Quantifying Crustal Fracture Heterogeneity by Seismic Scattering

Peter Leary

Published July 1995, SCEC Contribution #216

Seismic waves travelling through the crust generate trains of scattered waves called the seismic coda. Coda motion u(t) excited by a source pulse σ(t) and recorded at times t > 2t0 by a sensor located at traveltime t0 from the source can be written as the convolution of σ(t) with the gradient c'(r) = ∂c(r) of spatial velocity fluctuations c(r) encountered by the wave front at r tc0/2: u(t) 4πu0(t0/t)(c'(tc0/2)> * σ(t), where t0/t accounts for source wavelet spherical divergence, c0 is the mean seismic velocity of the medium, and > denotes the scattering amplitude c'(r) averaged over the 4π solid angle of the wave front. It follows that the coda frequency spectrum u(f) is proportional to vc(v)σ(f), where v is the spatial frequency v = 2f / c0, vc(v) the spectrum of c'(r) and σ(f) the source spectrum.

Sonic-velocity fluctuations cBH(r) logged at 15 cm intervals over a 1.5 km length of deep borehole in crustal rock show that the major cause of seismic velocity fluctuations c(r) in the brittle crust are fracture distributions with a power-law spectral dependence on fracture spacing, c(v) v−0.4. The resulting velocity gradient power-law spectrum is vc(v) v0.6. In accordance with the above scattering expressions, the power-law scattering amplitude vc(v) causes the high-frequency power-law enrichment of the coda spectrum relative to the source spectrum u(f)/σ(f) f0.6 as observed in coda waves recorded in a borehole at a depth of 2.5 km in the crust.

The coda wave phenomenology observed in deep crustal borehole data does not emerge from treatments of seismic scattering based on correlations between randomly distributed elastic heterogeneities. This is because scattering is proportional to the derivative of fluctuations in material properties; for power-law-distributed velocity fluctuations in crustal rock, differentiation enhances scattering while correlation smoothing of fluctuations reduces scattering.

Leary, P. (1995). Quantifying Crustal Fracture Heterogeneity by Seismic Scattering. Geophysical Journal International, 122(1), 125-142. doi: 10.1111/j.1365-246X.1995.tb03541.x .