Viterbi sparse spike detection

Accurate interpretation of seismic traveltimes and amplitudes in the exploration and global scales is complicated by the band-limited nature of seismic data. We discovered a stochastic method to reduce a seismic waveform into a most probable constituent spike train. Model waveforms were constructed from a set of candidate spike trains convolved with a source wavelet estimate. For each model waveform, a profile hidden Markov model (HMM) was constructed to represent the waveform as a stochastic generative model with a linear topology corresponding to a sequence of samples. Each match state in the HMM represented a sample in the model waveform, in which the amplitude was represented by a Gaussian distribution. Insert and delete states allowed the underlying source wavelet to dilate or contract, accounting for nonstationarity in the seismic data and errors in the source wavelet estimate. The Gaussian distribution characterizing each sample’s amplitude accounted for random noise. The Viterbi algorithm was employed to simultaneously find the optimal nonlinear alignment between a model waveform and the seismic data and to assign a score to each candidate spike train. The most probable traveltimes and amplitudes were inferred from the alignments of the highest scoring models. The method required no implicit assumptions regarding the distribution of traveltimes and amplitudes; however, in practice, the solution set may be limited to mitigate the nonuniqueness of solutions and to reduce the computational effort. Our analyses found that the method can resolve closely spaced arrivals below traditional resolution limits and that traveltime estimates are robust in the presence of random noise and source wavelet errors. The method was particularly well suited to fine-scale interpretation problems such as thin bed interpretation, tying seismic images to well logs, and the analysis of anomalous waveforms in global seismology.