FORWARD MODELING

A beam generated by a monochromatic point source (with angular frequency w and ray parameter p), and traveling through an horizontally homogeneous medium is described in cylindrical coordinates by  

$$\chi(\omega,p;r,z,t) = G(p,r) e^{i\omega(p r + \int_0^{z} s(z) dz - t)},$$ (1)

where

$$s(z) = \sqrt{1/v^2(z) - p^2},$$

is the vertical slowness of the medium at depth z, p is the horizontal slowness, r is the horizontal distance from the origin, and $G(p,r) = \sqrt{A_0/A(r)}$ is a geometrical factor corresponding to the ratio between the cross-sectional areas of the beam, at unit distance from the source and at point (r,z).

If we introduce a thin layer into the medium, the overall response of the reflected P wavefield beam (in the far-field approximation) at the surface, can be represented by the following expansion:

$$\begin{eqnarray} \chi^P_L(\omega,p;r,t) & = & G(p,r) \: e^{\textstyle i\omega(2 ... ...^{PP}_{L} e^{\textstyle i \omega 4 \Delta \! z_L s^P_L } + ... \}.\end{eqnarray}$$ (2)

In the above equation, indexes P and S refer to P and S waves, while indexes B and L correspond to the medium of the incident wave (background or perturbation layer respectively). R and T are the reflection and transmission coefficients, while the accents ($\acute{\mbox{up}}$ and $\grave{\mbox{down}}$) refer to the sense of propagation of the incident wave. z_L is the depth of the top of the layer and $\Delta \! z_L$ its thickness. Finally, s^P_L and s^S_L are the vertical slownesses of P and S waves inside the layer. For instance, $\acute{R}^{PS}_{L}$ would correspond to a P wave that is traveling upward inside the layer and is reflected back at the top interface as an S wave.

Figure  shows the rays corresponding to the contributions of the two leading terms of the expansion in equation (2); they represent the P wave reflected by the top of the layer and the mode that was transmitted to the layer, reflected in the bottom and transmitted back again, always as a compressional wave.

The first two terms of the expansion in equation (2) correspond to the two rays on the figure. The first is a P wave that is reflected as a P wave by the top of the layer, and the other corresponds to the same wave, transmitted into the layer as a P wave, reflect by the bottom still as P and then transmitted back to the background medium once again as a P wave.

Beneath the P wave's critical angle (and for geologically feasible elastic contrasts), the major contribution for the recorded P wave comes from the two first terms of the expansion in equation (2), and for angles close to the critical angle, the first-order conversion terms would be enough to suitably approximate the layer's reflection response. The approximation corresponding to the first two terms of the expansion is

 

$$\chi^P_L(\omega,p;r,t_p) \approx G(p,r) \: e^{i\omega t_p} \: \{ {\cal R}_L(\omega,p) \},$$ (3)

$\mbox{ \vspace{0.2in} \hspace{-0.1in} where \hspace{0.5in}} t_p = 2 p r - t,$

$$\mbox{ \hspace{-0.67in}and \hspace{0.5in}} {\cal R}_L(\omega... ...}^{PP}_{L} e^{\textstyle i \omega 2 \Delta \! z_L s^P_L } \}.$$

If we add other thin layers into the model, the P wave response of each layer can be expressed in the same way except for a multiplicative factor, which accounts for the background transmission losses up to that layer. It is important to emphasize that in the same way as the perturbation layer does not change traveltimes (which are kinematically-related), nor does it influence average transmission losses (because of the combined effect of all the overburden above any layer). For this reason, I assume that the smoothly varying transmission (and absorption) function ${\cal T}(z_L,p)$ must also be estimated by an independent method.

The overall contribution from all layers is

 

$$\chi^P(\omega,p;r,t_p) \approx G(p,r) \: e^{i\omega t_p} \su... ..._L,p) {\cal R}_L(\omega,p) = e^{i\omega t_p} {\cal R}(\omega,p)$$ (4)

$\mbox{\hspace{-0.1in} where \hspace{0.5in}} {\cal R}(\omega,p) = \sum_L {\cal T}(z_L,p) {\cal R}_L(\omega,p).$

Summing over time gives the P wave impulsive-source response of the perturbed medium, as the time Fourier-transform of ${\cal R}$:

 

$$\chi^P(p;r,pr-t) = G(p,r) \: \int_{-\infty}^{+\infty} e^{i\omega t_p} {\cal R}(\omega,p) d\omega dp.$$ (5)

For a given beam (p) the smooth background model can be used to find r(t), and the final result is achieved by summing the contributions from all beams. In my modeling program I used a superposition of one-half between adjacent beams and I smoothed around the central p of each beam with a triangle function.

A simple model (Figure ) was used to generate the synthetic data shown in Figure . It is important to notice that the seismic response from the two adjacent perturbation layers does not correspond to adjacent layers. The reason for this discrepancy is that the kinematics inside the layer is different from the kinematics of the background. To simulate the response of two adjacent layers in the real world, the perturbation model should have an overlap between layers (if the top layer is faster than the background) or a gap between them (top layer slower than background). On the other hand, to transform the model space into the real world model, a variable depth-shift (given by the difference between the depth integrals of the background and perturbed model) must be applied.

Model used to generate the synthetic data in Figure . The first 400 meters correspond to water. The model is composed of a smoothly varying background and some sparsely distributed layers.

Synthetic data generated with the described method, corresponding to the model of Figure . (a) p^1/2 - t₀ domain, and (b) x - t domain.