Assumptions and goals for least squares migration
Daniel O. Trad, Samuel H. Gray
Least squares migration (LSMIG) uses the assumption that, if we have an operator that can create data from a reflectivity function, the optimal image will predict the actual recorded data with minimum square error. For this assumption to be true, it is also required that: a) the prediction operator must be error-free; b) model elements not seen by the operator should be constrained by other means; c) data weakly predicted by the operator should make limited contribution to the solution. Under these conditions, LSMIG has the advantage over simple migration of being able to remove interference between different model components. LSMIG does that by deconvolving or inverting the so-called Hessian operator. The Hessian is the cascade of forward modeling and migration; for each image point, it computes the effects of interference from other image points (point-spread function) given the actual recording geometry and the subsurface velocity model. Because the Hessian contains illumination information (along its diagonal), and information about the model cross-correlation produced by non-orthogonality of basis functions, its inversion produces illumination compensation and increases resolution. In addition, sampling deficiencies in the recording geometry map to the Hessian (both diagonal and non-diagonal elements), so LSMIG has the potential to remove sampling artifacts as well. These (illumination compensation, resolution, mitigating recording deficiencies) are the three main goals of LSMIG, although the first one can be achieved by cheaper techniques. To invert the Hessian, LSMIG relies on the residual errors during iterations. Iterative algorithms, like conjugate gradient and others, use the residuals to calculate the direction and amplitudes (gradient and step size), of the necessary corrections to the reflectivity function or model. Failure of conditions a), b) or c) leads the inversion to calculate incorrect model updates, which translate to noise in the final image. In this paper we will discuss these conditions for Kirchhoff migration and RTM.