Inverse Spectral Decomposition

This paper introduces a method which spectrally decomposes a seismic trace by solving an inverse problem. In our technique, the reverse wavelet transform with a library of complex wavelets serves as a forward operator. The inversion reconstructs the wavelet coefficients that represent the seismic trace and satisfy an additional constraint. The constraint is needed as the inverse problem is non-unique. We show synthetic and real examples with three different types of constraints: 1) minimum L2 norm, 2) minimum L1 norm, and 3) sparse spike, or minimum support constraint. The sparse-spike constraint has the best temporal and frequency resolution. While the inverse approach to spectral decomposition is slow compared to other techniques, it produces solutions with better time inversion.