Abstract

Chaohu Lake is a large shallow lake in eastern China, and few eutrophication model studies have been conducted there. We present practical sensitivity indices based on the Morris method to compare the sensitivity of a parameter group on one model output with that of one parameter on multiple model outputs. The new sensitivity indices were employed to measure the parameter sensitivity of the Chaohu Lake eutrophication model. The results of the sensitivity analysis demonstrate that the most sensitive parameters on cyanobacteria biomass, NH4, NO3, and PO4 were BMR, KDN, Nitm, and KRP, and the most sensitive parameter groups were algae-related, nitrogen-related, and phosphorus-related, which all directly participate in their cycles. Furthermore, Nitm, KRP, KDN, KHP, BMR, KTB, KTHDR, and KTCOD were the most important for the Chaohu Lake eutrophication model. The water environment characteristics, such as the cyanobacteria life stage in the simulated period, significantly affected parameter sensitivity. The power-law relationship between the new sensitivity index and the standard deviation of model variables in the Chaohu Lake model were also determined. This finding allows us to estimate the interactions between parameters using their sensitivity index. The results provide a basis for further improvement of the Chaohu Lake eutrophication model.

INTRODUCTION

Chaohu Lake, located in the central Anhui Province of China, is one of the five largest fresh water lakes in China. The lake plays a key role as a water supply source for the Chaohu Lake district. Previous studies of Chaohu Lake have been based on in-situ observations, attempting to reveal eutrophication mechanisms (Zhang et al. 2008; Chen et al. 2011). Systematic studies with a focus on eutrophication models are rare. An effective eutrophication model is vital for water resources management, environmental engineering and design, and pollution control, thereby necessitating the building of a eutrophication model for Chaohu Lake.

Complex eutrophication models (Fennel et al. 2001; Zheng et al. 2012) often contain dozens or even hundreds of sophisticated and difficult-to-measure parameters. However, only a few input parameters significantly affect the model output (Saltelli et al. 2000). Therefore, it is necessary to measure the sensitivity of input parameters on model results to differentiate the most important parameters for model outputs (Borgonovo 2007) and properly establish a eutrophication model.

In general, sensitivity analysis (SA) methods can be divided into two classes: local sensitivity analysis (LSA) and global sensitivity analysis (GSA) (Cariboni et al. 2007; Pianosi et al. 2016). The LSA method is based on the partial derivative, which is determined by the ‘local’ character of model outputs. It reflects the influence of one parameter on the model output, while keeping other parameters fixed; therefore, it does not allow for an assessment of the interactions between the parameters influencing the output (Saltelli et al. 2008; Saltelli & Annoni 2010). GSA methods, which allow all parameters to change simultaneously across the whole input space (Campolongo et al. 2000; Saltelli et al. 2000), can overcome the limitations of the LSA method. GSA methods mainly include variance-based methods, elementary effect methods, and derivative-based global sensitivity methods (Sobol 1990, 2001; Morris 1991; Kucherenko et al. 2009; Nossent et al. 2011). Elementary effect methods (King & Perera 2013; Touhami et al. 2013; Qin et al. 2016) are the most widely used methods.

The elementary effect method was originally proposed by Morris (Morris 1991). It was revised by Campolongo et al. (2007) to avoid the compensating effects of opposite signs. The advantages of this method are simple in theory and require fewer model evaluations. However, the Morris method can only provide sensitivity for one parameter on one output. As a result, new normalised sensitivity indices based on the Morris method were proposed. The indices allow a comparison of the effect of the sensitivity of a parameter group on one model output with that of one parameter on multiple model outputs.

Many eutrophication models have been developed and applied worldwide (Jørgensen et al. 1978; Toro et al. 1983; Arhonditsis & Breet 2005). Among these models, the Environmental Fluid Dynamic Code (EFDC) model has been successfully used in many typical lakes in China (Wang et al. 2015; Gong et al. 2016; Qi et al. 2016), especially for shallow lakes and with relation to hydrodynamics, water age, and COD (Li et al. 2011; Hua et al. 2013; Huang et al. 2016). Studies on parameter sensitivity regarding EFDC have mainly involved hydrodynamic parameters (Li et al. 2012, 2015), and they seldom focus on parameters in a eutrophication kinetics module. Furthermore, no parameter sensitivity analysis has been performed on the eutrophication model for Chaohu Lake. Therefore, the sensitivity of eutrophication kinetics parameters is still a problem to be solved, especially in Chaohu Lake.

Due to the very high computational cost of the sensitivity analysis of the Chaohu Lake eutrophication model, we re-implemented the eutrophication module of the EFDC model. The spatial heterogeneity was ignored, as it does not substantially influence the results of the model, in order to reduce the partial differential equations (PDEs) to ordinary differential equations (ODEs). The new sensitivity indices presented in this paper were employed to estimate the parameter sensitivity of the Chaohu Lake eutrophication model. Furthermore, the model parameters were divided into seven types according to their role in the modelling processes, and the sensitivity of the parameters was computed. The most sensitive parameters and parameter types regarding the main water quality indicators were also obtained. Our research is valuable in that it can help establish a well-designed eutrophication model for Chaohu Lake.

