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The forecast sensitivity to observations (FSO) is embedded into a new optimization framework for improving the observation performance in atmospheric data assimilation. Key ingredients are introduced as follows: the innovation-weight parametrization of the analysis equation, the FSO-based evaluation of the forecast error gradient to parameters, a line search approach to optimization, and an efficient mechanism for step length specification. This methodology is tested in preliminary numerical experiments with the Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) and the U.S. Navy’s Global Environmental Model (NAVGEM) at a T425L60 resolution. The experimental setup relies on a verification state produced by the European Centre for Medium-Range Weather Forecasts (ECMWF) to estimate the analysis and short-range forecast errors. Parameter tuning is implemented in a training stage valid for April 1-14 of 2018 and aimed at improving the use of assimilated observations in reducing the initial-condition errors. Assessment is carried out for April 15 – May 31 of 2018 to investigate the performance of the weighted assimilation system in reducing the errors in analyses and 24-hour model forecasts. In average, as compared with the control run and verified against the ECMWF analyses, the weighted approach provided 17.4% reduction in analysis errors and 3.1% reduction in 24-hour forecast errors, measured in a dry total energy norm. Observation impacts are calculated to assess the use of various observation types in reducing the analysis and forecast errors. In particular, assimilation of satellite wind data is significantly improved through the innovation-weighting procedure.

This article presents a novel approach to couple a deterministic four‐dimensional variational (4DVAR) assimilation method with the particle filter (PF) ensemble data assimilation system, to produce a robust approach for dual‐state‐parameter estimation. In our proposed method, the Hybrid Ensemble and Variational Data Assimilation framework for Environmental systems (HEAVEN), we characterize the model structural uncertainty in addition to model parameter and input uncertainties. The sequential PF is formulated within the 4DVAR system to design a computationally efficient feedback mechanism throughout the assimilation period. In this framework, the 4DVAR optimization produces the maximum a posteriori estimate of state variables at the beginning of the assimilation window without the need to develop the adjoint of the forecast model. The 4DVAR solution is then perturbed by a newly defined prior error covariance matrix to generate an initial condition ensemble for the PF system to provide more accurate and reliable posterior distributions within the same assimilation window. The prior error covariance matrix is updated from one cycle to another over the main assimilation period to account for model structural uncertainty resulting in an improved estimation of posterior distribution. The premise of the presented approach is that it (1) accounts for all sources of uncertainties involved in hydrologic predictions, (2) uses a small ensemble size, and (3) precludes the particle degeneracy and sample impoverishment. The proposed method is applied on a nonlinear hydrologic model and the effectiveness, robustness, and reliability of the method is demonstrated for several river basins across the United States., Key Points A joint sequential and variational data assimilation method was developed for superior and robust dual‐state‐parameter estimationThe proposed HEAVEN approach accounts for all sources of uncertainties involved in model predictionsThe effectiveness and usefulness of HEAVEN was evaluated by both deterministic and probabilistic measures

An innovation-weight parametrization is introduced as a practical approach to account for deficiencies in the representation of both background error and observation error covariance in a variational data assimilation system. The adjoint-based evaluation of the forecast error sensitivity provides a computationally efficient diagnosis to observation-space distributed parameters and guidance for tuning the analysis Kalman gain operator. Theoretical aspects are discussed and preliminary results are presented with the adjoint versions of the Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) and the Navy’s Global Environmental Model (NAVGEM).

Novel applications of the adjoint-based sensitivity tools are investigated to obtain a priori guidance on the forecast impact of modeling correlated observational errors in a four-dimensional variational data assimilation system (4D-Var DAS) . A synergistic framework is considered that combines a posteriori estimates to the observation error covariance (R) and derivative information extracted from the adjoint-DAS forecast error R-sensitivity (FSR ). It is explained that the FSR approach allows the analysis of structured error correlation models and estimation of their potential impact on reducing the forecast errors. Theoretical aspects are discussed and a proof-of-concept is provided with Lorenz’s 40-variable model. The practical ability to exercise these new adjoint capabilities is shown in experiments performed with the Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) and the Navy’s Global Environmental Model (NAVGEM). In particular, the FSR analysis of radiances assimilated from the Infrared Atmospheric Sounding Interferometer (IASI) indicates that modeling inter-channel observation error correlations may provide an increased benefit to the forecasts, as compared with tuning procedures that ignore the error correlations and only adjust the assigned observation error variance parameters.

This article presents the adjoint-data assimilation system (adjoint-DAS) approach to evaluate the forecast sensitivity with respect to the specification of the observation-error covariance (R-sensitivity) and background-error covariance (B-sensitivity) in a four-dimensional variational (4D-Var) DAS with a single outer-loop iteration. Computationally efficient estimates to the forecast impact of adjustments in the error covariance models are obtained by exploiting the mathematical properties of the R- and B-sensitivity matrices and their relationship with the observation sensitivity vector. An additional contribution of this work is that it establishes a synergistic link between various methodologies to analyze the DAS performance: observation sensitivity and impact assessment, error covariance sensitivity, and a posteriori diagnosis. The practical ability to obtain sensitivity information with respect to R- and B-parameters is presented with the adjoint versions of the Naval Research Laboratory Atmospheric Variational Data Assimilation System–Accelerated Representer (NAVDAS-AR) and the Navy Operational Global Atmospheric Prediction System (NOGAPS). The adjoint approach is used to provide guidance on the forecast impact of weighting the radiance data in the DAS according to observation-error variance estimates derived from an a posteriori diagnosis. The results indicate that information extracted from both error covariance diagnosis and sensitivity analysis is necessary to design parameter tuning procedures that are effective in reducing the forecast errors. Copyright © 2012 Royal Meteorological Society

SUMMARY An observation sensitivity (OS) method to identify targeted observations is implemented in the context of four-dimensional variational (4D-Var) data assimilation. This methodology is compared with the well-established adjoint sensitivity (AS) method using a nonlinear Burgers equation as a test model. Automatic differentiation software is used to implement the first-order adjoint model (ADM) to calculate the gradient of the cost function required in the 4D-Var minimization algorithm and in the AS computations and the second-order ADM to obtain information on the Hessian matrix of the 4D-Var cost that is necessary in the OS computations. Numerical results indicate that the observation-targeting is particularly successful in reducing the forecast error for moderate Reynolds numbers. The potential benefits of the OS targeting approach over the AS are investigated. The effect of random perturbations on the performance of these adaptive observation techniques is also analyzed. Copyright © 2011 John Wiley & Sons, Ltd.

The efficiency of current adjoint-based observations targeting strategies in variational data assimilation is closely determined by the underlying assumption of a linear propagation of initial condition errors into the model forecasts. A novel targeting strategy is proposed in the context of four-dimensional variational data assimilation (4D-Var) to account for nonlinear error growth as the forecast lead time increases. A quadratic error growth model is shown to maintain the accuracy in tracking the nonlinear evolution of initial condition perturbations, as compared to the first-order approximation. A second-order adjoint model is used to provide the derivative information that is necessary in the higher-order Taylor series approximation. The observation targeting approach relies on the dominant eigenvectors of the Hessian matrix associated with a specific forecast error aspect as an indicator of the directions of largest quadratic error growth. A comparative qualitative analysis between observation targeting based on first- and second-order adjoint information is presented in idealized 4D-Var experiments with a two-dimensional global shallow-water model. The results indicate that accounting for the quadratic error growth in the targeting strategy is of particular benefit as the forecast lead time increases.

The development of the adjoint of the forecast model and of the adjoint of the data assimilation system (adjoint-DAS) makes feasible the evaluation of the local sensitivity of a model forecast aspect with respect to a large number of parameters in the DAS. In this study it is shown that, by exploiting sensitivity properties that are intrinsic to the analyses derived from a minimization principle, the adjoint-DAS software tools developed at numerical weather prediction centres for observation and background sensitivity may be used to estimate the forecast sensitivity to observation- and background-error covariance parameters and for forecast impact assessment. All-at-once sensitivity to error covariance weighting coefficients and first-order impact estimates are derived as a particular case of the error covariance perturbation analysis. The use of the sensitivity information as a DAS diagnostic tool and for implementing gradient-based error covariance tuning algorithms is illustrated in idealized data assimilation experiments with the Lorenz 40-variable model. Preliminary results of forecast sensitivity to observation- and background-error covariance weight parameters are presented using the fifth-generation NASA Goddard Earth Observing System (GEOS-5) atmospheric DAS and its adjoint developed at the Global Modeling and Assimilation Office. Copyright © 2010 Royal Meteorological Society

