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The benchmark problem g-function (Sobol’, 1990) is widely used as a test function in sensitivity analysis because of its strong nonlinearity and nonmonotonicity. The interaction terms of this function are defined as non-zero.
The function
$ f $ is defined in a$ k $ -dimensional unit cube,where
$ {a_i} \geqslant 0 $ are the parameters, and$ {x_i} $ are independent and uniformly distributed in the n-dimensional unit cube ($ {x_i} \in [0,1] $ ,$ \forall i $ ). The value of$ {a_i} $ determines the importance of the variable$ {x_i} $ since$ {a_i} $ determines the variation range of$ {g_i}({x_i}) $ :Thus, the higher the value of
$ {a_i} $ , the lower the importance of the corresponding variable$ {x_i} $ .The analytical formulae of the conditional variance and total variance of the g-function [Eq. (10)] are given by Saltelli and Sobol’ (1995). The first order partial variance
$ {V_i} $ iswhile the higher-order conditional variance is the product of the lower ones, i.e.,
$ {V_{12}}{\text{ = }}{V_1}{V_2} $ . The total variance$ V(f) $ can be expressed as the product of the first-order terms:Here we explore the sensitivity of variables
$ {x_i} $ using the CNOPSA method and the variance-based approach. Let$ k = 7 $ , and$({a_1},{a_2},...,{a_7}) = (0,1,4.5,9,99,99,99)$ . For the variance-based method, the values of$ N $ are 1024, 2048, 4096, 8192, 16 384, and 32 768, respectively. In the CNOPSA approach, the number of iterations is 100. The total sensitivity index (${S_{{\text{T}}_i}}$ ) of the variable$ {x_i} $ converges with the increase of the number of samples$ N $ (Fig. 1a). The comparison of analytical and numerical results (Table 1) shows that the variance-based method is accurate and feasible when evaluating the sensitivity of variable$ {x_i} $ with samples.Figure 1. Sensitivity analysis of seven variables in g-function. (a) Convergence of the numerical estimate of
${S_{{\text{T}}_i}}$ ; (b) Distribution of the optimal variables (CNOPSA method) and the parameter samples (Variance-based approach) in the two dimensions of variables$ {x_1} $ and$ {x_5} $ ; (c) Changes in the cost function$ f $ ($ \Delta f $ ) caused by the uncertainties in variables$ {x_i} $ ,$ i = 1,2,...,7 $ .Variable Sensitivity index (${S{\text{e} }_i}$) Total sensitivity index (${ { { S_{{\text{T} }_i}} } }$) Analytical Numerical $ {x_1} $ 0.999995 0.7872 0.787659 $ {x_2} $ 0.444444 0.24222 0.242377 $ {x_3} $ 0.094674 0.03432 0.034342 $ {x_4} $ 0.033058 0.01046 0.010466 $ {x_5} $ 0.000392 0.00011 0.000105 $ {x_6} $ 0.000392 0.00011 0.000105 $ {x_7} $ 0.000392 0.00011 0.000105 Table 1. The sensitivity index of the CNOPSA method (
${S{\text{e}}_i}$ ) and the analytical and numerical results of the total sensitivity index${S_{{\text{T}}_i}}$ using variance-based method.Although the sensitivity rankings of the variables
$ {x_i} $ ($ i = 1,2,...,7 $ ) obtained by the sensitivity index${S_{{\text{T}}_i}}$ and$ S{{\text{e}}_i} $ are consistent (Table 1), there are some differences between the results of the two methods. Firstly, if the uncertainty range of ith variable$ {x_i} \in [0,1] $ among seven variables is regarded as the line segment from 0 to 1, the optimized values for ith variable obtained by the CNOPSA method may not be located on both endpoints of the line segment. For example, when the sensitivity of the first variable$ {x_1} $ is judged by using the CNOPSA method [Eq. (7)], the optimized values for seven variables are (0.5,1,1,1,0,0,1) and (1,1,1,1,0,0,1), respectively. It is found that one of the optimized values is not located on the endpoint of the line segment for$ {x_1} $ (Fig. 1b). Thus, the optimized values obtained by the CNOPSA method that cause the maximum value of the cost function$ f $ is not simply the combination of the endpoints of the line segments of seven variables (that is, the boundary of the uncertainty ranges of the variables). Secondly, the maximal uncertainties due to the variables could be estimated deterministically using the CNOPSA method. However, the uncertainties may be underestimated using the variance-based method. Figure 1c shows that it is important to explore the maximum effect of the variable uncertainty, which includes the impact of variable uncertainty quantified by samples. The greater the influence of variable uncertainty on the function$ f $ , the more important and sensitive the variable is. If the maximum impact of variable uncertainty is small, the variables are insensitive, such as variables$ {x_5} $ ,$ {x_6} $ , and$ {x_7} $ . Finally, we also evaluated the computational efficiency of the two methods. The number of the function calls in the CNOPSA method is 16 000, which is less than 294 912 in the variance-based method. For a simple test g-function, the CNOPSA method is feasible to consider the maximum uncertainty of the variables$ {x_i} $ to the variable$ f $ . -
In this section the CNOPSA method is applied to a five-variable ecological-hydrological model, which includes the interaction among the atmosphere, vegetation and soil, to explore the sensitivity of physical parameters. The model considers one species of grass and divides the system into three layers: vegetation, soil surface, and the root zone (Zeng et al., 2004, 2005a, b, 2006). The first layer contains the living biomass (
