A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. T...A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. The optimization problem related to the determination of nonlinear singular vectors and singular values is formulated. The general idea of this approach is demonstrated by a simple two-dimensional quasigeostrophic model in the atmospheric and oceanic sciences. The advantage and its applications of the new method to the predictability, ensemble forecast and finite-time nonlinear instability are discussed. This paper makes a necessary preparation for further theoretical and numerical investigations.展开更多
Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional q...Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional quasigeostrophic model in this paper. The difference between the linear and nonlinear fastest growing perturbations is demonstrated. Moreover, local nonlinear fastest growing perturbations are also found numerically. This is one of the essential differences between linear and nonlinear theories, since in former case there is no local fastest growing perturbation. The results show that the nonlinear local fastest growing perturbations play a more important role in the study of the first kind of predictability than the nonlinear global fastest growing perturbation.展开更多
We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results ...We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results are proven for differential operators.展开更多
The orthogonal conditional nonlinear optimal perturbations (CNOPs) method, orthogonal singular vectors (SVs)method and CNOP+SVs method, which is similar to the orthogonal SVs method but replaces the leading SV (LSV) w...The orthogonal conditional nonlinear optimal perturbations (CNOPs) method, orthogonal singular vectors (SVs)method and CNOP+SVs method, which is similar to the orthogonal SVs method but replaces the leading SV (LSV) with the first CNOP, are adopted in both the Lorenz-96 model and Pennsylvania State University/National Center for Atmospheric Research (PSU/NCAR) Fifth-Generation Mesoscale Model (MM5) for ensemble forecasts. Using the MM5, typhoon track ensemble forecasting experiments are conducted for strong Typhoon Matsa in 2005. The results of the Lorenz-96 model show that the CNOP+SVs method has a higher ensemble forecast skill than the orthogonal SVs method, but ensemble forecasts using the orthogonal CNOPs method have the highest forecast skill. The results from the MM5 show that orthogonal CNOPs have a wider horizontal distribution and better describe the forecast uncertainties compared with SVs. When generating the ensemble mean forecast, equally averaging the ensemble members in addition to the anomalously perturbed forecast members may contribute to a higher forecast skill than equally averaging all of the ensemble members. Furthermore, for given initial perturbation amplitudes, the CNOP+SVs method may not have an ensemble forecast skill greater than that of the orthogonal SVs method, but the orthogonal CNOPs method is likely to have the highest forecast skill. Compared with SVs, orthogonal CNOPs fully consider the influence of nonlinear physical processes on the forecast results; therefore, considering the influence of nonlinearity may be important when generating fast-growing initial ensemble perturbations. All of the results show that the orthogonal CNOP method may be a potential new approach for ensemble forecasting.展开更多
Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial pertur...Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.展开更多
An adjoint sensitivity analysis of one mesoscale low on the mei-yu Front is presented in this paper. The sensitivity gradient of simulation error dry energy with respect to initial analysis is calculated. And after ve...An adjoint sensitivity analysis of one mesoscale low on the mei-yu Front is presented in this paper. The sensitivity gradient of simulation error dry energy with respect to initial analysis is calculated. And after verifying the ability of a tangent linear and adjoint model to describe small perturbations in the nonlinear model, the sensitivity gradient analysis is implemented in detail. The sensitivity gradient with respect to different physical fields are not uniform in intensity, simulation error is most sensitive to the vapor mixed ratio. The localization and consistency are obvious characters of horizontal distribution of the sensitivity gradient, which is useful for the practical implementation of adaptive observation. The sensitivity region tilts to the northwest with height increasing; the singular vector calculation proves that this tilting characterizes a quick-growing structure, which denotes that using the leading singular vectors to decide the adaptive observation region is proper. When connected with simulation of a mesoscale low on the mei-yu Front, the sensitivity gradient has the following physical characters： the obvious sensitive region is mesoscale, concentrated in the middle-upper troposphere, and locates around the key system; and the sensitivity gradient of different physical fields correlates dynamically.展开更多
