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Subsurface imaging via fully coupled elastic wavefield extrapolation

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2007 Inverse Problems 23 73

(http://iopscience.iop.org/0266-5611/23/1/004)

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INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 23 (2007) 73–98 doi:10.1088/0266-5611/23/1/004

Subsurface imaging via fully coupled elastic wavefieldextrapolation

Murthy N Guddati1 and A Homayoun Heidari2

1 Department of Civil, Construction and Environmental Engineering, North Carolina StateUniversity, Raleigh, NC, USA2 Atlantia Offshore Limited, Houston, TX, USA

E-mail: mnguddat@ncsu.edu and hheidari@atlantia.com

Received 9 June 2006, in final form 20 September 2006Published 5 December 2006Online at stacks.iop.org/IP/23/73

AbstractWe develop a new space-domain wavefield extrapolation technique forsubsurface imaging in heterogeneous elastic media. Unlike the existing space-domain techniques, which separately propagate pressure and shear waves, theproposed method simulates one-way propagation with strong coupling betweenthe pressure and shear waves, thus resulting in more accurate images. This isachieved by combining downward continuation ideas with the concepts ofthe recently developed arbitrarily wide-angle wave equations (Guddati M N2006 Comput. Methods Appl. Mech. Eng. 195 63–93). In this method, one-way propagation is modelled by attaching a virtual halfspace to each depthstep, which is represented by special complex-length finite elements. Theparameters of the method, namely the lengths of the special finite elements,are chosen such that the wavefield extrapolation is stable and accurate. Theaccuracy of the proposed method is illustrated by its impulse response, whichaccurately captures the pressure, shear and head wave fronts. Application tovarious synthetic imaging problems, in both homogeneous and heterogeneousdomains, further confirms the effectiveness of the method.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Imaging by sounding the subsurface with high frequency waves is common in various fieldssuch as geophysical prospecting (seismic imaging) and ultrasonic nondestructive evaluation(NDE). In many situations, the surface of the object to be imaged is excited to generate elasticwaves that propagate towards inhomogeneities and hidden objects (e.g. flaws, geophysicalevents) and reflect back to the surface. The wavefield measured at the surface, often referred toas the surface trace, is processed to map the material properties of the subsurface. In general,

0266-5611/07/010073+26$30.00 © 2007 IOP Publishing Ltd Printed in the UK 73

74 M N Guddati and A H Heidari

this imaging process consists of up to two interlaced inverse problems. The first probleminvolves the estimation of the singular (discontinuous) part of material properties, with knownor assumed regular part (background material properties). Cases with unknown regular part(e.g. seismic imaging) require the solution of an additional inverse problem, which involvesiterative estimation of the regular part, utilizing the solution operator to the first problem (forprecise mathematical exposition of the overall inversion process see e.g. Stolk and de Hoop(2001)). Our paper is limited to the first inverse problem, namely estimation of the singularpart based on a given regular part, such as the known background material properties in NDEproblems, or the estimated background velocity field in seismic imaging.

The first step in estimating the singular part is to perform Born approximation of thescattering operator about the regular part, which results in a linear relationship between thesingular part and the surface trace. This relationship, often called the modelling operator,needs to be inverted to obtain the singular part from the data. For imaging applications, theinverse can be replaced by the adjoint of the modelling operator, since only the location of thesingular part is of importance, and not the magnitude. The adjoint operator essentially involvesthe propagation of sources and receivers simultaneously into the subsurface and applying someform of correlation (an imaging condition) to obtain the location of the discontinuities. Thisprocess is called migration in the context of seismic imaging and synthetic aperture focusingtechnique (SAFT) in the context of NDE.

Ideally, the migration/SAFT techniques must consider the propagation of pressure, shearand interfacial waves that are strongly coupled due to the heterogeneity of the elastic solidbeing imaged (such as the earth). However, most of the migration/SAFT techniques considerthe wave propagation to be acoustic (scalar) by assuming (1) decoupling of the modes,(2) treating only pressure waves as useful data and (3) considering data from other sourcesas noise (see e.g. Mora (1987)). The main point of attraction in acoustic migration is itsbeing a scalar problem, which simplifies many things, from concepts to formulation toimplementation. This is the reason why acoustic migration is still the dominant industrystandard. However, because of this approximation, much useful information in the surface traceis lost. Using all the data from pressure, shear, and converted waves, and solving the migrationin a coupled manner, provides more accurate images and additional useful information aboutthe subsurface (see e.g. Yilmaz (2001) for a thorough description of the points of strength inelastic migration). In this paper, we propose an elastic migration technique that accuratelyconsiders the strong coupling between various wavemodes that occur in heterogeneous elasticmedia.

