An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian condit... more An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian conditional probability estimation, along with the well known ensemble Kalman filter. A Gaussian resampling method is proposed here to generate the posterior analysis ensemble in an effective and efficient way. As a result the ensemble particle filter has good stability and potential applicability to large-scale problems. The Lorenz model is used here to test the proposed method. Multi-modal probability distributions can appear either with state dependent stochastic model errors or nonlinear observations. Ensemble Kalman filter (EnKF)is known to have a difficulty in tracking state transitions accurately. Current implementations of EnKF have not taken non-Gaussian contributions into account. With the posterior Gaussian resampling method the ensemble particle filter can track state transitions more accurately. Moreover, it is applicable to systems with typical multi-modal behavior, provided that certain prior knowledge becomes available about the general structure of posterior probability distribution. A simple scenario is considered to illustrate this point based on Lorenz model attractors. The present work demonstrates that the proposed ensemble particle filter can provide an accurate estimation of multi-modal distribution and is potentially applicable to large-scale data assimilation problems.
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational... more This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for Proper Orthogonal Decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
The pointwise estimation of heat conduction solution as a function of truncation error of a finit... more The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The contribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error correction and that an asymptotic error bound may be found.
Advances in Geosciences Volume 12 Ocean Science, Mar 1, 2009
The "equation-free" (EF) method is often used in complex, multi-scale problems. In such cases it ... more The "equation-free" (EF) method is often used in complex, multi-scale problems. In such cases it is necessary to know the closed form of the required evolution equations about macroscopic variables within some applied fields. Conceptually such equations exist, however, they are not available in closed form. The EF method can bypass this difficulty. This method can obtain macroscopic information by implementing models at a microscopic level. Given an initial macroscopic variable, through lifting we can obtain the associated microscopic variable, which may be evolved using Direct Numerical Simulations (DNS) and by restriction, we can obtain the necessary macroscopic information and the projective integration to obtain the desired quantities. In this paper we apply the EF POD-assisted method to the reduced modeling of a large-scale upper ocean circulation in the tropical Pacific domain. The computation cost is reduced dramatically. Compared with the POD method, the method provided more accurate results and it did not require the availability of any explicit equations or the right-handside (RHS) of the evolution equation. †
A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using... more A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using both a standard method and an incremental method in an identical twin framework. The full physics adjoint model of the Florida State University global spectral model (FSUGSM) was used in the standard 4D-Var, while the adjoint of only a few selected physical parameterizations was used in the incremental method. The impact of physical processes on 4D-Var was examined in detail by comparing the results of these experiments. The inclusion of full physics turned out to be significantly beneficial in terms of assimilation error to the lower troposphere during the entire minimization process. The beneficial impact was found to be primarily related to boundary layer physics. The precipitation physics in the adjoint model also tended to have a beneficial impact after an intermediate number (50) of minimization iterations. Experiment results confirmed that the forecast from assimilation analyses with the full physics adjoint model displays a shorter precipitation spinup period. The beneficial impact on precipitation spinup did not result solely from the inclusion of the precipitation physics in the adjoint model, but rather from the combined impact of several physical processes. The inclusion of full physics in the adjoint model exhibited a detrimental impact on the rate of convergence at an early stage of the minimization process, but did not affect the final convergence.
ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a ... more ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a view to calculate the gradient of a cost functional in the framework of using this model to carry out variational data assimilation (VDA) experiments using optimal control of partial differential equations. The finite-element model employs a triangular finite-element Galerkin scheme and serves as a prototype of 2D shallow-water equation models with a view of tackling problems related to VDA with finite-element numerical weather prediction models. The derivation of the adjoint of this finite-element model involves overcoming specific computational problems related to obtaining the adjoint of iterative procedures for solving systems of nonsymmetric linear equations arising from the finite-element discretization and dealing with irregularly ordered discrete variables at each time step. The correctness of the adjoint model was verified at the subroutine level and was followed by a gradient check conducted once the full adjoint model was assembled. VDA experiments were performed using model-generated observations. In our experiments, assimilation was carried out assuming that observations consisting of a full-model-state vector are available at every time step in the window of assimilation. Successful retrieval was obtained using the initial conditions as control variables, involving the minimization of a cost function consisting of the weighted sum of difference between model solution and model-generated observations. An additional set of experiments was carried out aiming at evaluating the impact of carrying out VDA involving variable mesh resolution in the finite-element model over the entire assimilation period. Several conclusions are drawn related to the efficiency of VDA with variable horizontal mesh resolution finite-element discretization and the transfer of information between coarse and fine meshes. 51 refs., 17 figs., 5 tabs.
