Controlling balance in an ensemble Kalman filter

Gottwald, Georg A.

doi:10.5194/npg-21-417-2014

Cited by 10 publications

(15 citation statements)

References 47 publications

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“…Short of rigorous justification, the issue of efficient implementation, in terms of run-time and in terms of coding effort, is of considerable practical relevance. For the toy model considered here, we were able to solve the optimal balance system problem (7) by simple shooting. However, this might fail or become excessively expensive in higher dimensions.…”

Section: Discussionmentioning

confidence: 99%

“…Sophisticated boundary value solvers may be needed but are hard to implement and computationally costly. We remark that we have only provided an approximate iterative solution of the boundary-value-problem (7) and the issue of well-posedness of the original boundary-value problem was not addressed. Viúdez and Dritschel [26] suggest an iterative procedure where one integrates back and forth, resetting to the correct boundary condition at each end.…”

Section: Discussionmentioning

confidence: 99%

“…We shall see that the asymptotic construction of the slow vector field for the ramped system (7) contains additional terms at O(ε k+1 ) unless all derivatives of ρ up to order k vanish at the final time. Similarly, the description of the trivial slow manifold p = 0 differs from the description of the slow manifold for the ramped system (7) at O(ε k+1 ) unless all derivatives of ρ up to order k vanish at the intial time. Thus, the order of accuracy of optimal balance is limited by the order of vanishing of derivatives of the ramp function at the end points.…”

Section: Optimal Balancementioning

confidence: 95%

“…Then set p * = p(T ). We note that when the ramp parameter is frozen at t = 0, (7) reduces to the trivial linear fast-slow systemq = p andṗ = Jp, where p is fast and q is slow. This justifies the initial-time boundary condition p(0) = 0.…”

Section: Optimal Balancementioning

confidence: 99%

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Optimal Balance via Adiabatic Invariance of Approximate Slow Manifolds

Gottwald¹,

Mohamad²,

Oliver³

2017

Multiscale Model. Simul.

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Abstract. We analyze the method of optimal balance which was introduced by Viúdez and Dritschel (J. Fluid Mech. 521, 2004, pp. 343-352) to provide balanced initializations for two-dimensional and three-dimensional geophysical flows, here in the simpler context of a finite dimensional Hamiltonian two-scale system with strong gyroscopic forces. It is well known that when the potential is analytic, such systems have an approximate slow manifold that is defined up to terms that are exponentially small with respect to the scale separation parameter. The method of optimal balance relies on the observation that the approximate slow manifold remains an adiabatic invariant under slow deformations of the nonlinear interactions. The method is formulated as a boundary value problem for a homotopic deformation of the system from a linear regime, where the slow-fast splitting is known exactly, to the full nonlinear regime. We show that, providing the ramp function which defines the homotopy is of Gevrey class 2 and satisfies vanishing conditions to all orders at the temporal end points, the solution of the optimal balance boundary value problem yields a point on the approximate slow manifold that is exponentially close to the approximation to the slow manifold via exponential asymptotics, albeit with a smaller power of the small parameter in the exponent. In general, the order of accuracy of optimal balance is limited by the order of vanishing derivatives of the ramp function at the temporal end points. We also give a numerical demonstration of the efficacy of optimal balance, showing the dependence of accuracy on the ramp time and the ramp function.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Optimal Balancementioning

confidence: 95%

Section: Optimal Balancementioning

confidence: 99%

See 2 more Smart Citations

Optimal Balance via Adiabatic Invariance of Approximate Slow Manifolds

Gottwald¹,

Mohamad²,

Oliver³

2017

Multiscale Model. Simul.

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“…In modern weather forecasting, the initial state is estimated by correcting the output from the forecast model, which may contain model error as well as instabilities, using information from noisy observations in a procedure called data assimilation. This procedure, however, typically does not respect balance, with the consequence that it may produce highly imbalanced initial states (Bloom et al, 1996;Mitchell et al, 2002;Ourmiéres et al, 2006;Kepert, 2009;Greybush et al, 2011;Gottwald, 2014).…”

Section: Introductionmentioning

confidence: 99%

Comparison of variational balance models for the rotating shallow water equations

2017

Self Cite

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We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (2006) for small Rossby numbers Ro. This family of generalized large-scale semi-geostrophic (GLSG) models contains the L 1 -model introduced by Salmon (1983) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow water flow. It is shown that, whereas the L 1 -member of the GLSG family is able to reproduce the balanced dynamics of the full shallow water equations on time scales of O(1/Ro) very well, all other members develop significant unphysical high wavenumber contributions in the ageostrophic vorticity which spoil the dynamics.

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Study of conservation laws with the Local Ensemble Transform Kalman Filter

Zeng

Janjić

2016

Quart J Royal Meteoro Soc

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Numerical discretization schemes have a long history of incorporating the most important conservation properties of the continuous system in order to improve the prediction of the nonlinear flow. The question arises whether data assimilation algorithms should follow a similar approach. To address this issue, we explore the conservation properties during data assimilation using perfect model experiments with a 2D shallow‐water model preserving important properties of the true nonlinear flow. The data assimilation scheme used here is the Local Ensemble Transform Kalman Filter with varying observed variables, inflation, localization radius and thinning interval. It is found that, during the assimilation, the total energy of the analysis ensemble mean converges with time towards the nature run value. However, enstrophy, divergence and the energy spectra are strongly affected by the data assimilation settings. Having in mind that the conservation of both the kinetic energy and enstrophy by the momentum advection schemes in the case of non‐divergent flow prevents a systematic and unrealistic energy cascade towards the high wave numbers, we test the effects on the prediction depending on the type of error in the initial condition. During the assimilation, we assess the downward nonlinear energy cascade through a scalar, domain‐averaged noise measure. We show that the accumulated noise during assimilation and the error of analysis are good indicators of the quality of the prediction.

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Controlling balance in an ensemble Kalman filter

Cited by 10 publications

References 47 publications

Optimal Balance via Adiabatic Invariance of Approximate Slow Manifolds

Optimal Balance via Adiabatic Invariance of Approximate Slow Manifolds

Comparison of variational balance models for the rotating shallow water equations

Study of conservation laws with the Local Ensemble Transform Kalman Filter

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