76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014

We E106 04Multiparameter Full-waveform Inversion forAcoustic VTI Medium with Surface Seismic DataX. Cheng* (Schlumberger), K. Jiao (Schlumberger), D. Sun (Schlumberger)& D. Vigh (Schlumberger)

SUMMARYIn this study, we develop a strategy for multiparameter FWI for acoustic VTI medium with surface seismicdata. Through parameterization analysis and synthetic tests, we find that it is more feasible to invert for theparameterization as vertical and horizontal velocities instead of inverting for the parameterization asvertical velocity and anisotropy fields. We develop a hierarchical approach to inverting vertical velocityfirst, but we keep anisotropy fields unchanged and only switch to joint inversion when vertical velocityinversion are approaching convergence. We demonstrate the success of our strategy for VTI FWI usingsynthetic and real data examples from the Gulf of Mexico. Our results show that incorporation ofmultiparameter FWI improves migration of large offset full azimuth broad band acquisition data andproduces better focused migration images.

76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014

Introduction

Full-waveform inversion (FWI) as a data-driven minimization problem aims to directly fit the observed

and simulated seismic waveform in either time or frequency domain. The inversion is performed by

iteratively updating the velocity fields to reduce the difference between the two. It has been shown that

the inversion is very sensitive to the starting velocity fields, and data with long offsets and low frequen-

cies is crucial for the success of FWI to overcome this sensitivity (Vigh et al., 2011). Considering the

importance of data with long offsets and low frequencies, in most geologic environments, anisotropy is

an unavoidable topic for FWI, especially at long offsets, since anisotropy tends to have more pronounced

effects on waves traveled for a great distance (Prieux et al., 2011). In VTI medium, this means more

horizontal velocity will be registered in middle- to long-offset data, while more vertical velocity will be

registered in near- to middle-offset data. To date, most real-world applications of FWI still remain in

isotropic medium, and only a few studies have been shown to account for anisotropy. And most of those

studies only account for anisotropy in waveform simulation but do not invert for those anisotropy fields

(Wang et al., 2012; Warner et al., 2013). Multiparameter inversion for anisotropy fields, even in VTI

medium, remains a hot topic in the field (Plessix and Cao, 2011; Gholami et al., 2013a,b).

In this study, we develop a strategy for multiparameter FWI for acoustic VTI medium with surface seis-

mic data. Through parameterization analysis and synthetic tests, we find that it is more feasible to invert

for the parameterization as vertical and horizontal velocities instead of inverting for the parameteriza-

tion as vertical velocity and anisotropy fields. We develop a hierarchical approach to inverting vertical

velocity first, but we keep anisotropy fields unchanged and only switch to joint inversion when vertical

velocity inversion are approaching convergence. We demonstrate the success of our strategy for VTI

FWI using synthetic and real data examples from the Gulf of Mexico.

Method and Theory

By setting the shear velocity to zero, we can derive a set of first-order coupled VTI acoustic wave

equation that correctly describes the kinematics of compressional waves propagation (Alkhalifah, 2000;

Duveneck et al., 2008):

ρ∂v∂ t

= Dσ ,

∂σ∂ t

=CDv+ f,(1)

where v and σ stand for the particle velocity and stress wavefield, respectively. ρ is the density, f is the

source term, and C and D are defined as

C := ρv2p

⎛⎝ 1+2ε 1+2ε

√1+2δ

1+2ε 1+2ε√

1+2δ√1+2δ

√1+2δ 1

⎞⎠ , D :=

⎛⎜⎝

∂∂x 0 0

0 ∂∂y 0

0 0 ∂∂ z

⎞⎟⎠ , (2)

where VTI anisotropy is described using Thomson parameters ε and δ , and vp is the P-wave velocity

along symmetry axis.

Our multiscale time domain implementation of FWI iteratively updates the velocity fields (m) to re-

duce the misfit between the observed data (d) and the forward simulated data (−→ψ (m)), which can be

formulated as a standard least-squares problem:

minm

J(m) =1

2

∥∥W(S−→ψ (m)−d

)∥∥2

2, (3)

where S stands for the sampling/preprocessing operator, W represents the weighting operator (to appro-

priately scale multicomponent/multioffset data), and−→ψ =

(−→v−→σ)

solves the wave equation system, i.e.,

the VTI acoustic wave equation system (1).

