next up previous [pdf]

Next: Stationary-phase analysis of conventional Up: De Ridder and Papanicolaou: Previous: Conventional versus iterative interferometry

Green's function retrieval by correlation

We define the temporal correlation function between two time signals $ F_A(t)$ and $ F_B(t)$ measured at stations A and B as

$\displaystyle C^{(2)}_{B,A}(t) = \int_{-\infty}^{\infty}F_{B}(\tau+t)F_{A}(\tau...
...(\omega)F_{A}^*(\omega) \mathrm{exp}\left\{ i\omega t\right\}\mathrm{d}\omega,$ (1)

where $ \omega$ 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 $ A$ and $ B$ 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):

$\displaystyle G(\mathbf{x}_B,\mathbf{x}_A,\omega) - G^*(\mathbf{x}_A,\mathbf{x}...
...thbf{x}_s,\omega) G^*(\mathbf{x}_A,\mathbf{x}_s,\omega) \mathrm{d}\mathbf{x}_s,$ (2)

where $ \mathbf{x}_A$, $ \mathbf{x}_B$ and $ \mathbf{x}_s$ denote positions of stations $ A$ and $ B$ 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:

$\displaystyle G(\mathbf{x},\mathbf{x}_s,\omega) = G_0(\mathbf{x},\mathbf{x}_s,\omega) + G_1(\mathbf{x},\mathbf{x}_s,\omega),$ (3)

where $ G_0$ is the contribution of the direct wave, and $ G_1$ is the contribution of the scattered wave. In the Born approximation, the contribution of the scatterer is included to order $ \alpha$. The correlation product between measurements made at stations $ A$ and $ B$ therefore is composed of $ 2^2=4$ terms
$\displaystyle C^{(2)}_{B,A}(\omega) =
G_0(\mathbf{x}_B,\mathbf{x}_s,\omega)G_0^...
...mathbf{x}_B,\mathbf{x}_s,\omega)G_1^*(\mathbf{x}_A,\mathbf{x}_s,\omega) +\notag$     (4)
$\displaystyle G_1(\mathbf{x}_B,\mathbf{x}_s,\omega)G_0^*(\mathbf{x}_A,\mathbf{x...
...x}_B,\mathbf{x}_s,\omega)G_1^*(\mathbf{x}_A,\mathbf{x}_s,\omega), \hspace{.4cm}$     (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 $ C^{(2)}_{B,A}$ in a homogeneous medium containing one scatterer. For three source positions, $ a$, $ b$ and $ c$, two ray paths are shown for stationary phases; see text. $ \mathbf{[ER]}$
geomC2
[pdf] [png]

corrC2 resultC2
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. $ \mathbf{[ER]}$
[pdf] [pdf] [png] [png]

The second and third terms are of order $ \alpha$, and the fourth term is of order $ \alpha^2$. 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 $ C^{(2)}_{B,A}$ over the source coordinate and multiplication by the phase-modifying factor as follows:

$\displaystyle \tilde{C}^{(2)}_{B,A}(\omega) = -\frac{2i\omega}{c_0} \oint_{\partial\mathbf{D}} C^{(2)}_{B,A}(\omega) \mathrm{d}\mathbf{x}_s,$ (6)

where $ C^{(2)}_{B,A}(\omega)$ is an implicit function of source position $ \mathbf{x}_s$, according to equation 4.


next up previous [pdf]

Next: Stationary-phase analysis of conventional Up: De Ridder and Papanicolaou: Previous: Conventional versus iterative interferometry

2009-05-05