$ {M_{\text{c}}} $ ), wilted biomass ($ {M_{\text{d}}} $ ), and water content in the vegetation canopy ($ {W_{\text{c}}} $ ); the latter two layers are described by the water content in a thin soil surface ($ {W_{\text{s}}} $ ) and water content in the root layer ($ {W_{\text{r}}} $ ). The only input of this model is atmospheric precipitation. The processes of evaporation, transpiration and runoff are considered in vegetation and non-vegetation areas. The model’s equations are:where
$ {\alpha ^*} $ is the maximum growth rate,$ {\alpha _{\text{r}}} $ is a fraction used to describe the portion of$ {R_{\text{s}}} $ falling from the s-layer to the r-layer. The terms$ G $ ,$ {D_{\text{c}}} $ , and$ {C_{\text{c}}} $ are the growth, wilting, and consumption of the living biomass respectively;$ \beta '{D_{\text{c}}} $ ,$ {D_{\text{d}}} $ , and$ {C_{\text{d}}} $ are the accumulation, decomposition and consumption of the wilted biomass; Atmospheric precipitation$ P $ is divided into three parts$ {P_{\text{c}}} $ ,$ {P_{\text{s}}} $ , and$ {P_{\text{r}}} $ , where${P_{\text{c}}}{\text{ = }}\min (P_{\text{c}}^{\text{*}},\;P)$ ,${P_{\text{s}}}{\text{ = min(}}P_{\text{s}}^{\text{*}}{\text{,}}\;P - {P_{\text{c}}}{\text{)}}$ ,$ {P_{\text{r}}}{\text{ = }}P - ({P_{\text{c}}} + {P_{\text{s}}}) $ represent the interception of atmospheric precipitation by the canopy, soil surface and root zone respectively,$ P_{\text{c}}^{\text{*}} $ and$ P_{\text{s}}^{\text{*}} $ represent the maximum precipitation input that can be intercepted by the canopy and soil surface;$ {E_{\text{s}}} $ is the pure evaporation from the soil surface,$ {E_{\text{r}}} $ is the water flux drawn up by roots and transported to the canopy,$ {E_{\text{c}}} $ is the sum of transpiration and the part of water evaporation accumulated on leaves;$ {R_{\text{c}}} $ ,$ {R_s} $ , and$ {R_{\text{r}}} $ are the runoff of the canopy, surface soil, and root zone, respectively;$ {Q_{{\text{sr}}}} $ is the conductive transport from the soil surface to root zone. This model is a self-organization ecosystem that only depends on the initial grassland state. Despite its simplicity, the model can clearly reveal the essential features of the complex system in Inner Mongolia, China. To facilitate mathematical analysis, the state variables and physical parameters are dimensionless. More details on the description of this model are available in Zeng et al. (2004, 2005a, b, 2006) and Sun and Mu (2009).According to a previous hypothesis (Zeng et al., 2005a, b, 2006), the effect of Eq. (13c) on
$ {W_{\text{s}}} $ and$ {W_{\text{r}}} $ is equivalent to reducing the precipitation reaching the soil surface, Eq. (13c) satisfies:Thus, Eqs. (10a), (10b), (10d), and (10e) form a new set of four-dimensional ordinary differential equations (ODEs). Referring to the values of model parameters in Zeng et al. (2004), the wilted biomass [Eq. (13b)] can be expressed as:
where
$ {\alpha ^*} $ ,$ \beta ' $ ,$ \beta $ ,${\varepsilon _{\rm{d}}}$ ,${\varepsilon '_{\rm{d}}}$ ,${\beta _{\rm{z}}}$ , and${\varepsilon _{{\rm{dz}}}}$ are the physical parameters (Table 2).We use the fourth-order Runge-Kutta method to solve the model, with a time step of$ {\text{d}}t = {1 \mathord{\left/ {\vphantom {1 {24}}} \right. } {24}} $ , which corresponds to half a month.ID Parameter Default Minimum Maximum The physical description 1 $ {\alpha ^*} $ 0.4 0.32 0.48 The maximum growth rate 2 $ \beta ' $ 0.5 0.40 0.60 The rate of accumulation of the wilted biomass 3 $ \beta $ 0.1 0.08 0.12 The characteristic wilting rate 4 ${\varepsilon _{\rm{d}}}$ 1.0 0.80 1.20 The exponential attenuation coefficient of the living biomass 5 ${\varepsilon '_{\rm{d}}}$ 1.0 0.80 1.20 The exponential attenuation coefficient of water content in rooting layer 6 ${\beta _{\rm{z}}}$ 0.1 0.08 0.12 The characteristic rate of the wilted biomass decomposition 7 ${\varepsilon _{{\rm{dz}}} }$ 1.0 0.80 1.20 The exponential attenuation coefficient of the wilted biomass Table 2. Symbols, default values, ranges, and physical meanings of the seven parameters used in the grassland ecosystem model.
In this ecological-hydrological model, the physical parameters are determined by the observation data regarding Inner Mongolia grassland (Zeng et al., 2004, 2005a, 5b). Multiple equilibrium states (such as the grassland and desert state) coexist in semi-arid areas, and the transition from grassland to desert is usually abrupt at the boundary (Zeng et al., 2004, 2005a, b, 2006). The wilted biomass is distributed uniformly over the soil surface, and its “shading effect” is also considered (Appendix B). Zeng et al. (2004) found that the shading of soil by the wilted biomass can effectively reduce evaporation from the soil surface, so as to conserve enough soil water to maintain vegetation growth. Zeng et al. (2006) also emphasized that the shading of the soil by wilted biomass reduces the sunlight reaching the soil surface and thus reduces the soil surface temperature and the evaporation from the soil surface. Related theoretical studies (Zeng et al., 2004, 2006) clearly show that the decrease of shading effect can lead to rapid desertification. Therefore, the wilted biomass is crucial for this land surface process model. It is necessary to reduce the influence of the simulated wilted biomass caused by the uncertainties in physical parameters in the future.
