Attenuation Quality Factor Calculator
Find the seismic quality factor Q from measured amplitude decay, or predict how much wave amplitude survives a given travel time.
📉 What is the Attenuation Quality Factor?
The attenuation quality factor, written Q, is a dimensionless number that tells you how quickly a seismic wave loses energy as it travels through rock. It is defined as 2π times the energy stored in a wave divided by the energy dissipated per cycle. A large Q means the rock is nearly elastic and a wave rings for many cycles before dying away. A small Q means the rock is lossy and the wave fades within a few cycles. Its reciprocal, 1/Q, is often called the attenuation or internal friction of the medium.
Seismologists and exploration geophysicists use Q constantly. Earthquake engineers use it to predict how shaking decays with distance and to build ground-motion models. Seismologists invert for 3-D maps of Q to find hot, partially molten or fluid-rich zones under volcanoes and subduction zones, because those regions attenuate waves strongly. Reservoir geophysicists measure Q from vertical seismic profiles and well logs to flag gas-bearing sands and fractured rock, and processing geophysicists apply inverse-Q filters to restore the high frequencies that attenuation removes from reflection data.
It helps to separate three things that all reduce amplitude. Geometric spreading lowers amplitude as the wavefront grows, but it conserves energy. Scattering redirects energy without destroying it. Only anelastic attenuation, described by Q, turns wave energy into heat. A common mistake is to compute Q from raw amplitudes without first correcting for spreading, which makes Q look far too low. Another is to treat Q as a single universal number: it depends on the wave type (Qp versus Qs), the frequency band, the rock type, temperature and fluid content, and Q for the shallow crust can differ from Q for the mantle by an order of magnitude.
This calculator covers both directions of the same law. In the first mode you supply a measured amplitude decay over a known travel time and frequency and it returns Q, 1/Q and the spatial attenuation coefficient. In the second mode you supply Q and a path (as travel time or as distance) and it returns the fraction of amplitude that survives. Charts show how the answer moves if the true Q were half or double your estimate, which is useful for judging uncertainty.
📐 Formula
The relation follows from the definition of Q for a wave of angular frequency ω = 2πf: the amplitude decays as exp(−ω t / 2Q), which simplifies to exp(−π f t / Q). It is the standard form given in seismology texts such as Aki and Richards and in the Wikipedia article on seismic attenuation.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Upper Crust Coda Decay
f = 5 Hz, t = 20 s, A0 = 1000, A = 208, v = 3500 m/s
Example 2 — Soft Sediment Basin
f = 10 Hz, t = 2 s, A0 = 100, A = 35, v = 2000 m/s
Example 3 — Mantle Shear Wave
f = 1 Hz, t = 300 s, A0 = 100, A = 40, v = 6000 m/s
Example 4 — Amplitude Left After a 200 km Crustal Path
Amplitude mode: f = 2 Hz, Q = 500, v = 3500 m/s, path distance = 200 km
❓ Frequently Asked Questions
🔗 Related Calculators
What is the seismic quality factor Q?
Q is a dimensionless measure of how efficiently rock transmits seismic waves. It is defined as 2 pi times the stored energy divided by the energy lost per cycle, so a high Q means little attenuation and a low Q means strong attenuation. Typical values run from about 10 for soft sediments to over 1000 for cold mantle rock.
How do you calculate Q from amplitude decay?
Use Q = pi f t / ln(A0/A), where f is the dominant frequency in Hz, t is the travel time in seconds, A0 is the starting amplitude and A is the amplitude after time t. For f = 5 Hz, t = 20 s and an amplitude drop from 1000 to 208, Q is about 200.
What is the formula for seismic attenuation?
The amplitude of a wave of frequency f traveling for time t in a medium of quality factor Q is A(t) = A0 exp(-pi f t / Q). Equivalently, in space the amplitude falls as exp(-alpha x) with alpha = pi f / (Q v), where v is the wave velocity and x the path length.
What is a good Q value for the Earth's crust?
Crustal Q for S waves is commonly 100 to 600 at 1 Hz in stable regions and lower, around 50 to 200, in tectonically active or sedimentary regions. Coda-Q studies of California and Japan often report values near 150 to 250 at 1 to 5 Hz, rising with frequency.
What is 1/Q and why do seismologists use it?
1/Q is the attenuation, or internal friction, of the medium. It adds up naturally when several mechanisms act together and is directly proportional to the fractional energy lost per cycle (delta E / E = 2 pi / Q). Maps of 1/Q (often written Qinv) are common in tomography.
How is the attenuation coefficient alpha related to Q?
The spatial attenuation coefficient is alpha = pi f / (Q v), in nepers per unit distance. It is the rate at which amplitude decays with distance, as opposed to the 1/Q-based decay per unit time. A higher frequency or lower Q or lower velocity all give larger alpha.
Why does amplitude fall faster at high frequency?
The exponent in A = A0 exp(-pi f t / Q) grows linearly with frequency, so a 10 Hz wave loses amplitude twice as fast as a 5 Hz wave over the same time in the same rock. Attenuation therefore acts as a low-pass filter on seismic signals.
What is the e-folding time of a seismic wave?
The e-folding time is the time for amplitude to drop to 1/e, about 36.8 percent of its starting value. It equals Q / (pi f). For Q = 200 and f = 5 Hz it is about 12.7 seconds, which also means the amplitude halves roughly every 8.8 seconds.
Can Q be infinite or negative?
An infinite Q means a perfectly elastic medium with no energy loss, so A equals A0. A negative Q would imply amplitude growth, which is not physical for passive rock. This calculator therefore requires the measured amplitude A to be smaller than A0 in the Q-from-decay mode.
Is this the same as geometric spreading?
No. Geometric spreading is the drop in amplitude caused by the wavefront expanding over a larger area, roughly 1/r for body waves, and it conserves energy. Anelastic attenuation, described by Q, converts wave energy into heat. Real amplitude measurements combine both and must be corrected for spreading first.
What is the difference between Qp and Qs?
Qp describes the attenuation of compressional waves and Qs that of shear waves. In most of the crust and mantle Qs is smaller than Qp, often by a factor of about 2 (Qp is roughly 2.25 times Qs when attenuation is purely in shear). Use the Q appropriate to the wave you are measuring.