the total-order effects of each input factor to the model output, and the standard deviation  i describes the variability through the factor space and/or the factor interaction.

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The overall evaluation number of Morris is equal to r(n + 1). Campolongo et al. (2007) improved this method by estimating the mean of the absolute values of the elementary effect of the set of r trajectories of the ith factor, ∗i : 1 j |EEi | N N

∗i =

(3)

j=1 j

where EEi represents the jth trajectory of ith factors. 2.2.2. Robustness evaluation of the method of Morris For the method of Morris, sampling number r determines the number of evaluations, which was found to have a relationship to the convergence of sensitivity measures (Ciric et al., 2012; Gamerith et al., 2013; Wang et al., 2013). To balance the evaluation times (computational burden) and the convergence of sensitivity measures, the size of the sample is important. Previous experiments used sample sizes that ranged from 10 to 40 for dozens to thousands of factors (Ciric et al., 2012; Herman et al., 2013b; Ahmadi et al., 2014). Therefore, to assess the influence of sample size r and determine a suitable r value, 20, 40, and 60 samples (Ciric et al., 2012; Gamerith et al., 2013; Herman et al., 2013a) were used in our study, and the corresponding numbers of model evaluations were 1100, 2200, and 3300 with a ±30% factor value range, respectively. In addition, the impact of factor perturbation ranges on sensitivity analyses was determined also for a sample size of r = 40, where the perturbation range was amplified from ±20% to ±50%, except for wind velocity (Wser) and optimal temperature bounds for algal growth (TMc1, TMc2, TMd1, TMd2, TMg1, TMg2, TMp1, and TMp2). In addition, a p = 8 levels and a grid jump of four was chosen for this work. 2.2.3. Sensitivity analysis for different model metrics Because the output of the EFDC model is spatially and temporally variable, it is desirable to pre-process the model outputs to form appropriate metrics to represent model behaviors for sensitivity analysis (Cariboni et al., 2007; Herman et al., 2013c). Considering the purpose of the modeling, four output metrics were derived and used for the sensitivity analysis, and the results were compared to investigate how different output metrics would impact the results of the sensitivity analysis. The four metrics included the annual average concentration of the entire lake to reflect the gross level of water quality in the lake, the peak values to reflect a critical condition such as algal blooms (Liu et al., 2014), root mean square error (RMSE), and relative error (RE) between the observed data and the model output (Herman et al., 2013a, 2013c; Ahmadi et al., 2014). The reason of choosing the annual average water quality concentrations and peak algal blooms was that these metrics are important indicators usually used in practical water quality management; and the reason of choosing the RMSE and RE was that these metrics are important in evaluating model calibration performance. With these metrics, the corresponding sensitivity analysis would generate insights about the controlling factors contributing to these important metrics which are import either for management scenario analysis or for model calibration. The sensitivity analyses were performed using the cluster at Cornell University (thehigh-performance cube.cac.cornell.edu), which contains 32 computer nodes with Dual 8-core E5-2680 CPUs @ 2.7 GHz, 128 GB of RAM. The method of Morris was calculated in the software Python 2.7.6 using the Sensitivity Analysis Library in Python (SALib, https://github.com/ jdherman/SALib). Approximately 1800 computing hours were required for the model evaluations for all the executions of the Morris analysis.

Fig. 3. Pair-wise compare of sample sizes, x and y axis represents the sensitive ranks of each factor, rank 1 is the most sensitive and 54 is least.