An optimal use of the atmospheric data in numerical weather prediction requires an objective assessment of the value added by observations to improve the analyses and forecasts of a specific data assimilation system (DAS). This research brings forward the issue of uncertainties in the assessment of observation values based on deterministic observation impact (OBSI) estimations using observing system experiments (OSEs) and the adjoint-DAS framework. The state-to-observation space uncertainty propagation as a result of the errors in the verification state is investigated. For a quadratic forecast error measure, a geometrical perspective is used to provide insight and to convey some of the key aspects of this research. The study is specialized to a DAS implementing a linear analysis scheme and numerical experiments are presented using the Lorenz 40-variable model.

We describe the global behavior of the dynamics of a particle bouncing down an inclined staircase. For small inclinations all orbits eventually stop (independent of the initial condition). For large enough inclinations all orbits end up accelerating indefinitely (also independent of the initial conditions). There is an interval of inclinations of positive length between these two. In that interval the behavior of an orbit depends on its initial condition. In addition to stopping and accelerating orbits, there are also orbits with speeds bounded away from both zero and infinity. A second hallmark of the dynamics is that the orbits going at a finite (but non-zero) average speed tend to have close to constant speed. In the setting of this model these phenomena are robust in the sense that they are independent of the ‘ruggedness’ of the staircase and of the coefficients of restitution that govern the energy loss at each bounce. The behavior just described matches up well with physical observations of single particles falling down a rough slope as well as measurements in laboratory controlled avalanches. This (and the robustness of the results) suggests that many-particle systems (avalanches) behave in similar ways as our low-dimensional model.

A parametric approach to the adjoint estimation of the variation in model functional output due to the assimilation of data is considered as a tool to analyze and develop observation impact measures. The parametric approach is specialized to a linear analysis scheme and it is used to derive various high-order approximation equations. This framework includes the Kalman filter and incremental three-and four-dimensional variational data assimilation schemes implementing a single outer loop iteration. Distinction is made between Taylor series methods and numerical quadrature methods. The novel quadrature approximations require minimal additional software development and are suitable for testing and implementation at operational numerical weather prediction centers where a data assimilation system (DAS) and the associated adjoint DAS are in place. Their potential use as tools for observation impact estimates needs to be further investigated. Preliminary numerical experiments are provided using the fifth-generation NASA Goddard Earth Observing System (GEOS-5) atmospheric DAS.

The equations of the forecast sensitivity to observations and to the background estimate in a four-dimensional variational data assimilation system (4D-Var DAS) are derived from the first-order optimality condition in unconstrained minimization. Estimation of the impact of uncertainties in the specification of the error statistics is considered by evaluating the sensitivity to the observation and background error covariance matrices. The information provided by the error covariance sensitivity analysis is used to identify the input components for which improved estimates of the statistical properties of the errors are of most benefit to the analysis and forecast. A close relationship is established between the sensitivities within each input pair data/error covariance such that once the observation and background sensitivities are available the evaluation of the sensitivity to the specification of the corresponding error statistics requires little additional computational effort. The relevance of the 4D-Var sensitivity equations to assess the data impact in practical applications is discussed. Computational issues are addressed and idealized 4D-Var experiments are set up with a finite-volume shallow-water model to illustrate the theoretical concepts. Time-dependent observation sensitivity and potential applications to improve the model forecast are presented. Guidance provided by the sensitivity fields is used to adjust a 4D-Var DAS to achieve forecast error reduction through assimilation of supplementary data and through an accurate specification of a few of the background error variances.