SENSITIVITY INDEX

We assume that the model is and the input parameter is . For a given X, the elementary effect of the ith input parameter for the jth baseline point is defined as:  
formula
(1)
where , p is the number of sampling points of the parameter. should be determined with a special sampling strategy, and the definition of the two sensitivity indices, and , from the Morris method (Campolongo et al. 2007) are and , where r is the number of repeated samplings, estimates the impact of the ith input parameter on the model output, and estimates the interaction of the ith parameter with other parameters.
The original sensitivity index of the Morris method depends on the scale of the model measurements, which varies in magnitude in eutrophication models. Therefore, to remove the influence of these factors, the normalised sensitivity index shown below improves the Morris method. For the jth baseline point, the normalised is defined as:  
formula
(2)
where is the proportion of the ith parameter elementary effect of the total elementary effects of the input parameters. It is easier to compare the sensitivity of the parameters with each other at any baseline point after normalisation.
Similarly, the normalised global sensitivity index for the ith input parameter repeated r times is defined as:  
formula
(3)
where the value of the normalised global sensitivity index is between 0 and 1. It is easily obtained using .

The normalised global sensitivity index shows that the total sensitivity of all parameters on one model output is equal to 1. Therefore, is accounted for by the percentage of the total sensitivity. The sensitivity of the parameter group on one model output can be calculated by adding the values of all parameters in the group.

The sensitivity of the same parameter on all model output variables is difficult to compare using the original index of the Morris method. However, another index proposed in this study has unified comparison standards. It compares the average sensitivity of each parameter on multiple model output variables. The sensitivity index of the ith parameter on the mth variable can be written as . The index , defining the average sensitivity of the ith parameter on multiple model output variables, is defined as:  
formula
(4)
where , n is the number of variables and reflects the average sensitivity of the ith parameter, which is accounted for by the percentage of the total average sensitivity of model variables. The new sensitivity index overcomes the scale limitations of the original Morris method and can compare the sensitivity of both parameters and parameter groups on model output variables.

CHAOHU LAKE EUTROPHICATION MODEL

Study area

Chaohu Lake is approximately 54.4 km long from east to west and 21 km long from north to south. Its area is approximately 780 km2, with a mean depth of 3 m. The lake plays a key role in eastern China, and is an important water supply source for the Chaohu Lake district. However, pollution has become a serious problem in recent decades, and it is now one of the most eutrophic lakes in China. Therefore, the eutrophication of Chaohu Lake is of significant concern (Peimin et al. 1991; Xie 2009; Yang et al. 2013).

There are approximately 33 rivers around the Chaohu Lake, of which nine are major rivers (Figure 1). The flow of these rivers accounts for approximately 95% of the total flow. All modelling data were collected from observations (Figure 1). The monitoring points spread throughout Chaohu Lake are representative of the water quality conditions of Chaohu Lake; therefore, the mean values are representative of the overall water quality in Chaohu Lake.

Figure 1

Schematic diagram of Chaohu Lake and rivers.

Figure 1

Schematic diagram of Chaohu Lake and rivers.

Water was sampled at two depths for every observation point in Chaohu Lake and on boundary rivers to obtain water quality data. One position was at 0.5 m depth and the other was 0.5 m from the bottom. Every water sample was measured, and measurements showed that the water in Chaohu Lake was well mixed through its depth during the simulation period. The sampling frequency of monitoring sites in Chaohu Lake (Figure 1) was roughly weekly. The flow and water quality data in the boundary rivers of Chaohu Lake were sampled once a day. The driving force data were sampled by automatic observation stations once every hour.

The average ratio of total nitrogen to total phosphorus was 27, and the ratio of the sum of ammonia and nitrate to phosphate was approximately 53, which indicated that the growth of the algae was limited by phosphorus in Chaohu Lake during the sampling period.

Model description

The Chaohu Lake eutrophication model presented here was constructed based on the EFDC eutrophication module (Tetra Tech 2007), including five main cycles of carbon, nitrogen, phosphorus, oxygen, and silica. Four kinds of algae, diatom algae, green algae, stationary algae, and cyanobacteria, were included in the model. Three state variables of organic carbon, organic nitrogen, and organic phosphorus were also considered, as well as inorganic nutrients, ammonium (NH4), nitrate (NO3), and total phosphate (PO4). There are a total of 21 water quality variables and approximately 110 parameters in the model.

Due to algal blooms in the summer, mainly caused by cyanobacteria in Chaohu Lake, and the sum of other algal biomass being less than 10% of the total algal biomass, only cyanobacteria were considered in the model and other algae were ignored. Some variables, parameters, and processes, which only related to the other algal types, such as particulate biogenic silica, were also removed from the model. A schematic diagram of the Chaohu Lake eutrophication model is shown in Figure 2. In total, 15 variables, 68 parameters, and 30 processes (shown in Tables 1, 2, and 3, respectively) were involved. The main processes were still included in the model and the equations for the cycles of variables were the same as in the EFDC.

Table 1

Fifteen variables used in the Chaohu Lake eutrophication model

ShorthandVariableShorthandVariable
Algae cyanobacteria RPON refractory particulate organic nitrogen 
RPOC refractory particulate organic carbon LPON labile particulate organic nitrogen 
LPOC labile particulate organic carbon DON dissolved organic nitrogen 
DOC dissolved organic carbon NH4 ammonia nitrogen 
RPOP refractory particulate organic phosphorus NO3 nitrate nitrogen 
LPOP labile particulate organic phosphorus COD chemical oxygen demand 
DOP dissolved organic phosphorus DO dissolved oxygen 
PO4 total phosphate   
ShorthandVariableShorthandVariable
Algae cyanobacteria RPON refractory particulate organic nitrogen 
RPOC refractory particulate organic carbon LPON labile particulate organic nitrogen 
LPOC labile particulate organic carbon DON dissolved organic nitrogen 
DOC dissolved organic carbon NH4 ammonia nitrogen 
RPOP refractory particulate organic phosphorus NO3 nitrate nitrogen 
LPOP labile particulate organic phosphorus COD chemical oxygen demand 
DOP dissolved organic phosphorus DO dissolved oxygen 
PO4 total phosphate   
Table 2

Parameters in the Chaohu Lake eutrophication model

 ParameterValueUnitDescriptionType
PM d−1 maximum growth rate under optimal conditions for cyanobacteria 
KHN 0.03 gNm−3 half-saturation constant for nitrogen uptake for cyanobacteria IV 
KHP 0.005 gPm−3 half-saturation constant for phosphorus uptake for cyanobacteria II 
KTG1 0.004 °C−2 effect of temperature below temperature 1 on growth for cyanobacteria VII 
KTG2 0.012 °C−2 effect of temperature above temperature 2 on growth for cyanobacteria VII 
BMR 0.1 d−1 basal metabolic rate at reference temperature for cyanobacteria 
KTB 0.0322 °C−1 effect of temperature on metabolism for cyanobacteria VII 
PRR 0.02 d−1 reference predation rate for cyanobacteria 
ALPH  exponential dependence factor 
10 KTP 0.0001 °C−1 effect of temperature on predation for cyanobacteria VII 
11 KTHDR 0.069 °C−1 effect of temperature on hydrolysis of particulate organic matter VII 
12 KRC 0.005 d−1 minimum dissolution rate of RPOC III 
13 KLC 0.02 d−1 minimum dissolution rate of LPOC III 
14 KDC 0.01 d−1 minimum respiration rate of DOC III 
15 KRCALG 0.0001 m3(gC)−1d−1 constant that relates dissolution of RPOC to algal biomass III 
16 KLCALG 0.0001 m3(gC)−1d−1 constant that relates dissolution of LPOC to algal biomass III 
17 KDCALG 0.001 m3(gC)−1d−1 constant that relates respiration to algal biomass III 
18 FCRP 0.25  fraction of predated carbon produced as RPOC III 
19 FCLP 0.5  fraction of predated carbon produced as LPOC III 
20 FCDP 0.25  fraction of predated carbon produced as DOC III 
21 KRORDO 0.001 gO2m−3 denitrification half-saturation constant for DO 
22 KHDNN 0.001 gNm−3 denitrification half-saturation constant for NO3 IV 
23 KHORDO 0.5 gO2m−3 oxic respiration half-saturation constant for DO 
24 CP1 60 gC(gP)−1 minimum carbon-to-phosphorus ratio 
25 CP2 0.01 gC(gP)−1 difference between minimum and maximum carbon-to-phosphorus ratio 
26 CP3 0.01 (gP)−1 m3 effect of dissolved phosphate concentration on carbon-to-phosphorus ratio 
27 KRP 0.005 d−1 minimum hydrolysis rate of RPOP II 
28 KLP 0.12 d−1 minimum hydrolysis rate of LPOP II 
29 KDP 0.2 d−1 minimum mineralization rate of DOP II 
30 KRPALG 0.2 m3(gC)−1d−1 constant that relates hydrolysis of RPOP to algal biomass II 
31 KLPALG 0.0001 m3(gC)−1d−1 constant that relates hydrolysis of LPOP to algal biomass II 
32 KDPALG 0.001 m3(gC)−1d−1 constant that relates mineralization to algal biomass II 
33 FPR 0.0001  fraction of metabolized phosphorus by cyanobacteria produced as RPOP II 
34 FPL 0.0001  fraction of metabolized phosphorus by cyanobacteria produced as LPOP II 
35 FPD 0.5  fraction of metabolized phosphorus by cyanobacteria produced as DOP II 
36 FPI 0.4998  fraction of metabolized phosphorus by cyanobacteria produced as PO4 II 
37 FPRP 0.03  fraction of predated phosphorus produced as RPOP II 
38 FPLP 0.07  fraction of predated phosphorus produced as LPOP II 
39 FPDP 0.4  fraction of predated phosphorus produced as DOP II 
40 FPIP 0.4  fraction of predated phosphorus produced as PO4 II 
41 KRN 0.005 d−1 minimum hydrolysis rate of RPON IV 
42 KLN 0.03 d−1 minimum hydrolysis rate of LPON IV 
43 KDN 0.01 d−1 minimum mineralization rate of DON IV 
44 KRNALG 0.0001 m3(gC)−1d−1 constant that relates hydrolysis of RPON to algal biomass IV 
45 KLNALG 0.0001 m3(gC)−1d−1 constant that relates hydrolysis of LPON to algal biomass IV 
46 KDNALG 0.001 m3(gC)−1d−1 constant that relates mineralization to algal biomass IV 
47 FNR 0.15  fraction of metabolized nitrogen by cyanobacteria produced as RPON IV 
48 FNL 0.25  fraction of metabolized nitrogen by cyanobacteria produced as LPON IV 
49 FND 0.5  fraction of metabolized nitrogen by cyanobacteria produced as DON IV 
50 FNI 0.1  fraction of metabolized nitrogen by cyanobacteria produced as NH4 IV 
51 FNRP 0.15  fraction of predated nitrogen produced as RPON IV 
52 FNLP 0.25  fraction of predated nitrogen produced as LPON IV 
53 FNDP 0.5  fraction of predated nitrogen produced as DON IV 
54 FNIP 0.1  fraction of predated nitrogen produced as NH4 IV 
55 KNit1 0.003 °C−2 effect of temperature below temperature 3 on nitrification rate VII 
56 KNit2 0.003 °C−2 effect of temperature above temperature 4 on nitrification rate VII 
57 KHNitDO gO2m−3 nitrification half-saturation constant for DO 
58 KHNitN gNm−3 nitrification half-saturation constant for NH4 IV 
59 Nitm 0.01 gNm−3d−1 maximum nitrification rate IV 
60 ANC 0.175 gN(gC)−1 nitrogen-to-carbon ratio in cyanobacteria 
61 KCD 20 d−1 oxidation rate of COD at reference temperature 
62 KTCOD 0.069 °C−1 effect of temperature on oxidation of COD VII 
63 KHCOD 0.5 gO2m−3 half-saturation constant of DO required for oxidation of COD 
64 KR 0.2 d−1 re-aeration coefficient 
65 AANOX 0.01  ratio of denitrification rate to oxic DOC respiration rate III 
66 KPO4T2D 0.8  distribution coefficient of particulate phosphate and dissolved phosphate II 
67 KHI 60 Wm−2 half-saturation for light limitation VI 
68 KESS 2.87 m−1 light extinction coefficient VI 