The known properties of equivalence between four-dimensional variational (4D-Var) data assimilati... more The known properties of equivalence between four-dimensional variational (4D-Var) data assimilation and the Kalman filter as well as the fixed-interval Kalman smoother point to particular optimal properties of 4D-Var. In the linear context, the 4D-Var solution is optimal, not only with respect to the model trajectory segment over the assimilation time interval, but also with respect to any model state at a single observation time level; in the batch processing (cycling 4D-Var) method, the information in 4D-Var is fully transferred from one batch to the next by the background term; 4D-Var allows the processing of observations in subsets, while the final solution is optimal as all observations are processed simultaneously. These properties hold even for models that are imperfect, as well as not invertible. Various properties of equivalence of 4D-Var to the Kalman filter and smoother result from these optimality properties of 4D-Var. Further, we show that the fixed-lag Kalman smoother may also be constructed in an optimal way using a multiple batch-processing 4D-Var approach. While error covariances are crucial for the equivalence, practical techniques for evaluating error covariances in the framework of cycling 4D-Var are discussed.
The Kuramoto-Sivashinsky equation plays an important role as a low-dimensional prototype for comp... more The Kuramoto-Sivashinsky equation plays an important role as a low-dimensional prototype for complicated fluid dynamics systems having been studied due to its chaotic pattern forming behavior. Up to now, efforts to carry out data assimilation with this 1-d model were restricted to variational adjoint methods domain and only Chorin and Krause [26] tested it using a sequential Bayesian filter approach. In this work we compare three sequential data assimilation methods namely the Kalman filter (EnKF) approach the sequential Monte-Carlo particle filter approach (PF) and the Maximum Likelihood Ensemble Filter methods (MLEF). This comparison is to the best of our knowledge novel. We compare in detail their relative performance for both linear and nonlinear observation operators. The results of these sequential data assimilation tests are discussed and conclusions are drawn as to the suitability of these data assimilation methods in the presence of linear and nonlinear observation operators.
Linearization and adjoint-model derivation for the solar radiation transfer codes in the NMC spec... more Linearization and adjoint-model derivation for the solar radiation transfer codes in the NMC spectral model have been carried out. Verification of the validity of resulting tangent linear model and the correctness of the corresponding adjoint have been performed. Applications of derived adjoint model are considered, including parameter estimation for inputs to solar radiation codes with aid of the physics (i.e., the solar radiation codes) and a sensitivity study of the downward solar radiation flux at the earth surface with respect to water vapor amount at various heights. 194 J. Zou and 1. M. Navon
This work describes the dynamics of adjoint sensitivity perturbations that excite block onsets ov... more This work describes the dynamics of adjoint sensitivity perturbations that excite block onsets over the Pacific and Atlantic Oceans. Appropriate functions are derived for the blocking indices for these two regions and the model basic flow is constructed from Northern Hemisphere climatological data. The concepts of sensitivity analysis are extended to forced problems. This tool is used to investigate block onset due to atmospheric forcing, such as that resulting from tropical sea surface temperature anomalies. These linear studies are carried out in a hemispherical, primitive equations, -coordinate, two-layer model.
A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using... more A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using both a standard method and an incremental method in an identical twin framework. The full physics adjoint model of the Florida State University global spectral model (FSUGSM) was used in the standard 4D-Var, while the adjoint of only a few selected physical parameterizations was used in the incremental method. The impact of physical processes on 4D-Var was examined in detail by comparing the results of these experiments. The inclusion of full physics turned out to be significantly beneficial in terms of assimilation error to the lower troposphere during the entire minimization process. The beneficial impact was found to be primarily related to boundary layer physics. The precipitation physics in the adjoint model also tended to have a beneficial impact after an intermediate number (50) of minimization iterations. Experiment results confirmed that the forecast from assimilation analyses with the full physics adjoint model displays a shorter precipitation spinup period. The beneficial impact on precipitation spinup did not result solely from the inclusion of the precipitation physics in the adjoint model, but rather from the combined impact of several physical processes. The inclusion of full physics in the adjoint model exhibited a detrimental impact on the rate of convergence at an early stage of the minimization process, but did not affect the final convergence.