3. Results and discussions 3.1. Impacts of different sensitivity configurations 3.1.1. The impact of sample size on convergence of sensitivity measures The goal of a global sensitivity analysis is to obtain stable results at a reasonable computational cost. As mentioned above, this study tested three different sample sizes (i.e., 20, 40, and 60). Fig. 3 and Fig. S.1 in the supporting materials show the pair-wise comparisons of factor sensitivity ranks for different sample sizes, where the results for Chla, DO, TN, and TP were analyzed separately. As shown, the linear correlation coefficient R2 and Spearman coefficients were high (>0.95) between both the 20 versus 40, and 40 versus 60 comparisons, it indicates that the convergence might have been achieved at the smaller size of 20. On the other hand, it showed that there was a slight increase in R2 (Fig. 3 and Appendix Fig. A.1 (a) and (b)) from 20 versus 40 to 40 versus 60, which suggested that the results of the sensitivity analysis approached closer to the point of convergence with the increase in sample size. Fig. 3 and Appendix Fig. A.1 also shows that near the two ends of the sensitivity rank (i.e., those parameters with the greatest and least sensitivities), the results of different sample sizes were much more consistent, which suggested that the method of Morris was capable of isolating the most and least sensitive factors in a relatively small sample size with high reliability. This finding agrees with the results in Herman et al. (2013a). To further verify the consistency in model result, we applied the Chi-square test to assess whether sensitivity rank series vary significantly between the different sampling sizes. As shown in Table 2, the Chi-square value is smaller than 20.05 = 70/99 which indicates that sensitivity ranks are consistent between the different sample sizes. Based on both the R2 and Chi-square test, we chose to use a sample size = 40 for further model runs because that sample size was sufficient to generate results that converged reasonably. 3.1.2. The impact of factor value ranges Morris sensitivity analysis was run with sample size of 40 for different ranges of factor perturbations (Table 2; Fig. 4 and Appendix Fig. A.2). The small changes were ignored by assuming they came from the aleatory component (Ciric et al., 2012). The R2 of the sensitivity ranks between the 20% and 30% perturbation ranges are 0.9626, 0.9287, 0.9623, and 0.9799, while those between the 20% and 50% are 0.894, 0.8101, 0.8588, and 0.9196 for Chla, DO, TN, and TP, respectively. Table 2 shows the Chi-square test results. As shown, when the perturbation range change from 20% to 30%, the results are consistent with each other; however, when the perturbation range is too large, i.e. 50%, the obtained sensitivity

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Table 2 2 values of different sensitivity configurations (2 0.05 = 70.99), the bold values represent significantly differences. Sensitivity configurations

Comparison groups

Chla

DO

TN

TP

Sample sizes

20 vs. 40 40 vs. 60 20% vs. 30% 30% vs. 50% 20% vs. 50%

21.42 15.36 53.40 29.15 105.15

22.58 20.84 34.01 81.51 116.54

19.77 17.28 38.26 67.13 156.74

15.57 15.28 25.59 32.55 86.41

Range perturbations

the perturbation ranges of the parameters. Also, for those factors that were not amplified proportionally, their sensitivities showed significant variability to the change in perturbation ranges of other factors, which agrees with the conclusion of Wang et al. (2013) that sensitivity of some factors is related largely to the interactions between other factors.

3.2. Impacts of different output metrics

Fig. 4. Pair-wise compare of different value range perturbations. Rank = 1 is the most sensitive and rank = 54 is the least sensitive.

ranks appear to be significantly different from those obtained from the 20% range. It is also noted that between the 30% and 50% ranges, the sensitivity ranks are in general consistent, except for DO that demonstrates significant difference. Judging by the correlation coefficient and Chi-square value, the sensitivity ranks for TP appeared to be the most robust, which was likely because phosphorus was least sensitive to variations in model parameters due to its small portion in algal biomass. The perturbation range analysis results suggest that the sensitivity results are robust within a reasonably small range of perturbation, i.e. 20–30%. However, with the increase in perturbation ranges, the sensitivity results can become significantly different. Therefore, one should be careful in applying the sensitivity analysis results in guiding future model calibration if certain parameters might take values beyond the perturbation ranges used in the sensitivity analysis. It is interesting to note from Fig. 4 that the sensitivity ranks for the most sensitive factors (i.e., those with low rank numbers), and the least sensitive factors were more robust than those of moderate sensitivity. This is consistent with the sample size analysis, which suggested that the method of Morris was highly reliable in identifying the most important parameters that govern the behavior of the water quality model, even when there was uncertainty in

