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Crustal Thickness From Receiver Functions Calculator

Free Crustal thickness receiver functions Calculator for geology & geophysics. Enter variables to compute results with formulas and detailed steps.

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Earth Science & Geology

Crustal Thickness From Receiver Functions Calculator

Calculate Moho depth and crustal thickness from teleseismic receiver function Ps delay times, P-wave velocity, and Vp/Vs ratio.

Last updated: December 2025Reviewed by NovaCalculator Mathematics Team

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Formula

H = tPs / (sqrt(1/Vsยฒ - pยฒ) - sqrt(1/Vpยฒ - pยฒ))

Crustal thickness H is calculated from the Ps conversion delay time (tPs), average crustal P-wave (Vp) and S-wave (Vs) velocities, and the teleseismic ray parameter (p). The Vp/Vs ratio determines Vs from Vp.

Last reviewed: December 2025

Worked Examples

Example 1: Continental Crust Estimation

A seismic station records a Ps delay of 4.5 seconds. Average crustal Vp is 6.3 km/s, Vp/Vs = 1.73, and ray parameter is 0.06 s/km. Calculate crustal thickness.
Solution:
Vs = 6.3 / 1.73 = 3.642 km/s termVs = sqrt(1/3.642ยฒ - 0.06ยฒ) = sqrt(0.07539 - 0.0036) = 0.2679 termVp = sqrt(1/6.3ยฒ - 0.06ยฒ) = sqrt(0.02520 - 0.0036) = 0.1471 H = 4.5 / (0.2679 - 0.1471) = 4.5 / 0.1208 = 37.25 km
Result: Crustal thickness H = 37.25 km

Example 2: Thin Oceanic Crust

An ocean-bottom seismometer shows tPs = 1.0 s with Vp = 6.8 km/s, Vp/Vs = 1.80, and p = 0.05 s/km.
Solution:
Vs = 6.8 / 1.80 = 3.778 km/s termVs = sqrt(1/3.778ยฒ - 0.05ยฒ) = sqrt(0.07007 - 0.0025) = 0.2600 termVp = sqrt(1/6.8ยฒ - 0.05ยฒ) = sqrt(0.02163 - 0.0025) = 0.1384 H = 1.0 / (0.2600 - 0.1384) = 1.0 / 0.1216 = 8.22 km
Result: Crustal thickness H = 8.22 km (typical oceanic crust)
Expert Insights

Background & Theory

The Crustal Thickness From Receiver Functions 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) โˆ’ 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 (0โ€“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 (1โ€“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 5โ€“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 ร— exp(โˆ’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 Crustal Thickness From Receiver Functions 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 โ€” 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.

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Frequently Asked Questions

Receiver functions are time series computed from teleseismic earthquake recordings that isolate the response of Earth structure directly beneath a seismometer. They are created by deconvolving the vertical component from the radial (or transverse) component of a three-component seismogram. This process removes the effects of the earthquake source and instrument response, leaving only the P-to-S converted waves and their reverberations generated at subsurface discontinuities such as the Moho. The most prominent phase in a receiver function is the Ps conversion at the crust-mantle boundary, which arrives after the direct P wave by a time delay that depends on crustal thickness and seismic velocities.
Crustal thickness is calculated from the time delay between the direct P-wave arrival and the P-to-S (Ps) converted phase at the Mohorovicic discontinuity. The fundamental equation is H = tPs / (sqrt(1/Vs^2 - p^2) - sqrt(1/Vp^2 - p^2)), where H is crustal thickness, tPs is the Ps delay time, Vs and Vp are average crustal S-wave and P-wave velocities, and p is the ray parameter. Additional constraints come from the PpPs and PsPs+PpSs reverberation phases, which provide independent thickness estimates. The H-kappa stacking method by Zhu and Kanamori (2000) simultaneously estimates both crustal thickness (H) and Vp/Vs ratio (kappa) by stacking amplitudes at predicted arrival times.
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
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.
Educational Note: This calculator is provided for educational and informational purposes. Results are based on the formulas and inputs provided. Always verify important calculations independently. NovaCalculator processes calculator inputs client-side; optional analytics follow visitor consent settings.Reviewed by: NovaCalculator Mathematics Team โ€” Verified against standard mathematical and scientific references. Last reviewed: December 2025. ยฉ 2024โ€“2026 NovaCalculator.