As with the case of acoustic migration, there are various ways of performing elasticmigration. It can be performed either by using ray-based methods (e.g. Kirchhoff migration),or by wave-equation-based methods, with the latter being more robust and economical,especially in highly heterogeneous media (Ritchie 2003). Wave-equation-based elasticmigration may be performed in two forms: (1) reverse time migration (based on the fullelastic wave equation, see e.g. Chang and McMechan (1987)), or (2) using elastic one-waywave equations (OWWEs). OWWEs are hyperbolic equations in the depth variable, henceamenable to efficient implementation as wavefield extrapolation (downward continuation usingdepth stepping). Moreover, they prevent unwanted reflections and artifacts in the imageas they only allow propagation in the direction of wavefield extrapolation. Although theformulation and implementation of exact elastic OWWEs in frequency–wavenumber domainare straightforward and relatively trouble-free, they are limited to laterally homogeneousmedia. In order to treat heterogeneities, the computation must be performed either in spacedomain or in dual domain (interlacing space and wavenumber domains). Wu and co-workersfollowed the latter approach, where they devised the so-called complex screen method (see

Subsurface imaging via fully coupled elastic wavefield extrapolation 75

e.g. Wu (1994), Xie and Wu (2001), Xie and Wu (2005)). The method attains high efficiencyby performing mode-conversion computations in space domain, followed by propagationcomputations in wavenumber domain. While wide-angle corrections can be incorporated intopropagation computations (Xie and Wu 2005), mode-conversion computation is limited tonarrow-angle approximation; the effect of this limitation on imaging process is unclear. Inthis paper, we attempt to circumvent this limitation, but by resorting to purely space-domaincomputations.

OWWEs, in order to be implemented in space domain, must be approximated. Whileeffective space-domain OWWEs are developed for acoustics and applied widely to migrationproblems, the development of accurate approximation of elastic OWWEs has been largelylimited to range-stepping problems, mostly in the context of underwater acoustics. Landersand Claerbout (1972), McCoy (1977), Wales and McCoy (1983), Hudson (1980), Coroneset al (1982) and Greene (1985) are among the researchers who have developed differentversions of elastic OWWEs. These formulations were later enhanced for stability andaccuracy by Wetton and Brooke (1990), Collins (1991), Milinazzo et al (1997) and Frederickset al (2000). Except for the low-order elastic OWWE developed by Hudson (1980), allthe elastic OWWEs mentioned above are formulated in terms of displacement potentials ordisplacement derivatives, and tend to be extremely complex when applied to heterogeneousmedia. Furthermore, these OWWEs are limited to range-stepping problems and not readilyapplicable to migration problems.

With the goal of obtaining OWWEs that are amenable to space-domain migration ingeneral heterogeneous anisotropic elastic media, Guddati (2006) developed a new class ofhigh-order one-way wave equations called arbitrarily wide-angle wave equations (AWWEs).The basic idea behind the derivation of AWWE is to emulate one-way propagation aswave propagation in a fictitious halfspace that is efficiently discretized using special finiteelements. While AWWEs’ approximability is comparable to that of existing elastic OWWEs,the important difference is that AWWEs have broader applicability. As opposed to theexisting elastic OWWEs, AWWEs are completely in terms of displacement components andare applicable to virtually any complex media where the full wave equation is second order inspace (e.g., elastic, poro-elastic and visco-elastic media). We have already utilized AWWEs todevelop a new acoustic migration technique (Guddati and Heidari 2005, Heidari 2005). In thispaper, we extend the concepts of AWWEs and develop a wavefield extrapolation algorithmfor frequency–space elastic migration.

Since the derivation of the AWWE relies on the discretization of a fictitious halfspace,our approach in developing an effective AWWE-based wavefield extrapolation method is tointegrate the discretization required for wavefield extrapolation with that used in derivingthe equations themselves. This integrated approach avoids possible complications associatedwith inconsistent discretizations. The details of this integrated approach are discussed insection 2.

Another complication arising in the adaptation of AWWE to elastic migration is stability.It is well known from ocean acoustics literature that elastic OWWEs are generally unstablein their original form (see e.g. Yevick and Thomson (2000)). Different methods have beenproposed for the stabilization of high-order parabolic equations developed in the contextof range-stepping problems (see e.g. Wetton and Brooke (1990), Collins (1991), Milinazzoet al (1997)). In section 3, we study the stability properties of elastic AWWE extrapolationand develop a simple approach for adapting the stabilization method proposed by Milinazzoet al (1997). We show the effectiveness of this stabilization method for high-order elasticAWWE by directly studying the AWWE approximation of wavenumbers for propagating andevanescent wavemodes.

76 M N Guddati and A H Heidari

(a) (b) (c)

(d) (e)

(g)

(f)

Figure 1. Schematic diagram of steps involved in formulation of wavefield extrapolation usingelastic AWWE.

Finally, in section 4, we illustrate the performance of stabilized high-order elastic AWWEextrapolation using numerical impulse response of the method, along with various syntheticmigration problems including single flat and dipping reflectors in homogeneous media and adipping interface in a heterogeneous media.

2. Formulation

The methodology for formulating and implementing the wavefield extrapolation using elasticAWWE is schematically summarized in figure 1, and can be explained as follows. Weconsider an elastic halfspace, as shown in figure 1(a), with the preferred direction of one-way propagation being the positive z axis (pointing downwards). In order to implement thedownward continuation, we discretize the domain into finite element layers in the z direction,as shown in figure 1(b). At this point, the goal is to propagate the wavefield from one depth,

Subsurface imaging via fully coupled elastic wavefield extrapolation 77

e.g. z1, to the next, e.g. z2, using one-way propagation. In order to simulate one-waypropagation of the wavefield from z1 to z2, we attach a halfspace to the [z1, z2] layer, as shown infigure 1(c). The material properties of the halfspace are chosen identical to those of the layer,implying that there are no reflections at the interface between the layer and the halfspace,thus ensuring one-way propagation. The key idea of AWWE wavefield extrapolation is tosolve this modified problem, still governed by the full wave equation, in an efficient manner.Here, we closely follow the ideas behind the original development of AWWE (see Guddati(2006), which contains proofs of various claims made in the rest of the paragraph). The stepsbehind developing the proposed wavefield extrapolation method can be summarized as follows.(1) The halfspace is discretized into an infinite number of finite-element layers with linearshape functions, henceforth referred to as half-space elements (figure 1(d)). This discretizationresults in errors in representing the halfspace, causing reflections at the element interfaces,shown by small upward arrows. (2) By using midpoint integration rule for evaluating thediscretized matrices for these half-space elements, we completely eliminate the discretizationerror in that there are no upward reflections. Thus, the discretized halfspace in figure 1(e) isnow equivalent to the actual half-space figure 1(c). Note that, due to the elimination of thediscretization error, we can use arbitrary thickness for the half-space elements, yet achieveperfect one-way propagation. However, there are still an infinite number of elements torepresent the halfspace and the problem is not computationally tractable. The third (3) stepwould be to truncate the discretized halfspace and apply the Dirichlet boundary condition atthe truncation point, as shown in figure 1(f). This truncation results in strong reflections, thuscompletely spoiling the desired one-way propagation property. This problem is fixed in thefinal step (4), where the lengths of the half-space elements are made imaginary (or complex),which aids in significantly reducing the amount of reflection, thus representing the one-waypropagation more accurately. In fact, it can be shown that each element of length Lj is aperfect propagator for wave modes with wavenumber equal to kj = 2i/Lj . Based on thisobservation, one can immediately conclude that real-length half-space elements accuratelyrepresent evanescent waves (since kj is imaginary). However, our goal is to represent thepropagating modes, which is achieved by choosing Lj to be imaginary. These imaginary- orcomplex-length elements, integrated using midpoint rule, can be designed to represent one-way propagation in an arbitrarily wide range of angles and are called the AWWE elements.The AWWE elements are essentially constructed by choosing reference wavenumbers kj forwhich we want exact propagation, and assigning Lj = 2i/kj for these elements as shownin figure 1. It should be noted that, in addition to being exact when the wavenumber equalskj , AWWE effectively approximates other wavenumbers in the neighbourhood of kj . Hence,by having a small number of AWWE elements with their parameters scattered along thedesired range of wavenumbers, one can accurately represent the one-way wave propagation.In summary, wavefield extrapolation using AWWE is achieved by solving the problem infigure 1(g), governed by the full wave equation, where the displacement at z1 is given and thegoal is to obtain the displacement at z2. The details of the solution procedure are presented inthe rest of the section.

