Explicit analytical expression for the multiparameter LSRTM Hessian

Derive an explicit analytical expression for the multiparameter imaging Hessian H = L^H[m0] L[m0] in acoustic least-squares reverse time migration, including all diagonal and off-diagonal (cross-parameter) blocks for simultaneous inversion of logarithmic P-wave velocity and logarithmic impedance, to enable accurate migration deconvolution without relying on sparse point-spread-function sampling and interpolation.

Background

Image-domain LSRTM requires access to the Hessian matrix that maps reflectivity perturbations to remigrated images. In practice, full storage is infeasible, and approximations using point spread functions (PSFs) or nonstationary filtering are employed, which can degrade multiparameter inversion quality due to incomplete sampling and strong parameter crosstalk.

The authors note that while a monoparameter analytical Hessian (for velocity perturbation) exists and is widely used, there is no explicit analytical form for the multiparameter case. This absence hinders robust preconditioning and accurate image-domain inversion for simultaneous parameter estimation (e.g., velocity and impedance).