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Due to the importance of the wilted biomass (
$ {M_{\text{d}}} $ ), we select its seven parameters for examination [Eq. (14)]. Table 2 shows the default values, ranges, and physical meanings for each parameter. The parameters are assumed to follow a uniform distribution during sampling as their distribution is unknown. Considering the existence of multiple equilibrium states in the model, two different initial grassland states, i.e., states A (0.066, 0.452, 0.637, 0.564) and B (0.345, 0.37, 0.630, 0.513) under the same meteorological forcing are selected for four-dimensional ODEs in this study. Grassland state A evolves into a desert state over time. The living biomass increases rapidly and then decreases slowly, and the wilted biomass shows a rapid and gradual decline (Fig. 2a). For grassland state B, the living biomass and the wilted biomass increase gradually with time and finally develop into a grassland equilibrium state (Fig. 2b). To explore the importance and sensitivity of seven physical parameters to the wilted biomass, two optimization times ($ t = 5 $ years and$ t = 10 $ years) are selected. The variations of the wilted biomass affected by seven parameters are selected as the cost function. For the variance-based method, the value of$ N $ is 32 768. For the CNOPSA approach, the number of iterations is 100. -
To analyze how parameter uncertainty affects the simulation and prediction of the wilted biomass for initial grassland state A (Fig. 2a), we explored the sensitivity of seven parameters for different optimization times using the CNOPSA method and the variance-based approach.
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We first discuss the sensitivity of single parameters. The sensitivity indices of the parameters (Fig. 3a) show that the sensitivity rankings obtained by the two methods are different. For the CNOPSA method (
$ {S{\rm{e}}_i} $ ), the ranking from large to small is$ {\varepsilon _{\text{d}}},{\text{ }}{\varepsilon _{{\text{dz}}}},{\text{ }}\beta ,{\text{ }}{\alpha ^*},{\text{ }}\beta ',{\text{ }}{\beta _{\text{z}}} $ , and$ {\varepsilon '_{\text{d}}} $ , while it is$ {\varepsilon _{{\text{dz}}}},{\text{ }}{\beta _{\text{z}}},{\text{ }}{\varepsilon _{\text{d}}},{\text{ }}\beta ,{\text{ }}\beta ',{\text{ }}{\varepsilon '_{\text{d}}}{\text{, and }}{\alpha ^*} $ for the variance-based approach ($ {S_{ {{\rm{T}}_i}}} $ ). The most sensitive parameters identified by the two methods are$ {\varepsilon _{\text{d}}} $ and$ {\varepsilon _{{\text{dz}}}} $ , respectively. Thus, which parameter is more important for the wilted biomass arises.Figure 3. The sensitivity indices of the parameters identified by the CNOPSA method and the variance-based approach for grassland state A. (a) and (b) represent single parameter at 5 years and 10 years, respectively; (c) and (d) represent two-parameter combination at 5 years and 10 years, respectively.
Subsequently, we selected sensitive parameters
$ {\varepsilon _{\text{d}}} $ ,$ {\beta _{\text{z}}} $ , and$ {\varepsilon _{{\text{dz}}}} $ to investigate the reasons for the differences in the sensitivity rankings. Firstly, Figs. 4a-c show that the influences of the wilted biomass caused by the uncertainties in the parameters$ {\varepsilon _{\text{d}}} $ ,$ {\beta _{\text{z}}} $ , and$ {\varepsilon _{{\text{dz}}}} $ obtained from the optimal parameters is greater than that of parameter samples, especially for$ {\varepsilon _{\text{d}}} $ (Fig. 4a). Secondly, similar to the numerical results in section 3, the uncertainty ranges of the parameters are expressed as the line segments. It is shown that the optimized values are (0.48, 0.6, 0.12, 1.2, 1.0335, 0.08, 0.8) and (0.48, 0.6, 0.12, 0.8, 1.0335, 0.08, 0.8) when the sensitivity of$ {\varepsilon _{\text{d}}} $ is judged by using the CNOPSA method (Table 3, and Fig. 5a). The optimized value for$ {\varepsilon '_{\text{d}}} $ (1.0335) is not located on the endpoints of the line segment from 0.8 to 1.2. Consequently, the optimized values of the parameters that lead to the greatest uncertainty in the simulated wilted biomass using the CNOPSA method is not the combination of endpoints of the line segments (that is, the boundary values of the uncertainty ranges of the parameters). Finally, compared with$ {\beta _{\text{z}}} $ and$ {\varepsilon _{{\text{dz}}}} $ , the parameter samples of$ {\varepsilon _{\text{d}}} $ significantly underestimates the variations of four physical variables in the variance-based method (Figs. 6a-d), especially for the wilting of living biomass$ {D_{\text{c}}} $ (Fig. 6a). In contrast, the CNOPSA method fully considers the possibilities of all parameters in the whole parameter space. Thus, the optimization parameters of the CNOPSA approach are important for the simulation of the wilted biomass, which can easily cause a shift from desert to grassland state in the model.Figure 4. Relative changes of the wilted biomass affected by the optimal parameters of the CNOPSA method and parameter samples in the variance-based approach within different optimization times for grassland state A. (a), (b) and (c) are the results of
