Differential image perturbation

Various methods can be used to improve images created with inaccurate, reference velocity models. Residual migration Al-Yahya (1989); Etgen (1990); Stolt (1996) is one such option, although we could use other methods like residual moveout or image continuation.

If image enhancement is done with a Stolt-type residual migration operator ${\bf S}$Sava (2000, 2003); Stolt (1996), we can write a relation for an improved image $\mathcal R$ derived from a reference image $\mathcal R_o$

$$\mathcal R= {\bf S}\left(\r \right)\left[\mathcal R_o\right]\;,$$ (18)

where $\r$ is a spatially varying scalar parameter indicating the magnitude of residual migration at every image point. We can compute a linearized image perturbation by a simple first-order expansion relative to the parameter $\r$

$$\Delta \mathcal R_a= \left. \frac{d{\bf S}}{d\r} \right\vert _{\r=\r_o} \left[\mathcal R_o\right]\; \Delta \r\;,$$ (19)

from which we can compute a wavefield perturbation $\Delta \mathcal U_a$ using the adjoint of the imaging operator.

The operator $\left. \frac{d{\bf S}}{d\r} \right\vert _{\r=\r_o}$can be computed analytically, since it only depends on the background image, while $\Delta \r$ can be picked at every location from a suite of images computed using different values of $\r$ Sava and Biondi (2003). Similar formulations are possible for other kinds of operators (e.g., normal residual moveout), and are not restricted to residual migration, in general, or to Stolt residual migration, in particular.

With this definition of the wavefield perturbation, we can compute another slowness perturbation:  

$$\Delta s_a= {\bf B}^* \left(\mathcal U_o\right)\left[\Delta \mathcal U_a\right]\;.$$ (20)