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
Is the definition of SPAC wrong in Aki (1957)?
Is the definition of SPAC wrong in Aki (1957)?
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Re: Is the definition of SPAC wrong in Aki (1957)?
As far as I can remember the Aki formulation compares the signals with a zerolag 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)?
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 timedomain noise correlation and spatial autocorrelation (SPAC) results"
In equation (21) of this paper, we can find the relationship which you mentioned before.
Paper: Victor C, (2010), GJI, "An explicit relationship between timedomain noise correlation and spatial autocorrelation (SPAC) results"
In equation (21) of this paper, we can find the relationship which you mentioned before.
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