1

Chapter 2

Wavefield representation, propagation and

imaging using localized waves: Beamlet, Curvelet

and Dreamlet

Ru-Shan Wu and Jinghuai Gao

This Chapter is a review on the application of wavelet transform to wave propagation and

imaging. The Authors review the development of phase-space localized propagators in

heterogeneous media and the application to geophysical, especially seismic imaging. In

the first part, phase-space localization, mainly along the line of time-frequency

localization is reviewed. Then phase-space localization using generalized wavelet

transform applied to wave field and one-way propagator decompositions are reviewed

and analyzed. Physically the phase-space localized propagators are beamlet or

wavepacket propagators which are propagator matrices for short-range iterative

propagation. When asymptotic solutions are applied to the beamlet for long-range

propagation, beamlets evolve into global beams. Various asymptotic beam propagation

methods have been developed in the past, such as the Gaussian beam, complex ray,

Gaussian packet, coherent state, and more recently the curvelet and dreamlet (tensor

product of drumbeat and beamlet) methods. Local perturbation method for propagation in

strongly heterogeneous media is also briefly described. Curvelet transform and its

application to propagation and imaging is reviewed in comparison with the beamlet

approach. Finally physical wavelet is introduced and dreamlet as a type of physical

wavelet on the observation surface is discussed with the recent applications to seismic

data decomposition, compression, extrapolation and imaging.

Based on the review and analysis, some conclusions are reached as follows. For

wavefield decomposition, beamlet, dreamlet and curvelet transforms have elementary

functions of directional wavelets. Beamlet and dreamlet belong to a type of physical

wavelet, representing an elementary wave (satisfying wave equation) in various

wavefield decomposition schemes using localized building elements (wavelet atoms),

such as coherent state, Gabor atom, Gabor-Daubechies frame vector, local trigonometric

basis function. Curvelet transform is a specifically defined mathematical transform,

characterized by the parabolic scaling. The parabolic scaling law, width length2

or its

generalization width wavelength2 is similar to the beam-aperture requirement for

asymptotic beam solution: the beamwidth must be smaller than the scale of heterogeneity

2

and much greater than the wavelength, i.e. a > β >> λ where is the wavelength, is

the beamwidth, and a is scale of heterogeneity. Optimal beamwidth is reached by

balancing the beam geometric spreading which is controlled by the ratio / , and the

beam-front distortion which depends on a / . Using optimal beamwidth,

beamlet/dreamlet or curvelet propagator will be sparse in smooth media for short range

propagation. For strong and rough heterogeneities, beamlet/dreamlet or curvelet

scattering will occur and asymptotic propagator may not work well. In this case, the local

perturbation method can be applied, in which the propagator is decomposed into a

background propagator and a perturbation operator for each forward marching step.

Numerical examples demonstrated the validity of the approach.

Key words: Beamlet, Curvelet, Deamlet, Wavelet transformation and decomposition,

Localized wave propagators, Gaussian beam method, Wavepacket method

2.1. Introduction

The development of the theory and methods in wavefield decomposition,

propagation, and imaging plays a fundamental and crucial role to seismic exploration, or

more general, to geophysical, including, acoustic, electromagnetic, and seismic,

exploration. In this Chapter, we concentrated on one-way propagation and its application

to imaging. In the past thirty years, various techniques and methods, including high-

frequency asymptotic method (ray method, Maslov method), the wave-equation based

phase-shift method for laterally homogeneous media, phase-screen method (or Split-step

Fourier method) for weak heterogeneous media, Hybrid Dual-domain method (Fourier-

finite-difference method and generalized screen method) for strong heterogeneous media,

have been developed for the applications in different environments. For reviews of recent

progress, see Wu and Maupin (2007). In these methods, the wavefield is expanded by

sets of basic functions such as spatial Fourier harmonics, modes, and spatial Green’s

functions. The common factor in these basic functions is their global nature in either the

space domain or the wavenumber domain. Basic functions in the wavenumber domain

are plane waves, which have the best localization in direction but no spatial localization.

Basic functions in the space domain are point sources (delta functions), which have the

best localization in space but have no direction localization. In highly heterogeneous

media, the global nature of these two kinds of basic functions creates some difficulties in

dealing with strong and fast-varying heterogeneities. To overcome this fundamental

limitation caused by the global nature of these propagators, efforts have been made to

investigate and develop wavefield decomposition and extrapolation methods with

localization in both space and direction.

The newly developed fast wavelet transform (WT), with its extension and

generalization, is considered to be a revolutionary breakthrough in signal

analysis/processing. Its impact to wave propagation and imaging has only started to be

noticed and could never be overestimated. On the other front, there has been significant

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progress in one-way wave propagation theory and algorithms, including the recently

developed multi-domain techniques, such as the fast acoustic and elastic generalized

screen propagators and their applications to imaging (Wu et al., 2012). The cross-

breeding of these two new developments has the potential of revolutionizing modeling

and imaging techniques for complex Earth media.

In this review paper, we will use the terms “wavelet”, “wavelet transform”, and

“wavelet decomposition” in a very general sense. “Wavelet transform” means a transform

using a phase-space localized, multi-scaled (single-scale is a special case) elementary

function. In the case of classical wavelet transform using a basis generated by translation

and dilation of a mother wavelet, we will refer it as the “classic” wavelet transform.

The distinctive and fundamental feature of wavelet transform (generalized) is so

called “the multi-scale phase-space localization”. In the context of wave propagation, the

phase-space localization means the localization in both the space (space-domain) and

propagation direction (wavenumber-domain). The phase-space localization gives the

possibility and flexibility in treating the fast-varying lateral heterogeneities in wave

propagation. We show the comparison between the global perturbation and the local

perturbation methods for the SEG-EAGE salt model (Fig. 2.1) to demonstrate the

advantages of localized propagators in Fig. 2.2.

The original velocity model is decomposed into a bi-scale model: A large-scale

reference medium with the scale of window-width defined for spatial localization (Fig.

2.2c), and the small-scale perturbations with respect to the reference medium (Fig. 2.2d).

Compared with the global reference medium (Fig. 2.2a) and the global perturbations (Fig.

2.2b), it is clear that the local perturbations are much weaker and are small-scale in

nature, since the large-scale, strong velocity variations are modeled by the reference

medium. In this example, we see the advantages of the space localization. Equally

important is the direction localization. With the direction localization, we can apply some

angle-related operations to the localized wavefield. We know that, the space-domain

representation of a wavefield has the maximum space localization; however, it does not

have any direction localization. The direction localization of a space-localized wavefield

can have a direction-resolution obeying the Heisenberg uncertainty principle: the location

and the direction of a wave field cannot be simultaneously given exactly, and the product

of location uncertainty and direction uncertainty must be equal or greater than a

minimum value: the Heisenberg cell. In contrast, under the h-f asymptotic theory (ray

theory), both the space and direction can be specified with infinite accuracy. However,

this assertion does not reflect the physical reality of wave phenomena and will give

erroneous predictions for wave propagation in complex media.

Fig. 2.1 2D SEG/EAGE velocity model used in this study.

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Fig. 2.2 Comparison between global perturbation and local perturbation schemes: (a) Global reference velocity

for global perturbation methods; (b) Velocity perturbation with respect to the global reference model; (c) Local

reference velocity for the beamlet method; (d) Local velocity perturbations for beamlet method.

2.2. Phase-Space Localization and Wavelet Transform

Historically, phase-space localization has been developing along two lines: the time-

frequency (or space-frequency) (x-ξ) localization and the time–scale (x-a) (or space-

scale) localization. Here “x” is the time or spatial axis, and “ξ” is the time or spatial

frequency, and “a” is the scale. In the context of wave propagation in this paper, we will

refer x as the variable (may be multidimensional) in space domain, and ξ as the spatial

frequency or wavenumber with the understanding that (x-ξ) means both (space-

frequency) and (time-frequency). Modern revolution in signal analysis has been ignited

by the introduction of “wavelet” and its fast transform algorithms. Wavelets were

introduced in the form of time-scale (x-a) localization, first by a geophysicist J. Morlet

(Morlet, 1982a, b), and later rigorously defined by a physicist A. Grossmann and J.

Morlet (Grossmann and Morlet, 1984). However, the time-frequency localization has

been developed much early in physics and engineering. They were popular in signal

processing and physics, however, as pointed by Jaffard, Meyer and Ryan (2001),

neglected by mathematicians for a long while:

“The former (time-frequency analysis) is a result of cross-fertilization between

signal processing and quantum mechanics, and mathematicians took little interest in

these ideas until recently. The latter (time-scale analysis) was pioneered by

mathematicians long before it was adopted in physics and signal processing.”

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First, let us review the concept and historical development of the time-frequency

localization since it is more related to our work on wavefield decomposition and

propagation.

2.2.1 Time-Frequency Localization

D. Gabor (1946) first introduced the concept of decomposing signals with localized

Fourier transforms for nonstationary signals. Originally, the localization proposed by

Gabor is realized by the Gaussian window and the sampling rate is critical (no

redundancy). His idea is to divide a wave into segments and to use one of these segments

as the analyzing element. Here we follow the exposition of Jaffard-Mayer-Ryan (2001)

for the mathematical definition. Assume the signal is segmented with window function

( ),w x lb l ∈Z, where b is the nominal length of the window. Then the Fourier

transform is applied to the segments with the calculation of the coefficient

( ) ( )i xe w x bl s x , where ( )s x is the signal. This action is the same as taking the

scalar products of the signal s with the “wavelets”

, ( ) ( )ikax

k lw x e w x bl (2.1)

Where a = Δ Therefore, the Gabor decomposition equation (2.1) can be thought as the

early proposal of wavelet decomposition using the time-frequency atoms. However, the

reconstruction of the Gabor expansion was not well studied at that time. In late 1940’s,

after Gabor’s proposal, many scientists, including L. Brillouin and J. von Neumann,

though that the system defined in equation (2.1) could be used as a basis to decompose

any function in L2(R). Two physicists, Balian (1981) and Low (1985) proved

independently that this is not true. Later G. Battle (1988) proved this theory from the

Heisenberg uncertainty principle (see also Daubechies and Janssen, 1993; Chapter 2 of

Feichtinger and Strohmer, 1998). Mathematically the Balian-Low obstruction can be

stated as the following (Mayer, 2001). If the two integrals 22(1 ) ( )x w x dx and

22(1 ) ( )w d are both finite (the window function is sufficiently regular and

well-localized), the functions , ( ),k lw x k, l∈Z, cannot be an orthonormal basis of L2(R).

In other words, if the wavelet w(x) is an orthogonal basis, then the Heisenberg product,

i.e. the time-frequency resolution-cell (uncertainty), must be infinite. In the case of

Gabor’s critical sampling, the frequency spreading is infinite: no frequency localization at

all. This seemingly insurmountable obstacle gave a blow to the enthusiasm of the search

for orthonormal local Fourier basis, and had blocked the way of finding efficient time-

frequency localization method for many years. To overcome the instability of Gabor’s

reconstruction, studies were focused on two approaches: one is to develop orthonormal

bases that can evade the Balian-Low obstruction; the other is to develop frame

representations which are not orthogonal but still efficient.

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2.2.1.1 Wilson-Malvar Bases and Trigonometric Bases

This approach was initiated by a physicist Wilson (Nobel laureate) (Wilson, 1971)

when working in quantum mechanics. He observed that one does not need basis functions

that distinguish between positive and negative frequencies of the same order when

studying kinetic operator. He proposed to replace the exponential function with the

cosine and sine functions alternatively. Later Malvar (1990), a scientist in signal

processing, discovered a similar approach independently. Significant improvement along

this line has been made (Daubechies et al., 1991; Coifman and Meyer, 1991;

Wickerhauser, 1993, 1994) and the local cosine/sine bases with adaptive window lengths

were constructed. The local trigonometric (cosine/sine) bases are orthonormal bases, and

are also called local Fourier bases (Auscher, 1994). The evasion of Bailian-Low theorem

through the replacement of exponential basis function with trigonometric functions can

be explained as following (Auscher, 1994). The trigonometric functions can form an

orthonormal basis if and only if { , ( )k lw x } is a tight frame with redundancy 2

(Daubechies et al., 1991). “A tight frame with redundancy 2 contains twice as many

vectors needed to form a basis and the linear combinations eliminate this redundancy to

actually yield an orthonormal basis”. However, trigonometric bases do not have the

complete directivity localization as the local Fourier decomposition (WFT or windowed

Fourier frame). The local cosine basis function has always two symmetric spectral lobes:

a positive lobe and mirror-symmetric negative lobe. This can be seen from equation

(2.14) in the next section. Therefore the directivity is not uniquely specified. Although a

local exponential function can be formed by combination of cosine basis function in the

real part and sine basis function in the imaginary part, there is always some spectral

leakage occurs at the window overlapping areas, as shown by Wickerhauser (1994,

Chapter 4.1). That is to say, there is always some false energy in the negative mirror

direction if the real energy has only a positive lobe. The amount of leakage is

proportional to the overlapping area and therefore inversely proportional to the steepness

of the bell side-slope. With steeper side-slope, the leakage will be smaller. On the other

hand, the Heisenberg uncertainty cell-size is proportional to the steepness of the bell

slope. With steeper side-slope, however, the Heisenberg cell-size will be greater, so the

directivity resolution will be poor. When the steepness is infinite, there will be no spectral

leakage, but the spectral spreading is infinite and there is no direction localization!

Therefore, the trigonometric bases are not ideal for directivity representation. There is a

tradeoff between the leakage and resolution. As for direction localization, the “lunch” is

not totally free and you have to pay something. The trick is to trade the important thing

you want with things not valuable to you. Even though not ideal for direction

localization, the trigonometric bases have been successfully applied to wave propagation

as localized propagators for imaging due to the efficiency, which will be reviewed in the

next section.

2.2.1.2. Windowed Fourier Frames

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Traditional windowed Fourier transform (WFT) uses densely overlapping windows,

which can be any type of smooth function. (e.g. Daubechies, 1992; Kaiser, 1994).

However, the implementation of the WFT is time-consuming and prevented it to be

useful in practical applications. The development of the frame theory provided an

efficient approach for discretizing the WFT. Frame representation is complete but not

orthogonal, and therefore has redundancy in the representation. Daubechies (1990) has

proved that the reconstruction using critical sampling is unstable, and for stable

reconstruction over sampling (x < 2) must hold, where x measures the size of

windowed Fourier atoms (latticed coherent states). The necessary and sufficient

conditions for the stable reconstruction have been derived based on the frame theory

(Daubechies, 1990; 1992). The dual frame vector for reconstruction (synthesis frame

vector), which in general is different from the original frame vector for decomposition

(decomposition frame vector) can be calculated efficiently (Mallat, 1998; Qian and Chen,

1996). The frame vector with a Gaussian window is called the Gabor frame or windowed

Fourier frame. It is also called as Weyl-Heisenberg coherent state frame (Daubechies,

1990) because of its relation with the Weyl-Heisenberg group in quantum field theory

(Klauder and Skagerstam,1985; Foster and Huang, 1991) or Gabor-Daubechies (G-D)

frame (Wu and Chen, 2001, 2002a, Chen et al., 2006).

Ville’s work interprets the time-frequency localization in terms of pseudo-

differential operators. This started from Weyl’s formulism and the work in quantum

mechanics of the physicist E. Wigner in the 1930’s. The technology transfer to signal

processing begins with Ville’s work in 1950’s.

2.2.2 Time-Scale Localization

Fourier transform was and still is considered to be a revolution in the history of

signal analysis. Any (or almost “any”) signal (function) can be decomposed into various

harmonic (wave-like) components. However, Fourier analysis, namely the frequency

analysis, is not convenient for scale analysis, especially for multi-scale analysis. The

Fourier decomposition is not efficient for signals with discontinuities or abrupt changes.

There was a need for another revolution for efficient multi-scale analysis. The first multi-

scale analysis with an orthogonal basis was introduced by Haar in 1909, more than 100

years after Fourier invented his analysis (1807). From Fourier to Haar, total new concepts

were introduced to mathematics, and total new weapons of frequency and scale analysis

were brought to the scientists and engineers for attacking different application problems.

However, Haar’s basis is the most discontinuous basis, composed of positive and

negative box functions with different widths (scales), and is not efficient, even not

appropriate to decompose continuous functions. In comparison, the Fourier basis is

composed of most continuous functions, the sine and cosine functions. The revolution of

multi-scale analysis can only be said is really succeeded until Daubechies orthonormal

bases were discovered. Daubechies basis function is regular and has continuous

derivatives up to certain degree. With the new-generation basis functions, the multi-scale

analysis becomes an efficient and powerful tool for scientists and engineers and many

tough problems which are untreatable before can be attacked now.

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Jaffard et al. (2001) have an excellent review on the historical development along the

line of time-scale localization. The authors (JMR) divided the development into three

periods. In the first period, wavelets had been studied individually by some

mathematicians, from Lévy, Littlewood and Paley, Franklin, Lusin, Calderόn, Coifman

and Weis, to Strőmberg. In this period, those mathematician’s works are unknown to the

physics and signal-analysis communities and did not form a unified approach. The

second period started with a more general definition and unified approach of wavelet

(Grossman and Morlet, 1984) and a broad application of the theory to physics and signal

processing. In fact the term “wavelet” was first introduced by Morlet et al. (1982a,b). As

JMR pointed out, although the basic theory of Grossman and Morlet (1984) was a

rediscovery of Calderόn’s identity (1960), but the synthesis and unification of the scale

analysis by physicist Grossman and geophysicist Morlet, had a powerful push for the

development of the mathematics of wavelets. JMR called this as the first synthesis. The

third period, they called the second synthesis, is marked by the unification of the works

by not only the mathematicians but also the works by scientists/engineers working in

signal/image processing, digital communication, and physics. This is the period of

modern wavelet theory, the most flourishing period of the theory. Daubechies discovery

was inspired by the relation between wavelets and the quadrature mirror filtering.

In the conclusion of their historical review, JMR has an interesting comment, which

we cited here:

“Today the boundaries between mathematics and signal and image processing have

faded, and mathematics has benefited from the rediscovery of wavelets by experts

from other disciplines. The detour through signal and image processing was the most

direct path leading from the Haar basis to Daubechies’s wavelets.”

2.2.3 Extension and Generalization of Time-Frequency, Time-Scale

Localizations

In spite of the success of discrete wavelet transform based on time-scale localization

and its outstanding features in signal analysis, it has some shortcomings in dealing with

long oscillatory signals, such as the textured images, fingerprints, and wave phenomena.

Meyer (1998) has an excellent comment on the strong and weak points of time-scale

approaches:

“Wavelets, however, are ill suited to represent oscillatory patterns. Rapid variations

of intensity can only be described by the small-scale wavelet coefficients. Long

oscillatory patterns thus require many such fine scale coefficients. Unfortunately

those small scale coefficients carry very little energy, and are often quantized to

zero, even at low compression rates. “

Here “wavelet” is referred to the classical wavelet in our terminology. On the other hand,

time-frequency localization has some disadvantages in representing the multi-scale

structures with sharp, irregular boundaries, such as those encountered in image

processing. It was the consensus in the community of sciences and engineering that

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neither of the two approaches can be optimally decomposes all the signals/images in

different fields. Realizing the limitations and restrictions of two approaches, investigators

tried to generalize the two categories of phase-space localizations.

Wavelets as representations of groups based on the group theory:

From the viewpoint of group theory, the classic wavelet transform based on time-

scale localization is related to the affine group, which is invariant of translation and

dilation. The windowed Fourier transform and windowed Fourier frame based on time-

frequency localization are related to the Weyl-Heisenberg group, which is in variant with

time and frequency shifts. Wavelets can be considered as the canonical representation

(square-integrable) of the corresponding groups.

The extension can be done by adding frequency shift in the affine group or adding

dilation in the Weyl-Heisenberg group, resulting in an affine Weyl-Heisenberg group

(Torrésani, 1991). In this way, the flexibility of selecting best bases is increased,

resulting in more efficient representations for complex image structures. In wavefield

decomposition, this generalization provides the possibility of more efficient

representation of complicated wavefield and wave propagator (See section 2.3 for

details).

Another direction of extension is proposed by Antoine et al. (1996) by adding the

rotation action to the affine group. Taking all the actions together, these transformations

are represented by a unitary operator ( , , )x a in the space L2(R

2,d

2x) of signals. The

three operations of translation, rotation and dilation generate the similitude group SIM(2)

of R2, i.e. the two-dimensional Euclidean group with dilations.

The scale-angle pair may be interpreted as spatial frequency variables in polar

coordinates. Directional wavelets were introduced as wavelets whose Fourier transform is

(essentially) supported in a convex cone in spatial frequency space, with apex at the

origin. In physical terminology, the 2-D wavelets are the coherent states associated to the

representation of SIM(2) (Torrésani, 1991). Fig. 2.3 shows the tiling of the spatial

frequency plane with the effective support of the wavelet.

Definition of the unitary operator:

1 1

, ,( ( , , ) )( ) ( ) ( ( ))aa s x s x a s a r x

bb b (2.2)

Or in frequency (k) domain

, ,

ˆ ˆ ˆ ˆ( ( , , ) )( ) ( ) ( ( ))i

aa s s ae s ar

b k

bb k k k (2.3)

And the directional wavelet:

2

22

2ˆ(2 ) ( )

dc

kk

k

(2.4)

The selection of the spatial frequency vector k: The best angular selectivity is

obtained by taking the vector perpendicular to the largest axis of the modulus, thus

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0(0, )kk . This selection is same as the definition of curvelet and wave atom discussed

in section 2.4.

Fig. 2.3 Tiling the spatial frequency plane with the effective support of the wavelet (From Torrésani, 1991)

The above extensions are defined for continuous wavelet transform (CWT), the

numerically implementation and fast algorithm have been searched from two directions.

One is the projections method with a tree structure from a dictionary or library (optimum

wavelet packets) (Coifman and Wickerhauser’s best basis algorithm). The other is the

matching pursuit (or other pursuit schemes) to find the “best” approximate representation

(Mallat, 1989; Mallat and Zhang, 1998). The third method is the Wigner-Ville transform

(see Chapter 5 of JMR) (its relation to the Cohen class, see Cohen, 1995).

From the above summary on the extensions of the two kinds of phase-space

localization, we see that the generalizations from different approaches have not

converged to a common terminology. This can be seen from the increasingly large

number of terms introduced by different investigators under different schemes for

different applications, such as ridgelets (Candės, 1999; Donoho, 1995), brushlet (Meyer

and Coifman, 1997), vaguelette (Donoho, 1995), curvelet (Candes and Donoho, 1999),

chirplet (Baraniuk and Jones, 1990; Coifman et al., 1997), seislet (Fomal, 2006), wave-

atom (Demanet and Ying, 2006), physical wavelet (acoustic wavelet) (Kaiser, 1994),

eigen-wavelet (Kaiser, 2004)), among others. Things can be even more divergent, since

different researchers can design their own “some-lets” by a lifting algorithm (e.g.

Sweldens and Shröder, 1996) to fit their special signals. Borrowing one sentence from

Jaffard, Meyer and Ryan (2001), “it appears that things are in a state of disorder and

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confusion”. This appears also natural, since signals are so different in nature, no single

type of bases or algorithms can be optimum to the decomposition of all the signals. In the

following section, we will concentrate on the decomposition of wavefield and the related

propagators (operators), and summarize the recent progress in this field.

2.3 Localized Wave Propagators: From Beam to Beamlet

Despite the confusion and divergence in the terminology relating to localization

elements of phase-space in the field of signal processing, however, the phase-space

localization has a very clear and well-defined physical meaning in wavefield

decomposition and propagation. In order to decompose wavefields efficiently, the

elementary function needs to have the ability of representing oscillating and directional

signals. This is the property of “wave beams”. Beam physics has been studied since the

early time for directional, localized wavefield in seismology, such as Gaussian beam

(Popov, 1982; Červený et al., 1982; Červený, 1983, 1985, 2001; Hill, 1990, 2000;

Albertin et al., 2001, 2002, Nowack et al. 2006) in radar, optics, and other fields of

physics and engineering. The beam properties are also studied under the term of complex

ray, complex source (Kravtsov, 1967; Deschamps, 1971; Keller and Streifer, 1971;

Felsen, 1975, 1976, 1984; Wu, 1985), coherent states (Klauder, 1987; Foster and Huang,

1991; Albertin et al., 2001; Foster et al., 2002). However, the decomposition becomes an

efficient tool for wavefield representation and propagation can only be achieved after

wavelet transform was introduced into physics and geophysics (wave physics).

The term “beamlet” is a general physical term, first introduced by Mosher, Foster

and Wu (1996). Originally it meant to represent any elementary function of decomposing

a wavefield by wavelet transform (in the generalized sense) along the spatial axes.

Therefore it is a space-wavenumber (spatial frequency) localization atom with scaling

capability (dilation). Beamlet representation of a wavefield along a decomposition plane

differs from the traditional space-domain or wavenumber-domain representations at the

dual localizations in both spatial location and propagation direction (phase space

localization): each element of the representation is a beamlet, i.e. a “small beam”.

Beamlet decomposition is also different from the decomposition using Gaussian beams

which are approximate in decomposition and high-frequency asymptotic in propagation.

Different wavelet atoms have been tested for wavefield decomposition and propagation

(Wu and Yang, 1997; Wu and Wang, 1998, 1999; Wang and Wu, 1998, 2000, 2002;

Chen et al., 2001; Chen and Wu, 2002; Foster et al., 2002; Mosher et al., 2002). It

became apparent that only smooth wavelet atoms, either orthonormal bases or frame

vectors, can efficiently represent wavefields and their propagation in heterogeneous

media. Many popular wavelet bases in imaging processing, such as the Daubechies

wavelets, do not work well for wave propagation. The beamlet atoms need to be well

localized in both the space and frequency domains. From the viewpoint of the group

theory as we discussed in the previous section, beamlets can be defined as space-

wavenumber (time-frequency) atoms related to the finite-energy (or square-integrable)

representation of the affine Weyl-Heisenberg group. In the case of CWT (continuous

wavelet transform), the beamlets are space-shift, wavenumber-shit and dilation invariant.

