Diving-Wave Transmission

Migration of the diving-wave residual along the yellow transmission wavepath in Figure 4.4a provides the low-wavenumber velocity update for waveform inversion (Zhou et al., 1995; Mora, 1989), or wave equation traveltime inversion (Woodward, 1992; Luo and Schuster, 1991; Woodward, 1989) if the trace residual is replaced by the recorded trace weighted by the traveltime residual. The boundary of the first Fresnel-zone wavepath^4.4is defined by values of ${\bf {x}}$ for the delayed diving wave time $\tau_{sg}^{\textrm{dive}}+T/2=\tau_{sx}+\tau_{xg}$ , where $\tau_{sg}^{\textrm{dive}}$ is the diving wave traveltime at the geophone location $\bf g$ . As illustrated in Figure 4.2b, the minimum width and height of the intersecting fat ellipses defines the effective resolution limits of transmission tomography (Williamson, 1991) or transmission migration (Sheley and Schuster, 2003).

More rigorously, Appendix F shows that the model resolution equation 4.6 for diving waves can be transformed into the Fourier integral

$$\delta m({\bf {x}})^{\textrm{mig}}=\alpha\!\!\int_{ {{\mathcal D}... ...!\!\!\!e^{-i {\bf {k}}\cdot {\bf {x}}}\delta M({\bf {k}}) J^{-1} dk_{x} dk_{z},$$ (4.10)

where $\alpha$ is a term related to geometrical spreading and ${\mathcal D}_{r_o}$ defines the range of source-geophone pairs whose first Fresnel zone wavepaths visit the scatterer localized at ${\bf {r}}_o$ . The formulas for resolution limits are the same as in equation 4.2, except $\mathcal D$ is replaced by ${\mathcal D}_{r_o}$ .

The range of allowable source-geophone pairs (see Sheng and Schuster (2003)) in ${\mathcal D}_{r_o}$ is illustrated in Figure 4.6b, where only the sources between the blue and red stars will contribute to the slowness update around the scatterer point at ${\bf {r}}_o$ . This differs from the Fourier integral 4.9 for diffraction imaging where $all$ source-geophone pairs contribute to the integration domain in $\mathcal D$ for a recorded diffraction. Hence, the resolution limits for migrating transmission residuals with the kernel $[G({\bf {g}}\vert{\bf {x}})^{\textrm{dir}}G({\bf {x}}\vert{\bf {s}})^{\textrm{dir}}]^*$ should be worse than migrating diffraction residuals with the same kernel.

Figure 4.6: Range of sources (blue and red stars) that generate a) reflection and b) transmission wavepaths that visit the scatterer (filled circle). Here, the wavepath is approximated by the first Fresnel zone for the specified source and geophone pair.

The precise connection between intersecting wavepaths in Figure 4.2b, the range of available wavenumbers, and resolution limits in equation 4.2 can be made by assuming a homogeneous medium. In this case, Figure 4.7 shows that the half-width $\Delta z$ of the first Fresnel zone at the point midway between the source and geophone is equal to

$$\Delta z = \sqrt{L \lambda/4},$$ (4.11)

where $L$ is the distance between the source and geophone, which is equal to that given by equation 4.2. It also shows that $\Delta z$ is inversely proportional to the sum of the source-scatterer and geophone-scatterer wavenumbers, implying that $\min 1/k_z$ is equivalent to finding the width of the wavepath intersections in Figure 4.2b.

For a single source-geophone pair, the best direction of transmission spatial resolution for a slowness anomaly midway between the source and geophone is perpendicular to the central ray. This means that a slowness anomaly moved perpendicular to the ray from the central ray will lead to the most noticeable change in the transmission arrival. The worst direction of spatial resolution is along the ray itself because the slowness anomaly can be slid along it without changing the traveltime; moreover, the model wavenumber ${\bf {k}}={\bf {k}}_{sx}+{\bf {k}}_{xg}$ is zero all along the transmission central ray.