256
I. Gankevich and A. Degtyarev
Here coefficients í µí»© are calculated from back to front: from i = M to i = í µí¿. White
noise variance is estimated by
í µí¼
2
í µí¼ =
í µí»¾ í µí¿
1 +
∑ M
j=í µí¿ í µí»©
2
j
.
Authors of [7] suggest using Newton–Raphson method to solve this equation with
higher precision, however, this method does not work in three dimensions. Using
slower method does not have dramatic effect on the overall programme performance,
because the number of coefficients is small and most of the time is spent generating
wavy surface.
Mixed Autoregressive Moving Average (ARMA) Process
Generally speaking, ARMA process is obtained by plugging MA generated wavy
surface as random impulse to AR process, however, in order to get the process with
desired ACF one should re-compute AR coefficients before plugging. There are several approaches to “mix” AR and MA processes.
∙ The approach proposed in [7] which involves dividing ACF into MA and AR part
along each dimension is not applicable here, because in three dimensions such
division is not possible: there always be parts of the ACF that are not taken into
account by AR and MA process.
∙ The alternative approach is to use the same (undivided) ACF for both AR and
MA processes but use different process order, however, then realisation characteristics (mean, variance etc.) become skewed: these are characteristics of the two
overlapped processes.
For the first approach there is a formula to re-compute ACF for AR process, but there
is no such formula for the second approach. So, the best solution for now is to simply
use AR and MA process exclusively for different types of waves.
Process Selection Criteria for Different Wave Profiles
One problem of ARMA model application to ocean wave generation is that for different types of wave profiles different processes must be used: standing waves are
modelled by AR process, and propagating waves by MA process. This statement
comes from practice: if one tries to use the processes the other way round, the resulting realisation either diverges or does not correspond to real ocean waves. So, the
best way to apply ARMA model to ocean wave generation is to use AR process for
standing waves and MA process for progressive waves.
I. Gankevich and A. Degtyarev
Here coefficients í µí»© are calculated from back to front: from i = M to i = í µí¿. White
noise variance is estimated by
í µí¼
2
í µí¼ =
í µí»¾ í µí¿
1 +
∑ M
j=í µí¿ í µí»©
2
j
.
Authors of [7] suggest using Newton–Raphson method to solve this equation with
higher precision, however, this method does not work in three dimensions. Using
slower method does not have dramatic effect on the overall programme performance,
because the number of coefficients is small and most of the time is spent generating
wavy surface.
Mixed Autoregressive Moving Average (ARMA) Process
Generally speaking, ARMA process is obtained by plugging MA generated wavy
surface as random impulse to AR process, however, in order to get the process with
desired ACF one should re-compute AR coefficients before plugging. There are several approaches to “mix” AR and MA processes.
∙ The approach proposed in [7] which involves dividing ACF into MA and AR part
along each dimension is not applicable here, because in three dimensions such
division is not possible: there always be parts of the ACF that are not taken into
account by AR and MA process.
∙ The alternative approach is to use the same (undivided) ACF for both AR and
MA processes but use different process order, however, then realisation characteristics (mean, variance etc.) become skewed: these are characteristics of the two
overlapped processes.
For the first approach there is a formula to re-compute ACF for AR process, but there
is no such formula for the second approach. So, the best solution for now is to simply
use AR and MA process exclusively for different types of waves.
Process Selection Criteria for Different Wave Profiles
One problem of ARMA model application to ocean wave generation is that for different types of wave profiles different processes must be used: standing waves are
modelled by AR process, and propagating waves by MA process. This statement
comes from practice: if one tries to use the processes the other way round, the resulting realisation either diverges or does not correspond to real ocean waves. So, the
best way to apply ARMA model to ocean wave generation is to use AR process for
standing waves and MA process for progressive waves.