The earthquake seismology and seismic exploration communities have developed a variety of seismic imaging methods for passive- and active-source data. Despite the seemingly different approaches and underlying principles, many of those methods are based in some way or another on Green's theorem. The aim of this paper is to discuss a variety of imaging methods in a systematic way, using a specific form of Green's theorem (the homogeneous Green's function representation) as a common starting point. The imaging methods we cover are time-reversal acoustics, seismic interferometry, back propagation, source–receiver redatuming and imaging by double focusing. We review classical approaches and discuss recent developments that fully account for multiple scattering, using the Marchenko method. We briefly indicate new applications for monitoring and forecasting of responses to induced seismic sources, which are discussed in detail in a companion paper.

Through the years, the earthquake seismology and seismic exploration communities have developed a variety
of seismic imaging methods for passive- and active-source data, based on a wide range of principles such as
time-reversal acoustics, Green's function retrieval by noise correlation (a form of seismic interferometry),
back propagation (also known as holography) and source–receiver redatuming.
Many of these methods are rooted in some way or another in Green's theorem

We start by reviewing a specific form of Green's theorem, namely the
classical representation of the homogeneous Green's function, originally
developed for optical holography

Although the solid Earth supports elastodynamic (vectorial) waves, to facilitate the comparison of the different methods discussed in this paper we have chosen to consider scalar waves only. Scalar waves, which obey the acoustic wave equation, serve as an approximation for compressional body waves propagating through the solid Earth, or for the fundamental mode of surface waves propagating along the Earth's surface, depending on the application. In several places we give references to extensions of the methods that account for the full elastodynamic wave field.

We consider an inhomogeneous lossless acoustic
medium, with mass density

The time-reversal of the Green's function,

Configuration for the homogeneous Green's function
representation (Eq.

In the following sections we discuss different imaging methods.
Each time we first introduce the specific method in an intuitive way, after
which we present a more quantitative derivation based on Eq. (

Time-reversal acoustics has been pioneered by Fink and co-workers

Principle of time-reversal acoustics.

The time-reversal principle can be made more quantitative using Green's theorem

It should be noted that the integration in Eq. (

Figure

Time-reversal acoustics in a layered medium.

Whereas in Fig.

Under certain conditions, the cross-correlation of passive ambient-noise
recordings at two receivers converges to the response that would be measured
at one of the receivers if there were an impulsive source at the position of
the other. This methodology, which creates a virtual source at the position
of an actual receiver, is known as Green's function retrieval by noise
correlation (a form of seismic interferometry). At the ultrasonic scale it has
been pioneered by Weaver and co-workers

Figure

Principle of seismic interferometry.

We use the homogeneous Green's function representation of Eq. (

Seismic interferometry with body waves in a layered medium.
The upper boundary is a free surface.

According to Eqs. (

Figure

For both methods discussed here (surface-wave interferometry and body-wave
interferometry) we assumed that the noise sources are regularly distributed
along a part of

Given a wave field observed at the surface of a medium, the field inside the
medium can be obtained by back propagation

Principle of back propagation.

Back propagation is conceptually different from time-reversal acoustics. In time-reversal acoustics the observed wave field is reversed in time and (physically or numerically) emitted into the medium, whereas in back propagation the original observed wave field is numerically back-propagated through the medium by a time-reversed Green's function. Despite this conceptual difference (time reversal of the wave field versus time reversal of the propagation operator), it is not surprising that these methods are very similar from a mathematical point of view.

A quantitative discussion of back propagation follows from Eq. (

In the previous section we discussed back propagation of

The redatumed response

The applications of Green's theorem, discussed in Sect.

The focusing function

Numerical example of the focusing function.

The classical homogeneous Green's function representations (Eqs.

The focusing function is illustrated using a numerical example in Fig.

Given the focusing function for a focal point at

The representations of Eqs. (

Time-reversal acoustics in a layered medium.

We discuss a modification of time-reversal acoustics. Assuming the focusing
functions are available for

Obtaining an accurate focus as in Fig.

We modify the approach for back propagation.
By interchanging

Principle of modified back propagation.

This back propagation method has an interesting application in the monitoring of induced
seismicity. Assuming

Reflectivity images obtained using the double-focusing method.

We modify the approach for source–receiver redatuming. First, in Eq. (

If we applied imaging to the retrieved response

Other methods exist that deal with internal multiple reflections in imaging.

The classical homogeneous Green's function representation, originally developed for optical image formation by holograms, expresses the Green's function plus its time-reversal between two arbitrary points in terms of an integral along a surface enclosing these points. It forms a unified basis for a variety of seismic imaging methods, such as time-reversal acoustics, seismic interferometry, back propagation, source–receiver redatuming and imaging by double focusing. We have derived each of these methods by applying some simple manipulations to the classical homogeneous Green's function representation, which implies that these methods are all very similar. As a consequence, they share the same advantages and limitations. Because the underlying representation is exact, it accounts for all orders of multiple scattering. This property is exploited by seismic interferometry in a layered medium below a free surface and, to some extent, by time-reversal acoustics in a medium with random scatterers. However, in most cases multiple scattering is not correctly handled because in practical situations data are not available on a closed surface. We also discussed a single-sided homogeneous Green's function representation, which requires access to the medium from one side only, say from the Earth's surface. This single-sided representation ignores evanescent waves, but it accounts for all orders of multiple scattering, similar as the classical closed-surface representation. We used the single-sided representation as the basis for deriving modifications of time-reversal acoustics, back propagation, source–receiver redatuming and imaging by double focusing. These methods account for multiple scattering and can be used to obtain accurate images of the source or the subsurface, without artefacts related to multiple scattering. Another interesting application is the monitoring and forecasting of responses to induced seismic sources, which is discussed in detail in a companion paper.

The modeling and imaging software that was used to generate the numerical examples in this paper
can be downloaded from

The supplement related to this article is available online at:

JB and JT developed the software and generated the numerical examples. KW wrote the paper. All authors reviewed the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Advances in seismic imaging across the scales”. It is a result of the EGU General Assembly 2018, Vienna, Austria, 8–13 April 2018.

We thank the two reviewers, Andreas Fichtner and Robert Nowack, for their valuable feedback, which helped us to improve the paper. This work has received funding from the European Union's Horizon 2020 research and innovation programme: European Research Council (grant agreement no. 742703).

This paper was edited by Nicholas Rawlinson and reviewed by Andreas Fichtner and Robert Nowack.