Lowrank one-step wave extrapolation for reverse-time migration

Appendix A: Proof of the stability of one-step wave extrapolation operator

In this appendix, we prove the unconditional stability of one-step wave extrapolation linear operator $u = \L f$ in one-dimensional isotropic media defined by:

$$u(x) = \sum\limits_{\xi \in z} e^{ 2\pi i(x \xi+V(x)\vert\xi\vert t)} \hat{f}(\xi) \quad f(x) \in l^2[0,1] \; ,$$ (47)

where $2\pi\xi = \k$ , $f(x)$ is assumed to have periodic boundary condition, and $\hat{f}(\xi)$ is the Fourier transform of $f(x)$ as defined by equation 3. We treat $\xi$ as discrete and $x$ as continuous for the ease of derivation. Our argument can be viewed as a discrete version of the standard stationary phase method in the study of pseudodifferential operators Stein (1993); Grigis and Sjöstrand (1994). To show that the operator $\L$ is stable, a sufficient condition is that $\Vert\L\Vert _{2 \to 2} \leq 1+Ct$ , where $C$ is a bounded constant. From equation 47, we observe that operator $\L$ is the composition of two operators $\L = \mathbf{A}\mathbf{F}$ , where $F$ is the inverse Fourier transform and $\mathbf{A}$ is the operator defined by:

$$(\mathbf{A}\omega)(z) = \sum\limits_{\xi\in z}e^{ 2\pi i(x \xi+V(x)\vert\xi\vert t)}\omega(\xi) \quad \omega \in l^2(z) \; .$$ (48)

Let us consider $\mathbf{A}^\intercal\mathbf{A}: l^2 \to l^2$ where $\mathbf{A}$ corresponds to a matrix with $(x,\xi)$ entry given by $\mathbf{A}(x,\xi) = e^{ 2\pi i(x \xi+V(x)\vert\xi\vert t)}$ , and $\mathbf{A}^\intercal$ corresponds to a matrix with $(\eta,x)$ entry given by $\mathbf{A}^\intercal(\eta,x) = e^{ -2\pi i(x \eta+V(x)\vert\eta\vert t)}$ . $\mathbf{A}^\intercal\mathbf{A}$ represents a matrix with $(\eta,\xi)$ entry given by:

$$\mathbf{A}^\intercal\mathbf{A}(\eta,\xi) = \int e^{ 2\pi i[x(\xi-\eta)+V(x)(\vert\xi\vert-\vert\eta\vert)t]} dx\; .$$ (49)

In order to bound the $l^2 \to l^2$ norm of $\mathbf{A}^\intercal\mathbf{A}$ we estimate the $(\eta,\xi)$ entry of $\mathbf{A}^\intercal\mathbf{A}$ . For $\xi = \eta$ , we have $\mathbf{A}^\intercal\mathbf{A}(\eta,\xi) = 1$ . For $\xi \neq \eta$ :

$$x(\xi-\eta)+V(x)(\vert\xi\vert-\vert\eta\vert)t = x(\xi-\eta)+V(x)\alpha(\xi-\eta)t \; ,$$ (50)

with $\alpha = (\vert\xi\vert-\vert\eta\vert)/(\xi-\eta)$ . Clearly, $\vert\alpha\vert \leq 1$ . Then $\mathbf{A}^\intercal\mathbf{A}$ can be expressed as

$$\mathbf{A}^\intercal\mathbf{A}(\eta,\xi) = \int e^{2\pi i(\xi-\eta)[x+V(x)\alpha t]}dx$$ (51)

For sufficiently small $t$ , $x+V(x)\alpha t$ satisfies

$$\frac{1}{2} \leq \nabla_x\left[x+V(x)\alpha t\right] \leq \frac{3}{2} \; .$$ (52)

