Waveform tomography of two-dimensional three-component ......Pure and Applied Geophysics (2018) Gao...

Pure and Applied Geophysics (2018) Gao et al. Waveform tomography of two-dimensional three-component seismic data

1

Waveform tomography of two-dimensional

three-component seismic data for HTI anisotropic

media

Fengxia Gao 1,2, Yanghua Wang 2 and Yun Wang 1

1 School of Geophysics and Information Technology, China University of Geosciences (Beijing), 100083, China

2 Centre for Reservoir Geophysics, Department of Earth Science and Engineering, Imperial College London, UK.

Abstract–Reservoirs with vertically aligned fractures can be represented equivalently by HTI

(horizontal transverse isotropy) media. But inverting for the anisotropic parameters of HTI media is a

challenging inverse problem, because of difficulties inherent in a multiple parameter inversion. In

this paper, when we invert for the anisotropic parameters, we consider for the first time the azimuthal

rotation of a two-dimensional seismic survey line from the symmetry of HTI. The established wave

equations for the HTI media with azimuthal rotation consist of nine elastic coefficients, expressed in

terms of five modified Thomsen parameters. The latter are parallel to the Thomsen parameters for

describing velocity characteristics of weak VTI (vertical transverse isotropy) media. We analyze the

sensitivity differences of the five modified Thomsen parameters from their radiation patterns, and

attempt to balance the magnitude and sensitivity differences between the parameters through

normalization and tuning factors which help to update the model parameters properly. We

demonstrate an effective inversion strategy by inverting velocity parameters in the first stage and

updates the five modified Thomsen parameters simultaneously in the second stage, for generating

reliably reconstructed models.

Key words: HTI media, seismic anisotropy, seismic tomography, waveform inversion.

1. Introduction

Fractures in reservoirs act as migration channels and storage spaces, and therefore the description

and recognition of fractures plays a key role in hydrocarbon exploration. Considering the compaction

effect from the overlying strata, horizontal fractures or fractures with lower angle nearly disappear,

while vertical or near-vertical fractures are relatively easy to conserve. Theoretically, a model with

one set of vertically aligned fractures can be equivalent to a horizontal transverse isotropic (HTI)

model. The most common physical explanation for HTI media is a system of parallel vertical cracks

(Figure 1a), with quasi-circular shapes (like pennies), embedded in an isotropic background

(Crampin 1985; Thomsen 1988; Tsvankin 1997; Grechka et al. 2006). The composite seismic

response of the fractured model is equivalent to the response for an HTI model, which may be simply

described as an anisotropic anomaly embedded within an isotropic background (Figure 1b).

Considering fracture characteristics can be evaluated using anisotropic parameters, the estimation of

the anisotropic parameters is consequently necessary for fractured reservoirs.

Azimuthal variation of seismic reflection waves can provide valuable information about the

anisotropy associated with natural fracture systems. P-wave azimuthal moveout analysis based on the

normal-moveout ellipse (Grechka and Tsvankin 1998, 1999; Al-Dajani and Alkhalifah 2000) is

effective in predicting the dominant fracture orientation (Lynn et al. 1999; Tod et al. 2007).


2

Amplitude variation with offset/angle and azimuth analyses can achieve a much higher vertical

resolution than a traveltime related method, since reflection coefficients are determined by the elastic

properties on both sides of an interface (Tsvankin et al. 2010). To avoid the non-uniqueness in the

inversion, converted S-wave data of different azimuth angles can be combined with the P-wave data

for parameter estimation, such as crack density and fluid indicator in HTI media (Liu et al. 2012;

Zhao et al. 2012; Pan et al. 2016). But the P-wave and S-wave separation as well as relative S-wave

data processing make it difficult in field data application. Ultimately, the multi-component seismic

data with different azimuth angles is helpful to suppress the non-uniqueness in multiple parameter

inversion.

Figure 1. (a) An anisotropic model with vertically aligned fractures. (b) An equivalent HTI model.

An efficient method of involving the multi-component data in multiple parameter inversion is

seismic waveform tomography (Tarantola 1984, 1986; Gauthier et al. 1986; Ravaut et al. 2004; Wang

and Rao 2006; Brossier et al. 2009; Sourbier et al. 2009; Wang and Rao 2009; Rao et al. 2016). After

years of development, waveform tomography has been extended to include the characterization of

anisotropic media. Most relative research studies have concerned the VTI media with an acoustic

assumption (Rao and Wang 2009; Plessix and Cao 2011; Gholami et al. 2013; Cheng et al. 2016).

There are also some works related to VTI models without the acoustic constraint, including the

implementation of parameter sensitivity analysis for different parameterizations (Kamath and

Tsvankin 2013, 2016) as well as the inversion of the elastic coefficients in the stiffness matrix (Lee et

al. 2010). However, little waveform tomography literature has been published concerning the HTI

media, and the research that does exist was implemented in the intrinsic coordinate system without

considering the azimuthal influences (Pan et al. 2016).

There are at least two difficulties in waveform tomography regarding HTI media. One such

difficulty is that the wave equations for simulating wave propagation in HTI media are more

complex than in VTI media (Tsvankin 1997). It is because, for HTI model, the azimuth dependence

of velocities and amplitudes should be considered. When the azimuth angle is taken into account,

elastic stiffness coefficients in the survey coordinate system should be firstly derived from the

intrinsic coordinate system through Bond transform before wavefield simulation in 2D cases. In the

following section we show that the number of nonzero elastic coefficients in stiffness matrix

increases from five to nine after we apply the Bond transform. After the transform, the wave

equations in HTI media are also more complicated than those in an intrinsic coordinate system.

Another difficulty is the implementation of multiple parameter inversion. The influence between

the parameters can induce parameter crosstalk, wherein the parameters which play a dominant role in

the simultaneous inversion will influence those not sensitive to the objective function (Operto et al.