We start with the equations of two-dimensional elastodynamics in the frequency domain:

∂

∂z

(Gzz

∂u∂z

)+

∂

∂x

(Gxz

∂u∂z

)+

∂

∂z

(GT

xz

∂u∂x

)+

∂

∂x

(Gxx

∂u∂x

)+ ρω2Iu = 0, (1)

where u = {u v}T is the displacement vector, Gij are coefficient matrices associated withmaterial moduli, ρ is the material density and I2 is the 2 × 2 identity matrix. The material

78 M N Guddati and A H Heidari

property matrices are defined as

Gzz =[E33 E23

E23 E22

], Gxz =

[E13 E12

E33 E23

]and Gxx =

[E11 E13

E13 E33

], (2)

where Eij are the material moduli in various directions. We limit our treatment to isotropicmedium, for which the components of the above matrices are

E11 = E22 = λ + 2µ, E12 = λ, E13 = E23 = 0, E33 = µ, (3)

where λ and µ are Lame moduli, which are related to pressure and shear wave velocities asfollows:

cp =√

λ + 2µ

ρand cs =

õ

ρ. (4)

Since the AWWE procedure is based on the finite element method involving a variationalformulation, we start by multiplying equation (1) by a virtual displacement δu and integratingin the z direction:∫ zmax

z1

δuT

[∂

∂z

(Gzz

∂u∂z

)+

∂

∂x

(Gxz

∂u∂z

)+

∂

∂z

(GT

xz

∂u∂x

)

+∂

∂x

(Gxx

∂u∂x

)+ ρω2Iu

]dz = 0. (5)

The above must be true for any δu vanishing at z = z1 and z = zmax (because Dirichletboundary conditions are applied on the displacement u). Integrating the first and third termsby parts, and noting that δu = 0 on the boundaries, we obtain∫ zmax

z1

[−∂δu

∂z

T

Gzz

∂u∂z

+ δuT ∂

∂x

(Gxz

∂u∂z

)− ∂δu

∂z

T

GTxz

∂u∂x

+ δuT ∂

∂x

(Gxx

∂u∂x

)+ δuT ρω2Iu

]dz = 0. (6)

The domain is now discretized into finite elements, with the ith element defined as [zi, zi+1].The elements are numbered such that i = 1 denotes the physical element and i = 2, . . . , m+ 1correspond to the m AWWE elements. With such discretization, the integral in (6) can bedivided into integrals over each of the finite elements:

m+1∑i=1

{∫ zi+1

zi

[−∂δu

∂z

T

Gzz

∂u∂z

+ δuT ∂

∂x

(Gxz

∂u∂z

)− ∂δu

∂z

T

GTxz

∂u∂x

+ δuT ∂

∂x

(Gxx

∂u∂x

)+ δuT ρω2Iu

]dz

}= 0. (7)

Following the Bobnov–Galerkin approach (see e.g. Becker et al (1981)), we approximateu and δu using shape functions,

u(z) ≈ N(z)

{ui

ui+1

}, δu(z) ≈ N(z)

{δui

δui+1

}, z ∈ [zi, zi+1]. (8)

In the above, N is the shape function matrix associated with linear approximation within thefinite elements, i.e.,

N(z) = 1

zi+1 − zi

I2 ∗ [zi+1 − z z − zi], (9)

Subsurface imaging via fully coupled elastic wavefield extrapolation 79

where the ∗ operator is the reversed Kronecker product defined as follows: An×n ∗ Bm×m resultsin a matrix, Cmn×mn, which is obtained by replacing each element of B by a submatrix bij A,

e.g., assuming that B is a 2 × 2 matrix, we can write An×n ∗ B2×2 = [b11A b12Ab21A b22A

]2n×2n

.