 ParameterValueUnitDescriptionType
PM d−1 maximum growth rate under optimal conditions for cyanobacteria 
KHN 0.03 gNm−3 half-saturation constant for nitrogen uptake for cyanobacteria IV 
KHP 0.005 gPm−3 half-saturation constant for phosphorus uptake for cyanobacteria II 
KTG1 0.004 °C−2 effect of temperature below temperature 1 on growth for cyanobacteria VII 
KTG2 0.012 °C−2 effect of temperature above temperature 2 on growth for cyanobacteria VII 
BMR 0.1 d−1 basal metabolic rate at reference temperature for cyanobacteria 
KTB 0.0322 °C−1 effect of temperature on metabolism for cyanobacteria VII 
PRR 0.02 d−1 reference predation rate for cyanobacteria 
ALPH  exponential dependence factor 
10 KTP 0.0001 °C−1 effect of temperature on predation for cyanobacteria VII 
11 KTHDR 0.069 °C−1 effect of temperature on hydrolysis of particulate organic matter VII 
12 KRC 0.005 d−1 minimum dissolution rate of RPOC III 
13 KLC 0.02 d−1 minimum dissolution rate of LPOC III 
14 KDC 0.01 d−1 minimum respiration rate of DOC III 
15 KRCALG 0.0001 m3(gC)−1d−1 constant that relates dissolution of RPOC to algal biomass III 
16 KLCALG 0.0001 m3(gC)−1d−1 constant that relates dissolution of LPOC to algal biomass III 
17 KDCALG 0.001 m3(gC)−1d−1 constant that relates respiration to algal biomass III 
18 FCRP 0.25  fraction of predated carbon produced as RPOC III 
19 FCLP 0.5  fraction of predated carbon produced as LPOC III 
20 FCDP 0.25  fraction of predated carbon produced as DOC III 
21 KRORDO 0.001 gO2m−3 denitrification half-saturation constant for DO 
22 KHDNN 0.001 gNm−3 denitrification half-saturation constant for NO3 IV 
23 KHORDO 0.5 gO2m−3 oxic respiration half-saturation constant for DO 
24 CP1 60 gC(gP)−1 minimum carbon-to-phosphorus ratio 
25 CP2 0.01 gC(gP)−1 difference between minimum and maximum carbon-to-phosphorus ratio 
26 CP3 0.01 (gP)−1 m3 effect of dissolved phosphate concentration on carbon-to-phosphorus ratio 
27 KRP 0.005 d−1 minimum hydrolysis rate of RPOP II 
28 KLP 0.12 d−1 minimum hydrolysis rate of LPOP II 
29 KDP 0.2 d−1 minimum mineralization rate of DOP II 
30 KRPALG 0.2 m3(gC)−1d−1 constant that relates hydrolysis of RPOP to algal biomass II 
31 KLPALG 0.0001 m3(gC)−1d−1 constant that relates hydrolysis of LPOP to algal biomass II 
32 KDPALG 0.001 m3(gC)−1d−1 constant that relates mineralization to algal biomass II 
33 FPR 0.0001  fraction of metabolized phosphorus by cyanobacteria produced as RPOP II 
34 FPL 0.0001  fraction of metabolized phosphorus by cyanobacteria produced as LPOP II 
35 FPD 0.5  fraction of metabolized phosphorus by cyanobacteria produced as DOP II 
36 FPI 0.4998  fraction of metabolized phosphorus by cyanobacteria produced as PO4 II 
37 FPRP 0.03  fraction of predated phosphorus produced as RPOP II 
38 FPLP 0.07  fraction of predated phosphorus produced as LPOP II 
39 FPDP 0.4  fraction of predated phosphorus produced as DOP II 
40 FPIP 0.4  fraction of predated phosphorus produced as PO4 II 
41 KRN 0.005 d−1 minimum hydrolysis rate of RPON IV 
42 KLN 0.03 d−1 minimum hydrolysis rate of LPON IV 
43 KDN 0.01 d−1 minimum mineralization rate of DON IV 
44 KRNALG 0.0001 m3(gC)−1d−1 constant that relates hydrolysis of RPON to algal biomass IV 
45 KLNALG 0.0001 m3(gC)−1d−1 constant that relates hydrolysis of LPON to algal biomass IV 
46 KDNALG 0.001 m3(gC)−1d−1 constant that relates mineralization to algal biomass IV 
47 FNR 0.15  fraction of metabolized nitrogen by cyanobacteria produced as RPON IV 
48 FNL 0.25  fraction of metabolized nitrogen by cyanobacteria produced as LPON IV 
49 FND 0.5  fraction of metabolized nitrogen by cyanobacteria produced as DON IV 
50 FNI 0.1  fraction of metabolized nitrogen by cyanobacteria produced as NH4 IV 
51 FNRP 0.15  fraction of predated nitrogen produced as RPON IV 
52 FNLP 0.25  fraction of predated nitrogen produced as LPON IV 
53 FNDP 0.5  fraction of predated nitrogen produced as DON IV 
54 FNIP 0.1  fraction of predated nitrogen produced as NH4 IV 
55 KNit1 0.003 °C−2 effect of temperature below temperature 3 on nitrification rate VII 
56 KNit2 0.003 °C−2 effect of temperature above temperature 4 on nitrification rate VII 
57 KHNitDO gO2m−3 nitrification half-saturation constant for DO 
58 KHNitN gNm−3 nitrification half-saturation constant for NH4 IV 
59 Nitm 0.01 gNm−3d−1 maximum nitrification rate IV 
60 ANC 0.175 gN(gC)−1 nitrogen-to-carbon ratio in cyanobacteria 
61 KCD 20 d−1 oxidation rate of COD at reference temperature 
62 KTCOD 0.069 °C−1 effect of temperature on oxidation of COD VII 
63 KHCOD 0.5 gO2m−3 half-saturation constant of DO required for oxidation of COD 
64 KR 0.2 d−1 re-aeration coefficient 
65 AANOX 0.01  ratio of denitrification rate to oxic DOC respiration rate III 
66 KPO4T2D 0.8  distribution coefficient of particulate phosphate and dissolved phosphate II 
67 KHI 60 Wm−2 half-saturation for light limitation VI 
68 KESS 2.87 m−1 light extinction coefficient VI 
Table 3