2013; Pan et al. 2016). To suppress this crosstalk effect between different parameters, one may

choose proper parameterizations by analyzing the radiation patterns (Tarantola 1986; Alkhalifah and

Plessix 2014; Kamath and Tsvankin 2016; Oh and Alkhalifah 2016; Pan et al. 2016) or by


3

implementing singular value decomposition to the sensitivity matrix (Wang and Pratt 1997; Kamath

and Tsvankin 2013). One may also precondition the gradient using the approximate Hessian matrix

or exact Hessian matrix to decrease the influences between the parameters (Pratt et al. 1998; Métivier

et al. 2015; Pan et al. 2016). However, the computation cost for a Hessian matrix may be

unaffordable for processing seismic field data. Apart from the aforementioned methods, the subspace

method is also a choice for multiple parameter inversion (Kennett et al. 1988; Wang and Houseman

1994, 1995; Wang 2016) wherein the parameters are divided into different parameter classes. One

may also choose to balance the differences of different parameters using a tuning factor to make the

misfit function decrease along the optimal composite gradient direction (Wang 1998; Gao and Wang

2016). Adding constraints, such as Total Variation regularization, to the misfit function is also

helpful for reducing leakage of imprints between different parameters (Ramos-Martínez et al. 2017).

Meanwhile, proper inversion strategies, such as a multi-scale inversion strategy or inverting one

parameter by one parameter sequentially (Waheed et al. 2016), are also helpful for multiple

parameter inversions. To sum up, for multiple parameter inversion, selecting proper

parameterizations and inversion strategies will contribute to an effective reconstruction of the

anisotropic parameters, which leads to an investigation of combinations of the above-mentioned

strategies in this paper.

For the HTI media, we approximate the elastic stiffness coefficients by Thomsen parameters,

which are parallel to the Thomsen parameters for describing velocity characteristics in VTI media.

We use a time-domain waveform tomography from two-dimensional three-component (2D3C)

seismic data, to invert the Thomsen parameters for HTI media. We arrange this paper in the

following sequence. First, we derive the stiffness matrix in a survey coordinate system through Bond

transformation, and establish a set of wave equations for HTI media (section 2). Then, we present the

theory of a shot-encoded waveform tomography, including gradient calculation as well as schemes to

balance the differences of different parameters in the inversion (section 3). After the sensitivity

analysis of the radiation patterns (section 4), we compare two strategies for multiple parameter

inversion (section 5). Finally, we also discuss the influence of azimuth angels to the multiple

parameter inversion (section 6).

2. Wavefield Simulation in HTI Media

The elastic coefficients that are convenient to use in forward-modeling algorithms are not

necessarily well suited for application in seismic processing and inversion (Tsvankin et al. 2010). In

this study, we adopt Thomsen parameters for HTI media and derive the stiffness coefficients used in

forward modeling from these Thomsen parameters (Tsvankin 1997).

In multi-azimuth multi-component seismic data acquisition, the survey line may be not in

accordance with the symmetry axis for HTI media (Figure 2). For a two dimensional inversion, it is

difficult to invert seismic data on the survey line by using wave equations defined in the intrinsic

coordinate system (blue rectangle in Figure 2), when the survey line is not inside this system. It is

therefore necessary to establish wave equations in the survey coordinate system containing the

survey line and to implement the inversion using these equations. In Figure 2, we define the elastic

coefficients in stiffness matrix that describe ordinary HTI media in the intrinsic coordinate system

(x1, x2, x3), where the axis of symmetry of the HTI media is parallel to the x1 axis. The intrinsic

coordinate frame is related to the survey frame (x, y, z) by a clockwise rotation with the azimuth

angle about x3-axis. We obtain the elastic coefficients in the survey coordinate system from those

in intrinsic coordinate system via Bond transformation. Then, we are able to establish the relations

between Thomsen parameters and wave equations in HTI media.

Considering that in conventional HTI media, the relationships between the elastic coefficients ijc

in the intrinsic coordinate system and Thomsen parameters are (Rüger 1997; Tsvankin 1997)


4

( )( )

.

,21

,

,)21(

,)21(

55

)(44

33

)(

13

)(

11

vc

v

c

vc

vvvvvc

vc

2

vS

E

2

vS

2

vP

2

vS

2

vS

2

vP

2

vS

2

vP

E

2

vP

E

=

+=

=

−−−+=

+=

(1)

where ),,,,( )()()(vvEEE

SP vv are the Thomsen parameters for HTI media (Tsvankin 1997). Among these five parameters, vPv is the P-wave velocity polarized in the isotropic plane, with the

subscription ‘v’ indicating this plane perpendicular to the symmetrical axis, while vSv is the

SV-wave velocity polarized in the symmetry axis plane, with ‘v’ representing polarization normal to

the cracks plane. Therefore, they are fast velocities. The rest ),,( )()()( EEE , with superscript ‘(E)’

indicating the equivalence, describe the anisotropic velocity characteristics (Thomsen 1986).

Figure 2. The intrinsic coordinate system (x1, x2, x3) and the survey coordinate system (x, y, z). The azimuth angle, , is the rotation angle between the two coordinate systems. The blue rectangle represents the intrinsic coordinate system, and

two directions are depth direction and symmetrical direction, respectively. Thomsen parameters for HTI media is defined

in this coordinate system. The red rectangle is the survey coordinate system, where the survey line is paralleling to the

x-direction. Four survey lines are displayed at the surface. The blue pentagrams represent the receivers (R) and the red

complex shape is the source (S). In 2D3C seismic data inversion, only seismic data recorded at the purple line is

involved. The angle between the survey line and symmetry axis of the fractures is .

