Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model

A. M. Stuart; A. Shukla; K. J. H. Law

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Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model

K. J. H. Law, A. Shukla, A. M. Stuart

Dynamical Systems · Submitted 20 Dec 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.

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arXiv ID1212.4923 (version 2)

Pith IDWJ76OXFA, Pith's permanent record of this paper and its review

CitationA. M. Stuart, A. Shukla, K. J. H. Law. "Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model." arXiv:1212.4923 (2012). https://pith.science/paper/WJ76OXFA

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.14 · mentions code or data availability 0.02.

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The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics.	background 1.00	0.04	0.02	asserted 1.00
s1	It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting.	background 1.00	0.04	0.02	asserted 1.00
s2	The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used.	background 1.00	0.05	0.03	asserted 1.00
s3	Prototypical of these ad hoc filters is the 3DVAR method.	background 0.97	0.05	0.03	asserted 1.00
s4	The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas.	contribution 0.99	0.60	0.04	asserted 0.89
s5	The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered.	method 0.61	0.68	0.04	asserted 0.95
s6	The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.	result 0.92	0.87	0.77	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Original abstract

The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.

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3 arXiv papers cite this in 9 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 5 sentences without a confident label

Compares with it 1 paper

The same problem arises using 3DVAR as shown by [ref] , however they also give numerical results showing that the accuracy of the filter is fortunately a lot better than the theoretical bound implies.

Almost sure error bounds for data assimilation in dissipative systems with unbounded observation noise · Lorenz '63 model · 0.94

Cites it as background 2 papers

In [ref] , results are obtained in expectation for unbounded noise for the Lorenz '96 and '63 models, respectively.

Almost sure error bounds for data assimilation in dissipative systems with unbounded observation noise · Introduction · 0.84
Theoretical analysis of the continuous time 3DVAR algorithm [ref] may be found in [ref] , [ref] , [ref] , [ref] , [ref] , [ref] and [ref] .

Ensemble Kalman Methods: A Mean Field Perspective · Bibliographical Notes · 0.87

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Compares with 1 paper

The current work demonstrates the properties of the 3DVAR algorithm when applied to the partially observed Lorenz '63 model; it is analogous to the more involved theory developed for the 3DVAR filter applied to the partially observed Navier-Stokes equations in [ref] .

Accuracy and Stability of The Continuous-Time 3DVAR Filter for The Navier-Stokes Equation · Conclusions · 0.81

Cites as background 1 paper

We give a brief derivation, referring to [ref] for further details and to [ref] for a related analysis of continuous time limits in the context of the ensemble Kalman filter.

Accuracy and Stability of The Continuous-Time 3DVAR Filter for The Navier-Stokes Equation · 3DVAR: Continuous Time Data · 0.87

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