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


doi: 10.1007/s00376-022-1445-3

  • 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。
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  • 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.

    VariableSensitivity index (${S{\text{e} }_i}$)Total sensitivity index (${ { { S_{{\text{T} }_i}} } }$)
    AnalyticalNumerical
    $ {x_1} $0.9999950.78720.787659
    $ {x_2} $0.4444440.242220.242377
    $ {x_3} $0.0946740.034320.034342
    $ {x_4} $0.0330580.010460.010466
    $ {x_5} $0.0003920.000110.000105
    $ {x_6} $0.0003920.000110.000105
    $ {x_7} $0.0003920.000110.000105
    DownLoad: CSV

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

    IDParameterDefaultMinimumMaximumThe physical description
    1$ {\alpha ^*} $0.40.320.48The maximum growth rate
    2$ \beta ' $0.50.400.60The rate of accumulation of the wilted biomass
    3$ \beta $0.10.080.12The characteristic wilting rate
    4${\varepsilon _{\rm{d}}}$1.00.801.20The exponential attenuation coefficient of the living biomass
    5${\varepsilon '_{\rm{d}}}$1.00.801.20The exponential attenuation coefficient of water content in rooting layer
    6${\beta _{\rm{z}}}$0.10.080.12The characteristic rate of the wilted biomass decomposition
    7${\varepsilon _{{\rm{dz}}} }$1.00.801.20The exponential attenuation coefficient of the wilted biomass
    DownLoad: CSV

    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 $.
    DownLoad: CSV

    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 $.
    DownLoad: CSV

    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 stateMethodParameter sensitivity from large to small
    Grassland state ACNOPSA$ {\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 BCNOPSA$ {\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.
    DownLoad: CSV
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Manuscript received: 10 December 2021
Manuscript revised: 03 August 2022
Manuscript accepted: 12 August 2022
通讯作者: 陈斌, bchen63@163.com
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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.

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。

    • Currently, simulations and predictions of the atmosphere, land, and ocean using numerical models exhibit considerable uncertainties (Bonan and Doney, 2018; Sun and Mu, 2021). There are quite a few possible sources of these uncertainties (Sheffield and Wood, 2007; Boyle et al., 2015; Ye et al., 2017; Wu et al., 2020). One of the most important sources is the uncertainty of physical parameters that describe actual physical processes, which can be obtained by experience or observation. Due to the lack of observations and accurate estimation strategies, uncertainties in model parameters are often inevitable. Some studies indicated that reducing the uncertainties in physical parameters could significantly improve the simulation ability and prediction skills in numerical models (Rayner et al., 2005; Liu et al., 2016; Li et al., 2017). However, reducing the uncertainties of all physical parameters would be costly and impractical. Many studies have focused on the influence of the most important and sensitive physical parameters on model simulation and prediction (Sun and Mu, 2017; Ma et al., 2020; Chinta and Balaji, 2020). A naturally arising question is how to identify sensitive and important model parameters.

      Various sensitivity analysis (SA) methods have been developed to explore the uncertainty and sensitivity of physical parameters. The classical one-at-a-time (OAT) approach (Daniel, 1973) is common and easy to implement. The OAT method has been applied to analyze the relative importance of physical parameters in the Biosphere-Atmosphere Transfer Scheme model (Pitman, 1994) and semi-distributed hydrological model (Khalid et al., 2016). However, the OAT method ignores the nonlinear synergistic effects among physical parameters. The Morris approach (Morris, 1991) was proposed to overcome this limitation. Li et al. (2013) applied the Morris approach to evaluate the sensitivity of parameters qualitatively in the Common Land Model (CoLM). Di et al. (2015) also employed the Morris approach to explore sensitive factors for short-term precipitation forecast in the Weather Research and Forecasting (WRF) model. However, the Morris approach only provides qualitative results of parameter sensitivity and lacks a quantitative description.