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Formula

H = tPs / (sqrt(1/Vsยฒ - pยฒ) - sqrt(1/Vpยฒ - pยฒ))

Crustal thickness H is calculated from the Ps conversion delay time (tPs), average crustal P-wave (Vp) and S-wave (Vs) velocities, and the teleseismic ray parameter (p). The Vp/Vs ratio determines Vs from Vp.

Worked Examples

Example 1: Continental Crust Estimation

Problem: A seismic station records a Ps delay of 4.5 seconds. Average crustal Vp is 6.3 km/s, Vp/Vs = 1.73, and ray parameter is 0.06 s/km. Calculate crustal thickness.

Solution: Vs = 6.3 / 1.73 = 3.642 km/s\ntermVs = sqrt(1/3.642ยฒ - 0.06ยฒ) = sqrt(0.07539 - 0.0036) = 0.2679\ntermVp = sqrt(1/6.3ยฒ - 0.06ยฒ) = sqrt(0.02520 - 0.0036) = 0.1471\nH = 4.5 / (0.2679 - 0.1471) = 4.5 / 0.1208 = 37.25 km

Result: Crustal thickness H = 37.25 km

Example 2: Thin Oceanic Crust

Problem: An ocean-bottom seismometer shows tPs = 1.0 s with Vp = 6.8 km/s, Vp/Vs = 1.80, and p = 0.05 s/km.

Solution: Vs = 6.8 / 1.80 = 3.778 km/s\ntermVs = sqrt(1/3.778ยฒ - 0.05ยฒ) = sqrt(0.07007 - 0.0025) = 0.2600\ntermVp = sqrt(1/6.8ยฒ - 0.05ยฒ) = sqrt(0.02163 - 0.0025) = 0.1384\nH = 1.0 / (0.2600 - 0.1384) = 1.0 / 0.1216 = 8.22 km

Result: Crustal thickness H = 8.22 km (typical oceanic crust)

Frequently Asked Questions

What are receiver functions in seismology?

Receiver functions are time series computed from teleseismic earthquake recordings that isolate the response of Earth structure directly beneath a seismometer. They are created by deconvolving the vertical component from the radial (or transverse) component of a three-component seismogram. This process removes the effects of the earthquake source and instrument response, leaving only the P-to-S converted waves and their reverberations generated at subsurface discontinuities such as the Moho. The most prominent phase in a receiver function is the Ps conversion at the crust-mantle boundary, which arrives after the direct P wave by a time delay that depends on crustal thickness and seismic velocities.

How is crustal thickness calculated from receiver functions?

Crustal thickness is calculated from the time delay between the direct P-wave arrival and the P-to-S (Ps) converted phase at the Mohorovicic discontinuity. The fundamental equation is H = tPs / (sqrt(1/Vs^2 - p^2) - sqrt(1/Vp^2 - p^2)), where H is crustal thickness, tPs is the Ps delay time, Vs and Vp are average crustal S-wave and P-wave velocities, and p is the ray parameter. Additional constraints come from the PpPs and PsPs+PpSs reverberation phases, which provide independent thickness estimates. The H-kappa stacking method by Zhu and Kanamori (2000) simultaneously estimates both crustal thickness (H) and Vp/Vs ratio (kappa) by stacking amplitudes at predicted arrival times.

How do I get the most accurate result?

Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.

What inputs do I need to use Crustal Thickness From Receiver Functions Calculator accurately?

Each field is labelled with the required unit (metric or imperial). Gather your source values before starting โ€” for example, a weight measurement in kilograms, a distance in metres, or a dollar amount โ€” and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.

How accurate are the results from Crustal Thickness From Receiver Functions Calculator?

All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.

Can I use the results for professional or academic purposes?

You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.

References

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