As described in the Methodology section, we used four output metrics (AVER, RMSE, and RE) for Chla, DO, TN, and TP to analyze model sensitivities. Table 3 and Appendix Table A.1 show the Chisquare values and Spearman correlation coefficient between Morris indices of different output metrics and water quality variables. The ranges of Chi-square values and the correlation coefficients for the four constituents were: 2 = [0, 73.65],  = [0.923, 0.994] for Chla, 2 = [19.14, 220.59],  = [0.857, 0.981] for DO, 2 = [0, 23.95],  = [0.985, 0.997] for TN, and 2 = [8.90, 40 : 18],  = [0.950, 0.990] for TP where 20.05 = 70.99. The results indicated that for TN and TP, the sensitivity distribution was consistent between different output metrics, therefore, the key parameters identified from using one metric might be used in both model uncertainty analysis or model calibration without losing much information. In contrast, the sensitivity distribution for Chla is less robust with regard to the choice of metrics, where the sensitivity distribution obtained by using metric RE is significantly different from that obtained by using metric Max. Therefore, the key parameters identified by using the RE might not be applicable to parameter uncertainty analysis for maximum algal biomass. The sensitivity distributions for DO exhibit the least consistency between using the error related metrics and the concentration related metrics. Therefore, one should not directly use the key parameters derived with regard to RE or RMSE to conduct parameter uncertainty analysis of DO concentration, and vice versa. In such a case, sensitivity analysis should be designed targeting its specific purpose, i.e., model calibration, or uncertainty analysis in compliance evaluation, thus using the corresponding metrics as the basis.

Table 3 2 values of different output metrics (2 0.05 = 70.99), the bold values represent significantly differences. The AVER is the temporally-spatially average concentration, MAX represents the maximum concentration over year. Expected actual

RMSE

RE

AVER

MAX

Chla RMSE RE AVER MAX

0.00 0.00 9.23 70.05

Expected actual

RMSE

0.00 0.00 18.52 9.82

RE

AVER

MAX

20.11 0.00 114.81 220.59

150.60 153.30 0.00 174.33

126.60 160.08 121.65 0.00

DO 0.00 0.00 12.59 83.65

9.28 11.94 0.00 67.61

52.86 66.66 58.39 0.00

0.00 19.14 107.16 164.00 RMSE

RE

AVER

MAX

0.00 0.00 15.46 10.07

23.95 20.52 0.00 13.10

10.58 10.84 12.75 0.00

TN RMSE RE AVER MAX

RMSE

RE

AVER

MAX

8.90 0.00 13.39 46.56

13.42 12.66 0.00 40.18

18.07 39.92 37.50 0.00

TP 0.00 8.12 14.56 19.49

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Fig. 5. Morris sensitivity result for average concentration of four water quality constituents.

3.3. Factor sensitivity for different constituents The sensitivity ranks among four different constituents correlated with each other significantly ( > 0.7) while also showing some obvious differences (Appendix Table A.1). The significant correlation can be explained by the fact that water quality constituents including algae, DO, TN, and TP are highly interactive components in a eutrophic lake like Lake Dianchi, therefore, the parameters governing the dynamics of one constituent are likely to impact the others. On the other hand, each of the constituents is subjected to different kinetic processes, therefore, the dynamic behavior can be controlled by different sensitive parameters. Therefore, to identify sensitive parameters that govern the behavior of a complex eutrophication model (i.e., the Lake Dianchi model), all the concerned constituents should be included in the sensitivity analysis to obtain a robust set of parameters that are important for characterizing the entire system. In order to understand the mechanisms underlying the sensitivity analysis results, we might take a close look at the result and analyze the causes of certain sensitivity distributions. Fig. 5 summarizes the sensitivities of lake surface average concentrations for all the factors, where the horizontal axis represents the mean and vertical axis represents the standard deviation of the elementary effects calculated through the Morris process. The sensitive factors for the four constituents were quite similar, although clear disparities existed for a subset of the factors. Among the top 16 sensitive parameters for each constituent, 9 of them were the same. These factors included phosphorus-tocarbon ratio (CPprm1), max growth rates of algae groups 1 and 3 (Pc and Pg), basal respiration rates of algae groups 1 and 3 (Rc and Rg), optimal temperature for algae group 1 (TMc1), Chla-induced light extinction coefficient (KEc), LPOP hydrolysis rate (KLP), and the external driver wind velocity (Wser). Consequently, when calibrating the water quality model for Lake Dianchi, modelers should pay careful attention to these parameters, because they are critical in characterizing the dynamics of all the primary constituents. CPprm1, the P:C ratio for algae, was the most influential factor for all four state variables, because algal growth in Lake Dianchi is governed largely by the availability of phosphorus in the lake. When the P:C ratio is high, the growth of algae might deplete

