Seismic reservoir monitoring with encoded permanent seismic arrays

Data recording and imaging

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

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

where $\omega$ is frequency, ${m({\bf x})}$ is the reflectivity at image points ${\bf x}$ , $f_{s}(\omega)$ is the source wavelet, and $G ({\bf x_{s}, x,\omega})$ and $G ({\bf x, x_{r},\omega})$ are the GreenŐ's functions from ${\bf x_{s}}$ to ${\bf x}$ and from ${\bf x}$ to ${\bf x_{r}}$ , respectively. When there are multiple seismic sources, the recorded seismic data is due to a concatenation of phase-shifted sources. For example, the recorded data due to shots starting from ${\bf s=q}$ to ${\bf s=p}$ , is given by

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

where $a(\gamma_{\bf s})$ is given by

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

and $\gamma_{s}$ , the phase-shift function, depends on the delay time $t_{s}$ .

Randomized intermittent shooting of several shots is equivalent to repetition of equation 2 with a spatially and temporally varying encoding function. The recorded data are similar to passive data, except that all shot positions and timings are known. Therefore, the recording experiment can be regarded as a controlled-source continuous-recording experiment. Figures 1 and 2 show examples of idealized source waveforms at six shot positions. It is assumed that these sources are orders of magnitude weaker than conventional seismic sources and that data from a single sweep is insufficient to create a good-quality image.

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Figure 1. First $150$ seconds of the $320$ seconds long source waveforms at six source positions. Note that the relative delays between intermittent sweeps from different sources are determined by the encoding function in equation 3.

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Figure 2. First $12$ seconds of the source waveforms shown in Figure 1. Note that there are temporal and spatial differences in the starting times of the seismic sweep at all shot positions.

Direct imaging of the recorded data, the adjoint to equation 2, is a linear phase-encoding migration operator (Romero et al., 2000). This is equivalent to a single shot-profile migration of the recorded data with an areal source-function derived by concatenating delayed source waveforms from all shot positions.

Seismic reservoir monitoring with encoded permanent seismic arrays