Gradient Vector Definition

Methods for calculating the gradient without explicitly computing the partial derivatives of the data are well established Lailly (1983); Pratt et al. (1998); Pratt and Worthington (1989); Tarantola (1984). The main result in the time-domain inversion literature is that the gradient vector can be computed by a zero-lag correlation of the wavefield propagated forward from the source point, $P_f ({\bf s},{\bf x};\omega)$, and the residual wavefield propagated backwards from the farthest receiver location toward the source point past each successive receiver, $P_b({\bf x},{\bf r}; \omega)$. The frequency-domain equivalent to this zero-lag correlation is the multiplication of these two wavefields according to

$$g ({\bf x}; w) = -\omega^2 \sum_s \sum_r Re \left( P^{*}_f (... ...,{\bf x};\omega) P_b ({\bf s},{\bf x},{\bf r}; \omega) \right),$$ (6)

where Re indicates the real component of the multiplication result. The summation over sources and receivers is done for each non-linear iteration at each frequency. Following Sirgue and Pratt (2004) in assuming a point source of unit amplitude and zero phase, the forward-propagated wavefield P_f is given by

$$P_f ({\bf s},{\bf x};\omega) = G_0({\bf s},{\bf x};\omega),$$ (7)

while back-propagated wavefield P_b is defined by

$$P_b({\bf s},{\bf x},{\bf r};\omega) = G_0^{*}({\bf x},{\bf r};\omega) \Delta \Psi({\bf s},{\bf r};\omega),$$ (8)

where $G_0({\bf s},{\bf x}; \omega)$ and $G_0({\bf x},{\bf r}; \omega)$ represent the monochromatic Green's functions for an excitation at the source and receiver points in the medium, respectively. Hence, the full gradient vector expression is  

$$g ({\bf x}; \omega) = -\omega^2 \sum_s \sum_r Re \left( G_0^... ...,{\bf r}; \omega) \Delta \Psi({\bf s},{\bf r}; \omega) \right).$$ (9)

Note that $G_0^{*}({\bf x},{\bf r}; \omega) \Delta \Psi ({\bf s},{\bf r}; \omega)$ represents the back-projection of the data residuals and is similar to a "migration with the residual wavefield data" Mora (1987).