Maximum Likelihood Stacking in White Gaussian Noise with Unknown Variances

In seismic exploration, the process of estimating the source produced signal common to a collection of sampled data recordings contaminated with additive noise is known as stacking. Stacking is used when a low energy acoustic source can be repeated in the same physical location (or nearly so) and a collection of data recordings are gathered at the same electromechanical transducer from the multiple source initiations. This stacking process is primarily aimed at reducing environmental noise which frequently dominates each individual recording when low energy sources are used. This is in contrast to other techniques used to reduce or eliminate so called source generated noise which behaves as signal in the stacking process. Specifically, the problem we address is the simultaneous maximum likelihood (ML) estimation of signal values and unknown noise distribution parameters from a collection of sampled data recordings, each containing the same signal sequence. In this paper we restrict our consideration to independent, zero mean, Gaussian noise variates with unknown variance which may differ between data records. This model is justified when a broadband analog noise source is subsequently low pass filtered along with the signal and the result sampled at the Nyquist rate. The linear filtering tends to produce a Gaussian noise process and the Nyquist rate sampling produces approximately uncorrelated samples. The ML estimates of signal values and noise variances are obtained numerically by steepest ascent.