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

image-space wave-equation tomography

Image-space wave-equation tomography is a non-linear inverse problem that tries to find an optimal background slowness that minimizes the residual field, ${\bf\Delta I}$, defined in the image space. The residual field is derived from the background image, ${\bf I}$, which is computed with a background slowness (or the current estimate of the slowness). The residual field measures the correctness of the background slowness; its minimum (under some norm, e.g. $\ell_2$) is achieved when a correct background slowness has been used for migration. There are many choices of the residual field, such as residual moveout in the Angle-Domain Common-Image Gathers (ADCIGs), differential semblance in the ADCIGs, reflection-angle stacking power (in which case we have to maximize the residual field, or minimize the negative stacking power), etc.. Here we follow a definition similar to that in Biondi (2008), and define a general form of the residual field as follows:

$${\bf\Delta I} = {\bf I} - {\bf F}({\bf I}),$$ (A-1)

where ${\bf F}$ is a focusing operator, which measures the focusing of the migrated image. For example, in the Differential Semblance Optimization (DSO) method (Shen, 2004), the focusing operator takes the following form:

$${\bf F}({\bf I}) = \left( {\bf 1} - {\bf O} \right) {\bf I},$$ (A-2)

where ${\bf 1}$ is the identity operator and ${\bf O}$ is the DSO operator either in the subsurface offset domain or in the angle domain (Shen, 2004). The subsurface-offset-domain DSO focuses the energy at zero offset, whereas the angle-domain DSO flattens the ADCIGs.

In the wave-equation migration velocity analysis (WEMVA) method (Sava, 2004), the focusing operator is the linearized residual migration operator defined as follows:

$${\bf F}({\bf I}) = {\bf R}[\rho]{\bf I} \approx {\bf I} + {\bf K}[\Delta \rho]{\bf I},$$ (A-3)

where ${\bf\rho}$ is the ratio between the background slowness $\widehat{\bf s}$ and the true slowness ${\bf s}$, and $\Delta \rho = 1-\rho = 1-\frac{\widehat{\bf s}}{\bf s}$; ${\bf R}[\rho]$ is the residual migration operator (Sava, 2003), and ${\bf K}[\Delta \rho]$ is the differential residual migration operator defined as follows (Sava and Biondi, 2004a,b):

$${\bf K}[\Delta \rho] = \Delta \rho \left. \frac{\partial {\bf R}[\rho]}{\partial \rho} \right\vert _{\rho=1}.$$ (A-4)

The linear operator ${\bf K}[\Delta \rho]$ applies different phase rotations to the image for different reflection angles and geological dips (Biondi, 2008).

In general, if we choose $\ell_2$ norm, the tomography objective function to minimize can be written as follows:

$$J = \frac{1}{2}\vert\vert{\bf\Delta I}\vert\vert _2 = \frac{1}{2}\vert\vert{\bf I}-{\bf F}({\bf I})\vert\vert _2,$$ (A-5)

where $\vert\vert\cdot\vert\vert _2$ stands for the $\ell_2$ norm. Gradient-based optimization techniques such as the quasi-Newton method and the conjugate gradient method can be used to minimize the objective function $J$. The gradient of $J$ with respect to the slowness ${\bf s}$ reads as follows:

$${\bf\nabla} J = \Re \left( \left(\frac{\partial {\bf I}}{\partial... ...rtial {\bf s}} \right)^{\prime} \left( {\bf I}-{\bf F}({\bf I})\right) \right),$$ (A-6)

where $\Re$ denotes taking the real part of a complex value and $^{\prime}$ denotes the adjoint. For the DSO method, the linear operator ${\bf O}$ is independent of the slowness, so we have

$$\frac{\partial {\bf F}({\bf I})}{\partial {\bf s}} = ({\bf 1}-{\bf O})\frac{\partial {\bf I}}{\partial {\bf s}}.$$ (A-7)

Substituting Equations 2 and 7 into Equation 6 and evaluating the gradient at a background slowness yields

$${\bf\nabla} J_{\rm DSO} = \Re \left( \left.\left(\frac{\partial {... ...=\widehat{\bf s}}\right)^{\prime}{\bf O}^{\prime}{\bf O}\widehat{\bf I}\right),$$ (A-8)

where $\widehat {\bf I}$ is the background image computed using the background slowness $\widehat{\bf s}$.

For the WEMVA method, the gradient is slightly more complicated, because in this case, the focusing operator is also dependent on the slowness ${\bf s}$. However, one can simplify it by assuming that the focusing operator is applied on the background image $\widehat {\bf I}$ instead of ${\bf I}$, and $\widehat{\Delta \rho}$ is also picked from the background image $\widehat {\bf I}$, that is

$${\bf F}(\widehat{\bf I})=\widehat{\bf I}+{\bf K}[\widehat{\Delta \rho}]\widehat{\bf I}.$$ (A-9)

With these assumptions, we get the "classic" WEMVA gradient as follows:

$$\nabla J_{\rm WEMVA} = \Re \left( - \left.\left(\frac{\partial {\... ...t{\bf s}}\right)^{\prime}{\bf K}[\widehat{\Delta \rho}]\widehat{\bf I} \right).$$ (A-10)

The complete WEMVA gradient without the above assumptions can also be derived following the method described by Biondi (2008).

No matter which gradient we choose to back-project the slowness perturbation, we have to evaluate the adjoint of the linear operator $\left.\frac{\partial{\bf I}}{\partial{\bf s}}\right\vert _{{\bf s}=\widehat{\bf s}}$, which defines a linear mapping from the slowness perturbation ${\bf\Delta s}$ to the image perturbation ${\bf\Delta I}$. This is easy to see by expanding the image ${\bf I}$ around the background slowness ${\widehat {\bf s}}$ as follows:

$${\bf I} = {\widehat {\bf I}} + \left. \frac{\partial{\bf I}}{{\pa... ...}}}\right\vert _{{\bf s}=\widehat{\bf s}}({\bf s}-{\widehat {\bf s}}) + \cdots.$$ (A-11)

Keeping only the zero and first order terms, we get the linear operator $\left.\frac{\partial{\bf I}}{\partial{\bf s}}\right\vert _{{\bf s}=\widehat{\bf s}}$ as follows:

$${\bf\Delta I} = \left. \frac{\partial{\bf I}}{\partial{\bf s}}\right\vert _{{\bf s}=\widehat{\bf s}} {\bf\Delta s} = {\bf T}{\bf\Delta s},$$ (A-12)

where ${\bf\Delta I} = {\bf I} - {\widehat {\bf I}}$ and ${\bf\Delta s} = {\bf s} - {\widehat {\bf s}}$. ${\bf T} = \left. \frac{\partial{\bf I}}{\partial{\bf s}}\right\vert _{{\bf s}=\widehat{\bf s}}$ is the wave-equation tomographic operator. The tomographic operator can be evaluated either in the source and receiver domain (Sava, 2004) or in the shot-profile domain (Shen, 2004). In next section we follow an approach similar to that discussed by Shen (2004) and review the forward and adjoint tomographic operator in the shot-profile domain. In the subsequent sections, we generalize the expression of the tomographic operator to the generalized source domain.

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