Phase encoding with Gold codes for wave-equation migration

Conclusion

We showed that the use of Gold codes in phase encoding can virtually eliminate crosstalk if the codes are satisfactorily selected. Its advantage over conventional random codes is not only in the image quality but also cost. The elimination of the crosstalk can be achieved with only one realization of Gold codes, while for conventional random codes more realizations are needed to obtain reasonable images, but still of lower quality when compared to the ones obtained by using Gold codes. The need for more realizations increases the cost of modeling and migration.

We also showed that the phase of the cross-correlation of Gold codes is given by the lag in time where the cross-correlation peaks multiplied by frequency. This amounts to apply linear phase encoding to the encoded wavefields.

We used a simple example with just one reflector to illustrate the encoding techniques. The performance of the Gold codes in the presence of more than one reflector needs to be tested. However, as just one realization is sufficient to drastically reduce the crosstalk, they can be affordable to use in a horizon-based approach to the modeling in which each reflector is modeled separately. Additional research is needed to analyze the performance of Gold codes in more complex situations, where the depth shift of the crosstalk may not be a simple function of velocity.

Phase encoding with Gold codes for wave-equation migration