76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014

The gradient of the misfit function can be derived through adjoint-state method (Plessix, 2006). Vigh et

al. (2014) derived the gradient formula for general elastic medium. In the case of VTI anisotropy, the

gradient regarding to anisotropy model parameters are given by:

∇vpJvti(m) =−⟨∫ T

0

⎛⎝ ∂

←−λvx∂x +

∂←−λvy∂y

∂←−λvz∂ z

⎞⎠dt,2ρvp

((1+2ε)

√1+2δ√

1+2δ 1

)(∂−→vx∂x +

∂−→vy∂y

∂−→vz∂ z

)⟩T

,

∇εJvti(m) =−⟨∫ T

0

(∂←−λvx

∂x+

∂←−λvy

∂y

)dt,2ρv2

p

(∂−→vx

∂x+

∂−→vy

∂y

)⟩T

,

∇δ Jvti(m) =−⟨∫ T

0

⎛⎝ ∂

←−λvx∂x +

∂←−λvy∂y

∂←−λvz∂ z

⎞⎠dt,

ρv2p√

1+2δ

(∂−→vz∂ z

∂−→vx∂x +

∂−→vy∂y

)⟩T

,

(4)

where←−λvx ,

←−λvy and

←−λvz stand for x, y and z component of the back-propagating particle velocity wave-

fields.

The misfit function is minimized by iteratively updating the model parameters with line-search method:

mk+1 = mk +αkgk, (5)

where α is the step size resolved from line search, and g is the model updating direction.

In Equation (5), the model updating direction is computed by means of conjugate gradient where only

first-order information is used. Although it was widely applied in practice, interactions between different

model parameters are simply ignored. A strong preconditioner is usually required to accelerate the

inversion convergence. This is especially true for multiparameter anisotropic inversion, since velocity

and anisotropic parameters have different physical unit and very different strength of influence on surface

seismic data. Bringing in second-order information through Newton-like method will better balance the

contribution of different parameter classes and improve the inversion. During simultaneous inversion,

we observe significant acceleration in the convergence when incorporating second-order information

and preconditioning into inversion. Here, in this study, all the results are obtained with Equation (5).

Several studies have shown that there is an ambiguity between the depth and the anisotropy parameter

δ (Alkhalifah and Tsvankin, 1995; Plessix and Cao, 2011). From surface seismic alone, δ cannot be

recovered uniquely. Thus, we decide to hold the δ fields unchanged during our inversion, and invert

only for vertical velocity and ε fields.

Synthetic Examples

In our first synthetic example, we did a sensitive analysis of the misfit function for an VTI Marmousi

model. The model is 17 km wide and 3.5 km in depth. There is a total of 171 shots with shot interval of

100 m. Receivers are located 25 m apart at 10 m depth. In all of our synthetic tests, the δ fields were set

as zero and fixed. We first smooth both the velocity and ε model for 500 m (Figure 1). Then, fractions

of the difference between the smoothed model and the true model were used to generate different model

variations. The misfit evaluation is conducted at a band witdth centered at 3 Hz. As we can see in

Figure 1e, velocity variations clearly dominate the misfit function. On the contrary, ε only has very

small influence on the misfit function. A joint inversion starting with velocity fields far away from the

true velocity will easily cause wrong ε updates and lead the inversion into a local minimum (e.g., models

indicated as λvp = 0 and λε = 0). Thus, we developed a hierarchical strategy of inverting for velocity

first for VTI medium, and only starts joint velocity and ε inversion when velocity model is good enough.

In another words, we should only start joint inversion when λvp is close to 1.

Our second synthetic example is for the VTI SEAM model. The model is 40 km wide and 15 km in

depth. Similar as the Marmousi test, we used a total of 401 shots with shot interval of 100 m. Receivers

76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014

0.0 8.5 17.0

0.0

3.5

Distance (km)

Dep

th (

km

)

17.00.0

3.5

Dep

th (

km

)

0.0 8.5

0.0 8.5 17.00.0

3.5

Distance (km)

0.0

3.5

0.0 8.5 17.0

1500 5500 0.00 0.27

(a) (b)

(c) (d)

-0.5 0.0 0.5 1.0 1.50.0

1.0 1.0

2.0

3.0

4.0

5.0

λvp

λε

Mis

fit

( )

( )

500 500

500 500

k sm k true sm

p p vp p p

k sm k true sm

v v v v

ε

λ

ε ε λ ε ε

= + −

= + −

(e)

Figure 1 True velocity (a) and epsilon (b) model; 500 m smoothed velocity (c) and epsilon (d) model;(e) misfit function sensitivity analysis.

are located 25 m apart at 10 m depth. The starting models were the smoothed version of the true models

(500 m smoothing for velocity and 2,000 m smoothing for ε). In addition, velocity was reduced further

by 10%, and ε is further reduced by 20% in the starting model fields. We start the inversion from a

bandwidth centered at 3 Hz. Figure 2 shows the comparison between the inversion results derived from

simultaneous inversion and from the hierarchical inversion developed in this study. Consistent with

the sensitivity analysis that was previously discussed, simultenous inversion is quickly trapped into a

local minimum. The working inversion strategy is to use our hierarchical approach to start the inversion

with velocity only. Only starting to invert for ε when velocity model is approaching convergence. The

comparison of the depth profiles for the inversion results reveals reliable model updates are obtained

from our hierarchical multiparameter inversion approach.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Distance (km) Distance (km) Distance (km) Distance (km)

