GEOPHYSICS, VOL. 74, NO. 4 共JULY-AUGUST 2009兲; P. S85–S93, 12 FIGS.
10.1190/1.3131383
Full-wave directional illumination analysis in the frequency domain
Jun Cao1 and Ru-Shan Wu1
the rays. However, the high-frequency asymptotic approximation
and the caustics inherent in ray theory might limit severely its accuracy in complex regions 共e.g., Hoffmann, 2001兲. Furthermore, the
ray method cannot describe the finite-frequency effect of seismic
waves. To obtain reliable and frequency-dependent illumination, we
need a wave-theory-based method.
One-way wave-equation-based propagators are widely used in illumination analysis. Although they neglect multiples, they can handle multiple forward-scattering phenomena, including focusing/defocusing, and diffraction. Unlike the ray-based methods, the wavefield obtained from wave-equation-based methods do not explicitly
specify the directional information. Recently developed techniques,
such as local slant-stack 共LSS兲 共e.g., Xie and Wu, 2002兲 and beamlet
decomposition 共e.g., Wu et al., 2000兲, can decompose the wavefield
into local plane waves, from which we can compute LAD illuminations 共e.g., Wu and Chen, 2002, 2006; Xie et al., 2003, 2006; Mao
and Wu, 2007; Cao and Wu, 2009兲.
However, the amplitude of the one-way wave propagator is inaccurate in complex models with sharp contrasts, even after corrections 共e.g., Zhang, 1993; Zhang et al., 2003; Kiyashchenko et al.,
2005; Wu and Cao, 2005兲. Numerical implementations based on a
one-way wave equation with the z-axis as the preferred propagation
direction always have inherent limitations in wide-angle accuracy.
Some one-way propagation-based methods can model backscattered waves or turning waves 共e.g., Wu, 1996; Xie and Wu, 2001; Kiyashchenko et al., 2005; Jin et al. 2006; Wu and Jia, 2006; Xu and
Jin, 2006; Zhang et al., 2006兲; however the amplitude accuracy for
these propagators still is in question because of various factors.
Full two-way wave equations solved by the finite-difference or finite-element method can simulate accurate and complete wave behavior in complex media. Therefore, full-wave equation-based illumination analysis should provide full-angle true-amplitude illuminations of all arrivals for survey design and image amplitude corrections. Both full-wave modeling and LAD illumination analysis can
be implemented in the time or frequency domain. Xie and Yang
共2008兲 proposed an illumination method using a time-domain fullwave finite-difference method as the propagator and a time-domain
local-slowness analysis method to derive the directional informa-
ABSTRACT
Directional illumination analysis based on one-way wave
equations has been studied extensively; however, its inherent
limitations, e.g., one-way propagation, wide-angle error, and
amplitude inaccuracy, can severely hinder its applications for
accurate survey design and true-reflection imaging corrections in complex media. We have analyzed the illumination in
the frequency domain using full two-way wave propagators
considering the extensive computation and huge storage required for time-domain methods, and the fact that the illumination is frequency dependent. This full-wave analysis can
provide frequency-dependent full-angle true-amplitude illumination not only for the downgoing waves but also for the
upgoing waves, including turning waves and reflected
waves. Two methods were considered to decompose the full
wavefield into the local angle domain: a direct full-dimensional decomposition and more efficient split-step decomposition composed of three lower-dimensional decompositions.
The results of illumination analysis demonstrated the advantages of this method. The two decomposition methods produced similar results.
INTRODUCTION
Local angle-domain 共LAD兲 illumination analysis 共e.g., Wu and
Chen, 2006兲 studied the acquisition aperture and propagation-path
effects. It has many applications, including survey design 共e.g., Li
and Dong, 2006兲, studying the influences of acquisition geometry
and overburden structures on the image 共e.g., Jin and Walraven,
2003; Wu and Chen, 2006; Xie et al., 2006兲, and image amplitude
correction 共e.g., Wu et al., 2004; Cao and Wu, 2005, 2008兲.
