I have read some about Karhunen-Loeve Transform (KLT) and its application to the field of seismic data processing.
The method as I understand it based on decomposing the data (actually mostly used in image processing) using SVD on its covariance matrix and projecting the data back after manipulating one or some of the covariance matrix's eigenvalues.
One of the assumptions, when using on seismic data to remove random noise, is that the features to keep, should be aligned in time. If this is not the case, then one have to correct for the observed move-out.
For ex. in the top picture below 400 time samples (y-axis) for more than 50 data traces(x-axis) are shown. Each trace represents data recorded on one geophone. There are multiple "events" in the data represented by horizontal lines across the figure. There is no time delay for the observed event across the traces. The picture in the bottom is the result after applying KLT on the data, by zeroing the smallest eigenvalue of data's covariance matrix. My question is, why the assumption of time alignment of the "event" (horizontal features)? The method fails if the events occurred along a line with a slope, that is the same event was observed with a delay from one trace to the next.