Abstract: We propose a method based on the Poynting vector that combines angle-domain imaging and image amplitude correction to overcome the shortcomings of reverse-time migration that cannot handle different angles during wave propagation. First, the local image matrix (LIM) and local illumination matrix are constructed, and the wavefi eld propagation directions are decomposed. The angle-domain imaging conditions are established in the local imaging matrix to remove low-wavenumber artifacts. Next, the angle-domain common image gathers are extracted and the dip angle is calculated, and the amplitude-corrected factors in the dip angle domain are calculated. The partial images are corrected by factors corresponding to the different angles and then are superimposed to perform the amplitude correction of the fi nal image. Angle-domain imaging based on the Poynting vector improves the computation efficiency compared with local plane-wave decomposition. Finally, numerical simulations based on the SEG/EAGE velocity model are used to validate the proposed method.Keywords: Poynting vector, angle-domain imaging, local image matrix, illumination analysis, amplitude correction
Introduction
Reverse-time migration (RTM) was proposed by Hemon (1978) and compared with the ray-based Kirchhoff migration and one-way wave equation migration, the full-wave equation RTM offers high fidelity, no dip angle limitations, and well reproduces wave propagation in complex media, e.g., scattering, reflection, diffraction, and inhomogeneous wave propagation. Conventional cross-correlation imaging
superposes waves from all directions and consequently random noise is eliminated but the directional information of the waves is lost and low-wavenumber noise is introduced. Processing of seismic waves in the angle domain reconstructs the angle-domain imaging conditions, extracts the imaging gathers, analyzes the variation in the amplitude of a seismic reflection with angle of incidence or source–geophone distance (Yan and Xie, 2012a, 2012b; Wang et al., 2013) and provides directional illumination (Xie et al., 2006; Xie et al, 2013) and imaging resolution analysis and imaging correction
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(Xie et al., 2005; Wu and Chen, 2006; Wang et al., 2013).
When using wave equations to calculate the wavefield, the wave solutions do not explicitly contain the information of the wave propagation direction. There are three different approaches to extract the propagation direction.The fi rst method is based on direction vectors, such as the Poynting vector (Yoon et al., 2004), the polarization vector or the instantaneous wavenumber vector (Zhang and McMechan, 2011a; 2011b). Such methods are highly efficient and offer better angle resolution but yield one propagation direction per image point per time step; thus, it is diffi cult to acquire the propagation direction of every wavefront when overlapping wavefronts exist.The second set of methods are based on local plane-wave decomposition and use local slant stacking or local Fourier transform (Xie and Wu, 2002; Jia and Wu, 2009; Zhang et al., 2010; Xu et al., 2011) to decompose the local wavefi eld. The angle decomposition is independent of the imaging procedures. These methods are stable and can handle waves from different directions. However, they are computationally costly because local decomposition is performed at each spatial point. The third set of methods extract the direction of the propagating wavefield during imaging. The local subsurface-offset or time-shift images are computed and then are transformed into the angle domain by Fourier or Randon transforms (Sava and Fomel, 2003, 2005a, 2005b, 2005c; Sava and Vasconcelos, 2011). These methods are also computationally costly and only decompose the angles during imaging; therefore, they are unsuitable for illumination analysis. Yoon et al. (2004) used the Poynting vector to compute the propagation direction of the wavefields and constructed the angle-domain imaging conditions by introducing an angle-related fi lter to add weight to the waves from different directions and suppress low-wavenumber artifacts. Chen and He (2014) separated wavefields into propagating up or down and left or right and proposed a normalized separation cross-correlation imaging that minimizes the low-wavenumber artifacts and improves image quality.
