The document analyzes the reference velocity sensitivity of an elastic internal multiple attenuation algorithm. It first provides background on internal multiples and inverse scattering series methods for their attenuation. It then presents a 1.5D layered earth model and uses it to show that the elastic internal multiple algorithm can correctly predict arrival times without requiring an accurate reference velocity model, similarly to the acoustic case. The analysis involves deriving expressions for predicted internal multiples in the frequency-wavenumber domain and showing they depend only on the traveltimes of the primary reflections involved, not the reference velocities.
The Inverse Scattering Series (ISS) is a direct inversion method
for a multidimensional acoustic, elastic and anelastic earth. It
communicates that all inversion processing goals are able to
be achieved directly and without any subsurface information.
This task is reached through a task-specific subseries of the
ISS. Using primaries in the data as subevents of the first-order
internal multiples, the leading-order attenuator can predict the
time of all the first-order internal multiples and is able to attenuate
them.
However, the ISS internal multiple attenuation algorithm can
be a computationally demanding method specially in a complex
earth. By using an approach that is based on two angular
quantities and that was proposed in Terenghi et al. (2012), the
cost of the algorithm can be controlled. The idea is to use the
two angles as key-control parameters, by limiting their variation,
to disregard some calculated contributions of the algorithm
that are negligible. Moreover, the range of integration
can be chosen as a compromise of the required degree of accuracy
and the computational time saving.
This time-saving approach is presented
Accuracy of the internal multiple prediction when a time-saving method based ...Arthur Weglein
The inverse scattering series (ISS) is a direct inversion method for a multidimensional acoustic,
elastic and anelastic earth. It communicates that all inversion processing goals can be
achieved directly and without any subsurface information. This task is reached through a taskspecific
subseries of the ISS. Using primaries in the data as subevents of the first-order internal
multiples, the leading-order attenuator can predict the time of all the first-order internal multiples
and is able to attenuate them.
The Inverse Scattering Series (ISS) is a direct inversion method
for a multidimensional acoustic, elastic and anelastic earth. It
communicates that all inversion processing goals are able to
be achieved directly and without any subsurface information.
This task is reached through a task-specific subseries of the
ISS. Using primaries in the data as subevents of the first-order
internal multiples, the leading-order attenuator can predict the
time of all the first-order internal multiples and is able to attenuate
them.
However, the ISS internal multiple attenuation algorithm can
be a computationally demanding method specially in a complex
earth. By using an approach that is based on two angular
quantities and that was proposed in Terenghi et al. (2012), the
cost of the algorithm can be controlled. The idea is to use the
two angles as key-control parameters, by limiting their variation,
to disregard some calculated contributions of the algorithm
that are negligible. Moreover, the range of integration
can be chosen as a compromise of the required degree of accuracy
and the computational time saving.
This time-saving approach is presented
Accuracy of the internal multiple prediction when a time-saving method based ...Arthur Weglein
The inverse scattering series (ISS) is a direct inversion method for a multidimensional acoustic,
elastic and anelastic earth. It communicates that all inversion processing goals can be
achieved directly and without any subsurface information. This task is reached through a taskspecific
subseries of the ISS. Using primaries in the data as subevents of the first-order internal
multiples, the leading-order attenuator can predict the time of all the first-order internal multiples
and is able to attenuate them.
Multiscale methods for next generation graphene based nanocomposites is proposed. This approach combines atomistic finite element method and classical continuum finite element method.
Eh4 energy harvesting due to random excitations and optimal designUniversity of Glasgow
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In this report, we summarize the given scenario behind the “Quantum Interrogation" problem, and offer historical background of the development of quantum “interaction-free measurement" theory. We then describe some real-world applications of this procedure, whether already verified experimentally or as potential applications in the future. Finally, we exhibit our calculations that display the feasibility of this quantum effect, and summarize the key takeaways from this problem.
Multiscale methods for next generation graphene based nanocomposites is proposed. This approach combines atomistic finite element method and classical continuum finite element method.