To simplify equation (1), we scale the three anisotropic parameters among the five conventional

Thomsen parameters for HTI media to ,21~ )()( EE += ,21~ )()( EE += )()( 21~ EE += , and they are

referred to as modified Thomsen anisotropic parameters. In this way, all the five parameters will be

positive, while the original three anisotropic parameters are in the range of (–0.2, 0] in most weak

anisotropic cases (Rüger 1997; Tsvankin 1997). Then, the relationships between this given modified

Thomsen parameters and the five elastic coefficients for HTI media are


5

( )( )

.

,~

,

,~

,~

55

)(44

33

)(

13

)(

11

vc

v

c

vc

vvvvvc

vc

2

vS

E

2

vS

2

vP

2

vS

2

vS

2

vP

2

vS

2

vP

E

2

vP

E

=

=

=

−−−=

=

(2)

When the survey line is not in accordance with the symmetry axis of HTI media (Figure 2), for

2D3C inversion, we need to transform the stiffness matrix in the intrinsic coordinate system to the

survey line coordinate system by Bond transformation to simulate the seismic data recorded at

survey line.

After coordinate system rotation, we can express the rotated stiffness matrix as TMCMC =' ,

where C and 'C are the stiffness matrices in the intrinsic coordinate system and survey

coordinate system, respectively, and M is the Bond transform matrix (Bond 1943),

.

2cos0002sin2

12sin

2

10cossin000

0sincos000

000100

2sin000cossin

2sin000sincos22

22

−

−

−

=

M (3)

The relationships between different stiffness coefficients for HTI media are 31211312 cccc === ,

3322 cc = , 44223223 2cccc −== , 6655 cc = (Rao & Wang 2009). Therefore, the stiffness matrix C

before coordinate rotation is

−

−

=

55

55

44

33443313

44333313

131311

00000

00000

00000

0002

0002

000

c

c

c

cccc

cccc

ccc

C . (4)

Since TCC = , '' CMCMMCMC === TTTT )(][ , the rotated stiffness matrix is a symmetric

matrix,

=

66362616

5545

4544

36332313

26232212

16131211

00

0000

0000

00

00

00

c'c'c'c'

c'c'

c'c'

c'c'c'c'

c'c'c'c'

c'c'c'c'

'C (5)

where


6

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) .2cos2sin24

1

,cossin

,2sin2

12sin

2

1

,sincos

,2sin22

12sin

2

1

,

,4sin2

1cossin2sin

2

1

,cos2sin

,2sincoscossin2sin

,4sin2

1sincos2sin

2

1

,sin2cos

,2sincossincossin

,2sinsincossin2cos

255

233131166

255

24455

554445

255

24444

44331336

3333

552

33132

131126

24433

21323

255

433

2213

41122

552

33132

131116

24433

21313

255

4413

22331112

255

433

2213

41111

ccccc'

ccc'

ccc'

ccc'

cccc'

cc'

cccccc'

cccc'

ccccc'

cccccc'

cccc'

ccccc'

ccccc'

++−=

+=

+−=

+=

−−=

=

+−+−=

−+=

+++=

−−+−=

−+=

−+++=

+++=

(6)

Note that after coordinate rotation, the number of elastic coefficients for the 3D case is increased

from 5 to 13. For the 2D case, the number of elastic coefficients is increased from 5 to 9, which are

),,,,,,,,( 665545443633161311 c'c'c'c'c'c'c'c'c' in the survey coordinate system.

Then, we can establish 2D3C wave equations for HTI media in the survey coordinate system as

.~

,

~

,

~

,~

,~

,~~1

,

~~1

,~~1

4555

4544

663616

363313

161311

z

uc'

x

u

z

uc'

t

x

u

z

uc'

z

uc'

t

x

uc'

z

uc'

x

uc'

t

x

uc'

z

uc'

x

uc'

t

x

uc'

z

uc'

x

uc'

t

fzxt

u

fzxt

u

fzxt

u

yzxzx

zxyyz

yzxxy

yzxzz

yzxxx

zzzxzz

y

yzxyy

x

xzxxx

+

+

=

+

+

=

+

+

=

+

+

=

+

+

=

+

+

=

+

+

=

+

+

=

(7)


7

where ),,( zyx uuu are particle displacement components in the x-, y- and z-directions, )~,~( zzxx

indicate the integration of normal stresses in the x- and z-directions along time, )~,~,~( zxyzxy are

the integration of shear stresses along time, is the density, and ),,( zyx fff are the source

components in the x-, y- and z-directions.

For seismic wave simulation using the 2D3C wave equations above, we apply a high-order

finite-difference method (Crase 1990), and use a rotated staggered grid scheme in which we define

the particle displacements and density at one grid, and define the time integrated stresses and elastic

coefficients on the other grid (Saenger et al. 2000; Saenger and Bohlen 2004). We employ a

convolutional perfectly matched layer (CPML) method for the absorbing boundary condition

(Komatitsch and Martin 2007; Martin and Komatitsch 2009).

3. The Inverse Theory

3.1. The Objective Function and the Gradient Vector

We adopt a shot-encoding technique in waveform tomography. In the shot-encoded waveform

tomography, we sum up individual shots with random weighting coefficients as a supershot, and thus

significantly reduce the number of forward simulations needed (Krebs and Anderson 2009;

Schiemenz and Igel 2013; Castellanos et al. 2015; Rao and Wang 2017). We define the objective

function in shot-encoded waveform tomography as

−=s t r

rrt2

2calobs )(

~~d2

1)( muum ,

(8)

where r and s indicate the receivers and shots, respectively, robs~u denotes an encoded supershot,

r

iobs,u is a shot gather for the ith shot, )(~ murcal represents the calculated wavefied from a supershot,

T

321 )~,~,~(~ uuur =u represents the x-, y- and z-components of the encoded seismic data, T means

transpose, T)()()(vv )~

,~,~,,( EEESP vv =m are the modified Thomsen parameters for HTI media to be inverted in this paper, and t is the recording time.