available phosphorus in the water quickly and cause a phorophruslimiting condition in the lake, which would shape algal dynamics. Because Lake Dianchi is a hyper-eutrophic lake where algae are the dominant factors affecting water quality, the P:C ratio would impact the entire dynamics of water quality eventually through its effect on algae dynamics. This finding is consistent with previous results (Refsgaard et al., 2014). Similarly, Pc, Pg, Rc, Rg, and the other parameters that control algal growth and metabolism directly are also the common sensitive parameters for all the constituents, because they dominate algal dynamics and, hence, nutrient cycling and DO processes. The hydrolysis rate of particulate organic phosphorus (KLP), but not that of particulate organic nitrogen (KLN), was a sensitive factor for all the constituents, which can be explained using the same reasoning as for the P:C ratio. The hydrolysis of particulate organic phosphorus produced a bio-available form of phosphorus for supporting algal growth, which impacted all other nutrients and DO dynamics. It is also easy to understand the importance of wind speed on water quality in the lake. Wind speed can impact water circulation and the pollutant transport pattern in the lake, and it also controls directly the exchange of oxygen between the air and water. Higher wind speed would result in higher DO during periods when oxygen was depressed due to organic matter in the water column or by algal respiration, and it can cause lower DO during periods when super-saturation of DO existed due to algal photosynthesis. In addition, higher wind speed caused stronger vertical mixing of water, which enhanced vertical transport of all water quality constituents. All of these processes directly or indirectly caused a response in water quality, which explains why wind speed was one of the key common sensitive parameters for all the constituents. Algae settling velocity (Sc, Sg) was also an important factor that characterized water quality processes. It is clear that algae biomass was lost from the water column through settling, which consequently brought nutrients contained in the algae to the underlying bed. In the meantime, the nutrients in the algae that settled to the sediment bed experienced a sediment diagenetic process and produced inorganic nutrients that would re-enter the water column. Also, the carbon in the settled algae later contributed to the sediment oxygen demand (SOD), which impacted the DO concentration in the water column. Because algae groups 1 and 3 dominated the biomass in the lake most of the time, the temporal-spatial average Chla concentration in the lake was sensitive to those parameters that were related to these two algae groups. In addition, Chla was more sensitive to P loads than to N and C loads; this was due to the P limiting characteristic of Lake Dianchi, which often happens in freshwater systems (Conley et al., 2009). For DO, the organic carbon loadings (C N) from the watershed consumed the DO in water directly, therefore, they were the sensitive parameters for DO. For TN, nitrification rate (KN) and DON decay rate (KDN) were sensitive parameters through their direct impact on nitrogen kinetics. It is interesting to notice that TN was more sensitive to KLP than to KLN, which was most likely caused by the interactions among algae, phosphorus, and nitrogen in the lake. Spatially, it was clear that the N and P loadings from the northern part of the basin were the sensitive factors that contributed to water quality in the lake. This was not surprising, because the major pollutant loadings to Lake Dianchi are from those inflows from the northern part of the lake. Fig. 5 also shows both * and  of the Morris sensitivities. A high * represents the high sensitivity of a factor, and a high  represents a higher nonlinearity and interaction between factors. We found that a higher  generally corresponded to a higher * (Fig. 5), which indicated that the sensitive factors also had a higher