Substituting the approximations in equation (8) into equation (7), we obtain

m+1∑i=1

{δui

δui+1

}T

Si

{ui

ui+1

}= 0, (10)

where Si is the stiffness matrix of the ith element and is given by

Si =∫ zi+1

zi

−∂N∂z

T

Gzz

∂N∂z

+ NT ∂

∂x

(Gxz

∂N∂z

)− ∂N

∂z

T (GT

xz

∂

∂x

)N

+ NT ∂

∂x

(Gxx

∂

∂x

)N + ω2NT ρN

dz. (11)

It can be shown that for any arbitrary 2 × 2 matrix A,

∂NT

∂zA

∂N∂z

= 1

(zi+1 − zi)2A ∗

[+1 −1−1 +1

],

NT A∂N∂z

= 1

(zi+1 − zi)2A ∗

[−(zi+1 − z) (zi+1 − z)

−(z − zi) (z − zi)

],

∂N∂z

T

AN = 1

(zi+1 − zi)2A ∗

[−(zi+1 − z) −(z − zi)

(zi+1 − z) (z − zi)

],

NT AN = 1

(zi+1 − zi)2A ∗

[(zi+1 − z)2 (zi+1 − z)(z − zi)

(zi+1 − z)(z − zi) (z − zi)2

].

(12)

Nodal point integration is used for the physical element (i = 1), which results in the followingstiffness matrix:

Sphysical = − 1

�zGzz ∗

[+1 −1−1 +1

]+

1

2

(∂

∂xGxz

)∗

[−1 +1−1 +1

]− 1

2

(GT

xz

∂

∂x

)∗

[−1 −1+1 +1

]

+�z

2

(ρω2I2 +

∂

∂x

(Gxx

∂

∂x

))∗ I2. (13)

For the AWWE elements (i > 1) representing the halfspace, midpoint integration is used,which results in the following element stiffness matrix:

Sj

AWWE = − 1

Lj

Gzz ∗[

+1 −1−1 +1

]+

1

2

(∂

∂xGxz

)∗

[−1 +1−1 +1

]− 1

2

(GT

xz

∂

∂x

)∗

[−1 −1+1 +1

]

+Lj

4

(ρω2I2 +

∂

∂x

(Gxx

∂

∂x

))∗

[+1 +1+1 +1

], (14)

where Lj is the length of the AWWE element. Noting that the lengths are linked to thereference wavenumbers by Lj = 2i/kj , we have

Sj

AWWE = ikj

2Gzz ∗

[+1 −1−1 +1

]+

1

2

(∂

∂xGxz

)∗

[−1 +1−1 +1

]− 1

2

(GT

xz

∂

∂x

)∗

[−1 −1+1 +1

]

+i

2kj

(ρω2I2 +

∂

∂x

(Gxx

∂

∂x

))∗

[+1 +1+1 +1

]. (15)

Substituting the expressions for the stiffness matrices in equations (14) and (15) inequation (10), noting that δu1 = δum+1 = um+1 = 0, and following the manipulations

80 M N Guddati and A H Heidari

involved in finite element assembly (see e.g. Becker et al 1981), we have

δu2

δu3

...

δum

T

×

i

2Gzz ∗

−2i

�z

2i

�z+ k1 −k1

−k1 k1 + k2. . .

. . .. . . −km−1

−km−1 km−1 + km

+1

2

(GT

xz

∂

∂x+

∂

∂xGxz

)∗

−1 0 +1

−1 0. . .

. . .. . . +1−1 0

+i

2

(ρω2I2

+ ∂∂x

(Gxx

∂∂x

)) ∗

0 −i�z + 1/k1 1/k1

1/k1 1/k1 + 1/k2. . .

. . .. . . 1/km−1

1/km−1 1/km−1 + 1/km

×

u1

u2

u3

...

um

= 0. (16)

Since u1,2 are the physical variables, while u3,...,m+1 are auxiliary variables introduced byAWWE elements, we use the notation uz = u1 for the known displacement, uz+�z = u2

for the displacement to be computed, and u1,...,m−1 = u3,...,m+1 for the unknown auxiliaryvariables. With this notation, transferring the known displacement uz to the right-hand side,we rewrite equation (16) as

Gzz ∗ Λ1 +1

2

(GT

xz

∂

∂x+

∂

∂xGxz

)∗ Λ2

+

(ρω2I2 +

∂

∂x

(Gxx

∂

∂x

))∗ Λ3

U =

−1

�zGzz

+1

2

(GT

xz

∂

∂x+

∂

∂xGxz

) U′, (17)

where

Λ1 = i

2

2i

�z+ k1 −k1

−k1 k1 + k2. . .

