Correlation Amplitude Levels

In frequency domain the correlation can be written as the multiplication of uncorrelated data \(D_u\) by the complex conjugate of the reference \(S\) : \[D_c(\nu)=‎ D_u(\nu).\bar{S}(\nu)\]

The resulting scaling amplitude of the result is obviously linked to the reference power (length, drive etc…). Several options for the scaling :

Normalize the correlation according to the total energy of the reference used :

\[D_c(\nu) = \frac{D_u(\nu).\bar{S(\nu)}}{\sum{S_n^2}}\]

In this case the gain of the obtained “Earth Response” is independent of the reference : all the signal is at the same level.

Normalize the correlation according to the signal-to-noise ratio :

\[D_c(\nu) = \frac{D_u(\nu).\bar{S(\nu)}}{\sqrt{\sum{S_n^2}}}\]

In this case the gain of the obtained “Earth Response” id scaled according to the reference and all the noise is at the same level.

Hybrid Sercel Unite/428 version :

\[D_c(\nu) = \frac{2.D_u(\nu).\bar{S(\nu)}}{N.\sum{S_n^2}}\]

Here the obtained “Earth Response” is scaled on the amplitude part of the reference but not on its length. The sweep length has no impact on the output gain.