Impact Factor: 6.5

Feb.  2023

Article Contents

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

•  Bonan, G. B., and S. C. Doney, 2018: Climate, ecosystems, and planetary futures: the challenge to predict life in Earth system models. Science, 359, eaam8328, https://doi.org/10.1126/science.aam8328. Boyle, J. S., S. A. Klein, D. D. Lucas, H.-Y. Ma, J. Tannahill, and S. Xie, 2015: The parametric sensitivity of CAM5’s MJO. J. Geophys. Res.: Atmospheres, 120, 1424−1444, https://doi.org/10.1002/2014JD022507. Bratley, P., and B. L. Fox, 1988: Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Transactions on Mathematical Software, 14(1), 88−100, https://doi.org/10.1145/42288.214372. Chinta, S., and C. Balaji, 2020: Calibration of WRF model parameters using multiobjective adaptive surrogate model-based optimization to improve the prediction of the Indian summer monsoon. Climate Dyn., 55, 631−650, https://doi.org/10.1007/s00382-020-05288-1. Daniel, C., 1973: One-at-a-time plans. Journal of the American Statistical Association, 68(342), 353−360, https://doi.org/10.1080/01621459.1973.10482433. Di, Z. H., and Coauthors, 2015: Assessing WRF model parameter sensitivity: A case study with 5 day summer precipitation forecasting in the Greater Beijing Area. Geophys. Res. Lett., 42, 579−587, https://doi.org/10.1002/2014GL061623. Duan, W. S., and R. Zhang, 2010: Is model parameter error related to a significant spring predictability barrier for El Niño events? Results from a theoretical model. Adv. Atmos. Sci., 27, 1003−1013, https://doi.org/10.1007/s00376-009-9166-4. Homma, T., and A. Saltelli, 1996: Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering and System Safety, 52(1), 1−17, https://doi.org/10.1016/0951-8320(96)00002-6. Khalid, K., M. F. Ali, N. F. Abd Rahman, and M. R. Mispan, 2016: Application on one-at-a-time sensitivity analysis of semi-distributed hydrological model in tropical watershed. International Journal of Engineering and Technology, 8(2), 132−136, https://doi.org/10.7763/IJET.2016.V8.872. Lamboni, M., 2018: Global sensitivity analysis: a generalized, unbiased and optimal estimator of total-effect variance. Statistical Papers, 59, 361−386, https://doi.org/10.1007/s00362-016-0768-5. Li, J. D., and Coauthors, 2013: Assessing parameter importance of the common land model based on qualitative and quantitative sensitivity analysis. Hydrology and Earth System Sciences, 17(8), 3279−3293, https://doi.org/10.5194/hess-17-3279-2013. Li, Y., and Coauthors, 2017: Reducing the uncertainty of parameters controlling seasonal carbon and water fluxes in Chinese forests and its implication for simulated climate sensitivities. Global Biogeochemical Cycles, 31, 1344−1366, https://doi.org/10.1002/2017GB005714. Liu, Y., W. D. Guo, and Y. M. Song, 2016: Estimation of key surface parameters in semi-arid region and their impacts on improvement of surface fluxes simulation. Science China Earth Sciences, 59, 307−319, https://doi.org/10.1007/s11430-015-5140-4. Ma, H. Q., C. F. Ma, X. Li, W. P. Yuan, Z. J. Liu, and G. F. Zhu, 2020: Sensitivity and uncertainty analyses of flux-based ecosystem model towards improvement of forest GPP Simulation. Sustainability, 12(7), 2584, https://doi.org/10.3390/su12072584. Morris, M. D., 1991: Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2), 161−174, https://doi.org/10.1080/00401706.1991.10484804. Mu, M., W. S. Duan, and B. Wang, 2003: Conditional nonlinear optimal perturbation and its applications. Nonlinear Processes in Geophysics, 10, 493−501, https://doi.org/10.5194/npg-10-493-2003. Mu, M., W. Duan, Q. Wang, and R. Zhang, 2010: An extension of conditional nonlinear optimal perturbation approach and its applications. Nonlinear Processes