Simulation of Standing and Propagating Sea Waves . . .
259
For the empirical method the wave profile is simply multiplied by a decaying exponent without need to adapt the maximum value of ACF (as it is required for standing
wave).
Comparison of Studied Methods
To summarise, the analytic method of finding ocean wave’s ACF reduces to the following steps.
∙ Make wave profile decay when approaching ±∞ by multiplying it by a decaying
exponent.
∙ Apply Fourier transform to the absolute square of the resulting equation using
symbolic computation programme.
∙ Fit the resulting polynomial to the appropriate ACF approximation.
Two examples in this section showed that in case of standing and propagating
waves their decaying profiles resemble the corresponding ACFs with the exception
that the ACF’s maximum should be moved to the origin to preserve simulated process variance. Empirical method of finding ACF reduces to the following steps.
∙ Make wave profile decay when approaching ±∞ by multiplying it by a decaying
exponent.
∙ Move maximum value of the resulting function to the origin by using trigonometric identities to shift the phase.
Evaluation and Discussion
In [18–20] for AR model the following items were verified experimentally:
∙ probability distributions of different wave characteristics (wave heights, lengths,
crests, periods, slopes, three-dimensionality),
∙ dispersion relation,
∙ retention of integral characteristics for mixed wave sea state.
In this work we repeat probability distribution tests for three-dimensional AR and
MA model.
In [9] the authors show that several ocean wave characteristics (listed in Table 1)
have Weibull distribution, and wavy surface elevation has Gaussian distribution. In
order to verify that distributions corresponding to generated realisation are correct,
quantile-quantile plots are used (plots where analytic quantile values are used for
OX axis and estimated quantile values for OY axis). If the estimated distribution
matches analytic then the graph has the form of the straight line. Tails of the graph
may diverge from the straight line, because they can not be reliably estimated from
the finite-size realisation. Different methods of extracting waves from realisation produce variations in quantile function tails, it is probably impractical to extract every
possible wave from realisation since they may (and often) overlap.
259
For the empirical method the wave profile is simply multiplied by a decaying exponent without need to adapt the maximum value of ACF (as it is required for standing
wave).
Comparison of Studied Methods
To summarise, the analytic method of finding ocean wave’s ACF reduces to the following steps.
∙ Make wave profile decay when approaching ±∞ by multiplying it by a decaying
exponent.
∙ Apply Fourier transform to the absolute square of the resulting equation using
symbolic computation programme.
∙ Fit the resulting polynomial to the appropriate ACF approximation.
Two examples in this section showed that in case of standing and propagating
waves their decaying profiles resemble the corresponding ACFs with the exception
that the ACF’s maximum should be moved to the origin to preserve simulated process variance. Empirical method of finding ACF reduces to the following steps.
∙ Make wave profile decay when approaching ±∞ by multiplying it by a decaying
exponent.
∙ Move maximum value of the resulting function to the origin by using trigonometric identities to shift the phase.
Evaluation and Discussion
In [18–20] for AR model the following items were verified experimentally:
∙ probability distributions of different wave characteristics (wave heights, lengths,
crests, periods, slopes, three-dimensionality),
∙ dispersion relation,
∙ retention of integral characteristics for mixed wave sea state.
In this work we repeat probability distribution tests for three-dimensional AR and
MA model.
In [9] the authors show that several ocean wave characteristics (listed in Table 1)
have Weibull distribution, and wavy surface elevation has Gaussian distribution. In
order to verify that distributions corresponding to generated realisation are correct,
quantile-quantile plots are used (plots where analytic quantile values are used for
OX axis and estimated quantile values for OY axis). If the estimated distribution
matches analytic then the graph has the form of the straight line. Tails of the graph
may diverge from the straight line, because they can not be reliably estimated from
the finite-size realisation. Different methods of extracting waves from realisation produce variations in quantile function tails, it is probably impractical to extract every
possible wave from realisation since they may (and often) overlap.