Migration methods are used to image the subsurface geometry and obtain the physical parameters of subsurface media. Nevertheless, factors, such as limited acquisition aperture and complex velocity structure and refl ector dipping angle, typically produce irregular illumination of subsurface regions. It is difficult to produce correct true-amplitude images of complex regions even using accurate propagators. Target-oriented seismic directional illumination analysis is an effective
quantitative tool and used in image correction.Traditional illumination analysis is based on ray theory (Schneider and Winbow, 1999; Bear et al., 2000; Muerdter and Ratcliff, 2001); however, the high-frequency asymptotic approximation and the singularity of ray theory do not allow to handle scattering, diffraction, and defocusing in complex media. Based on the relation between migration and the physical characteristics of the subsurface media, Gelius et al. (2002) established the point-spread function (PSF) in the wavenumber domain for correcting images. Lecomte (2008) analyzed the illumination and resolution of depth-migration images based on ray theory. Based on the one-way wave equation, Wu and Chen (2002, 2006), Wu et al. (2003), and Xie et al. (2003, 2005, 2006) proposed illumination analysis in the angle domain and used it to correct the acquisition aperture. Du et al. (2012) corrected RTM images by analyzing the illumination of the source and receiver wavefi elds. Xie and Yang (2008) and Cao and Wu (2009) investigated the full-wave equation illumination analysis and applied it to RTM. Based on inverse theory and the local-angle domain decomposition of Green’sfunctions, Wu et al. (2006) analyzed the imaging resolution and its relation to the limited frequency band of the source, the complex overlaying structure, the limited acquisition aperture, and the approximate migration operator.Two image correction methods are used: image deconvolution by the point-spread function (Gelius et al., 2002) and local image compensation by angle-domain illumination (Gherasim et al., 2012; Yan et al., 2014). Wu and Luo (2005) discussed four different amplitude correction methods and concluded that dip angle domain correction yields the best results. Yan et al. (2014) developed a dip angle domain amplitude correction algorithm to retrieve the true impedance contrast of the subsurface structure by slant stacking.
The local image matrix (LIM) based on the local plane-wave decomposition in the frequency domain and angle-domain imaging can eliminate low-wavenumber artifacts and extract the angle gathers. LIM retrieves information related to the angles of incidence and reflection. However, LIM in the frequency domain is computationally costly. In this study, the Poynting vectors are used to calculate the wavefi eld propagation angle and analyze the illumination using the full-wave equation. To greatly reduce the computational cost, we use LIM and local illumination matrix in the time domain. Common dip-angle images (CDIs) are computed from regular RTM images. To improve the computational efficiency of illumination analysis, the propagation angles are only calculated at several
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moments corresponding to high wavefield amplitude rather at every moment. Numerical simulations using the 2D SEG model were used to test the proposed method.
Decomposing the wavefi eld propagation direction using the
Poynting vector
Poynting vector was fi rst used to describe the energy fl ux density in the electromagnetic fi eld and denote the passing energy per unit area per unit time along the direction perpendicular to the vector (Poynting, 1884).Yoon et al. (2004) first introduced the Poynting vector to RTM to extract the wave propagation direction. The two-dimensional Poynting vector is (Yoon et al., 2011)
( , ) ,x zp ptuP u (1)
where u is either the source wavefi eld us or the receiver wavefield ug, ∂u/∂t is the time differential of the wavefi eld, ∆u = (∂u/∂x, ∂u/∂z) is the spatial gradient, x and z are the horizontal and vertical axes respectively, and px and pz are the horizontal and vertical components of the Poynting vector, respectively. Then, the wave propagation angle is
arctan( ).x
z
pp
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Using equations (1) and (2), the propagation direction is calculated at each location and each time step of the wavefield. Therefore, we use the Poynting vector to decompose the wave propagation direction or angle. Intermediate results, such as ∂u/∂t and ∆u, are already calculated by staggered finite-difference methods and no extra computations are required in the case of the Poynting vector. Figure 1a shows a two-layer homogeneous medium at the time window of 1.12 s, in which the velocities of the layers are 2.0 km/s and 4.0 km/s. Figures 1b and 1c show the horizontal component px and vertical component pz of the Poynting vector and the white color denotes the positive direction and the black color denotes the negative direction. When an incident wave reaches the interface, reflected and transmitted waves are simultaneously generated at the interface. The wavefi eld and Poynting vector of the white square area in Figure 1a at three time steps are extracted and enlarged to show the propagation angles calculated by equation (2), as shown in Figures 1d−1f. At 1.07 s, the fi rst arrival of the incident wave has passed the interface. The incident angles below the interface are greater than those above the interface. There are large errors in the calculations of the incident angles at low-amplitude regions but the contributions of the low-amplitude waves to the fi nal image are negligible. At 1.094 s, the wavefi eld
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Fig.1 Wavefi eld snapshot of the two-layer model. The wave propagation direction is computed by the Poynting vector. The local snapshots of the wavefi eld overlap with the propagation direction. The horizontal black line represents the interface. (a) Snapshot of wavefi eld at t = 1.12 s; (b) horizontal Poynting vector; (c) vertical Poynting vector; (d)−(f) local wavefi eld snapshots of the white square in (a) at t = 1.07 s, 1.094 s, and 1.1 s that are overlap with the wave-propagating direction (white arrows) computed by using the Poynting vectors.
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propagation angles above the interface are actually the summation of the vectors of the incident and reflection waves. At 1.1 s, the incident waves have passed through the interface and both the incident and refl ection waves appear near the interface simultaneously.
Angle-domain images
Local image matrixThe high fi delity of the full-wave propagators allows
to use RTM to image complex reflectors. The zero-lag cross-correlation imaging of prestack RTM is expressed as
max
s max0( ) ( , ) ( , )d ,
T
gt T t tI r u r u r (3)
where us(t, r) and ug(t, r) are the wavefields of the source and receiver at time t and location r = (x, y, z), and Tmax is the maximal recording time. Applying equations (1) and (2) to the wavefield of source us(t, r) and the wavefi eld of receiver ug(t, r), the propagation direction θ is determined at each time–space point (t, r), which is introduced into the above wavefi elds to obtain the angle-domain wavefi elds of the source and receiver us(t, θs, r) and ug(t, θg, r). Substituting them into equation (3), the imaging condition is modifi ed to (Yan and Xie, 2012a)
g s
s g( ) ( , , ),I r I r (4)
where
max
s g s s g max g0( , , ) ( , , ) ( , , )d ,
Tt T t tI r u r u r (5)
is the partial image in the angle domain, and θs and θg are the propagation directions of the incident and scattering waves, respectively.
If the directions θs and θg are discretized into 24 sections respectively and each section corresponds to a 15° angle. Based on equation (5), I(θs, θg, r) corresponds to all the (θs, θg) pairs at the imaging location r and is denoted as I(θs, θg, r). Each element of the LIM represents a partial image and there are 24×24 partial images. The fi nal image is obtained by stacking up all the partial images. If the local refl ectors are approximated as planes, the observation coordinates (θs, θg) are converted to the target coordinates (θr, θd) based on Snell’s law (Xie and Wu, 2002), where θr is the refl ection angle and θd is the dip angle. For local planar refl ectors, the distribution of the wave energy in LIM is stripped, and different image gathers, such as common refl ection angle gathers, common dip-angle gathers, common scattering-angle gathers and common illumination gathers, are extracted.
