Resolution Limits for Imaging Diving Wave Residuals

The resolution limits for imaging diving wave residuals are rigorously derived by multiplying the migration kernel in Figure 4.4a by the expression for the diving-wave arrival

$$\delta d({\bf {g}}\vert{\bf {s}})=\omega^4 \int_{\Omega} G({\bf {g}}\vert{\bf {y}})^{\textrm{dir}} G({\bf {y}}\vert{\bf {s}})^{\textrm{dir}} \delta m({\bf {y}}) dy^2 ,$$

$$\approx \omega^4 \int_{\Omega_{gs}} \!\! G({\bf {g}}\vert{\bf {y}})^{\textrm{dir}} G({\bf {y}}\vert{\bf {s}})^{\textrm{dir}} \delta m({\bf {y}}) dy^2 ,$$ (11.1)

where the integral over the model-space region $\Omega$ is approximated by the one over the region $\Omega_{gs}$ . Here, $\Omega_{gs}$ coincides with the yellow first Fresnel zone of the diving wave in Figure 4.4a for the source-geophone pair denoted by $s$ and $g$ . This approximation recognizes that only model perturbations within the first Fresnel zone of the diving wave will strongly affect the timing and/or amplitude of the diving-wave arrival at ${\bf {g}}$ .

Plugging equation F.1 into equation 4.6 gives

$$\delta m({\bf {x}})^{\textrm{mig}}=\omega^4 \!\!\int_{{D_{r_o}}} ... ...}G({\bf {y}}\vert{\bf {s}})^{\textrm{dir}} \, \delta m({\bf {y}})dy^2dx_g dx_s.$$ (11.2)

We now assume a localized subwavelength perturbation $\delta m({\bf {y}})$ centered at ${\bf {r}}_o=(x_o,z_o)$ that is non-zero only within a fraction of a wavelength from ${\bf {r}}_o$ . In this case, the range of source-geophone pairs in $D$ is restricted to the set $D_{r_o}$ of source-geophone pairs that allow for first Fresnel diving wavepaths to visit the localized perturbation centered at ${\bf {r}}_o$ . These source-geophone pairs are the only ones whose transmitted diving waves^11.1 will be significantly influenced by the model perturbations centered at ${\bf {r}}_o$ . For example, if the image point is at ${\bf{y}}$ and the geophone is at $\bf C$ in Figure 4.6b, then $D_{r_o}$ is limited to the sources between $A$ and $B$ . If the wavepaths are those for a specular reflection, then the range of source locations in $\Omega_{gs}$ is between $A$ and $B$ in Figure 4.6a.

For a smooth background velocity we assume the following asymptotic Green's function for the migration and data kernels

$$G({\bf {x}}\vert{\bf {y}})^{\textrm{dir}} = A_{xy} e^{-i\omega \tau_{xy}},$$ (11.3)

so that equation F.2 becomes

$$\delta m({\bf {x}})^{\textrm{mig}}= \omega^4 \int_{ D_{r_o}} \int... ...(\tau_{gx}+\tau_{sx}-\tau_{gy}-\tau_{sy})} \,\delta m({\bf {y}})dy^2 dx_g dx_s.$$ (11.4)

Here, $\tau_{xy}$ is the traveltime for the transmitted wave to propagate from ${\bf{y}}$ to ${\bf {x}}$ , and $A_{xy}$ is its attendant geometrical spreading term that satisfies the transport equation.

Assuming that the subwavelength scatterer represented by $\delta m({\bf {y}})$ is located within a fraction of a wavelength from the trial image point at ${\bf {x}}$ , then $\tau_{sy}$ , $\tau_{gy}$ , $\tau_{sx}$ , and $\tau_{gx}$ , can be expanded about its center point ${\bf {r}}_o$ to give

$$\tau_{sy} \approx \,\tau_{sr_o}+ \nabla \tau_{sr_o} \cdot [{\bf {y}}-{\bf {r}}_o],$$

$$\tau_{gy} \approx \,\tau_{gr_o}+ \nabla \tau_{gr_o} \cdot [{\bf {y}}-{\bf {r}}_o],$$

$$\tau_{sx} \approx \,\tau_{sr_o}+ \nabla \tau_{sr_o} \cdot [{\bf {x}}-{\bf {r}}_o],$$

$$\tau_{gx} \approx \,\tau_{gr_o}+ \nabla \tau_{gr_o} \cdot [{\bf {x}}-{\bf {r}}_o].$$ (11.5)

Inserting these approximations into equation F.4 gives

$$\delta m({\bf {x}})^{\textrm{mig}} \approx \omega^4 \int_{D_{r_o}... ... \tau_{sr_o}) \cdot [{\bf {y}}-{\bf {x}}]} \,\delta m({\bf {y}})dy^2 dx_g dx_s.$$

Under the far-field approximation, the geometric spreading terms can be taken outside the integral to give

$$\delta m({\bf {x}})^{\textrm{mig}} = \omega^4 \int_{ D_{r_o}}\!\!... ... \tau_{sr_o}) \cdot [{\bf {y}}-{\bf {x}}]} \,\delta m({\bf {y}})dy^2 dx_g dx_s.$$ (11.6)

Here, the gradient of the traveltime field $\nabla \tau_{sr_o}$ is parallel to the direct wave's incident angle at ${\bf {r}}_o$ , so, according to the dispersion equation, $\omega \nabla \tau_{sr_o} = {\bf {k}}_{sr_o}$ can be identified as the source-to-scatterer point wavenumber vector ${\bf {k}}_{sr_o}$ ; similarly, the geophone-to-scatterer wavenumber is denoted as $\omega \nabla \tau_{gr_o} = {\bf {k}}_{gr_o}$ . This means that, by definition of the Fourier transform with a restricted domain of integration $\delta M({\bf {k}})=\int_{\Omega_{gs}} e^{-i {\bf {k}}\cdot {\bf {y}}} \delta m({\bf {y}})dy^2$ , equation F.6 becomes

$$\delta m({\bf {x}})^{\textrm{mig}}\approx \omega^4 A_{s_og_or_o}^... ...}) \cdot {\bf {x}}} \, \delta M({\bf {k}}_{gr_o}+{\bf {k}}_{sr_o})\, dx_g dx_s,$$ (11.7)

where $A_{s_og_or_o}$ approximates the geometrical spreading for the scatterer at ${\bf {r}}_o$ with the range of allowable source-geophone pairs centered around the pairs denoted by $s_og_o$ , the Fourier spectrum of the model is given by $\delta M({\bf {k}})$ , and the model wavenumber components ${\bf {k}}=(k_x,k_z)$ are

$$k_x=k_{sx_o}+k_{gx_o} = \omega s({\bf {r}}_o) (\sin \beta_{sr_o}+ \sin \beta_{gr_o}),$$

$$k_z=k_{sz_o}+k_{gz_o} = \omega s({\bf {r}}_o) (\cos \beta_{sr_o}+ \cos \beta_{gr_o}),$$ (11.8)

where $\beta_{sr_o}$ and $\beta_{gr_o}$ denote the incidence angles of the source and geophone rays, respectively, at the scatterer's location ${\bf {y}}=(x_o,z_o)$ . As shown in the appendix, these incidence angles are implicit functions of the source $(x_s,0)$ , geophone $(x_g,0)$ , and scatterer ${\bf {r}}_o=(x_o,z_o)$ coordinates.

The determinant of the Jacobian in equation G.2 (see Appendix G) can be used to map the $dx_g dx_s$ integration in equation F.7 to a $dk_x dk_z$ integration:

$$\delta m({\bf {x}})^{\textrm{mig}}= A_{s_og_or_o}\int_{ {{\mathcal D}_{r_o}}} \!\!\!\!\!e^{-i {\bf {k}}\cdot {\bf {x}}} ~\delta M({\bf {k}}) \, J^{-1} dk_{x} dk_{z} ,$$ (11.9)

where ${{\mathcal D}_{r_o}}$ is the set of wavenumbers that equation F.8 maps from the source-geophone pairs in $D_{r_o}$ for the scatterer at ${\bf {r}}_o$ , and $J$ is the determinant of the Jacobian matrix in equation G.2.