  # A New Sensitivity Analysis Approach Using Conditional Nonlinear Optimal Perturbations and Its Preliminary Application

• Simulations and predictions using numerical models show considerable uncertainties, and parameter uncertainty is one of the most important sources. It is impractical to improve the simulation and prediction abilities by reducing the uncertainties of all parameters. Therefore, identifying the sensitive parameters or parameter combinations is crucial. This study proposes a novel approach: conditional nonlinear optimal perturbations sensitivity analysis (CNOPSA) method. The CNOPSA method fully considers the nonlinear synergistic effects of parameters in the whole parameter space and quantitatively estimates the maximum effects of parameter uncertainties, prone to extreme events. Results of the analytical g-function test indicate that the CNOPSA method can effectively identify the sensitivity of variables. Numerical results of the theoretical five-variable grassland ecosystem model show that the maximum influence of the simulated wilted biomass caused by parameter uncertainty can be estimated and computed by employing the CNOPSA method. The identified sensitive parameters can easily change the simulation or prediction of the wilted biomass, which affects the transformation of the grassland state in the grassland ecosystem. The variance-based approach may underestimate the parameter sensitivity because it only considers the influence of limited parameter samples from a statistical view. This study verifies that the CNOPSA method is effective and feasible for exploring the important and sensitive physical parameters or parameter combinations in numerical models.
摘要: 数值模式中物理参数的不确定性是数值模拟和预测不确定性的重要来源之一。由于数值模式中包含大量的物理过程和参数，通过减少所有物理参数的不确定性以提高数值模式的模拟能力和预测技巧将花费大量的人力和物力。因此，识别敏感的参数或参数组合至关重要。本研究提出了一种识别物理参数敏感性的新方法：条件非线性最优扰动敏感性分析（CNOPSA）方法。该方法克服了传统方法的局限性，在参数不确定性范围内充分考虑了物理参数间的非线性协同效应，可识别出相对敏感和重要的物理参数和参数组合，并定量估计出由物理参数变化导致的数值模拟和预测不确定性的最大程度，因而适用于对极端事件的研究。利用理论的g-函数和五变量草原生态系统模型检验了CNOPSA方法的可行性和有效性，结果表明CNOPSA方法可以有效地识别物理变量和物理参数的敏感性。本文进一步利用该方法，定量地估计了由物理参数不确定性导致的该草原生态系统模型中枯草量模拟和预测不确定性的最大程度，识别出的敏感参数的变化易使得草原生态系统发生突变。然而，基于方差分析的参数敏感性分析方法，仅从统计的角度考虑有限的参数样本，易低估物理参数的敏感性。
• • 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 ${x_i}$ , $i = 1,2,...,7$ .

Figure 2.  The 100-years nonlinear evolution of the living and wilted biomass in the grassland ecosystem model for (a) grassland state A and (b) grassland state B.

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.

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.

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 B

Figure 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 (${\beta _{\text{z}}}$ , ${\varepsilon _{{\rm{dz}}}}$  ).

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 (${\varepsilon _{\rm{d}}}$   , ${\varepsilon _{{\rm{dz}}}}$  ).

Figure 8.  Same as in Fig. 3, but for grassland state B.

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.

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 (${\varepsilon _{\rm{d}}}$ , ${\varepsilon _{{\rm{dz}}}}$   ).

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 (${\varepsilon _{\rm{d}}}$  , ${\varepsilon _{{\rm{dz}}}}$   ).

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.

 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 2.  Symbols, default values, ranges, and physical meanings of the seven parameters used in the grassland ecosystem model.

 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 3.  The optimized parameter vectors and corresponding cost function values of the CNOPSA method at 5 years for grassland state A.

 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 4.  Same as in Table 3, but for grassland state B.

 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 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.

 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.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## A New Sensitivity Analysis Approach Using Conditional Nonlinear Optimal Perturbations and Its Preliminary Application

###### Corresponding author: Mu MU, mumu@fudan.edu.cn;
• 1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
• 2. Department of Atmospheric and Oceanic Sciences and Institute of Atmospheric Sciences, Fudan University, Shanghai 200438, China
• 3. University of Chinese Academy of Sciences, Beijing 100049, China
• 4. Key Laboratory of Marine Hazards Forecasting, Ministry of Natural Resources, Hohai University, Nanjing 210098, China
• 5. College of Oceanography, Hohai University, Nanjing 210098, China

Abstract: Simulations and predictions using numerical models show considerable uncertainties, and parameter uncertainty is one of the most important sources. It is impractical to improve the simulation and prediction abilities by reducing the uncertainties of all parameters. Therefore, identifying the sensitive parameters or parameter combinations is crucial. This study proposes a novel approach: conditional nonlinear optimal perturbations sensitivity analysis (CNOPSA) method. The CNOPSA method fully considers the nonlinear synergistic effects of parameters in the whole parameter space and quantitatively estimates the maximum effects of parameter uncertainties, prone to extreme events. Results of the analytical g-function test indicate that the CNOPSA method can effectively identify the sensitivity of variables. Numerical results of the theoretical five-variable grassland ecosystem model show that the maximum influence of the simulated wilted biomass caused by parameter uncertainty can be estimated and computed by employing the CNOPSA method. The identified sensitive parameters can easily change the simulation or prediction of the wilted biomass, which affects the transformation of the grassland state in the grassland ecosystem. The variance-based approach may underestimate the parameter sensitivity because it only considers the influence of limited parameter samples from a statistical view. This study verifies that the CNOPSA method is effective and feasible for exploring the important and sensitive physical parameters or parameter combinations in numerical models. followshare  DownLoad:  Full-Size Img  PowerPoint