Figures 2b−2e show the LIM of four points in the fi ve-layer model of Figure 2a. The wave propagation velocity in every layer is 3.5 km/s, 3.75 km/s, 4.0 km/s, 4.25 km/s, and 4.5 km/s, respectively. The model grid size is 512 × 208, where dx = dz = 12 m. A synthetic data set is generated using 92 explosion sources and 472 receivers. The sources are placed at 60 m intervals and the space between the receivers is 12 m. A fourth-order in space
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and second-order in time finite-difference method is used to calculate the wavefi eld propagation. The source time function is a 15 Hz Ricker wavelet. Figure 2b shows the LIM of location A without the refl ector. The energy is distributed at θr = ± /2 and produces low-wavenumber artifacts. Point B is on the horizontal refl ector and the dip angle θd is zero. The energy in the LIM is distributed along a strip and the intersection with the dipping axis is at zero, which exactly denotes the geometry of the refl ector, as shown as Figure 2c. Points C and D are located on reflectors with different dip angles. The energy distribution in the LIM of Figures 2d and 2e is still stripped but the intersections with the dipping axis deviate from the zero dip angle. The shift from the intersection to the dipping axis gives the dip angle. For example, there is approximately a −25° shift from the intersection to the zero dip angle in Figure 2e, which agrees well with the actual reflector dip angle. The local illumination aperture is also relatively small because of the dip angle. It can be seen from Figures 2c, 2d, and 2e that the energy range along the refl ection axis decreases with increasing dip angle and depth; therefore, the effective illumination aperture of these points also gradually decreases. Figure 3 shows the dip angle of the four locations in Figure 2a. The refl ectors at locations 1 and 4 are horizontal and all the energy peaks concentrate along the center line. The first reflectors at locations 2 and 3 are horizontal, and the dip angles of the other reflectors increase. The energy peaks shift from the center line and the offset distance denotes the corresponding dip angle value. For example, the offset (labeled as ‘dip’) of the fourth energy peak at location 2 corresponds to the refl ector dip angle.
shows the ADCIGs calculated by using the model in Figure 2a. Figures 4a–4c show the ADCIGs of the true velocity model and models with −5% and +5% velocity errors, respectively. The ADCIGs are calculated at four horizontal locations between 1.2 km and 4.8 km at intervals of 1.2 km. The angle range is from −50˚ to +50˚. As shown in the figure, the ADCIGs of thetrue velocity model are fl at, the gathers curve upward for the −5% velocity error and downward for the +5% velocity error. The actual angle range of the gathers depends on the effective illumination aperture and the refl ector dip angle of the image point, which decreases gradually with depth.
Fig.3 Dip angle estimation in the fi ve-layer model.
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As previously mentioned, angle-domain common image gathers (ADCIGs) can be extracted from the LIM. The gather events are flat in the true velocity model but they curve down or up in the fast or slow model.The curvature correlates with the velocity error, which can be used for updating the velocity model. Figure 4
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Imaging conditions in the angle domainBased on the angle transform, equation (4) is
transformed to
r d
r d( ) ( , , ).I r I r (6)
The waves from all the incident and scattering angles are thus imaged by using I(r) and low-wavenumber are introduced. The energy distribution of the artifacts is shown in LIM. As shown in Figure 2b, there is no reflector at point A and the opening angle between the
Fig.4 ADCIGs of the fi ve-layer model.(a) True velocity model; (b) model with −5% velocity error; and (c) model
with +5% velocity error.
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source and receiver wavefield is close to ; thus, the energy distribution mainly comes from the turning wave or the wide-angle reflected wave. Low-wavenumber artifacts appear in cross-correlation imaging (Díaz and Sava, 2015). Therefore, an angle-domain filter is introduced in the LIM to prevent the wide-angle wave energy from entering the image. The fi lter removes the low-wavenumber artifacts effectively. Then, the imaging conditions in the angle-domain are expressed by the following equation (Yan and Xie, 2012a)
r d
r d r d( ) ( , ) ( , , ),I r F I r (7)
where F(θr, θd) is the angle filter that attenuates the energy in wide-angle refl ections.
Considering the 2D SEG velocity model in Figure 5a, as an example, we use 350 shots at intervals of 48.77 m and each shot is matched to176 left-sided receivers at 24.38 m spacing. The source function is a 15 Hz Ricker wavelet. A second-order in time and fourth-order in
space finite-difference algorithm with PML boundary conditions is used to extrapolate the wavefi eld. The wave propagation angle is calculated by using the Poynting vector at every grid point and each time step during imaging. To decrease the computation time and memory use during illumination analysis, only the angle at the time that of the highest amplitude wavefi eld is calculated at each grid point.
Figures 5b and 5c show regular images and those after the application of the angle-domain fi lter F(θr, θd)
0
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Comparing Figures 5b and 5c, strong artifacts in the regular RTM mask the true image and the real is not readily recognized, especially at the shallow area. However, the artifacts are effectively removed by using the angle-domain imaging conditions that maintain the images at the interface.
Fig.5 Comparison of the migrated images.(a) 2D SEG salt model; (b) image without angle-domain fi ltering; (c) images with angle-domain fi ltering that eliminates refl ections with angle larger than 60˚.
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Amplitude correction for RTM in the angle domain
Local illumination matrixFiltering in the angle domain removes low-wavenumber
artifacts to some extent. However, the acquisition aperture, complex overburden structures, and reflector dip angle produce irregular illumination to the subsurface regions, leading to shadows and distorted amplitude in migration imaging. Seismic illumination analysis is used to optimize the acquisition system, evaluate image quality, correct the imaging amplitude, and recover the subsurface physical parameters more accurately. If the observation system is given, the illumination in the time domain from source rs to subsurface target point r along the direction θs is (Yang and Xie, 2008)
2
s s s s s s( , ; ) ( , , ; ) d .t tD r r u r r (9)
Similarly, the illumination in the time domain from receiver rg to target point r along direction θg is
2
g g g g g g( , ; ) ( , , ; ) d .t tD r r u r r (10)
Using equations (9) and (10), the local illumination matrix of the target location r in the time domainis
s g s g
s s s g g g
2 2s s s g g g
( , , ; , )
( , ; ) ( , ; )
[ ( , , ; )] d [ ( , , ; )] d ,t t t t
A r r rD r r D r r
u r r u r r (11)
where θs and θg are calculated with equation (2). If the observation system consists of multiple sources and receivers, the local illumination matrix of the entire acquisition on system adds the contributions from each source and receiver pairs
s g
s g s g s g( , , ) ( , , ; , ).A r A r r rr
(12)
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Figure 6 shows the local illumination matrix at six points in the SEG model. The observation system, source wavelet, wavefield extrapolation operator, and absorbing boundary are all the same as in Figure 5. The illumination range mainly concentrates in the upper left corner of every local illumination matrix because of the single-sided receivers. The white square in the matrix is the range of incident and scattering angles within ±
90˚, which denotes the illumination range of one-way propagators. The illumination range in the full-wave method is ±180˚ for both incident and scattering waves. The figure also shows that the illumination aperture is wide at shallow regions and narrows with increasing depth. Because the high-velocity salt focuses the seismic wave energy, the illumination at the subsalt region is distinctly weak, which affects the fi nal imaging quality.
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where I(θd, r) is CDI and Ic(r) is the corrected image. The workfl ow for the angle-domain correction is shown in Figure 7.
Model calculationThe Green function is not decomposed when using
the Poynting vector to calculate the angle because it has been computed during the wave propagation. This is time-consuming and strains the computer memory when calculating the angle at all grid points and at each time step in the model space. However, Hu et al. (2014) pointed out that it is more reliable than the
Fig.6 Local image matrices at different locations for the SEG model.
ADR and image correctionADR describes the total illumination of a target
with specific dip angle in space in the case of a given observation system, which can be calculated by the local illumination matrix that sums all A(θs, θg, r) satisfying the given dip angle condition. The RTM image is decomposed into different CDIs that are normalized using ADR as the image amplitude correction factor. Based on the angle transform, (θs, θg) is transformed to (θr, θd) and then ADR transforms to
d r d r( , ) ( , , )d .r A rADR (13)
The image correction by ADR is (Yan et al., 2014)
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wave amplitude in estimating the propagation angle, and Yang and Xie (2008) reported the similar results. Therefore, the wave propagation angle at each grid point is practically calculated only at the highest amplitudes of the wavefi eld rather than at each time step. In addition,
image correction uses both CDI I(θd, r) and ADR. The RTM image is decomposed and I(θd, r) is calculated by local slant stacking. ADR is a slowly varying function in the space domain and it is calculated with a coarse mesh and interpolated into the image grid (Yan et al., 2014).