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nonlinearity and had more interactions with other factors. The identified non-linearity or interaction of the factors can reduce the factor identifiability (Gamerith et al., 2013), because several combinations of parameter values can lead to the same results. In such a case, a single set of model parameter will not be able to generate reliable prediction for management scenarios to support reliable decision making (Zou and Lung, 2004). Since Fig. 5 shows that almost all the critical parameters in Lake Dianchi model are subjected to significant nonlinear interactions, therefore, it is very important to enhance the existing model using systematic parameter identification methods such as the multiple-pattern inverse modeling approach to identify those distinct parameter combinations that equally reproduce observed data and use them to drive models to analyze management scenarios (Zou et al., 2009, 2014). In our future research we are planning to use the sensitive parameters identified to formulate an uncertainty based robust modeling system to conduct decision support analysis for Lake Dianchi water quality management. 3.4. Spatiotemporal variability in sensitivity Multi-dimensional models have been used widely to simulate water quality responses at high spatial (and/or vertical) and Table 4 Factor ranks on spatial distribution. Top 10 factors of each state variables are listed. Factors

A1

A2

A3

A4

A5

A6

A7

A8

Maximum difference

Chla CPprm1 Pc Wser Pg KEc Rc TMc1 Sg Dc Rg

1 2 3 5 4 6 7 8 9 10

1 2 3 5 4 6 7 8 9 10

1 2 3 4 6 5 7 8 9 10

1 2 3 4 6 5 7 10 9 8

1 2 3 4 6 5 7 8 9 10

1 2 3 4 6 5 7 8 9 10

1 2 3 4 6 5 7 8 9 10

1 2 3 4 5 6 7 8 10 9

0 0 0 1 2 1 0 2 1 2

TN CPprm1 Pc KN Wser KDN KEc Pg NN TMc1 Rg

1 2 3 4 5 6 7 17 8 9

1 2 3 4 5 6 7 18 8 9

1 2 3 4 5 6 7 8 10 9

1 2 3 4 5 6 7 9 8 10

1 2 3 4 5 6 7 8 10 9

1 2 3 4 5 6 7 11 9 8

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 8 7 9 10

0 0 0 0 0 0 1 11 2 2

DO CPprm1 Pc Pg KEc Wser TMc1 Rg Rc CN KLP

3 1 4 2 5 6 7 8 23 11

1 2 4 3 5 7 6 8 26 11

1 2 4 5 3 6 8 7 14 9

4 1 2 3 8 7 5 6 12 18

1 2 3 5 4 6 8 7 9 11

1 2 4 5 3 6 7 8 14 9

1 2 3 4 5 7 6 8 9 13

1 2 3 4 5 7 6 8 9 15

3 1 2 3 5 1 3 2 17 9

TP CPprm1 Wser Rg Pg KEc Pc SLP Rc PN Sg

2 1 3 4 5 6 7 8 11 9

2 1 3 4 5 6 7 8 11 9

1 2 3 4 5 7 6 8 9 10

2 1 3 4 5 7 6 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 7 6 9 8 10

1 2 3 4 5 8 7 9 6 10

1 1 0 0 0 2 1 1 5 1

81

Table 5 Factor ranks on vertical domain. #1 to #6 is from bottom to surface water. Factors