International Journal For Numerical Methods in Fluids, Mar 19, 2009
A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced u... more A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced unstructured mesh finite-element ocean model, the Imperial College Ocean Model (ICOM, described in detail below), which includes many recent developments in ocean modelling and numerical analysis. The advantages of the POD model developed here over existing POD approaches are the ability:
A Sasaki variational approach is for the first time applied to enforce conservation of potential ... more A Sasaki variational approach is for the first time applied to enforce conservation of potential enstrophy and total mass in long term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a beta plane. The performance of the variational approach is compared with that of a modified Bayliss-Isaacson technique also designed to enforce conservation of potential enstrophy and total mass at each time step of the numerical integration. Both techniques yielded very satisfactory results after 20 days of numerical integration. It appears, however, that the Bayliss-Isaacson technique is more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the method used to update the Lagrange multiplier.
The specification of the initial ensemble for ensemble data assimilation is addressed. The presen... more The specification of the initial ensemble for ensemble data assimilation is addressed. The presented work examines the impact of ensemble initiation in the Maximum Likelihood Ensemble Filter (MLEF) framework, but is also applicable to other ensemble data assimilation algorithms. Two methods are considered: the first is based on the use of the Kardar-Parisi-Zhang (KPZ) equation to form sparse random perturbations, followed by spatial smoothing to enforce desired correlation structure, while the second is based on the spatial smoothing of initially uncorrelated random perturbations. Data assimilation experiments are conducted using a global shallow-water model and simulated observations. The two proposed methods are compared to the commonly used method of uncorrelated random perturbations. The results indicate that the impact of the initial correlations in ensemble data assimilation is beneficial. The root-mean-square error rate of convergence of the data assimilation is improved, and the positive impact of initial correlations is notable throughout the data assimilation cycles. The sensitivity to the choice of the correlation length scale exists, although it is not very high. The implied computational savings and improvement of the results may be important in future realistic applications of ensemble data assimilation.
This chapter presents the mathematical framework to evaluate the sensitivity of a model forecast ... more This chapter presents the mathematical framework to evaluate the sensitivity of a model forecast aspect to the input parameters of a nonlinear four-dimensional variational data assimilation system (4D-Var DAS): observations, prior state (background) estimate, and the error covariance specification. A fundamental relationship is established between the forecast sensitivity with respect to the information vector and the sensitivity with respect to the DAS representation of the information error covariance. Adjoint modeling is used to obtain first-and second-order derivative information and a reduced-order approach is formulated to alleviate the computational cost associated with the sensitivity estimation. Numerical results from idealized 4D-Var experiments performed with a global shallow water model are used to illustrate the theoretical concepts.
An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian condit... more An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian conditional probability estimation, along with the well known ensemble Kalman filter. A Gaussian resampling method is proposed here to generate the posterior analysis ensemble in an effective and efficient way. As a result the ensemble particle filter has good stability and potential applicability to large-scale problems. The Lorenz model is used here to test the proposed method. Multi-modal probability distributions can appear either with state dependent stochastic model errors or nonlinear observations. Ensemble Kalman filter (EnKF)is known to have a difficulty in tracking state transitions accurately. Current implementations of EnKF have not taken non-Gaussian contributions into account. With the posterior Gaussian resampling method the ensemble particle filter can track state transitions more accurately. Moreover, it is applicable to systems with typical multi-modal behavior, provided that certain prior knowledge becomes available about the general structure of posterior probability distribution. A simple scenario is considered to illustrate this point based on Lorenz model attractors. The present work demonstrates that the proposed ensemble particle filter can provide an accurate estimation of multi-modal distribution and is potentially applicable to large-scale data assimilation problems.
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational... more This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for Proper Orthogonal Decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
The pointwise estimation of heat conduction solution as a function of truncation error of a finit... more The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The contribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error correction and that an asymptotic error bound may be found.
Advances in Geosciences Volume 12 Ocean Science, Mar 1, 2009
The "equation-free" (EF) method is often used in complex, multi-scale problems. In such cases it ... more The "equation-free" (EF) method is often used in complex, multi-scale problems. In such cases it is necessary to know the closed form of the required evolution equations about macroscopic variables within some applied fields. Conceptually such equations exist, however, they are not available in closed form. The EF method can bypass this difficulty. This method can obtain macroscopic information by implementing models at a microscopic level. Given an initial macroscopic variable, through lifting we can obtain the associated microscopic variable, which may be evolved using Direct Numerical Simulations (DNS) and by restriction, we can obtain the necessary macroscopic information and the projective integration to obtain the desired quantities. In this paper we apply the EF POD-assisted method to the reduced modeling of a large-scale upper ocean circulation in the tropical Pacific domain. The computation cost is reduced dramatically. Compared with the POD method, the method provided more accurate results and it did not require the availability of any explicit equations or the right-handside (RHS) of the evolution equation. †
A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using... more A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using both a standard method and an incremental method in an identical twin framework. The full physics adjoint model of the Florida State University global spectral model (FSUGSM) was used in the standard 4D-Var, while the adjoint of only a few selected physical parameterizations was used in the incremental method. The impact of physical processes on 4D-Var was examined in detail by comparing the results of these experiments. The inclusion of full physics turned out to be significantly beneficial in terms of assimilation error to the lower troposphere during the entire minimization process. The beneficial impact was found to be primarily related to boundary layer physics. The precipitation physics in the adjoint model also tended to have a beneficial impact after an intermediate number (50) of minimization iterations. Experiment results confirmed that the forecast from assimilation analyses with the full physics adjoint model displays a shorter precipitation spinup period. The beneficial impact on precipitation spinup did not result solely from the inclusion of the precipitation physics in the adjoint model, but rather from the combined impact of several physical processes. The inclusion of full physics in the adjoint model exhibited a detrimental impact on the rate of convergence at an early stage of the minimization process, but did not affect the final convergence.
ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a ... more ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a view to calculate the gradient of a cost functional in the framework of using this model to carry out variational data assimilation (VDA) experiments using optimal control of partial differential equations. The finite-element model employs a triangular finite-element Galerkin scheme and serves as a prototype of 2D shallow-water equation models with a view of tackling problems related to VDA with finite-element numerical weather prediction models. The derivation of the adjoint of this finite-element model involves overcoming specific computational problems related to obtaining the adjoint of iterative procedures for solving systems of nonsymmetric linear equations arising from the finite-element discretization and dealing with irregularly ordered discrete variables at each time step. The correctness of the adjoint model was verified at the subroutine level and was followed by a gradient check conducted once the full adjoint model was assembled. VDA experiments were performed using model-generated observations. In our experiments, assimilation was carried out assuming that observations consisting of a full-model-state vector are available at every time step in the window of assimilation. Successful retrieval was obtained using the initial conditions as control variables, involving the minimization of a cost function consisting of the weighted sum of difference between model solution and model-generated observations. An additional set of experiments was carried out aiming at evaluating the impact of carrying out VDA involving variable mesh resolution in the finite-element model over the entire assimilation period. Several conclusions are drawn related to the efficiency of VDA with variable horizontal mesh resolution finite-element discretization and the transfer of information between coarse and fine meshes. 51 refs., 17 figs., 5 tabs.
The known properties of equivalence between four-dimensional variational (4D-Var) data assimilati... more The known properties of equivalence between four-dimensional variational (4D-Var) data assimilation and the Kalman filter as well as the fixed-interval Kalman smoother point to particular optimal properties of 4D-Var. In the linear context, the 4D-Var solution is optimal, not only with respect to the model trajectory segment over the assimilation time interval, but also with respect to any model state at a single observation time level; in the batch processing (cycling 4D-Var) method, the information in 4D-Var is fully transferred from one batch to the next by the background term; 4D-Var allows the processing of observations in subsets, while the final solution is optimal as all observations are processed simultaneously. These properties hold even for models that are imperfect, as well as not invertible. Various properties of equivalence of 4D-Var to the Kalman filter and smoother result from these optimality properties of 4D-Var. Further, we show that the fixed-lag Kalman smoother may also be constructed in an optimal way using a multiple batch-processing 4D-Var approach. While error covariances are crucial for the equivalence, practical techniques for evaluating error covariances in the framework of cycling 4D-Var are discussed.
The Kuramoto-Sivashinsky equation plays an important role as a low-dimensional prototype for comp... more The Kuramoto-Sivashinsky equation plays an important role as a low-dimensional prototype for complicated fluid dynamics systems having been studied due to its chaotic pattern forming behavior. Up to now, efforts to carry out data assimilation with this 1-d model were restricted to variational adjoint methods domain and only Chorin and Krause [26] tested it using a sequential Bayesian filter approach. In this work we compare three sequential data assimilation methods namely the Kalman filter (EnKF) approach the sequential Monte-Carlo particle filter approach (PF) and the Maximum Likelihood Ensemble Filter methods (MLEF). This comparison is to the best of our knowledge novel. We compare in detail their relative performance for both linear and nonlinear observation operators. The results of these sequential data assimilation tests are discussed and conclusions are drawn as to the suitability of these data assimilation methods in the presence of linear and nonlinear observation operators.