      At present, the most widely used quantitative SA approach is the variance-based method (Sobol’, 1990; Homma and Saltelli, 1996; Saltelli et al., 2010; Lamboni, 2018). Rosero et al. (2010) applied the variance-based SA to identify critical parameters and understand nonlinear synergistic parameter effects controlling land surface model (LSM) simulations in three versions of the Noah LSM. Ren et al. (2013) used this method to evaluate parameter uncertainties in the Simplified Photosynthesis and Evapotranspiration (SIPNET) model and found that carbon and water fluxes were highly sensitive to parameters related to photosynthesis. The variance-based method considers the influence of parameter uncertainty on model outputs from a statistical point of view. Thus, it may fail to consider extreme events, so it may not give the maximum effect of parameter uncertainty. Another challenge is the “curse of dimensionality” (Saltelli et al., 2010). As the number of uncertain parameters increases, the volume of the parameter space expands rapidly. Any attempt to describe it requires a large number of parameter samples (Sheikholeslami et al., 2019). As a result, Razavi and Gupta (2015, 2016) pointed out that the variance-based method might not able to provide a robust and reliable sensitivity evaluation.

      Another kind of approach to identify parameter sensitivity is optimization methods. Mu et al. (2003) proposed the conditional nonlinear optimal perturbation with initial uncertainties, and extended it to parameters (Mu et al., 2010, CNOP-P). The CNOP-P method can explore the maximum effects of uncertainties in physical parameters on the model output and consider the nonlinear synergistic effects among parameters. It has been applied to the simulation and prediction of the atmospheric, land, and ocean processes (Duan and Zhang, 2010; Sun and Xie, 2017; Wang et al., 2020). Subsequently, Sun and Mu (2017) developed a three-step method based on the CNOP-P method to determine the important and sensitive parameter combinations in the land surface process models (Sun et al., 2017, 2020; Peng and Sun, 2017). Wang et al. (2017) proposed the Optimization Parameter Sensitivity Analysis (OPSA) approach and applied it to explore the sensitivity of model simulation to uncertain parameters in ocean models (Wang et al., 2019). These optimization methods use model simulation results (i.e., a control run) obtained in advance under a given parameter reference state. As the control run may be uncertain in a numerical model, the important and sensitive physical parameters may differ for different control runs.

      Although reducing the uncertainties in the sensitive parameters or parameter combinations identified by the above methods can improve the numerical simulation ability and prediction skills, the limitations of these methods are also non-negligible. Hence, it is necessary to develop a new parameter sensitivity analysis method to compensate for the relevant limitations. In this paper, we consider the maximum effect of parameter uncertainty and propose the conditional nonlinear optimal perturbations sensitivity analysis (CNOPSA) approach to explore the importance and sensitivity of physical parameters in numerical models.

      The remainder of this paper is organized as following. Section 2 gives a description of the variance-based approach and the CNOPSA method. The analytical g-function test is used to measure the effectiveness of the CNOSPA method in section 3. Section 4 presents the numerical results in a theoretical grassland ecosystem model. A summary and discussion are given in section 5.

    2.   Methods
    • First, we briefly review the variance-based approach (Saltelli et al., 2008, 2010). Based on variance decomposition theory, this method considers the variability of the model outputs caused by each parameter as an indicator of parameter sensitivity. With the assumption that the parameters $ {p_i} $($ i = 1,2,...,n $) are independent, the total variance $ V(U) $ of model output $ U $ can be decomposed into $ {2^n} - 1 $ components as follows:

      where $ {V_i} $ is the portion of $ V(U) $ explained by the model parameter $ {p_i} $. $ {V_{ij}} $ is the contribution of the coupling effects of the parameters $ {p_i} $ and $ {p_j} $ to the total variance $ V(U) $. The sensitivity indices are defined as $ {S_i} = \frac{{{V_i}}}{{V(U)}} $, $ {S_{ij}} = \frac{{{V_{ij}}}}{{V(U)}} $, etc. These sensitivity indices measure the proportion of the variance contributed by individual parameters (i.e., first-order effects) and individual parts of the nonlinear synergistic effects among parameters (i.e., high-order effects).

