An Hermite Distributed Approximation Functional fitting method to augment reflection data down to zero frequency

We have developed a method to augment reflection data in the missing low-frequency range for acoustic Inverse Scattering Series. It relies on a least square fitting using Hermite Distributed Approximation Functionals (HDAF). This particular choice is based on the property that DAF has the ability to deliver a smooth approximation of the fitted data. The HDAF data fitting is applied to acoustic Volterra Inverse Scattering series (VISS) for non-compact velocity contrast potential. It uses a remarkable property of the first order term of the VISS (for compact support velocity profiles) which is that its Fourier transform at zero frequency equals the area under the velocity contrast. This property allows a coupling between the reflection data and its VISS inversion. The main advantage of our method is that it provides a data augmentation down to the zero-frequency. Numerical results on test cases with synthetic data illustrate that provided the data gap is not large compared to the Nyquist wavelength of the known data, the coupled fitting reconstructs well the full low frequency data down to the zero frequency. Thus it allows the VISS to produce satisfactory results missing low-frequency range. Our method represents a solution to augment seismic data in the low-frequency range.