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In physics, the Born approximation allows us to separate the wave-function in terms of electronic and nuclear components.
Why is this concept relevant in seismic inversion? For example in Colombi et al. (2014), Geophys. J. Int. (2014), doi 10.1093/gji/ggu112 .
The best reference on this is:
J.A. Hudson and J.R. Heritage, 1981, The use of the Born approximation in seismic scattering problems, Geophys. J. R. Astr. Soc., v.66, 221-240.
Basically, the Born approximation is a governing principle of the standard imaging condition that treats seismic wavefields from a single point scatterer. This is the justification for simplifying the full elastic wave equation into the scalar and vector potential wavefields and just working with the scalar wave equation for the purposes of building seismic images.
There is a very nice derivation of this in Aki & Richards (Quantitative Seismology, 1980).
