MarchenkoLippmannSchwinger inversion
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Date
15/06/2023Author
Cummings, Dominic Gerard
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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
nonlinear, 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 singlyscattered, 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 fullyscattering reflection data is called the LippmannSchwinger 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 LippmannSchwinger 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
LippmannSchwinger equation.
This thesis begins by deriving the linear LippmannSchwinger 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 leastsquares full waveform
inversion called MarchenkoLippmannSchwinger 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 frequencysampling density is required
to correctly estimate the amplitude of the velocity perturbation. More importantly,
even when the scattered wavefield is defined to be singlyscattering 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 illposed. 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
manylayered 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 underperformance of LippmannSchwinger inversion when using
Marchenkoderived 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 LippmannSchwinger 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 LippmannSchwinger
inversion results are found to be nearly identical. When Marchenkoderived Green’s
functions are introduced, the inversion results are worse than either the Born inversion
or the LippmannSchwinger 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.
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