in Geophysics, 17, 211−220, https://doi.org/10.5194/npg-17-211-2010. Peng, F., and G. D. Sun, 2017: A new climate scenario for assessing the climate change impacts on soil moisture over the Huang-Huai-Hai Plain region of China. Atmospheric and Oceanic Science Letters, 10(2), 105−113, https://doi.org/10.1080/16742834.2017.1255536. Pitman, A. J., 1994: Assessing the sensitivity of a Land-Surface scheme to the parameter values using a single column model. J. Climate, 7, 1856−1869, https://doi.org/10.1175/1520-0442(1994)007<1856:ATSOAL>2.0.CO;2. Rayner, P. J., M. Scholze, W. Knorr, T. Kaminski, R. Giering, and H. Widmann, 2005: Two decades of terrestrial carbon fluxes from a carbon cycle data assimilation system (CCDAS). Global Biogeochemical Cycles, 19, GB2026, https://doi.org/10.1029/2004GB002254. Razavi, S., and H. V. Gupta, 2015: What do we mean by sensitivity analysis? The need for comprehensive characterization of “global” sensitivity in Earth and Environmental systems models Water Resour. Res., 51, 3070−3092, https://doi.org/10.1002/2014WR016527. Razavi, S., and H. V. Gupta, 2016: A new framework for comprehensive, robust, and efficient global sensitivity analysis: 1. Theory. Water Resour. Res., 52, 423−439, https://doi.org/10.1002/2015WR017558. Ren, X. L., H. L. He, D. J. P. Moore, L. Zhang, M. Liu, F. Li, G. R. Yu, and H. M. Wang, 2013: Uncertainty analysis of modeled carbon and water fluxes in a subtropical coniferous plantation. J. Geophys. Res.: Biogeosci., 118, 1674−1688, https://doi.org/10.1002/2013JG002402. Rosero, E., Z. L. Yang, T. Wagener, L. E. Gulden, S. Yatheendradas, and G. Y. Niu, 2010: Quantifying parameter sensitivity, interaction, and transferability in hydrologically enhanced versions of the Noah land surface model over transition zones during the warm season. J. Geophys. Res.: Atmos., 115, D03106, https://doi.org/10.1029/2009JD012035. Saltelli, A., and I. M. Sobol’, 1995: About the use of rank transformation in sensitivity analysis of model output. Reliability Engineering and System Safety, 50, 225−239, https://doi.org/10.1016/0951-8320(95)00099-2. Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola, 2010: Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications, 181(2), 259−270, https://doi.org/10.1016/j.cpc.2009.09.018. Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, 2008: Global Sensitivity Analysis: The Primer. John Wiley & Sons, 285 pp. Sheffield, J., and E. F. Wood, 2007: Characteristics of global and regional drought, 1950–2000: Analysis of soil moisture data from off-line simulation of the terrestrial hydrologic cycle. J. Geophys. Res.: Atmos., 112, D17115, https://doi.org/10.1029/2006JD008288. Sheikholeslami, R., S. Razavi, H. V. Gupta, W. Becker, and A. Haghnegahdar, 2019: Global sensitivity analysis for high-dimensional problems: How to objectively group factors and measure robustness and convergence while reducing computational cost. Environmental Modelling and Software, 111, 282−299, https://doi.org/10.1016/j.envsoft.2018.09.002. Sobol’, I. M., 1990: On sensitivity estimation for nonlinear mathematical models. Matematicheskoe Modelirovanie, 2(1), 112−118. (in Russian) Sobol’, I. M., and S. S. Kucherenko, 2005: On global sensitivity analysis of quasi-Monte Carlo algorithms. Monte Carlo Methods and Applications, 11(1), 83−92, https://doi.org/10.1515/1569396054027274. Storn, R., and K. Price, 1997: Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4), 341−359, https://doi.org/10.1023/A:1008202821328. Sun, G. D., and M. Mu, 2009: Nonlinear feature of the abrupt transitions between multiple equilibria states of an ecosystem model. Adv. Atmos. Sci, 26(2), 293−304, https://doi.org/10.1007/s00376-009-0293-8. Sun, G. D., and