. . .. . . −km−1

−km−1 km−1 + km

, (18)

Subsurface imaging via fully coupled elastic wavefield extrapolation 81

Λ2 =

0 +1

−1 0. . .

. . .. . . +1−1 0

, (19)

Λ3 = i

2

−i�z + 1/k1 1/k1

1/k1 1/k1 + 1/k2. . .

. . .. . . 1/km−1

1/km−1 1/km−1 + 1/km

, (20)

U =

uz+�z

u1

u2

...

um

and U′ =

uz

0...

0

. (21)

Equation (17) represents wavefield extrapolation using the mth-order elastic AWWE, andis referred to as the elastic AWWEm extrapolation; it essentially extrapolates the wavefieldfrom depth z to depth z + �z by allowing only one-way propagation. The accuracy of theone-way approximation in (17) is determined by the number of AWWE layers (m), and thecorresponding reference wavenumbers (k1,...,m).

Equation (17) is continuous in x and must be discretized to facilitate numericalimplementation. We follow a finite element approach similar to that employed for thediscretization in z direction (for the physical layer). At this point, for problems involvingunbounded aperture such as in seismic imaging, we truncate the aperture at x = xmin andx = xmax to enable numerical computation, with Dirichlet boundary conditions applied at thetruncation points. The AWWE extrapolation equation (17) is then multiplied by the virtualdisplacement δU and integrated over the (truncated) aperture:

∫ xmax

xmin

δUT

(Gzz ∗ Λ1 + ρω2I2 ∗ Λ3) +

(1

2GT

xz

∂

∂x

)∗ Λ2

+

(1

2

∂

∂xGxz

)∗ Λ2 +

∂

∂x

(Gxx

∂

∂x

)∗ Λ3

U dx

=∫ xmax

xmin

δUT

− 1

�zGzz

+1

2

(GT

xz

∂

∂x+

∂

∂xGxz

) U′ dx. (22)

Performing integration by parts on the third and fourth terms on the left-hand side, as well asthe last term on the right-hand side, and noting that δU vanishes at the boundaries, we have

∫ xmax

xmin

δUT (Gzz ∗ Λ1 + ρω2I2 ∗ Λ3)U

+ δUT

(1

2GT

xz ∗ Λ2

)∂U∂x

− ∂δU∂x

T (1

2Gxz ∗ Λ2

)U

− ∂δU∂x

T

(Gxx ∗ Λ3)∂U∂x

dx

82 M N Guddati and A H Heidari

=∫ xmax

xmin

−δUT1

�zGzzU′

+ δUT1

2GT

xz

∂U′

∂x

− ∂δU∂x

T 1

2GxzU′

dx = 0. (23)

We now discretize in the x direction using the following piecewise linear approximation:

u(x) ≈ N(x)

{uq

uq+1

}, δu(x) ≈ N(x)

{δuq

δuq+1

}, x ∈ [xq, xq+1], (24)

where

N(x) = 1

LqI2 ∗ [xq+1 − x x − xq], (25)

and Lq is the length of the finite element q in the x direction.Repeating the procedure followed for the z direction, i.e. splitting the integral in

equation (23), substituting equation (24), utilizing nodal point integration, and eliminating thediscretized virtual field variable, we obtain the fully discrete version of AWWE extrapolation,which is formally written as

D1 BT1

B1 D2 BT2

B2. . .

. . .

. . .. . . BT

n−1

Bn−1 Dn

U1

U2

...

Un−1

Un

=

R10 R1

+1

R1−1

. . .. . .

. . .. . . Rn−1

+1

Rn−1−1 Rn

0

U′1

U′2...

U′n−1

U′n

. (26)

The left-hand side coefficient matrix is in a block tridiagonal form with the blocks defined as

Dj = 1

2

(Gzz

j + Gzzj+1

) ∗ Λ1 + ρω2I2 ∗ Λ3 +1

4�x

(GT

xzj − GT

xzj+1 − Gxz

j + Gxzj+1

) ∗ Λ2

− 1

�x2

(Gj

xx + Gj+1xx

) ∗ Λ3, (27)

Bj = − 1

4�x

(GT

xzj+1 + Gj+1

xz

) ∗ Λ2 +1

�x2Gj+1

xx ∗ Λ3.

