I establish a general theory on the true-amplitude migration with limited aperture. The key point of this theory is to find the true-amplitude weight function, eliminating the contribution from the boundary of the migration aperture, or edge effect. I show that the true-amplitude weight function is a singular generalized function defined by the migration aperture. If it is used without regularization, the singular weight function reduces a diffraction stack to the distribution of the amplitude at a given traveltime to every point along a corresponding isochron. After regularization, I obtain a formula for the true-amplitude weight function as a weak solution having the same global properties as the fundamental solution to the governing equation derived here. In comparison with other published weight functions that disregard migration aperture, this theory contains a taper function that makes the obtained weight function suitable for the migration with limited aperture. If the migration aperture goes to infinity, my weight function reduces to the one cited in the literature. Furthermore, I present a theoretical guideline for designing both the taper function and the migration aperture. Specifically, an optimal migration aperture should be larger than or equal to the region confined by the outer boundary of the second transformed Fresnel zone. The exact size of the optimal migration aperture is obtained when an equation describing the global property of the true-amplitude weight function is satisfied.
Using the true-amplitude weight function developed, together with a first-order perturbation of the traveltime surfaces of point-diffracted rays, I also obtain an asymptotic formula for the depth-migrated image. According to the formula, diffraction stack results in frequency scaling in the depth domain. This frequency scaling gives physical insight into the problem of pulse stretching in depth migration. If the formula is expanded into a Taylor series and if the Taylor series is truncated at its linear term, the formula published in the literature can be derived. Therefore, this paper also presents a general theory on the depth-migrated image.