M. Mu, 2017: A new approach to identify the sensitivity and importance of physical parameters combination within numerical models using the Lund-Potsdam-Jena (LPJ) model as an example. Theor. Appl. Climatol., 128, 587−601, https://doi.org/10.1007/s00704-015-1690-9. Sun, G. D., and D. D. Xie, 2017: A study of parameter uncertainties causing uncertainties in modeling a grassland ecosystem using the conditional nonlinear optimal perturbation method. Science China Earth Sciences, 60(9), 1674−1684, https://doi.org/10.1007/s11430-016-9065-9. Sun, G. D., and M. Mu, 2021: Impacts of two types of errors on the predictability of terrestrial carbon cycle. Ecosphere, 12(1), e03315, https://doi.org/10.1002/ecs2.3315. Sun, G. D., F. Peng, and M. Mu, 2017: Uncertainty assessment and sensitivity analysis of soil moisture based on model parameter errors - results from four regions in China. J. Hydrol., 555, 347−360, https://doi.org/10.1016/j.jhydrol.2017.09.059. Sun, G. D., M. Mu, and Q. L. You, 2020: Identification of key physical processes and improvements for simulating and predicting net primary production over the Tibetan Plateau. J. Geophys. Res.: Atmos., 125, e2020JD033128, https://doi.org/10.1029/2020JD033128. Wang, L., X. S. Shen, J. J. Liu, and B. Wang, 2020: Model uncertainty representation for a convection-allowing ensemble prediction system based on CNOP-P. Adv. Atmos. Sci., 37(8), 817−831, https://doi.org/10.1007/s00376-020-9262-z. Wang, Q., Y. M. Tang, and H. A. Dijkstra, 2017: An optimization strategy for identifying parameter sensitivity in atmospheric and oceanic models. Mon. Wea. Rev., 145(8), 3293−3305, https://doi.org/10.1175/MWR-D-16-0393.1. Wang, Q., S. Pierini, and Y. M. Tang, 2019: Parameter sensitivity analysis of the short-range prediction of Kuroshio extension transition processes using an optimization approach. Theor. Appl. Climatol., 138, 1481−1492, https://doi.org/10.1007/s00704-019-02911-y. Wu, H. H., C. S. Fu, H. W. Wu, and L. L. Zhang, 2020: Plant hydraulic stress strategy improves model predictions of the response of gross primary productivity to drought across China. J. Geophys. Res.: Atmos., 125, e2020JD033476, https://doi.org/10.1029/2020JD033476. Ye, D., S. W. Zhang, F. Y. Wang, F. P. Mao, and X. X. Yang, 2017: The applicability of different parameterization schemes in semi-arid region based on Noah-MP land surface model. Chinese Journal of Atmospheric Sciences, 41(1), 189−201, https://doi.org/10.3878/j.issn.1006-9895.1604.15226. (in Chinese with English abstract Zeng, X. D., S. S. P. Shen, X. B. Zeng, and R. E. Dickinson, 2004: Multiple equilibrium states and the abrupt transitions in a dynamical system of soil water interacting with vegetation. Geophys. Res. Lett., 31, L05501, https://doi.org/10.1029/2003GL018910. Zeng, X. D., X. B. Zeng, S. S. P. Shen, R. E. Dickingson, and Q. C. Zeng, 2005b: Vegetation-soil water interaction within a dynamical ecosystem model of grassland in semi-arid areas. Tellus B, 57, 189−202, https://doi.org/10.3402/tellusb.v57i3.16542. Zeng, X. D., A. H. Wang, G. Zhao, S. S. P. Shen, X. B. Zeng, and Q. C. Zeng, 2005a: Ecological dynamic model of grassland and its practical verification. Science in China. Series C, Life Sciences, 48, 41−48, https://doi.org/10.1360/03yc0219. Zeng, X. D., A. H. Wang, Q. C. Zeng, R. E. Dickinson, X. B. Zeng, and S. S. P. Shen, 2006: Intermediately complex models for the hydrological interactions in the atmosphere-vegetation-soil system. Adv. Atmos. Sci., 23(1), 127−140, https://doi.org/10.1007/s00376-006-0013-6.

Export:

## Manuscript History

Manuscript revised: 03 August 2022
Manuscript accepted: 12 August 2022
###### 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.

Reference

/