Seismic Reflection Coefficient Calculator
Compute seismic reflection coefficient using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
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The reflection coefficient R is the ratio of reflected to incident wave amplitude, determined by the acoustic impedance contrast between two layers. Z is acoustic impedance (density times velocity). T = 2Z1/(Z1+Z2) gives the transmission coefficient.
Last reviewed: December 2025
Worked Examples
Example 1: Sandstone over Limestone
Example 2: Shale over Gas Sand
Background & Theory
The Seismic Reflection Coefficient Calculator applies the following established principles and formulas. Earth science calculators draw on a wide range of measurement scales and physical principles that quantify natural phenomena across geological, atmospheric, and hydrological systems. Earthquake magnitude is most precisely described by the Moment Magnitude Scale (Mw), which replaced the original Richter scale for larger events. Mw is calculated as Mw = (2/3) log10(M0) − 10.7, where M0 is the seismic moment in dyne-centimeters. The Richter scale, while still referenced colloquially, is a local magnitude (ML) measurement derived from peak seismograph amplitude at a standard 100 km distance. Wind intensity is classified using the Beaufort Scale, a 13-point empirical scale (0–12) relating wind speed in knots to observable sea and land effects, with Beaufort 12 corresponding to hurricane-force winds above 64 knots. Tropical cyclone intensity is further categorized by the Saffir-Simpson Hurricane Wind Scale, which assigns Categories 1 through 5 based on sustained wind speed, correlating with expected structural damage. Mineral hardness is quantified on the Mohs scale (1–10), comparing scratch resistance relative to reference minerals from talc (1) to diamond (10). Soil composition analysis measures the proportions of sand, silt, and clay by particle size, alongside organic matter content, bulk density, and porosity, which together determine engineering and agricultural suitability. Seismic wave velocity in rock varies by material: P-waves travel at approximately 5–7 km/s in granite and 1.5 km/s in water, while S-waves travel at roughly 60% of P-wave speeds. Atmospheric pressure decreases with altitude according to the barometric formula: P = P0 × exp(−Mgh / RT), where M is molar mass of air, g is gravitational acceleration, h is altitude, R is the universal gas constant, and T is temperature in Kelvin. Standard sea-level pressure is 101,325 Pa. Tidal calculations use harmonic analysis of gravitational forcing by the Moon and Sun, with the principal lunar semidiurnal tidal constituent (M2) having a period of approximately 12.42 hours.
History
The history behind the Seismic Reflection Coefficient Calculator traces back through the following developments. The systematic study of Earth's structure and processes spans millennia, but the scientific foundations were laid in the seventeenth century. In 1669, Danish naturalist Nicolas Steno published his principles of stratigraphy, establishing the laws of superposition, original horizontality, and lateral continuity — foundational rules for reading rock layers that remain in use today. Scottish geologist James Hutton introduced the concept of uniformitarianism in 1788, proposing that geological processes observable in the present have operated throughout Earth's history at broadly consistent rates. This idea of deep time challenged prevailing biblical chronologies and set the stage for modern geology. Charles Lyell systematized these ideas in his landmark three-volume work Principles of Geology, published beginning in 1830, which directly influenced Charles Darwin's thinking on biological evolution during the voyage of the Beagle. The nineteenth century saw growing curiosity about continental shapes, but a coherent theory awaited Alfred Wegener, a German meteorologist who proposed continental drift in 1912, arguing that the continents had once formed a supercontinent he called Pangaea. His evidence included matching fossil records and geological formations across the Atlantic, but his mechanism was disputed for decades. The theory gained acceptance in the 1960s when seafloor spreading was confirmed through paleomagnetic studies, and plate tectonics emerged as the unifying framework of modern geoscience. The United States Geological Survey was established by Congress in 1879 to classify public lands and examine the geological structure, mineral resources, and products of the national domain. The twentieth century brought instrumental advances, including the global seismograph network deployed after World War II, initially to monitor nuclear tests, which dramatically improved earthquake detection and characterization. Satellite Earth observation began in earnest with the Landsat program launched in 1972, enabling continuous global monitoring of land use, glacier retreat, and vegetation patterns. Today, GPS networks, LIDAR scanning, and ocean-floor mapping provide centimeter-scale precision for tracking tectonic motion, sea level rise, and volcanic deformation in near real time.
Frequently Asked Questions
Formula
R = (Z2 - Z1) / (Z2 + Z1) | Z = rho * V
The reflection coefficient R is the ratio of reflected to incident wave amplitude, determined by the acoustic impedance contrast between two layers. Z is acoustic impedance (density times velocity). T = 2Z1/(Z1+Z2) gives the transmission coefficient.
Worked Examples
Example 1: Sandstone over Limestone
Problem: Calculate the reflection coefficient at the boundary between sandstone (V=3000 m/s, rho=2300 kg/m^3) and limestone (V=5000 m/s, rho=2700 kg/m^3).
Solution: Z1 = 2300 * 3000 = 6.9e6\nZ2 = 2700 * 5000 = 1.35e7\nR = (1.35e7 - 6.9e6) / (1.35e7 + 6.9e6) = 6.6e6 / 2.04e7 = 0.3235\nCritical angle = arcsin(3000/5000) = 36.87 degrees
Result: R = 0.3235, Critical angle = 36.87 deg
Example 2: Shale over Gas Sand
Problem: Calculate R for shale (V=2500 m/s, rho=2350 kg/m^3) over gas sand (V=2000 m/s, rho=2100 kg/m^3).
Solution: Z1 = 2350 * 2500 = 5.875e6\nZ2 = 2100 * 2000 = 4.2e6\nR = (4.2e6 - 5.875e6) / (4.2e6 + 5.875e6) = -0.1664\nNegative R indicates polarity reversal (gas indicator)
Result: R = -0.1664 (negative polarity, potential gas indicator)
Frequently Asked Questions
What is a seismic reflection coefficient?
The seismic reflection coefficient describes the fraction of seismic wave amplitude that is reflected when the wave encounters a boundary between two rock layers with different acoustic properties. It ranges from -1 to +1, where positive values indicate the reflected wave has the same polarity as the incident wave, and negative values indicate a polarity reversal. The coefficient depends on the acoustic impedance contrast between the layers, which is the product of density and seismic velocity.
What is the critical angle in seismic reflection?
The critical angle occurs when a seismic wave traveling from a slower medium to a faster medium reaches an incidence angle where the refracted wave travels along the interface. It is calculated as arcsin(V1/V2), where V1 is the velocity in the upper layer and V2 in the lower. Beyond the critical angle, total internal reflection occurs and no energy is transmitted into the lower layer. The critical angle only exists when V2 is greater than V1. At the critical angle, head waves (refracted arrivals) are generated, which are used in seismic refraction surveys.
How does the reflection coefficient change with angle?
The reflection coefficient varies with the angle of incidence according to the Zoeppritz equations, which account for P-to-S wave conversions at the interface. The Shuey approximation simplifies this to R(theta) = R0 + G*sin^2(theta) + F*sin^2(theta)*tan^2(theta), where R0 is the normal incidence reflectivity and G is the AVO gradient. This angle-dependent behavior is the basis of AVO (Amplitude Versus Offset) analysis, a key technique in hydrocarbon exploration used to distinguish gas-bearing sands from other lithologies.
Why might my result differ from another tool or reference?
Differences typically arise from rounding conventions, the specific version of a formula (for example, simple vs compound interest), or unit inconsistencies between inputs. Check that both tools are using the same formula variant and the same units. The References section links to the authoritative source behind the formula used here.
What inputs do I need to use Seismic Reflection Coefficient Calculator accurately?
Each field is labelled with the required unit (metric or imperial). Gather your source values before starting — for example, a weight measurement in kilograms, a distance in metres, or a dollar amount — and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
How do I verify Seismic Reflection Coefficient Calculator's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy