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.
The equations are important in geophysics because they relate the amplitude of P-wave, incident upon a plane interface, and the amplitude of reflected and refracted P- and S-waves to the angle of incidence. 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 20 years earlier, in 1899, to derive Knott's equations. Both approaches are valid, but Zoeppritz's approach is more easily understood.
Equations.
The Zoeppritz equations consist of four equations with four unknowns
formula_1
RP, RS, TP, and TS, are the reflected P, reflected S, transmitted P, and transmitted S-wave amplitude coefficients, respectively, formula_2=angle of incidence, formula_3=angle of the transmitted P-wave, formula_4=angle of reflected S-wave and formula_5=angle of the transmitted S-wave. Inverting the matrix form of the Zoeppritz equations give the coefficients as a function of angle.
Although the four equations can be solved for the four unknowns, they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties involved (density, velocity etc.). Several attempts have been made to develop approximations to the Zoeppritz equations, such as Bortfeld's (1961) and Aki & Richards’ (1980), but the most successful of these is the Shuey's, which assumes Poisson's ratio to be the elastic property most directly related to the angular dependence of the reflection coefficient.
Shuey equation.
The 3-term Shuey equation can be written a number of ways, the following is a common form:
where
and
where formula_10=angle of incidence; formula_11 = P-wave velocity in medium; formula_12
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”:
Further reading.
A full derivation of these equations can be found in most exploration geophysics text books, such as: