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Introduction
To reduce wave-mode crosstalk artifacts in elastic imaging, a
classical workflow is to first separate com- pressional and shear
waves for extrapolated source and receiver wavefields, followed by
applying the zero-lag cross-correlation imaging condition to
produce PP and PS images. In isotropic media, the wave- mode
separation can be implemented using the divergence and curl
operators (Yan and Sava, 2008). In anisotropic media, however,
seismic wave polarization directions are neither parallel nor
perpendicular to propagation directions. Corresponding anisotropic
wavefield separation needs more complex algo- rithms, such as the
non-stationary filters (Yan and Sava, 2011) and low-rank
approximations (Cheng and Fomel, 2014).
To mitigate these problems, we present an elastic least-squares
imaging (LSM) method for the tilted transversely isotropic (TTI)
media. We parameterize the TTI elastic wave equation using the
perturbed stiffness parameters lnC33 and lnC55 as the reflectivity
models, and derive the Born modeling oper- ator for first-order
scattering. Here, C33 = ρv2
p and C55 = ρv2 s , ρ is the density, vp and vs are the axial
compressional (P) and shear (S) velocities. The reflections from
anisotropic parameters and tilt angles are neglected. Using the
Lagrange multiplier method, we derive the corresponding adjoint
wave equation and sensitivity kernels. Numerical experiments
illustrate that LSM can improve the spatial resolution and
amplitude fidelity compared with the adjoint-based migration and
enhance the contributions of weak S-wave to estimated subsurface
reflectivities, producing high-quality lnC33 and lnC55
images.
Method
The 3D elastic TTI wave equation can be written as
ρ∂tv−Pσ= 0, ∂tσ−DPT v = f, (1)
where ρ is the density, f is the source, ∂t denotes the time
partial derivative, superscript T denotes the transpose, v and σ
are the particle velocity and the stress tensor, respectively. P is
a spatial partial- derivative operator in a matrix form as
P =
∂x 0 0 0 ∂z ∂y 0 ∂y 0 ∂z 0 ∂x 0 0 ∂z ∂y ∂x 0
. (2)
D is the fourth-order stiffness tensor for elastic TTI media, which
can be computed using a Bond trans- formation (Carcione, 2007)
as
D = MCMT , (3)
where C is the unrotated stiffness matrix without density
normalization, and M is the Bond matrix. According to the Born
approximation (Aki and Richards, 1980), the stiffness parameters
C33 and C55 can be written as
C33 =C0 33 +C33, C55 =C0
55 +C55, (4)
where and superscript 0 denote the perturbed and background models,
respectively. We define the reflectivity models as the relative
perturbations of C33 and C55 as
lnC33 = C33
C0 33
, lnC55 = C55
C0 55
, (5)
Inserting the definitions in equation 4 into equation 3 and
applying the definitions of reflectivities in equation 5, the
rotated stiffness matrix D can be linearized as
D = D0 +M ∂C
Similarly, the wavefields can be linearized as
σ= σ0 +σ, v = v0 +v, (7)
where v0 and σ0 are the background particle velocity and stress
wavefields, v and σ are the perturbed particle velocity and stress
wavefields. Substituting equations 6 and 7 into equation 1 and
neglecting the high-order perturbation terms, we obtain the
following Born modeling operator
ρ0∂tv0−Pσ0 = 0,∂tσ0−D0PT v0 = f, ρ0∂tv−Pσ= 0,∂tσ−D0PT
v0 = fvir, (8)
where fvir is the virtual source introduced by the perturbations of
C33 and C55, and can be expressed as
fvir = M ∂C
At the receiver locations, the first-order multicomponent particle
velocities vi (i = x,y,z), or pressure p = 1
3(σxx +σyy +σzz) are saved as synthetic data to match the observed
data in reflectivity inver- sion.
Adjoint migration for elastic TTI media
The adjoint migration operator can be derived using the Lagrange
multiplier method (Liu and Tromp, 2006). The adjoint wave equation
can be written as
ρ0∂tv†−PDT 0 ξ
† = dmul(xr, tmax− t),∂tξ †−PT v† = dp(xr, tmax− t). (10)
and the sensitivity kernels of C33 and C55 are given by
K33(x) =− ∫ tmax
0 C0
dt, (11)
where ξ†(x, t) and v† is the adjoint strain and particle velocity
wavefields computed by solving the adjoint wave equation 10,
dmul(xr, t) and dp(xr, t) are the multicomponent and pressure data
residuals, and tmax is the record duration.
Least-squares inversion for lnC33 and lnC55
With the Born modeling (L) and adjoint migration (L†) operators,
the least-squares migration can be formulated as
m = (L†L)−1L†dobs, (12)
where m = [ lnC33, lnC55] T , and dobs is the observed data. L†L
denotes the Gauss-Newton Hessian.
Because it is prohibitive to directly compute the Hessian matrix
and its inverse, an alternative way is to iteratively solve such a
linear inverse problem as
m = argmin m Lm−dobs2
2 +νm2 2 +µ ∑
xi
∂xim1, (xi = x,y,z), (13)
where 2 and 1 denote the L2 and L1 norms, respectively. The first
term in equation 13 is used for data fitting. The second and third
terms are Tikhonov and total-variation (TV) regularizations,
respectively, which help to avoid data-overfitting problems and
remove swing artifacts in LSM. ν and µ are two scalar parameters to
balance the trade-off between data fitting and
regularizations.