      The total order sensitivity index $ {S_{{{\rm{T}}_i}}} $ (Homma and Saltelli, 1996) is defined as:

      where ${V_{{{\text{T}}_i}}} = V(U) - {V_{1,...,i - 1,i + 1,...,n}} = E(V(U|{p_1},{p_2},...,{p_{i - 1}}, {p_{i + 1}},...,{p_n}))$ represents the remaining unexplained part of the total variance $ V(U) $ when all parameters are fixed except parameter $ {p_i} $. The total order sensitivity index ${S_{{{\rm{T}}_i}}}$ denotes all effects of the parameter $ {p_i} $ considering the nonlinear synergistic effects between $ {p_i} $ and all other parameters. Thus, ${S_{{{\rm{T}}_i}}}$ is a comprehensive measure of the contribution of the uncertainty of parameter $ {p_i} $. Details of the calculations and multi-dimensional extensions of the total sensitivity index can be found in Saltelli et al. (2010) and Lamboni (2018). The acquisition of parameter samples is crucial to the variance-based method. Thus, Sobol’s quasi-random sequences (Bratley and Fox, 1988; Sobol’ and Kucherenko, 2005), characterized by low discrepancy properties, are used to generate a ($ N $, $ 2n $) matrix of parameter samples, where $ N $ is the sample size.

    • In the following we introduce our new conditional nonlinear optimal perturbations sensitivity analysis (CNOPSA) method. Assume that the model can be expressed as:

      where ${\boldsymbol{P}} = ({p_1},{p_2},...,{p_n})$ is the physical parameter vector, $ {U_{\text{0}}} $ is the initial conditions. Let $ {M_t} $ be the nonlinear propagator from the initial time 0 to $ t $. $ U(t) $ is the solution of the model [Eq. (3)] at time $ t $:

      For convenience, we assume that the initial condition $ {U_{\text{0}}} $ is fixed. The solution of the nonlinear model [Eq. (4)] can be described as

      The number and range values of the model physical parameters are generally limited and bounded. Let $ {C_\varepsilon } $ represent the constraint conditions of the parameters. For arbitrary parameter vectors ${{\boldsymbol{P}}_1} = ({p'_1}, {p'_2},...,{p'_n}) \in {C_\varepsilon }$ and ${{\boldsymbol{P}}_2} = ({p''_1}, {p''_2},...,{p''_n}) \in {C_\varepsilon }$, the influence of model output (simulation or prediction) caused by the uncertainty of all parameters at time $ t $ can be expressed as

      where $ \left\| \cdot \right\| $ represents a chosen norm measuring the magnitude of the deviation in model output. However, it is unrealistic to evaluate the effects of parameter uncertainty by considering all parameter values under the given constraint condition. Therefore, we search for the maximum value of the function ${J_{{\text{all}}}}({{\boldsymbol{P}}_1},{{\boldsymbol{P}}_2})$ over the whole parameter space $ {C_\varepsilon } $ by solving the constrained optimization problem:

      where ${\boldsymbol{P}}_1^\delta$ and ${\boldsymbol{P}}_{2}^\delta$ are the optimized parameter vectors that satisfy the given constraint condition ${\boldsymbol{P}}_1^\delta$, ${\boldsymbol{P}}_2^\delta \in {C_\varepsilon }$. The above equation means that the maximum influence of the cost function $ {J_{{\text{total}}}} $ at time $ t $ is obtained by the parameter vectors ${\boldsymbol{P}}_1^\delta$ and ${\boldsymbol{P}}_{2}^\delta$.