The main processes of the cycles and parameters included in the process

ProcessParametersProcessParameters
1. RPOC dissolution 11,12,15 16. DON produced by algae predated and metabolized 6–10,49,53,60 
2. LPOC dissolution 11,13,16 17. NH4 produced by algae predated and metabolized 6–10,50,54,60 
3. denitrification 14,15,21,22,65 18. RPOP hydrolysis 3,11,27,30 
4. heterotrophic respiration 14,17,23 19. LPOP hydrolysis 3,11,28,31 
5. RPOC produced by algae predated 8–10,18 20. DOP mineralization 3,11,29,32 
6. LPOC produced by algae predated 8–10,19 21. RPOP produced by algae predated and metabolized 6–10,24–26,33,37 
7. DOC produced by algae predated 8–10,20 22. LPOP produced by algae predated and metabolized 6–10,24–26,34,38 
8. RPON hydrolysis 2,11,41,44 23. DOP produced by algae predated and metabolized 6–10,24–26,35,39 
9. LPON hydrolysis 2,11,42,45 24. PO4 produced by algae predated and metabolized 6–10,24–26,36,40 
10. DON mineralization 2,11,43,46 25. PO4 uptake by algae 1–5,24–26 
11. nitrification 55–59,65 26. DO produced by photosynthesis 1–5,66 
12. nitrate uptake by algae 1–5,60 27. respiration 6–7 
13. NH4 uptake by algae 1–5,60 28. COD oxidation 61–63, 
14. RPON produced by algae predated and metabolized 6–10,47,51,60 29. re-aeration 64 
15. LPON produced by algae predated and metabolized 6–10,48,52,60 30. algal processes (growth, basal metabolism, predation) 1–10,67,68 
ProcessParametersProcessParameters
1. RPOC dissolution 11,12,15 16. DON produced by algae predated and metabolized 6–10,49,53,60 
2. LPOC dissolution 11,13,16 17. NH4 produced by algae predated and metabolized 6–10,50,54,60 
3. denitrification 14,15,21,22,65 18. RPOP hydrolysis 3,11,27,30 
4. heterotrophic respiration 14,17,23 19. LPOP hydrolysis 3,11,28,31 
5. RPOC produced by algae predated 8–10,18 20. DOP mineralization 3,11,29,32 
6. LPOC produced by algae predated 8–10,19 21. RPOP produced by algae predated and metabolized 6–10,24–26,33,37 
7. DOC produced by algae predated 8–10,20 22. LPOP produced by algae predated and metabolized 6–10,24–26,34,38 
8. RPON hydrolysis 2,11,41,44 23. DOP produced by algae predated and metabolized 6–10,24–26,35,39 
9. LPON hydrolysis 2,11,42,45 24. PO4 produced by algae predated and metabolized 6–10,24–26,36,40 
10. DON mineralization 2,11,43,46 25. PO4 uptake by algae 1–5,24–26 
11. nitrification 55–59,65 26. DO produced by photosynthesis 1–5,66 
12. nitrate uptake by algae 1–5,60 27. respiration 6–7 
13. NH4 uptake by algae 1–5,60 28. COD oxidation 61–63, 
14. RPON produced by algae predated and metabolized 6–10,47,51,60 29. re-aeration 64 
15. LPON produced by algae predated and metabolized 6–10,48,52,60 30. algal processes (growth, basal metabolism, predation) 1–10,67,68 
Figure 2

Schematic diagram of the Chaohu Lake eutrophication model structure.

