# Equation of Motion

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

**(1)**

For an *elastic continuum*, it is:

**(2)**

where is density, is a component of the displacement vector, is a component of the stress tensor, and the are the elements of the elastic stiffness tensor (see Wave equation for a derivation). Using chain-rule calculus, this becomes

**(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.

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

- ↑ Backus, G., 1962. Long-wave elastic anisotropy produced by horizontal layering: J. Geoph. Res., 67(11), 4427.
- ↑ O'Doherty, R. F. and Anstey, N. A., 1971. Reflections on Amplitudes: Geoph. Prospg., 19, 430.
- ↑ Banik, N.C., Lerche, I., & Shuey, R.T. 1985. Stratigraphic Filtering, Part I: Derivation of the O'Doherty-Anstey Formula. Geophysics, 50, 2768.
- ↑ 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.