Traditionally, illumination analyses have used ray-based methods
共e.g., Schneider and Winbow, 1999; Bear et al., 2000; Muerdter et
al., 2001; Muerdter and Ratcliff, 2001a, 2001b; Lecomte et al.,
2003兲, in which the directional information is inherently carried by
Manuscript received by the Editor 13 August 2008; revised manuscript received 23 October 2008; published online 5 June 2009.
1
University of California, Department of Earth and Planetary Sciences, Institute of Geophysics and Planetary Physics, Santa Cruz, California, U.S.A. E-mail:
jcaogeo@gmail.com; wrs@pmc.ucsc.edu.
© 2009 Society of Exploration Geophysicists. All rights reserved.
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Cao and Wu
tion. The method is particularly useful for providing illumination
analysis for reverse time migration.
We propose to analyze the full-wave illumination in the frequency
domain for several reasons. First, the illumination is frequency dependent. Second, the frequency-domain analysis method requires
less storage compared with the time-domain method. Third, the frequency-domain angle decomposition of the wavefield is more efficient than the time-domain method.
We illustrate the idea in 2D media. First we describe the basic concept for full-wave equation-based LAD illumination. Then we describe the two proposed wavefield decomposition methods for fullwave propagators. Following that, we show some examples demonstrating the advantages of full-wave illumination in the frequency
domain, including the illumination for upgoing waves, such as turning waves, and reflected waves. We also show the illumination analysis for the BP 2004 benchmark model 共Billette and BrandsbergDahl, 2005兲. In the discussion section, we show the influence of multiples on the illumination strength and compare the angle resolution
of the two proposed angle-decomposition methods.
FULL-WAVE EQUATION-BASED DIRECTIONAL
ILLUMINATION ANALYSIS
One-way wave-equation-based LAD illumination is discussed in
many papers 共e.g., Wu and Chen 2006; Xie et al., 2006兲. Here we
briefly describe the basic concepts of wave-equation-based LAD illumination. The computation procedure of LAD illumination can be
summarized as follows. First, in a given model we put a unit-strength
source at the shot/receiver location 共xs / xRs兲 and propagate the wavefield into the model space x ⳱ 共x,z兲. Second, by local plane-wave
decomposition techniques, e.g., LSS 共Xie and Wu, 2002兲 and beamlet decomposition 共Wu et al., 2000兲, we can obtain the Green’s
functions in the local source/receiving angle 共 ␪¯ s,␪¯ g兲 domain,
G共x,␪¯ s;xs兲, G共x,␪¯ g;xRs兲. Third, with the LAD Green’s functions
from all shot and receiver locations, we can compute the LAD illuminations for this acquisition configuration.
For a single frequency, the directional illumination is defined as
共e.g., Wu and Chen 2006兲
DI共x,␪¯ s兲 ⳱
冋
兩G共x,␪¯ s;xs兲兩2
兺
x
s
册
1/2
共1兲
,
冋兺 兺
xs ␪¯
r
DI2共x,␪¯ s;xS兲 兺 DI2共x,␪¯ g;xRs兲
xR
s
WAVEFIELD DECOMPOSITION FOR
FULL-WAVE PROPAGATORS
Local plane-wave decomposition techniques, such as the LSS
method and beamlet decomposition method, were applied along the
horizontal coordinate共s兲 共e.g., Xie and Wu, 2002; Wu and Chen,
2002; Xie et al., 2003兲, which is appropriate to one-way propagators.
First we summarize these two decomposition methods for one-way
propagators. Then we describe two decomposition methods for fullwave propagators: direct 2D decomposition and split-step decomposition using 1D decompositions.