Equation 51 can be expressed as:

$$\mathbf{A}^{\intercal}\mathbf{A}(\eta,\xi) = \int e^{2\pi i(\xi-\eta)[x+V(x)\alpha t]}\frac{1+V^{\prime}(x)\alpha t}{1+V^{\prime}(x)\alpha t} dx$$ (53)

$$= \int \frac{1}{1+V^{\prime}(x)\alpha t} e^{2\pi i(\xi-\eta)[x+V(x)\alpha t]} d\left[ x + V(x)\alpha t \right] \; .$$

Let us define $y=x+V(x)\alpha t$ . From equation 52, it is clear that the map $x \leftrightarrow y$ is one to one. Substituting $y$ into equation 53 gives

$$\mathbf{A}^{\intercal}\mathbf{A}(\eta,\xi) = \int \frac{1}{1+V^{\prime}(x(y))\alpha t} e^{2\pi i(\xi-\eta)y} dy \; ,$$ (54)

which is the inverse Fourier transform of $\frac{1}{1+V^{\prime}(x(y))\alpha t}$ . When $t$ is small,

$$\mathbf{A}^{\intercal}\mathbf{A}(\eta,\xi) = \int \left[ 1-V^{\prime}(x(y)\alpha t+\mathcal{O}(t^2)) \right]e^{2\pi i(\xi-\eta)y} dy$$ (55)

$$= -\alpha t \int V^{\prime}(x(y)) e^{2\pi i(\xi-\eta)y} dy + \mathcal{O}(t^2) \; .$$

To evaluate the norm of the integration term in the last equation, we perform integration by part for $k$ -times and apply periodic boundary condition:

$$\left\vert \int V^{\prime}(x(y)) e^{2\pi i(\xi-\eta)y} dy \right\... ...\pi i(\xi-\eta)y} \left \vert \partial_y^k V^{\prime}(x(y)) \right\vert dy \; .$$ (56)

Assuming sufficient smoothness on $V(x)$ , we have for $\xi \neq \eta$

$$\vert(\mathbf{A}^{\intercal}\mathbf{A}-\mathbf{I})(\eta,\xi)\vert \leq \frac{Ct}{\vert\xi-\eta\vert^k},$$ (57)

for a constant C. To estimate the $l^2 \to l^2$ norm of $\mathbf{A}^{\intercal}\mathbf{A}$ , we make use of the following lemma which can be derived from direct calculation.

Lemma 1   Suppose $\mathbf{G}: l^2\to l^2$ with $\sum\limits_\xi\vert\mathbf{G}(\xi,\eta)\vert\leq C$ and $\sum\limits_\eta\vert\mathbf{G}(\xi,\eta)\vert\leq C$ , then $\mathbf{G}$ is a bounded $l^2 \to l^2$ operator.

We now apply the above lemma to $\mathbf{A}^{\intercal}\mathbf{A}- \mathbf{I}$ . When $k \geq d$ where $d$ is the spatial dimension, we have $\sum\limits_{\xi\neq\eta}\frac{C}{\vert\xi-\eta\vert^k}$ bounded. Therefore, for sufficiently smooth $V$ , we have $\Vert\mathbf{A}^{\intercal}\mathbf{A} - \mathbf{I} \Vert _{2\to 2} \leq Ct$ , for suffciently small $t$ . Hence $\Vert\mathbf{A}^{\intercal}\mathbf{A}\Vert _{2\to 2} \leq 1 + Ct$ and $\Vert\mathbf{A}\Vert _{2\to 2} \leq 1 + Ct$ . Since $\L = \mathbf{A}\mathbf{F}$ and $\mathbf{F}$ as the Fourier transform is an isometry, we have $\Vert\L\Vert _{2 \to 2} \leq 1+Ct$ .

When performing wave extrapolation, fix a final time $T$ and propagate $T/dt$ steps, the operator is stable since

$$\Vert\L ^{T/t} \Vert _{2\to 2} \leq (1 + Ct)^{T/t} \leq e^{CT}.$$ (58)

Lowrank one-step wave extrapolation for reverse-time migration