The gradient vector is the first-order derivatives of the objective function with respect to the

modified Thomsen parameters. The derivative with respect to vPv , for example, is

==

=

−

−

−

−=

−

−=

s t r

Ns

i

Ns

ijj

r

jcal,

r

jobs,

vP

r

ical,

ji

s t r

Ns

i

r

ical,

r

iobs,

vP

r

ical,

i

s t r

r

cal

r

obs

vP

r

cal

vP

vtd

vtd

vtd

v

1 1

T

1

T

2

T

)]([)(

)]([)(

)](~~[)(~)(

muumu

muumu

muumum

. (9)

where the first term in the encoded gradient is the conventional gradient, and the second term is the

cross-talk term, which is the cross-correlation of wavefields from different shots, introducing

artefacts in the gradient.

Instead of calculating the derivatives of the objective function with respect to the modified

Thomsen parameters directly, we derive first the derivatives to the elastic coefficients ijc' .

Subsequently, taking into account the relationships between the elastic coefficients and the modified

Thomsen parameters, we can get the derivative with respect to vPv , for example, by following the

chain rule,


8

=

, ,

)()(

k ji ijk

ij

vP

k

vP c'c

c'

v

c

v

mm , (10)

where ijc' and kc )6 , ,2 ,1, ,,( =kji are the elastic coefficients in the survey coordinate system and the intrinsic coordinate system, respectively. We summarize the three-step calculation for

the gradients in Appendix A.

3.2. Upscaling Model-Updates

The five sets of parameters include two velocity parameters },{ vv SP vv and three dimensionless

parameters }~

,~,~{ )()()( EEE δγε . To balance their magnitude and units differences, we normalize these

parameters by

,~~

~~

,~~

~~

,~~

~~

,

,

minmax

min

)()(

minmax

min

)()(

minmax

min

)()(

minmax

minvv

minmax

minv

−

−=

−

−=

−

−=

−

−=

−

−=

EE

EE

EE

SS

SSS

PP

PPPv

vv

v

vv

v

δδ

γγ

εε

vv

vv

(11)

where ),,,,( )()()(vvEEE

SP δγεvv

are the normalized Thomsen parameter vectors, minm and maxm

are the minimum and maximum values of model m . Besides the magnitude and units differences, we also take account the sensitivity differences to the

objective function during the inversion. To balance these updates to the normalized models, the

gradient sub-vectors for the rest of the parameters can simply be amplified to the parameter exerting

a dominant role in the simultaneous inversion by their corresponding positive tuning factors (Wang

2009; Gao and Wang 2016). Subsequently, together with the step length, the objective function will

decrease toward the optimal solutions along the composite gradient vectors. We express the gradients

of the five parameters after applying tuning factors as

,mtune m

Im

=

(12)

where I is the identity matrix, m

/ represents the gradient vector for the normalized Thomsen

parameters, tune/ m

indicates the gradient vector for the normalized parameters after using the tuning factors, and m refers to ),,,,( vsvp , which are the tuning factors for the modified Thomsen parameters. We estimate the tuning factor by

2main

1

2

mm

=

−

m . (13)

Here m represents one of the parameter vectors ),,,,( )()()( EEESvPv δγεvv

, 2

/ m indicates the L2-norm of the gradient sub-vector for the normalized Thomsen parameters, and

2main/ m is

the L2-norm of the gradient sub-vector for the parameter playing a dominant role in the simultaneous

inversion. For example, in the five Thomsen parameter inversion, mainm represents the vertical

P-wave velocity vPv

.

Once we have tuned the gradient, we apply an energy scaling to the gradients, in order to suppress


9

the singularity values around the shot and receiver positions. In addition, we apply a smoothing

operator to make the gradients smooth.

Conjugate gradient method in P-wave and S-wave velocity inversion and steepest-descent method

in five-parameter inversion are applied to optimize the model updates, respectively. Each

parameter in the simultaneous inversion has its own step length. We design the step length by

utilizing a parabolic step-length searching method (Vigh et al. 2009). For the ith parameter we

calculate the step length by (Köhn 2012)

)(

max,

)(

max,)(

k

i

k

ik

im

m

= , (14)

where )(ki is the step length for the ith parameter in the kth iteration, denotes the step length coefficient, )( max,

k

im

indicates the maximum value of the gradient for the ith parameter, and )( max,k

im

represents the model value for the ith parameter corresponding to )( max,k

im

. We update the models by

)()()()1( k

i

k

i

k

i

k

i mmm

+=+ , (15)

where k represents the model at the kth iteration. After an inverse calculation of equation (11), we

can retrieve the updated models.

4. Radiation Pattern Analysis

We can evaluate sensitivities of the objective function with respect to the parameters by

computing the Fréchet kernel for a point scatterer in the subsurface (Eaton and Stewart 1994;

Alkhalifah and Plessix 2014). The amplitude of the kernel as a function of the scattering angle

reveals the sensitivity of full waveform tomography to a model parameter (Kamath and Tsvankin

2016). If there are overlaps between the radiation patterns over a range of scattering angles, the

crosstalk between these parameters will influence the model updates. Therefore, it is necessary to

implement radiation pattern analysis before attempting multi-parameter inversion.

Figure 3. Reflection from a horizontal reflector, where PP- and SVP- are the angles between the normal axis and

the reflected P wave and reflected SV wave.

Following the 3D radiation patterns for a general anisotropic media (Pan et al. 2016) and using

the chain rule, we derive in Appendix B the formulas of 3D radiation patterns for the modified

Thomsen parameters in the survey coordinate system. However, because this current study tries to

invert the parameters from reflected seismic data, we present in the following section only the 2D

radiation patterns under the reflection case. Here, we assume the reflection coming from a flat

reflector, and denote the reflection angle as (Figure 3). We present the P-P and P-SV wave

radiation patterns, with respect to the perturbation of the modified Thomsen parameters explicitly in

Appendix B.