In the case of discrete transform, the shift and dilation invariants are understood in the

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sense of latticed invariant. In the special case of wavefield representation, its similarity to

the representation of the similitude group can be seen from the relation of the local

wavenumber vector k to the local horizontal wavenumber , where the bar “ ” on top

of a parameter stands for the localized (phase-space localization) parameter. is the

localized frequency along the decomposition axis (in the 2D case) or decomposition

plane (in the 3D case):

( , ) k (2.5)

where ( , )x yk k is the local horizontal wavenumber in the decomposition plane, and

2

2(2 / ) (2.6)

is the local vertical wavenumber, and 2 /c is the wavelength of the localized

wavefield with c as the local wave speed and ω as the circular frequency of the

wavefield.

The beamlets defined in this way can be related to the directional wavelets (defined

as the representation of the similitude group) by the correspondence between the local

rotation angle and the local wavenumber

0sin /2

k

, (2.7)

with 0 /k c .

The decomposition plane is designed to perpendicular to the main (dominant)

propagation direction, which can be determined by some asymptotic methods, such as ray

tracing or Hamilton-Jaccobi equation. In case of short range propagation with data

acquired on the surface of the earth, such as in exploration seismology, the dominant

direction is often preset to the depth (z) direction. Along the main propagation direction,

the beamlet with 0 0(0, )kk is same as the directional wavelet defined by Antoine et al.

(1996). The other beamlets are obtained by wavenumber (ξ) shift from 0k . For the

directional wavelet they are generated by rotation of the 0k wavelet, and for each wavelet

the oscillating component is always perpendicular to the beam-width (phase-front).

Therefore, in the context of wave propagation, beamlets and directional wavelet have the

same capacity of presenting phase-space localized wavefield. This is due to the unique

feature of wavefield: the angle-wavenumber correspondence. However, there are some

differences between these two decomposition schemes which may have different

consequences for wave propagation:

1. Beamlets are decomposed on the data-plane (observation plane) or extrapolated

data-plane, while the directional wavelets are decomposed in the whole image

space. This difference can be seen more clearly from the comparison between

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dreamlet and curvelet (see Figure 2.20 and 2.21). These two decompositions

may have different noise-resistant properties. The decomposition of directional

wavelets may be less prone to the noise contamination.

2. The sampling in the angle-domain is different. Since

0

0

sin

cos

k

k

(2.8)

Therefore the sampling in the local angle-domain is nonuniform for the beamlet

decomposition.

3. Wave propagation will be formulated differently for these two kinds of

decompositions. Beamlets are very convenient for one-way wave propagation

(wavefield extrapolation), while the propagation of directional wavelets is

formulated by the movement of these wavelets according to the wave-equation

under certain approximation. This will become clear in the discussion of

curvelet propagation in the next section.

2.3.1. Frame Beamlets and Orthonormal Beamlets

Beamlet decomposition can be realized using either orthonormal bases or

overcomplete frame vectors. Windowed Fourier frame (WFF), especially the Gabor

frame which is a Gaussian-shaped WFF, is a widely used wavefield decomposition atom.

Gabor (1946) originally represented signals with sampling intervals satisfying ΔxΔξ = 2π,

where Δx and Δξ are the space (time) and wavenumber (frequency) sampling intervals,

which was later proven to be the most compact representation of this kind (Daubechies,

1990, 1992) and therefore is an orthogonal decomposition. However, Daubechies later

proved through the frame theory that the reconstruction under Gabor’s critical sampling

is unstable and that oversampling ΔxΔξ < 2π must hold for stable reconstruction

(Daubechies, 1990, 1992). Optimally localized in both space (time) and wavenumber

(frequency) domains under the Heisenberg uncertainty principle, the Gaussian window is

the most favorable for windowed Fourier analysis of signals (Mallat, 1998). Generally,

the reconstruction frame vectors (the dual frame vectors) are different from the

decomposition frame vectors, and therefore are no longer Gaussian. However, in the case

of tight frame or nearly tight frame, the dual frame vectors are the same or nearly same as

the decomposition vectors. The windowed Fourier frame with a Gaussian window was

named by Daubechies as the Weyl-Heisenberg coherent state frame. It was also called the

Gabor frame (Feichtinger and Strohmer, 1998) and the Gabor-Daubechies (G-D) frame

(Wu and Chen, 2001, 2002a, Chen et al., 2006).

The elementary functions (vectors) of the G-D frame decomposition are Gaussian

enveloped (Gaussian windowed) harmonic wavefields:

( ) ( ) ( ) mim x i x

mn x ng x g x n e g x x e (2.9)

14

where g(x) is a Gaussian window function, xn nx is the space locus and mm ,

the wavenumber locus of the beamlet. Applying G-D frame decomposition to the

wavefield in a local homogeneous region is like decomposing the wavefield into many

small Gaussian beams (beamlets) at different window locations and with different beam

directions. As shown in equation (2.5), the local wavenumber mm of a

decomposition atom can be associated to a horizontal wavenumber of a beamlet (a

directional wavelet atom), and a propagating beamlet can be expressed as

( )

( ) ( ) m mi x z

mn nb x g x x e

(2.10)

With as the vertical wavenumber defined by equation (2.6). Here z means a small

step of extrapolation. The direction defined by ( , )m m is the nominal direction of a

beam-lobe. The angular spectra of the lobe can be seen more clearly from the

wavenumber domain expression of the beamlet

( )

( , ) ( ) nn mi xi x x i z

mn mb x z d e g e

(2.11)

From the elementary wave expressed by (2.9) we can say that the G-D beamlet is

equivalent to the local Fourier beamlet defined by windowed Fourier transform

(Steinberg and McCoy, 1993; Steinberg, 1993; Steinberg and Birman, 1995) and coherent

state atom (Albertin, 2001a,b). However, when we consider the lattice structure of

representation and the reconstruction atoms, the differences appear. The reconstruction

atom of the G-D frame beamlet is the dual frame vector

( ) ( ) mi x

mn ng x g x x e

(2.12)

where g is the dual window function. The local Fourier atom for reconstruction is the

same as for decomposition but the reconstruction of windowed Fourier transform is very

time-consuming.

Beamlet decomposition can be also done with orthonormal bases. In the literature,

the only type of basis function used for wave propagation is the local trigonometric basis:

Local cosine basis (LCB) or local sine basis (LSB) (Wu and Wang, 1998, Wang and Wu,

2002; Luo and Wu, 2003; Wang et al., 2003, 2005). The basis function of LCB can be

written as

2 1

cos2

nmn n

n n

x xx B x m

L L

(2.13)

15

Where nnn xxL 1 is the nominal length of the window, nB x is a bell function

which is smooth and supported by the compact interval 1[ , ]n nx x for

1 ]n nx x , with , are the left and right overlapping radius, respectively.

Fig. 2.4 shows an example of bell functions and basis function (atom) of LCB, where the

left and right slopes are symmetric, i.e. = . The basis function of LCB in equation (2.13) can be also expressed in the Fourier

domain:

1 2

2n m n mix ix

mn n m n m

n

e B e BL

(2.14)

Similar to equation (2.10), we can associate a propagating LCB beamlet to the basis

function:

( ) ( )1{ }

2

n nm m mi x x i x x i z

mn n

n

b x B x e e eL

(2.15)

where

1

2m

n

mL

(2.16)

is the effective wavenumber. We see that, a LCB beamlet corresponds to a two-lobe

symmetric beam.

16

Fig. 2.4 Bell function and basis function of Local cosine basis (LCB).

2.3.2 Beamlet Spreading, Scattering and Wave Propagation in the

Beamlet Domain

Beamlet propagation observes wave equation. Historically, the propagation of

Gaussian beam is developed under the h-f asymptotic theory (Červený, 1983, 2001) for

smoothly varying media. In the case of complex, multi-scaled heterogeneous media,

perturbation methods can be applied in combination with the asymptotic methods. Beam

scattering has been studied in radar, acoustics, optics, quantum mechanics and other

branches of physics and engineering. For the purpose of wavefield extrapolation and

imaging, the forward-scattering problem, i.e. the one-way propagation problem, is more

relevant and useful. In the following we will concentrate on the forward scattering and

propagation problems.

17

The one-way propagator (wavefield extrapolator) can be expressed as surface

integral with space-varying kernel (Green’s function). In the frequency-space (f-x)

domain, the scalar wave equation (Helmholtz equation) can be written as,

2 2 2 2[ / ( , )] ( , ) 0x z v x z u x z (2.17)

where denotes frequency, ),( zxv is wave velocity at (x, z) and ),( zxu stands for

the frequency domain wave field. Here we consider the transversal coordinate x as the

horizontal coordinate in the 2D case, and as 1 2( , )x x in the 3D case. Therefore, in the 3D

case it is understood that1 2

2 2 2

x x x . If the field is known on a give surface, such as

the measurement surface (data surface), then the field at any point surrounding by the

surface can be calculated by the representation integral (Kirchhoff integral)

( ', ') ( , ; ' ')

( , ) ( , ; ' ') ( ', ')s

u x z g x z x zu x z ds g x z x z u x z

n n

(2.18)

where g is the Green’s function, / n is the normal derivative on the surface, and S is

integration surface. If the surface S is a flat surface, the integral can be reduced to a

Rayleigh integral

( , ; ' ')

( , ) 2 ( ', ')s

g x z x zu x z ds u x z

n

(2.19)

The representation integrals are formal solutions of the corresponding wave

equation. Since the integrals (2.18) and (2.19) represent the propagation effects ruled by

the wave equation, these integrals are also called the propagation integrals. These

integrals can be considered as a linear operator P

( , ) ' ( , ; ', ') ( ', ')Pu x z ds K x z x z u x z (2.20)

where ( , ; ', ')K x z x z is kernel and

( , ; ' ')

( , ; ', ') 2g x z x z

K x z x zn

(2.21)

in the case of propagation integral (2.19).

In principle, the propagation integrals can be applied to arbitrarily heterogeneous

media, if we use the exact Green’s function. However, in many cases the exact Green’s

function is not known or too complicated to compute. In many practical applications,

especially in wave-theory based imaging methods, we prefer to neglect the backscattering

with respect to the main propagation direction and the propagation integrals become

extrapolation integrals, which are one-way propagators.

18

If we consider the action of P on a beamlet (wavelet atom of a wavefield

decomposition), it becomes a scattering problem of beamlet. Gaussian beam scattering

has been studied since its introduction (Choudhary and Felsen, 1974; Felsen, 1976;

Červený, 1983; …). Scattering of elementary wavelet has also been studied by Young

(1993). For the extrapolation integrals, mainly the forward scattering problem is

interested. After propagation, the beamlet becomes spread and distorted due to geometric

spreading and scattering, and is no longer a beamlet. If the deformed beamlet is re-

decomposed into beamlets, we can get a propagator matrix in the beamlet domain:

, , ,jl mn jl mn jl mnP b Pb b dsKb (2.22)

where <b,a> stands for the projection of function a onto the wavelet vector b. This matrix

represents completely the linear operator (Candès and Demanet, 2003). In matrix form,

the operator decomposition can be written as (Wu and Yang, 1997; Wu and Wang, 1998)

TP BPB (2.23)

where P is the space domain propagation matrix, and P is the one in beamlet domain; B

is the beamlet vector and T

B is its transpose. Fig. 2.5 illustrates the concept of propagator

decomposition and wave propagation in the beamlet domain.

For beamlet scattering problem, two approaches widely used in wave propagation

theory can be adopted to this purpose: one is the asymptotic approximation for smooth

media; the other is the perturbation method for rough heterogeneities. Traditionally the

Gaussian beam method adopted almost exclusively the asymptotic approach (see

Červený, 1983, 2001); while the beamlet imaging uses the combination of asymptotic

method for the background propagation and perturbation approximation for rough

heterogeneities (Wu et al., 2000; Wu and Chen, 2001, 2002a; Chen et al., 2006; Wang

and Wu, 2002; Luo and Wu, 2003; Wang et al., 2003, 2005).

In the case of smooth media, the local homogeneous approximation or asymptotic

approximation can be applied to the propagator decomposition. Under these

approximations, the Green’s function is known or can be approximated by asymptotical

solutions. Then the propagator matrix in beamlet domain can be easily calculated.

Different wavelet atoms, such as Daubechies wavelet D4, Coifman wavelet C5, best

bases of wavelet packet, and local cosine basis have been tested for the propagator P

decomposition (Wu and Yang, 1997; Wu et al., 1997; Wu and Wang, 1998; Wang and

Wu, 1998a, b). Figs. 2.5 to 2.9 show the decomposition of propagation integral under

local homogeneous approximation into different beamlet domains and the comparison of

sparseness of the beamlet propagators. First, we see obviously that the propagator

matrices in beamlet domain are quite sparse compared to the dense space-domain ones. It

is also seen that the smooth wavelets (with well-localized wavenumber spectra) can

deliver better behaved propagator matrices than the popular wavelets in imaging

compression such as the orthonormal Daubechies wavelets. Especially the local cosine

basis yields very sparse and well-organized propagator matrices, which are important for

efficient implementation. Later in our research, we mostly adopted the smooth wavelets,

19

such as G-D (Gabor-Daubechies) frame and LCB beamlets for wave propagation and

imaging.

