Asymptotic pseudounitary stacking operators

Abstract

Stacking operators are widely used in seismic imaging and seismic data processing. Examples include Kirchhoff datuming, migration, offset continuation, dip moveout, and velocity transform. Two primary approaches exist for inverting such operators. The first approach is iterative least‐squares optimization, which involves the construction of the adjoint operator. The second approach is asymptotic inversion, where an approximate inverse operator is constructed in the high‐frequency asymptotics. Adjoint and asymptotic inverse operators share the same kinematic properties, but their amplitudes (weighting functions) are defined differently. This paper describes a theory for reconciling the two approaches. I introduce a pair of asymptotic pseudounitary operators, which possess both the property of being adjoint and the property of being asymptotically inverse. The weighting function of the asymptotic pseudounitary stacking operators is shown to be completely defined by the derivatives of the operator kinematics. I exemplify the general theory by considering several particular examples of stacking operators. Simple numerical experiments demonstrate a noticeable gain in efficiency when the asymptotic pseudounitary operators are applied for preconditioning iterative least‐squares optimization.