Eh4 energy harvesting due to random excitations and optimal designUniversity of Glasgow
This lecture is about vibration energy harvesting when both the excitation and the system have uncertainties. Two cases, namely, when the excitation is a random process and when the system parameters are described by random variables are described. Optimal design for both cases is discussed.
Slides of my talk at IISc Bangalore on nanomechanics and finite element analysis for statics and dynamics of nanoscale structures such as carbon nanotube, graphene, ZnO nanotube and BN nano sheet.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
Quantum Interrogation: Interaction-Free Determination of Existence (Physics 1...BenjaminKan4
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Reference velocity sensitivity for the marine internal multiple attenuation a...Arthur Weglein
In this note, we present a review of the inverse scattering series internal multiple attenuation
algorithm for land data (Matson, 1997). In particular, we look at internal multiple attenuation
since it is a dominant issue when processing land data. The requirements of four-component
data and reference velocity for a 2D earth is shown. Effects due to velocity errors includes
artifacts in the (P, S) data (Matson, 1997) and the algorithm’s sensitivity to reference velocity.
Two analytic examples for 1D normal incidence (Weglein and Matson, 1998) and 1D non-normal
incidence (Nita and Weglein, 2005) are used to demonstrate the inner working of this algorithm.
We extend their analysis to investigate velocity sensitivity. In conclusion, accurate near surface
reference velocities are essential to properly predict internal multiples except for the 1-D normal
incidence case.
Obtaining three-dimensional velocity information directly from reflection sei...Arthur Weglein
This paper present a formalism for obtaining the subsurface
velocity configuration directly from reflection seismic data.
Our approach is to apply the results obtained for inverse
problems in quantum scattering theory to the reflection
seismic problem. In particular, we extend the results of
Moses (1956) for inverse quantum scattering and Razavy
(1975) for the one-dimensional (1-D) identification of the
acoustic wave equation to the problem of identifying the
velocity in the three-dimensional (3-D) acoustic wave equation
from boundary value measurements. No a priori knowledge
of the subsurface velocity is assumed and all refraction,
diffraction, and multiple reflection phenomena are
taken into account. In addition, we explain how the idea of
slant stack in processing seismic data is an important part
of the proposed 3-D inverse scattering formalism.
ABSTRACT: In this paper, we proposed a new identification algorithm based on Kolmogorov–Zurbenko Periodogram (KZP) to separate motions in spatial motion image data. The concept of directional periodogram is utilized to sample the wave field and collect information of motion scales and directions. KZ Periodogram enables us detecting precise dominate frequency information of spatial waves covered by highly background noises. The computation of directional periodogram filters out most of the noise effects, and the procedure is robust for missing and fraud spikes caused by noise and measurement errors. This design is critical for the closure-based clustering method to find cluster structures of potential parameter solutions in the parameter space. An example based on simulation data is given to demonstrate the four steps in the procedure of this method. Related functions are implemented in our recent published R package {kzfs}.
Blind separation of complex-valued satellite-AIS data for marine surveillance...IJECEIAES
In this paper, the problem of the blind separation of complex-valued Satellite-AIS data for marine surveillance is addressed. Due to the specific properties of the sources under consideration: they are cyclo-stationary signals with two close cyclic frequencies, we opt for spatial quadratic time-frequency domain methods. The use of an additional diversity, the time delay, is aimed at making it possible to undo the mixing of signals at the multi-sensor receiver. The suggested method involves three main stages. First, the spatial generalized mean Ambiguity function of the observations across the array is constructed. Second, in the Ambiguity plane, Delay-Doppler regions of high magnitude are determined and Delay-Doppler points of peaky values are selected. Third, the mixing matrix is estimated from these Delay-Doppler regions using our proposed non-unitary joint zero-(block) diagonalization algorithms as to perform separation.
