Simulation of Standing and Propagating Sea Waves . . .
257
The other problem is inability to automatically determine optimal number of coefficients for three-dimensional AR and MA processes. For one-dimensional processes
this can be achieved via iterative methods [7], but they diverge in three-dimensional
case.
The final problem, which is discussed in section “Mixed Autoregressive
Moving Average (ARMA) Process”, is inability to “mix” AR and MA process in
three dimensions.
In practice some statements made for AR and MA processes in [7] should be
flipped for three-dimensional case. For example, the authors say that ACF of MA process cuts at q and ACF of AR process decays to nought infinitely, but in practice making ACF of 3-dimensional MA process not decay results in it being non-invertible
and producing realisation that does not look like real ocean waves, whereas doing
the same for ACF of AR process results in stationary process and adequate realisation. Also, the authors say that one should allocate the first q points of ACF to MA
process (as it often needed to describe the peaks in ACF) and leave the rest points
to AR process, but in practice in case of ACF of a propagating wave AR process is
stationary only for the first time slice of the ACF, and the rest is left to MA process.
To summarise, the only established scenario of applying ARMA model to ocean
wave generation is to use AR process for standing waves and MA process for propagating waves. With a new formulae for 3 dimensions a single mixed ARMA process
might increase model precision, which is one of the objectives of the future research.
The Shape of ACF for Different Types of Waves
Analytic Method of Finding the ACF
The straightforward way to find ACF for a given ocean wave profile is to apply
Wiener–Khinchin theorem. According to this theorem the autocorrelation K of a
function í µí¼ is given by the Fourier transform of the absolute square of the function:
K(t) = F
{
|í µí¼ (t)|
2
} .
(8)
When í µí¼ is replaced with actual wave profile, this formula gives you analytic formula
for the corresponding ACF.
For three-dimensional wave profile (2D in space and 1D in time) analytic formula is a polynomial of high order and is best obtained via symbolic computation
programme. Then for practical usage it can be approximated by superposition of
exponentially decaying cosines (which is how ACF of a stationary ARMA process
looks like [7]).
Empirical Method of Finding the ACF
However, for three-dimensional case there exists simpler empirical method which
does not require sophisticated software to determine shape of the ACF. It is known
257
The other problem is inability to automatically determine optimal number of coefficients for three-dimensional AR and MA processes. For one-dimensional processes
this can be achieved via iterative methods [7], but they diverge in three-dimensional
case.
The final problem, which is discussed in section “Mixed Autoregressive
Moving Average (ARMA) Process”, is inability to “mix” AR and MA process in
three dimensions.
In practice some statements made for AR and MA processes in [7] should be
flipped for three-dimensional case. For example, the authors say that ACF of MA process cuts at q and ACF of AR process decays to nought infinitely, but in practice making ACF of 3-dimensional MA process not decay results in it being non-invertible
and producing realisation that does not look like real ocean waves, whereas doing
the same for ACF of AR process results in stationary process and adequate realisation. Also, the authors say that one should allocate the first q points of ACF to MA
process (as it often needed to describe the peaks in ACF) and leave the rest points
to AR process, but in practice in case of ACF of a propagating wave AR process is
stationary only for the first time slice of the ACF, and the rest is left to MA process.
To summarise, the only established scenario of applying ARMA model to ocean
wave generation is to use AR process for standing waves and MA process for propagating waves. With a new formulae for 3 dimensions a single mixed ARMA process
might increase model precision, which is one of the objectives of the future research.
The Shape of ACF for Different Types of Waves
Analytic Method of Finding the ACF
The straightforward way to find ACF for a given ocean wave profile is to apply
Wiener–Khinchin theorem. According to this theorem the autocorrelation K of a
function í µí¼ is given by the Fourier transform of the absolute square of the function:
K(t) = F
{
|í µí¼ (t)|
2
} .
(8)
When í µí¼ is replaced with actual wave profile, this formula gives you analytic formula
for the corresponding ACF.
For three-dimensional wave profile (2D in space and 1D in time) analytic formula is a polynomial of high order and is best obtained via symbolic computation
programme. Then for practical usage it can be approximated by superposition of
exponentially decaying cosines (which is how ACF of a stationary ARMA process
looks like [7]).
Empirical Method of Finding the ACF
However, for three-dimensional case there exists simpler empirical method which
does not require sophisticated software to determine shape of the ACF. It is known