Sediment ColumnWater Column
VIII. Sediment Phosphorus 𝑑(𝑆𝑃)
𝑑𝑡 = 𝑟𝑝𝑠 − 𝑟𝑝𝑟𝑠 4. 112
4.8.2. Sensitivity analysis
Sensitivity analysis of a model can be broadly described as an effort to gage the sensitivity of the parameters, forcing functions or sub-models involved in an ecological model (Jørgensen and Bendoricchio, 2001). It is carried out a number of times during the modelling process of an ecological system to identify the most sensitive parts of the developed model. It can be also used to refine the most sensitive components of the model.
Sensitivity analysis is performed by incrementing or decrementing the components of the model by a certain amount, depending the uncertainty involved in the component, and sub- sequently recording the response of the state variables to such a change. The changes in the components can be carried out one-at-a-time (OAT) or all-at-a-time (AAT) (Pianosi et al., 2016). In the OAT method, the sensitivity analysis of the model is carried out by varying the value of only one of the components (generally a parameter) at a time while keeping the val- ues of other components fixed. On the other hand, all the components of the model are varied at the same time in the AAT method to understand the sensitivity of the all the components
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as well as the collective influence of the components on the model. In fact, based on the effect of collective influence of the components, the sensitivity analysis itself can be classified into two categories; local sensitivity analysis and global sensitivity analysis (Pianosi et al., 2016).
Local sensitivity analysis refers to that method of sensitivity analysis which neglects collec- tive influence of the components and typically uses OAT sampling method for estimation. On the other hand, global sensitivity analysis considers the effects of the joint interactions of the components on the model while utilizing either OAT or AAT sampling approaches for the es- timation, i.e., it considers simultaneous variation of all independent input parameters.
The sensitivity, S, of a parameter can be calculated using the following equation.
𝑆 =(𝜕𝑣 𝑣 ) (𝜕𝑃
𝑃 )
4. 113
where v is a state variable of the model and P denotes the parameters. The sensitivity of a sub-model can be evaluated by removing the sub-model entirely from the model or altering the mathematical equation involved in the sub-model. The consequent changes in the values of the state variables provides a qualitative measure of the sensitivity of that sub-model. Such sensitivity analysis can be helpful in structural modifications of the model.
Global sensitivity analysis of the one-dimensional ecological model was carried out by One-At-a-Time (OAT) sampling approach, following the method described by Morris (1991).
In the Morris method, it is assumed that if the input variables (parameters) are changed by the same relative amount, then the input variable that causes the highest variation in the out- put variable (state variable) is the most sensitive in the model. The Morris method is also known as the Elementary Effects method, as it calculates the elementary effect of change in the input variable (positive or negative) on the output variable. The elementary effect is cal- culated by Eq. 4. 114.
𝐸𝐸𝑖(𝑥) =[𝑦(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑖−1, 𝑥𝑖+ Δ, 𝑥𝑖+1, … , 𝑥𝑘) − 𝑦(𝑥)]
Δ 4. 114
where y is the output variable, 𝑥 is the input vector, and 𝑥𝑖 is an element in the input vec- tor. Δ is the change in 𝑥𝑖 and 𝑘 is the number of elements in the input vector. As pointed out by King and Perera (2013), the Morris method requires 2k simulations of the model to deter- mine the sensitivity of all the input variables.
In order to carry out sensitivity analysis using the Morris method, a trajectory of changes in the input variables of a k variable model was constructed. This was done by considering a normalized range of probable values for each input variable and dividing each range by equal
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intervals or levels. Base values of the input variables were randomly selected from 0 to ∆ to mark the starting point of the trajectory. The trajectory was then subsequently calculated by the final trajectory matrix B* as given by the following equations (Saltelli et al. 2008).
𝐵∗= {𝐽𝑚,1𝑥∗+ (Δ
2) [(2𝐵 − 𝐽𝑚,𝑘)𝐷∗+ 𝐽𝑚,𝑘]} 𝑃∗ 4. 115
where,
𝐵∗ is a strictly lower triangular matrix of 1’s 𝐽𝑚,𝑘 is a 𝑚 − 𝑘 matrix of 1’s where 𝑚 = 𝑘 + 1
𝑥∗ is the base input vector 𝐽𝑚−1 is a column vector of 1’s
𝐷∗ is a k-dimensional diagonal matrix in which each element is +1 or -1 with same probability
𝑃∗ is a k-by-k random permutation matrix of 0’s and 1’s in which each row contains a solitary 1 and varies from the other rows by the relative position of that 1.
The elementary effects were finally measured by two measures given by the following equations (Morris 1991; Campolongo et al. 2007) (Eq. 4. 115, 4. 116, and 4. 117).
𝜇𝑖∗=(∑𝑟𝑛=1|𝐸𝐸𝑛|)
𝑟 4. 116
𝜎𝑖= √1
𝑟∑(𝐸𝐸𝑛− 𝜇𝑖)2
𝑟
𝑛=1
4. 117
where, 𝜇𝑖 =∑𝑟𝑛=1𝐸𝐸𝑛
𝑟 4. 118
Moreover, r is the number of trajectories constructed during the analysis.
𝜇𝑖∗ is the absolute mean of all the elementary effects due to the ith input variable, free from non-monotonic behaviour of the elementary effects, which measures the degree of sensitivity of the input variables considered in the analysis. A high value of this sensitivity index indicates that the output variable is highly sensitive to the input variable considered. 𝜎𝑖 is the standard deviation of all the elementary effects due to ith input variable, higher value of which indicates non-linearity or interaction of the input variable with other variables of the model.
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4.8.3. Model calibration and validation
Calibration of the one-dimensional eutrophication model was carried out using the data ob- tained from October 2017 to October 2018, and the model was then validated using the data obtained during the period October 2018 to February 2019. Euler rectangular method of in- tegration was employed to solve all the differential equations involved in the model in MATLAB R2018b version.
4.9. Summary
This chapter aimed at providing to understand the limnology of wetlands in response to an- thropogenic interferences to the natural ecosystem. A framework for the research was de- signed, and this framework was further used to study different objectives. The changes oc- curring in the natural ecosystem were primarily attributed to heavy metal and nutrient con- tamination. Different novel methodologies adopted to study these changes have been pre- sented in this chapter and finally validated through a eutrophic-based ecological model.
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We never know the worth of water till the well is dry.
- Thomas Fuller
