A better way for calculating the approximation of the zero-offset trace is to calculate the spatial arithmetic mean of a low-rank approximated data matrix using principal component analysis (PCA). PCA is an important tool for multivariate analysis in statistics. The idea is to reduce the dimensionality of a data set while preserving as much variability of data variables as possible (Jolliffe, 2010).

Suppose the data matrix is composed of signal component , random noise , erratic noise , and mis-aligned data components :

For seismic stacking in this paper, is simply a common midpoint gather. If we assume the error components are composed of small random perturbations, an optimal estimate of can be acquired via the following optimization problem:

where denotes the rank constraint applied to the target signal components. The problem can be efficiently solved via singular value decomposition (SVD). The observed data matrix can be decomposed into a group of eigen-images via the SVD. The low-rank component can be described with a few eigen-images that are associated with the largest singular values. The other noise items , , , however, will have energy spread over all the eigen-images.

2020-07-18