# Dictionary:Zoeppritz equations

In geophysics and reflection seismology, the **Zoeppritz equations** are a set of equations that describe the partitioning of seismic wave energy at an interface, typically a boundary between two different layers of rock. They are named after their author, the German geophysicist Karl Bernhard Zoeppritz, who died before they were published in 1919.^{[1]}

The equations are important in geophysics because they relate the amplitude of plane waves, incident upon a planar interface separating two isotropic media, and the amplitude of reflected and refracted P- and S-waves to the angle of incidence.^{[2]} They are the basis for investigating the factors affecting the amplitude of a returning seismic wave when the angle of incidence is altered - also known as amplitude versus offset analysis - which is a helpful technique in the detection of petroleum reservoirs.

The Zoeppritz equations were not the first to describe the amplitudes of reflected and refracted waves at a plane interface. Cargill Gilston Knott used an approach in terms of potentials almost 30 years earlier, in 1899, to derive Knott's equations. Both approaches are valid and Zoeppritz's approach is more easily understood^{[2]}, although approximations are commonly made (see below).

## Contents

## Equations

### From Sheriff's *Dictionary*

(zō' pritz) Equations that express the partition of energy when a plane wave impinges on an acoustic-impedance contrast. In the general case for an interface between two solids when the incident angle is not zero, four waves are generated: reflected P-wave and S-wave and transmitted P-wave and S-wave. The partition of energy among these is found from four boundary conditions which require continuity of normal and tangential displacement and stress. Using the symbols given in Figure S-12, Snell's law states

and this defines all angles. For an incident plane P-wave of unity amplitude, the continuity conditions yield the four Zoeppritz equations shown here:

where *Z*_{i} = ρ_{i}*V*_{Pi}, *W*_{i} = ρ_{i}*V*_{Si}, and *R*_{P}, *R*_{S}, *T*_{P}, and *T*_{S} are respectively the amplitudes of the reflected P- and S-waves and the transmitted P- and S-waves. However, their derivation does not consider head waves and hence they do not yield reliable values at and beyond the critical angle.

Figure Z-1 also shows the variation of energy with angle for several sets of parameters. Beyond the critical angles for P- and S-waves, the respective refracted waves vanish. The increase in reflection energy near the critical angle is sometimes referred to as the **wide-angle phenomenon** and is sometimes exploited in seismic surveying. The same relationships in terms of potentials are called **Knott’s equations**.^{[3]} Because no provision was made in the equation’s derivation for the head waves, these equations do not give head-wave amplitudes or correct values beyond the critical angle.

The explicit solution of these four simultaneous equations was given by Aki and Richards (1980)^{[4]}, for the four outgoing amplitudes from an incident P-wave (and also the corresponding results for incident SV and SH waves). These solutions are exact (within their assumptions: plane wave incidence, planar interface between isotropic half-spaces).
However,they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties involved (density, velocity etc.), so approximations are useful.

### Shuey Equation

With the assumption (commonly valid) that the *contrast* in elastic properties (across the interface) is small, the "exact" simplify considerably. Several such approximations to the Zoeppritz equations exist, such as Bortfeld (1961) ^{[5]} and Aki & Richards. We follow here Shuey (1985)^{[6]}.

The 3-term Shuey Equation can be written a number of ways, the following is a common form:^{[7]}

where

and

- ;

where =angle of incidence; = P-wave velocity in medium; = P-wave velocity contrast across interface; = S-wave velocity in medium; = S-wave velocity contrast across interface; = density in medium; = density contrast across interface;

A proposed better approximation of Zoeppritz equations:

and

In the Shuey Equation, R(0) is the reflection coefficient at normal incidence and is controlled by the contrast in acoustic impedances. G, often referred to as the AVO gradient, describes the variation of reflection amplitudes at intermediate offsets and the third term, F, describes the behaviour at large angles/far offsets that are close to the critical angle. This equation can be further simplified by assuming that the angle of incidence is less than 30 degrees (i.e. the offset is relatively small), so the third term will tend to zero. This is the case in most seismic surveys and gives the “Shuey Approximation”:

## See also

- Amplitude versus offset, a practical application of the phenomenon described by these equations.

## Further reading

A full derivation of these equations can be found in most exploration geophysics text books, such as:

- Sheriff, R. E., Geldart, L. P., (1995), 2nd Edition. Exploration Seismology. Cambridge University Press.

## References

- ↑ Zoeppritz, Karl (1919). Erdbebenwellen VII. VIIb. Über Reflexion und Durchgang seismischer Wellen durch Unstetigkeitsflächen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, 66-84.
- ↑
^{2.0}^{2.1}Sheriff, R. E., Geldart, L. P., (1995), 2nd Edition. Exploration Seismology. Cambridge University Press. - ↑ Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.
- ↑ Aki, K. and Richards, P. G., 1980, Quantitative seismology: Theory and methods, v.1 : W.H. Freeman and Co.
- ↑ Bortfeld, R., 1961, Approximation to the reflection and transmission coefficients of plane longitudinal and transverse waves: Geophys. Prosp., 9, 485–502.
- ↑ Shuey, R. T. (April 1985). "A simplification of the Zoeppritz equations".
*Geophysics***50**(9): 609–614. doi:10.1190/1.1441936. - ↑ Avesth, P, T Mukerji and G Mavko (2005). Quantitative seismic interpretation. Cambridge University Press, Cambridge, UK