$ {\varepsilon _{\text{d}}}{\text{, }}{\beta _{\text{z}}}{\text{, and }}{\varepsilon _{{\text{dz}}}} $ within 5 years; (d), (e) and (f) are the results of$ \beta ,{\text{ }}{\varepsilon _{\text{d}}}{\text{,}} $ and$ {\varepsilon _{{\text{dz}}}} $ within 10 years.Optimized parameter vector ${\boldsymbol{P}}_1^\delta$ Optimized parameter vector ${\boldsymbol{P}}_2^\delta$ Cost function value (0.48, 0.4, 0.12, 1.2, 0.8, 0.12, 1.2) (0.48, 0.6, 0.08, 0.8, 1.2, 0.08, 0.8) ${\boldsymbol{J}_{ {\rm{total} } } }$ 0.5573 (0.32, 0.6, 0.08, 0.80, 1.2, 0.08, 0.8) (0.48, 0.6, 0.08, 0.80, 1.2, 0.08, 0.8) $ {J_{{p_1}}} $ 0.0748 (0.48, 0.4, 0.08, 0.8, 1.2, 0.08, 0.8) (0.48, 0.6, 0.08, 0.8, 1.2, 0.08, 0.8) $ {J_{{p_2}}} $ 0.0687 (0.48, 0.6, 0.12, 0.9895, 0.8, 0.08, 0.8) (0.48, 0.6, 0.08, 0.9895, 0.8, 0.08, 0.8) $ {J_{{p_3}}} $ 0.0831 (0.48, 0.6, 0.12, 1.2, 1.0335, 0.08, 0.8) (0.48, 0.6, 0.12, 0.8, 1.0335, 0.08, 0.8) $ {J_{{p_4}}} $ 0.0925 (0.48, 0.6, 0.12, 0.9438, 0.8, 0.08, 0.8) (0.48, 0.6, 0.12, 0.9438, 1.2, 0.08, 0.8) $ {J_{{p_5}}} $ 0.0472 (0.48, 0.6, 0.08, 0.9698, 1.2, 0.12, 0.8) (0.48, 0.6, 0.08, 0.9698, 1.2, 0.08, 0.8) $ {J_{{p_6}}} $ 0.0631 (0.48, 0.6, 0.08, 0.9171, 1.2, 0.08, 1.2) (0.48, 0.6, 0.08, 0.9171, 1.2, 0.08, 0.8) $ {J_{{p_7}}} $ 0.0849 Note: The simulated wilted biomass caused by ${\boldsymbol{P}}_1^\delta$ is lower than that caused by $ {\boldsymbol{P}}_2^\delta $. Table 3. The optimized parameter vectors and corresponding cost function values of the CNOPSA method at 5 years for grassland state A.
Figure 5. Distribution of the optimal parameters (CNOPSA approach) at 5 years and parameter samples (Variance-based method) in the two dimensions. (a) Parameters
$ {\varepsilon _{\text{d}}} $ and$ {\varepsilon _{{\text{dz}}}} $ for grassland state A; (b) Parameters$ \beta ' $ and${\varepsilon _{\rm{d}}}$ for grassland state BFigure 6. The variations of
$ \Delta {D_{\rm{c}}} $ ,$ \Delta {D_{\rm{d}}} $ ,$ \Delta {M_{\rm{c}}} $ , and$ \Delta {M_{\rm{d}}} $ caused by single parameters and two-parameter combinations using the CNOPSA method and the variance-based approach for grassland state A at 5 years. (a), (b), (c) and (d) represent parameters$ {\varepsilon _{\rm{d}}} $ ,$ {\beta _{\rm{z}}} $ , and$ {\varepsilon _{{\rm{dz}}}} $ ; (e), (f), (g) and (h) represent two-parameter combinations ($ \beta $ ,$ {\varepsilon _{\rm{d}}} $ ) and ($ {\beta _{\text{z}}} $ ,$ {\varepsilon _{{\rm{dz}}}} $ ). -
Although we identified the sensitivity of single parameters to the wilted biomass above, the importance and sensitivity of the parameter combinations may be different from those of a single parameter due to the nonlinear synergistic effects among parameters. Consequently, the sensitivities of two-parameter combinations were identified using the two methods (Fig. 3c). For the variance-based approach (
${S_{{{\rm{T}}_{ij}}}}$ ), the most sensitive parameter combination is ($ {\beta _{\text{z}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ), corresponding to the two most sensitive parameters in subsection 4.2.1. Interestingly, for the CNOPSA method (${{{{S\rm e}}}_{ij}}$ ), the most sensitive two-parameter combination is ($ \beta $ ,$ {\varepsilon _{\text{d}}} $ ), which is not composed of the two most sensitive parameters$ {\varepsilon _{\text{d}}} $ and$ {\varepsilon _{{\text{dz}}}} $ . Figures 6e-h show that the variance-based approach significantly underestimates the variation of four variables caused by the uncertainties in parameter combinations using the limited parameter samples, especially for$ {D_{\text{c}}} $ (Fig. 6e). More importantly, according to the results of the CNOPSA method, the maximum influence of the wilted biomass caused by the uncertainty of the two-parameter combination ($ \beta $ ,$ {\varepsilon _{\text{d}}} $ ) is greater than that induced by the uncertainty of ($ {\beta _{\text{z}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ). Therefore, the uncertainties in the two-parameter combination ($ \beta $ ,$ {\varepsilon _{\text{d}}} $ ) are more likely affects the grassland state than that in ($ {\beta _{\text{z}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ). This implies that the two-parameter combination ($ \beta $ ,$ {\varepsilon _{\rm{d}}} $ ) is critical for the simulation and prediction of the wilted biomass and the grassland ecosystem state. -
Since the rapid and slow decline of the wilted biomass occurred within the first 10 years, we explored the sensitivity of the seven parameters at the optimization time of 10 years. Figures 3b and 3d illustrate the sensitivity indices of single parameter and two-parameter combinations quantified by the two methods. For the CNOPSA method, the two most sensitive parameters are