Figure 2

Schematic diagram of the Chaohu Lake eutrophication model structure.

The original EFDC was implemented as a large PDEs system. The typical run time for lakes as large as Chaohu Lake is days, and the cost of the computational time is very expensive. Indeed, it is impossible to research parameter sensitivity on a personal computer.

The annual material input for some large lakes does not change significantly; thus, the lake can be considered a quasi-balanced system (Jørgensen & Bendoricchio 2001; Chen 2003) and can be studied using the box model (Asaeda & Van Bon 1997; Havens et al. 2001; Arhonditsis & Breet 2005; Mao et al. 2006). Previous studies have used this model to study Chaohu Lake, but they did not consider the spatial heterogeneity of Chaohu Lake, and reliable results were still obtained (Xu et al. 1999). Regardless, the ratio of extreme differences to minimum values of cyanobacteria, NH4, NO3, PO4, DO, and COD determined from simultaneous monitoring of the 25 observation points in Chaohu Lake during the simulation period were 33.1%, 25.2%, 28.4%, 35.6%, 32.5%, and 29.3%. Therefore, the spatial heterogeneity of Chaohu Lake is relatively weak (Schwarzenbach et al. 2002). We conclude that the box model can be used in Chaohu Lake and that spatial heterogeneity in the eutrophication model can be ignored to simplify the reduction of PDEs to ODEs.

Table 3 shows the main processes of the cycles involved in the model, and presents the serial numbers of parameters included in these processes. The corresponding parameters can be found in Table 2. The sum of the four distribution coefficients (FPR, FPL, FPD, and FPI) for basal metabolism should be unity; therefore, FPI was not directly involved in the parameter sensitivity calculation. Similarly, FPIP, FNI, FNIP, and FCLP were also not directly involved in the calculation; therefore, there were 63 free parameters involved in the modelling.

FORTRAN language was applied to solve the ODEs of the Chaohu Lake eutrophication model, and other steps were completed using R language (R Development Core Team 2009; Soetaert & Petzoldt 2010). The FORTRAN language program was compiled into a dynamic link library for R call. The computational speed of joint programming was 80 times faster than using the R language to finish all operations. This facilitated the sensitivity analysis, which had a typical run time of minutes. As shown in the following section, this module can simulate the average concentrations of water quality indicators in Chaohu Lake.

Results of the model simulation

The simulation time lasted from July 9 to September 27 in 2009, which was summer in Chaohu Lake. As mentioned, the boundary conditions were sampled once a day. The flow and water quality of nine major rivers were considered in the simulation. The driving force data, such as water temperature and light intensity, were sampled once an hour. The average values of all water quality samples collected simultaneously from 25 sites in Chaohu Lake were used for model validation. Concentrations of cyanobacteria, ammonia (NH4), nitrate (NO3), phosphate (PO4), dissolved oxygen (DO), and chemical oxygen demand (COD) in Chaohu Lake were selected for the model. The parameters used for simulation are shown in Table 2. One box model was built and used for the simulation and parameter sensitivity analysis in Chaohu Lake. The time step was 0.5 hour for calculation. The linear interpolations of average temperature and light intensity from automatic observation stations were listed in a table for the calculation program call.

Figure 3 shows the simulation results of water quality indicators in Chaohu Lake. The maximum simulated error for cyanobacteria was 59%. The average error for cyanobacteria was 29%. The maximum and average errors for NH4, NO3, and PO4 were 76% and 35%, 38% and 27%, and 32% and 24%, respectively. The maximum simulated error for DO was 41% and the average error was 28%. The average simulated error for COD was 25% and the maximum error was 40%.

Figure 3

Trends in the average values of cyanobacteria, NH4, NO3, PO4, DO, and COD in Chaohu Lake.

Figure 3

Trends in the average values of cyanobacteria, NH4, NO3, PO4, DO, and COD in Chaohu Lake.

The Pearson correlation coefficients for the simulated and observed values for cyanobacteria, NH4, NO3, PO4, DO, and COD were 0.59, 0.56, 0.66, 0.62, 0.64, and 0.68, respectively. The positive coefficients indicate that the box model simulation results had similar trends to the observed values. These correlation coefficients were not very large, suggesting the possibility of a non-linear relationship between simulated and observed values, and errors loaded by one box model neglected the spatial heterogeneity in Chaohu Lake. Despite this, the box model simulation results agreed with the observed values and trends in Chaohu Lake. This suggests that the ecological dynamics in Chaohu Lake are correctly reflected by the model. The results also indicate that, although the box model is not perfect, it is acceptable for the parameter sensitivity analysis.

The box model is suitable for long-term simulation; however, this is not necessary for a parameter sensitivity analysis. Technical and funding limitations make it difficult to collect long-term data in large shallow lakes, especially in China. In parameter sensitivity analysis studies in China, the simulation time is generally no longer than one year (Li et al. 2012; Zheng et al. 2012; Yi et al. 2016), and some studies only last several days (Li et al. 2015). In general, the simulation results indicate that one box model can be used for a parameter sensitivity analysis of the Chaohu Lake eutrophication model. The parameter values in Table 2 could be used as baselines to calculate the sensitivity of nutrient and cyanobacteria parameters.

RESULTS AND DISCUSSION OF SENSITIVITY ANALYSIS

The following section does not include a sensitivity analysis of DO and COD outputs because these processes are simple and have fewer direct parameters.

The sensitivity of the 63 parameters was determined using the new sensitivity index. The probability distributions of parameters were unknown so the probability distribution of the parameters was assumed to be uniform, with ±25% variation. The baselines of parameters are shown in Table 2. The results were calculated using R = (63 + 1) × 2,000 = 128,000, where the model was run with 11 levels, and 2,000 was the number of total repetitions of the Morris design. The model was also run using 2,500 total repetitions of the Morris design, but the calculation results were similar to the model with 2,000 total repetitions, indicating that the latter was sufficiently accurate for the sensitivity analysis of the model parameters. The results from some of the parameter groups were excluded from the model results, because they were unreasonable. For example, in real-world physics, the maximum values should be less than three times the maximum historical observation data for Chaohu Lake. The rule helped legitimize the results of the sensitivity analysis. A total of 843 of the 2,000 total repetitions met these selection requirements.

The 63 parameters in the Chaohu Lake eutrophication model were divided into seven types (see Table 2) according to their roles in the different processes, namely, I: algae-related, II: phosphorus-related, III: carbon-related, IV: nitrogen-related, V: dissolved-oxygen-related, VI: light-related, and VII: temperature-related. The sensitivity analysis results of the new index are presented in Figure 4. The top ten most sensitive parameters are marked with labels in the figures.

Figure 4

Results of the sensitivity analysis: (a) sensitivity analysis for cyanobacteria; (b) sensitivity analysis for NH4; (c) sensitivity analysis for NO3; (d) sensitivity analysis for PO4.

Figure 4

Results of the sensitivity analysis: (a) sensitivity analysis for cyanobacteria; (b) sensitivity analysis for NH4; (c) sensitivity analysis for NO3; (d) sensitivity analysis for PO4.

In Figure 4(a), parameters 6 (BMR) and 7 (KTB) of the cyanobacteria metabolic process are the two most sensitive parameters for cyanobacteria biomass. Parameters 3 (KHP) and 8 (PRR), which represented the cyanobacteria half-saturation constant for phosphorus uptake and the reference predation rate, also had strong influences on cyanobacteria biomass. Some parameters of the cyanobacteria growth process, such as 1 (PM), 2 (KHN), 4 (KTG1), and 5 (KTG2), had less effect on cyanobacteria biomass. The sum of the sensitivity index of the top ten rankings from the results accounted for 97.5% of the total sensitivity. These results showed that the sensitivity of the top ten parameters had a dominant influence on cyanobacteria biomass.

The parameters in type I, which were associated with algae-related processes, had a significant effect on cyanobacteria biomass. The sum of the sensitivity index for type I accounted for 43% of the total sensitivity. The types related to phosphorus and temperature processes accounted for 20% and 32% of the total sensitivity, respectively. This indicates the key roles of phosphorus and temperature in the cyanobacteria cycle. The ratios of parameter sensitivity for other types were all less than 2%, suggesting that they did not affect the cyanobacteria cycle.

In Figure 4(b) and 4(c), the results of sensitivity parameters are similar to the results of cyanobacteria biomass. The most sensitive parameters for NH4 and NO3 were Nitm and KRP, respectively. The most influential parameter group for NH4 and NO3 was the nitrogen-related type, which accounted for 80.2% and 66% of the total sensitivity. The sum of the sensitivity index of the top ten rankings accounted for 92.6% and 84% of the total sensitivity for NH4 and NO3. The parameters denoting temperature effects also had a significant influence on NH4. The sensitivity of type VII accounted for 9.4% of the total sensitivity for NH4. The sum of the sensitivity index for NO3 for temperature-related parameters accounted for 12.3% of the total sensitivity.

Figure 4(d) shows only 26 parameters because the for some parameters was almost zero. These parameters belonged to four types: I, II, VI, and VII so that only four types of parameters affected the cycle of PO4. The cycle of PO4 was hardly affected by other parameters for the growth of the algae was limited by phosphorus in Chaohu Lake.

In the results, the sum of the sensitivity index of the top ten rankings reached 99%. The two most sensitive parameters for PO4 were 27 (KRP) and 3 (KHP), which were directly related to the phosphorus cycle. In the top ten rankings, there were four other parameters (KDP, KRPALG, KLP, and KPO4T2D) that also directly participated in the phosphorus cycle. The sum of the sensitivity index for the phosphorus-related parameters accounted for 91% of the total sensitivity. These results suggest that type II parameters had the most influence on PO4 concentration.

In the top ten rankings, the sensitivity index for temperature-related (KTHDR, KTB) and algae-related parameters (BMR, PRR) accounted for 5.7% and 2.9% of the total sensitivity, respectively. These results suggest that the impact of these parameters was also significant, particularly considering that the sum of the sensitivity index for type VI was 0.4% and for other parameter types (III, IV and V) was almost zero.

Based on the above results, the most sensitive parameters and important parameter types for the main water quality indicators directly participate in their own cycles. Parameter types associated with all main water quality indicators, such as temperature-related parameters, were also important.

The rankings based on the for the most sensitive parameters in the Chaohu Lake eutrophication model are presented in Table 4. For the four water quality indicators, the of eight parameters (Nitm, KRP, KDN, KHP, BMR, KTB, KTHDR, and KTCOD) reached 85% of the sensitivity of all parameters. These parameters belonged to types I, II, IV, and VII, and the remaining parameter types can be ignored. The parameters directly related to environmental indicators or with the most processes are the most important for modelling.

