Accurate state and parameter estimation in nonlinear systems with sparse observations

Rey, D.; Eldridge, Michael; Kostuk, Mark; Schumann-Bischoff, Jan; Parlitz, Ulrich

doi:10.1016/j.physleta.2014.01.027

Cited by 23 publications

(40 citation statements)

References 22 publications

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“…In this paper, we do not address the typical situation where the number of measurements actually available is less than that needed to allow the ground level of the action A 0 (X 0 ) to lie well below the next level. For a solution to that problem, we have used information from the waveform of the measurements as shown in some detail in Rey et al (2014).…”

Section: Discussionmentioning

confidence: 99%

Improved variational methods in statistical data assimilation

Kadakia

Rozdeba

et al. 2015

Nonlin. Processes Geophys.

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Abstract. Data assimilation transfers information from an observed system to a physically based model system with state variables x(t). The observations are typically noisy, the model has errors, and the initial state x(t 0 ) is uncertain: the data assimilation is statistical. One can ask about expected values of functions G(X) on the path X = {x(t 0 ), . . ., x(t m )} of the model state through the observation window t n = {t 0 , . . ., t m }. The conditional (on the measurements) probability distribution P (X) = exp[−A 0 (X)] determines these expected values. Variational methods using saddle points of the "action" A 0 (X), known as 4DVar (Talagrand and Courtier, 1987;Evensen, 2009), are utilized for estimating G(X) . In a path integral formulation of statistical data assimilation, we consider variational approximations in a realization of the action where measurement errors and model errors are Gaussian. We (a) discuss an annealing method for locating the path X 0 giving a consistent minimum of the action A 0 (X 0 ), (b) consider the explicit role of the number of measurements at each t n in determining A 0 (X 0 ), and (c) identify a parameter regime for the scale of model errors, which allows X 0 to give a precise estimate of G(X 0 ) with computable, small higher-order corrections.

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Section: Discussionmentioning

confidence: 99%

Improved variational methods in statistical data assimilation

Kadakia

Rozdeba

et al. 2015

Nonlin. Processes Geophys.

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“…In particular, we have recently discovered that the method discussed here and in Ref. [17] (also proposed earlier in Ref. [25]) is fundamentally equivalent to a control theoretic construct known as a Newton observer, which was first introduced by Moraal and Grizzle in Refs.…”

Section: Direct Estimation Of L Cmentioning

confidence: 84%

“…This idea was previously discussed in Ref. [17], but here we give a more detailed explanation of the method and provide additional numerical results.…”

Section: Using Time Delayed Measurements To Further Stabilize the mentioning

confidence: 96%

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Using waveform information in nonlinear data assimilation

et al. 2014

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Information in measurements of a nonlinear dynamical system can be transferred to a quantitative model of the observed system to establish its fixed parameters and unobserved state variables. After this learning period is complete, one may predict the model response to new forces and, when successful, these predictions will match additional observations. This adjustment process encounters problems when the model is nonlinear and chaotic because dynamical instability impedes the transfer of information from the data to the model when the number of measurements at each observation time is insufficient. We discuss the use of information in the waveform of the data, realized through a time delayed collection of measurements, to provide additional stability and accuracy to this search procedure. Several examples are explored, including a few familiar nonlinear dynamical systems and small networks of Colpitts oscillators.

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“…An analysis by Whartenby et al (2013) found that in the context of the nonlinear shallow water equations on a β plane, standard nudging techniques require observing approximately 70 % of the full set of state variables. Here we examine the same system using a method introduced by Rey et al (2014a), which generalizes standard nudging methods to utilize time delayed measurements. We show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions.…”

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confidence: 99%

Estimating the state of a geophysical system with sparse observations: time delay methods to achieve accurate initial states for prediction

Rey

2017

Nonlin. Processes Geophys.

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Abstract. The problem of forecasting the behavior of a complex dynamical system through analysis of observational time-series data becomes difficult when the system expresses chaotic behavior and the measurements are sparse, in both space and/or time. Despite the fact that this situation is quite typical across many fields, including numerical weather prediction, the issue of whether the available observations are "sufficient" for generating successful forecasts is still not well understood. An analysis by Whartenby et al. (2013) found that in the context of the nonlinear shallow water equations on a β plane, standard nudging techniques require observing approximately 70 % of the full set of state variables. Here we examine the same system using a method introduced by Rey et al. (2014a), which generalizes standard nudging methods to utilize time delayed measurements. We show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. In particular, we find that this estimate of 70 % can be reduced to about 33 % using time delays, and even further if Lagrangian drifter locations are also used as measurements.

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Accurate state and parameter estimation in nonlinear systems with sparse observations

Cited by 23 publications

References 22 publications

Improved variational methods in statistical data assimilation

Improved variational methods in statistical data assimilation

Using waveform information in nonlinear data assimilation

Estimating the state of a geophysical system with sparse observations: time delay methods to achieve accurate initial states for prediction

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