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Figure 4. The 2D radiation patterns. The blue curve is the P-P radiation pattern and red curve is the P-SV radiation

pattern. The columns from left to right correspond to the perturbation of vPv , vSv , )(~ E , )(~ E , and

)(~ E , respectively. (a) Radiation patterns at azimuth 0°. (b) Radiation patterns at azimuth 20°. (c) Radiation patterns at azimuth

45°.

We calculate the radiation patterns of the modified Thomsen parameters for a scatterer embedded

in the isotropic homogeneous background. The ratio of S-wave velocity to P-wave velocity for the

background is 0.55. Hence, following Snell’s law, for the P-P wave, the reflection angle PP- is in

the range of [0o, 90o], while for the P-SV wave in this study, the reflection angle SVP- is in the

range of [0o, 33.3o).

Figure 4 shows the radiation patterns with azimuth angles 0°, 20°, and 45°, respectively. An

azimuth angle of 0° means the survey line is parallel to the symmetry axis, and thus we can analyze

the radiation patterns in the symmetry axis plane.

From these radiation patterns, we have made a number of observations:

1) The radiation patterns with respect to two velocities, vPv and vSv , are independent from azimuthal variation. For two anisotropic parameters, )(~ E and )(~ E , the geometric shape of the radiation patterns seems independent from the azimuth angle, but the magnitude of )(~ E becomes smaller when the azimuth angle increases whereas the magnitude of )(~ E shows a

opposite trend.

2) For the third anisotropic parameter )(~ E , the radiation pattern varies with the azimuth angle. This

parameter is related to both the P-wave and S-wave velocities and is more sensitive to the azimuth

angle. This result is consistent with the conclusion of Thomsen (1986) insomuch that the

parameter controls most anisotropic phenomena, some of which are not negligible even when the

anisotropy is weak.

3) The P-P radiation pattern with respect to the P-wave velocity vPv is independent of reflection angle, and thus the overlap between the radiation pattern of vPv and the radiation pattern of other

four parameters is unavoidable.


11

4) In both P-P and P-SV radiation patterns, there is also an overlap between the radiation patterns of

vSv and )(~ E in middle offsets. The radiation energy of )(~ E is mainly concentrated in far

offset, and the overlap between the radiation patterns of )(~ E and )(~ E is not obvious. 5) As for )(

~ E , the P-P radiation energy distribution changes from middle offsets in small azimuth angle to far offset in large azimuth angle, while the energy in P-SV radiation pattern is mainly in

near and middle offsets for different azimuth angles. In both P-P and P-SV radiation patterns,

there is an overlap between the radiation patterns of )(~ E and )(~ E , whereas the overlap

between the radiation patterns of )(~ E and )(~ E only shows in large azimuth angle.

In summary, there are overlaps between the radiation patterns of the modified Thomsen parameters.

These overlaps indicate crosstalks between different parameters. Therefore, it is necessary to design

effective inversion strategies to update parameters properly in multi-parameter inversion, as

discussed in the following sections.

Figure 5. True models. (a) The P-wave velocity model. (b) The S-wave velocity model. (c) The )(~ E model. (d) The

)(~ E model. (e) The )(~ E model.

Figure 6. The starting models. (a) The P-wave velocity model. (b) The )(~ E model.


12

Figure 7. The two velocity models inverted from the first round of waveform tomography. (a) The P-wave velocity

model. (b) The S-wave velocity model.

Figure 8. The modified Thomsen models inverted by waveform tomography. The azimuth angle is 20°. (a) The P-wave

velocity model. (b) The S-wave velocity model. (c) The ( )E~ model. (d) The ( )Eγ~ model. (e) The

)(~ E model.

5. Inversion Strategies and Model Tests

For an effective time-domain waveform tomography, we adopt the multi-scale inversion strategy,

to deal with different frequencies of seismic data, and design a two-stage strategy to handle the

simultaneous inversion for multiple parameters. We demonstrate the feasibility of the inversion

strategies using the SEG/EAGE overthrust model (Aminzadeh et al. 1997; Mulder et at. 2006).

P-wave and S-wave velocities have some empirical relations (Castagna et al. 1985), so we assume

they have similar geological structures as shown in Figures 5a-b. We relate the density to the P-wave

velocity according to Gardner et al. (1974). As for the three anisotropic parameters, they are all

related to fractures, and thus we set similar geological structures in this test as shown in Figures 5c-f.


13

However, their structures are different from those for the two velocity models.

The model size is 110×200 with a space interval of 10 m. Thirty-six shots are evenly distributed at

the depth of 40 m. In this shot-encoded waveform tomography, the thirty-six shots are encoded to a

single supershot. Ninety-six receivers are set at the near surface with an interval of 20 m, and the

receivers are same for all the shots. The wavelet is a Ricker wavelet with a peak frequency of 15 Hz.

The time length for each shot gather is 1 s and the time interval is 0.001 s.

Figure 6 is the starting models for the inversion, obtained by smoothing the true model with a

smooth operator. Only the starting P-wave velocity and )(~ E models are shown as examples, since the others are obtained in the same way.

According to the sensitivity analysis, we adopt a framework of sequential multiple parameter

inversion strategy, and execute the inversion in two stages sequentially. In the first round (Figure 7),

we invert for the two velocity parameters, while we keep the anisotropic parameters unchanged as

per the starting models. In the second stage (Figure 8), we invert all the five parameters

simultaneously, by using the inverted velocity models from the first stage as the starting ones.

Figure 9. The work flowchart for the inversion process.

Figure 10. Misfit function values versus iterations in the second stage for five parameter inversion with the azimuth

angle 20°.


