## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Extension of wavefronts and rays

We extended the concept of wavefronts and rays proposed by Courant and Hilbert.

Our extension is based on the theory of hyperbolic partial differential equations, together with Goursat’s nonuniqueness theorem.

Previously, it had been known to theoretical seismologists that the concept of characteristics of differential equations and wavefronts are connected. However, proceeding from formal definitions obscured the physical meaning of subsequent interpretations.

To avoid such definitions, we extended the Courant and Hilbert formulation to anisotropic inhomogeneous media.

Presently, the general relation between the Cauchy problem of partial differential equations and its interpretation aswavefronts and rays is intimately, not only formally, connected.

This provides a seismologists with a plethora of methods available in applied mathematics.

Bos, L.P., Slawinski, M.A. (2010) Elastodynamic equations: Characteristics, wavefronts and rays. The Quarterly Journal of Mechanics and Applied Mathematics 63(1), 23–37