Dep

th (

km

)D

epth

(k

m)

0.0

0.015.0

15.0

0.0 30.0 0.0 30.0 0.0 30.0 0.0 30.0

1500 4500 0.00 0.14Velocity (m/s) Epsilon

2106

2808

3510

4212

4914

Vel

oci

ty (

m/s

)

0 1500 3000 4500 6000 7500 9000 10500 12000 13500 15000

Depth (m)

0.00

0.04

0.08

0.12

0.16

Ep

silo

n

0 1500 3000 4500 6000 7500 9000 10500 12000 13500 15000

Depth (m)

(i)

(j)

Figure 2 True velocity (a) and epsilon (b) model; starting velocity (c) and epsilon (d) model; finalvelocity (e) and epsilon (f) model derived from simultaneous inversion; final velocity (g) and epsilon(h) model derived from our hierarchical multiparameter inversion. Depth profiles for velocity (i) andepsilon (j) models; green - true model, blue - starting model, red - simultaneous inversion, and purple -hierarchical approach.

Real Data Examples

Our real data example is from Gulf of Mexico. The data is acquired with a dual-coil marine acquisition

technique that provides full azimuth and long offset (up to 14 km) coverage of the survey area. The

source is at 10 m depth. Receivers are slanted from 12 m to 40 m depth. The initial velocity model used

76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014

in this study was built using simple velocity analysis instead of a full tomographic inversion. Smooth

anisotropic parameter fields were generated based on knowledge of regional geology. Figure 3 shows

the overlay of the depth slice of the starting migration image on top of the starting ε model and the final

migration image on top of the inverted ε model. Significant image improvements are achieved after

running our multiscale FWI inversion. Detailed structures revealed in ε model are in lined with the final

migration image.

(a) (b)Epsilon

0.10

0.01

Figure 3 Depth slice of the starting migration image overlaying on top of the starting epsilon model (a)and the final migration image overlaying on top of the final epsilon model (b).

Conclusions

We developed a hierarchical multiparameter inversion strategy for acoustic VTI medium with surface

seismic data. We demonstrate our inversion strategy using synthetic and real data examples. Our results

show that incorporation of multiparameter FWI improves migration of large offset full azimuth broad

band acquisition data and produces better focused migration images.

References

Alkhalifah, T. [2000] An acoustic wave equation for anisotropic media. Geophysics, 65(4), 1239–1250.Alkhalifah, T. and Tsvankin, I. [1995] Velocity analysis for transversely isotropic media. Geophysics, 60(5), 1550–

1566.Duveneck, E., Milcik, P., Bakker, P.M. and Perkins, C. [2008] Acoustic VTI wave equations and their application

for anisotropic reverse-time migration. 78th SEG Annual Meeting, Expanded Abstract, 27, 2186–2190.Gholami, Y., Brossier, R., Operto, S., Ribodetti, A. and Virieux, J. [2013a] Which parameterization is suitable

for acoustic vertical transverse isotropic full waveform inversion? part 1: Sensitivity and trade-off analysis.Geophysics, 78(2), R81–R105.

Gholami, Y., Brossier, R., Operto, S., Prieux, V., Ribodetti, A. and Virieux, J. [2013b] Which parameterization issuitable for acoustic vertical transverse isotropic full waveform inversion? part 2: Synthetic and real data casestudies from valhall. Geophysics, 78(2), R107–R124.

Plessix, R.E. [2006] A review of the adjoint-state method for computing the gradient of a functional with geo-physical applications. Geophysical Journal International, 167(2), 495–503.

Plessix, R.E. and Cao, Q. [2011] A parametrization study for surface seismic full waveform inversion in an acous-tic vertical transversely isotropic medium. Geophysical Journal International, 185(1), 539–556.

Prieux, V. et al. [2011] On the footprint of anisotropy on isotropic full waveform inversion: the valhall case study.Geophysical Journal International, 187(3), 1495–1515.

Vigh, D., Jiao, K., Watts, D. and Sun, D. [2014] Elastic full waveform inversion application utilizing multi-component measurements of seismic data collection. Geophysics, accepted.

Vigh, D., Kapoor, J., Moldoveanu, N. and Li, H. [2011] Breakthrough acquisition and technologies for subsaltimaging. Geophysics, 76(5), WB41–WB51.

Wang, C., Yingst, D., Bloor, R. and Leveille, J. [2012] VTI waveform inversion with practical strategies: Appli-cation to 3D real data. 82th SEG Annual Meeting, Expanded Abstract, 31, 1–6.

Warner, M. et al. [2013] Anisotropic 3D full-waveform inversion. Geophysics, 78(2), R59–R80.