Strategies to achieve order reduction in four-dimensional variational data assimilation (4DVAR) search for an optimal low-rank state subspace for the analysis update. A common feature of the reduction methods proposed in atmospheric and oceanographic studies is that the identification of the basis functions relies on the model dynamics only, without properly accounting for the specific details of the data assimilation system (DAS). In this study a general framework of the proper orthogonal decomposition (POD) method is considered and a cost-effective approach is proposed to incorporate DAS information into the order-reduction procedure. The sensitivities of the cost functional in 4DVAR data assimilation with respect to the time-varying model state are obtained from a backward integration of the adjoint model. This information is further used to define appropriate weights and to implement a dual-weighted proper orthogonal decomposition (DWPOD) method for order reduction. The use of a weighted ensemble data mean and weighted snapshots using the adjoint DAS is a novel element in reduced-order 4DVAR data assimilation. Numerical results are presented with a global shallow-water model based on the Lin–Rood flux-form semi-Lagrangian scheme. A simplified 4DVAR DAS is considered in the twin-experiment framework with initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets. A comparative analysis with the standard POD method shows that the reduced DWPOD basis may provide an increased efficiency in representing an a priori specified forecast aspect and as a tool to perform reduced-order optimal control. This approach represents a first step toward the development of an order-reduction methodology that combines in an optimal fashion the model dynamics and the characteristics of the 4DVAR DAS.

SUMMARY Order reduction strategies aim to alleviate the computational burden of the four-dimensional variational data assimilation by performing the optimization in a low-order control space. The proper orthogonal decomposition (POD) approach to model reduction is used to identify a reduced-order control space for a two-dimensional global shallow water model. A reduced second-order adjoint (SOA) model is developed and used to facilitate the implementation of a Hessian-free truncated-Newton (HFTN) minimization algorithm in the POD-based space. The efficiency of the SOA/HFTN implementation is analysed by comparison with the quasi-Newton BFGS and a nonlinear conjugate gradient algorithm. Several data assimilation experiments that differ only in the optimization algorithm employed are performed in the reduced control space. Numerical results indicate that first-order derivative methods are effective during the initial stages of the assimilation; in the later stages, the use of second-order derivative information is of benefit and HFTN provided significant CPU time savings when compared to the BFGS and CG algorithms. A comparison with data assimilation experiments in the full model space shows that with an appropriate selection of the basis functions the optimization in the POD space is able to provide accurate results at a reduced computational cost. The HFTN algorithm benefited most from the order reduction since computational savings were achieved both in the outer and inner iterations of the method. Further experiments are required to validate the approach for comprehensive global circulation models. Copyright q 2006 John Wiley & Sons, Ltd.

The task of providing an optimal analysis of the state of the atmosphere requires the development of efficient computational tools that facilitate an efficient integration of observational data into models. In a variational approach the data assimilation problem is posed as a minimization problem, which requires the sensitivity (derivatives) of a cost functional with respect to problem parameters. The direct decoupled method has been extensively applied for sensitivity studies of air pollution. Adjoint sensitivity is a complementary approach which efficiently calculates the derivatives of a functional with respect to a large number of parameters. In this paper, we discuss the mathematical foundations of the adjoint sensitivity method applied to air pollution models, and present a complete set of computational tools for performing three-dimensional adjoint sensitivity studies. Numerical examples show that three-dimensional adjoint sensitivity analysis provides information on influence areas, which cannot be obtained solely by an inverse analysis of the meteorological fields. Several illustrative data assimilation results in a twin experiments framework, as well as the assimilation of a real data set are also presented.

