# Equation of Motion

The equation of motion for a point mass is Newton's Second Law:

 ${\displaystyle ma=f}$ (1)

For an elastic continuum, it is:

 ${\displaystyle \rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\sum _{j,k,l=1}^{3}{\frac {\partial \sigma _{ij}}{\partial x_{j}}}=\sum _{j,k,l=1}^{3}{\frac {\partial }{\partial x_{j}}}\left(C_{ijkl}{\frac {\partial u_{k}}{\partial x_{l}}}\right)}$ (2)

where ${\displaystyle \rho }$ is density, ${\displaystyle u_{i}}$ is a component of the displacement vector, ${\displaystyle \sigma _{ij}}$ is a component of the stress tensor, and the ${\displaystyle C_{ijkl}}$ are the elements of the elastic stiffness tensor (see Wave equation for a derivation). Using chain-rule calculus, this becomes

 ${\displaystyle \rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\sum _{j,k,l=1}^{3}C_{ijkl}{\frac {\partial ^{2}u_{k}}{\partial x_{j}\partial x_{l}}}+\sum _{j,k,l=1}^{3}{\frac {\partial C_{ijkl}}{\partial x_{j}}}{\frac {\partial u_{k}}{\partial x_{l}}}}$ (3)

This is the equation of motion, governing wave propagation in the heterogeneous subsurface. If the medium is locally uniform, then the second term above is zero, and what is left is the wave equation. Since the subsurface is not uniform, we usually model it as coarse regions (often layers) which are uniform on the scale of the seismic wavelength, with elastic properties changing discontinuously at the boundaries. Within each such region, the wave equation does apply, and solutions in each uniform region are matched, via the boundary conditions.

Heterogeneity at all scales

But is there any scale at which the the subsurface is uniform? Obviously, the answer depends upon the subsurface context. The figure at right is from a borehole televiewer, data taken in a sedimentary context. It shows layering persisting at finer and finer scales, spanning 4 orders of magnitude. At still finer scale (below the resolution of the instrument) is heterogeneity at the grain scale, which may or may not have preferred orientation, as in the figure.

At a place like this in the subsurface, it clearly makes no sense to model the subsurface as coarse layers of uniform elasticity. So, the equation of motion, rather than the wave equation, governs wave propagation in such regions.

Some consequences of the inclusion of the second term in equation (3) above include:
* Thin-bed anisotropy[1]
* "Friendly multiples"[2]
* Stratigraphic Filtering[3][4]

## References

1. Backus, G., 1962. Long-wave elastic anisotropy produced by horizontal layering: J. Geoph. Res., 67(11), 4427.
2. O'Doherty, R. F. and Anstey, N. A., 1971. Reflections on Amplitudes: Geoph. Prospg., 19, 430.
3. Banik, N.C., Lerche, I., & Shuey, R.T. 1985. Stratigraphic Filtering, Part I: Derivation of the O'Doherty-Anstey Formula. Geophysics, 50, 2768.
4. Banik, N. C. , I. Lerche, J. R. Resnick, and R. T. Shuey, 1985. Stratigraphic filtering, Part II: Model spectra. GEOPHYSICS. VOL. 50(12) P. 2775-2783.