Table 4

Top eight rankings of for the influence of parameters on the four water quality indicators

ParameterCyanobacteria τiNH4τiNO3τiPO4τiβi
59(Nitm) 0.00 0.13 0.57 0.00 17.53% 
27(KRP) 0.01 0.00 0.01 0.67 17.31% 
43(KDN) 0.00 0.56 0.05 0.00 15.21% 
3(KHP) 0.17 0.01 0.03 0.24 11.29% 
6(BMR) 0.38 0.01 0.02 0.01 10.27% 
7(KTB) 0.31 0.01 0.03 0.01 8.77% 
11(KTHDR) 0.01 0.06 0.01 0.05 3.27% 
62(KTCOD) 0.00 0.02 0.06 0.00 2.04% 
ParameterCyanobacteria τiNH4τiNO3τiPO4τiβi
59(Nitm) 0.00 0.13 0.57 0.00 17.53% 
27(KRP) 0.01 0.00 0.01 0.67 17.31% 
43(KDN) 0.00 0.56 0.05 0.00 15.21% 
3(KHP) 0.17 0.01 0.03 0.24 11.29% 
6(BMR) 0.38 0.01 0.02 0.01 10.27% 
7(KTB) 0.31 0.01 0.03 0.01 8.77% 
11(KTHDR) 0.01 0.06 0.01 0.05 3.27% 
62(KTCOD) 0.00 0.02 0.06 0.00 2.04% 

The environmental characteristics of Chaohu Lake had a significant effect on the sensitivity analysis of parameters. For example, that the growth of the algae was limited by phosphorus during the Chaohu Lake simulation period suggested that the growth of cyanobacteria was controlled by phosphorus rather than nitrogen. Therefore, the effect of nitrogen-related parameters on PO4 concentration through cyanobacteria growth, metabolism, and other related processes was removed, and type IV parameters had little effect on PO4. Similarly, the parameters related to DO and carbon were not involved in the PO4 cycle; therefore, parameters related to carbon and DO, such as 11 (KRC), 13 (KLC), and 23 (KHORDO), had almost no effect on PO4.

The cyanobacteria life stage during the simulation period also affected parameter sensitivity. The sensitivity of BMR and KTB, related to cyanobacteria metabolic processes, was larger than that of the cyanobacteria growth parameters (PM, KHP, KHN, KTG1, and KTG2) because the cyanobacteria biomass in Chaohu Lake was at or near its peak during that time. Therefore, cyanobacteria metabolic processes were more dominant than growth processes.

Moreover, a power-law relationship was found between the global sensitivity indices and for each water quality indicator in the Chaohu Lake model. This relationship is expressed by Equation (5), the coefficients of which are shown in Table 5:  
formula
(5)
The significance value of these relationships is p < 0.01 at a 95% confidence level, which indicates that the power-law relationship between and for each indicator is statistically reliable. This finding allows us to estimate the interactions between parameters using their sensitivity index.
Table 5

Coefficients used in Equation (5) for cyanobacteria, NH4, NO3, and PO4

 αCR2F
Cyanobacteria 0.87 2.39 0.99 334.10 
NH4 0.74 0.97 2,158.00 
NO3 0.81 2.11 0.90 546.20 
PO4 0.94 0.98 1,451.00 
 αCR2F
Cyanobacteria 0.87 2.39 0.99 334.10 
NH4 0.74 0.97 2,158.00 
NO3 0.81 2.11 0.90 546.20 
PO4 0.94 0.98 1,451.00 

CONCLUSIONS

New sensitivity indices, which can compare the sensitivity of parameters and parameter groups on model output variables, were proposed based on the Morris method. They were employed for a parameter sensitivity analysis of the Chaohu Lake eutrophication model using the re-implemented eutrophication module of EFDC. The model parameters were divided into seven types according to their roles in different processes. The sensitivity of the parameters and parameter types for the cycles of four water quality indicators were discussed. Then, the most sensitive parameters for the Chaohu Lake eutrophication model were proposed.

For the Chaohu Lake eutrophication model, the most sensitive parameters for cyanobacteria biomass, NH4, NO3, and PO4 were BMR, KDN, Nitm, and KRP, respectively, and the most sensitive parameter groups were algae-related, nitrogen-related, and phosphorus-related. Furthermore, the most important parameters for the model global outputs were Nitm, KRP, KDN, KHP, BMR, KTB, KTHDR, and KTCOD. The average sensitivity of these parameters accounted for 85% of the total average sensitivity. We also found that the sensitivity indices and had a power-law relationship for the state variables in the Chaohu Lake eutrophication model to estimate the interactions between parameters.

The analysis showed that the most sensitive parameters and parameter types for the water quality indicators in the Chaohu Lake eutrophication model directly participated in their own cycles. The results also showed that water environmental characteristics, such as the growth of the algae being limited by phosphorus and the life stage of cyanobacteria during the simulation period, significantly affected the sensitivity of parameters.

The new sensitivity indices are practical, particularly for complex eutrophication models. The results of the parameter sensitivity analysis reflect the characteristics of Chaohu Lake and provide a basis for further improvement of the Chaohu Lake eutrophication model. These benefits can also be extended to the study and management of other lakes.

ACKNOWLEDGEMENTS

This work was financially supported by the Major Science and Technology Program for Water Pollution Control and Treatment(2012ZX07103-005), National Natural Science Foundation of China (Grant Nos 51379060, 51739002), Qing Lan Project and PAPD Project, Jiangsu Province Water Conservancy Project(2016011), Jiangsu postgraduate scientific research and innovation projects (CXZZ13_0271,KYLX15_0474) and the Fundamental Research Funds for the Central Universities (2016B21214, 2015B2471, 2015B26614, 2015B36314).

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