      When all parameters except parameter $ {p_i} $ in the two parameter vectors are consistent, that is ${{\boldsymbol{P}}_{1i}} = ({p'_1},{p'_2},..., {p'_{i - 1}}, {p'_i},{p'_{i + 1}},...,{p'_n})$ and ${{\boldsymbol{P}}_{2i}} = ({p'_1},{p'_2},...,{p'_{i - 1}},{p''_i},{p'_{i + 1}},...,{p'_n})$, the influence of model output caused by the uncertainty in the parameter $ {p_i} $ and its nonlinear synergistic effects with other parameters can be expressed as

      Accordingly, the optimization problems similar to Eq. (6), is as follows:

      The cost function $ {J_{{p_i}}} $ measures the maximum influence of model output ($ U $) yielded by the uncertainty in parameter $ {p_i} $ and its nonlinear synergistic effects with other parameters at time $ t $, and ${\boldsymbol{P}}_{1i}^\delta ,{\boldsymbol{P}}_{2i}^\delta \in {C_\varepsilon }$ are the corresponding optimized parameter vectors.

      Similar to the total sensitivity index ${S_{ {{\text{T}}_i}}}$ of the variance-based approach, the sensitivity index ${S{\rm{e}}_i}$ for measuring the total effect of the parameter $ {p_i} $ can be defined as the ratio of $ {J_{{p_i}}} $ in Eq. (7) and $ {J_{{\text{total}}}} $ in Eq. (6):

      For the multi-parameter case, we can also define the sensitivity index of the parameter combinations. In the two-parameter case, the total influence of the uncertainty in parameters $ {p_i} $ and $ {p_j} $ on model output can be measured as follows:

      where ${J_{ij}}({{\boldsymbol{P}}_{1ij}},{{\boldsymbol{P}}_{2ij}}) = \left\| {{M_t}({{\boldsymbol{P}}_{1ij}}) - {M_t}({{\boldsymbol{P}}_{2ij}})} \right\|$, ${{\boldsymbol{P}}_{1ij}} = ({p'_1}, {p'_2},\,...,\,{p'_{i - 1}},\,{p'_i},\,{p'_{i + 1}},\,...,\,{p'_{j - 1}},\,{p'_j},\,{p'_{j + 1}},\,...,\,{p'_n}) \in {C_\varepsilon }$, ${{\boldsymbol{P}}_{2ij}} = ({p'_1}, {p'_2},\,...,\,{p'_{i - 1}},\,{p''_i},\,{p'_{i + 1}},\,...,\,{p'_{j - 1}},\,{p''_j},\,{p'_{j + 1}},\,...,\,{p'_n}) \in {C_\varepsilon }$. $ {J_{{p_{ij}}}} $ represents the maximum influence of the model output ($ U $) caused by all uncertainties in the parameters $ {p_i} $ and $ {p_j} $ at time $ t $, and ${\boldsymbol{P}}_{1ij}^\delta ,\,{\boldsymbol{P}}_{2ij}^\delta \in {C_\varepsilon }$ are the corresponding optimized parameter vectors. The sensitivity index ${ {S\text{e}}_{ij}}$ can be represented as the ratio of $ {J_{{p_{ij}}}} $ in Eq. (8) and $ {J_{{\text{total}}}} $ in Eq. (6):

      for which the inequality can be proven (see Appendix A):

      meaning that the sensitivity indices satisfy:

      For the CNOPSA method, the differential evolution (DE; Storn and Price, 1997) algorithm is applied to solve the optimization problem. The optimal values are obtained by continuous iteration based on 200 groups of initial parameter vectors.

      According to the definitions of the sensitivity in the above two methods, both consider the nonlinear synergistic effects among parameters and evaluate the important and sensitive parameters by calculating the influence of parameter uncertainty on numerical simulation. Of course, there are some differences between the two methods. The variance-based method yields the influence of parameter uncertainty on numerical simulation from a perspective of statistics. For example, the total effect of the parameter $ {p_i} $ ($ {V_{{{\text{T}}_i}}} $ in Eq. (2)) can be obtained by evaluating the average of the conditional variance of all the parameters except $ {p_i} $ (Saltelli et al., 2008). The calculation of the total order sensitivity index, using sample-based methods, often needs to consider the distribution of parameters and requires many parameter samples. In contrast, the CNOPSA method quantifies the influence of parameter uncertainty on numerical simulation from the deterministic point of view. Therefore, CNOPSA can clearly yield the maximum influence of parameter uncertainty on model outputs. The greater the maximum effect of parameter uncertainty, the more important and sensitive is the parameter. Since the CNOPSA method solves constrained optimization problems within the reasonable uncertainty ranges of the parameters using an optimization algorithm, the distribution of parameters does not need to be considered in this approach.