Fig. 2.5 Beamlet decomposition and propagation of wavefield.

Fig. 2.6 Matrix representations of Kirchhoff propagation operators in space domain: dx = dz = 25m, v = 2000m/s, N = 128. Left: 5.9Hz, Right: 25Hz. Only real parts of the complex operators are plotted.

20

Fig. 2.7 Beamlet propagators (propagation operators in beamlet domain) (Daub4: Daubechies-4 wavelet): The left panel is for DWT (discrete wavelet transform), and the right panel is for best basis wavelet-packets. The top

panel is for 5.9Hz, and the bottom panel is for 25Hz. Only real parts of the complex operators are plotted.

21

Fig. 2.8 Beamlet propagators (propagation operators in beamlet domain) (Coif5: Coifman-5 wavelet): The left panel is for DWT (discrete wavelet transform), and the right panel is for best basis wavelet-packets. The top

panel is for 5.9Hz, and the bottom panel is for 25Hz. Only real parts of the complex operators are plotted.

22

Fig. 2.9 Propagator matrices of the LCB background propagator in the beamlet domain (Top panel: f = 5.9 Hz;

Middle panel: f = 25.0 Hz) (Left panel: Real part; Right panel: imaginary part) with operator aperture of Nx=256.

2.3.3 Beam Propagation in Smooth media with High-Frequency

Asymptotic Solutions

Equation (2.22) formally defines the beamlet propagator matrix. There is no essential

difficulty in calculating propagation matrices for smooth media under the local

homogeneous approximation or h-f asymptotic approximation. However, the calculation

of the beamlet propagator in generally heterogeneous media is quite complicated. Various

approximations have been obtained for different applications. Historically, there are

basically two approaches. One is the beam propagation methods which calculate the

evolution of the elementary wave (a beamlet) globally using asymptotic solution of the

wave equation in smooth media, and synthesize the wavefield at the end point by

reconstruction (superposition of all the arriving beams). In this approach, the propagator

23

matrices do not enter into the computation and exist only in the definition. Each beamlet,

the elementary wave, evolves into a global beam in the propagating space and arrives at

the receiving point, contributing to the final summation. The other is the perturbation

approach for wave propagation in the beamlet domain. The propagation of a wavefield is

a step-by-step beamlet propagation and coupling using propagator matrices. At each

step, the laterally varying velocity profile is decomposed into a background velocity

profile and local perturbations. Each beamlet will be spread (in the background media)

and scattered (by local perturbations) into other beamlets. No global beam solution is

used in this approach. The details of this perturbation approach will be summarized in the

next subsection.

2.3.3.1 Conditions for the Application of Asymptotic Analysis

Before going reviewing the global beam approach, let us discuss the conditions of

applicability for asymptotic solutions in inhomogeneous media.

The one-way evolution of a beamlet can be formally written as

xbexa mn

ziA

mnn

(2.24)

where An is the square-root operator (for the case of scalar wave).

),(/ 222 zxvA xn (2.25)

Note that amn is no longer a beamlet due to diffraction and scattering.

Redecomposition of amn into beamlets, leads to the formulation of propagator matrix, as

seen in equation (2.22). For general heterogeneous media, the solution of equation (2.24)

can be quite involved and sometimes only numerical methods are applicable. However,

when the media is smooth in comparison to the wavelength, h-f asymptotic

approximations may be applied to the solution. In equations (2.22) or (2.24), if we

transform the propagator into wavenumber (ξ) domain, the application of propagator

operator to the beamlet can be written as

, , ( , ) ,niA z i x

mn mn mna x z e b x z d P x z e b z (2.26)

Standard h-f asymptotic analysis (Garding, 1987; Červený, 1983, 2001; Candès and

Demanet, 2005; Demanet, 2006) assumes that ( , ) i xP x z e can be approximated

( , , )( , ) ( , , )i x i x zP x z e e x z

(2.27)

where ( , , )x z is the phase term and ( , , )x z , the amplitude term. The phase Ф is

homogeneous of degree one in ξ; the amplitude 0 1 ... with

m

24

homogeneous of order –m in ξ. Physically the above conditions mean that the amplitude

term is a much slowly varying function in space, and the phase term is dominated by the

linear dependence in frequency (wavenumber). With this asymptotic approximation, the

one-way beamlet propagator can be represented by a Fourier integral operator (FIO) If

the phase term in (2.27) can be expressed exactly the linear function of frequency, i.e.

( , ) ( , , )i x i xP x z e e x z (2.28)

and ( , , )x z obeys some constraints

,( ( , ) (1 )m

x x C

(2.29)

for multi-indices α and β. Then the propagator belongs to a type of pseudodifferential

operator (PsDO) and ( , )x is its symbol of order m (type (1, 0) (Grossman, 2005;

Demanet, 2006). The space domain expression of the operator is ( , )x D with D i .

The condition in equation (2.29) requires the fast decay in high frequency of the operator

spectrum, implying smooth variation of the wavefield along x. Here z in equation (2.26)

stands for the coordinate along the propagation direction. For a long distance propagation

using asymptotic methods, it can be replaced with the curved-linear coordinates along the

ray which obeys the eikonal equation for the phase. The amplitude can be determined by

the transport equation.

The mathematical requirements for asymptotic analysis can be translate into wave

propagation regimes in terms of beam and medium parameters. For beam evolution in

inhomogeneous media, there three basic parameters that determine the propagation

regimes: wavelength λ, beam width β, and the scale of the heterogeneity a. First, a >>

and >> must hold for the validity of asymptotic approach. For short range

propagation, the beam asymptotic solution requires also a , namely the beamwidth

must be smaller than the scale of heterogeneities (Steinberg, 1993). This is the geometric

optics (GO) regime of wave propagation (Flatté et al., 1979; Wu and Aki, 1988). If

a , the wave front will be distorted and it is in the diffraction regime (ibid). For long

range beam propagation, there are two more parameters enter into the game: the range R

and the total perturbation strength . The requirement of a >> is replaced by

/ 2Fa r R where Fr is the Fresnel radius along the beam path. This is

understandable, since diffraction regime will be entered if the scale of heterogeneity is

smaller than the Fresnel radius. The other requirement that the total r.m.s. perturbation

strength must not exceed certain value is for the case long range propagation in randomly

heterogeneous media (Flatté et al., 1979, Ch. 8; Aki and Richards, 1980, Ch. 13; Wu and

Aki, 1988), which we will not discuss here.

Therefore, for strongly and roughly heterogeneous media, the h-f asymptotic

approximation may be not applicable and the beamlet propagator (or more general the

linear operators of wave propagation in equations (2.18) and (2.20) cannot be represented

by Fourier integral operators and pseudodifferential operators.

25

There are mainly two approaches related to beam propagation: the Gaussian beam

method and the coherent state method. In fact they are close relatives, but with different

historical origins and different approximations to the asymptotic solution.

2.3.3.2 Gaussian Beam Method

Gaussian beam can be considered as evolved from a parabolic approximation of the

Gabor beamlet (or Gabor-Daubechies beamlet) in equation (2.30).

2

00

( )2

( ) ( )

mmi x i k z

kg

mn nb x g x x e

(2.30)

where Δz implies that the wave front is defined in the vicinity of the reference point.

Gaussian beam is a wave-beam with a Gaussian envelope and a parabolic wave front. Its

evolution in smoothly varying media has been extensively studied and documented

(Červený et al., 1982; Popov, 1982; see Červený 2001, and Popov 2002 for detailed

expositions; Červený et al., 2007 for a review of recent progress). The beam propagates

along a ray-path. The spatial trajectories and the travel times are determined by the

eikonal equation (ray-tracing) and change of the beam shape and amplitude is governed

by the dynamic ray-tracing equation which calculates the complex-valued second

derivatives of the travel time along the central ray. In this way, the traveltime and

amplitude in the paraxial region about the ray can be determined. The evolution of a

Gaussian beam depends on two initial parameters, the ray curvature and the beam width.

The Gaussian beam summation provides regular wavefield everywhere even in media

where caustics and shadow zones may be present. The solution for a single beam can be

constructed in f-x domain (e.g., Červený et al., 1982) as well as in t-x domain using a

Hilbert transform (e.g., Hill, 1990, 2000). In seismic imaging, the recorded wavefield is

first decomposed using a set of overlapping Gaussian windows. For each windowed data

section, the local slant stack is performed to form individual beams (e.g., Raz, 1987;

Hale, 1994). Such decomposition is closely related to the decomposition using windowed

Fourier transform and efficient reconstructions can be obtained based on the “frame”

theory (e.g., Daubechies, 1992; Kaiser, 1994) in wavelet analysis. The Gabor frame-

based overcomplete representation provides redundant yet stable reconstruction of the

wavefield (e.g., Einziger, 1986; Nowack et al. 2006). The accuracy of the beam solution

depends on the beam width. High-frequency asymptotic analysis can be also applied to

the non-approximated Gabor beamlet, and the resulted propagating beam is the

windowed Fourier beam (Steinberg, 1993; Steinberg and Birman, 1995), the coherent

state beam (Klauder, 1987; Foster and Huang, 1991; Thomson, 2001; Albertin et al.,

2001; Foster et al., 2002) or the Gabor-frame beam (Gao et al., 2006). Červený et al. also

called the parabolic-approximated Gaussian beam as paraxial Gaussian beam, and the

non-approximated beam as strict Gaussian beam (Červený, 2007).

2.3.3.3 Asymptotic Coherent-State Solutions

26

The original definition of CST (coherent state transform) is a windowed Fourier

transform with Gaussian windows (Klauder and Skagerstam, 1985; Klauder, 1987; Foster

and Huang, 1991). Each Gabor atom defined by equation (2.9) is a coherent state. Later

Thomson (2001) proposed to call the propagating coherent state after making stationary

phase approximation (saddle point approximation) to the h-f asymptotic solution integral

of wave equation as “elementary coherent state” even the solution may not have a

Gaussian envelope. Therefore, the term coherent state may have different meanings in the

literature. Here we use the term in the general sense and refer to asymptotic solution of

coherent state (CS) as asymptotic CS. The coherent state transform (CST) is defined as

p,x u x x exp 1

2 x 2

eip x d x , (2.31)

where u is the wavefield in space domain, p is the horizontal slowness (ray parameter)

and Ω is inversely proportional to the beamwidth of the coherent state (width parameter).

The inverse transform is

u x

2

2

p,x dp . (2.32)

The coherent-state transform is a complete but non-orthogonal representation. A

property of the coherent-state transform is that the Heisenberg uncertainty cell is

minimized due to the Gaussian windowing. A coherent state can be considered as sum of

weighted plane waves (wavenumber domain summation) or a Gaussian windowed beam

(Gaussian beam) with certain width. In a smooth media, the h-f asymptotic solution (or

semi-classical solution, as called by Klauder, 1987) can be obtained for the coherent state

propagation. The phase is calculated by solving the complex eikonal equation (Hamilton

Jacobi equation), and the amplitude, by the transport equation. Due to the special inverse

transform defined by equation (2.32), the final wavefield is formed by a summation of

coherent states with different directions (p) at the same spatial location without

contributions from neighboring beams (coherent states). This is different from the

Gaussian beam summation which is a summation over neighboring beams with different

directions. Of course, there are other alternative reconstruction formulas can be defined.

However, the inverse transform defined by equation (2.32) is used in all the relevant

articles. In this sense, asymptotic CS (coherent state) modeling bears more similarity with

the Maslov method than with the Gaussian beam method. The Asymptotic CS method

can be considered as a windowed Maslov method; while the Maslov method (see

Chapman and Drummond, 1982; Chapman, 2004) can be said to be the limiting case of

the Asymptotic CS method when the beamwidth is set to infinity (i.e., Ω = 0). The other

limiting case of the Asymptotic CS method is when Ω approaches to infinity, i.e. when

the beamwidth is close to zero, it becomes the classic ray method. Remember that the

wavelength must be approaching zero faster than the beamwidth in order to apply the h-f

asymptotic solution. From the calculations of the eikonal and transport equations, we see

that there are three parameters to consider for the applicability of the method: beamwidth

27

(1/ Ω), wavefront curvature and wavelength. Wavelength being much smaller than the

beamwidth and wavefront curvature is the prerequisite of the asymptotic solution. Under

the asymptotic regime, different approximations can be applied for the cases of large or

small curvature/beamwidth (Klauder, 1987; Thomson, 2001, 2004).

As for numerical modeling schemes, there are different implementations. Foster and

Huang (1991, 2002) apply directly the inverse transform equation (2.32) to synthesize the

space domain wavefield, which is a summation of a bundle of asymptotic CS solutions.

The other way is apply further a stationary phase approximation (or saddle-point

approximation) to the integral and the final solution is similar to a single CS (Klauder,

1987; Thomson, 2001, 2004). In this way, the asymptotic CS ray tracing is similar to the

classic ray tracing with advantage of avoiding caustics. The beam width parameter Ω

enters only into the amplitude calculation as well as the boundary condition of the ray

tracing. The actual beamwidth is irrelevant to the final results. Therefore, unlike the

Gaussian beam dynamic ray tracing where the beamwidth changes along the propagation

path, the asymptotic CS solution assumes a fixed Ω along the ray.

The other difference of coherent state from the Gaussian beam is the inability of

modeling frequency-dependent wave phenomena. The asymptotic CS solution is a global,

uniform asymptotic solution and can handle all caustics, including the pseudo caustics for

which the Maslov method is invalid. It can also give some approximated results even in

shadow zones. However, the solution losses the frequency-dependent wave information

and can only model primary waves. On the other hand, the Gaussian beam approximates

the wave field in the paraxial region of the ray with a parabolic approximation. In the

original frequency domain formulation, the Gaussian beam method can explicitly

simulate frequency-dependent wave phenomena with certain degree of approximation.

Later, time (complex time) domain versions (Hill, 1990, 2001) were also derived for

seismic imaging with high efficiency but with the loss of frequency-dependent wavefield

information. The asymptotic CS method avoids the ray-centered coordinates and each CS

has an asymptotic solution.

Besides some canonical examples for demonstration purpose in the literature,

Albertin et al. (2001) did some comparison of impulse responses and imaging results

between Maslov, Gaussian beam and the asymptotic CS migration methods for a

complex geological model (the Marmousi model).

2.3.4 Beamlet Propagation in Heterogeneous Media by the Local

Perturbation Approach

In non-smooth heterogeneous media, the h-f asymptotic solution for long range

propagation has very limited applications. Wu et al. (2000) developed a local

perturbation theory for wave propagation in the beamlet domain. The propagation of a

wavefield is not synthesized by the superposition of a collection of globally evolving

beams, but by step-by-step beamlet propagation using propagator matrices. At each step,

the laterally varying velocity profile is decomposed into a background velocity profile

and local perturbations. The decomposition in fact is bi-scale decomposition. The large-

scale component is a piecewise homogeneous medium with the scale of window-width

defined for the spatial localization (see Fig. 2.2d); the small-scale component is the local

28

perturbations with respect to the local reference velocities (see Figure 1e). In comparison,

the standard (global) perturbation scheme has a global background velocity (see Fig.

2.2b) and the perturbations are spreading to all scales (see Fig.2.2c).

In the local perturbation approach, the propagator P is decomposed into a

background propagator 0

P and a perturbation term1

P . The background propagation uses

one of the two approximations: the local homogeneous approximation and the average

slowness approximation. Both belong to the h-f asymptotic solution for small-step

propagation. Since the local perturbations are much weaker than the global perturbations,

the perturbation correction operator can be approximated by a phase-screen correction. It

should be understood that the propagation of large-angle waves is mainly controlled by

the background propagator and can be quite accurate due to the wide operator aperture.

Therefore, the beamlet propagator in the local perturbation approach is a hybrid solver

with the asymptotic method for the large scale background media, and the perturbation

method for the small-scale fluctuations with respect to the background (see Wu et al.

2000, Wu and Chen 2001, 2002a, Wang and Wu, 2002, 2003; Luo and Wu, 2003; Chen

et al. 2006, Wu et al., 2007, for detailed derivation and discussions).

One special feature of the beamlet decomposition and propagation is the availability

of local angle information during the propagation. Of course, the wavefield

decomposition into the local angle domain can be performed by local slant stack or local

Fourier transform (e.g., Xu and Lambare, 1998; Sava and Fomel, 2002; Xie and Wu,

2002) during propagation using other types of wave extrapolator. However, the beamlet

propagator is formulated in the local angle domain and therefore is more efficient to get

the angle-related information. Directional illumination analysis has been developed using

the wavefield information in the local angle domain to study the influence of the

acquisition configuration and overburden structures to the illumination of subsurface

targets with different dip-angles (Wu and Chen, 2002b, 2003, 2006; Xie and Wu, 2002,

2003; Xie et al., 2004, 2006). Based on the illumination analysis, Amplitude correction

theory and methods to compensate the acquisition aperture effects have been also

developed (Wu et al., 2004, Wu and Luo, 2005; Jin et al., 2005; Cao and Wu, 2005,

2008, 2009; Mao and Wu, 2010, 2013; Ren et al., 2011).

2.3.4.1 Beamlet Evolution and Wave Propagation in the Beamlet Domain

Wavefield and propagator decompositions:

Substituting the wavefield decomposition equation into the wave

equation (2.17), the beamlets will also propagate obeying the scalar wave

equation,

2 2 2 2ˆ( , ; )[ / ( , )] ( ) 0n m x z mn

n m

u x z v x z b x . (2.33)

Note that in the above equation, ˆ( , ; )n mu x z is a set of coefficients with z as the

labeling parameter, not a variable. The propagation effect of the wavefield now is

29

included in the evolution of beamlets. For a local beam evolution problem, invoking the

one-way wave approximation (neglecting interactions between the forward-scattered and

backscattered waves), we can write a formal solution for the evolution of beamlets

niA zmn mna x e b x

. (2.34)

Where amn is a function evolved from a beamlet bmn propagating in the heterogeneous

medium, and An is the square-root operator

2 2 2/ ( , )n xA v x z . (2.35)

As we mentioned above, amn is no longer a beamlet due to distortion after

propagation. Decomposing amn with the same beamlet basis functions (or with the dual

frame atom in the case of frame beamlet)

,mn mn jl jl

l j

a x a b b x . (2.36)

The propagator matrix P̂ in beamlet domain is as we defined in (2.22) and (2.23)

,ˆ ˆ , ; , ,jl mn l j n m mn jlx x a x b x . (2.37)

The beamlet domain wavefield at depth z z can be obtained as

,ˆ ˆˆ ˆ ˆ( , ; ) ( , ; , ) ( , ; ) ( , ; )l j l j n m n m jl mn n m

n m n m

u x z z P x x u x z P u x z . (2.38)

Here ,ˆ

jl mnP are the matrix elements of the beamlet propagator matrix P̂ , which governs

the beamlet propagation and cross-coupling.

Wavefield reconstruction:

Here we perform the beamlet decomposition using orthogonal bases so the

reconstruction atoms are the same as the decomposition atoms. The wavefield at depth

z z after extrapolation can be reconstructed from the beamlet domain wavefield

through

ˆ ˆ( , ) , ; ( , ; ) ( )n m mn l j jl

n m l j

u x z z u x z a x u x z z b x (2.39)

2.3.4.2 Beamlet Propagator with Local Perturbation Approximation

30

The main task for beamlet imaging (migration) is to derive efficient propagators in

the chosen beamlet domain. The evolution of beamlets is governed by an operator

equation (2.35) which involves a square-root operator and there is no exact solution

available for a general problem. Various approximations are invoked to make the

calculation practical. Exploring the efficiency of fast wavelet transforms and the

sparseness of the propagator matrix, Wu et al. (2000) applied a local perturbation

approximation to the beamlet propagator resulting in a split-step implementation of wave

propagation in the beamlet domain. The local perturbation approach can keep all the

wave phenomena of forward propagation, such as diffraction, interference, scattering and

cross-coupling between beamlets, in the propagator. One successful example of applying

the local perturbation theory is the LCB (local cosine basis) beamlet propagator and

imaging algorithm.

In the local perturbation theory, a local reference velocity 0( , )nv x z is selected for

each window nx , and the local perturbation is calculated from the local reference

velocity. Due to the adaptability of local reference velocities to the lateral variations of

velocity model, generally the local perturbations are small, so that the first order

approximation, i.e. the phase-screen approximation can be adopted for the perturbation-

correction in each window. This leads to the approximation of the square-root operator as

(see Wu et al., 2000; Chen et al., 2006)

2 2 2 2 2 20/ ( , ) / ( , ) ( )n x x n nA v x z v x z k x (2.40)

where 0( ) (1/ ( , ) 1/ ( , ))n nk x v x z v x z denotes the local perturbations.

Therefore, the beamlet evolution equation can be approximated by

1

2n n

i k x z i zi xmn mna x e d e e b

(2.41)

where

i xmn mnb dxe b x (2.42)

is the basis vector in the wavenumber domain, ξ is the horizontal wavenumber and

2 2 20/ ( , )n nv x z (2.43)

is the vertical wavenumber with the local reference velocity. Equation (2.41) is a dual-

domain implementation of an operator split-step approximation. The first factor in the

right-hand side is a phase-screen term in the space-domain; the second factor is a phase-

shift in the wavenumber domain using the local reference velocity and localized by a

beamlet projection. Efficient algorithms of propagation and imaging in the beamlet

domain based on equation (2.41) have been developed using frame or basis beamlets (Wu

and Chen, 2001, 2002; Wang and Wu, 2002; Wu et al, 2003, 2008; Chen et al., 2006).

31

Fig. 2.10 gives a schematic illustration of the decomposition of a lateral velocity

section into a background velocity profile and local perturbations. We see that the

decomposition in fact is a bi-scale decomposition. The large-scale component is a

piecewise homogeneous medium with the scale of window-width; the small-scale

component is the local perturbations with respect to the local reference velocities. In

comparison, the standard perturbation scheme has a global background velocity and

global perturbations spreading to all scales (see the example of SEG salt model in Figure

2.2). We see that the local perturbations are in general much smaller than the global

perturbations, and therefore can reach higher accuracy in extrapolation and imaging in

strongly heterogeneous media.

Fig. 2.10 Background velocity profile (using local reference velocities) and local perturbations in the local

perturbation theory, compared with the global reference velocity (dashed line) and global perturbations in standard perturbation methods.

2.3.4.3. Beamlet Imaging in Strongly Heterogeneous Media

As a numerical example, Figure 3.6 shows the image of the prestack depth migration

using the LCB (local cosine basis) beamlet propagator for the 2D SEG/EAGE salt model.

The minimum velocity of the model is 5000 feet/s and the maximum velocity is

14700feet/s. The salt boundary is sharp and very irregular, especially on the top.

Therefore the model is a strong-contrast and rough heterogeneous medium and h-f

asymptotic method alone will not work. The acquisition system of this model consists of

325 shots (sources) with 176 left-hand-side receivers with receiver interval of 40 feet.

Fig. 2.11a is image using LCB beamlet propagator without acquisition aperture

correction. The aperture-effect corrected image is shown in Fig. 2.11b. We see the

32

excellent quality of the image and the significant improvement of the acquisition-aperture

compensation in the local angle domain.

Fig. 2.11 Images of prestack depth migration using LCB (local cosine basis) beamlet propagator for the 2D

SEG/EAGE salt model (see Fig. 2.1): (a) the image before the acquisition aperture correction, (b) image after

the acquisition-aperture correction. (From Cao and Wu, 2006).

2.4 Curvelet and Wave Propagation

2.4.1 Curvelet and its Generalization

Curvelet transform is originally developed for efficient representation of images with

sharp and curved edges (Candès and Donoho, 2000, 2005). Curvelets are elementary

oscillatory patterns that are highly anisotropic at fine scales, with effective support

obeying the parabolic scaling: width length2. Later the scaling law was loosened and

modified to width wavelength2

in order to let its cousin “wave atom’ and other family

members to live in the same kingdom (Demanet and Ying, 2006). The latter will be

33

discussed later in this section. Fig. 2.11 gives the definition of curvelets and the original

meaning of parabolic scaling. A curvelet is a directional wavelet, indexed by three

parameters: a scale a, 0 < a < 1; an orientation θ, / 2, / 2 and a location b, b ∈

R2. At scale a, the family of curvelets is generated by translation and rotation of a basic

element a

, , ( ) ( ( ))a b ax R x b (2.44)

Here, ( )a x is some kind of directional wavelet with spatial width ~a (perpendicular to

the oscillatory direction) and spatial length (along the oscillatory direction) ~ a :

1/ 0

( ) ( ), ;0 1/

a a a

ax D x D

a

(2.45)

where aD is the parabolic scaling matrix, R is a rotation operator. The essential spatial

support is plotted in the right-hand side of Fig. 2.12. Each flat disk specifies the effective

support of a curvelet in space domain. For two-dimensional patterns or wavefields, the

short axis is along the oscillatory direction and the long axis is perpendicular to that

direction. In beam wave language, the long axis is the effective beam width and the short

axis is along the propagation direction. Showing in the left-hand side of Fig. 2.12 is the

construction of curvelets in the frequency domain (for rigorous definition, see Candès

and Demanet, 2005).

The sampling in the frequency plane (polar coordinates) is also called second dyadic

decomposition (SDD). In the frequency domain, curvelets are supported near a

“parabolic” wedge. The shaded area represents such a generic wedge. Note the duality of

the frequency-domain and space-domain representations. A wide aperture (curvelet

width) corresponds to a narrow wavenumber band, i.e. a narrow lobe of direction, and a

short length in space domain corresponds to a wide frequency-band in the dual domain.

