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Is the definition of SPAC wrong in Aki (1957)?

Posted: Sat Apr 14, 2012 8:36 am
by Luosong
As we know, The autocorrelation function is defined in this way (Aki, 1957),
(see attachment - Aki)
but I think may defined in this way is wrong, it should be the following expression,
(see attachment - Me)
where the bar stands for mean with respect to the time variable s.
Cho (2008) defined the autocorrelation function in the following way:
(see attachment - Cho)
I thought the definition by Cho is reasonable, and the one defined by Aki (1957) may
do some modifications (see attachment - Me)

Is what I tell right?

Thanks for all,

Luosong

Re: Is the definition of SPAC wrong in Aki (1957)?

Posted: Wed Apr 18, 2012 4:27 pm
by admin
As far as I can remember the Aki formulation compares the signals with a zero-lag correlation. This is the way it is implemented in geopsy. It is certainly not wrong, but maybe not optimum? Current implementation has been tested with success on many datasets including synthetic ones where the true phase velocity is not disputable. If you can suggest papers that discuss this topic in details, they are pretty welcome.

Re: Is the definition of SPAC wrong in Aki (1957)?

Posted: Mon Apr 30, 2012 12:02 pm
by Luosong
I should say what you mentioned above is right, I find a paper which mentioned about this topic.
Paper: Victor C, (2010), GJI, "An explicit relationship between time-domain noise correlation and spatial autocorrelation (SPAC) results"
In equation (21) of this paper, we can find the relationship which you mentioned before.