Time-domain Modeling of Constant-Q Seismic Waves Using Fractional Derivatives

Kjartansson’s constant-Q model is solved in the time-domain using a new modeling algorithm based on fractional derivatives. Instead of time derivatives of order 2, Kjartansson’s model requires derivatives of order 2c, with 0 < c < 1=2, in the dilatation-stress formulation. The derivatives are computed with the Gru¨ nwald-Letnikov and central-difference approximations, which are finite-difference extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries. A synthetic cross-well seismic experiment illustrates the capabilities of this novel modeling algorithm.