The design of adaptive observations strategies must account for the particular properties of the data assimilation method. A new adjoint sensitivity approach to the targeted observations problem is proposed in the context of four-dimensional variational data assimilation (4D-Var). The method is based on a periodic update of the adjoint sensitivity field that takes into account the interaction between time distributed adaptive and routine observations. Information provided by all previously located observations is used to identify best locations for new targeted observations. Adaptive observations at distinct instants in time are selected in a sequential manner such that the method is only suboptimal. The selection algorithm proceeds backward in time and requires only one additional adjoint model integration in the assimilation window. Therefore, the method is very efficient and is suitable for practical applications. A comparative performance analysis is presented using the traditional adjoint sensitivity method as well as the total energy singular vectors technique as alternative adaptive strategies. Numerical experiments are performed in the twin experiments framework using a two-dimensional global shallow water model in spherical coordinates and an explicit Turkel-Zwas discretization scheme. Data from a NASA 500 mb analysis valid for 00Z 16 Mar 2001 6 h obtained with the GEOS-3 model was used to specify the geopotential height at the initial time and the initial velocities were obtained from a geostrophic balance. Numerical results show that the new adaptive observations approach is a promising method for targeted observations and its implementation is feasible for large scale atmospheric models.

The analysis of comprehensive chemical reactions mechanisms, parameter estimation techniques, and variational chemical data assimilation applications require the development of efficient sensitivity methods for chemical kinetics systems. The new release (KPP-1.2) of the kinetic preprocessor (KPP) contains software tools that facilitate direct and adjoint sensitivity analysis. The direct-decoupled method, built using BDF formulas, has been the method of choice for direct sensitivity studies. In this work, we extend the direct-decoupled approach to Rosenbrock stiff integration methods. The need for Jacobian derivatives prevented Rosenbrock methods to be used extensively in direct sensitivity calculations; however, the new automatic and symbolic differentiation technologies make the computation of these derivatives feasible. The direct-decoupled method is known to be efficient for computing the sensitivities of a large number of output parameters with respect to a small number of input parameters. The adjoint modeling is presented as an efficient tool to evaluate the sensitivity of a scalar response function with respect to the initial conditions and model parameters. In addition, sensitivity with respect to time-dependent model parameters may be obtained through a single backward integration of the adjoint model. KPP software may be used to completely generate the continuous and discrete adjoint models taking full advantage of the sparsity of the chemical mechanism. Flexible direct-decoupled and adjoint sensitivity code implementations are achieved with minimal user intervention. In a companion paper, we present an extensive set of numerical experiments that validate the KPP software tools for several direct/adjoint sensitivity applications, and demonstrate the efficiency of KPP-generated sensitivity code implementations.

The Kinetic PreProcessor KPP was extended to generate the building blocks needed for the direct and adjoint sensitivity analysis of chemical kinetic systems. An overview of the theoretical aspects of sensitivity calculations and a discussion of the KPP software tools is presented in the companion paper. In this work the correctness and efficiency of the KPP generated code for direct and adjoint sensitivity studies are analyzed through an extensive set of numerical experiments. Direct-decoupled Rosenbrock methods are shown to be cost-effective for providing sensitivities at low and medium accuracies. A validation of the discrete–adjoint evaluated gradients is performed against the finite difference estimates. The accuracy of the adjoint gradients is measured using a reference gradient value obtained with a standard direct-decoupled method. The accuracy is studied for both constant step size and variable step size integration of the forward/adjoint model and the consistency between the discrete and continuous adjoint models is analyzed. Applications of the KPP-1.2 software package to direct and adjoint sensitivity studies, variational data assimilation, and parameter identification are considered for the comprehensive chemical mechanism SAPRC-99.

In four-dimensional variational data assimilation (4D-Var) an optimal estimate of the initial state of a dynamical system is obtained by solving a large-scale unconstrained minimization problem. The gradient of the cost functional may be efficiently computed using the adjoint modeling, at the expense equivalent to a few forward model integrations; for most practical applications, the evaluation of the Hessian matrix is not feasible due to the large dimension of the discrete state vector. Hybrid methods aim to provide an improved optimization algorithm by dynamically interlacing inexpensive L-BFGS iterations with fast convergent Hessian-free Newton (HFN) iterations. In this paper, a comparative analysis of the performance of a hybrid method vs. L-BFGS and HFN optimization methods is presented in the 4D-Var context. Numerical results presented for a two-dimensional shallow-water model show that the performance of the hybrid method is sensitive to the selection of the method parameters such as the length of ...

