Conclusion

I presented the general theory of super-virtual diffraction interferometry where the signal-to-noise ratio (SNR) of diffraction arrivals can be theoretically increased by the factor $\sqrt {N}$ , where $N$ is the number of receiver and source positions associated with the recording of the diffractions. There are two steps to this methodology: correlation and summation of the data to generate traces with virtual diffraction arrivals, followed by the convolution and stacking of the data with the virtual traces to create super-virtual diffractions. This method is valid for any medium that generates diffraction arrivals due to isolated subwavelength scatterers. There are at least three benefits with this methodology: 1). the diffraction arrivals can be used as migration operators (Schuster, 2002; Brandsberg-Dahl et al., 2007; Sinha et al., 2009); 2). the diffraction arrivals can be used for estimating source and receiver statics; 3). estimation of velocities by traveltime tomography or MVA.

The problem with this method is that there will be artifacts associated with coherent events and quality degradation due to a limited recording aperture and a coarse spacing of the source and receivers.

Figure: The steps for creating super-virtual diffraction arrivals. (a) Correlation of the recorded trace at with that at for a source at to give the correlated trace with the virtual diffraction having traveltime denoted by $\tau _{A'B}-\tau _{A'A}$ . This arrival time will be the same for all source positions $x$ , so stacking will enhance the SNR of the virtual diffraction by $\sqrt {N}$ . (b) Similar to that in (a) except the virtual diffraction traces are convolved with the actual diffraction traces and stacked for different geophone positions $x'$ to give the (c) super-virtual trace with an enhanced SNR. Here, $N$ denotes the number of coincident source and receiver positions.

Figure 4.2: (a) Geometry for computing virtual Green's functions $G({\bf B}\vert{\bf A})$ from the recorded data $G({\bf A}\vert{\bf x})$ and $G({\bf B}\vert{\bf x})$ using the reciprocity theorem of correlation type in an arbitrary acoustic medium of constant density. (b) Geometry for computing super-virtual Green's functions $G({\bf B}\vert{\bf A})^{super}$ from the recorded data $G({\bf A}\vert{\bf x'})$ and the virtual data $G({\bf B}\vert{\bf x'})^{virt.}$ using the reciprocity theorem of convolution type.

Figure 4.3: Part of the BP2004 velocity model with three diffractors below the salt body.

Figure 4.4: Synthetic data results for part of the BP2004 model. (a) A common shot gather with a source at offset 6 km. Red lines indicate the time window and the moveout of the diffraction event. (b) The diffraction event within a small time window. (c) The result after median filtering and (d) after processing the median filtered data to get the super-virtual diffraction.

Figure 4.5: Velocity model with a fault and two diffractors.

Figure 4.6: Synthetic data results for the fault model. (a) A common shot gather with a source at offset 36 m. Red lines indicate the time window of the diffraction event. (b) The diffraction event within a small time window. (c) The result after median filtering and (d) after processing the raw data to get the super-virtual diffraction.

Figure 4.7: The super-virtual diffraction. In this figure, the red line indicates the predicted diffraction arrival times and the blue line indicates the picked arrival times.

Figure 4.8: Friendswood cross-well data example. (a) A common shot gather with a source at depth of 36.6 m. Red lines indicate the time window and the moveout of the diffraction event. (b) The diffraction event within a small time window. (c) The result after median filtering and (d) the super-virtual diffraction.