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Decomposition techniques for the ocean wave identification problem

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Abstract

This paper deals with the solution of the ocean wave identification problem by means of decomposition techniques. A discrete formulation is assumed. An ocean test structure is considered, and wave elevation and velocities are assumed to be measured with a number of sensors. Within the frame of linear wave theory, a Fourier series model is chosen for the wave elevation and velocities. Then, the following problem is posed (Problem P): Find the amplitudes of the various wave components of specified frequency and direction, so that the assumed model of wave elevation and velocities provides the best fit to the measured data. Here, the term best fit is employed in the least-square sense over a given time interval.

Problem P is numerically difficult because of its large size 2MN, whereM is the number of frequencies andN is the number of directions. Therefore, both the CPU time and the memory requirements are considerable (Refs. 7–12).

In order to offset the above difficulties, decomposition techniques are employed in order to replace the solution of Problem P with the sequential solution of two groups of smaller subproblems. The first group (Problem F) involvesS subproblems, having size 2M, whereS is the number of sensors andM is the number of frequencies; theseS subproblems are least-square problems in the frequency domain. The second group (Problem D) involvesM subproblems, having size 2N, whereM is the number of frequencies andN is the number of directions; theseM subproblems are least-square problems in the direction domain.

In the resulting algorithm, called the discrete formulation decomposition algorithm (DFDA, Ref. 2), the linear equations are solved with the help of the Householder transformation in both the frequency domain and the direction domain. By contrast, in the continuous formulation decomposition algorithm (CFDA, Ref. 1), the linear equations are solved with Gaussian elimination in the frequency domain and with the help of the Householder transformation in the direction domain.

Mathematically speaking, there are three cases in which the solution of the decomposed problem and the solution of the original, undecomposed problem are identical: (a) the case where the number of sensors equals the number of directions; (b) the case where Problem P is characterized by a vanishing value of the functional being minimized; and (c) the case where the wave component periods are harmonically related to the sampling time.

Numerical experiments concerning the OTS platform and the Hondo-A platform show that the decomposed scheme is considerably superior to the undecomposed scheme; that the discrete formulation is considerably superior to the continuous formulation; and that the accuracy can be improved by proper selection of the sampling time as well as by proper choice of the number and the location of the sensors. In particular, the choice of the sensor location for the Hondo-A platform is discussed.

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References

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Authors and Affiliations

  1. Aero-Astronautics Group, Rice University, Houston, Texas

    A. Miele (Professor of Astronautics and Mathematical Sciences), T. Wang (Senior Research Associate) & Z. G. Zhao (Graduate Student)

  2. Offshore Structures Division, Exxon Production Research Company, Houston, Texas

    J. C. Heideman (Research Associate)

Authors
  1. A. Miele
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  2. T. Wang
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  3. Z. G. Zhao
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  4. J. C. Heideman
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Additional information

This work was supported by Exxon Production Research Company, Houston, Texas. This paper is based on Refs. 1–6 and is a continuation of the work presented in Refs. 7–12.

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Miele, A., Wang, T., Zhao, Z.G. et al. Decomposition techniques for the ocean wave identification problem. J Optim Theory Appl 51, 1–39 (1986). https://doi.org/10.1007/BF00938601

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