Inverse Modeling

Nakamura, Gen; Potthast, Roland

doi:10.1088/978-0-7503-1218-9

Cited by 81 publications

(78 citation statements)

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“…The theoretical conclusions indicate how correlated error statistics in the observation and background can be expected to interact, and highlight areas where reconditioning and similar techniques could be used to reduce the increased computational cost associated with using correlated observation errors operationally. Although the primary motivation for the investigation of the impact of correlated observation errors arises from their application in meteorology, the theory and conclusions presented here are very general and apply to any other application of variational data assimilation such as in neuroscience 33,34 and ecology. 35,36 2 VARIATIONAL ASSIMILATION AND CONDITION NUMBER…”

Section: Discussionmentioning

confidence: 99%

“…We emphasise that the results presented here are general and are relevant for any application of variational data assimilation. Improved knowledge of the role of correlated observation error covariances will be of use in the context of engineering, 33 neuroscience, 33,34 and ecology. 35,36 In this work, we developed theoretical bounds on the condition number of the Hessian of the 3D-Var objective function, which can be studied as a bound on the speed of convergence of the minimisation.…”

Section: Discussionmentioning

confidence: 99%

“…This means that as λ 1 (S 1 ) changes from being controlled by λ 1 (R −1 ) to λ 1 (B −1 ), the change appears smooth, as the wavenumber associated with the frequency of the largest eigenvalue remains constant. It is clear that increasing L B will have a significant effect on the value of this bound, as changing L B increases λ 1 (B −1 ) significantly, and hence the upper bound given by (33). Therefore, for both regimes, changing L B has a large impact on both bounds for λ 1 (S).…”

Section: 6mentioning

confidence: 95%

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The conditioning of least‐squares problems in variational data assimilation

Tabeart

Dance

Haben

et al. 2018

Numerical Linear Algebra App

103

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SummaryIn variational data assimilation a least-squares objective function is minimised to obtain the most likely state of a dynamical system. This objective function combines observation and prior (or background) data weighted by their respective error statistics. In numerical weather prediction, data assimilation is used to estimate the current atmospheric state, which then serves as an initial condition for a forecast. New developments in the treatment of observation uncertainties have recently been shown to cause convergence problems for this least-squares minimisation. This is important for operational numerical weather prediction centres due to the time constraints of producing regular forecasts. The condition number of the Hessian of the objective function can be used as a proxy to investigate the speed of convergence of the least-squares minimisation. In this paper we develop novel theoretical bounds on the condition number of the Hessian. These new bounds depend on the minimum eigenvalue of the observation error covariance matrix and the ratio of background error variance to observation error variance. Numerical tests in a linear setting show that the location of observation measurements has an important effect on the condition number of the Hessian.We identify that the conditioning of the problem is related to the complex interactions between observation error covariance and background error covariance matrices. Increased understanding of the role of each constituent matrix in the conditioning of the Hessian will prove useful for informing the choice of correlated observation error covariance matrix and observation location, particularly for practical applications.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: 6mentioning

confidence: 95%

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The conditioning of least‐squares problems in variational data assimilation

Tabeart

Dance

Haben

et al. 2018

Numerical Linear Algebra App

103

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“…(1) is not continuous. Thus, estimating the unknown input from real data remains an ill posed problem [51,52], even if the system is invertible and Φ −1 exists.…”

Section: G Practical Invertibility and Robustnessmentioning

confidence: 99%

“…(19)) is a well known theorem from the theory of inverse problems (see e.g. [51,52]): Linear compact operators with an infinite dimensional range cannot have a continuous inverse. The linear input-output operator Eq.…”

Section: G Practical Invertibility and Robustnessmentioning

confidence: 99%

Structural Invertibility and Optimal Sensor Node Placement for Error and Input Reconstruction in Dynamic Systems

et al. 2019

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Despite recent progress in our understanding of complex dynamic networks, it remains challenging to devise sufficiently accurate models to observe, control or predict the state of real systems in biology, economics or other fields. A largely overlooked fact is that these systems are typically open and receive unknown inputs from their environment. A further fundamental obstacle are structural model errors caused by insufficient or inaccurate knowledge about the quantitative interactions in the real system.Here, we show that unknown inputs to open systems and model errors can be treated under the common framework of invertibility, which is a requirement for reconstructing these disturbances from output measurements. By exploiting the fact that invertibility can be decided from the influence graph of the system, we analyse the relationship between structural network properties and invertibility under different realistic scenarios. We show that sparsely connected scale free networks are the most difficult to invert. We introduce a new sensor node placement algorithm to select a minimum set of measurement positions in the network required for invertibility. This algorithm facilitates optimal experimental design for the reconstruction of inputs or model errors from output measurements. Our results have both fundamental and practical implications for nonlinear systems analysis, modelling and design.

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Data Assimilation in the Solar Wind: Challenges and First Results

et al. 2017

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Data assimilation (DA) is used extensively in numerical weather prediction (NWP) to improve forecast skill. Indeed, improvements in forecast skill in NWP models over the past 30 years have directly coincided with improvements in DA schemes. At present, due to data availability and technical challenges, DA is underused in space weather applications, particularly for solar wind prediction. This paper investigates the potential of advanced DA methods currently used in operational NWP centers to improve solar wind prediction. To develop the technical capability, as well as quantify the potential benefit, twin experiments are conducted to assess the performance of the Local Ensemble Transform Kalman Filter (LETKF) in the solar wind model ENLIL. Boundary conditions are provided by the Wang‐Sheeley‐Arge coronal model and synthetic observations of density, temperature, and momentum generated every 4.5 h at 0.6 AU. While in situ spacecraft observations are unlikely to be routinely available at 0.6 AU, these techniques can be applied to remote sensing of the solar wind, such as with Heliospheric Imagers or interplanetary scintillation. The LETKF can be seen to improve the state at the observation location and advect that improvement toward the Earth, leading to an improvement in forecast skill in near‐Earth space for both the observed and unobserved variables. However, sharp gradients caused by the analysis of a single observation in space resulted in artificial wavelike structures being advected toward Earth. This paper is the first attempt to apply DA to solar wind prediction and provides the first in‐depth analysis of the challenges and potential solutions.

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Inverse Modeling

Cited by 81 publications

References 0 publications

The conditioning of least‐squares problems in variational data assimilation

The conditioning of least‐squares problems in variational data assimilation

Structural Invertibility and Optimal Sensor Node Placement for Error and Input Reconstruction in Dynamic Systems

Data Assimilation in the Solar Wind: Challenges and First Results

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