The blocks on the right-hand side coefficient matrix are given as

Rj

0 = −1

2�z

(Gzz

j + Gzzj+1

)+

1

4�x

(GT

xzj − GT

xzj+1 − Gxz

j + Gxzj+1

),

Rj

+1 = 1

4�x

(GT

xzj + Gxz

j),

Rj

−1 = −1

4�x

(GT

xzj + Gxz

j).

(28)

The fully discrete wavefield extrapolation in equation (26) can be used either in prestackor poststack migration settings. In this paper, we incorporate the extrapolation into standardfrequency domain post-stack migration setting using the following steps: (a) the surface traceis Fourier transformed in time; (b) the trace is then downwards continued, to the bottom ofthe domain, for all the desired frequencies using equation (26); (c) once the wavefield isobtained for all the frequencies and depths, the image, which is the displacement at zero time,

Subsurface imaging via fully coupled elastic wavefield extrapolation 83

is computed using

u(x, z; 0) =∫ ∞

−∞u(x, z;ω) dω. (29)

Note that the above is simply the inverse Fourier transform specialized for t = 0. Qualitatively,the above integral is equivalent to the sum of the displacements resulting from downwardcontinuation at different frequencies3.

3. Stability and accuracy

Since evanescent waves are not critical to seismic migration, and because choosing complexreference wavenumbers (kj ) tends to degrade AWWE’s accuracy for propagating waves (seethe numerical experiments in Guddati (2006)), it appears best to choose real kj . However,when real kj were used for AWWE wavefield extrapolation in equation (26), instabilitieswere observed in the form of exponential growth. These instabilities were not grid-related,but stem from the approximations associated with AWWE. In this section, we stabilize theelastic AWWE extrapolation and investigate possible implications for the accuracy of one-waypropagation.

Due to the Dirichlet boundary conditions employed at the aperture truncation points, thehalfspace shown in figure 1(c) turns into a waveguide fixed on both sides, in which the one-way propagation must be implemented. Many researchers in the ocean acoustics communityhave studied one-way propagation in elastic waveguides extensively, with most one-wayapproximations being based on rational approximation of the square-root operator. They foundthat typical rational approximations lead to instabilities, and suggested various techniques torectify the problem (see e.g. Yevick and Thomson (2000)). As AWWE approximation isclosely linked to the rational approximation of the square-root operator (Guddati 2006), weborrow the ideas from the ocean acoustics community, specifically the branch-cut rotationidea of Milinazzo et al (1997), to stabilize the elastic AWWE extrapolation technique. Thefollowing are the details of the stabilization procedure.

Accurate explanation of the waveguide instability lies in the understanding of the modalproperties of an elastic waveguide. It is well known that an elastic waveguide supportsa finite number of propagating modes with real wavenumbers, and an infinite number ofevanescent modes with imaginary or complex wavenumbers. The wavenumbers can beobtained by solving the transcendental dispersion equation associated with the waveguide(see e.g. Achenbach (1987)). Alternatively, for a waveguide discretized in the directionperpendicular to the propagation direction, the wavenumbers can be obtained by solving thequadratic eigenvalue problem (see e.g. Tassoulas and Kausel (1983)). Figure 2 shows anexample of the distribution of wavenumbers for a certain elastic waveguide (in the complexwavenumber (k)-plane), which are obtained by solving the quadratic eigenvalue problem. Theinset of the figure shows the portion of the graph around the origin to illustrate the distributionof complex wavenumbers.

It can be seen from figure 2 that the full elastic wave equation has wavemodes thatpropagate and decay in both directions. The same waveguide governed by an exact elasticone-way wave equation should ideally contain only one side of the real axis in figure 2 alongwith all the complex wavenumbers on either top or bottom half-plane, depending on the

3 This might not result in accurate amplitude in the image. However, imaging methods based on one-way waveequations generally do not propagate the amplitude accurately to begin with, hence any concern about the amplitudeof the image at this point is immaterial.

