# Seismic interferometry

Explanation of seismic interferometry from Curtis et al (2006)

Seismic interferometry is a method used to cancel near-subsurface noise in seismic data by cross-correlating wavefields between receivers that are activated from a source either passive or active, taking the time component into account. As a mathematical simplification, seismic interferometry assumes that the data processing correlates two horizontal boundaries: one at depth and one generally taken to be the ground surface. [1] Seismic interferometry has also been used in construction to measure building response to a seismic event. [2].

## History [3]

Prior to the 1960s, seismometers were only capable of extracting information for deep within the subsurface. Seismic data in the first few meters beneath the surface had a tendency to have too much noise to correctly interpret the structures. In 1968, Clearbout first proposed the notion of “daylight imaging” where we used all of the nearby light sources to create an image of the world around us. In 2001, Lobkis and Weaver introduced Green’s function, a mathematical concept capable of extracting the 3-dimensional signals for the near-subsurface.

## How it is done

When a seismic data is first obtained, it is based upon a time-domain function. While useful, this data does not permit by itself a comparison with other seismic sources. To achieve this, the seismic data undergoes a Fourier transform, (the Green’s function). [1]
Seismic interferometry is done by using Green’s function to identify sources from individual receivers. Green’s function refers to theoretical signals that would be received if the source was at the location of the other receiver of interest[4]:

G*im(rA,rB)-Gim(rA,rB)= {|${\displaystyle \int _{b}}$|}{Gin(rA,r)njcnjkiδkG*ml(rB,r)-njcnjklδkGil(rA,r)G*mn(rB,r)}dS

• Gim(rA,rB) = Green's function for ith component with m direction
• rA = impulse force
• rB = monopole source
• k= direction
• cnjki = elastic tensor
• * = complex conjugation
• nj = outward normal
• S = surface
• njcnjkiδkG*ml(rB,r) = particle displacement
• r = dipole source

If the earth is horizontal, then the function can be simplified as[4]:

njcnjklδkFml(rA,rB)=Σvpvm(ZA,φ)Tnv*(ZB,φ)(e(i(kvX+(π)/(4))/(sqrt(((π)/(2))kX))

Example of seismic interferometry from source and receiver (Halliday et al, 2008)

where ZA and ZB are depths for rA and rB, Tnv = nth component of the traction vector. φ = azimuth of horizontal path.

Green’s function takes into account the source-type and the numerical equations change depending on whether the source is active or passive. Seismic interferometry is best done by first isolate parts of the seismic data with confidence and then applying the seismic interferometry. Correct isolation of surface modes is done with using bandpass filters, phase matched filtering, or mode-branch stripping, although the most efficient method is by frequency-wavenumber analysis.

Combining the Green’s function from the different receivers then permit to isolate the different signals[4]:

G*im(rA,rB)-Gim(rA,rB)=
 ${\displaystyle \sum _{v,v'}\int _{S}{\frac {e^{i(k_{v}X_{A}-k_{v'}X_{B})}}{{\frac {\pi }{2}}{\sqrt {k_{v}k_{v'}X_{A}X_{B}}}}}p_{i}^{v}(Z_{A},\phi _{A})p_{m}^{v'}(Z_{B},\phi _{B})*p_{n}^{v*}(Z,\phi _{A})T_{n}^{v'}(Z,\phi _{B})-T_{n}^{v*}(Z,\phi _{A})p_{n}^{v'}(z,\phi _{B})]dS)}$ ()

If we then include signals at depth, then the equation modifies to:

 ${\displaystyle G_{im}^{*}(r_{A},r_{B})=ik^{v}\int _{S}M^{v}(\omega )G_{in}(r_{B},r)G_{m}^{*}n(r_{A},r)dS}$ ()

where the scale factor when depth is not known is:

 ${\displaystyle M^{v}(\omega )=2n_{j}c^{v}U^{v}(\rho )}$ ()

ρ = near-surface density, ω= frequency, v= surface wave mode, U = complex vector[5].

If instead there are no source signals at depth, then the equation needs to include an additional term based on x-and y-coordinates[4]:

 ${\displaystyle G_{im}^{estimated}(r_{A},r_{B})=ik_{v}A^{v}(\omega )\int _{y1}^{y2}\int _{x1}^{x2}G_{i}n(r_{B},r)G_{m}^{*}n(r_{A},r)dxdy}$ ()

Seismic interferometry from two sources at different depths. (Halliday et al., 2008)

Based on signals collected up to today, the best resolution is found perpendicular to the sources of noise.

## Active Sources

Active sources are used primarily in exploration seismology. Active sources are easier to monitor as they original from a single source and can thus be used to cancel out much of the general subsurface noise[4].
Examples of active sources include[2]:

• Man-made explosions- used to "determine the temporal change in the medium from these waves."
• Earthquakes: dependent upon the lithology where the earthquake occurs.

## Passive Sources

In passive sources, signals are set off randomly and so a correct time estimation between each source is required[4].

## Limitations

Correct applications of seismic interferometry relies on the assumption that the subsurface is homogeneous and noises are spread out equally in all directions. This assumption, however, is not necessary if we are able to isolate different portions of the seismic data with confidence. Interferometry also relies on the knowledge of how seismic velocity change with depth[4]. For a more precise seismic interpretation, the lithology passed by the seismic waves have to be non-attenuating[2].