3. 1.1 Seismic Interferometry
Optical interferometry

7. Mathematical Description

8. Assuming an impulse source δ(t),
Receiver Source
location location
The exact wavefield excited by an impulsive point source at B with initiation time t
and an observer at A is described by the Green’s function g(A, tB│B, ts).
Fourier transform

9. Shifting the seismograms by 𝜏 𝐴𝑦𝐴 is equivalent to multiplying the spectrum D (A │ A) by 𝑒 −𝑖𝜔𝜏𝐴𝑦𝐴 .
This means that |D(A|A)’|2 will be sensitive to any irregularities in the shape of the sand lens. Moreover, this spectral interferogram is kinematically equivalent to one recorded with source and receivers redatumed to the top of the sand lens.

11. ` ` `

13. Time Shift  Trace Correlation
Equivalent to the squared spectrum of the shifted trace
Autocorrelation of the trace is equivalent to removing the distorted effects of the overburden & redatuming the source and receivers to be just above the target body.
The redatumed data summed over N buried sources with depths denoted by zA give
Source position
A = (xA, yA, zA)
 F-1
Receiver position
B = (xA, yA, 0)

14. 1.1.1 Multidimensional seismic interferometry
Good approximation!
Exact: the reciprocity equation
Green’s theorem!

15. 1.1.2 Generalization of seismic interferometry
Interferometric Image
Velocity Model
4.2 km
13 km
13 km

19. 1.2 Benefits and liabilities
Better image resolution of the target due to avoidance of distorting
effects of the uninteresting parts of the medium
2. No velocity model or statistics are needed for redatuming
3. Enlarging the aperture of the recording array if many scatterers
exist in the medium  super-resolution image
4. Enhancement of S/N ratios by super-stacking
5. Others including surface-wave inversion of earthquake data,
surface-wave prediction and elimination, trace interpolation and
extrapolation, and super-illumination of the subsurface

21. Key Problems:
The assumption for a wide aperture of source and receivers may not be realizable,
which leads incomplete cancellation of coherent noise
 filtering out unwanted events prior to cross-correlation and deconvolution of
source wavelet
2. Low S/N ratios due to severer intrinsic attenuation

22. 1.3 Historical development of seismic interferometry
Claebout (1968) showed how the Green’s function (i.e., impulse response of a point source) on the earth’s surface could be obtained by autocorrelating traces generated by buried sources
Claebout and his students postulated that correlation redatuming could be extended to multidimensional models  named as ‘the daylight imaging method’  later renamed as ‘seismic interferometry’ by Schuster (2001)
Claebout conjectured that cross-correlating traces generated by a random distribution of deeply buried sources but recorded on the earth’s surface could produce virtual SSP (surface seismic profile) data
Solar physicists independently discovered a procedure using the cross-correlation of solar vibration data to infer information about the Sun’s internal structure (Duvall et al., 1993)
Rickett and Claebout (1999) presented virtual shot gathers obtained by correlating vibration records of the Sun’s surface to empirically demonstrate the validity of Claebout’s conjecture.
Schuster and Rickett (2000) showed by a stationary phase argument that the summation of cross-correlated seismic data followed by migration will lead to imaging of the earth’s reflectivity.
Deterministic interferometry (Schuster, 2001) vs. Diffuse interferometry (Wapenaar et al., 2006)

23. Extended to migration of surface seismic data (Sheng, 2001), migration of converted waves, etc.
Snieder et al. (2002) extracted subsurface information from earthquake coda
Wapenaar et al. (2002) provided a solid mathematical foundation to prove Claebout’s conjecture using Green’s theorem
Earthquake seismologists correlated earthquake records across a large recording array to estimate the shear-wave velocity distribution (Campillo and Paul, 2003; Shaporo et al., 2005)
Calvert and Bakulin (2004) introduced the virtual source method for redatuming VSP data to single well profile (SWP) data.
Seismic interferometry, daylight imaging, virtual source imaging, and reverse time acoustics (Lobkis and Weaver, 2001; Weaver and Lobkis, 2006; Larose et al., 2006; Fink, 2006) are intimately connected by the summation of correlated data and based on the reciprocity theorem.