On The Fundamental Aspects of DemodulationCSCJournals
When the instantaneous amplitude, phase and frequency of a carrier wave are modulated with the information signal for transmission, it is known that the receiver works on the basis of the received signal and a knowledge of the carrier frequency. The question is: If the receiver does not have the a priori information about the carrier frequency, is it possible to carry out the demodulation process? This tutorial lecture answers this question by looking into the very fundamental process by which the modulated wave is generated. It critically looks into the energy separation algorithm for signal analysis and suggests modification for distortionless demodulation of an FM signal, and recovery of sub-carrier signals
This paper studies an approximate dynamic programming (ADP) strategy of a group of nonlinear switched systems, where the external disturbances are considered. The neural network (NN) technique is regarded to estimate the unknown part of actor as well as critic to deal with the corresponding nominal system. The training technique is simul-taneously carried out based on the solution of minimizing the square error Hamilton function. The closed system’s tracking error is analyzed to converge to an attraction region of origin point with the uniformly ultimately bounded (UUB) description. The simulation results are implemented to determine the effectiveness of the ADP based controller.
IT IS ABOUT FUSION OF TWO NATURE INSPIRED OPTIMIZATION ALGORITHM(S).THE FIRST ONE IS GRAVITATIONAL SEARCH ALGORITHM(GSA) BASED ON NEWTONS UNIVERSAL LAW OF GRAVITATION AND OTHER ONE i.e; BIOGEOGRAPHY BASED OPTIMIZATION(BBO) BASED ON BIOGEOGRAPGY (THE STUDY OF SPECIES IN A PARTICULAR HABITAT).
Wavelet estimation for a multidimensional acoustic or elastic earthArthur Weglein
A new and general wave theoretical wavelet estimation
method is derived. Knowing the seismic wavelet
is important both for processing seismic data and for
modeling the seismic response. To obtain the wavelet,
both statistical (e.g., Wiener-Levinson) and deterministic
(matching surface seismic to well-log data) methods
are generally used. In the marine case, a far-field
signature is often obtained with a deep-towed hydrophone.
The statistical methods do not allow obtaining
the phase of the wavelet, whereas the deterministic
method obviously requires data from a well. The
deep-towed hydrophone requires that the water be
deep enough for the hydrophone to be in the far field
and in addition that the reflections from the water
bottom and structure do not corrupt the measured
wavelet. None of the methods address the source
array pattern, which is important for amplitude-versus-
offset (AVO) studies.
The inverse scattering series for tasks associated with primaries: direct non...Arthur Weglein
The inverse scattering series for tasks associated with primaries: direct non-linear inversion of 1D elastic media. In this paper, research on direct inversion for two pa-
rameter acoustic media (Zhang and Weglein, 2005) is
extended to the three parameter elastic case. We present
the first set of direct non-linear inversion equations for
1D elastic media (i.e., depth varying P-velocity, shear
velocity and density). The terms for moving mislocated
reflectors are shown to be separable from amplitude
correction terms. Although in principle this direct
inversion approach requires all four components of elastic
data, synthetic tests indicate that consistent value-added
results may be achieved given only ˆDPP measurements.
We can reasonably infer that further value would derive
from actually measuring ˆDPP , ˆD PS, ˆDSP and ˆDSS as
the method requires. The method is direct with neither
a model matching nor cost function minimization.
Internal multiple attenuation using inverse scattering: Results from prestack 1 & 2D acoustic and
elastic synthetics
R. T. Coates*, Schlumberger Cambridge Research, A. B. Weglein, Arco Exploration and Production Technology
Summary
The attenuation of internal multiples in a multidimensional
earth is an important and longstanding problem in exploration
seismics. In this paper we report the results of applying
an attenuation algorithm based on the inverse scattering
series to synthetic prestack data sets generated in on
and two dimensional earth models. The attenuation algorithm
requires no information about the subsurface structure
or the velocity field. However, detailed information about
the source wavelet is a prerequisite. An attractive feature of:
the attenuation algorithm is the preservation of the amplitude
(and phase) of primary events in the data; thus allowing for
subsequent AVO and other true amplitude processing.
Inverse scattering series for multiple attenuation: An example with surface a...Arthur Weglein
A multiple attenuation method derived from an inverse scattering
series is described. The inversion series approach allows a
separation of multiple attenuation subseries from the full series.
