Low-frequency reflection-data augmentation by an inpainting method: 1D acoustic media

The augmentation of low frequencies is an important problem for inverting seismic reflection data. Through an acoustic-scattering approach, we have characterized how the data down to zero frequency grows in importance with increasing depth and the size of the velocity contrast. We have developed a Hermite distributed approximation functional (HDAF) inpainting method that provides a least-squares fitting of the acoustic-reflection data down to zero frequency. The HDAF data fitting is applied to the first Born approximation for compact velocity contrasts (the top and bottom depth velocities are equal). Several fitting constraints are derived using a Taylor expansion of the reflection data near zero frequency. They provide a coupling with the first Born approximation of the velocity contrast. The method can be extended to the general geophysical case, in which the final velocity i.e., is different from the starting velocity (by use of the distorted-wave Born approximation). Numerical results for 1D test cases with synthetic data showed that, if the shortest wavelength for reflection data is larger than the range of missing wavelengths, then the coupled fitting reconstructs well the full low-wavenumber data down to the zero wavenumber. Thus, it allows the first Born approximation to produce satisfactory results for data that are missing the low-frequency range. The method can be extended to higher dimensional problems and is a viable technique to augment seismic data in the low-frequency range.