True-amplitude weight functions in 3D limited-aperture migration revisited

The true-amplitude weight function in 3D limited-aperture migration is obtained by extending its formula at an actual reflection point to any arbitrary subsurface point. This implies that the recorded seismic signal is a delta impulse. When the weight function is used in depth migration, it results in an amplitude distortion depending on the vertical distance from the target reflector. This distortion exists even if the correct velocity model is used. If the image point lies at a depth shallower than the half-offset, the distortion cannot be ignored, even for a spatial wavelet having a short length. Using paraxial ray theory, I find a formula for the true-amplitude weight function causing no amplitude distortion, under the condition that the earth's surface is smoothly curved. However, the formula is reflector dependent. As a result, amplitude distortion, in parallel with pulse distortion, is an intrinsic effect in depth migration, and true-amplitude migration without amplitude distortion is possible only when the position of the target reflector is known. If this is the case, true-amplitude migration without amplitude distortion can be realized by filtering the output of a simple unweighted diffraction stack with the weight function presented here. Also, using Taylor expansions with respect to the vertical, I derive an alternative formula for the true-amplitude weight function that causes no amplitude distortion. Starting from this formula, I show that the previously published reflector-independent true-amplitude weight function is a zero-order approximation to the one given here.