Using Poynting vectors to compute thepropagation angles s of source wavefields.
us(t, r) is denoted as us( s, t, r)
Using Poynting vectors to compute the propagation angles g of receiver wavefields.ug(Tmax–t, r) is denoted as ug( g, Tmax–t, r)
Imaging conditionin angle domain
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Illumination computation:
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Correct imaged
c dd
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Fig.7 The workfl ow of RTM with angle-domain correction.
Figure 8 shows the four CDIs shown in Figure 5c. Figure 9 shows the corresponding normalized ADRs.
Comparing the CDIs and ADRs, it can be seen that the amplitude of CDIs is consistent with the intensity of
Fig.8 CDIs; the dip angles are (a) −15°, (b)15°, (c) −45°, and (d) 45°.
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ADRs. The corrected CDIs are shown in Figure 10. All the corrected CDIs form the fi nal corrected image.
Figure 11 shows the image before and after correction, where the image amplitude of the different reflectors becomes more balanced after the correction, especially the imaging quality of the subsalt regions with steep structures has clearly improved. Figure 12 shows the theoretical reflectivity (red lines) and the image amplitude (blue lines) before and after the correction. The image amplitudes are consistent with the theoretical reflectivity in shallow regions regardless of correction. However, in deep areas, especially in subsalt regions, the amplitude of regular RTM images is very weak and differs from the theoretical reflectivity, whereas the amplitude after the correction matches very well with the theoretical values. Fig.11 RTM image (a) before correction and (b) after
correction with ADRs.
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Discussion
Hu et al. (2014) summarized and compared the precision of the different methods for computing the wave propagation direction. To improve the computation stability by the Poynting vector, Yoon et al. (2011) adopted a Gaussian-weighted function in the time domain to calculate the Poynting vector. Thomas and Graham. (2011) proposed to smooth the Poynting vector in the space or time domain. For the SEG salt model, Figure 13 shows the wavefield propagation directions
calculated by the local plane-wave decomposition and Poynting vector methods, respectively. Figure 13a shows the velocity model for overlapping wavefi elds, where the source is denoted by the asterisk. The wave propagation directions of the four wavefront locations, which are calculated by the local plane-wave decomposition and Poynting vector methods, are shown in Figures 13b−13e. The orientations of the energy peaks in the polar coordinates point to the propagation directions calculated by the local plane-wave decomposition method, and the red arrows denote the propagation direction calculated by the Poynting vectors. The directions estimated by both methods are almost the same if there is only one wavefront, as shown in Figure 13b−d for the three different wavefronts. When there are more than one wavefront arriving simultaneously, for instance in Figure 13e with two energy peaks, the local plane-wave decomposition method points to the direction of each wavefront. However, the method of Poynting vector calculates the sum of vectors of the two wavefronts and can only estimate the propagation direction denoted by the red arrow in Figure 13e.
Although the Poynting vector method cannot separate the propagation directions when multiple wavefronts overlap, it decreases the number of calculations for the angle decomposition and has enormous potential. Considering the model in Figure 13a, as an example and at the same computing conditions, the methods of local plane-wave decomposition and Poynting vectors are used to calculate the propagation directions of the single-shot wavefield generated. Forward modeling is used to
0 2.4 4.8 7.2 9.6 12.0 14.4 16.8
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0
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Fig.12 Theoretical refl ectivity (red lines) with image amplitude (blue lines) (a) before and (b) after correction with ADRs.
0 2.4 4.8 7.2 9.6 12.0 14.40
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00Horizontal slowness
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00Horizontal slowness (d)
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Fig.13 Wave propagation directions calculated by the local plane-wave decomposition and Poynting vector methods.The orientation of the energy peaks in the polar coordinates denotes the propagation directions calculated by the local plane-wave decomposition
method and the red arrows denote the propagation directions calculated by the Poynting vectors (snapshot time is t = 1.7 s).
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Liu et al.
compare the angle calculation time and the wavefield data are transformed into the frequency domain before using the local plane-wave decomposition method. The
calculation time is 16269.09 s for the local plane-wave decomposition and 238.091 s forthe Poynting vector (Table 1).
