Seismic Reflection Coefficient Calculator

When a seismic wave crosses a boundary between two rock layers, the fraction of amplitude that returns is set by the contrast in acoustic impedance, Z = rho x V, the product of density and velocity. At normal incidence the reflection coefficient is R = (Z2 - Z1) / (Z2 + Z1) and the transmission coefficient is T = 2 x Z1 / (Z2 + Z1). The sign carries information: a negative R means the wave returns with reversed polarity because the lower layer is softer, the classic signature of a gas-charged sand beneath shale. Energy partitions as R squared, which is precisely why reflection seismology works. Most sedimentary interfaces give R between 0.05 and 0.2, returning under 4 percent of the incident energy, so nearly everything continues downward and remains available to image deeper targets. Strong contrasts such as shale over salt, or sediment over basalt, can reach R of 0.3 to 0.4 and produce bright, energy-robbing reflections that shadow the structure beneath them. When the lower layer is faster there is also a critical angle, arcsin(V1/V2), beyond which no energy is transmitted and head waves are generated, which is the basis of refraction surveying. These expressions assume plane waves striking a welded, isotropic, elastic contact between thick homogeneous layers. Where bed thickness approaches a quarter of the seismic wavelength, reflections from top and base interfere and tuning effects dominate the measured amplitude, and shale anisotropy modifies how the coefficient varies with angle.

The elastic theory of partitioning at an interface was worked out well before anyone tried to use it commercially. Cargill Gilston Knott published equations for the energy division between reflected and transmitted P and S waves in 1899, working from the seismological problem of how earthquake waves behave at internal boundaries within the Earth. Karl Bernhard Zoeppritz derived the equivalent set in terms of displacement amplitudes, expressed as four boundary conditions on continuity of displacement and traction yielding four coefficients. He died in 1908 at the age of 26, and his results were published posthumously in 1919 through the efforts of Ludwig Geiger and Beno Gutenberg. The equations went largely unused in exploration because solving them by hand for every case was impractical. Morris Muskat and Milton Meres changed that in 1940 by publishing extensive numerical tables of Zoeppritz solutions in the journal Geophysics. Linear approximations followed as digital processing arrived: Reinhard Bortfeld in 1961, and the widely quoted form given by Keiiti Aki and Paul Richards in their 1980 textbook. R. T. Shuey rearranged the approximation in 1985 into an intercept plus a gradient term in sine squared of the incidence angle, the version routinely fitted to prestack gathers. Interest expanded sharply after William Ostrander showed in 1984 that gas sands produce a characteristic increase of reflection amplitude with offset, turning the Zoeppritz angle dependence into the practical technique known as AVO analysis.