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On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations

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Abstract

The shallow water equations (SWE) are a widely used model for the propagation of surface waves on the oceans. We consider the problem of optimally determining the initial conditions for the one-dimensional SWE in an unbounded domain from a small set of observations of the sea surface height. In the linear case, we prove a theorem that gives sufficient conditions for convergence to the true initial conditions. At least two observation points must be used and at least one pair of observation points must be spaced more closely than half the effective minimum wavelength of the energy spectrum of the initial conditions. This result also applies to the linear wave equation. Our analysis is confirmed by numerical experiments for both the linear and nonlinear SWE data assimilation problems. These results show that convergence rates improve with increasing numbers of observation points and that at least three observation points are required for practically useful results. Better results are obtained for the nonlinear equations provided more than two observation points are used. This paper is a first step towards understanding the conditions for observability of the SWE for small numbers of observation points in more physically realistic settings.

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Acknowledgements

We are grateful to our colleague S. Alama for his suggestion to use the Fourier transform to analyze the fixed point of the gradient descent algorithm.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    N. K.-R. Kevlahan, R. Khan & B. Protas

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  1. N. K.-R. Kevlahan
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  2. R. Khan
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  3. B. Protas
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Correspondence to N. K.-R. Kevlahan.

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Communicated by: Youssef Marzouk

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This work was funded by the NSERC Discovery Grants of N.K.-R. Kevlahan and B. Protas

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Kevlahan, N.KR., Khan, R. & Protas, B. On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations. Adv Comput Math 45, 3195–3216 (2019). https://doi.org/10.1007/s10444-019-09733-6

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  • DOI: https://doi.org/10.1007/s10444-019-09733-6

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