Wavefield decomposition for one-way propagators
1D local slant stack for one-way propagators
which measures the incident-angle response of the source aperture in
a given model. The acquisition dip response 共ADR兲 is defined as
共e.g., Wu and Chen 2006兲
ADR共x,␪¯ n兲 ⳱
For full-wave equation-based illumination, the first step, generating the space-domain wavefield, has been studied extensively. To
obtain the frequency-domain wavefield, the frequency-domain
modeling 共e.g., Marfurt, 1984; Operto et al., 2007兲 or time-domain
modeling plus the running discrete Fourier transform 共DFT兲 sum
can be used 共e.g., Luo et al., 2004; Nihei and Li, 2007; Sirgue et al.,
2008兲. Nihei and Li 共2007兲 compared the requirements of storage
and floating-point operations of the time-domain and frequency-domain finite-difference methods for 2D and 3D multiple-source frequency-response modeling. 共In the comparison, the time-domain
method is an explicit scheme, and the frequency-domain method
uses direct solution of the linear system equations by LU-factorization with the nested dissection reordering.兲 Their comparison shows
that, for most 2D problems, when there is ample memory the frequency-domain method can efficiently provide the frequency responses for multisource problems; however, for 3D problems, a better choice is time-domain modeling plus the running DFT sum over
time marching 共see also Sirgue et al., 2008兲.
For the third step, illumination formulas originally defined for the
one-way propagator can be used for the full-wave case, except that
the local angle ranges are different 共full-wave illumination can obtain the illumination not only for downgoing waves, but also for upgoing waves兲. Therefore, the main task here is the second step: to decompose the space-domain full wavefield into LAD.
册
1/2
,
共2兲
where ␪¯ n ⳱ 共 ␪¯ s Ⳮ ␪¯ g兲 / 2,␪¯ r ⳱ 共 ␪¯ s ⳮ ␪¯ g兲 / 2 represent the local dip
and reflection angle, respectively. The ADR measures the dip-angle
response of the whole acquisition system, including the source and
receiver apertures.All contributions from the various source-scattering angle pairs for the same dip are summed together to obtain the
ADR for that dip. Because the Green’s function is calculated by
wave-theory-based one-way propagators, the illumination includes
the path effects, including all forward-scattering phenomena. However, backscattering is excluded because of the one-way approximation of the propagator.
In 1D LSS, using a windowed Fourier transform along a horizontal coordinate, we can decompose the wavefield uz共x,␻ 兲 at depth z
for frequency ␻ , obtained from any extrapolator, into local plane
waves 共Figure 1兲,
uz共x,␪¯ ,␻ 兲 ⳱
冕
w共x⬘ ⳮ x兲uz共x⬘,␻ 兲eⳮi共x⬘ⳮx兲·k共x兲sin ␪ dx⬘,
¯
共3兲
where ␪¯ is the local plane-wave propagating angle with respect to
the vertical direction, w is a 1D window in the horizontal direction
and centered at x, and k共x兲 ⳱ ␻ / V共x兲; V共x兲 is the local velocity.
1D beamlet decomposition for one-way propagators
The wavefield uz共x,␻ 兲 also can be decomposed into beamlets by
the following formula,
Full-wave directional illumination
uz共x,␻ 兲 ⳱ 兺 兺 ûz共x̄n,¯␰ m,␻ 兲bmn共x兲,
m
共4兲
u共x,␪¯ ,␻ 兲 ⳱
n
where bmn are the beamlets 共decomposition basis vectors兲 located
at space window x̄n and wavenumber window ¯␰ m, and where
ûz共x̄n,¯␰ m,␻ 兲 are the corresponding decomposition coefficients. References in the introduction provide details on beamlet decomposition, including the formula to obtain the decomposition coefficients.
We can obtain the local plane waves uz共x,¯␰ m,␻ 兲 by partial reconstruction of the beamlet-domain wavefield,
uz共x,¯␰ m,␻ 兲 ⳱ 兺 ûz共x̄n,¯␰ m,␻ 兲bmn共x兲.
共5兲
n
By the dispersion relation of the wave equation, we also can convert
these local plane waves from the wavenumber domain to the angle
domain.
