Born Adjoint Modeling

Equation 4.3 can be inverted by the iterative steepest descent formula

$$\delta m({\bf {x}})^{k+1}=\delta m({\bf {x}})^{k}-\alpha ~\delta m({\bf {x}})^{\textrm{mig}},$$ (4.4)

where the misfit gradient $\delta m({\bf {x}})^{\textrm{mig}}$ is given by the Born adjoint modeling equation

$$\delta m({\bf {x}})^{\textrm{mig}}= \,\omega^2\int_{D} \!\!G({\bf {g}}\vert{\bf {x}})^* G({\bf {x}}\vert{\bf {s}})^* \delta d({\bf {g}}\vert{\bf {s}}) dx_g dx_s$$

$$\rightarrow \delta {\bf {m}}^{\textrm{mig}} ={\bf L}^{\dagger} \delta {\bf {d}},$$ (4.5)

and the integration of points in $D$ is over the range of horizontal source and receiver coordinates along the horizontal recording line at $z=0$ . Here, $\delta d({\bf {g}}\vert{\bf {s}})=d({\bf {g}}\vert{\bf {s}})-d({\bf {g}}\vert{\bf {s}})^{obs}$ , ${\bf L}^{\dagger}$ represents the adjoint of the modeling operator ${\bf L}$ , the step length is denoted by $\alpha$ , $d({\bf {g}}\vert{\bf {s}})$ is the trace predicted from the estimated slowness model, and the observed trace is represented by $d({\bf {g}}\vert{\bf {s}})^{obs}$ . The misfit gradient symbol $\delta m({\bf {x}})^{\textrm{mig}}$ is superscripted by $mig$ because it also represents the migration of the residual. In fact, the first iteration $k=0$ of equation 4.5 represents the reverse time migration of the scattered data recorded at the surface.

If the windowed event is the reflection, equation 4.5 says that the velocity model is updated by smearing the residual^4.2along the yellow-colored rabbit ears in Figure 4.4b-c and the yellow ellipse in Figure 4.4e. Smearing residuals along the rabbit ears (ellipse) with the b-c (e) migration kernel updates the low-wavenumber (high-wavenumber) portion of the velocity model (Liu et al., 2011; Zhou et al., 1995; Mora, 1989). The spatial resolution limits $\Delta x$ and $\Delta z$ associated with any point along the central rays are determined by the, respectively, horizontal and vertical widths of the first-Fresnel zone.