One-way versions of the Kirchhoff integral

The conventional Kirchhoff integral, based on the two-way wave equation, states how the acoustic pressure at a point A inside a closed surface S can be calculated when the acoustic wave field is known on S. In its general form, the integrand consists of two terms: one term contains the gradient of a Green's function and the acoustic pressure; the other term contains a Green's function and the gradient of the acoustic pressure.The integrand can be simplified by choosing reflecting boundary conditions for the two-way Green's functions in such a way that either the first term or the second term vanishes on S. This conventional approach to deriving Rayleigh-type integrals has practical value only for media with small contrasts, so that the two-way Green's functions do not contain significant multiple reflections. We present a modified approach for simplifying the integrand of the Kirchhoff integral by choosing absorbing boundary conditions for the one-way Green's functions. The resulting Rayleigh-type integrals are the theoretical basis for true amplitude one-way wave-field extrapolation techniques in inhomogeneous media with significant contrasts.