Wavefield decomposition for full-wave propagators
When the local Fourier transform or beamlet decomposition is applied only along the horizontal coordinate, the local plane waves
uz共x,␪¯ ,␻ 兲 or uz共x,¯␰ m,␻ 兲 include not only the waves with positive
vertical wavenumbers 共propagating downward兲 but also corresponding negative vertical wavenumbers 共propagating upward兲. In
one-way propagators, the waves propagate along only one primary
direction; therefore decomposition along only the horizontal coordinate is appropriate. However, the full-wave propagators usually include both the downgoing and upgoing waves. The 1D decomposition techniques above will mix the downgoing and upgoing waves,
resulting in incorrect illumination amplitude and artificial interference patterns in the illumination map 共see, e.g., Figure 2; and Luo et
al., 2004兲. We decompose the full waves using direct 2D decomposition and an efficient split-step decomposition.
冕
w共x⬘ ⳮ x兲u共x⬘,␻ 兲
⫻ eⳮik共x兲·共共x⬘ⳮx兲sin ␪ Ⳮ共z⬘ⳮz兲cos ␪ 兲dx⬘,
¯
¯
共6兲
where w is a 2D spatial window 共Figure 3兲.
Split-step decomposition for full-wave propagators
The 2D decomposition discussed above can directly obtain the
wavefield for all directions; however, it costs much more than 1D decomposition. Here, we propose an efficient split-step decomposition
method to obtain the LAD wavefield for full-wave propagators using 1D decomposition techniques. First we decompose the full
wavefield along the vertical direction using the 1D technique to separate the downgoing and upgoing waves. Then we apply 1D decomposition along the horizontal direction to the downgoing and upgoing waves to obtain the wavefields in all directions. We need three
1D decompositions to decompose the full wavefields. This method
can be extended to the 3D case with 1D decomposition along the vertical direction and 2D decomposition along the horizontal coordinates.
For the first step, we need to separate the waves with positive and
negative vertical wavenumbers. The Gabor-Daubechies frame and
local exponential frame beamlet-decomposition methods are very
efficient in providing the local wavenumber-domain wavefield with
uniquely defined directional localization; hence, they can be used for
this step. We use Gabor-Daubechies frame beamlet decomposition
in this step because it can provide a more accurate directional wavefield than the local exponential frame beamlet decomposition 共Cao
and Wu, 2009兲. Appendix A summarizes the Gabor-Daubechies
frame beamlet decomposition and partial reconstruction to obtain
the local plane waves. For the second step, we can use either the LSS
method or the more efficient method with the Gabor-Daubechies
frame beamlet decomposition proposed by Cao and Wu 共2009兲.
EXAMPLES OF LOCAL-ANGLE-DOMAIN
FULL-WAVE ILLUMINATION ANALYSIS
2D decomposition for full-wave propagators
One direct way to obtain the LAD wavefield for full waves is by
using 2D local plane-wave decomposition for 2D problems. Similarly to 1D LSS, we can decompose the wavefield for a given frequency
into local plane waves using 2D LSS,
We demonstrate the advantages of full-wave equation-based illumination in the frequency domain, i.e., providing frequency-dependent full-angle true-amplitude illuminations for all arrivals. We
Depth (km)
a)
x
2
4
Surface location (km)
6
8
10
12
14
1
2
3
1
2
3
In
ci
de
nt
w
av
e
θ
0
b)
Depth (km)
1D window
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z
Figure 1. Basic geometry of 1D local angle-domain analysis for the
2D model.
0
Max
Figure 2. Acquisition dip-response maps for 15 Hz using the fullwave equation with 1D LSS applied along horizontal direction for
different dips: 共a兲 Ⳮ30°; 共b兲 ⳮ30°. Note the interference patterns
caused by the interaction between upgoing and downgoing waves.
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Cao and Wu
show illumination for downgoing waves with the 2D SEG/EAGE
salt model 共Aminzadeh et al., 1994; Aminzadeh et al., 1995兲, illumination for turning waves with a V共z兲 model and reflected waves with
a two-layer model, frequency-dependent illumination with a lens
model, and final application in the BP 2004 benchmark velocity
model 共Billette and Brandsberg-Dahl, 2005兲. We discuss the illumination of multiples in the following section. In examples below, we
consider only a frequency of 15 Hz for illumination calculation except where otherwise specified.
Downgoing-wave illumination in the 2D SEG/EAGE
salt model
The acquisition geometry of synthetic data for this model consists
of 325 shots with 176 left-side trailing receivers for each shot. The
shot and receiver intervals are 160 feet 共50 m兲 and 80 feet 共25 m兲,
respectively. The ADR results for dips Ⳮ30° and ⳮ30° from the
split-step method are very similar to those obtained by the much
more expensive 2D LSS method 共Figure 4兲. Interference patterns in
the illumination maps using 1D decomposition along the horizontal
direction 共Figure 2兲 do not appear in these results.
Turning-wave and reflected-wave illumination in
simple models
e
av
W
2
θ2
x
W
av
e
1
θ1
2D window
z
Figure 3. Basic geometry for 2D local plane-wave decomposition
for the 2D model.
0
1
12
14
Frequency-dependent illumination
3
To demonstrate frequency-dependent illumination, we use a lens
model consisting of a homogeneous elliptical lens 共2000 m / s兲 embedded in a homogeneous background 共3000 m / s兲. For 15-Hz
Split-step
1
2
a)
3
Depth (km)
Depth (km)
Surface location (km)
6
8
10
2
c)
Depth (km)
4
Split-step
b)
2D LSS
1
2
0
2
4
6
Surface location (km)
8
10
12
14
16
18
1
2
3
4
3
d)
Depth (km)
2
b)
2D LSS
Depth (km)
Depth (km)
a)
Turning waves and reflected waves can image overhung or vertical structures 共e.g., Jin et al. 2006; Xu and Jin, 2006; Zhang et al.,
2006; Jia and Wu, 2007兲, which downgoing waves in traditional oneway propagators cannot image. The examples here are for a single
shot. First we use a V共z兲 共 ⳱ 1.5Ⳮ 0.625z km/ s兲 model to demonstrate the illumination by turning waves. The directional illumination for incident angle 135° from the split-step method is similar to
that obtained by the 2D LSS method 共Figure 5兲. We notice that the illumination from the split-step method has a lower resolution; we discuss this result in the following section. The interference pattern is
caused by the interaction of turning waves and a weak upgoing reflected wave. This reflection is produced by the sharp velocity-gradient change at the model bottom, where we pad the V共z兲 model with a
constant velocity in the full-wave modeling.
In the next example, we use a two-layer model to show the illumination by reflected waves. It consists of a 5-km-thick homogeneous
layer with a velocity of 3.0 km/ s and a half-space with a velocity of
4.5 km/ s. The directional illumination for an incident angle of 135°
from the split-step method is very similar to that obtained by the 2D
LSS method 共Figure 6兲. The interference pattern in the lower-right
corner of the model is caused by the interaction of reflected waves
and head waves.
1
2
3
0
Max
Figure 4. Acquisition dip-response maps for 15 Hz using the fullwave equation from the split-step decomposition method 共a-b兲 and
the 2D LSS method 共c-d兲 for different dips: 共a, c兲 Ⳮ30°; 共b, d兲
ⳮ30°.
1
2
3
4
0
Max
Figure 5. Directional illumination at 15 Hz for an incident angle of
135° in a V共z兲 共 ⳱ 1.5Ⳮ 0.625z km/ s兲 model: 共a兲 from the split-step
method; 共b兲 from the 2D LSS method. In this and following plots, the
star represents the source location.
Full-wave directional illumination
DISCUSSION
Influence of multiples on the illumination strength
The LAD wavefield-decomposition methods for the full wavefield can separate the downgoing and upgoing waves. However, the
Depth (km)
b)
0
2
4
6
Surface location (km)
8
10
12
14
16
2
3
4
12
16
20
2
6
10
b)
2
6
10
c)
2
6
1
10
2
d)
3
2
4
Max
Figure 6. Directional illumination at 15 Hz for an incident angle of
135° in a two-layer model: 共a兲 from the split-step method; 共b兲 from
the 2D LSS method.