    3.   Analytical test g-function case
    • 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} $ is

      while 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 $.

      VariableSensitivity index (${S{\text{e} }_i}$)Total sensitivity index (${ { { S_{{\text{T} }_i}} } }$)
      AnalyticalNumerical
      $ {x_1} $0.9999950.78720.787659
      $ {x_2} $0.4444440.242220.242377
      $ {x_3} $0.0946740.034320.034342
      $ {x_4} $0.0330580.010460.010466
      $ {x_5} $0.0003920.000110.000105
      $ {x_6} $0.0003920.000110.000105
      $ {x_7} $0.0003920.000110.000105

      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 $.

    4.   Theoretical five-variable grassland ecosystem model
    • 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.

      IDParameterDefaultMinimumMaximumThe physical description
      1$ {\alpha ^*} $0.40.320.48The maximum growth rate
      2$ \beta ' $0.50.400.60The rate of accumulation of the wilted biomass
      3$ \beta $0.10.080.12The characteristic wilting rate
      4${\varepsilon _{\rm{d}}}$1.00.801.20The exponential attenuation coefficient of the living biomass
      5${\varepsilon '_{\rm{d}}}$1.00.801.20The exponential attenuation coefficient of water content in rooting layer
      6${\beta _{\rm{z}}}$0.10.080.12The characteristic rate of the wilted biomass decomposition
      7${\varepsilon _{{\rm{dz}}} }$1.00.801.20The 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.

    • 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.

      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.

    • 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.

    • 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 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}}}} $).

    • 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).

    • 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 ^*} $.

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

      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.

    5.   Summary and Discussion
    • 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.

    • 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 stateMethodParameter sensitivity from large to small
      Grassland state ACNOPSA$ {\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 BCNOPSA$ {\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.

    APPENDIX A
    • Proof of Inequality (9)

      We first prove that $ {J_{{p_i}}} \leqslant {J_{{p_{ij}}}} $. The objective function $ {J_{{p_i}}} $ and $ {J_{{p_{ij}}}} $ can be expressed as:

      $ {J_{{p_i}}} $ measures the changes of model outputs caused by different values of $ {p_i} $, the changes of other parameters $ {p_1},{p_2},...,{p_{i - 1}},{p_{i + 1}},...,{p_n} $ are consistent. Thus, the feasible region of parameters in $ {J_{{p_i}}} $ can be regarded as $ n + 1 $ dimensions, including $ {C_\varepsilon } $ and the variable range of parameter $ {p_i} $. Correspondingly, $ {J_{{p_{ij}}}} $ measures the variations of model outputs caused by the different values of the parameters $ {p_i} $ and $ {p_j} $, and the changes of the parameters $ {p_1},{p_2},... $, ${p_{i - 1}}, {p_{i + 1}}, ...,{p_{j - 1}},{p_{j + 1}},...,{p_n}$ are consistent. The feasible region of the parameters in $ {J_{{p_{ij}}}} $ can be regarded as $ n + 2 $ dimensions, including $ {C_\varepsilon } $ and the variable range of the additional parameters $ {p_i} $ and $ {p_j} $.

      Obviously, the parameter feasible region in $ {J_{{p_i}}} $ is a subset of the parameter feasible region in $ {J_{{p_{ij}}}} $, so the inequality $ {J_{{p_i}}} \leqslant {J_{{p_{ij}}}} $ holds.

      Thus, it can be proved that

      Similarly, the inequality ${J_{{p_{ij}}}} \leqslant {J_{{\text{total}}}}(i = 1,2,...,n; j = i + 1,...,n)$ also holds.

    APPENDIX B
    • The Important Relationship between the Wilted Biomass and Evapotranspiration

      The five-variable ecological hydrological system is developed based on the three-variable ecological-hydrological model [Eq. (B1)]:

      where $ W $ is soil surface water content, $ P $ is the total precipitation from atmosphere, $ R $ is the net runoff. This means that the soil surface water content characterized by $ W $ is an open system with input $ P $ and output $ {E_{\text{s}}}{\text{ + }}{E_{\text{r}}} + R $.

      Let terms $x \equiv {{{M_{\text{c}}}}}/{{M_{\text{c}}^{\text{*}}}},y \equiv {W}/{{{W^*}}},z \equiv {{{M_{\text{d}}}}}/{{M_{\text{d}}^{\text{*}}}}$, $ M_{\text{c}}^{\text{*}} $, $ {W^*} $, and $ M_{\text{d}}^{\text{*}} $ represent the maximum living biomass, the maximum water content, and the maximum accumulation of the wilted biomass, respectively. The evaporation from bare soil can be expressed as (Zeng et al., 2005b)

      where $ E_{\text{s}}^{\text{*}} $ is potential evaporation. The attenuation of the solar radiation by the wilted biomass follows the exponential law (Beer’s law). As with the transmittance, let

      where $ {\varepsilon _3} $ characterizes the shading effect of the wilted biomass (resulting in the decrease of soil surface temperature).

      The evaporation rate from the soil surface covered only by the wilted biomass is

      and then the net soil evaporation $ {E_{\text{s}}} $ can be expressed as

      where $ {\sigma _f} $ is the fraction of living grass coverage, $ {\kappa _1} $ is the amplitude of shading effect influenced by living leaves, $ \varepsilon $’s with different subscripts are the exponential attenuation coefficients. The determination of model parameter $ {\kappa _1} $ is affected by air temperature, relative humidity (RH), etc. The sum of $ {E_{\text{s}}} $ and $ {E_{\text{r}}} $ (Zeng et al., 2005a, b, 2006) can be expressed as

      where $ {\varphi _{{\text{rs}}}} $ is the ratio of potential transpiration to $ E_{\text{s}}^{\text{*}} $. Therefore, the wilted biomass indirectly affects the soil temperature by affecting the sunlight reaching the soil surface, and 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., 2004, 2006). In the five-variable ecological-hydrological model, the above physical process expressions are still established (Zeng et al., 2004, 2005a, 2006).

    APPENDIX C
    • Conditional Nonlinear Optimal Perturbation Related to Parameter (CNOP-P)

      For the model Eq. (3) described in subsection 2.2, let $U(T;{U_0},{\boldsymbol{P}})$ and $U(T;{U_0},{\boldsymbol{P}}){\text{ + }}u(T;{U_0},{\boldsymbol{p}})$ be the solutions of Eq. (3) with parameters vectors ${\boldsymbol{P}}$ and ${\boldsymbol{P}} + {\boldsymbol{p}}$ at time $ T $, respectively. The following relationships can be established:

      For a chosen norm $ \left\| \cdot \right\| $, a parameter perturbation $ {p_\delta } $ is called a CNOP-P if and only if

      where

      $ {\boldsymbol{P }}$ is a reference state of the parameters, ${\boldsymbol{p}} \in \Omega$ is a constraint condition, ${\boldsymbol{P}} + {\boldsymbol{p}} \in {C_\varepsilon }$. The CNOP-P is the parameter perturbation whose nonlinear evolution attains the maximum value of the cost function $ J $ [Eq. (C2)] at time $ T $.

      Acknowledgements. The authors thanks O. ZACCHEUS and S. KUCHERENKO for providing the software used in generating the Sobol’s quasi-random sequences matrix. This work was supported by the National Nature Science Foundation of China (41975132) and the Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2020B0301030004). The authors would like to express their gratitude to EditSprings (https://www.editsprings.cn/) for the expert linguistic services provided.

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