#1

#2

#3

#4

#5

#6

Maximum difference

Chla CPprm1 Pc Wser Pg KEc Rc TMc1 Sg Dc Rg

1 2 17 3 5 4 7 9 8 6

1 2 11 3 5 4 6 8 9 7

1 2 6 3 5 4 7 9 10 8

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 6 5 7 8 9 10

0 0 14 1 1 2 1 1 2 4

TN CPprm1 Pc KN Wser KDN KEc Pg NN TMc1 Rg

1 3 4 2 5 6 7 8 9 11

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

0 1 1 2 0 0 0 0 0 1

DO CPprm1 Pc Pg KEc Wser TMc1 Rg Rc CN KLP

8 3 2 9 1 6 4 5 7 29

6 3 2 4 1 7 5 8 9 21

4 1 2 3 6 8 5 7 9 16

3 1 2 4 8 6 5 7 9 14

2 1 3 4 5 6 7 8 9 12

1 2 3 4 5 6 7 8 9 10

7 2 1 6 7 2 3 3 2 19

TP CPprm1 Wser Rg Pg KEc Pc SLP Rc PN Sg

2 1 3 4 5 9 6 8 7 10

2 1 3 4 5 7 6 8 9 10

2 1 3 4 5 7 6 8 9 10

2 1 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 1 0 0 0 3 1 0 2 0

temporal resolution, although in practice it is common to evaluate model results in an aggregated manner as was done in this study; we evaluated temporal-spatial average of the model results in the sensitivity analysis. However, considering that the processes that affect water quality can vary over time and space, it is desirable to analyze spatiotemporal variability in the sensitivity analysis to acquire additional insights. 3.4.1. Horizontal and vertical distribution of sensitivities The ten most sensitive parameters identified based on the aggregate measures were chosen for further analysis of the horizontal and vertical distributions of sensitivities (Tables 4 and 5). Sensitivity ranks were generally similar among the eight sites, except for external pollutant loadings (C N to DO, N N to TN, and P N to TP), wind speed (Wser for DO), and parameters that were related to light extinction (KEc for DO) (Table 4, see Fig. 1 for the locations of monitoring sites). Because Lake Dianchi receives most of its external pollutant loadings from the north, water quality at the sites located at the northern part of the lake naturally were more sensitive (with lower rank number) to the external loadings from the northern part of the basin than were other sites. There is no clear pattern for the spatial variability in sensitivities regarding wind

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Fig. 6. Temporal variability of Morris results. Month #0 represents the aggregate values of the year.

speed (Wser), Chla-induced light extinction (KEc), and LPOP hydrolysis rate (KLP) for DO, because they are the outcomes of extremely complex interactions among sources, nutrients, algae, and DO in the water column, as well as the diagenetic process in the sediment bed. It is difficult, if not impossible, to isolate all the interacting factors to explain exactly why the DO at a specific site was more sensitive to KLP than at another site. For Chla, wind velocity (Wser) varied from insensitive (rank 17) in the bottom layer to sensitive (rank 3) in the surface layer (Table 4). Wind is the primary driver of the hydrodynamics, which affect the distribution of nutrients in the water column. Because different distributions of nutrients impact algal dynamics directly in the surface layers where light is not limiting, algae in the top few layers were very sensitive to wind speed. In the bottom layer, the distribution of nutrients had no impact on algae, because light becomes limiting for a large area of the lake at the bottom. Therefore, Chla in the bottom layer was insensitive to wind speed. The sensitivity of DO to wind speed also demonstrated significant vertical variability. Although DO was sensitive to Wser from the top to the bottom of the water column, it was clear that the bottom (rank 1) was more sensitive than the surface (rank 5). DO was sensitive to wind speed due to its controlling effect on the reaeration coefficient and vertical mixing. With increased wind speed, more oxygen entered the surface layer when it was undersaturated or exited it when it was super-saturated and, hence, this changed the oxygen balance in the surface layer. In addition, higher wind speed enhanced vertical mixing, propagating the changed oxygen balance in the surface layer to subsurface layers until it reached the bottom layer. Because DO is significantly lower in the bottom layer than at the surface layer during extended periods of time, any change in DO mass balance can result in a large percentage change in DO concentration at the bottom, which resulted in a large sensitivity coefficient in the Morris analysis. At the bottom, other processes such as algal activities were weak, and they became much less a contributor to DO. All of these factors in combination caused DO at the bottom to have a high sensitivity to wind speed.

