$\lambda$ Wavelength Imaging at the Diffractor

Figures 4.4d and 4.8a illustrate that the width of the diffraction-transmission wavepath is proportional to $\lambda$ at the diffractor location. This can be mathematically proven by locating the point $\bf E$ on the Figure 4.1b ellipse so that the line through it and the focus at $\bf g$ is perpendicular to the elliptical axis. The distance between $\bf E$ and $\bf g$ is denoted as $\overline{Eg}$ . In the farfield approximation, $L\gg\overline{Eg}=z_o$ so we can approximate the ellipse formula for the first Fresnel zone centering about ${\bf {g}}$ as

$$\lambda/2 =\sqrt{z_o^2+L^2} + z_o - L \approx z_o .$$ (4.16)

This suggests that the resolution limit of the updated velocity model is about $\lambda$ near the scatterer, which is much finer than the resolution limit of $\sqrt{L\lambda/4}$ along the middle of the Figure 4.8a wavepath. This unexpectedly high resolution limit near the reflector boundaries can be observed in wave equation reflection traveltime (Zhang et al., 2012) and migration velocity analysis (Zhang and Schuster, 2013) tomograms.

To illustrate the range of wavenumbers estimated from diffraction and transmission migration, Figures 4.9 depicts the low wavenumbers (magneta dots) of the model recovered with transmission migration (see Figure 4.7) and the higher wavenumbers (gray dots) recovered by diffraction migration (Figure 4.5). Note the large gap between the recovered low- and high- wavenumber spectra, which will be denoted as the missing intermediate wavenumbers. The abscence of such intermediate model wavenumbers is a serious challenge for waveform inversion (Jannane et al., 1989), which will be addressed in the next section.

Figure 4.9: Wavenumbers (represented by gray dots) recovered by (i) diffraction migration have much higher values than those (represented by thick magenta curve) recovered by (ii) specular transmission migration. Blue, green and red dots represent the wavenumbers recovered based on (iii) interbed multiples originating from a diffractor, of multiples order 0, 1, and 2, respectively. These wavenumbers somewhat bridge the gap between cases (i) and (ii). The acquisition geometry is a 4 km long line of geophones and sources on the top interface, with the trial image point, denoted by $\bullet$ , at depth 1 km. The diffractor in case (iii) is 80 m below the trial image point.