Mathematical Derivation with Adjoint State Method

In Chapter 4, the physical meaning of prism wave migration was explained with a simple geometrical interpretation. From the mathematical point of view, the migration of prism waves can be thought of as the adjoint operation of modeling a prism wave. To show this, I will derive the forward modeling operator of a prism wave and apply the adjoint state method to derive its corresponding migration operator. Given a background slowness model ${s}_o(\textbf{x})$ and a reflectivity model ${m}_1(\textbf{x})$ , the reflection data for a shot at $\textbf{x}_s$ can be modeled with the Born approximation using the following equations (Dai et al., 2012)

$$(\bigtriangledown^2+\omega^2{s}^2_o(\textbf{x})){P}_o(\textbf{x}) = {W}(\omega)\delta(\textbf{x}-\textbf{x}_s),$$ (85)

$$(\bigtriangledown^2+\omega^2{s}^2_o(\textbf{x})){P}_1(\textbf{x}) = \omega^2 {m}_1(\textbf{x}){P}_o(\textbf{x}).$$ (86)

By introducing a perturbation to the slowness model ${s}_o\rightarrow {s}_o+\delta{s}$ , the wavefields become ${P}_o\rightarrow {P}_o+\delta{P}_o$ , ${P}_1\rightarrow {P}_1+\delta{P}_1$ . Expanding the slowness term as

$$({s}_o+\delta{s})^2 \approx {s}^2_o + 2{s}_o\delta{s},$$ (87)

equations D.1 and D.2 become

$$(\bigtriangledown^2+\omega^2{s}^2_o+2\omega^2{s}_o\delta{s})({P}_o(\textbf{x})+\delta{P}_o(\textbf{x})) = {W}(\omega)\delta(\textbf{x}-\textbf{x}_s),$$ (88)

$$(\bigtriangledown^2+\omega^2{s}^2_o+2\omega^2{s}_o\delta{s})({P}_1(\textbf{x})+\delta{P}_1(\textbf{x})) = \omega^2 {m}_1(\textbf{x})({P}_o(\textbf{x})+\delta{P}_o(\textbf{x})).$$ (89)

Assuming ${m}_2(\textbf{x})=-2{s}_o(\textbf{x})\delta{s}(\textbf{x})$ , and subtracting equation D.1 from equation D.4, I get

$$(\bigtriangledown^2+\omega^2{s}^2_o)\delta{P}_o(\textbf{x}) = \omega^2 {m}_2(\textbf{x}){P}_o(\textbf{x}).$$ (90)

Similarly, subtracting equation D.2 from equation D.5, I get

$$(\bigtriangledown^2+\omega^2{s}^2_o)\delta{P}_1(\textbf{x}) = \om... ...\textbf{x})\delta{P}_o(\textbf{x})+\omega^2 {m}_2(\textbf{x}){P}_1(\textbf{x}),$$ (91)

where the higher order terms are neglected. Equation D.7 represents the modeling operator for the prism wave $\delta{P}_1(\textbf{x})$ , which requires solving equations D.1, D.2, and D.6. Calculation of the prism wave needs four finite-difference simulations.

The above equations can be expressed with Green's functions ${G}_o$ calculated with the slowness $s_o$ , so equation D.2 becomes

$${P}_1(\textbf{x}\vert\textbf{x}_s)=\int_{\textbf{x}'} \omega^2{W}... ...textbf{x}_s) {m}_1(\textbf{x}'){G}_o (\textbf{x}'\vert\textbf{x}) d\textbf{x}',$$ (92)

and equation D.6 becomes

$$\delta{P}_o(\textbf{x}\vert\textbf{x}_s)=\int_{\textbf{x}''} \ome... ...tbf{x}_s) {m}_2(\textbf{x}''){G}_o (\textbf{x}''\vert\textbf{x}) d\textbf{x}'',$$ (93)

where $\textbf{x}'$ and $\textbf{x}''$ are dummy variables. Thus, the modeling operator of the doubly scattered prism wave can be expressed as

$$\delta{P}_1(\textbf{x}\vert\textbf{x}_s)=\int_{\textbf{x}'''} \om... ...){m}_1(\textbf{x}''')\delta{P}_o(\textbf{x}'''\vert\textbf{x}_s) d\textbf{x}'''$$

$$+ \int_{\textbf{x}'''} \omega^2{G}_o(\textbf{x}'''\vert\textbf{x}){m}_2(\textbf{x}'''){P}_1(\textbf{x}'''\vert\textbf{x}_s) d\textbf{x}'''$$

