|
|
|
| Kinematics in iterated correlations of a passive acoustic experiment | |
|
Next: Stationary-phase analysis of conventional
Up: De Ridder and Papanicolaou:
Previous: Conventional versus iterative interferometry
We define the temporal correlation function between two time signals and measured at stations A and B as
|
(1) |
where denotes angular frequency. The right-hand side of equation 1 shows that through the inverse Fourier transformation of equation A-3, a correlation integral in the time domain is a direct product in the frequency domain. We can retrieve the Green's function between two stations and by independently measuring responses of sources positioned on a boundary surrounding the two stations, and summing the correlation between the measurements at the two stations. This property can be expressed as (Wapenaar and Fokkema, 2006):
|
(2) |
where
,
and
denote positions of stations and and the sources respectively.
We investigate the terms within this this integral for a medium containing a scatterer. The Green's function under the Born approximation in a scattering medium is composed of two terms:
|
(3) |
where is the contribution of the direct wave, and is the contribution of the scattered wave. In the Born approximation, the contribution of the scatterer is included to order . The correlation product between measurements made at stations and therefore is composed of terms
|
|
|
(4) |
|
|
|
(5) |
which will be referred to as 4.1, 4.2, 4.3 and 4.4 respectively.
geomC2
Figure 3. Geometry for the evaluation of
in a homogeneous medium containing one scatterer. For three source positions, , and , two ray paths are shown for stationary phases; see text.
|
|
|
---|
|
---|
corrC2,resultC2
Figure 4. a) Correlogram displaying correlations of source responses measured at stations A and B for sources as a function of position angle. b) Comparison of retrieved and true Green's functions.
|
---|
|
---|
The second and third terms are of order , and the fourth term is of order . Therefore, we should exclude the fourth term when we evaluate the right-hand side of equation 2 and compare it to the left-hand side of equation 2. See Snieder et al. (2008) for a more general discussion of the fourth term for exact Green's functions (without Born approximation). We denote the integration of
over the source coordinate and multiplication by the phase-modifying factor as follows:
|
(6) |
where
is an implicit function of source position
, according to equation 4.
|
|
|
| Kinematics in iterated correlations of a passive acoustic experiment | |
|
Next: Stationary-phase analysis of conventional
Up: De Ridder and Papanicolaou:
Previous: Conventional versus iterative interferometry
2009-05-05