The spatiotemporal distribution of observations plays an essential role in the data assimilation process. An adjoint sensitivity method is applied to the problem of adaptive location of the observational system for a nonlinear transport-chemistry model in the context of 4D variational data assimilation. The method is presented in a general framework and it is shown that in addition to the initial state of the model, sensitivity with respect to emission and deposition rates and certain types of boundary values may be obtained at a minimal additional cost. The adjoint modeling is used to evaluate the influence function and to identify the domain of influence associated with the observations. These essential tools are further used to develop a novel algorithm for targeting observations that takes into account the interaction among observations at different instants in time and spatial locations. Issues related to the case of multiple observations are addressed and it is shown that by using the adjoint modeling an efficient implementation may be achieved. Computational and practical aspects are discussed and this analysis indicates that it is feasible to implement the proposed method for comprehensive air quality models. Numerical experiments performed with a two-dimensional test model show promising results.

PAQMSG is an MPI-based, Fortran 90 communication library for the parallelization of air quality models (AQMs) on structured grids. It consists of distribution, gathering and repartitioning routines for different domain decompositions implementing a master–worker strategy. The library is architecture and application independent and includes optimization strategies for different architectures. This paper presents the library from a user perspective. Results are shown from the parallelization of STEM-III on Beowulf clusters. The PAQMSG library is available on the web. The communication routines are easy to use, and should allow for an immediate parallelization of existing AQMs. PAQMSG can also be used for constructing new models.

In the past decade the variational method has been successfully applied in data assimilation problems for atmospheric chemistry models. In 4D-var data assimilation, a minimization algorithm is used to find the set of control variables which minimizes the weighted least squares distance between model predictions and observations over the assimilation window. Using the adjoint method, the gradient of the cost function can be computed fast, at the expense of few function evaluations, making the optimization process very efficient. For large-scale models, the high storage requirements and the difficulty of implementing the adjoint code when sophisticated integrators are used to solve the stiff chemistry make the assimilation a very intensive computational process. If the sparse structure of the chemical models is carefully exploited, Rosenbrock methods have been proved to be reliable chemistry solvers because of their outstanding stability properties and conservation of the linear invariants of the system. In this paper we present an efficient implementation of the adjoint code for the Rosenbrock type methods, which can reduce the storage requirements of the forward model and is suitable for automatization. The adjoint code is completely generated using symbolic preprocessing and automatic differentiation tools which allow flexibility and require minimal user intervention.

Discusses computational challenges in air quality modelling (as viewed by the authors). The focus of the paper will be on Di, the “current” state‐of‐affairs. Owing to limitation of space the discussion will focus on only a few aspects of air quality modelling: i.e. chemical integration, sensitivity analysis and computational framework, with particular emphasis on aerosol issues.

Adjoint techniques are effective tools for the analysis and optimization of the observation performance on reducing the errors in the forecasts produced by atmospheric data assimilation systems (DASs). This chapter provides a detailed exposure of the equations that allow the extension of the adjoint-DAS applications from observation sensitivity and forecast impact assessment to diagnosis and tuning of parameters in the observation and background error covariance representation. The error covariance sensitivity analysis allows the identification of those parameters of potentially large impact on the forecast error reduction and provides a first-order diagnostic to parameter specification. A proof-of-concept is presented together with comparative results of observation impact assessment and sensitivity analysis obtained with the adjoint versions of the Naval Research Laboratory Atmospheric Variational Data Assimilation System – Accelerated Representer (NAVDAS-AR) and the Navy Operational Global Atmospheric Prediction System (NOGAPS).

The task of providing an optimal analysis of the state of the atmosphere requires the development of dynamic data-driven systems (DDDAS) that efficiently integrate the observational data and the models. Data assimilation, the dynamic incorporation of additional data into an executing application, is an essential DDDAS concept with wide applicability. In this paper we discuss practical aspects of nonlinear ensemble Kalman data assimilation applied to atmospheric chemical transport models. We highlight the challenges encountered in this approach such as filter divergence and spurious corrections, and propose solutions to overcome them, such as background covariance inflation and filter localization. The predictability is further improved by including model parameters in the assimilation process. Results for a large scale simulation of air pollution in North-East United States illustrate the potential of nonlinear ensemble techniques to assimilate chemical observations.

