# Imaging Seismic Reflections

T.J.P.M. op 't Root, Timotheus Johannes Petrus Maria Op 't Root

Research output: ThesisPhD Thesis - Research UT, graduation UT

### Abstract

The goal of reflection seismic imaging is making images of the Earth subsurface using surface measurements of reflected seismic waves. Besides the position and orientation of subsurface reflecting interfaces it is a challenge to recover the size or amplitude of the discontinuities. We investigate two methods or techniques of reflection seismic imaging in order to improve the amplitude. First we study the \textsl{one-way wave equation} with attention to the amplitude of the waves. In our study of \textsl{reverse-time migration} the amplitude refers to the amplitude of the image. Though the approach in the thesis is formal, our results have practical implications for seismic imaging algorithms, which improve or correct the amplitudes. The one-way wave equation is a $1^{\rm st}$-order equation that describes wave propagation in a predetermined direction. We derive the equation and identify the steps that determine the wave amplitude. We introduce a \textsl{symmetric square root operator} and a wave field normalization operator and show that they provide the correct amplitude. The idea to use a symmetric square root is generally applicable. Our amplitude claims are numerically verified. The one-way wave equation is an application of \textsl{pseudo-differential operators}. \textsl{Reverse-time migration} (RTM) is an imaging method that uses simulations of the source and receiver wave fields through a slowly varying estimate of the subsurface medium. The receiver wave is an in reverse time continued field that matches the measurements. An \textsl{imaging condition} transforms the fields into an image of the small scale medium contrast. We investigate the linearized inverse problem and use the RTM procedure to reconstruct the perturbation of the medium. We model the scattering event by the scattering operator, which maps the medium perturbation to the scattered wave. We propose an approximate inverse of the scattering operator, derive a novel imaging condition and show that it yields a reconstruction of the perturbation with correct amplitude. The study extensively uses the theory of \textsl{Fourier integral operators} (FIO). Besides that we globally characterize the scattering operator as a FIO, we also obtain a local expression to calculate the amplitude explicitly. The claims are confirmed and illustrated by numerical simulations.
Original language Undefined University of Twente van Groesen, Embrecht W.C., SupervisorStolk, C.C., Advisor Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) 6 Apr 2011 Zutphen University of Twente 978-90-365-3150-4 https://doi.org/10.3990/1.9789036531504 Published - 6 Apr 2011

### Keywords

• MSC-00A69
• Reverse-time migration
• One-way wave modeling
• EWI-19953
• Seismic inversion
• IR-76619
• METIS-277600
• NWO-VIDI 639.032.509

### Cite this

op 't Root, T. J. P. M., & Op 't Root, T. J. P. M. (2011). Imaging Seismic Reflections. Zutphen: University of Twente. https://doi.org/10.3990/1.9789036531504
op 't Root, T.J.P.M. ; Op 't Root, Timotheus Johannes Petrus Maria. / Imaging Seismic Reflections. Zutphen : University of Twente, 2011. 110 p.
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title = "Imaging Seismic Reflections",
abstract = "The goal of reflection seismic imaging is making images of the Earth subsurface using surface measurements of reflected seismic waves. Besides the position and orientation of subsurface reflecting interfaces it is a challenge to recover the size or amplitude of the discontinuities. We investigate two methods or techniques of reflection seismic imaging in order to improve the amplitude. First we study the \textsl{one-way wave equation} with attention to the amplitude of the waves. In our study of \textsl{reverse-time migration} the amplitude refers to the amplitude of the image. Though the approach in the thesis is formal, our results have practical implications for seismic imaging algorithms, which improve or correct the amplitudes. The one-way wave equation is a $1^{\rm st}$-order equation that describes wave propagation in a predetermined direction. We derive the equation and identify the steps that determine the wave amplitude. We introduce a \textsl{symmetric square root operator} and a wave field normalization operator and show that they provide the correct amplitude. The idea to use a symmetric square root is generally applicable. Our amplitude claims are numerically verified. The one-way wave equation is an application of \textsl{pseudo-differential operators}. \textsl{Reverse-time migration} (RTM) is an imaging method that uses simulations of the source and receiver wave fields through a slowly varying estimate of the subsurface medium. The receiver wave is an in reverse time continued field that matches the measurements. An \textsl{imaging condition} transforms the fields into an image of the small scale medium contrast. We investigate the linearized inverse problem and use the RTM procedure to reconstruct the perturbation of the medium. We model the scattering event by the scattering operator, which maps the medium perturbation to the scattered wave. We propose an approximate inverse of the scattering operator, derive a novel imaging condition and show that it yields a reconstruction of the perturbation with correct amplitude. The study extensively uses the theory of \textsl{Fourier integral operators} (FIO). Besides that we globally characterize the scattering operator as a FIO, we also obtain a local expression to calculate the amplitude explicitly. The claims are confirmed and illustrated by numerical simulations.",
keywords = "MSC-00A69, Reverse-time migration, One-way wave modeling, EWI-19953, Seismic inversion, IR-76619, METIS-277600, NWO-VIDI 639.032.509",
author = "{op 't Root}, T.J.P.M. and {Op 't Root}, {Timotheus Johannes Petrus Maria}",
note = "NWO-VIDI 639.032.509",
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doi = "10.3990/1.9789036531504",
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isbn = "978-90-365-3150-4",
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op 't Root, TJPM & Op 't Root, TJPM 2011, 'Imaging Seismic Reflections', University of Twente, Zutphen. https://doi.org/10.3990/1.9789036531504

Imaging Seismic Reflections. / op 't Root, T.J.P.M.; Op 't Root, Timotheus Johannes Petrus Maria.

