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D. N. Daescu
and
I. M. Navon

Abstract

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.

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X. Zou
,
I. M. Navon
, and
J. Sela

Abstract

Variational four-dimensional data assimilation, combined with a penalty method constraining time derivatives of the surface pressure, the divergence, and the gravity-wave components is implemented on an adiabatic version of the National Meteorological Center's 18-level primitive equation spectral model with surface drag and horizontal diffusion. Experiments combining the Machenhauer nonlinear normal-mode initialization procedure and its adjoint with the variational data assimilation are also presented. The modified variational data-assimilation schemes are tested to assess how well they control gravity-wave oscillations.

The gradient of a penalized cost function can be obtained by a single integration of the adjoint model. A detailed derivation of the gradient calculation of different penalized cost functions is presented, which is not restricted to a specific model.

Numerical results indicate that the inclusion of penalty terms into the cost function will change the model solution as desired. The advantages of the use of simple penalty terms over penalty terms including the model normal modes results in a simplification of the procedure, allowing a more direct control over the model variables and the possibility of using weak constraints to eliminate the high-frequency gravity-wave oscillations. This approach does not require direct information about the model normal modes. One of the encouraging results obtained is that the introduction of the penalty terms does not slow the convergence rate of the minimization process.

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David M. Legler
,
I. M. Navon
, and
James J. O'Brien

Abstract

A variational approach is used to develop an objective analysis technique which produces monthly average 1-deg pseudostress vector fields over the Indian Ocean. A, cost functional is constructed which consists of five terms, each expressing a lack of fit to prescribed conditions. The first expresses the proximity to the input (first-guess) field. The second deals with the closeness of fit to the climatological value for that month. The third is a measure of data roughness, and the fourth and fifth are kinematic constraints on agreement of the curl and divergence of the results to the curl and divergence of the climatology. Each term also has a coefficient (weight) which determines how closely the minimization fits each lack of fit. These weights are determined by comparing the results using various weight combinations to an independent subjective analysis of the same dataset. The cost functional is minimized using the conjugate-gradient method.

Results from various weight combinations are presented for the months of January and July 1984 and the results examined in terms of thee selections. Quantitative and qualitative comparisons to the subjective analysis are made to find which weight combination provides the best results. It was found that the weight on the second term balances the influence of the original (first-guess) field and climatology. The smoothing term weight determines how wide an area deviations of the first guess from climatology is affected. The weights on the kinematic terms are fine-tuning parameters.

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I. M. Navon
,
B. Neta
, and
M. Y. Hussaini

Abstract

A limited-area model of linearized shallow water equations (SWE) on an f plane for a rectangular domain is considered. The rectangular domain is extended to include the so-called perfectly matched layer (PML) as an absorbing boundary condition. Following the proponent of the original method, the equations are obtained in this layer by splitting the shallow water equations in the coordinate directions and introducing the absorption coefficients. The performance of the PML as an absorbing boundary treatment is demonstrated using a commonly employed bell-shaped Gaussian initially introduced at the center of the rectangular physical domain.

Three typical cases are studied:

  • A stationary Gaussian where adjustment waves radiate out of the area.

  • A geostrophically balanced disturbance being advected through the boundary parallel to the PML. This advective case has an analytical solution allowing one to compare forecasts.

  • The same bell being advected at an angle of 45° so that it leaves the domain through a corner.

For the purpose of comparison, a reference solution is obtained on a fine grid on the extended domain with the characteristic boundary conditions. Also computed are the rms difference between the 48-h forecast and the analytical solution as well as the 48-h evolution of the mean absolute divergence, which is related to geostrophic balance. The authors found that the PML equations for the linearized shallow water equations on an f plane support unstable solutions when the mean flow is not unidirectional. Use of a damping term consisting of a 9-point smoother added to the discretized PML equations stabilizes the PML equations. The reflection/transmission is analyzed along with the case of instability for glancing propagation of the bell disturbance. A numerical illustration is provided showing that the stabilized PML for glancing bell propagation performs well with the addition of the damping term.

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Richard L. Pfeffer
,
I. M. Navon
, and
Xiaolei Zou

Abstract

This paper presents evidence of the sensitivity of a general circulation Model (GCM) to the time-differencing scheme employed when the physical parameterizations and space discretization are not changed. For this purpose, two time-marching schemes-the leapfrog and the Matsuno schemes-are analyzed and tested on the National Aeronautics and Space Administration–Goddard Laboratory for Atmospheric Studies (NASA-GLAS) fourth-order GCM in terms of the stability and behavior of 2-month-averaged fields. Linear analysis suggests that Rossby waves are slightly damped and slightly accelerated when the Matsuno scheme is used and that these effects are scale selective, being smallest for the longest waves. It also suggests that such waves are accelerated less and are not damped when the leapfrog scheme is used. An empirical orthogonal function analysis of the meridional component of velocity at 46°N, keeping at least 70% of the variance, reveals less shortwave activity in the numerical solution with the Matsuno scheme but does not lend support to the conclusion that the waves are accelerated less in the solution with the leapfrog scheme.

The two-dimensional Eliassen-Palm (E-P) flux divergence and the eddy-induced mean meridional circulation are found to be stronger in the simulation with the leapfrog time-differencing scheme than in the one with the Matsuno scheme, suggesting that the transient-wave activity is damped by the Matsuno scheme. On the other hand, the three-dimensional stationary-wave activity flux in the Northern Hemisphere simulated with the Matsuno scheme is more intense than that simulated with the leapfrog scheme, indicating that the stationary waves are more robust in the integration with the Matsuno scheme.

