Horizontal Reflector Model

The Green's function for the horizontal reflector model can be decomposed into downgoing transmitted $D^{t}({\bf {x}}\vert{\bf {s}})$ and upgoing reflected $\mathcal RU^{r}({\bf {x}}\vert{\bf {s}})$ wave components in the first layer as illustrated in Figure :

$$G({\bf {x}}\vert{\bf {s}}) = D^{t}({\bf {x}}\vert{\bf {s}}) + \mathcal RU^{r}({\bf {x}}\vert{\bf {s}}); G({\bf {x}}\vert{\bf {r}}) = D^{t}({\bf {x}}\vert{\bf {r}}) + \mathcal RU^{r}({\bf {x}}\vert{\bf {r}}),$$ (6)

where ${\bf {x}}$ is the trial image point anywhere along the specular portion of the reflection ray and $\mathcal R$ is the reflection coefficient. The superscripts $t$ and $r$ are associated with transmittion and reflection, respectively.

Figure: Horizontal and vertical reflector models on the left along with their associated wavepaths (Woodward, 1992) to the right. a) shows the horizontal reflector model, and its wavepaths in b) take the shapes of a cigar, rabbit ears, and an elliptical smile. c) and d) are the vertical reflector model and its corresponding wavepaths. The wavepaths were computed by migrating a single trace excited by a wideband point source (star) and recorded at the receiver (quadrilateral).

Figure: Raypaths associated with the products in $[D^{t}({\bf{x}}\vert{\bf{r}})+U^{r}({\bf{x}}\vert{\bf{r}})]$ $[D^{t}({\bf{x}}\vert{\bf{s}})+U^{r}({\bf{x}}\vert{\bf{s}})]$ . The phase of these products at certain trial image points ${\bf {x}}$ will annihilate the phase in the data for either $\tilde d^{t}({\bf{r}}\vert{\bf{s}})$ or $\tilde d^{r}({\bf{r}}\vert{\bf{s}})$ , but not both. The diving ray in a) can lead to strong artifacts in the RTM image.

The dominant contributions to the migration image $m_{mig}({\bf {x}})$ described by equation  will be along the raypaths where the phase of $G^*({\bf {x}}\vert{\bf {s}})G^*({\bf {x}}\vert{\bf {r}})$ cancels that for events in the data $\tilde d({\bf {r}}\vert{\bf {s}}) = \tilde d^{t}({\bf {r}}\vert{\bf {s}}) + \tilde d^{r}({\bf {r}}\vert{\bf {s}})$ , i.e., inserting equation  into equation  gives

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

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

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

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

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

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

$$+~ \overbrace{\mathcal RD^{t}({\bf {x}}\vert{\bf {r}})^* D^{t}({\... ...\bf {s}})}^{strong~refl.~mig.}~] ~~~{\bf {x}}~\in~ specular~refl.~pt.+ellipse,$$ (7)

where unimportant terms are ignored and ${\bf {x}}$ is the trial image point where the phase of the migration kernel is equal and opposite to that of the reflection data.

The amplitude of the transmitted arrival $\tilde d^{t}({\bf{r}}\vert{\bf{s}})$ is $O(1)$ , so the strongest part of the migration image $D^{t}({\bf {x}}\vert{\bf {r}})^* D^{t}({\bf {x}}\vert{\bf {s}})^* \tilde d^{t}({\bf {r}}\vert{\bf {s}})$ is for ${\bf {x}}$ to be along the direct ray shown in Figure a, which coincides with the central part of the transmission wavepath in Figure b. The weakest contribution in the above approximation is $\mathcal R^3 U^{r}({\bf {x}}\vert{\bf {r}})^* U^{r}({\bf {x}}\vert{\bf {s}})^* \cdot \tilde d^{r}({\bf {r}}\vert{\bf {s}})$ with strength $O(\mathcal R^3)$ , and contributes at the specular reflection point shown in Figure b. The undesirable contributions to the migration image are along the interbed raypaths with strength $O(\mathcal R^2)$ shown in Figure c, which coincide with the central portions of the rabbit-ear wavepaths in Figure b. Finally, the most desirable contribution to the migration image is the Kirchhoff-like image $\mathcal RD^{t}({\bf {x}}\vert{\bf {r}})^* D^{t}({\bf {x}}\vert{\bf {s}})^* \cdot d^{r}({\bf {r}}\vert{\bf {s}})$ with strength $O(\mathcal R)$ . It contributes to the image at both the specular reflection point in Figure d, but also to the thick ellipse in Figure b.

For reflection migration, only the Kirchhoff-like term should be used and contributions from all other terms should be filtered out. This goal can be accomplished by dip filtering the Green's functions to separate upgoing and downgoing waves, and so only the Kirchhoff-like kernel $D^{t}({\bf {x}}\vert{\bf {r}})^* D^{t}({\bf {x}}\vert{\bf {s}})^*$ should be used for GDM.

As an example, the Green's functions associated with the GDM image in Figure b can be filtered to give the separate components in Figure . Here, the desirable image is the ellipse in Figure a (the last term in equation ), and the undesirable parts are the smile in Figure b (the 1st term in equation ) and the rabbit ears in Figures c-d (the 3rd and 4th terms in equation ).

Applying a dip filter to the migration kernels separates the migration image into the different portions shown in Figures a-d. Since we are only interested in imaging the reflector boundary, then only the migration kernel associated with the ellipse should be used.

Figure: Wavepaths in Figure b separated by dip filtering the migration kernel $G({\bf{x}}\vert{\bf{r}})G({\bf{x}}\vert{\bf{s}})$ associated with the horizontal reflector model in Figure a (see the kernels in equation ).