The acoustic Lagrangian density and the full waveform inversion gradient

Kristopher A. H. Innanen

ABSTRACT

The Euler-Lagrange equations relate the Lagrangian density L for a system of particles or fields with the associated equations of motion or field equations. A central problem of field theory is to postulate an L from which the correct equations derive. The problem may be posed in reverse also: known equations of motion can be used as a starting point from which to deduce the associated L. This is useful primarily as a pedagogical exercise. However, the L for acoustic continua is proportional to the acoustic Fréchet derivative, a crucial quantity in seismic full waveform inversion which often must be laboriously calculated. If Fréchet derivatives and thereby FWI gradients are derivable directly from the appropriate continuum mechanical Lagrangian densities, in addition to opening an avenue for physical interpretation of inversion iterates, a considerable savings in calculation would likely be available.

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