Two methods for initialization of ensemble forecasts are compared, namely, singular vector (SV) and conditional nonlinear optimal perturbation (CNOP). The comparison is done for forecast lengths of up to 10 days with ...Two methods for initialization of ensemble forecasts are compared, namely, singular vector (SV) and conditional nonlinear optimal perturbation (CNOP). The comparison is done for forecast lengths of up to 10 days with a three-level quasi-geostrophic (QG) atmospheric model in a perfect model scenario. Ten cases are randomly selected from 1982/1983 winter to 1993/1994 winter (from December to the following February). Anomaly correlation coefficient (ACC) is adopted as a tool to measure the quality of the predicted ensembles on the Northern Hemisphere 500 hPa geopotential height. The results show that the forecast quality of ensemble samples in which the first SV is replaced by CNOP is higher than that of samples composed of only SVs in the medium range, based on the occurrence of weather re-gime transitions in Northern Hemisphere after about four days. Besides, the reliability of ensemble forecasts is evaluated by the Rank Histograms. The above conclusions confirm and extend those reached earlier by the authors, which stated that the introduction of CNOP improves the forecast skill under the condition that the analysis error belongs to a kind of fast-growing error by using a barotropic QG model.展开更多
Conditional nonlinear optimal perturbation(CNOP) is an extension of the linear singular vector technique in the nonlinear regime.It represents the initial perturbation that is subjected to a given physical constraint,...Conditional nonlinear optimal perturbation(CNOP) is an extension of the linear singular vector technique in the nonlinear regime.It represents the initial perturbation that is subjected to a given physical constraint,and results in the largest nonlinear evolution at the prediction time.CNOP-type errors play an important role in the predictability of weather and climate.Generally,when calculating CNOP in a complicated numerical model,we need the gradient of the objective function with respect to the initial perturbations to provide the descent direction for searching the phase space.The adjoint technique is widely used to calculate the gradient of the objective function.However,it is difficult and cumbersome to construct the adjoint model of a complicated numerical model,which imposes a limitation on the application of CNOP.Based on previous research,this study proposes a new ensemble projection algorithm based on singular vector decomposition(SVD).The new algorithm avoids the localization procedure of previous ensemble projection algorithms,and overcomes the uncertainty caused by choosing the localization radius empirically.The new algorithm is applied to calculate the CNOP in an intermediate forecasting model.The results show that the CNOP obtained by the new ensemble-based algorithm can effectively approximate that calculated by the adjoint algorithm,and retains the general spatial characteristics of the latter.Hence,the new SVD-based ensemble projection algorithm proposed in this study is an effective method of approximating the CNOP.展开更多
Rainfall forecasting is becoming more and more significant and precipitation anomalies would lead to droughts and floods disasters.However,because of the complexity and non-stationary of rainfall data,it is difficult ...Rainfall forecasting is becoming more and more significant and precipitation anomalies would lead to droughts and floods disasters.However,because of the complexity and non-stationary of rainfall data,it is difficult to forecast.In this paper,a novel hybrid model to forecast rainfall is developed by incorporating singular spectrum analysis (SSA) and dragonfly algorithm (DA) into support vector regression (SVR) method.Firstly,SSA is used for extracting the trend components of the hydrological data.Then,SVR is utilized to deal with the volatility and irregularity of the precipitation series.Finally,the parameter of SVR is optimized by DA.The proposed SSA-DA-SVR method is used to forecast the monthly precipitation for Songbai,Panshui,Lanma and Jiulongchi stations.To validate the efficiency of the method,four compared models,DA-SVR,SSA-GWO-SVR,SSA-PSO-SVR and SSA-CS-SVR are established.The result shows that the proposed method has the best performance among all five models,and its prediction has high precision and accuracy.展开更多
The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with...The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.展开更多
文摘A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. The optimization problem related to the determination of nonlinear singular vectors and singular values is formulated. The general idea of this approach is demonstrated by a simple two-dimensional quasigeostrophic model in the atmospheric and oceanic sciences. The advantage and its applications of the new method to the predictability, ensemble forecast and finite-time nonlinear instability are discussed. This paper makes a necessary preparation for further theoretical and numerical investigations.
