Joint inversion of simultaneous source time-lapse seismic data sets

Linear phase-encoded Born modeling

Within limits of the Born approximation of the acoustic wave equation, the seismic data ${\bf d}$ recorded by a receiver at ${\bf x_{r}}$ due to a shot at ${\bf x_{s}}$ is given by

$${ d}{(\bf x_{s}, x_{r},\omega})=\omega^{2} \sum_{\bf x}f_{\bf s}(\omega)G ({\bf x_{s}, x,\omega}) G ( {\bf x, x_{r},\omega}) m(\bf x),$$ (1)

where $\omega$ is frequency, ${m(x)}$ is the reflectivity at image points ${\bf x}$, $f_{\bf s}(\omega)$ is the source waveform, and $G ({\bf x_{s}, x,\omega})$ and $G ({\bf x, x_{r},\omega})$ are respectively the GreenŐ's functions from shot ${\bf x_{s}}$ to ${\bf x}$ and from ${\bf x}$ to ${\bf x_{r}}$.

By considering randomized simultaneous source data as a special case of linear phase-encoded shot gathers, equation (1) is modified to include a concatenation of phase-shifted shots, from ${\bf s}={\bf q}$ to ${\bf s}={\bf p}$:

$${d}{(\bf x_{s_{pq}}, x_{r},\omega})=\sum_{\bf s=p}^{\bf q} a(\gam... ...\omega) G ({\bf x}_{\bf s}{\bf , x,\omega}) G ({\bf x, x_{r},\omega}) m(\bf x),$$ (2)

where ${\bf x_{s_{pq}}}$ defines the positions of the encoded sources, and

$$a(\gamma_{\bf s})=e^{i\gamma_{\bf s}}=e^{i \omega t_{\bf s}},$$ (3)

and $\gamma_{\bf s}$, the linear time-delay function, depends on the delay time $t_{\bf s}$ at shot ${\bf s}$.

Relative shot-timing non-repeatability arises due to the uncertainty (Folland and Sitaram, 1997) associated with correct positioning of shots and receivers while maintaining the correct time delays $t_{\bf s}$ between shots. This is particularly true for the blended acquisition geometry (Berkhout, 2008), where several (20 or more) shots are encoded into a single record.

Joint inversion of simultaneous source time-lapse seismic data sets