Marchenko-Lippmann-Schwinger inversion

Abstract

Seismic wave reflections recorded at the Earth’s surface provide a rich source of information about the structure of the subsurface. These reflections occur due to changes in the material properties of the Earth; in the acoustic approximation, these are the density of the Earth and the velocity of seismic waves travelling through it. Therefore, there is a physical relationship between the material properties of the Earth and the reflected seismic waves that we observe at the surface. This relationship is non-linear, due to the highly scattering nature of the Earth, and to our inability to accurately reproduce these scattered waves with the low resolution velocity models that are usually available to us. Typically, we linearize the scattering problem by assuming that the waves are singly-scattered, requiring multiple reflections to be removed from recorded data at great effort and with varying degrees of success. This assumption is called the Born approximation.

The equation that describes the relationship between the Earth’s properties and the fully-scattering reflection data is called the Lippmann-Schwinger equation, and this equation is linear if the full scattering wavefield inside the Earth could be known. The development of Marchenko methods makes such wavefields possible to estimate using only the surface reflection data and an estimate of the direct wave from the surface to each point in the Earth. Substituting the results from a Marchenko method into the Lippmann-Schwinger equation results in a linear equation that includes all orders of scattering. The aim of this thesis is to determine whether higher orders of scattering improve the linear inverse problem from data to velocities, by comparing linearized inversion under the Born approximation to the inversion of the linear Lippmann-Schwinger equation.

This thesis begins by deriving the linear Lippmann-Schwinger and Born inverse problems, and reviewing the theoretical basis for Marchenko methods. By deriving the derivative of the full scattering Green’s function with respect to the model parameters of the Earth, the gradient direction for a new type of least-squares full waveform inversion called Marchenko-Lippmann-Schwinger full waveform inversion is defined that uses all orders of scattering.

By recreating the analytical 1D Born inversion of a boxcar perturbation by Beydoun and Tarantola (1988), it is shown that high frequency-sampling density is required to correctly estimate the amplitude of the velocity perturbation. More importantly, even when the scattered wavefield is defined to be singly-scattering and the velocity model perturbation can be found without matrix inversion, Born inversion cannot reproduce the true velocity structure exactly. When the results of analytical inversion are compared to inversions where the inverse matrices have been explicitly calculated, the analytical inversion is found to be superior. All three matrix inversion methods are found to be extremely ill-posed. With regularisation, it is possible to accurately determine the edges of the perturbation, but not the amplitude.

Moving from a boxcar perturbation with a homogeneous starting velocity to a many-layered 1D model and a smooth representation of this model as the starting point, it is found that the inversion solution is highly dependent on the starting model. By optimising an iterative inversion in both the model and data domains, it is found that optimising the velocity model misfit does not guarantee improvement in the resulting data misfit, and vice versa. Comparing unregularised inversion to inversions with Tikhonov damping or smoothing applied to the kernel matrix, it is found that strong Tikhonov damping results in the most accurate velocity models. From the consistent under-performance of Lippmann-Schwinger inversion when using Marchenko-derived Green’s functions compared to inversions carried out with true Green’s functions, it is concluded that the fallibility of Marchenko methods results in inferior inversion results.

Born and Lippmann-Schwinger inversion are tested on a 2D syncline model. Due to computational limitations, using all sources and receivers in the inversion required limiting the number of frequencies to 5. Without regularisation, the model update is uninterpretable due to the presence of strong oscillations across the model. With strong Tikhonov damping, the model updates obtained are poorly scaled, have low resolution, and low amplitude oscillatory noise remains.

By replacing the inversion of all sources simultaneously with single source inversions, it is possible to reinstate all frequencies within our limited computational resources. These single source model updates can be stacked similarly to migration images to improve the overall model update. As predicted by the 1D analytical inversion, restoring the full frequency bandwidth eliminates the oscillatory noise from the inverse solution. With or without regularisation, Born and Lippmann-Schwinger inversion results are found to be nearly identical. When Marchenko-derived Green’s functions are introduced, the inversion results are worse than either the Born inversion or the Lippmann-Schwinger inversion without Marchenko methods. On this basis, one concludes that the inclusion of higher order scattering does not improve the outcome of solving the linear inverse scattering problem using currently available methods. Nevertheless, some recent developments in the methods used to solve the Marchenko equation hold some promise for improving solutions in future.