基金the National Key Basic Research Project, "Research on the FormationMechanism and Prediction Theory of Severe Synoptic Disasters in China" (Grand No. G1998040910), the National Natural Science Foundation of China (Grand Nos. 49775262 and 49823002) and t
文摘Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional quasigeostrophic model in this paper. The difference between the linear and nonlinear fastest growing perturbations is demonstrated. Moreover, local nonlinear fastest growing perturbations are also found numerically. This is one of the essential differences between linear and nonlinear theories, since in former case there is no local fastest growing perturbation. The results show that the nonlinear local fastest growing perturbations play a more important role in the study of the first kind of predictability than the nonlinear global fastest growing perturbation.
基金supported by National Natural Science Foundation of China(Grant Nos.11671308 and 11431011)Ministerio de Economia y Competitividad/al Fondo Europeo de Desarrollo Regional(Grant No.MTM2015-66157-C2-1-P)
文摘We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results are proven for differential operators.
基金sponsored by the National Natural Science Foundation of China (Grant Nos. 41525017 & 41475100)the National Programme on Global Change and Air-Sea Interaction (Grant No. GASI-IPOVAI-06)the GRAPES Development Program of China Meteorological Administration (Grant No. GRAPES-FZZX-2018)
文摘The orthogonal conditional nonlinear optimal perturbations (CNOPs) method, orthogonal singular vectors (SVs)method and CNOP+SVs method, which is similar to the orthogonal SVs method but replaces the leading SV (LSV) with the first CNOP, are adopted in both the Lorenz-96 model and Pennsylvania State University/National Center for Atmospheric Research (PSU/NCAR) Fifth-Generation Mesoscale Model (MM5) for ensemble forecasts. Using the MM5, typhoon track ensemble forecasting experiments are conducted for strong Typhoon Matsa in 2005. The results of the Lorenz-96 model show that the CNOP+SVs method has a higher ensemble forecast skill than the orthogonal SVs method, but ensemble forecasts using the orthogonal CNOPs method have the highest forecast skill. The results from the MM5 show that orthogonal CNOPs have a wider horizontal distribution and better describe the forecast uncertainties compared with SVs. When generating the ensemble mean forecast, equally averaging the ensemble members in addition to the anomalously perturbed forecast members may contribute to a higher forecast skill than equally averaging all of the ensemble members. Furthermore, for given initial perturbation amplitudes, the CNOP+SVs method may not have an ensemble forecast skill greater than that of the orthogonal SVs method, but the orthogonal CNOPs method is likely to have the highest forecast skill. Compared with SVs, orthogonal CNOPs fully consider the influence of nonlinear physical processes on the forecast results; therefore, considering the influence of nonlinearity may be important when generating fast-growing initial ensemble perturbations. All of the results show that the orthogonal CNOP method may be a potential new approach for ensemble forecasting.
基金supported by the Joint Funds of the Chinese National Natural Science Foundation (NSFC)(Grant No.U2242213)the National Key Research and Development (R&D)Program of the Ministry of Science and Technology of China(Grant No. 2021YFC3000902)the National Science Foundation for Young Scholars (Grant No. 42205166)。
文摘Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.
基金supported by the National Natural Science Foundation of China under Grant No.40405020.
文摘An adjoint sensitivity analysis of one mesoscale low on the mei-yu Front is presented in this paper. The sensitivity gradient of simulation error dry energy with respect to initial analysis is calculated. And after verifying the ability of a tangent linear and adjoint model to describe small perturbations in the nonlinear model, the sensitivity gradient analysis is implemented in detail. The sensitivity gradient with respect to different physical fields are not uniform in intensity, simulation error is most sensitive to the vapor mixed ratio. The localization and consistency are obvious characters of horizontal distribution of the sensitivity gradient, which is useful for the practical implementation of adaptive observation. The sensitivity region tilts to the northwest with height increasing; the singular vector calculation proves that this tilting characterizes a quick-growing structure, which denotes that using the leading singular vectors to decide the adaptive observation region is proper. When connected with simulation of a mesoscale low on the mei-yu Front, the sensitivity gradient has the following physical characters： the obvious sensitive region is mesoscale, concentrated in the middle-upper troposphere, and locates around the key system; and the sensitivity gradient of different physical fields correlates dynamically.
