My PhD Thesis, PDF 3MB - Stanford University

Here, the index i represents the ith ray and j the jth i
pixel of the medium; l j is the ray
length within the jth pixel. In practice we can measure f R from recorded seismograms,
2
but may not directly measure the source centroid frequency fS and the variance . For
s
the constant Q-model described by equation (4.2) and Gaussian spectrum given by
equation (4.9) the source spectrum S(f) and receiver spectrum R(f) exhibit the same
2
2
variance . Therefore, we may choose the average of variances at the receivers as
s
R
2
the estimate of the source variance . While the source spectral frequency f is also
s
S
unknown, I include it along with the matrix of unknown attenuation values. Then, I
i
simultaneously invert for both the attenuation coefficients j
f S as follows. Let
i
where f = max{ f S R
- 98 -
and the source frequency
f S f S f , (4.16)
} is an initial estimation of fs, and f is a static correction. Then
i
f f S R
2 f S f f i
R
2 f S f i
R
2
s
Equation (4.15) can now be written as
i
where j
i i
l j j
j
f
s
s
2 f S f R
s
s
and f are the unknowns to be determined.
i
f
2
s
. (4.17)
2 , (4.18)
In order to obtain more stable and precise measurement of the centroid frequency,
we need do some data processing. The main purpose of data processing is to extract the
direct wave and reduce the interference due to scattering. To do this I first pick and align