I. M. Navon

Variational data assimilation with an adiabatic version of the NMC spectral model

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 ...

Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography

The present paper has two aims. One is to survey briefly the state of the art of parameter estimation in meteorology and oceanography in view of applications of 4-D variational data assimilation techniques to inverse parameter estimation problems, which bear promise of serious positive impact on improving model prediction. The other aim is to present crucial aspects of identifiability and stability essential for validating results of optimal parameter estimation and which have not been addressed so far in either the meteorological or the oceanographic literature. As noted by Yeh (1986, Water Resour. Res. 22, 95–108) in the context of ground water flow parameter estimation the inverse or parameter estimation problem is often ill-posed and beset by instability and nonuniqueness, particularly if one seeks parameters distributed in space-time domain. This approach will allow one to assess and rigorously validate results of parameter estimation, i.e. do they indeed represent a real identification of physical model parameters or just compensate model errors? A brief survey of other approaches for solving the problem of optimal parameter estimation in meteorology and oceanography is finally presented.

Conjugate-Gradient Methods for Large-Scale Minimization in Meteorology

Abstract During the last few years new meteorological variational analysis methods have evolved, requiring large-scale minimization of a nonlinear objective function described in terms of discrete variables. The conjugate-gradient method was found to represent a good compromise in convergence rates and computer memory requirements between simpler and more complex methods of nonlinear optimization. In this study different available conjugate-gradient algorithms are presented with the aim of assessing their use in large-scale typical minimization problems in meteorology. Computational efficiency and accuracy are our principal criteria. Four different conjugate-gradient methods, representative of up-to-date available scientific software, were compared by applying them to two different meteorological problems of interest using criteria of computational economy and accuracy. Conclusions are presented as to the adequacy of the different conjugate algorithms for large-scale minimization problems in different met...

VARIATM—A FORTRAN program for objective analysis of pseudostress wind fields using large-scale conjugate-gradient minimization

Abstract A FORTRAN computer program is presented and documented which implements a new approach to objective analysis of pseudostress data over the Indian Ocean. (A pseudostress vector is defined as the wind components multiplied by the wind magnitude.) This method is a direct large-scale minimization approach of a cost functional expressed as a weighted sum of lack of fit to data as well as constraints on proximity to original observations and climatology, on a smoothing parameter and on kinematic equivalence to climatological patterns. Each of the constraints was weighted by selected coefficients controlling how closely the minimizing analysis fits each type of data or constraint. The functional operates on 7330 variables (i.e. two wind components at each grid location) and was minimized using a highly efficient memoryless quasi-Newton-like conjugate-gradient method. Use of an independent subjective analysis of the same data provide for a direct quantitative comparison and confirm the adequacy of the objective analysis. This scheme now has been adopted operationally to generate monthly average pseudostress wind values on a 1°-grid over the Indian Ocean.

Second-Order Information in 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.

NUMERICAL EXPERIENCE WITH LIMITED-MEMORY QUASI-NEWTON AND TRUNCATED NEWTON METHODS*

Computational experience with several limited-memory quasi-Newton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a well-known test library [J. J. More, B. S. Gaxbow, and K. E. Hillstrom, ACM Trans. Math. Software, 7 (1981), pp. 17–41], on several synthetic problems allowing control of the clustering of eigenvalues in the Hessian spectrum, and on some large-scale problems in oceanography and meteorology. The results indicate that among the tested limited-memory quasi-Newton methods, the L-BFGS method [D. C. Liu and J. Nocedal, Math. Programming, 45 (1989), pp. 503–528] has the best overall performance for the problems examined. The numerical performance of two truncated Newton methods, differing in the inner-loop solution for the search vector, is competitive with that of L-BFGS.

Objective Analysis of Pseudostress over the Indian Ocean Using a Direct-Minimization Approach

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 present...

Reduced-Order Modeling of the Upper Tropical Pacific Ocean Model using Proper Orthogonal Decomposition

The proper orthogonal decomposition (POD) is shown to be an efficient model reduction technique for simulating physical processes governed by partial differential equations. In this paper, we make an initial effort to investigate problems related to POD reduced modeling of a large- scale upper ocean circulation in the tropic Pacific domain. We construct different POD models with different choices of snapshots and different number of POD basis functions. The results from these different POD models are compared with that of the original model. The main findings are: (1) the large-scale seasonal variability of the tropic Pacific obtained by the original model is well captured by a low dimensional system of order 22, which is constructed using 20 snapshots and 7 leading POD basis functions. (2) the RMS errors for the upper ocean layer thickness of the POD model of order 22 are less than 1m that is less than 1% of the average thickness and the correlation between the upper ocean layer thickness with that from the POD model is around 0.99. (3) Retaining modes that capture 99% energy is necessary in order to construct POD models yielding a high accuracy.

The second order adjoint analysis: Theory and applications

SummaryThe adjoint method application in variational data assimilation provides a way of obtaining the exact gradient of the cost functionj with respect to the control variables. Additional information may be obtained by using second order information. This paper presents a second order adjoint model (SOA) for a shallow-water equation model on a limited-area domain. One integration of such a model yields a value of the Hessian (the matrix of second partial derivatives, ∇2J) multiplied by a vector or a column of the Hessian of the cost function with respect to the initial conditions. The SOA model was then used to conduct a sensitivity analysis of the cost function with respect to distributed observations and to study the evolution of the condition number (the ratio of the largest to smallest eigenvalues) of the Hessian during the course of the minimization. The condition number is strongly related to the convergence rate of the minimization. It is proved that the Hessian is positive definite during the process of the minimization, which in turn proves the uniqueness of the optimal solution for the test problem. Numerical results show that the sensitivity of the response increases with time and that the sensitivity to the geopotential field is larger by an order of magnitude than that to theu andv components of the velocity field. Experiments using data from an ECMWF analysis of the First Global Geophysical Experiment (FGGE) show that the cost functionJ is more sensitive to observations at points where meteorologically intensive events occur. Using the second order adjoint shows that most changes in the value of the condition number of the Hessian occur during the first few iterations of the minimization and are strongly correlated to major large-scale changes in the reconstructed initial conditions fields.

Optimal control of cylinder wakes via suction and blowing

Abstract A general method based on adjoint formulation is discussed for the optimal control of distributed parameter systems (including boundary parameter) which is especially suitable for large dimensional control problems. Strategies for efficient and robust implementation of the method are described. The method is applied to the problem of controlling vortex shedding behind a cylinder (through suction/blowing on the cylinder surface) governed by the unsteady two-dimensional incompressible Navier–Stokes equations space discretized by finite-volume approximation with time-dependent boundary conditions. Three types of objective functions are considered, with regularization to circumvent ill-posedness. These objective functions involve integration over a space–time domain. The minimization of the cost function uses a quasi-Newton DFP method. A complete control of vortex shedding is demonstrated for Reynolds numbers up to 110. The optimal values of the suction/blowing parameters are found to be insensitive to initial conditions of the model when the time window of control is larger than the vortex shedding period, the inverse of the Strouhal frequency. Although this condition is necessary for robust control, it is observed that a shorter window of control may suffice to suppress vortex shedding.