Depth (km)
1.5
2.0
2.5
d)
Surface location (km)
2
4
6
e)
1.5
2.0
2.5
f)
1.5
2.0
2.5
0
6
10
e)
Depth (km)
Surface location (km)
2
4
6
a)
c)
8
1
0
b)
a)
Depth (km)
Depth (km)
a)
Surface location (km)
4
Depth (km)
Finally we apply the full-wave-equation illumination to the complicated 2004 BP benchmark model. One challenge in this model is
to delineate the vertical and overhung salt flanks 共e.g., circled area in
Figure 8a兲. These targets can be imaged by the turning/reflected
waves. The ADR map from a single source-receiver acquisition on
the surface 共see Figure 8a兲 shows that this simple acquisition system
can illuminate the vertical salt flank well 共Figure 8d兲, although it
does not illuminate the overhung salt flank well 共Figure 8e兲. For
comparison, we also compute the ADRs in the BP model without salt
共Figure 9兲. The illuminations are more uniform in space and very different from those in the exact model. However, the wave can still illuminate the potential vertical structures very well 共Figure 9d兲.
Depth (km)
Illumination in the 2004 BP benchmark velocity model
downgoing waves include not only the primary incident waves but
also multiples; the primary incident waves include the first arrival
and multiarrivals. The ADR maps for the most energetic waves by
the split-step method show some illumination holes 共Figure 10兲.
Comparison with the ADRs for the full downgoing waves 共Figure 4a
and b兲 shows that the other arrivals 共i.e., multiarrivals and multiples兲
provide extra illumination to the subsurface. The migration methods
based on one-way propagators use not only the first arrival but also
Depth (km)
waves, the directional illumination from one shot right above the
lens 共Figure 7a and d兲 shows obvious illumination shadows and
wavefield focusing features below the lens because of the low-velocity anomaly. For lower frequencies 共5 and 2.5 Hz here兲, the illuminations below the lens are quite different from the illumination for
15 Hz: they seem to provide more even illumination, and the shadow
zones shrink with decreasing frequency 共Figure 7兲.
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2
6
10
Max
Figure 7. Frequency-dependent single-shot directional illumination
in a low-velocity lens 共black ellipse area in the figure兲 model. The
left and right columns are for 0° and 20° incidence angles, respectively. The rows from top to bottom are for frequencies of 15 Hz 共a,
d兲, 5 Hz 共b, e兲, and 2.5 Hz 共c, f兲, respectively.
0
Max
Figure 8. Acquisition dip response from a single source-receiver acquisition for 15-Hz waves in the 2004 BP benchmark model: 共a兲 part
of the exact model used; 共b-e兲 ADR for different dips: 共b兲 0°, 共c兲 40°,
共d兲 90°, 共e兲 130°. In this and following plots, the triangle represents
the receiver location.
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Cao and Wu
multiarrivals; therefore the multiarrivals should be included in the illumination analysis for survey design and true-reflection imaging
corrections. However, multiples, especially internal multiples,
should be eliminated because most migration methods based on the
one-way propagator do not use them. We cannot separate multiples
from other arrivals in the frequency domain. In the time domain, it is
also hard to do because both might have similar traveltimes in complex media. Therefore, it is difficult to evaluate the relative illumination strength from multiples and multiarrivals in general media.
Surface location (km)
Depth (km)
a)
4
8
12
16
20
2
6
10
Depth (km)
b)
2
The one-way and one-return boundary element method in the frequency domain 共He and Wu, 2007兲 can calculate the primary transmitted waves and multiples for a layered model or an inclusion model. This method can handle strong velocity contrasts. Here, we investigate the influence of internal multiples on illumination strength
with a simplified SEG/EAGE salt model, in which a homogeneous
salt body 共4480 m / s兲 with the same shape as that of the original 2D
SEG/EAGE salt model is embedded in a homogeneous background
medium 共2380 m / s兲. This is a scalar-wave model. For the acousticwave model, the salt internal multiples will be weaker because the
density of the salt usually is lower than that of the background sediment.
