Target-oriented joint inversion of incomplete time-lapse seismic data sets

Linear least-squares modeling/inversion

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

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

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

Taking the true recorded data at ${\bf x_{r}}$ to be ${d}^{t}$, the quadratic cost function is given by

$$\begin{array}{cc} S({\bf m})=\Vert {d}^{s}{(\bf x_{s}, x_{r},... ... {d}^{t}{(\bf x_{s}, x_{r}, \omega}) \Vert^2_{2}. \end{array}$$ (A-2)

As shown by previous authors (Valenciano, 2008; Plessix and Mulder, 2004), the gradient ${g(x)}$ of this cost function (summed over all frequencies, sources and receivers) with respect to reflectivity is the real part of

$${ g({\bf x})}=\sum_{w}\omega^{2} \sum_{\bf x_{s}} \sum_{\bf x_{r}... ... x_{s}, x,\omega}) G ({\bf x, x_{r},\omega}) \left ( {d}^{s} - {d}^{t} \right),$$ (A-3)

and the Hessian (second derivatives) is the real part of

$${H} \left ({\bf x}, {\bf x'} \right ) =\sum_{w}\omega^{4} \sum_{\... ... \sum_{\bf x_{r}} G ({\bf x, x_{r}, \omega}) \bar G ({\bf x', x_{r}, \omega}) ,$$ (A-4)

where ${\bf x'}$ denotes all image points and $\bar G$ is the complex conjugate of $G$. Plessix and Mulder (2004) and Valenciano (2008) discuss this derivation in detail.

Target-oriented joint inversion of incomplete time-lapse seismic data sets