Linearization and adjoint-model derivation for the solar radiation transfer codes in the NMC spec... more Linearization and adjoint-model derivation for the solar radiation transfer codes in the NMC spectral model have been carried out. Verification of the validity of resulting tangent linear model and the correctness of the corresponding adjoint have been performed. Applications of derived adjoint model are considered, including parameter estimation for inputs to solar radiation codes with aid of the physics (i.e., the solar radiation codes) and a sensitivity study of the downward solar radiation flux at the earth surface with respect to water vapor amount at various heights. 194 J. Zou and 1. M. Navon
This work describes the dynamics of adjoint sensitivity perturbations that excite block onsets ov... more This work describes the dynamics of adjoint sensitivity perturbations that excite block onsets over the Pacific and Atlantic Oceans. Appropriate functions are derived for the blocking indices for these two regions and the model basic flow is constructed from Northern Hemisphere climatological data. The concepts of sensitivity analysis are extended to forced problems. This tool is used to investigate block onset due to atmospheric forcing, such as that resulting from tropical sea surface temperature anomalies. These linear studies are carried out in a hemispherical, primitive equations, -coordinate, two-layer model.
A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using... more A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using both a standard method and an incremental method in an identical twin framework. The full physics adjoint model of the Florida State University global spectral model (FSUGSM) was used in the standard 4D-Var, while the adjoint of only a few selected physical parameterizations was used in the incremental method. The impact of physical processes on 4D-Var was examined in detail by comparing the results of these experiments. The inclusion of full physics turned out to be significantly beneficial in terms of assimilation error to the lower troposphere during the entire minimization process. The beneficial impact was found to be primarily related to boundary layer physics. The precipitation physics in the adjoint model also tended to have a beneficial impact after an intermediate number (50) of minimization iterations. Experiment results confirmed that the forecast from assimilation analyses with the full physics adjoint model displays a shorter precipitation spinup period. The beneficial impact on precipitation spinup did not result solely from the inclusion of the precipitation physics in the adjoint model, but rather from the combined impact of several physical processes. The inclusion of full physics in the adjoint model exhibited a detrimental impact on the rate of convergence at an early stage of the minimization process, but did not affect the final convergence.
International Journal For Numerical Methods in Fluids, Mar 19, 2009
A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced u... more A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced unstructured mesh finite-element ocean model, the Imperial College Ocean Model (ICOM, described in detail below), which includes many recent developments in ocean modelling and numerical analysis. The advantages of the POD model developed here over existing POD approaches are the ability:
A Sasaki variational approach is for the first time applied to enforce conservation of potential ... more A Sasaki variational approach is for the first time applied to enforce conservation of potential enstrophy and total mass in long term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a beta plane. The performance of the variational approach is compared with that of a modified Bayliss-Isaacson technique also designed to enforce conservation of potential enstrophy and total mass at each time step of the numerical integration. Both techniques yielded very satisfactory results after 20 days of numerical integration. It appears, however, that the Bayliss-Isaacson technique is more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the method used to update the Lagrange multiplier.
The specification of the initial ensemble for ensemble data assimilation is addressed. The presen... more The specification of the initial ensemble for ensemble data assimilation is addressed. The presented work examines the impact of ensemble initiation in the Maximum Likelihood Ensemble Filter (MLEF) framework, but is also applicable to other ensemble data assimilation algorithms. Two methods are considered: the first is based on the use of the Kardar-Parisi-Zhang (KPZ) equation to form sparse random perturbations, followed by spatial smoothing to enforce desired correlation structure, while the second is based on the spatial smoothing of initially uncorrelated random perturbations. Data assimilation experiments are conducted using a global shallow-water model and simulated observations. The two proposed methods are compared to the commonly used method of uncorrelated random perturbations. The results indicate that the impact of the initial correlations in ensemble data assimilation is beneficial. The root-mean-square error rate of convergence of the data assimilation is improved, and the positive impact of initial correlations is notable throughout the data assimilation cycles. The sensitivity to the choice of the correlation length scale exists, although it is not very high. The implied computational savings and improvement of the results may be important in future realistic applications of ensemble data assimilation.
This chapter presents the mathematical framework to evaluate the sensitivity of a model forecast ... more This chapter presents the mathematical framework to evaluate the sensitivity of a model forecast aspect to the input parameters of a nonlinear four-dimensional variational data assimilation system (4D-Var DAS): observations, prior state (background) estimate, and the error covariance specification. A fundamental relationship is established between the forecast sensitivity with respect to the information vector and the sensitivity with respect to the DAS representation of the information error covariance. Adjoint modeling is used to obtain first-and second-order derivative information and a reduced-order approach is formulated to alleviate the computational cost associated with the sensitivity estimation. Numerical results from idealized 4D-Var experiments performed with a global shallow water model are used to illustrate the theoretical concepts.
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Papers by Ionel M Navon