14

Figure 11. The modified Thomsen models inverted by conventional multi-parameter inversion method. The azimuth

angle is 20°. (a) The P-wave velocity model. (b) The S-wave velocity model. (c) The ( )E~ model. (d) The ( )Eγ~ model.

(e) The )(~ E model.

During the inversion of each stage, we divide seismic data into different frequency bands by

band-pass filtering, and implement the inversion on data from low-frequency bands to

high-frequency bands hierarchically (Wang 2011). In this multi-scale strategy, first the

low-frequency data helps to recover the large-scale background of the models, and then taking

inverted model from low-frequency band data as the starting model, the inversion of higher

frequency band data will refine the model. This strategy is similar to the discrete frequency group

inversion in frequency-domain full-waveform tomography (Brossier et al. 2009; Wang and Rao

2009; Wang 2011). In the hierarchical frequency band strategy mentioned above, there is a frequency

overlap between the nearby frequency groups. The five frequency bands are [0, 6], [5, 11], [10, 16],

[15, 21], and [20, 26] Hz in the test. The inversion process will terminate at the iteration loop when

the iteration and frequency group exceed the maximum iteration and the maximum frequency band.

Also, the cases that the relative difference of the misfit function and the step-length are smaller than

their corresponding threshold values ( ) will lead to a termination. Figure 9 shows the workflow of the inversion process. Note that the processes are same for the two inversion rounds except that only

P-wave and S-wave velocities are inverted in the first round while five parameters are updated

simultaneously in the second round. We implement 100 iterations to update the velocity models and

200 iterations for the five-parameter updating at each frequency band, respectively.

Figure 10 show the variation of the misfit function with iterations in the second stage for the

five-parameter inversion. Because of the encoded waveform tomography, the values of the misfit

function are fluctuated, but it shows a descend trend with iterations.

The conventional simultaneous inversion can simply skip the first stage in this strategy and invert


15

all the five parameters straightaway from the given initial models. If compared to the conventional

simultaneous inversion (Figure 11), we have found that this two-stage inversion strategy is able to

produce the velocity models close to the true models to some extent, and consequently to better

recover the modified Thomsen parameters in the second stage of simultaneous inversion.

Moreover, to show the feasibility of the two stage inversion strategy, Figure 12 presents the

differences of the field data and synthetic data generated from the inverted models (Figure 8).

Figures 12a, 12d, 12g are the x-component of the field data, synthetic data and their differences, and

they are plotted under the same scale. Figures 12b, 12e, 12h and Figures 12c, 12f, 12i are for the y-

and z-components, respectively, and they are also plotted under their own scales. It can be seen that

the synthetic data matches well to the field data, which means the five parameters have been properly

recovered.

Figure 12. (a, b, c) Seismic data generated from the true models. The azimuth angle is 20°. These data are the input for

waveform tomography test. (d, e, f) Calculated seismic data from inverted models. (g, h, i) Data differences. The panels

from left to right represent the x-, y-, and z-components, respectively. Note that x-component data are plotted under the

same scale. So as to y- and z-component data.


16

Figure 13. Inverted modified Thomsen models. The azimuth angle is 45°, and the inversion is implemented with the

correct azimuth angle. The description of the figures are the same as that in Figure 8.

6. Discussion on the Influence of Azimuthal Angles

Figure 13 illustrates the inverted modified Thomsen models with azimuth angle of 45°. Compared

to Figure 8, we can find that the inverted velocity models are nearly the same, while the )(~ E is contaminated by more artefacts in 45° case (model error 8.16%) than that in 20°case (model error

8.08%). We evaluate the model errors by 2true2trueinv ||||/|||| mmm − , where invm represents the

inverted model, and truem is the true model.

In Figure 13, the deep part of the inverted )(~ E model with azimuth angle of 45° is recovered better than that with 20° (Figure 8d). )(

~ E model in both azimuth angles are failed, since it is affected by P-wave and S-wave anisotropy. Single parameter, i.e. )(

~ E only, inversion with the other four parameter correct shows that )(

~ E parameter is quite sensitive to data errors, which means errors in seismic data will lead to a failure of the )(

~ E inversion. In multiple parameter inversion, the inaccuracy of the other four parameter during the inversion affects the proper update of )(

~ E model, leading to the improperly inverted )(

~ E models in Figures 8e and 13e. We also test the case when the azimuth angle used in the inversion is not correct. In this test, the

field data is generated with an azimuth angle of 45°, however, during the inversion, the azimuth

angle is set to 10°. Figure 14 illustrates the inverted Thomsen models. It is obvious that the inverted

models are poorly reconstructed (Figure 14) compared with those inverted using the correct azimuth

angle (Figures 8 and 13).


17

Figure 14. The modified Thomsen models obtained by inversion using an incorrect azimuth angle. The true azimuth

angle is 45°, but the inversion is implemented with the azimuth angle equal to 10°. Description of the figures are the

same as that in Figure 8.

7. Conclusions

Considering the angles between the intrinsic coordinate system and the survey coordinate system,

we have derived the elastic parameters in the survey coordinate system through a Bond

transformation, and constructed the 2D3C wave equations in the survey coordinate system.

Thereafter, we can invert the recorded seismic data with the proper wave equations in the survey

coordinate system.

As indicated by the radiation pattern analysis, there are crosstalks between )(~ Eδ parameter and

the other four parameters. If the velocity parameters as well as )(~ Eε and

)(~ Eγ are not accurate, the updates for )(

~ Eδ will be improper, leading to )(

~ Eδ being recovered badly.

After sensitivity analysis of the radiation patterns, we have proposed the two-stage strategy for

multiple parameter inversion. We have inverted the two velocity parameters at the first stage, and

recovered all the five modified Thomsen parameters simultaneously in the second stage. Inversion

results have demonstrated its effectiveness.