The Chla-induced light extinction KEc had a significant impact on DO, and the sensitivity rank was similar for all the layers except for the bottom layer. The vertical sensitivity variability to KEc might be due to the vertical light structure, which controlled the vertical distribution of algal dynamics. Because KEc modifies vertical light structure dependent on algae biomass, it is likely to have had less impact on the bottom layer where light was always limiting. Using light as the auxiliary variable, it might also help to explain why DO in the bottom layer had less sensitivity to KLP than at the surface. 3.4.2. Temporal distribution of sensitivities In addition to spatial variability, the sensitivities also demonstrated clear temporal variability. The Morris sensitivity indices changed over time for most factors, which reflected the temporal variation of dominant kinetics in the lake. To understand how the sensitivity distributions of all the parameters evolve over time, we plot the time series of sensitivity ranks of the parameters in Fig. 6. Throughout the year, temperature changed with the seasons, which might be the dominant factor that underlayed the seasonal variability in sensitivities to many parameters. For example, the ranges of optimal temperatures for the three algae groups used in the sensitivity analysis were: from 22 to 35 ◦ C for group 1, from 7 to 18 ◦ C for group 2, and from 17 to 27 ◦ C for group 3 (Table 1). Therefore, in winter and spring, the water temperature was low (below 20 ◦ C) and group 2 algae constituted the major primary productivity in the lake, and the parameters related to group 2 (Pd, Rd) controlled model behavior during that period. However, temperature increased from spring to summer, and it gradually reached the optimal temperature for groups 1 and 3, which caused parameters related to groups 1 and 3 (Pc, Pg, Rc, Rg) to dominate as sensitive parameters; at the same time, the sensitivity for parameters related to group 2 diminished. Although Pd and Rd were neglected as insensitive parameters in earlier analyses when the aggregate measures were considered, they were found to be significant players during certain periods. Therefore, in conducting a sensitivity analysis for a dynamic model,

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it is necessary to consider the temporal variable characteristics of parameter sensitivity for the complete picture. A similar conclusion was found by other investigators (Lamboni et al., 2009; Wang et al., 2013; Herman et al., 2013a, 2013c), which emphasizes the point that when performing sensitivity analysis on a dynamic model, it is important to use the output for time variables in addition to the aggregate measures. 4. Summary and conclusion In this study, Morris screening sensitivity analysis was applied to conduct a global sensitivity analysis for a three dimensional water quality model of Lake Dianchi. This method is a global sensitivity analysis which is suitable for complex, non-linear model due to the less computation burden, and it provides a qualitative evaluation on factor sensitivity and interaction/nonlinearity. The analysis focused on the response of four water quality constituents, including Chla, DO, TN, and TP, to 47 parameters and 7 external driving forces. We developed the following conclusions: 1) The sensitivity results indicated that the sensitivity ranks were reasonably consistent between the runs using different sample sizes, which suggested that robust results can be achieved using relatively small sample sizes. Particularly, our results show that the sensitivity rank results were most reliable for the least and the most sensitive factors. 2) Similarly, the Morris analysis results were robust generally with regard to the perturbation ranges when the perturbation range is reasonably constrained, i.e. 20–30%. However, when the perturbation range becomes too large such as the 50% as in this study, the sensitivity distribution is significantly different. This suggests that in applying Morris analysis or likely other sensitivity analysis method, it is important to constrain the perturbation range within a reasonable range. In the meantime, the sensitivity results must be interpreted in the context of the corresponding perturbation range. 3) We found a high correlation between the results using different output metrics, and different constituents showed slightly different correlations. Among the four constituents, DO showed the largest variability range with regard to output metrics; therefore, one should be cautious when using the sensitivity results obtained by one metric to a modeling analysis relevant to another metric. 4) The sensitivity characteristics for the four water quality constituents were moderately correlated with each other, while their differences were also significant. Water quality constituents typically are highly correlated in a eutrophic lake, and each of them is subjected to different kinetic processes. CPprm1, Pc, Pg, Rc, Rg, Wser, KEc, TMc1, and KLP were found to be sensitive for all four constituents, and CPprm1, Pc, Pg, Rc, and Rg were among the most sensitive factors. TN was more sensitive to KLP than to KLN, which may have been due to the interactions among algae, phosphorus, and nitrogen in the lake. This counter-intuitive phenomenon suggests the importance of conducting systematic sensitivity analysis for a complicated water quality model to identify the parameters truly control a specific response of concern. 5) The analysis demonstrated clear horizontal and vertical variability in sensitivity characteristics for several major sensitive factors, including external pollutant loads, wind speed, and light extinction, among others. This highlighted the importance of the multi-dimensional model and the need for high resolution analysis in sensitivity analysis. 6) The results also demonstrated clear temporal variability in factor sensitivities, which appeared to be dominated by the seasonality

