Vertical Reflector Model

If the reflector is vertically oriented and the source and receiver are to the right of the reflector, then a dip filter can be used to separate leftgoing and rightgoing terms. Migrating a single trace associated with the vertical reflector model in Figure c gives the migration image in Figure d. Only the ellipse-like feature is desired, which can be isolated by replacing the upgoing reflection Green's function $U({\bf {x}}\vert{\bf {s}})$ by the rightgoing one $R({\bf {x}}\vert{\bf {s}})$ in equation  to give

$$m_{mig}({\bf {x}}) = \int \int_{B} d \omega dx_r~\omega^2~[\tilde d^{t}({\bf {r}}\vert... ...bf {r}})]^*~[D^{t}({\bf {x}}\vert{\bf {s}})+R^{r}({\bf {x}}\vert{\bf {s}})]^* ,$$

$$\approx \int \int_{B} d \omega dx_r ~ \omega^2~[$$

$$~~~~~~ \overbrace{D^{t}({\bf {x}}\vert{\bf {r}})^* D^{t}({\bf {x... ... {r}}\vert{\bf {s}})}^{strongest~trans.~mig.} ~~~~{\bf {x}}~\in~ direct~raypath$$

$$+~ \overbrace{\mathcal R^3 R^{r}({\bf {x}}\vert{\bf {r}})^* R^{r... ...}}\vert{\bf {s}})}^{weakest~refl. ~mig.} ~~~{\bf {x}}~\in~ specular~refl.~point$$

$$+~ \overbrace{\mathcal R^2 R^{r}({\bf {x}}\vert{\bf {r}})^* D^{t... ... {s}})}^{weak~sou-side~trans.~mig.} ~~~{\bf {x}}~\in~ sou-side~interbed~raypath$$

$$+~ \overbrace{\mathcal R^2 D^{t}({\bf {x}}\vert{\bf {r}})^* R^{r... ... {s}})}^{weak~rec-side~trans.~mig.} ~~~{\bf {x}}~\in~ rec-side~interbed~raypath$$

$$+~ \overbrace{\mathcal RD^{t}({\bf {x}}\vert{\bf {r}})^* D^{t}({\... ...\bf {s}})}^{strong~refl.~mig.}~] ~~~{\bf {x}}~\in~ specular~refl.~pt.+ellipse.$$ (8)

Dip filtering can be used to separate the leftgoing and rightgoing waves in the migration kernel, and the image of the desired Kirchhoff kernel is shown in Figure a. The dot-product of this kernel with the data can be used to get the desirable image of the vertical reflector. Figures b-d respectively correspond to the 1st, 3rd, and 4th terms in equation , which are the undesirable parts for migration.

Figure: Wavepaths in Figure d separated by dip filtering the migration kernel $G({\bf{x}}\vert{\bf{r}})G({\bf{x}}\vert{\bf{s}})$ associated with the vertical reflector model in Figure c.