Identifying parameter sensitivity in a water quality model of a reservoir

Sensitivity analysis can provide useful insights into how a model responds to the variations in its parameter values (i.e. coefficients). The results can be very helpful for model calibration, refinement and application. A one-dimensional model has been set up to simulate the hydrothermal and water quality conditions of Cannonsville Reservoir, which provides water supply for New York City. This paper aims at identifying the most influential parameters in the model through sensitivity analysis. Firstly, the Morris method (a screening method) is used to identify influential parameters. It is found that 18 parameters are important in simulations of variables that include temperature, dissolved oxygen (DO), total phosphorus (TP) and chlorophyll a (Chla). Secondly, the method is enhanced to investigate the global sensitivity of the parameters. It highlights 20 parameters that are sensitive in the simulations of the above-mentioned variables. The 18 parameters identified by the original Morris method are among the 20 parameters and the other two parameters are not very sensitive. The results show that similar results can be obtained through the original and enhanced Morris methods, although they each have their own strengths and weaknesses. doi: 10.2166/wqrjc.2012.116 s://iwaponline.com/wqrj/article-pdf/47/3-4/451/163555/451.pdf Yongtai Huang (corresponding author) CUNY Institute for Sustainable Cities, Hunter College, The City University of New York, 71 Smith Ave, Kingston, NY 12401, New York, USA E-mail: yhu0011@hunter.cuny.edu Don Pierson New York City Department of Environmental Protection, Kingston, New York, USA


INTRODUCTION
Water quality models are usually developed to simulate a large number of variables, such as water temperature, dissolved and particulate nutrients, phytoplankton and dissolved oxygen (DO). Such models often contain many parameters (also called coefficients) associated with the many processes and state variables simulated. Identifying  (Saltelli et al. ). This method has been used in several fields, such as water quality modeling (Huang & Liu ), watershed modeling (Francos et al. ), climate change prediction (Campolongo & Braddock ) and laboratory ground water flow and solute transport modeling (Larsbo & Jarvis ).
In this paper sensitivity analysis is applied to a onedimensional model which was set up to simulate the hydrothermal and water quality conditions of Cannonsville Reservoir, which is located in New York State and is part of the New York City water supply. The model is being applied to the evaluation of the effects of climate change and human activities on the hydrothermal structure and water quality of the reservoir. The main purpose of this study is to identify the sensitive parameters in the model.
First of all, the Morris method is used to identify the sensitive parameters affecting the simulations of four output variables including temperature, DO, total phosphorus (TP) and chlorophyll a (Chla). Secondly, the method is enhanced to find the overall sensitivity of the parameters in the simulations of the above-mentioned variables as a whole. The sensitive parameters are the ones to which most of the calibration efforts should be directed. is predicted in terms of algal carbon and is a balance between growth (photosynthesis) and losses due to respiration, grazing, sedimentation and outflow. Phytoplankton growth is limited by temperature, light and nutrient availability. storm water and septic system infrastructure improvement has greatly reduced these problems in recent times.

Model configuration
To set up the one-dimensional model to simulate the hydrothermal and water quality conditions, the reservoir is discretized into 35 vertical layers with the average thickness of 1.5 m per layer. The model inputs include daily meteorological data (air temperature, dew-point temperature, wind speed and solar radiation) and other data related to the water balance (water elevation, discharge, dam spill and aqueduct outflow).
In addition, the following time series input data (which are generated by a separate watershed model) are also input for the model: (1) streamflow, (2) dissolved phosphorus and nitrogen from non-point and point sources, (3) particulate phosphorus from non-point and point sources, (4) dissolved organic carbon from non-point sources, (5) total suspended solids (TSS), (6) silicon load and (7)   Assume that the model output y ¼ y(x) is a scalar function of the vector x of parameters. The vector x ¼ (x 1 , …, x i …, x N ) has N parameters. The range of each x i is normalized to be within 0 and 1. Assume each x i may have a number of discrete values in the set {0, 1/(p À 1), 2/(p À 1), …, 1}, where p is the number of levels. A perturbation factor Δ is defined as a multiple of 1/(p-1). The elementary effect of the ith parameter is defined as: where M represents the total number of simulated values at all locations and time points; y k (x) is the kth output.
A five-level grid in a two-dimensional input space (N ¼ 2) is provided as an example to demonstrate how elementary effects are calculated (see Figure 2). To evaluate the elementary effect of x 1 between A and B on the output, the outputs at points A and B need to be evaluated. In the same way, to evaluate the elementary effect of x 2 between C and D on the output, the outputs at points C and D need also to be In this sampling method, the total computational effort required for obtaining R elementary effects for each of N parameters is (N þ 1)R model runs. Although a characteristic of this sampling method is that the points belonging to the same trajectory are not independent, the R elementary effects for each parameter are from different trajectories and are independent. The mean of elementary effects for parameter x i can be calculated by: where σ(x i ) represents the standard deviation of d j (x i ).
where μ*(x i ) represents the absolute mean of d j (x i ).
where c is the concentration of DO in the vessel (mg/L); c s is the saturated concentration of DO in the vessel (mg/L); k d is the effective deoxygenation rate (T À1 ); L 0 is the initial dissolved and particulate carbonaceous biochemical oxygen demand (CBOD) concentrations (mg/L); k r is the overall loss rate of CBOD from the water column due to both settling and oxidation of soluble BOD; and t is time (T ).
Equation (6) can be taken as a model with two parameters and one output: c ¼ f(k d , k r ). Assume c s ¼ 16 mg/L,  (6) as

Enhanced Morris method
The Morris method is enhanced to evaluate the overall sensitivity of model output to the combined affects of multiple parameters as follows: By adding a subscript to indicate the variable, the elementary effect in Equation (2) can be rewritten as the following equation: where d s (x i ) represents the elementary effect of the ith component of x on the model output of the sth variable; y sk (x) is the kth output of the sth variable, where s ¼ 1, …, V and V is the total number of variables on which the overall sensitivity of the model output to the parameter are to be investigated.
The elementary effect, d s (x i ), in Equation (7) where e s (x i ) is the normalized elementary effect of parameter x i on the model output of the sth variable.
The overall elementary effect can then be calculated by: The mean of R overall elementary effects of parameter x i can be calculated by: where μ a (x i ) represents the mean of e j (x i ).
The mean of R absolute overall elementary effects of parameter x i can be calculated by: where μ a * represents the absolute mean of e j (x i ).

The standard deviation of the overall elementary effects
for parameter x i is given by: where σ a (x i ) represents the standard deviation of e j (x i ).
The statistic μ a * reveals the influence of parameter x i on the overall model output. The larger the value of μ a *, the more sensitive the parameter is.
The statistic σ a (x i ) indicates the relationship between parameter x i with overall model output and other parameters. A high value of the standard deviation, σ a (x i ), indicates that the parameter has a non-linear effect on the overall model output, or a parameter that is involved in interaction with other parameters, or both.

Sensitivity of parameters by Morris method
In the  As shown in Figure 4(b), the parameter, rz (diffusion exponent), has a high μ * and a low σ value. That is to say,  Table 1, where, in total, 18 parameters are found to be sensitive in the simulations of the above variables. They therefore become the focus in model calibration. and σ a are presented in Figure 8. It can be observed that 20 parameters are well separated from the others. They are  the most sensitive parameters in the simulation of temp, DO, TP and Chla. In addition, by comparing the parameters in Figure 8 and

SUMMARY AND CONCLUSION
In this study, the original and enhanced Morris methods  about the water quality model is needed to understand and explain the results.