Making the most out of the least (squares migration)

Abstract

SUMMARYStandard migration images can suffer from migration artifacts due to 1) poor source-receiver sampling, 2) weak amplitudes caused by geometric spreading, 3) attenuation, 4) defocusing, 5) poor resolution due to limited source-receiver aperture, and 6) ringiness caused by a ringy source wavelet. To partly remedy these problems, least-squares migration (LSM), also known as linearized seismic inversion or migration deconvolution (MD), proposes to linearly invert seismic data for the reflectivity distribution. If the migration velocity model is sufficiently accurate, then LSM can mitigate many of the above problems and lead to a more resolved migration image, sometimes with twice the spatial resolution. However, there are two problems with LSM: the cost can be an order of magnitude more than standard migration and the quality of the LSM image is no better than the standard image for velocity errors of 5% or more. We now show how to get the most from leastsquares migration by reducing the cost and velocity sensitivity of LSM. LEAST-SQUARES MIGRATION THEORYThe theory for least-squares migration is described in Nemeth et al. (1999) and Duquet et al. (2000), where the smooth background does not change with iteration number. Only the reflectivity distribution is updated at each iteration. This algorithm is equivalent to linearized waveform inversion (Lailly, 1984), and can be described as iteratively updating the reflectivity vector m bywhere α is the step length, P is the preconditioning matrix that approximates the inverse to the Hessian matrix, d is the recorded reflection data, k represents the iteration index, and L represents the linearized forward modeling operator that uses the smooth background velocity model 1 . If the preconditioner is inadequate, a conjugate gradient or quasi-Newton method is used to iteratively update the solution. In practice, the algorithm is often implemented in the time-space domain.The implementation of LSM is described in Nemeth et al. (1999) and Duquet et al. (2000) for diffraction stack migration and in Plessix and Mulder (2004) for reverse time migration. Typically, diffraction stack migration provides images with fewer artifacts because it only smears reflections along the migration ellipses. In contrast, RTM automatically generates upgoing reflections from reflectors, so residuals are also smeared between 1 A smooth velocity model is typically used so as to avoid smearing residuals along the rabbit-ear wavepaths. The smearing of residuals should only be along the migration ellipses that are tangent to the reflector boundaries (Zhan et al., 2014). reflecting interfaces (Guitton, 2006; Liu et al., 2011;Zhan et al., 2014) to give rise to unwanted migration artifacts. To avoid such updates, we smooth the migration velocity model (McMechan, 1983;Loewenthal et al., 1987;Fletcher et al., 2005). To mitigate problems with an inaccurate migration velocity model, regularization terms or constraints (Sacchi et al., 2006;Guitton, 2006; Wang et al., 2011;Dong et al., 2012;Dai, 2013;Dai an...

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