Acoustic and elastic least-squares two-way wave equation migration with exact adjoint operator

• Author / Creator

  Xu, Linan

• A major problem in exploration seismology entails estimating subsurface structures and properties via linearized inversion. The problem is often called ''least-squares migration" where seismic imaging is posed as an iterative least-squares problem. The iterative solution employs the method of conjugate gradients which requires two operators: the forward operator and the adjoint operator. This thesis investigates the design of the forward operator and its associated exact adjoint for both acoustic and elastic least-squares migration. The forward operator is derived using the Born approximation and Green's functions of the two-way wave equation. A few key steps were followed to obtain an adjoint operator that has the exact adjoint formulation of the forward operator. I first derive the Born approximation and discretize Green's functions using the finite difference method, where I have adopted a staggered grid algorithm with stepping in the time domain. Then, a simple workflow was used to describe the discrete forward operator in terms of the concatenated multiplication of matrices. Finally, the exact adjoint operator is obtained by taking the transpose of the discrete forward operator. The adjointness of the forward and adjoint operators can be verified by the dot product test. Unlike the conventional adjoint operator derived via the discretization of continuous kernels, I observe that the proposed exact adjoint operator can pass the dot product test in machine precision, which implies that the forward and the proposed adjoint operator achieve sufficient accuracy to use conjugate gradients to solve the aforementioned problem of least-squares migration. While the forward and exact adjoint operators are derived in the form of matrices, creating explicit matrices in numerical solvers does not provide a memory-efficient implementation. To deal with this memory issue, a matrix-free programming process is applied to develop the forward operator and its exact adjoint operator. Preconditioning operators are also investigated to solve the least-squares migration for extended shot-index images efficiently. Finally, the proposed method is tested via synthetic examples. Compared with conventional migration techniques, least-squares migration (both acoustic and elastic) is capable of attenuating low-wavenumber artifacts, compensating for insufficient illumination, and increasing the resolution of seismic images. Elastic least-squares migration provides an additional benefit of suppressing multi-parameter cross-talk.

• Subjects / Keywords

  • Elastic
  • Exact adjoint
  • Inversion
  • Acoustic
  • Conjugate gradients
  • Least-squares migration
  • Wave equation
  • Thesis

• Graduation date

  Spring 2017

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R34X54V3M

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.