Physically, the parabolic scaling principle for curvelets or the generalized one for its

family represents the requirement of optimum beam-aperture for h-f beam-wave

propagation, which reaches an optimum compromise between the wave spreading and

wave front distortion. A wide aperture, which should contain a few wavelengths, is

necessary for better focusing in order to reduce wide-angle wave spreading; however, the

aperture should not be too large to avoid severe wave front distortion due to

heterogeneities. A good compromise will result in sparse beam (curvelet) propagator

matrix. The beams represented by curvelets may be viewed as coherent waveforms with

enough frequency (wavenumber) localization so that they behave like waves but at the

same time, with enough spatial localization so that they simultaneously behave like

particles. This gives the special advantages of curvelet transform for solving wave

equation in smooth media using asymptotic solutions.

It has been proved that the matrix representation of the Green's function in time-

space domain under curvelet transform is sparse in the sense that the matrix entries decay

34

nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in

the sense that the very few nonnegligible entries occur near a few shifted diagonals,

whose location is predicted by geometrical optics.

Fig. 2.12 Curvelet tiling of phase-space. The figure on the left represents the sampling in the frequency plane,

also called second dyadic decomposition (SDD). In the frequency domain, curvelets are supported near a

“parabolic” wedge. The shaded area represents such a generic wedge. The figure on the right schematically

represents the spatial Cartesian grid associated with a given scale and orientation. (From Candès and Demanet,

2005)

Fig. 2.13 shows the generalization of the curvelets to a family of directional

wavelets. Since they are constructed as tight-frames, they are a family of tight-frame

directional wavelets. From the analysis of optimum beam-aperture for sparse propagator

matrix in smooth media, we can see the extension of curvelet to a family with same

optimum beam-aperture but flexible length requirement (in the oscillatory direction) is a

natural generalization. The extension is realized by generalizing the scaling 2j vs. 2

j/2 to

2αj

vs. 2βj/2

in frequency plane or 2-αj

vs. 2-βj/2

in space plane. From Figure 2.13 we see that

the scaling parameters α and β control the flatness of the support in space, or the

thickness of the wedge in the frequency localization. It is shown that the wave packet

families defined by the horizontal segment at β =1/2 (Fig. 2.12) satisfy the scaling

requirement to yield sparse decompositions of Fourier Integral Operators (Demanet and

Ying, 2006). Along the segment of β =1/2, α =1 corresponds to curvelet, and α =1/2 is the

case of wave atoms. All the transforms specified on the segment can be realized as tight

frames. The other scaling parameters for Gabor atoms, wavelets and ridgelets are also

marked in the figure.

35

Fig. 2.13 Identification of various transforms as (α, β) families of wave packets. The horizontal segment at

β=1/2 indicates the only wave packet families that yield sparse decompositions of Fourier Integral Operators.

(From Demanet and Ying, 2006).

Fig. 2.14 shows the comparison between a curvelet and a wave atom. We can see

clearly the different scaling rules for these two atoms: curvelets obey width = length2 and

wave atoms obey width = wavelength2. Note that for the case of wave atoms, the support

is isotropic. However, for general directional wavelets, the supports are not necessarily

isotropic.

Fig. 2.14 A curvelet (left) and a wave atom (right). Curvelets obey width = length2 and wave atoms obey width

= wavelength2. Note that for the case of wave atoms, the support is isotropic. However, for general directional wavelets, the supports are not necessarily isotropic. (From Demanet and Yin., 2006).

36

2.4.2 Fast digital Transforms for Curvelets and Wave Atoms

Efficient tight-frame constructions and fast digital transform of order O(N2logN) for

curvelets and wave atoms have been developed (Candès et al., 2005; Ying et al., 2006).

As we mentioned before, with tight-frame representation, the decomposition and

reconstruction atoms are the same, and the signal energy is conserved in the frame

domain. In curvelet and wave-atom representations with the fast transform, the

redundancy is 2, i.e. the curvelet or wave-atom coefficients are twice as many as those by

orthonormal transforms.

The construction of the transform can be summarized as (see also Demanet 2006,

§1.3; Chauris, 2006): 1. Fast Fourier transform (FFT) (2D or 3D) the image into

wavenumber domain; 2. For all the scales (1 → N) perform wavenumber filtering

(windowing) using the curvelet function, and inverse FFT the filtered image to get the

curvelet coefficients. For reconstruction, just sum up the contributions from all the

curvelets in the Fourier domain and then transform back to get the space-domain results.

2.4.3 Wave Propagation in Curvelet Domain and the Application to

Seismic Imaging

It has been proved that the matrix representation of the Green's function to wave

equations in time-space domain, or more generally to hyperbolic system of differential

equations, under curvelet transform is sparse in the sense that the matrix entries decay

nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in

the sense that the very few nonnegligible entries occur near a few shifted diagonals,

whose location is predicted by geometrical optics. (See Demanet 2006, §1.1.1 §, 1.2.2;

Candes and Demanet, 2003).

Curvelet transform has been explored and applied to seismic imaging due to its

sparseness in representing multi-dimensional wavefield data and asymptotic propagators

in smooth media (Herrmann, 2003; Douma and de Hoop, 2004, 2005, 2006; Chauris,

2006). The sparseness of seismic data in the curvelet domain is well understood since it is

very similar to the case of image compression. To demonstrate numerically the

sparseness of one-way wave propagators, Douma and de Hoop (2004) gives an example

of curvelet spreading in 2D Kirchhoff migration (Fig. 2.15). In the figure, a curvelet with

dominant frequency 30 Hz (top-left) with coefficients on the spatial lattice (top-middle),

and its spectrum (top-right), is selected to represent an elementary function in space-time

domain excited on the surface. A 2D Kirchhoff migration operator, similar to equation

(2.18) or (2.19), is applied to the curvelet. For a point source in the space domain (a delta

function), the migration (backpropagation) results should be the point spreading function,

which is an elliptic isochrones in this common-offset (fixed source-receiver distance)

imaging configuration. In contrast, the curvelet spreading is rather limited only to its

neighboring cells both in the space and spectral domains (see the middle and right panels

of the bottom row of the figure). This is due to the coherent beam-forming of the spatial

aperture and the time localization of the wide frequency-band signal.

37

Fig. 2.15 The top row shows a curvelet with a dominant frequency of about 30 Hz (left, shown in depth,

2z vt ), the normalized absolute values of the coefficients on the spatial lattice (middle), and its

spectrum (right). The bottom row shows the common-offset Kirchhoff migration of the curvelet in the top row.

The middle panel in this row shows the coefficients on the spatial lattice in the lower left quadrant of the left panel (indicated with the dotted lines) for each of the numbered wedges (labeled `1' to `4') in the spectrum

(right). The middle and right panels of the bottom row show the spreading of a curvelet in space and spectrum,

respectively, during one-way propagation in homogeneous media. Here kx and kz are the horizontal and vertical

wavenumbers, respectively.(from Douma and de Hoop, 2004).

The simplest version of curvelet migration is to perform seismic map migration (For

the original definition and methods of map migration, see Weber 1955, Kleyn, 1977).

Seismic data is first transformed into frequency-wavenumber domain, and then a curvelet

transform is conducted to get the curvelet coefficients. Each curvelet has an initial

propagation direction and location (on the surface) determined by its parameters (x, p)

and the new direction and location after back propagation will be obtained by a rotation

and a time (or space) shift according to the solution of eikonal equation (Douma and de

Hoop, 2004). This is a total geometric mapping. Later a dilation (or stretch) operator is

added to the operation (Douma and de Hoop, 2005; Chauris, 2006) and the imaging

process is called TRD (translation, rotation and dilation) transform. The Algorithm of

Douma and de Hoop (2004, 2005) uses only the largest coefficients after the transform

(migration), resulting in a very efficient migration process (two orders of magnitude fast

than the ray-Kirchhoff migration). Chauris (2006) showed that interpolation (in his case,

the Shannon interpolation scheme) is necessary to keep the appropriate accuracy. He

showed an example of time migration for complex reflection structure taken from the

Marmousi model but with a uniform background. The data were regenerated by a ray-

Born method with the original reflectivity embedded in a homogeneous medium. The

imaging result is as good as the Kirchhoff migration. In Douma and de Hoop (2006), an

38

extra dilation term was introduced based on a stationary phase approximation to the

asymptotic solution to account for the focusing/defocusing effects and the geometric

spreading of curvelets. Numerical tests for simple models in homogeneous background

show the similarity of its image quality to that of the ray-Kirchhoff migration.

2.5 Wave Packet: Dreamlets and Gaussian Packets

2.5.1 Physical wavelet and wave-packets

Space-domain wavefields or seismic data recorded on a surface are special data sets.

They are not arbitrary pictures but rather the results of wave propagation and scattering.

They observe certain rules and cannot fill the 4-D space-time in arbitrary ways. The time-

space distributions must satisfy the causality relation which is dictated by the wave

equation. In the scalar case the wave equation is

(2.46)

The solution of the wave equation in fact is very sparse in the 4D volume. It can

only occupy a hyper-surface. For example, in the case of homogeneous media, the wave

solution can only exist on the “light cone” in the 4D space-time (Fig. 2.16a) or its

Fourier space (Fig. 2.16b). There are immense amount of points in the 4D

continuum outside the light cone that is excluded from the wave solutions because they

do not satisfy the causality relation (they propagate too fast or too slow). For a

inhomogeneous media with smoothly varying velocity, we can attach a local “light cone”

to each point, and the causality relation has to be satisfied locally at each point. The “light

cone” is a critical constraint to effectively represent wavefields. That is why physical

wavelet can greatly sparsify seismic and other wavefield data.

The light cone along the positive time-axis is the “causal cone”, and the one along

the negative time-axis is the “anti-causal cone”. Regular seismic data are situated on the

causal cone; Only backpropagated or time-reversal wavefields are located on the anti-

causal cone.

“Physical wavelet” was introduced by Kaiser (1993, 1994, and 2003) as localized

wave solutions to the wave equation by extending the solution to the complex space-time.

By introducing properly the imaginary-axis in time or space, wave solutions can be

localized, i.e. only exist around a space-time point and attenuating very quickly when

away from the central space-time point; But these wave-packets still satisfy the wave

equation and can be used as elementary wavefields (wavefield atoms) to synthesize any

complicated wavefields. In fact, the idea of localizing wave solution by complex

extension has been discussed by many authors (Felsen, 1976; Enziger and Raz, 1987;

Heyman and Steinberg, 1987). Kaiser systematically developed the theory and termed it

as “physical wavelet”. The wavelets derived in this way not only possess the properties of

the wavelet, but also satisfy automatically the wave equation, which is a distinctive

feature different from mathematical wavelets. This feature is very desirable when applied

to physical problems such as wave propagation and imaging.

2 2( ) ( , ) 0t u t x

( , )u tx

( , )tx

( , )p

39

Fig. 2.16 The light cones: (a) The causality hyper-surface in the 4D space-time; (b) The dispersion hyper-

surface in the 4D Fourier space.

A wave solution can be expressed in a form of plane wave superposition in the 4D

Fourier domain. In order to express the wave solution as a superposition of localized

waves, Kaiser (e.g. 1994) applied an AST (analytic signal transform) to , resulting in

an extension of from the light cone in real space-time to a causal tube in

complex space-time

(2.47)

where is the integration weight (Lorentz-invariant measure) on the

light cone, is the 3D wavenumber vector, and

(2.48)

is an acoustic wavelet of order (Kaiser, 1994), with denoting the unit step

function. In equation (2.47) the inner product of a wavefield with a physical wavelet

(2.48) is the decomposition of wavefield into localized solutions, the physical wavelets.

The localization is realized by the windowing process in the 4D Fourier domain. The

window function is defined in the causal tube. Points in the causal tube have projections

on the light cone in the form of windowing. From equation (2.48) we see that the pulse

width (waveform) is controlled by and the beamwidth-steering is parameterized by

. In Fig. 2.17 we show the wavepacket (physical wavelet) on the light

( )u x

( )u x ( , )x t x

( )u x iy

*

1( ) ( , ) ( , )zC

dpu x iy u

k

p p

3 3( /16 )dp d c p

p

1( , ) 2 ( ))exp ( ) ( )z k i t p p y p x p y

(.)

1 2 3( , )y y yy

40

cone as yellow spots. Since we take only the causal solution (for positive time), there

exist a pair of yellow spots on the light cone in the Fourier space. The windowing on the

light cone will have projected windows in the frequency-axis and the wavenumber plane.

Fig. 2.18 shows some examples of the time-domain waveform (for a fixed space location)

of different orders ( =3, 10, 15, 50) of a physical wavelet. In the same way, we show

the space localization at different times in Fig. 2.19 for the same physical wavelet. In this

example, the wavelets are isotropic ones, so the plot is for the r-dependent.

Fig. 2.17 Windowing (localization) on the light cone in the Fourier space ( ), where p is the space

wavenumber vector and is the frequency.

Fig. 2.18 Physical wavelets along the time-axis (at r = 0) for . Solid lines are the real part,

and dotted lines are the imaginary part.

,p

3,10,15,50

41

Fig. 2.19 Physical wavelets along the space-axis (r-axis) (at t = 0) for isotropic wavelets at .

2.5.2. Dreamlet as a Type of Physical Wavelet

A space-time physical wavelet is a localized solution of the corresponding wave

equation. As we discussed in the previous section, it can be formed by analytical

extension to a complex space-time domain, so the imaginary part of the wavefield

controls the localization of the solution. In fact, there are other ways to obtain localized

solutions of wave equation, such as the Gaussian packet introduced by Kiselev and Perel

(1999) and Perel and Sidorenko (2007).

Wu et al. introduced a space-time wavelet formed by a tensor product of wavelet in

time-domain (drumbeat) and beamlet in space domain, and termed it as dreamlet

(drumbeat beamlet) (Wu et al., 2008, 2009; Wu and Wu, 2010). The relation between

the dreamlet and physical wavelet is discussed theoretically in Wu et al. (2011). Here we

explain conceptually the relation between physical wavelet, dreamlet, and other wavelets

defined in a subspace of the 4D space-time.

Time Slice and Depth Slice of 4D Wavefield:

In Fig. 2.20 and Fig. 2.21 we show the different features of “time-slice” (snapshot)

and “depth-slice” (data section) of a 4D wavefield. We see that a “snapshot” is the

wavefield distribution in the 3D space domain for a given time t. Due to the causality, the

solution can only exist in a hollow sphere (in 2D, a circle) (Fig. 2.20); While for a depth-

slice (data on a plane), it becomes a “paraboloid” (Fig. 2.21). They appear to have quite

different picture structures, but are in fact from the same object. Therefore, the

3,10,15,50

42

x z

decomposition atoms for efficiently representing a wavefield (snapshot) or a data set

(seismograms) could be very different. For the snapshots in the space-domain, local

Fourier atom, Gabor atom, curvelet, or other directional wavelet can be used due to the

symmetry property of the time-slice; On the other hand, the data decomposition (depth-

slice) need different atoms to take into account of the paraboloidal structure. “Dreamlet”

and different wavepacket decomposition are designed for this purpose.

Fig. 2.20 A time-slice (snapshot) cut through a light cone.

Fig. 2.21 A depth-slice (data section) cut through a light cone.

43

Fig. 2.22 Examples of time-slices (snapshots) of the wavefield from the 2D acoustic SEG/EAGE salt model. The source is an 18 Hz Ricker wavelet.

Fig. 2.23 Examples of depth slice of the wavefield: the seismic data (acquired on the earth suface) of SEG 2D

salt model (coincided source and receirver)

In Fig. 2.22 we show two snapshots of the propagated wavefields for the 2D acoustic

SEG/EAGE salt model. The source is an 18 Hz Ricker wavelet. We see that the

wavefields are basically composed of different circles of various radii (the curvature of

the circle changes with the propagation velocity). In contrast, the seismic data set

Distance (km)

44

acquired on the earth surface (depth-slice) shown in Fig. 2.23 (post-stack data) for the 2D

acoustic SEG/EAGE salt model, resembles a collections of parabolas of different sizes.

This is consistent with the concepts demonstrated in Fig. 2.20 and Fig. 2.21.

Dreamlet =DrumbeatBeamlet:

A dreamlet atom is in the form

( , ) ( ) ( )tt x xd x t g t b x (2.49)

where

i ttg t W t t e

(2.50)

is a t f atom (drumbeat) with W t as a smooth window function, and

( ) ( ) i xx

b x B x x e (2.51)

is a x atom (beamlet) with ( )B x as a bell function. In this section, x represents a

horizontal position on the observation plane. The bars over letters signify the variables

are local variables related to the window centers. Therefore, ( , , , )t x marks the

coordinates of local time, local frequency, local distance, and local wavenumber,

respectively. Note that the phase terms in the t f atom and the x atom have

opposite signs. This is consistent with the causality relation imposed by the wave

equation. Although the beamlet transform is applied on the observation plane, however,

the beamlets are defined in the full wavenumber space through the dispersion relation.

The window ( )B x x on the observation plane (here the x-axis for the 2D case) is a

cross-section of the whole space window.

Note that in the definition (2.50) and (2.51), the phase terms are still using global

space-time. An alternative definition of the dreamlet atom is to use the local space time:

( )

( )( ) ( )

i t tt

i x xx

g t W t t e

b x B x x e

(2.52)

This definition is more consistent with the definition of the local cosine bases. It is only a

matter of phase-shift of the atoms, and all localization properties are kept same.

2.5.3. Seismic Data Decomposition and Imaging/Migration using

Dreamlets

45

2.5.3.1. Seismic Data Compression and Data Recovery in Compressive

Sensing using Dreamlets:

A space-time domain seismic data field or wavefield on an extrapolation plane can

be represented as superposition of the phase space atoms (dreamlets)

( , ) ( , )t x t x

t x

u x t u d x t

(2.53)

where t x

u are the dreamlet coefficients. As we discussed in previous sections, seismic

data are not ordinary pictures, but are produced by physical process satisfying the wave

equation. They can only situate on the depth-slices of the “light cone” in the 4D time-

space. Therefore, the representation by physical wavelets is perhaps the sparsest one. In

the following we show two examples demonstrating the representation efficiency and

sparseness of dreamlets in comparison with out methods, such as curvelets. Fig. 2.24

shows the compression ratios of the SEG 2D salt model data ( see Fig. 2.23) by different

decomposition methods (including the beamlet and curvelet) as a function of threshold

(Wu et al., 2008; Geng et al., 2009). We see that Dreamlet, or physical wavelet in

general, has the most efficient representation to this seismic data set, especially for high

compression ratio.

Fig. 2.24 Compression ratios of different decomposition methods for the SEG/EAGE salt model poststack data.

The second example is on the data recovery in compressive sensing (Donoho, 2006;

Herrmann, 2010). The compressive-sensed data set with 50% missing traces is simulated

by removing half of the geophone records randomly in the post-stack data set of

46

SEG/EAGE 2D salt model. Data recovery (recovery of the missing traces) is done by a

basis pursuit method based on l1-norm optimization to search for the most efficient

representation (least residual with minimum coefficients). As shown in Fig. 2.25,

Dreamlet method takes around 60 iterations to get the solution while Curvelet method

takes around 950 iterations with similar recovery quality (signal-to-noise ratio) (Wu et

al., 2013). From the Figure we see also that dreamlet has less number of coefficients to

represent the recovered data.

Fig. 2.25 Data recovery in compressive sensing by dreamlet and curvelet: L2-norm of the residuals (top panel)

and number of coefficients (bottom panel) as function of iteration number.

2.5.3.2 Dreamlet evolution

The evolution of dreamlets in heterogeneous media can be derived from the wave

equation. Assuming a wave propagator, here a one-way wave propagator, , is applied

to a dreamlet, the propagator matrix in the dreamlet domain is obtained as

t x

t x t x t xd d

(2.54)

where f g stands for the inner product of f and g, t x

d is the dreamlet atom at the

current level (depth), and t x

d is the dreamlet at the next level during propagation. d

is the dual frame vector of d. As in the case of beamlet propagator (see section 2.3.4), a

dreamlet propagator can be derived by the local perturbation theory. The use of local

background velocities and local perturbations results in a two-scale decomposition of the

dreamlet propagator: a background propagator for large-scale structures and a local

phase-screen correction for small-scale local perturbations (for detailed derivation, see

Wu et al., 2008, 2009; Wu and Wu, 2010).

47

Fig. 2.26 shows some examples of the Gabor dreamlet atoms and their snapshots

during propagation. Gabor atom is characterized by the use of Gaussian window and

exponential phase function. In the example, both the space and time windows are of L=32

points Gaussian window. Compared with the beamlet propagation, we see that the

dreamlet propagation maintains both the space and time localizations in homogeneous or

smoothly inhomogeneous media. For strongly heterogeneous media, the dreamlet will

split and spread due to diffraction and scattering. Similar to the beamlet propagation, we

apply the local perturbation method to calculate the propagation effect of small-scale

perturbations in each window (under one-way approximation).

We can also use orthogonal bases, such as the LCB (local cosine basis) as the

dreamlet atoms. Both decomposition and propagation using LCB dreamlets are more

efficient due to the orthogonality of representation. However, similar to the LCB beamlet

propagation, the LCB dreamlet has always two symmetric lobes.

Fig. 2.26 Gabor dreamlet atoms (space-time representation) (Top panels) and their snapshots (for 3 time-

instances) during propagation (Bottom panels). Window lengths for both time and space are Gaussian windows

with L=32 points. We see that atoms with different space-time patterns propagate in different localized directions.

2.5.3.3 Application of Dreamlet to Seismic Imaging

Dreamlet migration/imaging is composed of three major steps:

1. Data decomposition: On the surface, the data set is decomposed into dreamlets.

48

2. Wavefield extrapolation (downward propagation): in the dreamlet domain: For each

step z , apply the propagator matrix to the coefficient and obtain a new set of

coefficient. This step include both the propagation in local homogeneous reference

media and the local phase-screen corrections.

3. Apply the imaging condition at each step to obtain the image field: This can be done

either in the dreamlet domain, or by transforming back to the space-time domain by

the inverse dreamlet transform. Step 2 and 3 are implemented iteratively until the

bottom of the image space.

Fig. 2.27 Number variation of dreamlet coefficients during migration. The black and green lines are for the

survey sinking dreamlet coefficients using sunken data and the full data, respectively (See Wu et al., 2013).

The salient feature of the dreamlet migration is the ability to implement

“migration/imaging in compressed domain”. Due to the high efficiency in representing

the seismic data and wavefield, the dreamlet coefficients becomes very sparse, and can

stay sparse during imaging process such as the prestack depth migration. From Fig. 2.25,

we see that in general, the dreamlet coefficients will stay sparse during

migration/imaging process. The example shown here is for case of survey-sinking

migration, in which both the source array and receiver array are downward-extrapolated

step by step in depth, and apply the imaging condition locally at each depth (see, Wu et

al.,2009, 2013). After sinking the surveying system to the depth, imaging process

(determining the scattering points) is realized locally by coinciding incident and scattered

waves at the imaging points (zero-traveltime imaging condition). The green line shows

the variation of coefficients during migration using the full data, which is similar to the

case of shot-domain prestack depth migration; while the black line shows the variation

when using the sunken data during survey sinking imaging. Because of the time

localization in dreamlet decomposition, the data with negative traveltime which will have

no contribution to the imaging process beneath the current depth will be discarded. In this

way, the already sparse coefficients become sparser during the migration process (Fig.

49

2.27). Sparse data representation, recovery, imaging and inversion form an active area of

research.

2.5.4 Gaussian Packet Migration and Paraxial Approximation of

Dreamlet

When we use the Gabor frame atom (also called Gabor-Daubechies frame) as both

the drumbeat (t-f atom) and the beamlet atoms ( x atom) to form the dreamlet, the

atom is called Gabor dreamlet (section 2.5.3.2). Since the window in both time and space

is of Gaussian shape, it is also called Gaussian wave packet or simply Gaussian packet

(Norris et al.,1987; Klimeš, 1989; Kiselev and Perel, 1999; Červený, 2001; Perel and

Sidorenko, 2007; Qian and Ying, 2010). In case of smooth media, h-f asymptotic solution

exists for the Gaussian packet and the packet will travel along the central ray determined

by the eikonal equation (see, Červený et al., 2007). The asymptotic solution of

wavepacket is also called “quasiphotons (Babich and Ulin, 1981), “space-time Gaussian

beam” (Ralston, 1983), “coherent state” (Klauder, 1987).

Dreamlet evolution in arbitrarily heterogeneous media can be calculated by dreamlet

propagator matrix (2.54). However, in smooth media, the scattering is weak, and

dreamlet evolution only involves shift, rotation, wavefront spreading, envelope

broadening and other distortions. These forward-scattering phenomena can be calculated

approximately by h-f asymptotic method. In this approach, once the data set is

decomposed on the surface, the wavepacket (dreamlets) can propagate globally and

individually without mutual coupling. This method of asymptotic solution is convenient

and efficient, though less accurate. It has great potential in many applications.

High-Frequency Asymptotic Solution and Paraxial Approximation:

Traditional high-asymptotic solution (e.g., Červený et al., 2007) can be applied to the

dreamlet evolution

( , , ) , , exp , ,t x

d x z t A x z t i x z t (2.55)

where can be understood as the central frequency of the wave packet, and both phase

function ( , , )x z t and amplitude function ( , , )A x z t are complex-valued functions of

space coordinates ( , )x z and time t . For simplicity, we use instead of ( , , )x z t and

A instead of ( , , )A x z t in the following equations. From the definition of (2.49) –

(2.51), we see that dreamlet atoms with Gaussian windows can be treated as Gaussian

wavepacket. In the following we summarize the method of paraxial Gaussian packet and

establish the link of seismic data decomposition using Gaussian packet to the dreamlet

decomposition.

For the convenience of derivation and efficiency of propagation algorithm, the

accurate complex phase function (traveltime field) of a wavepacket can be expanded into

second order of Taylor series in the vicinity of its center along a ray. Under this

50

approximation, not only the amplitude envelope, but also the phase front become

Gaussian (parabolic approximation). That is the origin of the name of Gaussian beam and

Gaussian packet. Červený (2007) call this approximation as “paraxial Gaussian packet”,

and call the original asymptotic solution as “strict” Gaussian beam or Gaussian packet.

