Image-space wave-equation tomography in the generalized source domain

APPENDIX C

This appendix demonstrates a matrix representation of the adjoint tomographic operator ${\bf T}^{'}$. Since the slowness perturbation ${\bf\Delta s}$ is linearly related to the perturbed wavefields, ${\bf\Delta D}$ and ${\bf\Delta U}$, to obtain the back-projected slowness perturbation, we first must get the back-projected perturbed wavefields from the perturbed image ${\bf\Delta I}$. From Equation B-15, the back-projected perturbed source and receiver wavefields are obtained as follows:

$${\bf\Delta D} = \overline{{\rm diag}\left( {\widehat {\bf U}} \right)} {\bf\Delta I}$$ (C-1)

and

$${\bf\Delta U} = \overline{{\rm diag}\left( {\widehat {\bf D}} \right)} {\bf\Delta I}.$$ (C-2)

Then the adjoint equations of Equations B-10 and B-14 are used to get the back-projected slowness perturbation ${\bf\Delta s}$. Let us first look at the adjoint equation of Equation B-10, which can be written as follows:

$${\bf\Delta s}_{D} = {\bf G}^{\prime}({\widehat {\bf D}},{\widehat... ...\left( {\bf 1} - {\bf E}^{\prime}({\widehat {\bf s}})\right)^{-1}{\bf\Delta D}.$$ (C-3)

We can define a temporary wavefield ${\bf\Delta P}_{D}$ that satisfies the following equation:

$${\bf\Delta P}_{D} = {\bf E}^{\prime}({\widehat {\bf s}}) \left( {\bf 1} - {\bf E}^{\prime}({\widehat {\bf s}})\right)^{-1}{\bf\Delta D}.$$ (C-4)

After some simple algebra, the above equation can be rewritten as follows:

$${\bf\Delta P}_{D} = {\bf E}^{\prime}({\widehat {\bf s}}) {\bf\Delta P}_{D} + {\bf E}^{\prime}({\widehat {\bf s}}) {\bf\Delta D}.$$ (C-5)

Substituting Equation C-1 into equation C-5 yields

$${\bf\Delta P}_{D} = {\bf E}^{\prime}({\widehat {\bf s}}) {\bf\Del... ...{\bf s}}) \overline{{\rm diag}\left( {\widehat {\bf U}} \right)} {\bf\Delta I}.$$ (C-6)

Therefore, ${\bf\Delta P}_{D}$ can be obtained by recursive upward continuation, where ${\bf\Delta D} = \overline{{\rm diag}\left( {\widehat {\bf U}} \right)} {\bf\Delta I}$ serves as the initial condition. The back-projected slowness perturbation from the perturbed source wavefield is then obtained by applying the adjoint of the scattering operator ${\bf G}({\widehat {\bf D}},{\widehat {\bf s}})$ to the wavefield ${\bf\Delta P}_{D}$ as follows:

$${\bf\Delta s}_{D} = {\bf G}^{\prime}({\widehat {\bf D}},{\widehat {\bf s}}) {\bf\Delta P}_{D}.$$ (C-7)

Similarly, the adjoint equation of Equation B-14 reads as follows:

$${\bf\Delta s}_{U} = {\bf G}^{\prime}({\widehat {\bf U}},{\widehat... ...\left( {\bf 1} - {\bf E}({\widehat {\bf s}})^{\prime}\right)^{-1}{\bf\Delta U}.$$ (C-8)

We can also define a temporary wavefield ${\bf\Delta P}_{U}$ that satisfies the following equation:

$${\bf\Delta P}_{U} = {\bf E}({\widehat {\bf s}})^{\prime} \left( {\bf 1} - {\bf E}({\widehat {\bf s}})^{\prime}\right)^{-1}{\bf\Delta U}.$$ (C-9)

After rewriting it, we get the following recursive form:

$${\bf\Delta P}_{U} = {\bf E}({\widehat {\bf s}})^{\prime} {\bf\Delta P}_{U}+ {\bf E}({\widehat {\bf s}})^{\prime} {\bf\Delta U}$$

$$= {\bf E}({\widehat {\bf s}})^{\prime} {\bf\Delta P}_{U}+ {\bf E}({... ...^{\prime} \overline{{\rm diag}\left( {\widehat {\bf D}} \right)} {\bf\Delta I}.$$ (C-10)

The back-projected slowness perturbation from the perturbed receiver wavefield is then obtained by applying the adjoint of the scattering operator ${\bf G}({\widehat{\bf U}},{\widehat{\bf s}})$ to the wavefield ${\bf\Delta P}_{U}$ as follows:

$${\bf\Delta s}_{U} = {\bf G}^{\prime}({\widehat {\bf U}},{\widehat {\bf s}}) {\bf\Delta P}_{U}.$$ (C-11)

The total back-projected slowness perturbation is obtained by adding ${\bf\Delta s}_D$ and ${\bf\Delta s}_U$ together:

$${\bf\Delta s} = {\bf\Delta s}_D + {\bf\Delta s}_U.$$ (C-12)

Image-space wave-equation tomography in the generalized source domain