Velocity and Q from reflection seismic data

This study has resulted in the discovery of an exact method for the theoretical formulation of the effects of intrinsic damping where the attenuation coefficient, a(v), is an arbitrary function of the frequency, v. Absorption-dispersion pairs are computed using numerical Hilbert transformation; approximate analytical expressions that require the selection of arbitrary constants and cutoff frequencies are no longer necessary. For constant Q, the dispersive body wave velocity, p(v), is found to be

p(v) = (p(v_N)/(1+(1/2Q H(-v)/v))

where H denotes numerical Hilbert transformation, p(v) is the phase velocity at the frequency v, and p(v_N) is the phase velocity at Nyquist. From (1) it is possible to estimate Q in the time domain by measuring the amount of increase, ΔW, of the wavelet breadth after a traveltime,

Q=(2Δ𝛕)/(𝝅ΔW)

The inverse problem, i.e., the determination of Q and velocity is also investigated using singular value decomposition (SVD). The sparse matrices encountered in the acquisition of conventional reflection seismology data result in a system of linear equations of the form AX = B, with A the design matrix, X the solution vector, and B the data vector. The system of normal equations is AᵀAX = AᵀB where the least-squares estimate of X = X = V(1/S)UᵀB and the SVD of A is A = USVᵀ. A technique to improve the sparsity pattern prior to decomposition is described.

From an application of equation (2) using reference reflections from shallower reflectors, crystalline rocks in South Carolina over the depth interval from about 5 km to 10 km yield values of Qin the range Q = 250 - 300.

Non-standard recording geometries ( "Q-spreads") and vibroseis recording procedures are suggested to minimize matrix sparseness and increase the usable frequency bandwidth between zero and Nyquist. The direct detection of body wave dispersion by conventional vibroseis techniques may be useful to distinguish between those crustal volumes that are potentially seismogenic and those that are not. Such differences may be due to variations in fracture density and therefore water content in the crust.