The surface multiple attenuation subseries was described and illustrated
in Carvalho et al. (1991, 1992). The internal multiple
attenuation method consists of selecting the parts of the odd
terms that are associated with removing only multiply reflected
energy. The method, for both types of multiples, is multidimensional
and does not rely on periodicity or differential moveout,
nor does it require a model of the reflectors generating the multiples.
An example with internal and surface multiples will be
presented.
In this paper we present a multidimensional method for attenuating internal multiples that derives from
an inverse scattering series . The method doesn't depend on periodicity or differential moveout, nor does it
require a model for the multiple generating reflectors.
Summary
Methods for removal of free-surface and internal multiples have been developed from bath a feedback model approach and inverse scatterin g theory. White these two formulations derive from different mathematica) viewpoints,
the resulting algorithm s for free-surface multiple are very similar. By contrast , the feedback and inverse scattering
method for internal multiple are totally different and have different requirements for sub surface information or
interpretive intervention . The former removes all multiple related to a certain boundary with the a of a surface
integral along this boundary ; the alter wilt predict and attenuate a ll internal multiple a t the same time . In this paper, we continue our comparison study of these internal multiple attenuation method ; specifically , we examine two
different realizations of the feedback method and the inverse scattering technique .
Internal multiple attenuation using inverse scattering: Results from prestack...Arthur Weglein
The attenuation of internal multiples in a multidimensional
earth is an important and longstanding problem in exploration
seismics. In this paper we report the results of applying
an attenuation algorithm based on the inverse scattering
series to synthetic prestack data sets generated in on
and two dimensional earth models. The attenuation algorithm
requires no information about the subsurface structure
or the velocity field. However, detailed information about
the source wavelet is a prerequisite. An attractive feature of:
the attenuation algorithm is the preservation of the amplitude
(and phase) of primary events in the data; thus allowing for
subsequent AVO and other true amplitude processing.
Wavelet estimation for a multidimensional acoustic or elastic earth- Arthur W...Arthur Weglein
A new and general wave theoretical wavelet estimation
method is derived. Knowing the seismic wavelet
is important both for processing seismic data and for
modeling the seismic response. To obtain the wavelet,
both statistical (e.g., Wiener-Levinson) and deterministic
(matching surface seismic to well-log data) methods
are generally used. In the marine case, a far-field
signature is often obtained with a deep-towed hydrophone.
The statistical methods do not allow obtaining
the phase of the wavelet, whereas the deterministic
method obviously requires data from a well. The
deep-towed hydrophone requires that the water be
deep enough for the hydrophone to be in the far field
and in addition that the reflections from the water
bottom and structure do not corrupt the measured
wavelet. None of the methods address the source
array pattern, which is important for amplitude-versus-
offset (AVO) studies
Inverse scattering series for multiple attenuation: An example with surface a...Arthur Weglein
A multiple attenuation method derived from an inverse scattering
series is described. The inversion series approach allows a
separation of multiple attenuation subseries from the full series.
The surface multiple attenuation subseries was described and illustrated
in Carvalho et al. (1991, 1992). The internal multiple
attenuation method consists of selecting the parts of the odd
terms that are associated with removing only multiply reflected
energy. The method, for both types of multiples, is multidimensional
and does not rely on periodicity or differential moveout,
nor does it require a model of the reflectors generating the multiples.
An example with internal and surface multiples will be
presented.
Deghosting is a longstanding seismic objective and problem that has received considerable renewed attention due to : (1). an interest in so-called "broadband seismology" and the low frequency /low vertical wave number.
1. An analytic example examining the reference velocity sensitivity
of the elastic internal multiple attenuation algorithm
S.-Y. Hsu, S. Jiang and A. B. Weglein, M-OSRP, University of Houston
Abstract
Internal multiple attenuation is a pre-processing step for seismic imaging and amplitude analysis.