$ {\varepsilon _{\rm{d}}} $ and$ \beta $ , and the most sensitive two-parameter combination is ($ \beta ,{\text{ }}{\varepsilon _{\rm{d}}} $ ), while the corresponding results of the variance-based method are$ {\varepsilon _{\rm{d}}} $ ,$ {\varepsilon _{{{\rm{dz}}} }} $ , and ($ {\varepsilon _{\rm{d}}} $ ,$ {\varepsilon _{{\rm{dz}}}} $ ). It can be seen that$ {\varepsilon _{{\rm{d}}} } $ is one of the most sensitive parameters according to both methods. Thus, we focus on parameters with different sensitivity rankings between the two methods$ \beta $ and$ {\varepsilon _{{\rm{dz}}}} $ . Figures 4d-f show that although enough parameter samples are used in the variance-based approach, some parameter values that lead to significant changes in the wilted biomass were still missed. Since parameter samples fail to consider the nonlinear synergistic effects among parameters sufficiently, the degrees of underestimation of the parameters$ \beta $ and$ {\varepsilon _{\rm{d}}} $ are higher than that of$ {\varepsilon _{{\rm{dz}}}} $ when evaluating the changes of the four physical variables with discrete parameter samples (Figs. 7a-d). The underestimation of the two-parameter combination ($ \beta $ ,$ {\varepsilon _{\rm{d}}} $ ) is also greater than that of ($ {\varepsilon _{{\rm{d}}} } $ ,$ {\varepsilon _{{\rm{dz}}}} $ ) (Figs. 7e-h). In contrast, the results of the CNOPSA show that the accumulation of wilted biomass$ {D_{\rm{c}}} $ affected by$ {\varepsilon _{\rm{d}}} $ or two parameter combination ($ \beta $ ,$ {\varepsilon _{\rm{d}}} $ ) is the main factor to the simulation of the wilted biomass, and even affect the development of the grassland state.Figure 7. Same as in Fig. 6, but for 10 years. (a), (b), (c) and (d) represent single parameters
$ \beta $ ,$ {\varepsilon _{\rm{d}}} $ , and$ {\varepsilon _{{\rm{dz}}}} $ ; (e), (f), (g) and (h) represent two-parameter combinations ($ \beta $ ,$ {\varepsilon _{\rm{d}}} $ ) and ($ {\varepsilon _{\rm{d}}} $ ,$ {\varepsilon _{{\rm{dz}}}} $ ). -
To analyze how parameter uncertainty affects the simulation and prediction of the wilted biomass for different initial grassland states, we also investigated the sensitivity of seven parameters for different optimization times using the two methods in initial grassland state B (Fig. 2b).
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Here we discuss the sensitivity of the seven parameters to the simulated wilted biomass when the optimization time is five years. The sensitivity indices of seven parameters identified by the two methods are sorted differently (Fig. 8a). The sensitivity ranking of the seven parameters from large to small identified by the CNOPSA method (
${S{\rm{e}}_i}$ ) is$ {\varepsilon _{{\text{dz}}}},{\text{ }}{\varepsilon _{\text{d}}},{\text{ }}\beta ',{\text{ }}\beta ,{\text{ }}{\beta _{z} },{\text{ }}{\varepsilon '_{\text{d}}}{\text{ , and }}\;{\alpha ^*}, $ while the result of the variance-based approach (${S_{ {{\rm{T}}_i}}}$ ) is$ {\varepsilon _{{\text{dz}}}},{\text{ }}\beta ',{\text{ }}{\beta _{\text{z}}},{\text{ }}{\varepsilon _{\text{d}}},{\text{ }}\beta ,{\text{ }}{\varepsilon '_{\text{d}}} $ , and$ {\alpha ^*} $ .Similar to subsection 4.2.1, parameters
$ \beta ' $ ,$ {\varepsilon _{\text{d}}} $ , and$ {\varepsilon _{{\text{dz}}}} $ are chosen to analyze the differences between the two methods. Figures 9a-c show that the influence of the wilted biomass caused by the uncertainties in the parameters$ \beta ' $ ,$ {\varepsilon _{\text{d}}} $ , and$ {\varepsilon _{{\rm{dz}}}} $ obtained from the optimal parameters is greater than that of parameter samples, especially for$ {\varepsilon _{\text{d}}} $ (Fig. 9b). Table 4 and Fig. 5b also show that the optimized values using the CNOPSA method may be difficult to sample in the variance-based method when estimating the sensitivity of each parameter using the CNOPSA method. The reason is that the optimized values may not be a simple combination of endpoints of the line segments (that is, the boundary values of the uncertainty ranges of the parameters). Compared with$ \beta ' $ and$ {\varepsilon _{{\text{dz}}}} $ , the parameter samples of the parameter$ {\varepsilon _{\text{d}}} $ significantly underestimate the influence of parameter uncertainty on the changes of four variables (Figs. 10a-d). In contrast, the results of the CNOPSA method show that the uncertainty of the parameter$ {\varepsilon _{\text{d}}} $ leads to the most significant changes in the four variables than$ \beta ' $ . Thus, the limited discrete samples of$ {\varepsilon _{\text{d}}} $ cannot fully measure the changes of the four physical variables. The two most sensitive parameters$ {\varepsilon _{\text{d}}} $ and$ {\varepsilon _{{\text{dz}}}} $ obtained by the CNOPSA method are important and need special attention when exploring the development of the grassland ecosystem in this grassland state.Figure 9. Same as in Fig. 4, but for grassland state B. (a), (b) and (c) are the results of