Directional illumination maps from the primary transmitted
waves and salt internal multiples in the subsalt region for incident
angles ⳮ10° and ⳮ60° show that the maximum amplitude of the illumination of primaries is more than six times stronger than that of
the multiples for these two angles 共Figure 11兲. Therefore the contribution of the multiples to illumination strength could be considered
as a secondary effect here. However, we can also notice the difference in the spatial distribution of the illumination between results for
multiples and those for primaries. For example, for the ⳮ60° incident angle, the primaries strongly illuminate only the left part of the
subsalt area; however, the multiples illuminate the whole subsalt
area more evenly and provide extra illumination to the shadow in the
illumination by primaries 共right part in subsalt area兲.
6
Comparison of the 1D and 2D local slant-stack methods
c)
2
Depth (km)
10
6
Previous results show that the split-step method based on 1D decompositions might produce a lower resolution result than the 2D
LSS method 共Figure 5兲. Here we show a theoretical analysis and numerical investigation. For 1D decomposition, from the dispersion
relation we have
兩d¯␰ 兩 ⳱ 兩d␪¯ 兩k0 cos ␪¯ ,
10
6
10
a)
2
Depth (km)
Depth (km)
e)
where k0 is the wavenumber. We can further obtain the angular resolution as a function of wavelength ␭, window length Lwin, and the angle ␪¯ ,
2
6
0
2
4
Surface location (km)
6
8
10
11
12
1
2
3
b)
Depth (km)
Depth (km)
d)
共7兲
10
0
1
2
3
Max
0
Figure 9. Acquisition dip response from a single source-receiver acquisition for 15-Hz waves in the 2004 BP benchmark model without
salts: 共a兲 part of the model used; 共b-e兲 ADR for different dips: 共b兲 0°,
共c兲 40°, 共d兲 90°, 共e兲 130°.
Max
Figure 10. Acquisition dip-response maps for the most energetic
waves at 15 Hz with the split-step method for different dips: 共a兲
Ⳮ30°; 共b兲 ⳮ30°.
Full-wave directional illumination
兩d␪¯ 兩 ⳱
1
␭
.
Lwin cos ␪¯
The numerical results show that for waves propagating within about
Ⳳ30° from the decomposition direction, the angular resolution of
the 1D LSS method is similar to that of the 2D LSS method. The direction of the local plane wave having maximum energy from the 1D
LSS decomposition still is consistent with the true incident direction
of the global plane wave from any direction.
共8兲
Therefore, the angular resolution is angle and window-length dependent for a given ␭. For the 2D LSS method,
兩d␪¯ 兩 ⳱
␭
,
Lwin
共9兲
CONCLUSIONS
which is angle independent. This can explain previous differences of
angular resolution in illumination results.
Next we compare the methods numerically. We investigate the decomposed local plane waves for incident global plane waves along
different directions, using the 2D LSS method and the 1D LSS method along the horizontal direction. Results show that the 2D LSS
method gives the same angular resolution for the plane-wave incident along any direction 共Figure 12a兲. The 1D LSS method yields
angular resolution decreasing with the increase of the propagating
angle of the global plane wave 共Figure 12b兲, and it gives the best resolution 共red line兲 when the decomposition direction is parallel to the
wavefront.
Depth (km)
a)
Depth (km)
c)
6
Surface location (km)
8
10
12
b)
We have analyzed the local-angle domain 共LAD兲 illumination in
the frequency domain with full-wave propagators using two wavefield decomposition methods: one is the direct 2D/3D local planewave decomposition; the other is the more efficient split-step decomposition. The methods can provide frequency-dependent fullangle true-amplitude illumination analysis for all arrivals. They are
more efficient and storage saving compared with the time-domain
angle analysis method. They can be used for accurate survey design
and true-reflection imaging corrections. Results of illumination
analysis from both decomposition methods are very similar, although the split-step method might produce a lower resolution result
at wide angles. We conclude that the proposed methods provide efficient and accurate tools for the LAD wave-theory-based illumination analysis in complex models.