Hsu and Weglein (2007) showed that the internal multiple algorithm from the inverse scattering
series is independent of reference velocity for an 1D earth with acoustic background. In this
study, we followed the analysis of Nita and Weglein (2005) and used the elastic internal multiple
algorithm (Matson, 1997) to investigate velocity sensitivity for an 1D earth with elastic reference
medium. The result suggests that the algorithm is also independent of reference velocity for an
1D earth with elastic background.
1 Introduction
Seismic processing is used to estimate subsurface properties from the reflected wave-fields. The
seismic data are a set of reflected waves including primaries and multiples. Primaries are events
that have only one upward reflection before arriving at the receiver. Multiply reflected events
(multiples) are classified as free-surface or internal multiples depending on the location of their
downward reflections. Free-surface multiples have at least one downward reflection at the air-
water or air-land surface (free surface). Internal multiples have all downward reflections below the
measurement surface (Weglein et al., 1997; Weglein and Matson, 1998).
Methods for seismic imaging and amplitude analysis usually assume that the reflection data are
primaries-only. To accommodate this assumption, multiple removal/attenuation is a prerequisite
for seismic processing. Conventional methods successfully attenuate multiples by assuming simple
or known subsurface geology. However, in geologically complex areas those methods may become
inadequate (Otnes et al., 2004). To overcome the limitations of conventional methods, the inverse
scattering series (ISS) methods were proposed to perform multiple removal/suppression for acoustic
data without subsurface information or assumptions (Carvalho, 1992; Ara´ujo, 1994; Weglein et al.,
1997). Following this framework, Matson (1997) extended the multiple attenuation algorithm from
acoustic to elastic.
In order to keep the perturbation below the measurement surface, the ISS multiple removal/attenuation
methods usually require a known source wavelet and information about the near surface (Matson,
1997). To obtain the source wavelet, Wang and Weglein (2008) has shown how Green’s theorem is
used to fulfill the requirement. The need of the near surface properties is a practical obstacle for on-
shore/ocean bottom application. Here, we present an analytic example to show the inner workings
of the elastic internal multiple algorithm. In particular, we demonstrate how this scattering-based
algorithm can predict exact arrival times without the above assumption being true.
32
2. Internal multiple reference velocity sensitivity MOSRP08
2 Theory
We start with the forward scattering series derived from the Lippmann-Schwinger equation
G = G0 + G0V G, (1)
which can be expanded in a forward series
G = G0 + G0V G0 + G0V G0V G0 + · · · , (2)
where G and G0 are the actual and reference Green’s functions. The perturbation V can be written
as V = n Vn where Vn is n-th order in the data.
Define D = G − G0 as the measurement of the scattered field. Substituting into the forward series
gives the inverse scattering series in terms of data
D = G0V1G0,
0 = G0V2G0 + G0V1G0V1G0,
0 = G0V3G0 + G0V2G0V1G0 + G0V1G0V2G0 + G0V1G0V1G0V1G0,
....
(3)
The term G0V1G0V1G0V1G0 is the first term in the inverse series of first order internal multiples.
Following the method developed by Weglein et al. (1997) and Ara´ujo (1994), the equation for first
order internal multiples in 2-D acoustic data is
b3IM (kg, ks, qg + qs) =
1
(2π)2
∞
−∞
dk1e−iq1(zg−zs)
∞
−∞
dk2eiq2(zg−zs)
×
∞
−∞
dz1b1(kg, k1, z1)ei(qg+q1)z1
×
z1−
−∞
dz2b1(k1, k2, z2)e−i(q1+q2)z2
×
∞
z2+
dz3b1(k2, ks, z3)ei(q2+qs)z3
(4)
The parameter > 0 ensures z1 > z2 and z3 > z2 which satisfy the geometric relationship between
reflections of internal multiples (lower-higher-lower). b1 is defined in terms of the original pre-stack
data without free surface multiples. The data can be written as
D(kg, ks, ω) = (−2iqs)−1
b1(kg, ks, qg + qs), (5)
where b1(kg, ks, qg + qs) represents the data result from a single-frequency incident plane wave.
33
3. Internal multiple reference velocity sensitivity MOSRP08
The adapted elastic version of (4) given by Matson (1997) is
B3IM
ij (kg, ks, qi
g + qj
s) =
1
(2π)2
∞
−∞
dk3e−iql
1(zg−zs)
∞
−∞
dk2eiqm
2 (zg−zs)
×
∞
−∞
dz1Bil(kg, k1, z1)ei(qi
g+ql
1)z1
×
z1−
−∞
dz2Blm(k1, k2, z2)e−i(ql
1+qm
2 )z2
×
∞
z2+
dz3Bmj(k2, ks, z3)ei(qm
2 +qj
s)z3
(6)
where qi
1 indicate P and S vertical wavenumbers for i = P and S, respectively.
Similarly, Bij is defined in terms of the original pre-stack data, that is
Dij(k1, k2, ω) = (−2iqj
2)−1
Bij(kg, ks, qi
1 + qj
2). (7)
Hence,
DPP (k1, k2, ω) =(−2iqP
2 )−1
BPP (kg, ks, qP
1 + qP
2 ),
DPS(k1, k2, ω) =(−2iqS
2 )−1
BPS(kg, ks, qP
1 + qS
2 ),
DSP (k1, k2, ω) =(−2iqP
2 )−1
BSP (kg, ks, qS
1 + qP
2 ),
DSS(k1, k2, ω) =(−2iqS
2 )−1
BSS(kg, ks, qS
1 + qS
2 ).
(8)
One can see that if the converted waves do not exist, equation (6) becomes equation (4), which is
the first order internal multiple attenuator in acoustic form.
3 Attenuation of elastic internal multiples : a 1.5D example
We consider a layering model given by Nita and Weglein (2005) (see Figure 1). The velocity changes
across the interfaces located at z = za and z = zb where the velocities are c0, c1, and c2, respectively.
The sources and receivers are located at the measurement surface where the depth z = 0. The
reference vertical speeds are defined as
ci
v =
α0/cosθi, if i = P
β0/cosθi, if i = S
(9)
The reference horizontal speed is defined as
ci
h =
α0/sinθi, if i = P
β0/sinθi, if i = S
(10)
where α0 and β0 are P-wave and S-wave velocity in the reference medium, respectively.
34
4. Internal multiple reference velocity sensitivity MOSRP08
Figure 1: The model for the 1.5D example
Figure 2: The geometry of the first primary in the data
The horizontal wave numbers are
ki
s = ω/ci
h,
kj
g = ω/cj
h.
(11)
For media with depth-dependent velocity, the horizontal slowness (sinθi/c(z) ≡ p) is constant along
35
5. Internal multiple reference velocity sensitivity MOSRP08
a ray path whether or not converted waves are present (Aki and Richards, 2002). Therefore,
ki
s = kj
g = ω/ch,
ki
s + kj
g = 2ω/ch.
(12)
The vertical wave numbers are
qi
s =
( ω
α0
)2 − k2
s, if i = P
( ω
β0
)2 − k2
s, ifi = S
(13)
qj
g =
( ω
α0
)2 − k2
g, if j = P
( ω
β0
)2 − k2
g, ifj = S
(14)
The total travel time for the n-th primary can be written as
Tn = τn + th, (15)
where τn and th are the vertical and horizontal travel times, respectively. The horizontal travel
time is defined as
th =
xg − xs
ch
=
2xh
ch
, (16)
and the vertical travel time corresponding to the n-th event is
τn =
zn
ci
v
+
zn
cj
v
, (17)
where zn is the pseudo depths for the n-th event. Therefore,
ωτn = zn(
ω
ci
v
+
ω
cj
v
)
= zn(qi
s + qj
g) = znkij
z ,
ωth = ω(
2xh
ch
) = khxh.
(18)
The data in the frequency ω domain can be written as
Dij(xh, 0; ω) =
1
2π
∞
−∞
dkh
n
Rij
n eikij
z zn
2iqi
s
eikhxh
. (19)
where kij
z = qi
s + qj
g, kh = kg + ks ,and xh = (xg − xs)/2.
