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Wave parameter identification problem for ocean test structure data, part 2, discrete formulation

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Abstract

This paper deals with the solution of the wave parameter identification problem for ocean test structure data. 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: 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.

At each time instant, the wave representation involves four indexes (frequency, direction, instrument, time); hence, four-dimensional arrays are required. This formal difficulty can be avoided by switching to an alternative representation involving only two indexes (frequency-direction, instrument-time); hence, standard vector-matrix notation can be used. Within this frame, optimality conditions are derived for the amplitudes of the assumed wave model.

A characteristic of the wave parameter identification problem is that the condition number of the system matrix can be large. Therefore, the numerical solution is not an easy task and special procedures must be employed. Specifically, Gaussian elimination is avoided and advantageous use is made of the Householder transformation, in the light of the least-square nature of the problem and the discretized approach to the problem.

Numerical results are presented. The effect of various system parameters (number of frequencies, number of directions, sampling time, number of sensors, and location of sensors) is investigated in connection with global or strong accuracy, local or weak accuracy, integral accuracy, and condition number of the system matrix.

From the numerical experiments, it appears that the wave parameter identification problem has a unique solution if the number of directions is smaller than or equal to the number of sensors; it has an infinite number of solutions otherwise. In the case where a unique solution exists, the condition number of the system matrix increases as the size of the system increases, and this has a detrimental effect on the accuracy. However, the accuracy can be improved by proper selection of the sampling time and by proper choice of the number and location of the sensors.

Generally speaking, the computations done for the discrete case exhibit better accuracy than the computations done for the continuous case (Ref. 5). This improved accuracy is a direct consequence of having used advantageously the Householder transformation and is obtained at the expense of increased memory requirements and increased CPU time.

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References

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Author notes

  1. J. N. Sharma (Senior Research Engineer)

    Present address: Civil Engineering Department, University of Houston, Houston, Texas

Authors and Affiliations

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

    A. Miele (Professor of Astronautics and Mathematical Sciences) & T. Wang (Graduate Student)

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

    J. C. Heideman (Research Associate) & J. N. Sharma (Senior Research Engineer)

Authors
  1. A. Miele
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  2. T. Wang
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  3. J. C. Heideman
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  4. J. N. Sharma
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Additional information

This work was supported by Exxon Production Research Company, Houston, Texas. This paper is based partly on Refs. 1–4.

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Miele, A., Wang, T., Heideman, J.C. et al. Wave parameter identification problem for ocean test structure data, part 2, discrete formulation. J Optim Theory Appl 44, 453–484 (1984). https://doi.org/10.1007/BF00935462

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