Reflection-Transmission

Migrating the reflection arrival with any of the kernels in the first column of Figure 4.4b-c leads to the low-wavenumber velocity update along the rabbit-ear wavepaths^4.5 in Figure 4.4b-c or Figure 4.1a.

The corresponding model resolution formula for the rightmost rabbit-ear wavepath is

$$\delta m({\bf {x}})^{\textrm{mig}}=\omega^4 \!\!\int_{D_{r_o}}\!\... ...G({\bf {s}}\vert{\bf {y}})^{\textrm{refl}}\,\delta m({\bf {y}})dy^2 dx_g dx_s ,$$ (4.12)

and, as before, can be analyzed for the resolution limits. However, now the asymptotic Green's functions for the transmitted arrival $G({\bf {g}}\vert{\bf {x}})^{\textrm{dir}}$ and the reflection field

$$G({\bf {x}}\vert{\bf {s}})^{\textrm{refl}} =A_{sx}^{\textrm{refl}} \,e^{-i \omega \tau_{sx}^{\textrm{refl}}} ,$$ (4.13)

are plugged into equation 4.12 to give the resolution limits for updating the velocity by smearing the reflection residual along the rabbit ears. Here, $A_{sx}^{\textrm{refl}}$ accounts for amplitude and phase effects from geometrical spreading and the reflection coefficient, and $\tau_{sx}^{\textrm{refl}}$ is the time it takes reflection energy to propagate from the source at ${\bf {s}}$ to the listener at ${\bf {x}}={\bf {r}}_o$ along the specular dashed raypath in Figure 4.1a.

Estimating the resolution limits for the rabbit-ear wavepaths will result in model resolution formulas similar to that given in Figure 4.4a for transmission imaging. This can be understood without going through the detailed algebra by recognizing that the upgoing reflection wavepath (rightmost rabbit ear in Figure 4.1a) is identical to the transmission wavepath in Figure 4.1b above the interface. This is denoted as a mirror transmission wavepath because it coincides with the first Fresnel zone for a source at the mirror position $(0,2d)$ in a homogeneous velocity. Thus, the reflection traveltime in a) is identical to the transmission traveltime in b) for any receiver at ${\bf {r}}_o$ . This means that the resolution limits defined by equation 4.2 are applicable to the transmission wavepaths in Figure 4.1b and the reflection wavepaths in Figure 4.1a. However, the range of available wavenumbers for the traces recorded at ${\bf {g}}$ is determined by the limited range of sources in Figure 4.6a that allow for the intersection of their first Fresnel zones with the scatterer. For example, the resolution limit $2 \Delta r$ perpendicular to the ray at the midpoint should be equal to the $2 \Delta r=\sqrt{\lambda L}$ in Figure 4.1a, except the total length of the reflection ray is $L=\sqrt{X^2+4d^2}$ .