Table 1 Performance of the local plane-wave decomposition and Poynting vector Calculation method If FFT needed Extra memory Computation time (s)Local plane-wave
decompositionNeed FFT (9×9 local window is adopted)
No additional memory 16269.09
Poynting vector FFT not needed 8.27 G 238.091
Conclusions
The Poynting vector is used to calculate the wave propagation directions. The LIM is used to analyze the wave propagation and the angle problem during imaging. Angle fi ltering suppresses the low wave-number artifacts effectively. The local illumination matrix based on the Poynting vector is used to calculate ADRs. The RTM images are decomposed to common dip partial images by the slant stacking method and the common dip partial images are compensated by the corresponding ADRs.The RTM image amplitude is corrected in the dip angle domain. Compared with the local plane-wave decomposition based on slant stacking or windowed FFT, the Poynting vector method offers improved computational effi ciency. Although there are problems, such as stability and accuracy in estimating the angle, the Poynting vector method can yields reasonably accurate results.
References
Bear, G., Lu, C., Lu, R., et al., 2000, The construction of subsurface illumination and amplitude maps via ray tracing:The Leading Edge, 19(7), 726–728.
Cao, J., and Wu, R. S., 2009, Full-wave directional i l lumination analysis in the frequency domain: Geophysics, 74(4), S85–S93.
Chen, T., and He, B. S., 2014, A normalized wavefield separation cross-correlation imaging condition for reverse time migration based on Poynting vector: Applied Geophysics, 11(2), 158–166.
Díaz, E., and Sava, P., 2015, Understanding the reverse time migration backscattering: noise or signal?: Geophysical Prospecting, 64(3), 581−594.
Du, X., Fletcher, R., Mobley, E., et al., 2012, Source and receiver illumination compensation for reverse-time migration: 74th Annual International Conference and
Exhibition, EAGE, Extended Abstracts, X046.Gelius, L. J., Lecomte, I., and Tabti, H., 2002, Analysis
of the resolution function in seismic prestack depth imaging: Geophysical Prospecting, 50(5), 505–515.
Gherasim, M., Albertin, U., Nolte, B., et al., 2012, Wave-equation angle-based illumination weighting for optimized subsalt imaging: 82th Ann. Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 3293–3296.
Hu, J., McMechan, G. A., and Guan, H. M., 2014, Comparison of methods for extracting ADCIGs from RTM: Geophysics, 79(3), 89–103.
Jia, X. F., and Wu, R. S., 2009, Calculations of wavefi eld gradients and wave propagation angles in complex media: application to turning wave simulations: Geophysical Journal International, 178(3), 1565–1573.
Lecomte, I., 2008, Resolution and illumination analyses in PSDM: A ray-based approach: The Leading Edge, 27(5), 650–663.
Muerdter, D., and Ratcliff, D., 2001, Understanding subsalt illumination through ray-trace modeling, Part 1: Simple 2-D salt models: The Leading Edge, 20(6), 578–594.
Poynting, J. H., 1884, On the transfer of energy in the electromagnetic fi eld: Philosophical Transactions of the Royal Society of London, 175, 343–361.
Sava, P., and Fomel, S., 2003, Angle-domain common image gathers by wavefield continuation methods: Geophysics, 68(3), 1065–1074.
Sava, P., and Fomel, S., 2005c, Wave-equation common angle gathers for converted waves: 75th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 947–950.
Sava, P., and Vasconcelos, I., 2011, Extended imaging conditions for wave equation migration: Geophysical
516
Reverse-time migration and amplitude correction
Prospecting, 59(1), 35–55.Schneider, W. A., and Winbow, G.A., 1999, Efficient and
accurate modeling of 3-D seismic illumination: 69th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 633–636.
Thomas, A. D., and Graham, Q. W., 2011, RTM angle gathers using Poynting vectors: 81th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 3109–3113.