Fourier transform over xg, xs,
Dij(kh, 0, ω) =
2π
2iqi
s n
Rij
n eiznkij
z
δ(kg − ks). (20)
36
6. Internal multiple reference velocity sensitivity MOSRP08
The data is
Bij(kh, 0, ω) = 2iqi
sDij(kh, 0, ω)
= 2π
n
Rij
n eikij
z zn
δ(kg − ks).
(21)
Inverse Fourier transforming into the pseudo depth domain gives
Bij(kh, z, ω) = 2π
n
Rij
n δ(z − zn)δ(kg − ks). (22)
Substituting equation (22) into equation (6), the integration over z3 gives
∞
z2+
dz3eik mj
z z3 Bmj(k2, ks, z3) =2π
∞
z2+
dz3eik mj
z z3
p
Rmj
p δ(z3 − zp)δ(k2 − ks)
=2π
p
Rmj
p eik mj
z zp H(zp − (z2 + ))δ(k2 − ks)
The integration over z2 is
z1−
−∞
dz2e−ik lm
z z2 Blm(k1, k2, z2) × 2π
p
Rmj
p eik mj
z zp H(zp − (z2 + ))δ(k2 − ks)
= (2π)2
z1−
−∞
dz2e−ik lm
z z2
q
Rlm
q δ(z2 − zq)δ(k1 − k2)
p
Rmj
p eik mj
z zp H(zp − (z2 + ))δ(k2 − ks)
= (2π)2
p,q
Rmj
p Rlm
q eik mj
z zp e−ik lm
z zq H(zp − (zq + ))H((z1 − ) − zq)δ(k2 − ks)δ(k1 − k2)
The integration over z1 is
∞
−∞
dz1eik jl
z z1 Bjl(kg, k1, z1)
×[(2π)2
p,q
Rmj
p Rlm
q eik mj
z zp e−ik lm
z zq H(zp − (zq + ))H((z1 − ) − zq)δ(k2 − ks)δ(k1 − k2)]
=(2π)3
∞
−∞
dz1eik jl
z z1
r
Rjl
r δ(z1 − zr)δ(kg − k1)
p,q
Rmj
p Rlm
q eik mj
z zp e−ik lm
z zq
×H(zp − (zq + ))H((z1 − ) − zq)δ(k2 − ks)δ(k1 − k2)
=(2π)3
p,q,r
Rmj
p Rlm
q Rjl
r ei(k mj
z zp−k lm
z zq+k jl
z zr)
H(zp − (zq + ))H(zr − (zq + ))
×δ(k2 − ks)δ(k1 − k2)δ(kg − k1)
(23)
Note that the parameter > 0 ensures zp > zq and zr > zq which satisfy a lower-higher-lower
relationship between the pseudo depths.
37
7. Internal multiple reference velocity sensitivity MOSRP08
The result for B3IM
ij is
B3IM
ij (kh, z, ω) = 2π
p,q,r
zp>zq
zr>zq
Rmj
p Rlm
q Rjl
r ei(k mj
z zp−k lm
z zq+k jl
z zr)
δ(kg − ks) (24)
Inverse Fourier transforming over km and kh gives
B3IM
ij (kh, kz, ω) =
1
2π
dkh
p,q,r
Rmj
p Rlm
q Rjl
r ei(k mj
z zp−k lm
z zq+k jl
z zr)
eikhxh
. (25)
The predicted first internal multiple in the space domain is
IM1st
predicted =
1
2π
dkh
p,q,r
Rmj
p Rlm
q Rjl
r ei(k mj
z zp−k lm
z zq+k jl
z zr)
eikhxh
2iqi
s
. (26)
Note that
M.S.
Figure 3: The geometry of pseudo depths
k mj
z zp = ωτp,
k lm
z zq = ωτq,
k jl
z zr = ωτr,
k hxh = ωth.