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Numerical Examples
In the first example, the Marmousi-II model is used to test the
feasibility of the proposed TTI-LSM method. The velocity and
anisotropy models in Fig. 1 are used to generate observed data, and
their smoothed versions are used in migration. There are 127
explosive sources with a spacing of 187.5 m. Each source is
recorded by 401 receivers with a 25 m spacing. Migrated images
using traditional wave-mode separation based elastic RTM and the
proposed LSM are presented in Fig. 2. The spatial derivatives for
the forward and adjoint wavefields in the sensitivity kernels
produce lnC33 and lnC55 images with higher resolution (Figs 2b and
e) than traditional elastic RTM images (Figs 2a and d). These
spatial derivatives give rise to stronger source aliasing artifacts
at shallow depths. Because of insufficient illumination beneath the
three dominant faults, the reflectors within the anticline have
weak amplitudes and are contaminated by migration noises,
especially in the PS and lnC55 images (Figs 2e). After fifteen
iterations, LSM reduces shallow source aliasing artifacts, enhances
the image amplitudes at great depths, and improves the spatial
resolution (Figs 2c and f).
(a) (b) (c)
(d) (e) (f)
Figure 1: Elastic TTI Marmousi-II model. Panels (a)-(f) are models
for P-wave velocity, S-wave veloc- ity, density, ε , δ and tilt
angle, respectively.
In the second example, we apply the elastic TTI-LSM to a marine
field dataset, which contains 300 common-source gathers. The P-wave
velocity model is built using ray-based tomography, and S-wave
velocity is derived by scaling P-wave velocity by 0.6. Anisotropic
models are built based on well and regional geology information.
LSM results including a gas pocket at the first and twentieth
iterations are presented in Fig. 3. Compared to the first iteration
results (Figs 3a and b), the lnC33 image amplitudes at great depths
are significantly improved, and shallow fine-scale reflectors are
resolved better at the 20th iteration (Figs 3c and d). But because
of not compensating attenuation effects in migration, the
reflectors in the middle gas pocket, i.e., greater than 1.5 km
depth in Fig. 3c, are still much weaker than in the shallow
sedimentary layers. For lnC55 images, its first iteration result
looks similar to lnC33 profile (Figs 3a and b). With increasing
iterations, LSM improves the amplitudes and spatial resolution,
especially for deep reflectors (Fig. 3d). It is notable that lnC55
image shows some different characteristics from the lnC33 image.
For examples, in the gas pocket region, the lnC55 image shows
reflectors with stronger amplitudes than the lnC33 image. The
possible reasons for these differences in lnC33 and lnC55 images
include (1) P-wave AVO effects and (2) enhanced converted S-wave
energy. The AVO effects of P-wave on lnC55 image at large offsets
can produce strong amplitudes than that in lnC33 image which are
mainly resolved from near-offset data. On the other hand, S-wave
propagation does not affected by pore fluids, which makes the
contribution of converted waves to lnC55 images not suffer from as
much fluid-associated attenuation as in the lnC33 image.
Conclusions
We present an elastic least-squares migration in TTI media in this
study. Unlike traditional wave-mode separation based elastic RTM,
the proposed elastic TTI LSM parameterizes the wave equation using
lnC33 and lnC55 as reflectivity models, and estimates them by
solving a linear inverse problem. The source illumination is used
as a preconditioner to accelerate convergence, and
TV-regularization is introduced into the LSM to reduce migration
artifacts. Compared with the adjoint migration results,
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(a) (b) (c)
(d) (e) (f)
Figure 2: Migrated images for the Marmousi-II model. (a, d) PP and
PS images from traditional elastic RTM. (b, e) LSM images at the
first iteration. (c, f) LSM results at the fifteenth
iteration.
(a) (b)
(c) (d)
Figure 3: A numerical experiment of TTI-LSM for a marine field
dataset. (a, b) I33 and I55 images at the first iteration. (c, d)
I33 and I55 images at the twentieth iteration.
LSM can enhance deep reflector amplitudes, improve the spatial
resolution and produce high-quality lnC33 and lnC55 images.
References
Aki, K., and P. Richards, 1980, Quantitative seismology: Theory and
methods: WH Freeman and Com- pany, San Francisco.
Carcione, J. M., 2007, Wave fields in real media: Wave propagation
in anisotropic, anelastic, porous and electromagnetic media:
Elsevier Scientific Publ. Co., In.
Cheng, J., and S. Fomel, 2014, Fast algorithms for
elastic-wave-mode separation and vector decompo- sition using
low-rank approximation for anisotropic media: Geophysics, 79,
C97–C110.
Liu, Q., and J. Tromp, 2006, Finite-frequency kernels based on
adjoint methods: Bulletin of the Seis- mological Society of
America, 96, 2383–2397.
Yan, J., and P. Sava, 2008, Isotropic angle-domain elastic
reverse-time migration: Geophysics, 73, S229– S239.
——–, 2011, Improving the efficiency of elastic wave-mode separation
for heterogeneous tilted trans- verse isotropic media: Geophysics,
76, T65–T78.
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