ACCURATE VISCOELASTIC MODELING BY FREQUENCY-DOMAIN FINITE-DIFFERENCESUSING ROTATED OPERATORS

The viscoelastic wave equation is an integro-differential equation tha t requires special methods when using time-domain numerical finite-dif ference methods. In the frequency domain, the integral terms are easil y represented by complex valued elastic media properties. There are fu rther significant advantages to using the frequency domain if the forw ard or the inverse problem requires modeling or inverting a large numb er of prestack source gathers. Numerical modeling is expensive for sei smic data because of the large number of wavelengths typically separat ing sources from receivers, which results in a need for a large number of grid points. A major obstacle to using frequency-domain methods is the consequent storage requirements. To reduce these, we maximize the accuracy and simultaneously minimize the spatial extent of the numeri cal operators. We achieve this by extending earlier published methods introduced for the viscoacoustic case to the viscoelastic case. This r equires the formulation of two new numerical operators: a differencing operator in a rotated coordinate frame and a lumped mass term. The ne w operators are combined with ordinary second-order, finite-difference operators in an optimal manner to minimize numerical errors without i ncreasing the size of the numerical operator. For a fixed number of gr id points, the resulting second-order differencing scheme is no more e xpensive than an ordinary second-order differencing scheme, but a nume rical dispersion analysis shows that the number of grid points require d per smallest wavelength is reduced from approximately 15 to approxim ately 4. The new scheme is also capable of handling embedded fluid lay ers without instability. We demonstrate that no further improvement in performance can be achieved using higher order spatial operators beca use of the associated computational overheads associated with the larg er differencing operators. The new viscoelastic modeling scheme is use d to study a crosshole data set in which the exact nature of the seism ic coda is unclear. The results of the modeling study indicate this co da is likely related to the generation of mode-converted shear waves w ithin the complicated, finely layered sediments at the site.