The extremely large size of typical seismic imaging problems has been a major stumbling block for iterative techniques to attain accurate migration amplitudes. These iterative methods are important because they complement theoretical approaches hampered by difficulties controlling problems such as finite-acquisition aperture, source-receiver frequency response, and directivity. To solve these problems, we apply preconditioning, which significantly improves convergence of least-squares migration. We discuss different levels of preconditioning: corrections for the order of the migration operator, corrections for spherical spreading, and position- and reflector-dip-dependent amplitude errors. Although the first two corrections correspond to simple scalings in the Fourier and physical domain, the third correction requires phase-space (space spanned by location and dip) scaling, which we carry out with curvelets. Our combined preconditioner significantly improves the convergence of least-squares wave-equation migration on a line from the SEG/EAGE AA′ salt model.

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