The propagation of the paraxial Gaussian packet using the asymptotic solution is well

documented in the literature (e.g. Klimeš, 1989; Červený, 2007). Here we summarize

some major points and explain the results in terms of its wave physics

Under the h-f asymptotic approximation, the wavepacket will move along a ray

satisfying the eikonal equation

2 2( ) 0v

t t

(2.56)

where v is the local velocity. The eikonal equation determines the trajectory of the

wavepacket movement, such as its shift and rotation. In fact, the eikonal equation is the

mathematical form of saying that the trajectory must stay on the local light cone, and is a

form of causality relation (dispersion relation) of wave physics. On the other hand, to

describe the distortion of the wavepacket, the phase function was expanded into in a

second order Taylor expansion in the vicinity of its central point along the ray,

21

( ) ( , ) ( ) ( )2

r rt x x p x x N x x (2.57)

where x is the 4-dimensional space vectors, ( , / )t p is 4-dimensional

slowness vector (gradient of the phase function), and N is the 4-dimensional curvature

matrix (second derivative of the phase function), respectively. rx is the central point of

the packet along the ray. For the 2D case, equation (2.57) can be written explicitly

2 2 22 2

2

( , ) ( ) ( )

1 1( ) ( ) ( )( ) ( )( )

2 2

r r

r r r r r r

t t tt

t t x x t t z z t tt x t zt

x p x x

N x x

(2.58)

where p , x and rx are 2D vectors (in 3D media: 3D vectors). Note that here p denotes

slowness vector (not to be confused with the case of physical wavelet, where it is

wavenumber vector). Due to the causality relation, it should hold in the above equation

1t

(2.59)

Under this paraxial approximation, equation (2.55) becomes

51

2

( , ) , , , , exp , ,

1exp ( ) ( ) ( )

2

pgp pgp pgpt x

pgpr r r

d t u x z t A x z t i x z t

A i t t

x

p x x N x x

(2.60)

Where pgpu ,

pgp , and pgpA are corresponding functions of “paraxial Gaussian

packet”.

The above equation describes the paraxial Gaussian Packet (PGP) evolution in the

Cartesian coordinate system. Changing to the ray-centered coordinate system ( , , )q s t ,

where q is the perpendicular distance from the central ray and s is the distance along the

central ray. The curvature matrix becomes

2 2 2

2

2 2 2

2

2 2 2

2

qq qs qt

sq ss st

tq ts tt

q s q tqM M M

M M Ms q s ts

M M M

t q t s t

M (2.61)

Then the curvature matrix and other quantities needed for the packet evolution in

equation (2.58) can be efficiently calculated by kinematic and dynamic ray tracings (see

Červený, 2007). Therefore, the main calculation task is the dynamic ray tracing for P

(slowness derivative) and Q (position derivative). Once the spatial curvature qqM is

determined by the DRT (dynamic ray-tracing), the temporal curvature and other

quantities can be determined by analytical relations (Klimeš, 1989; Geng et al., 2013).

This is also due to the causality relation which restricts these curvatures from arbitrary

variation. This can explain the efficiency of the wavepacket method compared with the

beam method. In order to calculate the frequency-dependent phenomena of wave

propagation, we need to compute beam propagation for many frequencies in a frequency-

band. The final waveform change is the result of interference between all the beams in

the band. However, for smooth media, this change can be predicted by the parabolic

approximation, which is the formulation of wavepacket propagation.

Wavepacket Decomposition of Seismic Data:

An important issue in the application of Gaussian packet in seismic imaging, the

decomposition of seismic data into Gaussian packet, was not well-solved until recently

and is still an active research area.

In the literature, some effort has been paid to the decomposition of seismic data

recorded on the surface into paraxial Gaussian packets (Zacek 2004, 2005, 2006a,b).

52

However, the proposed method is time-consuming and the imaging results are not

satisfactory. From the discussion on physical wavelet and dreamlet, we know that seismic

data are generated by exact wave equation, not by paraxial wave equation. Therefore, the

decomposition using paraxial Gaussian packets does not fit the task; while dreamlet atom

may be one of the optimum decomposition atoms for seismic data. From Fig. 2.24 and

Fig. 2.25 we see the high efficiency of dreamlet representation due to the property of

physical wavelets. The other reason of the efficient representation of seismic data by

dreamlets is the relatively narrow band of seismic signal in exploration seismology. In

Fig. 2.28 we show the fitting of a Ricker wavelet with central frequency at 15 Hz (solid

line) by a Gabor time atom (strict Gaussian packet) of different parameters. It can be seen

that with 2 2/ 20ttM t i , a single Gaussian packet can well represent the seismic

source time function.

After dreamlet decomposition, data compression can be done with thresholding.

Depending on the data sets, the compression ratio varies. In the migration example of

shown later in Figure 2.31, only 1% of coefficients are used for imaging. The conversion

from dreamlet parameters to the initial parameters of paraxial Gaussian packets has been

discussed (Geng et al., 2013).

Fig. 2.28 Comparison of the time-profile of a Gaussian packet of different parameters (2 2

33 /M t )

with the Ricker source wavelet. We see that with the choice of appropriate parameter (M33=20i), the seismic source wavelet can be simulated by a single Gaussian packet.

Examples of Guassian Packet Evolution, Modeling and Imaging:

Fig. 2.29 shows an example of the evolution of three Gaussian Packets of

20f Hz with different propagating directions in a vertically gradient media, which has

velocity of 2000 /m s at the surface and 2900 /m s at depth of 3000m . Black lines

are the three central rays for the corresponding Gaussian Packets. For comparison, the

evolution results using 2D fourth-order acoustic FD algorithm are shown at the bottom.

We see that the Gaussian Packet solution in this simple medium can provide fairly

accurate phase and amplitude information around the vicinity of the central point.

However, the phase begins to be inaccurate when it is too far from the central point.

53

Fig. 2.29 Evolution of an initial Gaussian packet in a linearly gradient medium. a) by paraxial Gaussian Packet

method; b) by full wave Finite Difference method.

Second example is for the case of complex model. In order to perform ray tracing,

the 2D SEG-EAGE velocity model is smoothed to avoid singular behavier of rays on the

salt boundaries. Even after smoothing, the velocity model is still strongly varying in

irregular ways. Fig. 2.30 shows the impulse response by the Gaussian packet summation

method. We shot rays with the same initial time from the surface for all the Gaussian

Packets centered at different time rt . The wavefield at time T in the subsurface can be

approximated by the summation of all Gaussian Packets traveled to the point ( , )r rx z

with traveltime rT t . The source time function used here is a Ricker wavelet with

dominant frequency 30f Hz and is decomposed into a superposition of Gabor frame

with redundancy 8 and window length of 64. Gaussian Packets are shot from the source

point with 200 directions. The comparison with FD method shows that the Gaussian

packet method can simulate most of the prominent features of the impulse response

outside the salt body or after traveling a short distance inside the salt body. However,

after long propagation inside body, the distortion becomes more severe.

Fig. 2.31 gives an example of Gaussian packet migration for the 2D SEG-EAGE

zero-offset data set. As mentioned earlier, the salt boundaries are smoothed for the sake

of ray-tracing. The minimum velocity is 1600 /m s and the maximum velocity is

4500 /m s . The data decomposition uses the Gabor dreamlet, with frame redundancy 8

and window length of 64 for both time and space. On the top is the image obtained by

beamlet migration for comparison. At the bottom shows the image by the Gaussian

packet migration. Only 1% of the largest dreamlet coefficients are used in the migration.

Compared to LCB (wide-angle one-way migration) method, Gaussian packet method can

54

recover almost all the important structures of the model, such as the boundary of the salt

body, the steep faults and most of the subsalt structures.

Fig. 2.30 Impulses responses (by Gaussian packet summation) in the 2D SEG-EAGE model calculated by the

paraxial Gaussian Packet method at 1.0 t s (b) and at 1.5 t s (d). For comparison plot on the left are

those calculated by full-wave finite-difference method (FD); The source location is 8550 x m .

Fig. 2.31 Comparison of imaging results for SEG-EAGE 2D model. Top: By wide-angle one-way imaging

method (LCB beamlet migration); Bottom: By paraxial Gaussian packet method with Gabor dreamlet

decomposition of redundancy 8 and 64N N .

However, some imaging artifacts appear at the irregular top salt boundary due to the

strong scattering effect. It is also noticeable that in some subsalt area, Gaussian Packet

55

method has weaker images due to the reduced number of effective packets which can

penetrate the thick and irregular salt body. However, Gaussian packet migration is much

faster, and has the flexibility to include large-angle and turning waves in migration.

Therefore, the research along this direction is still active.

2.6 Conclusions

1. For wavefield decomposition, both beamlet and curvelet transforms have elementary

functions of directional wavelets. Beamlet is a type of physical wavelet, representing

an elementary wave (satisfying wave equation) in various wavefield decomposition

schemes using localized building elements, such as coherent state, Gabor atom,

Gabor-Daubechies frame vector, local trigonometric basis function, etc.. Curvelet

transform is a specifically defined mathematical transform, characterized by the

parabolic scaling: width length2. A family of beamlets is built on the invariance

with space shift, dilation and frequency (wavenumber) shift; while curvelets are

based on the invariance with space shift, dilation and rotation. In curvelets the

directivity is directly defined by the rotation parameter; while in the case of beamlets

the directivity is related to the wavenumber by the wavenumber-angle relation (2.7).

2. Sparseness of propagator matrix and the parabolic scaling: The parabolic scaling law

originally is defined as width length2

for the elementary function in the space

domain. Later it is relaxed to width wavelength2

in order to let “wave atom’ and its

other cousins stay in the family. In this way, the length requirement is more flexible.

This extended parabolic scaling law is similar to the beam-aperture requirement for

asymptotic beam solution a > >> , namely the beamwidth must be smaller than

the scale of heterogeneity and much greater than the wavelength. Optimal

beamwidth is reached by balancing the beam geometric spreading which is

controlled by the ratio / , and the beam-front distortion which depends on a /

. Using optimal beamwidth, beamlet or curvelet propagator will be sparse in smooth

media for short range propagation, since the elementary function will propagate with

least distortion and cross-coupling. Even through the parabolic scaling is only a

special case of the general criterion, the efficient tight-frame construction of curvelet

transform based on this scaling is very useful. For long range beam propagation,

there are two more parameters enter into the game: the range R and the total

perturbation strength . The requirement of a >> for asymptotic solution is

replaced by / 2Fa r R where Fr is the Fresnel radius along the beam path

(see section 2.3.3)

3. Asymptotic methods for beam propagation: In smooth media, various asymptotic

theories and methods have been developed for propagation of beams – globally

evolved beamlets: The Gaussian beam, complex ray, pulsed beam, coherent state and

other asymptotic methods. These solutions are either in a form of wavenumber

integral or an explicit space-domain solution. These solutions have very similar

56

derivations, although they may have different degrees of approximation. Asymptotic

solutions for curvelets have also emerged recently.

4. Beamlet scattering in heterogeneous media: In case that the beam-aperture

requirement a > >> is not satisfied, or other criteria of asymptotic analysis are

violated, beamlet or curvelet scattering will occur and needs to be studied. For

beamlet propagation, a local perturbation theory and method have been developed to

handle strong and rough heterogeneities. The propagator is decomposed into a

background propagator and a perturbation operator for each forward marching step.

For background propagation asymptotic solutions can be applied to the piece-wise

background media, and a phase-screen correction is performed for the local

perturbations. Numerical examples demonstrated the superior image quality for

subsalt structures.

5. Dreamlet (Drumbeat Beamlet) is a fully localized wavefield atom, and is a type of

physical wavelet (satisfying the wave equation automatically). It is efficient in

representing seismic data and other wavefield data (with high compression ratio).

Dreamlet propagator is also sparse, and asymptotic approximation leads to a

Gaussian packet propagator which has been applied to seismic migration/imaging.

6. Future work on beamlet/curvelet and dreamlet scattering in generally heterogeneous

media is needed for the development of efficient methods on propagation and

imaging. The multi-scale nature and the advantages of phase-space localization have

to be further studied and put into applications in imaging and inversion.

Acknowledgement

We have benefited from the discussion with Yingcai Zheng, Jun Cao, Xiao-Bi Xie and

Chuck Mosher. Yingcai has contributed to the review of Gaussian beam and coherent

state. Yu Geng, Yaofeng He, Jian Mao, Lingling Wang, Bangyu Wu have participated in

the preparation of the draft and helped in drawing some figures. We are very grateful to

Laurent Demanet, Huub Douma and Martijn de Hoop for discussion and allowing the use

of their figures in this review article. We would like to acknowledge the support from

WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration)

Project and the DOE/Basic Energy Sciences project at University of California, Santa

Cruz, California, USA and the support from the China NSF and other projects of Xi’an

Jiaotong University.

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67

Author Information

Ru-Shan Wu

Modeling and Imaging Laboratory, Institute of Geophysics and Planetary

Physics/Department of Earth and Planetary Sciences, University of California, Santa

Cruz, CA 95064, USA

E-mail: rwu@ucsc.edu

Jinghuai Gao

School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an,

China