Seismic Attenuation Calculator

Seismic waves lose amplitude faster than geometric spreading alone predicts, and the surplus loss is described by the dimensionless quality factor Q, defined so that 1/Q is the fraction of stored energy lost per radian of oscillation. For a wave of frequency f travelling for time t through material of quality factor Q, amplitude decays as A = A0 x exp(-pi x f x t / Q). Because f and t enter as a product, attenuation acts as a low-pass filter: a 10 Hz signal is stripped out over the same path that leaves a 0.1 Hz signal almost untouched, which is why distant earthquakes look smooth and long-period on a seismogram. Where Q varies along the ray, the path integral t*, the sum of dt/Q, replaces t/Q and carries the whole path effect in one number. Two mechanisms contribute: intrinsic anelastic loss, largely grain-boundary and pore-fluid friction converting wave energy to heat, and scattering, which redistributes energy rather than destroying it. Typical Q is 20 to 100 in unconsolidated sediment, 300 to 1000 in consolidated crust, above 1000 in cold lithospheric mantle, and as low as 80 in the asthenospheric low-velocity zone. Shear waves attenuate more strongly than compressional waves, with Qs roughly 4/9 of Qp when the loss occurs in shear. Q is also mildly frequency dependent, often modelled as Q0 x f raised to a power between 0 and 0.5, so a value fitted in one frequency band should not be extrapolated far outside it.

The idea that the Earth is not perfectly elastic is old, but the specific bookkeeping was borrowed from electrical engineering, where Q had long described the sharpness of a resonant circuit. Seismologists adopted it during the 1950s and 1960s, once instruments were good enough to show that amplitude decay could not be explained by spreading and reflection alone. Walter Futterman set out the causality requirement in 1962, proving that absorption and velocity dispersion are inseparable: a medium that attenuates must also make velocity vary with frequency, which is why constant-Q models require a matching dispersion term. Leon Knopoff surveyed the field in a 1964 review that fixed the modern definition and terminology, and in the same year Don Anderson and Charles Archambeau used free oscillation and surface wave data to build the first depth-resolved Q models of the mantle. Hsi-Ping Liu, Anderson and Hiroo Kanamori proposed the absorption band model in 1976 to explain why Q is only weakly frequency dependent across the seismic band. From 1969 onward Keiiti Aki showed that the coda of local earthquakes could be used to measure a regional coda Q, which turned attenuation into a routine field measurement rather than a laboratory curiosity. The Preliminary Reference Earth Model of 1981 included radial Q profiles alongside velocity and density, and attenuation tomography has since become a standard tool for mapping partial melt and temperature at depth.