$ \beta ',{\text{ }}{\varepsilon _{\text{d}}}{\text{, and }}\;{\varepsilon _{{\text{dz}}}} $ within 5 years; (d), (e) and (f) are the results of$ \beta ',{\text{ }}{\varepsilon _{\text{d}}}{\text{, and }}\;{\varepsilon _{{\text{dz}}}} $ within 10 years.Optimized parameter vector ${\boldsymbol{P}}_1^\delta$ Optimized parameter vector ${\boldsymbol{P}}_2^\delta$ Cost function value (0.48, 0.4, 0.12, 1.2, 0.8, 0.12, 1.2) (0.48, 0.6, 0.08, 0.8696, 1.2, 0.08, 0.8) ${\boldsymbol{J}_{ {\rm{total} } } }$ 1.0277 (0.32, 0.6, 0.08, 0.8, 1.2, 0.08, 0.8) (0.48, 0.6, 0.08, 0.8, 1.2, 0.08, 0.8) $ {J_{{p_1}}} $ 0.0692 (0.48, 0.4, 0.08, 1.1451, 1.2, 0.08, 0.8) (0.48, 0.6, 0.08, 1.1451, 1.2, 0.08, 0.8) $ {J_{{p_2}}} $ 0.1741 (0.48, 0.6, 0.12, 1.2, 0.8, 0.08, 0.8) (0.48, 0.6, 0.08, 1.2, 0.8, 0.08, 0.8) $ {J_{{p_3}}} $ 0.1511 (0.48, 0.6, 0.12, 1.2, 0.8, 0.08, 0.8) (0.48, 0.6, 0.12, 0.8, 0.8, 0.08, 0.8) $ {J_{{p_4}}} $ 0.1869 (0.48, 0.6, 0.12, 1.2, 0.8, 0.08, 0.8) (0.48, 0.6, 0.12, 1.2, 1.2, 0.08, 0.8) $ {J_{{p_5}}} $ 0.0917 (0.48, 0.6, 0.12, 0.8, 0.8, 0.12, 0.8) (0.48, 0.6, 0.12, 0.8, 0.8, 0.08, 0.8) $ {J_{{p_6}}} $ 0.1296 (0.48, 0.6, 0.08, 1.2, 1.2, 0.08, 1.2) (0.48, 0.6, 0.08, 1.2, 1.2, 0.08, 0.8) $ {J_{{p_7}}} $ 0.1900 Note: The simulated wilted biomass caused by ${\boldsymbol{P}}_1^\delta$ is lower than that caused by $ {\boldsymbol{P}}_2^\delta $. Table 4. Same as in Table 3, but for grassland state B.
Figure 10. Same as in Fig. 6, but for grassland state B. (a), (b), (c) and (d) represent single parameters
$ \beta ' $ ,$ {\varepsilon _{\text{d}}} $ , and$ {\varepsilon _{{\rm{dz}}}} $ ; (e), (f), (g) and (h) represent two-parameter combinations ($ \beta ' $ ,$ {\varepsilon _{{\rm{dz}}}} $ ) and ($ {\varepsilon _{\rm{d}}} $ ,$ {\varepsilon _{{\rm{dz}}}} $ ). -
We also discuss the sensitivity of parameter combinations for this grassland state in this subsection. Fig. 8c shows the sensitivity of the two-parameter combinations identified by the two methods. According to
${S{\text{e}}_{ij}}$ , the two-parameter combination ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) is the most sensitive to the simulated wilted biomass, while the result of$ {S_{{{\text{T}}_{ij}}}} $ is ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ). Figures 10e-h show that the limited parameter samples are not sufficient to represent the whole parameter space. The changes in the four variables caused by the uncertainties of the two-parameter combinations ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ) and ($ {\varepsilon _d} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) are underestimated in the variance-based method, especially for$ {D_c} $ (Fig. 10e). According to the CNOPSA method, the underestimation of the four variables caused by the uncertainty of ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) is greater than that of ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ). Thus, the uncertainty of ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) optimized by the CNOPSA method more likely affects the grassland state than that of ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ). Parameter combination ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) can easily affect the simulation of the wilted biomass and then transform the grassland state into a desert state or another grassland equilibrium state. -
We investigated the importance of the previously selected seven parameters for grassland state B at different optimization times. Figures 8b and 8d show the sensitivity indices of single parameter and two-parameter combinations to the wilted biomass. The two most sensitive parameters and the most sensitive two-parameter combination in the CNOPSA approach are
$ {\varepsilon _{{\text{dz}}}},{\text{ }}{\varepsilon _{\text{d}}} $ , and ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ), while the results of the variance-based method are$ {\varepsilon _{{\text{dz}}}},{\text{ }}\beta ' $ , and ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ). Figures 9d-f show that the discrete parameter samples may fail to consider all parameters’ possibilities when estimating the relative changes in the wilted biomass affected by the uncertainty of the parameters$ \beta ',{\text{ }}{\varepsilon _{\text{d}}}, $ and$ {\varepsilon _{{\text{dz}}}} $ . Figures 11a-d show that the discrete parameter samples in the variance-based method underestimate the impact of the uncertainty in parameters$ \beta ' $ ,$ {\text{ }}{\varepsilon _{\text{d}}} $ , and$ {\varepsilon _{{\text{dz}}}} $ on the four physical variables, especially for$ {\varepsilon _{\text{d}}} $ . The influence of the four physical variables caused by the uncertainty of the two-parameter combinations ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ) and ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) are also underestimated using parameter samples (Figs. 11e-h). The degrees of undervaluation of the parameters$ {\varepsilon _{\text{d}}} $ and ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) are more obvious than the others. In contrast, the CNOPSA method fully considers the nonlinear synergistic effects among parameters. The uncertainty of the two-parameter combination ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) leads to larger influence on the wilted biomass than the uncertainty of ($ \beta ' $ ,$ {\varepsilon _{{\text{dz}}}} $ ), which may easily transform the grassland state into a desert equilibrium state or another grassland equilibrium state with more biomass. Therefore, the calibration of uncertain parameters$ {\varepsilon _{{\text{dz}}}} $ and ($ {\varepsilon _{\text{d}}} $ ,$ {\varepsilon _{{\text{dz}}}} $ ) is important to the simulation and prediction of this grassland state.Figure 11. Same as in Fig. 10, but the optimization time is 10 years. (a), (b), (c) and (d) represent single parameters
$ \beta ' $ ,$ {\varepsilon _{\rm{d}}} $ , and$ {\varepsilon _{{\rm{dz}}}} $ ; (e), (f), (g) and (h) represent two-parameter combinations ($ \beta ' $ ,$ {\varepsilon _{{\rm{dz}}}} $ ) and ($ {\varepsilon _{\rm{d}}} $ ,$ {\varepsilon _{{\rm{dz}}}} $ ). -
Since the physical mechanisms for the model parameters are similar under different initial grassland states and optimization times, we only explain the physical meaning and the importance of the sensitive parameters when the optimization time is five years for grassland state A. According to the initial grassland state, the wilted biomass is about 6.8 times higher than the living biomass. The numerical results in subsection 4.2.1 and 4.2.2 show that the most sensitive parameter and two-parameter combination of the CNOPSA method are
$ {\varepsilon _{\rm{d}}} $ and ($ \beta ,{\text{ }}{\varepsilon _{\rm{d}}} $ ), while the results of the variance-based approach are$ {\varepsilon _{{\rm{dz}}}} $ and ($ {\beta _{\text{z}}},{\text{ }}{\varepsilon _{{\text{dz}}}} $ ). According to the role of each parameter (Sun and Xie, 2017), parameters$ \beta $ and${\varepsilon _{\rm{d}}}$ have a positive impact on the wilting of living biomass$ {D_{\rm{c}}} $ , and then affect$ {M_{\rm{d}}} $ directly and indirectly. Finally,$ \beta $ and$ {\varepsilon _{\rm{d}}} $ have negative feedbacks on${M_{\rm{d}}}$ (Table 3). Parameters${\beta _{\rm{z}}}$ and$ {\varepsilon _{{\rm{dz}}}} $ have direct negative feedback on$ {M_{\rm{d}}} $ (Table 3). The variance-based method that relies on parameter samples mainly underestimates the changes in$ {D_{\rm{c}}} $ caused by the uncertainty of parameters with direct and indirect effects and, in turn, underestimates the variation in the biomass$ {M_{\rm{c}}} $ and$ {M_{\rm{d}}} $ . Instead, the CNOPSA method considers the physical processes and the nonlinear synergistic effects among the physical parameters. It can effectively identify important and sensitive parameters. The results demonstrate that parameters with direct and indirect effects are crucial to the simulation and development of the wilted biomass due to the nonlinear synergistic effects among parameters. -
In this study, a new optimization method CNOPSA has been proposed to determine the sensitivity of physical parameters in numerical models. This method has the following characteristics:
(a) It investigates the whole parameter space, compensating for the dependency of limited parameter samples in the variance-based method.
(b) It considers the nonlinear synergistic effects among parameters.
(c) It eliminates the dependence on control runs in the model simulations.
(d) It quantifies the maximum influence of the model outputs caused by parameter uncertainty from the deterministic point of view.
We used the g-function and the five-variable grassland ecosystem model to test the CNOPSA method. In addition, we also compared the CNOPSA method with the variance-based approach. The results of the g-function show that the CNOPSA method is feasible and the sensitivity ranking of the variables is consistent with that of the variance-based method. Numerical results of the grassland ecosystem model show that the sensitivities of each parameter and two-parameter combinations depend on the initial grassland state and the optimization time. The variance-based method may underestimate the sensitivities of the parameters and two-parameter combinations because the discrete parameter samples may fail to consider all possible in the parameter space. However, the CNOPSA method can fully consider the nonlinear synergistic effects among parameters. The influence of parameter uncertainty on the simulated wilted biomass optimized by the CNOPSA method is greater than that measured by parameter samples in the variance-based approach.
For grassland state A, the wilted biomass is most sensitive to the exponential attenuation coefficient of living biomass
$ {\varepsilon _{\rm{d}}} $ and its combination with the characteristic wilting rate$ \beta $ at two optimization times (${{t}} = 5$ years and${{t}} =$ 10 years). The sensitive parameters are important to the wilting of the living biomass$ {D_{\rm{c}}} $ , which is an important component of the wilting biomass accumulation$ \beta '{D_{\rm{c}}} $ . The wilting biomass accumulation directly affects the shading effect of the wilted biomass and then influences the simulation of the wilted biomass$ {M_{\rm{d}}} $ . For grassland state B, the exponential attenuation coefficient of wilted biomass$ {\varepsilon _{{{\rm{dz}}} }} $ and the two-parameter combination ($ {\varepsilon _{{\rm{d}}} } $ ,$ {\varepsilon _{{{\rm{dz}}} }} $ ) are most sensitive to the wilted biomass for the optimization times of 5 and 10 years. They mainly affect$ \beta '{D_{\rm{c}}} $ and the decomposition of the wilted biomass$ {D_{\rm{d}}} $ . The two physical processes determine the wilted biomass$ {M_{\rm{d}}} $ and affect the development of this grassland state.In this paper only the benchmark problem and a theoretical grassland ecosystem model were used to test the usefulness of the CNOPSA method. In the future, we will examine the parameter uncertainty and sensitivity in a more complicated earth system model. In addition, calibrating parameters to improve the simulation ability and prediction skills of the model focusing on the sensitive and important parameters is a great challenge. Relevant work is also currently under investigation.
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As an extension of the CNOP-P method (Appendix C), the CNOPSA method eliminates the control run of the CNOP-P method based on a parameter reference state. Thus, the parameter sensitivity relative to a parameter reference state is extended to the overall sensitivity of the parameters in the whole parameter space. Table 5 shows the sensitivity results of the CNOPSA method and the CNOP-P approach. The results suggest that the important and sensitive physical parameters identified by the CNOP-P method may be dependent on the parameter reference states. However, the CNOPSA method measures the sensitivity of parameters in the whole parameter space. The two methods have different research emphases, and we currently pay more attention to the sensitivity of parameters in the whole parameter space.
Initial state Method Parameter sensitivity from large to small Grassland state A CNOPSA $ {\varepsilon _{\rm {d}} } $, $ {\varepsilon _{\rm {dz}}} $, $ \beta $, $ {\alpha ^*} $, $ \beta ' $, $ {\beta _{\rm z}} $, $ {\varepsilon '_{\rm {d}}} $ CNOP-P (1*) $ {\varepsilon _{\rm {dz}}} $, $ {\beta _{\rm z} } $, $ {\varepsilon _{\rm {d}}} $, $ \beta $, $ {\varepsilon '_{\rm {d}}} $, $ \beta ' $, $ {\alpha ^*} $ CNOP-P (2*) $ {\varepsilon _{\rm {d}}} $, $ {\varepsilon _{\rm {dz}}} $, $ \beta $, $ \beta ' $, $ {\beta _{\rm z}} $, $ {\varepsilon '_{\rm {d}}} $, $ {\alpha ^*} $ Grassland state B CNOPSA $ {\varepsilon _{\rm {dz}}} $, $ {\varepsilon _{\rm{d}}} $, $ \beta ' $, $ \beta $, $ {\beta _{\rm z}} $, $ {\varepsilon '_{\rm{d}}} $, $ {\alpha ^*} $ CNOP-P (1*) $ {\varepsilon _{\rm {dz}}} $, $ {\beta _{\rm{z}}} $, $ \beta ' $, $ {\varepsilon _{\rm {d}}} $, $ \beta $, $ {\varepsilon '_{\rm {d}}} $, $ {\alpha ^*} $ CNOP-P (2*) $ {\varepsilon _{\rm {dz}}} $, $ {\varepsilon _{\rm{d}}} $, $ \beta ' $, $ \beta $, $ {\beta _{\rm{z}}} $, $ {\varepsilon '_{\rm {d}}} $, $ {\alpha ^*} $ Note: 1* represents parameter reference state 1, and 2* represents parameter reference state 2. Table 5. The sensitivity rankings of the parameters identified by the CNOPSA method and the CNOP-P approach for different initial grassland states when the optimization time is five years.
In this study, we explored the maximum influence of parameter uncertainty on numerical simulation or prediction from the deterministic point of view. The extreme case of the influence of parameter uncertainty on numerical simulation, which may represent different physical states (such as grass or desert states in grassland ecosystem model), was investigated. The applications of this method show that the greater the maximum influence of parameter uncertainty on numerical simulation, the more important and sensitive are the parameters. If the uncertainty ranges of the parameters can be expressed as line segments, the optimized values of the parameters obtained by the CNOSPA method may not always be located on the ends of the line segments (that is, the boundary of the uncertainty ranges of the parameters). The variance-based method measures the influence of parameter uncertainty from a statistical point of view. In most numerical models, it considers the general situation of the influence of parameter uncertainty on numerical simulation by using the sample-based method. It may be difficult to measure the maximum influence of parameter uncertainty on model outputs from the parameter samples, especially for large parameter dimensions, which may directly affect the evaluation of parameter sensitivity. We encourage future research to further explore the usefulness of the CNOPSA method in more numerical models. Perhaps more interesting studies could be found in the future by combining the advantages of the CNOPSA method and the variance-based approach.
Variable | Sensitivity index (${S{\text{e} }_i}$) | Total sensitivity index (${ { { S_{{\text{T} }_i}} } }$) | |
Analytical | Numerical | ||
$ {x_1} $ | 0.999995 | 0.7872 | 0.787659 |
$ {x_2} $ | 0.444444 | 0.24222 | 0.242377 |
$ {x_3} $ | 0.094674 | 0.03432 | 0.034342 |
$ {x_4} $ | 0.033058 | 0.01046 | 0.010466 |
$ {x_5} $ | 0.000392 | 0.00011 | 0.000105 |
$ {x_6} $ | 0.000392 | 0.00011 | 0.000105 |
$ {x_7} $ | 0.000392 | 0.00011 | 0.000105 |