Signal and traveltime parameter estimation using singular value decomposition

Signal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters. We used singular value decomposition of the data matrix, not eigendecomposition of a covariance matrix, to review the most commonly used coherence measures. This resulted in a new reduced semblance coefficient defined from the first eigenimage, assuming that the signal amplitude was the same on all data channels (as in classical semblance). In a second signal model, the time signal was constant on each channel, but the amplitude changed. Then, the semblance coefficient is the square of the first singular value divided by the energy of the data. Two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as a coherence measure: The normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one, and the normalized crosscorrelation of the temporal singular vector and the average time signal (the stacked trace). We defined a multiple signal classification (MUSIC) measure as the inverse of one minus any of the normalized coherence measures described above. To reduce the numerical range, we preferred to use $log10$ MUSIC. Numerical examples with different coherence measures applied to seismic velocity analysis of synthetic and real data revealed that the normalized crosscorrelation coefficients performed poorly and that log MUSIC gave no resolution enhancement on real data. The normalized eigenimage-energy coherence measure performed poorly on synthetic data but gave the best result for a simulated reflection with a polarity reversal. It also gave good time resolution on the real data. The classical semblance coefficient and the reduced semblance coefficient gave similar results with the reduced semblance coefficient having better resolution.