Assimilation studies of open‐ocean flows: 2. Error measures with strongly nonlinear dynamics

Gunson, James Reginald; Malanotte‐Rizzoli, Paola

doi:10.1029/96jc02780

Cited by 22 publications

(20 citation statements)

References 18 publications

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“…Both results (2.2.10) and (2.2.15) can be equivalently derived by application of the Gauss-Markov theorem, as shown explicitly by Gunson and Malanotte-Rizzoli (1996) and discussed further by Wunsch (2006, p. 129) …”

Section: Nonlinear Inverse Problemmentioning

confidence: 91%

“…the collection of the Hessian matrices of each of its scalar components (Thacker 1989), with the inverse covariance of observations and with the residual misfits vector. This second term is named the nonlinear term (Gunson and Malanotte-Rizzoli 1996). It appears only for nonlinear model M, since for linear model M the tensor of its second derivatives is zero.…”

Section: The Hessian Methodsmentioning

confidence: 99%

“…The first term is the backward projection of the inverse covariance of observation uncertainty, similar to (2.3.4) if the linear forward model M is substituted with the tangent linear model of M. This first term is commonly referred to as the linearized Hessian and was calculated in the previous studies of error covariance inversion (Thacker 1989, Gunson and Malanotte-Rizzoli 1996, Losch and Wunsch 2003. The second term is the product of the tensor of second derivatives of the multivariate model M, i.e.…”

Section: The Hessian Methodsmentioning

confidence: 99%

“…No machinery for its direct computation was previously available for large nonlinear ocean models. The previous efforts to calculate the Hessian were limited to linear (Thacker 1989) or linearized (Moore et al 2011) models, small systems and approximate Hessian estimation techniques (Gunson andMalanotte-Rizzoli 1996, Losch andWunsch 2003). Here, we develop a novel computational methodology for direct Hessian calculation of nonlinear GCMs with large dimensionality, scalable to realistic ocean state estimation problems.…”

Section: Uncertainty Quantification In Ocean State Estimationmentioning

confidence: 99%

“…He also heuristically suggested that "as long as the model is not too nonlinear, the inverse of the Hessian should provide a good approximation to the error-covariance matrix even in the nonlinear case". This heuristic linearized approach was continued in the nonlinear oceanographic uncertainty studies by Gunson and Malanotte-Rizzoli (1996) and Losch and Wunsch (2003). In the linearized framework of variational data assimilation (Moore et al 2011) the inverse of the Hessian of the strictly quadratic misfit function can be formally related to the estimated error covariance matrix (see Section 2.2).…”

Section: Uncertainty Quantification In Ocean State Estimationmentioning

confidence: 99%

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Uncertainty quantification in ocean state estimation

Kalmikov¹

2013

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Quantifying uncertainty and error bounds is a key outstanding challenge in ocean state estimation and climate research. It is particularly difficult due to the large dimensionality of this nonlinear estimation problem and the number of uncertain variables involved. The "Estimating the Circulation and Climate of the Oceans" (ECCO) consortium has developed a scalable system for dynamically consistent estimation of global timeevolving ocean state by optimal combination of ocean general circulation model (GCM) with diverse ocean observations. The estimation system is based on the "adjoint method" solution of an unconstrained least-squares optimization problem formulated with the method of Lagrange multipliers for fitting the dynamical ocean model to observations. The dynamical consistency requirement of ocean state estimation necessitates this approach over sequential data assimilation and reanalysis smoothing techniques. In addition, it is computationally advantageous because calculation and storage of large covariance matrices is not required. However, this is also a drawback of the adjoint method, which lacks a native formalism for error propagation and quantification of assimilated uncertainty. The objective of this dissertation is to resolve that limitation by developing a feasible computational methodology for uncertainty analysis in dynamically consistent state estimation, applicable to the large dimensionality of global ocean models.Hessian (second derivative-based) methodology is developed for Uncertainty Quantification (UQ) in large-scale ocean state estimation, extending the gradient-based adjoint method to employ the second order geometry information of the model-data misfit function in a high-dimensional control space. Large error covariance matrices are evaluated by inverting the Hessian matrix with the developed scalable matrix-free numerical linear algebra algorithms. Hessian-vector product and Jacobian derivative codes of the MIT general circulation model (MITgcm) are generated by means of algorithmic differentiation (AD). Computational complexity of the Hessian code is reduced by tangent linear differentiation of the adjoint code, which preserves the speedup of adjoint checkpointing schemes in the second derivative calculation. A Lanczos algorithm is applied for extracting the leading rank eigenvectors and eigenvalues of the Hessian matrix. The eigenvectors represent the constrained uncertainty patterns. The inverse eigenvalues are the corresponding uncertainties. The dimensionality of UQ calculations is reduced by eliminating the uncertainty null-space unconstrained by the 4 supplied observations. Inverse and forward uncertainty propagation schemes are designed for assimilating observation and control variable uncertainties, and for projecting these uncertainties onto oceanographic target quantities. Two versions of these schemes are developed: one evaluates reduction of prior uncertainties, while another does not require prior assumptions. The analysis of uncertainty propagation in the oc...

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Section: Nonlinear Inverse Problemmentioning

confidence: 91%

Section: The Hessian Methodsmentioning

confidence: 99%

Section: The Hessian Methodsmentioning

confidence: 99%

Section: Uncertainty Quantification In Ocean State Estimationmentioning

confidence: 99%

Section: Uncertainty Quantification In Ocean State Estimationmentioning

confidence: 99%

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Uncertainty quantification in ocean state estimation

Kalmikov¹

2013

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Variational Lagrangian data assimilation in open channel networks

Tinka

Weekly

et al. 2015

Water Resources Research

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This article presents a data assimilation method in a tidal system, where data from both Lagrangian drifters and Eulerian flow sensors were fused to estimate water velocity. The system is modeled by first-order, hyperbolic partial differential equations subject to periodic forcing. The estimation problem can then be formulated as the minimization of the difference between the observed variables and model outputs, and eventually provide the velocity and water stage of the hydrodynamic system. The governing equations are linearized and discretized using an implicit discretization scheme, resulting in linear equality constraints in the optimization program. Thus, the flow estimation can be formed as an optimization problem and efficiently solved. The effectiveness of the proposed method was substantiated by a large-scale field experiment in the Sacramento-San Joaquin River Delta in California. A fleet of 100 sensors developed at the University of California, Berkeley, were deployed in Walnut Grove, CA, to collect a set of Lagrangian data, a time series of positions as the sensors moved through the water. Measurements were also taken from Eulerian sensors in the region, provided by the United States Geological Survey. It is shown that the proposed method can effectively integrate Lagrangian and Eulerian measurement data, resulting in a suited estimation of the flow variables within the hydraulic system.

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On optimal solution error covariances in variational data assimilation problems

Gejadze¹,

Dimet²,

Shutyaev³

2010

Journal of Computational Physics

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The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection-diffusion model

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Assimilation studies of open‐ocean flows: 2. Error measures with strongly nonlinear dynamics

Cited by 22 publications

References 18 publications

Uncertainty quantification in ocean state estimation

Uncertainty quantification in ocean state estimation

Variational Lagrangian data assimilation in open channel networks

On optimal solution error covariances in variational data assimilation problems

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