Continuous case and seismic imaging

Of course, the linear theory is not limited to discrete grids. It is interesting to consider the continuous case because of its connection to the linear integral operators commonly used in seismic imaging. Indeed, in the continuous case, linear decomposition () takes the form of the integral operator  

$$f (y) = \int m (x) G (y; x) d x \;,$$ (50)

where x is a continuous analog of the discrete coefficient k in (), the continuous function m (x) is analogous to the coefficient c_k, and G (y; x) is analogous to one of the basis functions $\psi_k (x)$. The linear integral operator in () has a mathematical form similar to the form of well-known integral imaging operators, such as Kirchhoff migration or ``Kirchhoff'' DMO. Function G (y; x) in this case represents the Green's function (impulse response) of the imaging operator. Linear decomposition of the data into basis functions means decomposing it into the combination of impulse responses (``hyperbolas'').

In the continuous case, equation () transforms to  

$$W (y, n) = \int\!\!\int \Psi^{-1} (x_1, x_2) G (y;x_1) \bar{G} (n;x_2) dx_1\,dx_2\;,$$ (51)

where $\Psi^{-1} (x_1, x_2)$ refers to the inverse of the ``matrix'' operator  

$$\Psi (x_1, x_2) = \int G (y;x_1) \bar{G} (y;x_2) dy\;.$$ (52)

When the linear operator, defined by equation (), is unitary ,  

$$\Psi^{-1} (x_1, x_2) = \delta (x_1 - x_2)\;,$$ (53)

and equation () simplifies to the single integral  

$$W (y, n) = \int G (y;x) \bar{G} (n;x) dx \;.$$ (54)

With respect to seismic imaging operators, one can recognize in the interpolation operator () the generic form of azimuth moveout Biondi et al. (1996), which is derived either as a cascade of adjoint ($\bar{G}(n;y)$) and forward (G (x;y)) DMO or as a cascade of migration ($\bar{G}(n;y)$) and modeling (G (x;y)) Fomel and Biondi (1995a,b). In the first case, the intermediate variable y corresponds to the space of zero-offset data cube. In the second case, it corresponds to a point in the subsurface.