estimating subsurface p- and s-wave reflectivities using
TRANSCRIPT
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.
82nd EAGE Annual Conference & Exhibition
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,
82nd EAGE Annual Conference & Exhibition
(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.
82nd EAGE Annual Conference & Exhibition
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.
82nd EAGE Annual Conference & Exhibition
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,
82nd EAGE Annual Conference & Exhibition
(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.
82nd EAGE Annual Conference & Exhibition