The azimuth angle is a key parameter in multi-parameter inversion of HTI media. With a properly

estimated azimuth angle, we can reconstruct the modified Thomsen models. For field seismic data,

including azimuth angle as an unknown variable in multiple parameter inversion would further

complicate the problem, so we would better estimate the azimuth angle by preprocessing, and then

invert the five anisotropic parameters firmly using the estimated azimuth angle.

The observed seismic data generated using wave equations (7) is an approximate to the 2D field

seismic data extracted from three-dimensional wide azimuth data. Therefore, data corrections, such

as traveltimes, and processing are needed before inverting the seismic data using waveform

tomography method.


18

Appendix A: The Gradients with Respect to Thomsen Parameters

Following equation (10), the gradient calculation is divided into three steps. First, using the

differential of the objective function for general anisotropic media (Kamath & Tsvankin 2016), we

derive the derivatives of the objective function, with respect to each of the nine elastic coefficients

for HTI media in survey coordinate system, as

.

~

,~~

,

~~~

,

~

,~~

,~

,~~

,~~

,~

66

55

45

44

36

33

16

13

11

−=

+

+

−=

+

+

+

−=

−=

+

−=

−=

+

−=

+

−=

−=

t

yy

t

zxzx

t

zxyzxy

t

yy

t

zyyz

t

zz

t

xyyx

t

xzzx

t

xx

tdx

u

x

w

c'

tdx

u

z

u

x

w

z

w

c'

tdx

w

z

w

z

u

x

u

z

u

z

w

c'

tdz

u

z

w

c'

tdz

u

x

w

x

u

z

w

c'

tdz

u

z

w

c'

tdx

u

x

w

x

u

x

w

c'

tdx

u

z

w

z

u

x

w

c'

tdx

u

x

w

c'

(A1)

where u~ and w are the encoded forward and backward seismic wavefields, and the subscripts x,

y, z represent the x-, y-, and z-components of u~ and w . For simplicity, equation (10) only shows

the gradient calculated using one supershot. For the case with multiple supershots, it requires a sum

of gradients over the supershots.

Secondly, exploiting relations between the two sets of coefficients , , , ,( 33161311 c'c'c'c'

) , , , , 6655454436 c'c'c'c'c' and ) , , , ,( 5544331311 ccccc , we can obtain the derivatives of the objective

function, with respect to 11c , for example, in the intrinsic coordinate system, by

,2sin4

12sincos

2

1cos

66

2

16

2

11

4

, 1111

c'c'c'

c'c

c'

c ji ij

ij

+

+

=

=

(A2)

The derivatives for the rest of the elastic coefficients, )/ ,/ ,/ ,/( 55443313 cccc , can be derived

in the same way following ,

/ ( / )( / )ij k ij kk

c c' c c' = . Finally, based on equation (2), we derive the gradients of the objective function, with respect to

the modified Thomsen parameters, as


19

( )

( )

( )( )

.~2

1

~~

,~1

~~

,~~

,1~

~1

2

1

~1

2

,))(

~(

~1

2

1~

~2

13)(

,)()(

44

2

)(

,)()(

11,)()(

13)(

)(

5544

)(

,

13)(

)()(

3311

)(

,

cvv

vvv

c

c

c

v

c

c

c

vc

c

cvvvv

vv

ccv

cv

c

v

cvvvv

vv

ccv

cv

c

v

2

vS

2

vP

E

2

vS

2

vP2

vP

kkE

k

E

E

2

vS

kkE

k

E

2

vP

kkE

k

E

2

vS

2

vP

2

vS

2

vP

E

2

vS

2

vP

E

EvS

kk vS

k

vS

2

vS

2

vP

2

vS

2

vP

E

2

vS

E2

vP

E

E

vP

kk vP

k

vP

−

−=

=

−=

=

=

=

+−−

−+

−

+

=

=

−−

+−

+

+

=

=

(A3)

Appendix B: Formulas of Radiation Patterns

The radiation pattern due to perturbation of the model parameters is generally defined as (Pan et

al. 2016; Chapman 2004)

pm

Tg ˆ

ˆ]ˆ[),,,( T

= scscscininP-αR , (B1)

where is the inclination angle of the wave, departing from the z-axis, and is defined in the x-z

plane, is the angle departing from the x-axis, and is defined in the x-y plane, the subscript ‘in’ and

‘sc’ stand for incident and scattered waves, respectively, and indicates either P or SV mode of the

reflection wave. Hence, P-PR is the P-P wave radiation pattern, and P-SVR is the P-SV wave

radiation pattern. We focus on the case with a plane P-wave incidence in this paper.

On the right-hand side of equation (B1), T̂ is the reduced equivalent moment tensor,

321332313

232212

131211

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆ tttT =

=

. (B2)

For a plane P-wave incidence, three column vectors can be expressed as


20

+

++

++

=

31553245

21662336

2116

21162313

2111

1

ˆˆ2ˆˆ2

ˆˆ2ˆˆ

ˆˆ2ˆˆ

ˆ

ppc'ppc'

ppc'pc'pc'

ppc'pc'pc'

t , (B3)

+

++

=

31453244