Air quality prediction plays an important role in the management of our environment. As more atmospheric chemical observations become available chemical data assimilation is expected to play an essential role in air quality forecasting. In this paper the current status of air quality forecasting is discussed and illustrated by comparison of predictions with observations. The future directions are also discussed, with an emphasis on data assimilation. Applications of the four dimensional variational method (4D-Var) and the ensemble Kalman filter (EnKF) approach are presented and discussed.

[1] In this paper, the four-dimensional variational (4D-Var) technique is applied to assimilate aircraft measurements during the Transport and Chemical Evolution over the Pacific (TRACE-P) field experiment into a chemical transport model, Sulfur Transport Eulerian Model, version 2K1 (STEM-2K1). Whether data assimilation would produce better analyzed fields is examined. It is found that assimilating ozone observations from one of two independent flights improves model prediction of the other flight ozone measurements, which are withheld as validation data. The adjusted initial fields after only assimilating the total reactive nitrogen (NOy) observations lead to better predictions of NO, NO2, and PAN, based on their agreement with the withheld measurements. One experiment simultaneously assimilating the observations of O3, NO, NO2, HNO3, PAN, and RNO3 demonstrates that the model is able to match those measurements well by changing the initial fields. In addition, the model predictions of NOy improve significantly after assimilating the aforementioned multiple observation species, which are independent of the withheld NOy measurements. In the paper, we also show that the key species whose initial mixing ratios would significantly affect the agreement between model and measurements can be identified using adjoint sensitivity analysis. Such information can be used to reduce the number of control variables in the 4D-Var data assimilation. To speed up the optimization process in the 4D-Var, we enforce the concentration upper bounds through the limited memory–Broyden-Fletcher-Goldfarb-Shanno-B (L-BFGS-B) algorithm, and this proves to be effective.

This paper presents an application of adjoint sensitivity calculation to retrieve the initial distribution of the aerosol population from measurements at later times. A general framework is given for the discretization of particle dynamics equation by projection methods. The adjoint of the discrete model is constructed. Adjoint modeling successfully retrieves the initial distribution, even is the measurements are restricted to only specific size bins.

The spatial-temporal distribution of the observations plays an essential role in the data assimilation process. Strategies for the adaptive location of the observational system search for locations in space and time where additional observational resources should be deployed in order to minimize the forecast error. In this paper we present an application of the adjoint sensitivity analysis to the problem of adaptive location of the observational system and area targeting for a transport-chemistry model in the context of 4D variational data assimilation. Numerical experiments are presented for a two-dimensional test model.

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 KardarParisi-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.

In variational data assimilation (VDA) for meteorological and/or oceanic models, the assimilated fields are deduced by combining the model and the gradient of a cost functional measuring discrepancy between model solution and observation, via a first-order optimality system. However, existence and uniqueness of the VDA problem along with convergence of the algorithms for its implementation depend on the convexity of the cost function. Properties of local convexity can be deduced by studying the Hessian of the cost function in the vicinity of the optimum. This shows the necessity of second-order information to ensure a unique solution to the VDA problem. In this paper a comprehensive review of issues related to second-order analysis of the problem of VDA is presented along with many important issues closely connected to it. In particular issues of existence, uniqueness, and regularization through second-order properties are examined. The focus then shifts to second-order information related to statistical properties and to issues related to preconditioning and optimization methods and second-order VDA analysis. Predictability and its relation to the structure of the Hessian of the cost functional is then discussed along with issues of sensitivity analysis in the presence of data being assimilated. Computational complexity issues are also addressed and discussed. Automatic differentiation issues related to second-order information are also discussed along with the computational complexity of deriving the second-order adjoint. Finally an application aimed at illustrating the use of automatic differentiation for deriving the second-order adjoint as well as the Hessian/vector product applied to minimizing a cost functional of a meteorological problem using the truncated-Newton method is presented. Results verifying numerically the computational cost of deriving the second-order adjoint as well as results related to the spectrum of the Hessian of the cost functional are displayed and discussed.