Zutphen : University of Twente, 2011. 110 p.

Research output: ThesisPhD Thesis - Research UT, graduation UT

TY - THES

T1 - Imaging Seismic Reflections

AU - op 't Root, T.J.P.M.

AU - Op 't Root, Timotheus Johannes Petrus Maria

N1 - NWO-VIDI 639.032.509

PY - 2011/4/6

Y1 - 2011/4/6

N2 - The goal of reflection seismic imaging is making images of the Earth subsurface using surface measurements of reflected seismic waves. Besides the position and orientation of subsurface reflecting interfaces it is a challenge to recover the size or amplitude of the discontinuities. We investigate two methods or techniques of reflection seismic imaging in order to improve the amplitude. First we study the \textsl{one-way wave equation} with attention to the amplitude of the waves. In our study of \textsl{reverse-time migration} the amplitude refers to the amplitude of the image. Though the approach in the thesis is formal, our results have practical implications for seismic imaging algorithms, which improve or correct the amplitudes. The one-way wave equation is a $1^{\rm st}$-order equation that describes wave propagation in a predetermined direction. We derive the equation and identify the steps that determine the wave amplitude. We introduce a \textsl{symmetric square root operator} and a wave field normalization operator and show that they provide the correct amplitude. The idea to use a symmetric square root is generally applicable. Our amplitude claims are numerically verified. The one-way wave equation is an application of \textsl{pseudo-differential operators}. \textsl{Reverse-time migration} (RTM) is an imaging method that uses simulations of the source and receiver wave fields through a slowly varying estimate of the subsurface medium. The receiver wave is an in reverse time continued field that matches the measurements. An \textsl{imaging condition} transforms the fields into an image of the small scale medium contrast. We investigate the linearized inverse problem and use the RTM procedure to reconstruct the perturbation of the medium. We model the scattering event by the scattering operator, which maps the medium perturbation to the scattered wave. We propose an approximate inverse of the scattering operator, derive a novel imaging condition and show that it yields a reconstruction of the perturbation with correct amplitude. The study extensively uses the theory of \textsl{Fourier integral operators} (FIO). Besides that we globally characterize the scattering operator as a FIO, we also obtain a local expression to calculate the amplitude explicitly. The claims are confirmed and illustrated by numerical simulations.

AB - The goal of reflection seismic imaging is making images of the Earth subsurface using surface measurements of reflected seismic waves. Besides the position and orientation of subsurface reflecting interfaces it is a challenge to recover the size or amplitude of the discontinuities. We investigate two methods or techniques of reflection seismic imaging in order to improve the amplitude. First we study the \textsl{one-way wave equation} with attention to the amplitude of the waves. In our study of \textsl{reverse-time migration} the amplitude refers to the amplitude of the image. Though the approach in the thesis is formal, our results have practical implications for seismic imaging algorithms, which improve or correct the amplitudes. The one-way wave equation is a $1^{\rm st}$-order equation that describes wave propagation in a predetermined direction. We derive the equation and identify the steps that determine the wave amplitude. We introduce a \textsl{symmetric square root operator} and a wave field normalization operator and show that they provide the correct amplitude. The idea to use a symmetric square root is generally applicable. Our amplitude claims are numerically verified. The one-way wave equation is an application of \textsl{pseudo-differential operators}. \textsl{Reverse-time migration} (RTM) is an imaging method that uses simulations of the source and receiver wave fields through a slowly varying estimate of the subsurface medium. The receiver wave is an in reverse time continued field that matches the measurements. An \textsl{imaging condition} transforms the fields into an image of the small scale medium contrast. We investigate the linearized inverse problem and use the RTM procedure to reconstruct the perturbation of the medium. We model the scattering event by the scattering operator, which maps the medium perturbation to the scattered wave. We propose an approximate inverse of the scattering operator, derive a novel imaging condition and show that it yields a reconstruction of the perturbation with correct amplitude. The study extensively uses the theory of \textsl{Fourier integral operators} (FIO). Besides that we globally characterize the scattering operator as a FIO, we also obtain a local expression to calculate the amplitude explicitly. The claims are confirmed and illustrated by numerical simulations.

KW - MSC-00A69

KW - Reverse-time migration

KW - One-way wave modeling

KW - EWI-19953

KW - Seismic inversion

KW - IR-76619

KW - METIS-277600

KW - NWO-VIDI 639.032.509

U2 - 10.3990/1.9789036531504

DO - 10.3990/1.9789036531504

M3 - PhD Thesis - Research UT, graduation UT

SN - 978-90-365-3150-4

PB - University of Twente

CY - Zutphen

ER -

op 't Root TJPM, Op 't Root TJPM. Imaging Seismic Reflections. Zutphen: University of Twente, 2011. 110 p. https://doi.org/10.3990/1.9789036531504