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I. Gankevich and A. Degtyarev
surface point is represented as a weighted sum of previous in time and space points
plus weighted sum of previous in time and space normally distributed random
impulses. The governing equation for 3-D ARMA process is
𝜁 i =
N
∑
j=𝟎
𝛷 j 𝜁 i−j +
M
∑
j=𝟎
𝛩 j 𝜀 i−j ,
(4)
where 𝜁 —wave elevation, 𝛷—AR process coefficients, 𝛩—MA process coefficients, 𝜀—white noise with Gaussian distribution, N—AR process order, M—MA
process order, and 𝛷 𝟎 ≡ 0, 𝛩 𝟎 ≡ 0. Here arrows denote multi-component indices
with a component for each dimension. In general, any scalar quantity can be a component (temperature, salinity, concentration of some substance in water etc.). Equation parameters are AR and MA process coefficients and order.
Any ARMA process can be uniquely represented as either MA or AR process of
infinite order [10], and the parameters of the spectral representation are defined by
the rule of division of power series (in a rational factorized form [9]):
S(𝜔) =
𝛥𝜎 2
𝜋
∏
m
(1 − z m e −im𝜔𝛥 )(1 − z m e im𝜔𝛥 )
∏
n
(1 − p n e −in𝜔𝛥 )(1 − p n e in𝜔𝛥 )
,
where z m and p n are the zeros of numerator (MA), and denominator (AR), respectively, which form a pair of mutually conjugate numbers. If some of the zeros are
located near the unit circle, then the spectral density will have pronounced dips.
Autoregressive (AR) Process
AR process is ARMA process with only one random impulse instead of their
weighted sum:
𝜁 i =
N
∑
j=𝟎
𝛷 j 𝜁 i−j + 𝜀 i,j,k .
(5)
The coefficients 𝛷 are calculated from auto-covariate function (ACF) via threedimensional Yule–Walker (YW) equations, which are obtained after multiplying
both parts of the previous equation by 𝜁 i−k and computing the expected value.
Generic form of YW equations is
𝛾 k =
N
∑
j=𝟎
𝛷 j 𝛾 k−j + 𝜎
2
𝜀 𝛿 k ,
𝛿 k =
{
1, if k = 0
0, if k ≠ 0,
(6)