The GCM precipitation when integrated with the leapfrog scheme is much more intense over the tropical western Pacific and the northeastern Pacific and less intense over the western North Atlantic Ocean. The kinetic energy of waves with wavenumber greater than 9 simulated by the Matsuno scheme is consistently smaller than that obtained by the leapfrog scheme. These results give evidence that climate simulations are sensitive not only to physical parameterizations of subgrid-scale processes but also to the numerical methodology employed.

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Jieping Zou
,
William W. Hsieh
, and
I. M. Navon

Abstract

The feasibility of sequential open-boundary control by data assimilation in a regional ocean model has been investigated using a barotropic wind-driven ocean circulation model. A simple open-boundary scheme has been constructed based on the idea of optimal boundary control of a diagnostic equation and illustrated with the problem of modeling the subpolar gyre subject to an open southern boundary. The results show that use of such a scheme in conjunction with traditional radiation boundary conditions allows for a longer model integration that would otherwise be unstable when only the radiation boundary conditions are imposed due to presence of dispersive waves.

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I. M. Navon
,
X. Zou
,
J. Derber
, and
J. Sela

Abstract

Variational four-dimensional (4D) data assimilation is performed using an adiabatic version of the National Meteorological Center (NMC) baroclinic spectral primitive equation model with operationally analyzed fields as well as simulated datasets. Two limited-memory quasi-Newton minimization techniques were used to iteratively find the minimum of a cost function, with the NMC forecast as a constraint. The cost function consists of a weighted square sum of the differences between the model forecast and observations over a time interval. In all the experiments described in this paper, observations are available for all degrees of freedom of the model. The derivation of the adjoint of the discretized adiabatic NMC spectral model is presented. The creation of this adjoint model allows the gradient of the cost function with respect to the initial conditions to be computed using a single backward-in-time integration of the adjoint equations.

As an initial evaluation of the variational data-assimilation procedure, an assimilation system with a low-resolution version of the NMC spectral model was tested using fields from a Rossby-Haurwitz-wave solution as observations. The results were encouraging, with a significant reduction in the magnitudes of both the cost function and the norm of its gradient during the minimization process. In particular, the high-frequency noise exhibited in the rms of the divergence field, produced by random perturbation in the initial conditions, is largely eliminated after the variational data assimilation.

The performance of the assimilation scheme was examined in a more realistic configuration using the adiabatic NMC spectral model truncated at T40. Both operationally analyzed observations, consisting of vorticity, divergence, temperature, surface pressure and moisture fields (distributed at two time levels separated by a 6-h time interval), and model-generated data were variationally assimilated. The effect of the number of observation fields in time on the convergence rate of the minimization and the impacts due to the inclusion of the horizontal diffusion and the surface drag in the model and its adjoint on the convergence rate and the accuracy of the retrieval are addressed.

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Francois-Xavier Le Dimet
,
I. M. Navon
, and
Dacian N. Daescu

Abstract

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.

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Y. Li
,
I. M. Navon
,
P. Courtier
, and
P. Gauthier

Abstract

An adjoint model is developed for variational data assimilation using the 2D semi-Lagrangian semi-implicit (SLSI) shallow-water equation global model of Bates et al. with special attention being paid to the linearization of the interpolation routines. It is demonstrated that with larger time steps the limit of the validity of the tangent linear model will be curtailed due to the interpolations, especially in regions where sharp gradients in the interpolated variables coupled with strong advective wind occur, a synoptic situation common in the high latitudes. This effect is particularly evident near the pole in the Northern Hemisphere during the winter season. Variational data assimilation experiments of “identical twin” type with observations available only at the end of the assimilation period perform well with this adjoint model. It is confirmed that the computational efficiency of the semi-Lagrangian scheme is preserved during the minimization process, related to the variational data assimilation procedure.

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Yan Yang
,
I. M. Navon
,
Ricardo Todling
, and
Weiyu Yang

Abstract

An adjoint sensitivity analysis of the relaxed Arakawa–Schubert scheme in the National Aeronautics and Space Administration GEOS-1 GCM with respect to perturbations in large-scale environmental fields was conducted. The response functions were defined as measures of the strength of convective cloud precipitation, the cloud-induced heating and drying (moistening) in both the instantaneous and time-integrated sense. The roles of different variables in producing variations on the response functions were evaluated and the most sensitive vertical levels of the perturbations were identified with the gradient provided by the adjoint model.

It was found that the potential temperature perturbations had significant impact on all the response functionals analyzed, especially on the convective precipitation. The perturbations at subcloud layers and at midtroposphere from 500 to 600 hPa were found to be the most influential. The impact from the moisture fields was most significant on cloud heating and drying effects and the strongest influence came from the subcloud layers. The moisture perturbations at midtroposphere also significantly influenced the cloud drying (moistening) effect. On the other hand, the cloud-induced heating and drying at levels between 400 and 600 hPa felt the strongest impact from perturbations in large-scale fields. The influence of the perturbations in the wind field was weaker but still provided reasonable sensitivity patterns. The time-integrated and instantaneous sensitivities for the same response differ only in magnitude but not in the general distributions.

The impact of large-scale condensation and reevaporation on the sensitivity was also evaluated. Their effect was significant at the midtropospheric level and they enhanced the model sensitivity to perturbations in temperature and moisture fields.

The sensitivity analysis results obtained indicated that accurate gridscale vertical profile of temperature and moisture, especially at subcloud layers and midtroposphere between 500 and 600 hPa were essential for the accurate evaluation of the cumulus cloud effects. The implications of the results of this work for variational data assimilation were also discussed.

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