(27)
Therefore,
k
mj
z zp − k
lm
z zq + k
jl
z zr + khxh = ω(τp − τq + τr + th) (28)
38
8. Internal multiple reference velocity sensitivity MOSRP08
The travel time for the predicted first order internal multiple is
τp − τq + τr + th = (Tp − th) − (Tq − th) + (Tr − th) + th
= Tp − Tq + Tr
(29)
where Tp, Tq and Tr are travel times corresponding to the p-th, q-th, and r-th event, respectively.
The travel time for the actual first order internal multiple is
Ttotal = τp − τq + τr + th = (Tp − th) − (Tq − th) + (Tr − th) + th
= Tp − Tq + Tr.
(30)
The result Ttotal = Tp − Tq + Tr agrees with the actual arrival time for the first order internal
multiple.
One can see that reference velocity errors give incorrect vertical and horizontal travel times for each
event (see equation (16) and (17)). However, equation (29) shows that the errors due to wrong
reference velocities can be canceled, and the predicted arrival times remain correct.
To attenuate an internal multiple for a single frequency requires data at all frequencies. This
requirement comes from the integral over temporal frequency when transforming data from the
frequency domain to the pseudo-depth domain. The integral transformation is truncated with band-
limited data. The 1.5D example suggests that the algorithm is insensitive to reference velocities
and hence has potential for spectral extrapolation if we choose reasonable reference velocities.
4 Conclusion
In this paper, we have presented an elastic internal multiple attenuation algorithm (Matson, 1997)
following the analysis of Nita and Weglein (2005). Although the scattering-based methods usually
require known near surface properties, our study has suggested that the internal multiple attenu-
ation algorithm can tolerate errors in reference velocity for 1D earth. The capability of predicting
exact travel time without known reference velocities is useful for land application, where the near
surface properties are often complicated and ill-defined. Moreover, the freedom of choosing ref-
erence velocities has potential for the spectral extrapolation. The future direction of this study
includes extending the algorithm to multi-dimensional elastic case and extrapolation of band-limited
synthetic data.
Acknowledgments
We would like to thank all M-OSRP sponsors for constant support and encouragement. We espe-
cially thank all the members in M-OSRP for valuable suggestions.
References
Aki, K. and P. G. Richards. Quantitative Seismology. 2nd edition. University Science Books, 2002.
39
9. Internal multiple reference velocity sensitivity MOSRP08
Ara´ujo, F. V. Linear and non-linear methods derived from scattering theory: backscattered tomog-
raphy and internal multiple attenuation. PhD thesis, Universidade Federal da Bahia, 1994.
Carvalho, P. M. Free-surface multiple reflection elimination method based on nonlinear inversion
of seismic data. PhD thesis, Universidade Federal da Bahia, 1992.
Hsu, Shih-Ying and Arthur B. Weglein. On-shore project report I– Reference velocity sensitivity
for the marine internal multiple attenuation algorithm: analytic examples. Technical report,
Mission-Oriented Seismic Research Project, University of Houst on, 2007.
Matson, K. H. An inverse-scattering series method for attenuating elastic multiples from multi-
component land and ocean bottom seismic data. PhD thesis, University of British Columbia,
1997.
Nita, Bogdan G. and Arthur B. Weglein. Inverse scattering internal multiple attenuation algo-
rithm in complex multi-d media. Technical report, Mission-Oriented Seismic Research Project,
University of Houston, 2005.
Otnes, Einar, Ketil Hokstad, and Roger Sollie. “Attenuation of internal multiples for
multicomponent- and towed streamer data..” SEG Technical Program Expanded Abstracts 23
(2004): 1297–1300.
Wang, Min and Arthur B. Weglein. A short note about elastic wavelet estimation. Technical report,
Mission-Oriented Seismic Research Project, University of Houst on, 2008.
Weglein, A. B., F. A. Gasparotto, P. M. Carvalho, and R. H. Stolt. “An Inverse-Scattering Series
Method for Attenuating Multiples in Seismic Reflection Data.” Geophysics 62 (November-
December 1997): 1975–1989.
Weglein, Arthur B. and Ken H. Matson. “Inverse scattering internal multiple attenuation: an
analytic example and subevent interpretation.” SPIE, 1998, 108–117.
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