Modeling and inversion for seismic wavefields that include the attenuation and phase dispersion effects of Q can be implemented in the space‐frequency domain. The viscoacoustic wave equation is solved by the moment method. Absorbing boundary conditions are implemented by reducing Q and adjusting the complex velocity (to reduce Q‐dependent reflectivity) in a zone around the edges of the model grid. Nonlinear inversion is performed using iterative linearized inversions. The residual wavefield at a single frequency is back projected, using an anticausal Green’s function, along the viscoacoustic wavepath in an estimate of the model, to get updated velocity and Q distributions. The model obtained from data at one frequency becomes input to inversion at the next higher frequency. Velocity and Q are inverted simultaneously as they are interdependent. Both modeling and inversion algorithms are successfully tested with synthetic examples; data at two or three frequencies are sufficient to produce reliable images from noise‐free synthetic data.

The centroid frequency shift method is implemented, tested with synthetic data, and applied to field data from three contiguous crosswell seismic experiments at the Gypsy Pilot in northern Oklahoma. The similtaneous iterative reconstruction technique is used for tomographic estimations of both P‐wave velocity and Q. No amplitude corrections or spreading loss corrections are needed for the Q estimation. The estimated in‐situ velocity and Q distributions correlate well with log data and local lithology. The Q/velocity ratio appears to correlate with the sand/shale ratio (ranging from an average of ∼15 s/km for the sand‐dominated lithologies to an average of ∼8.5 s/km for the shale‐dominated ones), with the result that new information is provided on interwell connectivity.

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Two pseudo-spectral implementations of 2-D viscoacoustic modeling are developed in a distributed-memory multi-processor computing environment. The first involves simultaneous computation of the response of one model to many source locations and, as it requires no interprocessor communication, is perfectly parallel. The second involves computation of the response, to one source, of a large model that is distributed across all processors. In the latter, local rather than global, Fourier transforms are used to minimize interprocessor communication and to eliminate the need for matrix transposition. In both algorithms, absorbing boundaries are defined as zones of decreased Q as part of the model, and so require no extra computation. An empirical method of determining sets of relaxation times for a broad range of Q values eliminates the need for iterative fitting of Q-frequency curves.