基金Supported by Knowledge Innovation Project of the Chinese Academy of Sciences (Grant No. KZCX3-SW-230)National Natural Science Foundation of China (Grant Nos. 40675030, 40633016)
文摘Two methods for initialization of ensemble forecasts are compared, namely, singular vector (SV) and conditional nonlinear optimal perturbation (CNOP). The comparison is done for forecast lengths of up to 10 days with a three-level quasi-geostrophic (QG) atmospheric model in a perfect model scenario. Ten cases are randomly selected from 1982/1983 winter to 1993/1994 winter (from December to the following February). Anomaly correlation coefficient (ACC) is adopted as a tool to measure the quality of the predicted ensembles on the Northern Hemisphere 500 hPa geopotential height. The results show that the forecast quality of ensemble samples in which the first SV is replaced by CNOP is higher than that of samples composed of only SVs in the medium range, based on the occurrence of weather re-gime transitions in Northern Hemisphere after about four days. Besides, the reliability of ensemble forecasts is evaluated by the Rank Histograms. The above conclusions confirm and extend those reached earlier by the authors, which stated that the introduction of CNOP improves the forecast skill under the condition that the analysis error belongs to a kind of fast-growing error by using a barotropic QG model.
基金jointly sponsored by the National Natural Science Foundation of China(Grant Nos.41176013,41230420 and 41006007)
文摘Conditional nonlinear optimal perturbation(CNOP) is an extension of the linear singular vector technique in the nonlinear regime.It represents the initial perturbation that is subjected to a given physical constraint,and results in the largest nonlinear evolution at the prediction time.CNOP-type errors play an important role in the predictability of weather and climate.Generally,when calculating CNOP in a complicated numerical model,we need the gradient of the objective function with respect to the initial perturbations to provide the descent direction for searching the phase space.The adjoint technique is widely used to calculate the gradient of the objective function.However,it is difficult and cumbersome to construct the adjoint model of a complicated numerical model,which imposes a limitation on the application of CNOP.Based on previous research,this study proposes a new ensemble projection algorithm based on singular vector decomposition(SVD).The new algorithm avoids the localization procedure of previous ensemble projection algorithms,and overcomes the uncertainty caused by choosing the localization radius empirically.The new algorithm is applied to calculate the CNOP in an intermediate forecasting model.The results show that the CNOP obtained by the new ensemble-based algorithm can effectively approximate that calculated by the adjoint algorithm,and retains the general spatial characteristics of the latter.Hence,the new SVD-based ensemble projection algorithm proposed in this study is an effective method of approximating the CNOP.
文摘Rainfall forecasting is becoming more and more significant and precipitation anomalies would lead to droughts and floods disasters.However,because of the complexity and non-stationary of rainfall data,it is difficult to forecast.In this paper,a novel hybrid model to forecast rainfall is developed by incorporating singular spectrum analysis (SSA) and dragonfly algorithm (DA) into support vector regression (SVR) method.Firstly,SSA is used for extracting the trend components of the hydrological data.Then,SVR is utilized to deal with the volatility and irregularity of the precipitation series.Finally,the parameter of SVR is optimized by DA.The proposed SSA-DA-SVR method is used to forecast the monthly precipitation for Songbai,Panshui,Lanma and Jiulongchi stations.To validate the efficiency of the method,four compared models,DA-SVR,SSA-GWO-SVR,SSA-PSO-SVR and SSA-CS-SVR are established.The result shows that the proposed method has the best performance among all five models,and its prediction has high precision and accuracy.
基金Project supported by the National Natural Science Foundation of China(No.42207182)the Research Grants Council of the Hong Kong Special Administrative Region Government of China(Nos.HKU 17207518 and R5037-18)。
文摘The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.