Wang, B. L., Gao, J. H., and Chen, W. C., et al., 2013, Extracting Efficiently Angle Gathers Using Poynting Vector During Reverse Time Migration: Chinese Journal of Geophysics, 56(1), 262–268.
Wang, M. X., Yang, H., and Osen, A., 2013, Full-wave equation based illumination analysis by NAD method: 75th EAGE Conference & Exhibition Incorporation, SPE, EUROPEC, 10–13.
Wu, R. S., and Chen, L., 2002, Mapping directional illumination and acquisition-aperture effi cacy by beamlet propagators: 72th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 1352–1355.
Wu, R. S., Chen, L., and Xie, X. B., 2003, Directional illumination and acquisition dip-response: 65th Conference and Technical Exhibition, EAGE, Expanded abstracts, 1–4.
Wu, R. S., and Luo, M. Q., 2005, Comparison of different schemes of image amplitude correction in pre-stack depth migration: 75th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 2060–2063.
Wu, R. S., and Chen, L., 2006, Directional illumination analysis using beamlet decomposition and propagation: Geophysics, 71(4), S147–S159.
Wu, R. S., Xie, X. B., Fehler, M., et al., 2006, Resolution analysis of seismic imaging: 68th Annual International Conference and Exhibition, EAGE, Extended Abstracts, GO48.
Xie, X. B., and Wu, R. S., 2002, Extracting angle domain information from migrated wavefields: 72th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 1360–1363.
Xie, X. B., Jin, S. W., and Wu, R. S., 2003, Three-dimensional illumination analysis using wave-equation based propagator: 73th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 989–992.
Xie, X. B., Wu, R. S., Fehler, M., et al., 2005, Seismic resolution and illumination: A wave equation based analysis: 75th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 1862–1865.
Xie, X. B., Jin, S. W., and Wu, R. S., 2006, Wave-equation based seismic illumination analysis: Geophysics, 71(5), S169–S177.
Xie, X. B., and Yang, H., 2008, A full-wave equation based seismic illumination analysis methods: 70th
Annual International Conference and Exhibition, EAGE, Extended Abstracts, P284.
Xie, X. B., He, Y. Q., and Li, P. M., 2013, Seismic Illumination Analysis and Its Applications in Seismic Survey Design: Chinese Journal of Geophysics, 56(5), 1–14.
Xu, S., Zhang, Y., and Tang, B., 2011, 3D common image gathers from reverse time migration: Geophysics, 76(2), S77–S92.
Yan, R., and Xie, X. B., 2012a, An angle-domain imaging condition for elastic reverse time migration and its application to angle gather extraction: Geophysics, 77(5), 105–115.
Yan, R., and Xie, X. B., 2012b, AVA analysis based on RTM angle-domain common image gather: 82th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 1–6.
Yan, R., Guan, H. M., Xie, X. B., et al., 2014, Acquisition aperture correction in the angle domain toward true-reflection reverse time migration: Geophysics, 79(6), S241–S250.
Yoon, K , Marfurt, and Starr, W., 2004, Challenges in reverse-time migration: 74th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 1057–1060.
Yoon, K, Guo, M. H., Cai, J., et al., 2011, 3D RTM angle gathers from source wave propagation direction and dip of reflector: 81th Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, 3136–3140.
Zhang, Q. S., and McMechan, G. A., 2011a, Common-image gathers in the incident phase-angle domain from reverse time migration in 2D elastic VTI media: Geophysics,76(6), 197–206.
Zhang, Q. S., and McMechan, G. A., 2011b, Direct vector-field method to obtain angle-domain common-image gathers from isotropic acoustic and elastic reverse-time migration: Geophysics, 76(5), 135–149.
Zhang, Y., Xu, S., Tang, B., et al., 2010, Angle gathers from reverse time migration: The Leading Edge, 29(11), 1364–1371.
Liu Ji-Cheng, Ph.D., Professor at the School of Electric and Automation Engineering, Changshu Institute of Technology. His interests are non-stationary signal analysis, seismic imaging, and illumination analysis.
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