EM 1110-2-1100 (Part II)
30 Apr 02
(c) First, it would be necessary to assume that the process described by the wave record (i.e., a sea
state), say η(t), is stationary, which means that the statistical properties of η(t) are independent of the origin
of time measurement. Since the statistics of stationary processes are time-invariant, there is no drift with time
in the statistical behavior of η(t). The stationarity requirement is necessary as we shall see later for
developing a probability distribution for waves, which is the fraction or percentage of time an event or
process (say, the sea state depicted in time series of the wave surface profile) is not exceeded. The probability
distribution may be obtained by taking η1(t1), η2(t1), η3(t1),..., as variables, independent of the instant t1. If in
addition, η(t) can be measured at different locations and the properties of η(t) are invariant or do not depend
on location of measurements, the process may then be assumed homogenous. In reality, η(t) may be assumed
stationary and homogenous only for a limited duration at the location data are gathered. Wind waves may
be considered approximately stationary for only a few hours (3 hr or less), beyond which their properties are
expected to change.
(d) Second, the process η(t) is assumed to be ergodic, which means that any measured record of the
process say η1(t) is typical of all other possible realizations, and therefore, the average of a single record in
an ensemble is the same as the average across the ensemble. For an ergodic process, the sample mean from
the ensemble approaches the real mean , and the sample variance approaches the variance σ of the process
(sea state). The ergodicity of η(t) implies that the measured realization of η(t), say η1(t1) is typical of all other
possible realizations η2(t1), η3(t1), ...., all measured at one instant t1. The concept of ergodicity permits
derivation of various useful statistical information from a single record, eliminating the need for multiple
recordings at different sites. The assumptions of stationarity and ergodicity are the backbones of developing
wave statistics from wave measurements. It is implicitly assumed that such hypotheses exist in reality, and
are valid, particularly for the sea state.
(e) To apply these concepts to ocean waves, consider an ensemble of records representing the sea state
by η(t) over a finite time T. The mean or expected value of the sea state, denoted by η, or η, or E[η], is
defined as
τ
1 2
τ m& τ
η ' E [η(t)] '
η(t) dt
(II-1-119)
2
where the symbol E denotes the expected value of η(t). Similarly, the mean-square of η corresponds to the
second moment of η, denoted by E[η2]. The standard deviation ση or the root-mean-square value of the
process is the square root of this. The variance of η, represented by ση2 may be expressed in terms of the
variance of the process V as
2
2
ση ' V [η(t)] ' E [η2] & η
(II-1-120)
(f) The standard deviation ση is the square root of the variance, also called the second central moment
of η(t). The standard deviation characterizes the spread in the values of η(t) about its mean.
(g) The autocorrelation or autocovariance function of the sea state is denoted by Rη, relating the value
of η at time t to its value at a later time t+τ. This is defined as
Rη(t, t%τ) ' E [η(t) η(t%τ)]
(II-1-121)
(h) The value of Rη gives an indication of the correlation of the signal with itself for various time lags
τ, and so it is a measure of the temporal variation of η(t) with time. If the signal is perfectly correlated with
itself for zero lag τ, its autocorrelation coefficient, defined as
Water Wave Mechanics
II-1-69
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