Surface location (km)
8
10
12
6
S91
ACKNOWLEDGMENTS
1.8
The authors thank the associate editor and reviewers for their
valuable comments that greatly improved this manuscript. We acknowledge Xiao-Bi Xie for helpful discussion on LSS, and Yaofeng
He for generating the primary and multiple data with the one-way
and one-return boundary element method. This research is sponsored by the Wavelet Transform on Propagation and Imaging 共for
seismic exploration兲 Research Consortium 共WTOPI兲 at the University of California, Santa Cruz. We also thank BP and Frederic Billette
for providing the 2004 BP 2D benchmark velocity model.
2.6
3.4
0.2
0.4
0.6
d)
1.8
0.02
0.04
0.06
0.02
0.04
0.06
2.6
3.4
0
0.1
0.2
0.3
0.4
0
Figure 11. Directional illumination for the primary transmitted
waves and salt internal multiples at 15 Hz in a simplified SEG/
EAGE salt model: 共a兲 transmitted waves for ⳮ10°; 共b兲 multiples for
ⳮ10°; 共c兲 transmitted waves for ⳮ60°; 共d兲 multiples for ⳮ60°.
APPENDIX A
GABOR-DAUBECHIES FRAME
BEAMLET DECOMPOSITION
a)
180
210
b)
1
180
210
150
150
0.6
0.6
Wavefield amplitude
1
0.8
0.8
240
120
240
120
0.4
0.4
0.2
0.2
90 270
270
90
The Gabor-Daubechies frame 共GDF兲 beamlets 共e.g., Wu et al., 2000; Wu and Chen, 2001;
Chen et al., 2006兲 have uniquely defined and
good localization information available after decomposition. The GDF beamlets for the beamlet
decomposition equation 4 are Gaussian function
windowed exponential harmonics,
¯
300
60
330
30
0
Decomposed angle ( ° )
300
60
330
30
0
Decomposed angle ( ° )
Figure 12. Decomposed local plane waves for incident global plane waves from different
directions 共dashed lines show the true incident directions兲 using the 共a兲 2D LSS and
共b兲1D LSS methods.
bmn共x兲 ⳱ g共x ⳮ x̄n兲ei␰ mx,
共A-1兲
where ¯␰ m ⳱ m⌬␰ 共⌬␰ is the wavenumber sampling interval兲, and g共x兲 is a Gaussian window
function,
冉 冊
g共x兲 ⳱ 共␲ s2兲ⳮ1/4 exp ⳮ
x2
,
2s2
共A-2兲
S92
Cao and Wu
s2 ⳱
R · ⌬N2
,
2␲
共A-3兲
where s is the scale of the Gaussian window, R is the redundancy ratio, and ⌬N is the lateral sampling interval of the frame.
Substituting the GDF beamlet representation A-1 into the beamlet decomposition equation 4, we can obtain the partially reconstructed local plane waves,
¯
uz共x,¯␰ m,␻ 兲 ⳱ ei␰ mx 兺 g共x ⳮ x̄n兲ûz共x̄n,¯␰ m,␻ 兲,
共A-4兲
n
with the decomposition coefficients
ûz共x̄n,¯␰ m,␻ 兲 ⳱ 具uz共x,␻ 兲,b̃mn共x兲典 ⳱
冕
* 共x兲dx,
uz共x,␻ 兲 · b̃mn
共A-5兲
where 具·典 stands for inner product, * stands for complex conjugate,
and b̃mn共x兲 are the dual GD frame atoms
¯
b̃mn共x兲 ⳱ g̃共x ⳮ x̄n兲ei␰ mx,
共A-6兲
with g̃共x兲 being the dual-window function of g共x兲. The dual-window
function can be calculated by pseudoinversion of the original window function 共Qian and Chen, 1996; Mallat, 1998; Wu and Chen,
2001兲. From equation A-4, we can see that the local plane wave is a
weighted average of the windowed beamlets with the same wavenumber from neighboring windows. The total space-domain wavefield can be written as
uz共x,␻ 兲 ⳱ 兺 uz共x,¯␰ m,␻ 兲.
共A-7兲
m
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