# Two-way Wave-equation Migration

Friday, October 21, 2005 - 9:00am - 9:50am

EE/CS 3-180

Wim Mulder (The Shell Group)

Joint with R.-E. Plessix.

The goal of seismic surveying is the determination of the structure and properties of the subsurface. Oil and gas exploration is

restricted to the upper 5 to 10 kilometers. Seismic data are usually recorded at the earth's surface as a function of time. Creating a

subsurface image from these data is called migration.

Seismic data are band-limited with frequencies in the range from about 10 to 60 Hz. As a result, they are mainly generated by short-range

variations in the subsurface impedance, the product of velocity and density. The common approach towards migration is the construction of

a reflection-free background velocity model from the apparent travel times from source to receiver. In this background model, the

migration algorithm maps the data amplitudes to the impedance contrasts that generated them. Single scattering is implicitly assumed.

The wave propagation in the background model is usually described by an approximation to the wave equation to keep the required computer

time down to months. Ray tracing used to be a popular choice, but is gradually taken over by one-way (paraxial or parabolic) wave

equation approximations. We have investigated the use of the acoustic wave equation, which will we call the two-way wave equation in

order to distinguish it from the widely used one-way wave equation. The two-way approach provides a more accurate description of wave

propagation than the one-way method, particularly near underground structures that have steep interfaces. The one-way equation requires

considerably less computer time in 3D, but in 2D the one-way and two-way methods compete.

Migration algorithms can be derived from the least-squares error that measures the difference between observed and modeled data. The

gradient of this functional with respect to the model parameters is a migration image. This gradient can be used to minimize the error,

but leads to a problem that is nonlinear in the model parameters and has many local minima. Gradient-based optimization algorithms will

only provide meaningful results if the initial model is close to the global minimum. Because migration is computational very costly,

global searches are not an option.

An alternative is to return to the classic approach, where migration serves to map the impedance contrasts without changing the wave

propagation model. This can be achieved in the context of the two-way wave equation by assuming that the contrasts are small

perturbations, leading to a linearization with respect to the model parameters. This is the well-known Born approximation.

A disadvantage of the two-way method is that it models all waves, not only reflections. This may produce artifacts in the images. We will

discuss ways to remove them. In the nonlinear approach, these artifacts can actually be used to update the background model. Also,

multiple reflections can be included in the minimization. In general, however, the least-squares functional is not very well suited to

determine the background model and an alternative cost functional needs to be sought.

Examples on synthetic and real data will serve as illustrations.

The goal of seismic surveying is the determination of the structure and properties of the subsurface. Oil and gas exploration is

restricted to the upper 5 to 10 kilometers. Seismic data are usually recorded at the earth's surface as a function of time. Creating a

subsurface image from these data is called migration.

Seismic data are band-limited with frequencies in the range from about 10 to 60 Hz. As a result, they are mainly generated by short-range

variations in the subsurface impedance, the product of velocity and density. The common approach towards migration is the construction of

a reflection-free background velocity model from the apparent travel times from source to receiver. In this background model, the

migration algorithm maps the data amplitudes to the impedance contrasts that generated them. Single scattering is implicitly assumed.

The wave propagation in the background model is usually described by an approximation to the wave equation to keep the required computer

time down to months. Ray tracing used to be a popular choice, but is gradually taken over by one-way (paraxial or parabolic) wave

equation approximations. We have investigated the use of the acoustic wave equation, which will we call the two-way wave equation in

order to distinguish it from the widely used one-way wave equation. The two-way approach provides a more accurate description of wave

propagation than the one-way method, particularly near underground structures that have steep interfaces. The one-way equation requires

considerably less computer time in 3D, but in 2D the one-way and two-way methods compete.

Migration algorithms can be derived from the least-squares error that measures the difference between observed and modeled data. The

gradient of this functional with respect to the model parameters is a migration image. This gradient can be used to minimize the error,

but leads to a problem that is nonlinear in the model parameters and has many local minima. Gradient-based optimization algorithms will

only provide meaningful results if the initial model is close to the global minimum. Because migration is computational very costly,

global searches are not an option.

An alternative is to return to the classic approach, where migration serves to map the impedance contrasts without changing the wave

propagation model. This can be achieved in the context of the two-way wave equation by assuming that the contrasts are small

perturbations, leading to a linearization with respect to the model parameters. This is the well-known Born approximation.

A disadvantage of the two-way method is that it models all waves, not only reflections. This may produce artifacts in the images. We will

discuss ways to remove them. In the nonlinear approach, these artifacts can actually be used to update the background model. Also,

multiple reflections can be included in the minimization. In general, however, the least-squares functional is not very well suited to

determine the background model and an alternative cost functional needs to be sought.

Examples on synthetic and real data will serve as illustrations.

MSC Code:

35L05

Keywords: