Page 222 - Acquisition and Processing of Marine Seismic Data

4.1 AUTOCORRELATION 213

Angular frequency is defined as recurrence functions, as well as their graphics and important
speed of the signal in radians/s and is expressed properties are provided in Table 4.1.
as ω ¼ 2πf,where f is the frequency.
For the signals observed in the space domain,
however, wavelength and wavenumber terms 4.1 AUTOCORRELATION
are used instead of period and frequency param-
eters. Wavelength is the recurrence distance of a Autocorrelation is used to obtain the degree
periodic signal observed in the space domain of similarity of a time series with itself, which
and expressed as λ ¼ V/f,where V and f are provides to obtain periodical components
signal propagation velocity and frequency, embedded in the data. Autocorrelation of an
respectively. Vertical resolution of seismic data x(t) series is expressed analytically as
depends on the wavelength of the recorded sig- Z ∞
nal (Fig. 4.1B). Wavenumber is defined as the
ð
R xx τðÞ ¼ xtðÞxt + τÞdt (4.2)
number of recurrence of a periodic signal in a unit
1
distance. It is represented by k,and itsunitis m . ∞
Using the parameters defined here, a periodic where τ is the time lag. According to Eq. (4.2), x(t)
sinusoid in the time domain can be mathemati- analytical function is lagged by an amount of τ,
cally expressed as multipliedbyx(t),andsummedupviaintegration.
Autocorrelation function is even, which is
ð
ftðÞ ¼ A sin ω 0 t + ϕð Þ ¼ A sin 2πf 0 t + ϕÞ (4.1)
expressed as R xx ( τ) ¼ R xx (τ). When τ ¼ 0(lag
where A is the amplitude, ω 0 is the angular fre- 0), autocorrelation value is always maximum,
quency (ω 0 ¼ 2πf 0 ), and f 0 and ϕ are the funda- which corresponds to the total energy of the input
mental frequency and phase angle of the function. If function x(t) is periodic, then its
sinusoid, respectively. These terms are illus- autocorrelation function R xx (τ)isalsoperiodic,
trated on a periodical signal in Fig. 4.1. When and we can determine the periodic components
the signal is in time domain, we define the signal of the input by examining its autocorrelation.
using its period and frequency (Fig. 4.1A), Hence, autocorrelation function is used to deter-
whereas for a space domain signal, these param- mine the required parameters of the deconvolu-
eters correspond to wavelength and wavenum- tion process. Furthermore, deconvolution
ber, respectively (Fig. 4.1B). Phase is defined as operator is obtained by calculating the
the negative of the phase shift, and a negative autocorrelation of the input seismic trace
time shift corresponds to a positive phase value (Chapter 6). In addition, quality control and effec-
(Fig. 4.1C). tivenessofthedeconvolutionoutputinvolvesana-
There are some specific functions termed lyzing the autocorrelation of deconvolved output.
“generalizedfunctions”whichareoftenuseddur- Calculation of autocorrelation of an x(t) series
ingtheprocessingor toderivetheoreticalimplica- in time domain means squaring the amplitude
tionsoftheprocessingmethodology.Forinstance, spectrum in frequency domain. Phase spectrum
odd and even functions are used to compute the of the autocorrelation output becomes zero,
Fourier series approximation of a given function; which implies that autocorrelation function
box-car and sinc functions are important in R xx (τ) is not reversible. That is, we cannot recover
designing the band-pass filter operators; and x(t) function if we only know R xx (τ). Because a
Signum function is used to obtain Hilbert trans- time domain autocorrelation calculation equals
form of the signal, which is then used to compute squaring of the amplitude spectrum, time
complex trace attributes of the seismic data. The domain counterpart of the power spectrum is
mathematical and graphical expressions of these autocorrelation function, which is known as