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of temperature. Even though some factors were not influential when analyzed using the output from the aggregated results, they can be dominant during certain periods. Therefore, it is important to conduct sensitivity analysis with a temporal perspective to acquire additional insight into the dynamics of this system. In general we found that the Morris sensitivity analysis is computationally efficient in identifying sensitive factors of the Lake Dianchi model, and the results appear to be particularly robust for those most and least sensitive factors. However, since the perturbation range, sample sizes and different result metrics can affect the final results, it might be more meaningful to interpret results in terms of influential and non-influential distinction instead of the actual ranks. In our future work, we will identify one or multiple sets of sensitive parameters based on the sensitivity analysis results and formulate an uncertainty based inverse modeling system to enhance the current model for better supporting future management decisions in Lake Dianchi watershed. In addition, we might explore more sophisticated sensitivity analysis approaches and evaluate the results against that of the Morris sensitivity analysis to gain more insights about the system behavior. Acknowledgements The authors wish to express their gratitude to the China Scholarship Council (201406010221) for funding the visiting venture that generated this paper. This research is also funded by the Major Science and Technology Program for Water Pollution Control and Treatment of China (2013ZX07102). We thank Dr. Jonathan D. Herman, Assistant Professor, University of California, Davis for methodological and computational support and Thomas A. Gavin, Professor Emeritus, Cornell University, for help with editing the English in this paper. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ecolmodel.2016. 01.005. References Ahmadi, M., Ascough, J.C., DeJonge, K.C., Arabi, M., 2014. Multisite-multivariable sensitivity analysis of distributed watershed models: enhancing the perceptions from computationally frugal methods. Ecol. Model. 279, 54L 67. Campolongo, F., Saltelli, A., 1997. Sensitivity analysis of an environmental model an application of different analysis methods. Reliab. Eng. Syst. Safe 57, 49L 69. Campolongo, F., Cariboni, J., Saltelli, A., 2007. An effective screening design for sensitivity analysis of large models. Environ. Model. Softw. 22, 1509L 1518. Cariboni, J., Gatelli, D., Liska, R., Saltelli, A., 2007. The role of sensitivity analysis in ecological modelling. Ecol. Model. 203, 167L 182. Castelletti, A., Pianosi, F., Soncini-Sessa, R., Antenucci, J.P., 2010. A multiobjective response surface approach for improved water quality planning in lakes and reservoirs. Water Resour. Res., 46. Chapra, S.C., 1997. Surface Water-Quality Modeling. The McGraw-Hill Companies, Inc., New York, USA. Ciric, C., Ciffroy, P., Charles, S., 2012. Use of sensitivity analysis to identify influential and non-influential parameters within an aquatic ecosystem model. Ecol. Model. 246, 119L 130. Conley, D.J., Paerl, H.W., Howarth, R.W., Boesch, D.F., Seitzinger, S.P., Havens, K.E., Lancelot, C., Likens, G.E., 2009. Controlling Eutrophication: Nitrogen and Phosphorus. Science, 1014–1015. DeJonge, K.C., Ascough, J.C., Ahmadi, M., Andales, A.A., Arabi, M., 2012. Global sensitivity and uncertainty analysis of a dynamic agroecosystem model under different irrigation treatments. Ecol. Model. 231, 113L 125. Gamerith, V., Neumann, M.B., Muschalla, D., 2013. Applying global sensitivity analysis to the modelling of flow and water quality in sewers. Water Res. 47, 4600L 4611.

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