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Future Work

As mentioned previously, the nature of the nonlinear problem strongly affects our results. Thus, a good initial guess is needed to obtain a better sparse reflectivity. In most cases, data will resemble the Ricker wavelet, as is true for the band-limited marine seismic data with ghosts and the for the land response of an accelerometer. For this situation, we can use the Ricker wavelet to approximate the data and derive the initial filter from this wavelet. Since the Ricker wavelet vanishes at zero frequency and at the Nyquist frequency, it has no stable inverse. Therefore, we use the approximated Ricker wavelet instead of the true one.

Another potential solution is to do the preconditioning, which utilizes prior information. In this non-linear problem, we hope it can guide the gradient along sensible pathways thus avoiding potential local minima.