Full-waveform inversion (FWI) seeks to estimate subsurface elastic properties by iterative minimization of the difference between synthetic and observed data. Its industrial application suffers from several well-defined obstacles, including high computational cost, slow convergence rate, and the phenomenon of cycle skipping. We have developed an efficient domain waveform inversion aimed at reducing the computational burden of FWI with a phase-encoding technique. The gradient is constructed in the domain using linear phase-encoding, and a slant update strategy further reduces the computational burden. Poorly scaled and blurred gradient updates can be enhanced using exact or approximate versions of the inverse Hessian, which leads to a faster convergence rate. We developed a new chirp phase-encoding strategy for diagonal Hessian construction. Preconditioning the gradient using the diagonal phase-encoded approximate Hessian forms what we refer to as a pseudo-Gauss-Newton (PGN) step. To test the effectiveness of the -domain FWI, the strategies were enacted on a modified Marmousi model. We compared the computational cost of the PGN method with traditional methods, and we evaluated the quality of the inversion results. The PGN method can get a better inversion result with the same computational cost. We have also analyzed the effects of different ray parameter settings and the influence of source spacing, and we compared different preconditioning methods for -domain FWI.