$$= \int_{\textbf{x}'''} \omega^2{G}_o(\textbf{x}'''\vert\textbf{x}){... ...textbf{x}''){G}_o (\textbf{x}''\vert\textbf{x}''') d\textbf{x}'' d\textbf{x}'''$$

$$+ \int_{\textbf{x}'''} \omega^2{G}_o(\textbf{x}'''\vert\textbf{x}){... ...1(\textbf{x}'){G}_o (\textbf{x}'\vert\textbf{x}''') d\textbf{x}' d\textbf{x}'''$$ (94)

If I switch the order of integration for the first term, the above equation becomes

$$\delta{P}_1(\textbf{x}\vert\textbf{x}_s)$$

$$= \int_{\textbf{x}''} \omega^2{W}(\omega) {G}_o (\textbf{x}''\vert ... ...extbf{x}'''){G}_o (\textbf{x}''\vert\textbf{x}''') d\textbf{x}''' d\textbf{x}''$$

$$+ \int_{\textbf{x}'''} \omega^2{W}(\omega){G}_o(\textbf{x}'''\vert\... ...1(\textbf{x}'){G}_o (\textbf{x}'\vert\textbf{x}''') d\textbf{x}' d\textbf{x}'''$$

$$= \int_{\textbf{x}''} \omega^2{W}(\omega) {G}_o (\textbf{x}''\vert\textbf{x}_s) {m}_2(\textbf{x}''){G}_1(\textbf{x}\vert\textbf{x}'') d\textbf{x}''$$

$$+ \int_{\textbf{x}'''} \omega^2{W}(\omega){G}_o(\textbf{x}'''\vert\textbf{x}){m}_2(\textbf{x}''') {G}_1(\textbf{x}'''\vert s) d\textbf{x}''',$$ (95)

where ${G}_1$ represents the Green's function for reflection wave:

$${G}_1(\textbf{x}\vert\textbf{x}'') = \int_{\textbf{x}'''} \omega^2{G}_o(\textbf{x}'''\vert\textbf{x}){m}_1(\textbf{x}'''){G}_o (\textbf{x}''\vert\textbf{x}''') d\textbf{x}''',$$ (96)

$${G}_1(\textbf{x}'''\vert\textbf{x}_s) = \int_{\textbf{x}'} \omega^2 {G}_o (\textbf{x}'\vert\textbf{x}_s){m}_1(\textbf{x}'){G}_o (\textbf{x}'\vert\textbf{x}''') d\textbf{x}'.$$ (97)

When the wavefield $\delta P_1$ is recorded at the receiver location $\textbf{x}_g$ , the shot gather ${d}_2(\textbf{x}_g\vert\textbf{x}_s)$ of the prism wave can be expressed as

$$d_2(\textbf{x}_g\vert\textbf{x}_s) = \int_{\textbf{x}} \omega^2{W}(\omega) {G}_o (\textbf{x}\vert\textbf{x}_s) {m}_2(\textbf{x}){G}_1(\textbf{x}\vert\textbf{x}_g) d\textbf{x}$$

$$+ \int_{\textbf{x}} \omega^2{W}(\omega){G}_1(\textbf{x}\vert\textbf{x}_s){m}_2(\textbf{x}) {G}_1(\textbf{x}\vert\textbf{x}_g) d\textbf{x},$$ (98)

Equation D.14 is the forward modeling operator for the prism wave. By simply applying adjoint of the forward modeling (Plessix, 2006), the migration image of the shot gather ${d}_2(\textbf{x}_g\vert\textbf{x}_s)$ can be shown to be

$${m}_{mig}(\textbf{x}\vert\textbf{x}_s) = \sum_{\omega}\sum_g \omega^2{W}^*(\omega){G}^*_o (\textbf{x}\vert... ...f{x}_s) {G}^*_1(\textbf{x}\vert\textbf{x}_g) d_2(\textbf{x}_g\vert\textbf{x}_s)$$

$$+ \sum_{\omega}\sum_g \omega^2{W}^*(\omega){G}^*_1 (\textbf{x}\vert... ...}_s) {G}^*_o(\textbf{x}\vert\textbf{x}_g) {d}_2(\textbf{x}_g\vert\textbf{x}_s),$$ (99)

which are exactly the terms in equations and . The computation of these terms is described in the text.