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In numerous textbooks and papers, the phrase "high-frequency" approximation is used in regards to different types of seismic inversion. What exactly is meant by this? Also, what are the implications if this approximation is in no way satisfied?

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The propagation of seismic waves is described by the wave equation: $$\mathrm{\nabla \cdot \sigma = \frac{\partial ^2 u}{\partial t^2}}$$ Where $\sigma$ is the stress and $u$ is the displacement.

Geometric ray theory is an approximation to the full wave equation where the length scale of variation in seismic wavespeed is much larger than the seismic wavelength (i.e. a high frequency approximation). Typical seismic wavelengths are anything from $\mathrm{\text{~}50-5000~m}$ depending on the application so the ray approximation is generally valid, unless you are trying to resolve very high resolution structure. It is used widely because it is computationally much less expensive than solving the full wave equation.

This book has a very detailed explanation of many aspects of seismic inversion if you need more detail.

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