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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.

Thanks

enter image description here

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There is something that is not clear of what you have done with the data, and that is who do you form the random vectors to perform de SVD (or EVD) on the covariance matrix.

1 -The KLT can be succesfully used on a one dimensional signal (only one Geophone), taking frames of $M$ samples and estimating a covariance matrix from it, and the performing Eigenvalue decomposition, or with the SVD of the Data matrix (which supposedly is the same), then you sort the eigenvalues (or singular values) in descending order, and remove the smallest ones according to a particular criteria. With this you will remove part of the noise. Based on the process i described, there is no assumption about any correlation between the signaals on the geophones (and in reality i think there will be, but i am not an expert in geology), so if you perform the same processing to every signal, it doesnt matter what delay you have between particular events, the process will still remove noise.

2 - But, if you construct your random vector with one sample from each geophone taken at the same time instant, then the time delay might affect the correlation between each sample from a geophone and then applying KLT might not reduce the noise as it is expected.

The approach number 1 i described is called Temporal PCD (Principal component analysis) and the second one is called Spatial PCA, assuming you put the geophones in different locations.

Personally i think it should not fail if you do spatial PCA, because even though there will be time delay, the samples from different geophones at same time instants will have some degree of correlation, except if you put them hundreds of miles apart, but if they are all in the same region, then they will each sense approximately the same signal but contaminated with noise and probably convoluted with an impulse response of the geographical region, but as long as there is correlation, you can apply KLT.

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  • $\begingroup$ Thanks. in the example above the geophones are located on a straight line 50 meters apart. So, the entire spatial aperture is a linear array of geophones of about 2.5 km long. I think as you say, the premise is the correlation between data from different geophnes. If there is no correlation in the signals observed on the geophones, then it doesn't work. In the above, The signals are incident on the geophones vertically and from far field, so there is no observed delay and as such high correlation across the array. $\endgroup$ – user1641496 Mar 19 '14 at 7:05
  • $\begingroup$ Ohh, that is really unfortunate, to construct a 2.5km long microphone array, only to have the wave front impact perpendicularly to the line of the array. If the wave would come at an angle, then you could use beamforming, which doesn't even need KLT (but can still be used) $\endgroup$ – bone Mar 19 '14 at 11:15
  • $\begingroup$ Well, the above example is synthetic. Just to illustrate why KLT works on signals with no delays across the geophones. On the other hand if the wave arrives at an angle, one shifts the data traces using cross-correlation analysis, so that the signals are aligned horizontally, remove the random noise by means of KLT and finally shifts back the traces again. $\endgroup$ – user1641496 Mar 19 '14 at 13:10

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