Model Resolution Equation:  ${\bf {m}}^{\textrm {mig}} ={\bf L}^{\dagger } {\bf L} {\bf {m}}$

The forward and adjoint modeling equations can be combined to give the equation for model resolution, i.e., plugging equation 4.3 into equation 4.5 gives

$$\delta m({\bf {x}})^{\textrm{mig}}= \omega^4 \int_{D} \overbrace{... ...{y}}\vert{\bf {s}})}^{\textrm{data~kernel}} \delta m({\bf {y}})dy^2 dx_g dx_s ,$$ (4.6)

or in more compact notation

$${\bf\delta} {\bf {m}}^{\textrm{mig}} ={\bf L}^{\dagger} {\bf L} {\bf\delta} {\bf {m}}.$$ (4.7)

The kernel for the operator ${\bf L}^{\dagger} {\bf L}$ is related to the model resolution matrix (Menke, 1989) and is interpreted as the point spread function (Schuster and Hu, 2000) similar to that used in optics, except here, if $\delta m({\bf {y}})=\delta({\bf {y}}-{\bf {r}}_o)$ , it is the migration response to a point slowness perturbation in the model at ${\bf {r}}_o$ . The ideal response to a point slowness anomaly is the same point with perfect resolution.

For a two-layer medium, the above Green's function can be decomposed into its direct and reflection components:

$${G}({\bf {g}}\vert{\bf {x}}) ={G}({\bf {g}}\vert{\bf {x}})^{\textrm{dir}}+{G}({\bf {g}}\vert{\bf {x}})^{\textrm{rfl}},$$

$${G}({\bf {s}}\vert{\bf {x}}) ={G}({\bf {s}}\vert{\bf {x}})^{\textrm{dir}}+{G}({\bf {s}}\vert{\bf {x}})^{\textrm{rfl}},$$ (4.8)

where ${ G}({\bf {g}}\vert{\bf {x}})^{\textrm{dir}}$ and ${G}({\bf {g}}\vert{\bf {x}})^{\textrm{rfl}}$ are the, respectively, Green's function for the direct wave and upgoing reflection in the upper layer. Inserting equation 4.8 into the migration kernel in equation 4.6 results in the five migration kernels shown in Figure 4.4, each of which is used to smear residuals along one of the five yellow-colored wavepaths (Liu et al., 2011; Zhan et al., 2013).