21662336

2116

2

ˆˆ2ˆˆ2

0

ˆˆ2ˆˆ

ˆ

ppc'ppc'

ppc'pc'pc'

t , (B4)

++

+

+

=

21362333

2113

31453244

31553245

3

ˆˆ2ˆˆ

ˆˆ2ˆˆ2

ˆˆ2ˆˆ2

ˆ

ppc'pc'pc'

ppc'ppc'

ppc'ppc'

t . (B5)

where p̂ is the slowness vector, which is in the propagation direction (Chapman, 2004),

=

=

in

inin

inin

p

p

p

cos

sinsin

cossin

ˆ

ˆ

ˆ

ˆ

3

2

1

p . (B6)

The vectors Pscĝ and SV

scĝ are polarization vectors for scattered P- and SV-wave in equation (B1)

=

=P

sc

P

sc

P

sc

P

sc

P

sc

P

P

P

P

sc

g

g

g

cos

sinsin

cossin

ˆ

ˆ

ˆ

ˆ

3

2

1

g , (B7)

−

=

=SV

sc

SV

sc

SV

sc

SV

sc

SV

sc

SV

SV

SV

SV

sc

g

g

g

sin

sincos

coscos

ˆ

ˆ

ˆ

ˆ

3

2

1

g . (B8)

Substituting equations (B2)-(B8) into equation (B1), we obtain the P-P wave radiation pattern

with respect to elastic coefficients ijc' as

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )( )

( )

( )

( ) ,ˆˆˆˆ4

,ˆˆˆˆ4

,ˆˆˆˆ4ˆˆˆˆ4

,ˆˆˆˆ4

,ˆˆˆ2ˆˆˆ2

,ˆˆ

,ˆˆˆ2ˆˆˆ2

,ˆˆˆˆ

,ˆˆ

212166

313155

3132323145

323244

21

2

3

2

32136

2

3

2

333

2

12121

2

116

2

1

2

3

2

3

2

113

2

1

2

111

ppggc'R

ppggc'R

ppggppggc'R

ppggc'R

ppgpggc'R

pgc'R

pggppgc'R

pgpgc'R

pgc'R

PP

P-P

PP

P-P

PPPP

P-P

PP

P-P

PPP

P-P

P

P-P

PPP

P-P

PP

P-P

P

P-P

=

=

+=

=

+=

=

+=

+=

=

(B9)

and the P-SV radiation pattern as


21

( )

( )

( )

( )

( )

( ) ( )( )

( ) ( )( ) ( ).ˆˆˆˆ2ˆˆˆˆ2

,ˆˆˆˆˆˆˆˆ2

),ˆˆˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆ(2

,ˆˆˆˆˆˆˆˆ2

,ˆˆˆˆ2ˆˆˆˆˆˆ

,ˆˆˆ

,ˆˆˆˆˆˆˆˆˆˆ2

,ˆˆˆˆˆˆ

,ˆˆˆ

22121266

31333155

31323231

3123321345

3232322344

2133

2

321

2

31236

2

33333

2

121

2

112211116

2

133

2

31113

2

11111

ppggppggc'R

ppggppggc'R

ppggppgg

ppggppggc'R

ppggppggc'R

ppggpggpggc'R

pggc'R

pggpggppggc'R

pggpggc'R

pggc'R

1

SVP

1

SVP

P-SV

1

SVP

1

SVP

P-SV

SVPSVP

SVPSVP

P-SV

SVPSVP

P-SV

SVPSVPSVP

P-SV

SVP

P-SV

SVPSVPSVP

P-SV

SVPSVP

P-SV

SVP

P-SV

+=

+=

++

+=

+=

++=

=

++=

+=

=

(B10)

Once we obtain the radiation patterns for coefficients ijc' , we can derive the radiation patterns for

coefficients ijc , using the chain rule, as

( )( )( )( )( )

( )( )( )( )( )( )( )( )( )

,

66

55

45

44

36

33

16

13

11

55

44

33

13

11

=

c'R

c'R

c'R

c'R

c'R

c'R

c'R

c'R

c'R

cR

cR

cR

cR

cR

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

P-α

Θ (B11)

where Θ is a 95 matrix, ][ 921 θθθΘ = . Each column vectors are

=

)2(sin

0

sin

cossin2

cos

2

4

22

4

1

θ ,

−

=

0

sin2

sin

cos

0

2

2

2

2

θ ,

( )( )

( )

( )

−

−

−

=

4sin

0

2sinsin

4sin

2sincos

21

2

21

41

2

21

3θ ,

=

0

0

1

0

0

4θ ,

−=

0

)2sin(

)2sin(

)2sin(

0

21

21

5

θ ,

=

2

2

6

sin

cos

0

0

0

θ ,

−

=

)2sin(

)2sin(

0

0

0

21

21

7

θ ,

=

2

2

8

cos

sin

0

0

0

θ ,

−

=

)2(cos

0

)2(sin

)2(sin

)2(sin

2

2

41

2

21

2

41

9

θ . (B12)


22

Then we can derive the radiation patterns for modified Thomsen parameters. Assuming the

background is an isotropic media with ( ) ,1~ =E ( ) ,1~ =E ( ) 1~

=E (Kamath & Tsvankin 2016), we

obtain the P-P and P-SV radiation patterns for the modified Thomsen parameters as

( )

( )

( )( )

( )

,)(

)(

)(

00010

01000

00001

11020

00111

)~

(

)~(

~

55

44

33

13

11

)(

)(

)(

cR

cR

cR

cR

cR

-

D

R

R

R

vR

vR

P-α

P-α

P-α

P-α

P-α

E

P-α

E

P-α

E

P-α

vSP-α

vPP-α

−

=

(B13)

where } , , ,2 ,2diag{ 2v212

v2

vvv PSPSP vvvvvD = . The subscript ‘ -P ’ in equations (B11) and (B13) represents either the P-P mode or the P-SV mode.

Focusing on the effect of the azimuth angles (shown in Figure 4), the radiation patterns in

equation (B13) are normalized by D . Therefore, only the S-wave to P-wave velocity ratio is needed

for the calculation of radiation patterns which depends on the incident/reflection angles. The

radiation patterns shown in Figure 4 is the 2D case with in sc 0 = = .

Acknowledgements

This research is partly funded by China Postdoctoral Science Foundation (no. 2016M601080),

and the National Natural Science Foundation of China (no. 41704136 and 41425017). The authors

are also grateful to the sponsors of the Centre for Reservoir Geophysics, Imperial College London,

for supporting this research.

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