We carry out high-frequency analyses of Claerbout's double-square-root equation and its (numerical) solution procedures in heterogeneous media. We show that the double-square-root equation generates the adjoint of the single-scattering modelling operator upon substituting the leading term of the generalized Bremmer series for the background Green function. This adjoint operator yields the process of 'wave-equation' imaging. We finally decompose the wave-equation imaging process into common image point gathers in accordance with the characteristic strips in the wavefront set of the data.
de Hoop, M. V., Le Rousseau, J. H., & Biondi, B. L. (2003). Symplectic structure of wave-equation imaging: A path-integral approach based on the double-square-root equation. Geophysical Journal International, 153(1), 52–74. https://doi.org/10.1046/j.1365-246X.2003.01877.x