84 M N Guddati and A H Heidari

Figure 2. An example distribution of wavenumbers of an elastic waveguide in the complex plane.The inset shows the distribution of complex wavenumbers around the origin. The waveguideproperties, i.e., wave guide width, pressure wave velocity, Poisson’s ratio, and number of elementsin horizontal discretization, are l = 2000 m, cp = 4000 m s−1, ν = 0.35, n = 200, respectively.The frequency is f = 20 Hz.

preferred direction of the one-way wave equation. For example, the exact downward (towardspositive z) propagating one-way wave equation should only allow wave modes for which

0 � θk = tan−1

( �(k)

�(k)

)< π, (30)

where �(k) and �(k) are real and imaginary parts of the wave mode k, respectively. Notethat the above range does not include θk = π , which corresponds to upward propagatingwaves. The wavenumber distribution for the exact one-way wave equation is shown infigure 3. In contrast with the exact OWWE, many approximate OWWEs, including the AWWE,do not effectively capture these wavenumbers. This is illustrated by plotting the wavenumbersassociated with typical AWWEs along with the wavenumbers of exact OWWE (see figure 4).Clearly, there are significant errors in the approximation of imaginary and complexwavenumbers. This is not a significant concern as evanescent energy is often unimportantin migration. A more problematic issue, however, is that some of the wavenumbers mappedby AWWE have negative imaginary part, indicating spurious exponential growth with depth.Similar problems exist for OWWEs used in ocean acoustics, which are based on rationalapproximations of the square-root operator (see e.g. Wetton and Brooke (1990)). Again,this similarity between AWWE and rational-approximation-based OWWEs is not surprisingas AWWEs can be considered as generalization of rational-approximation-based OWWEs(Guddati 2006), which forms our basis for stabilization of AWWE extrapolation.

With the ultimate goal of stabilizing the elastic AWWE, we start by reviewing the existingstabilization techniques for rational-approximation-based OWWEs. The reason for unstable

Subsurface imaging via fully coupled elastic wavefield extrapolation 85

Figure 3. Wavenumber distribution for the same waveguide as in figure 2, governed by the exactdownward (towards positive z) propagating one-way wave equation.

mapping can be explained by examining the wavenumbers in the k2-plane (see e.g. Wettonand Brooke (1990)). Rational approximations with real parameters, which are employed inthe original development of acoustic OWWE, tend to map the bottom half of the k2-plane tothe bottom half of the k-plane, resulting in instabilities. This problem has been addressed ina number of research works (e.g. Wetton and Brooke (1990), Collins (1991), Milinazzo et al(1997), Lu (1998), Yevick and Thomson (2000)). Considering its simplicity, and a reasonabletradeoff between efficiency and accuracy, we employ the method proposed by Milinazzo et al(1997) to stabilize elastic AWWE. Milanazzo et al’s method is based on the observation thatreal-parameter rational approximations effectively use the negative real line in the k2-planeas the branch cut of the square-root operator, causing the bottom half of the k2-plane to bemapped onto the bottom half of the k-plane. In order to ensure that the wavenumbers in thek2-plane get mapped onto the top half of the k-plane, they suggested a simple rotation ofthe branch cut. Specifically, they proposed the following approximation:

ApproxMilinazzo

(√k2

) = eiα/2 · Approxreal

(√e−iαk2

), (31)

where the Approxreal refers to real-parameter rational approximation. Equation (31) has thesimple effect of rotating the branch cut in the k2-plane by α in the counterclockwise direction.The angle α is thus chosen such that all the wavenumbers in the third quadrant of the k2-planelie above the branch cut (see figures 5 and 6). While Milinazzo et al’s idea is simple andelegant, there is a complication associated with the fact that the square root is approximated.Due to this approximation, the branch cut is no longer a straight line, but a wavy line as shownin figure 7. On a more encouraging note, the real branch cut becomes more and more straightas the order of approximation is increased, indicating that the wavenumber distribution ink2-plane can be used as a guide for choosing α. Ultimately, however, it is important to chooseα such that the real (wavy) branch cut falls below all the wavenumbers in the third quadrantof the complex k2-plane. Of course, the higher the rotation angle, the higher the order ofapproximation needed to accurately map the propagating wave modes.

86 M N Guddati and A H Heidari

(a) (b)

(d)(c)

Figure 4. Exact and AWWE-approximated wavenumbers in k-plane for f = 5 Hz and realAWWE coefficients. (a) AWWE2, (b) AWWE6, (c) AWWE10 and (d) AWWE20. Dots are theexact wavenumbers while circles are the approximations.

Applying Milinazzo et al’s stabilization to AWWE, we have

AWWEstabilized(√

k2) = eiα/2 · AWWEreal kj

(√e−iαk2

), (32)

where AWWEstabilized is the desired stabilization after branch-cut rotation in the k2-plane,while AWWEreal kj

is the approximation with real reference wavenumbers kj . Note thatAWWEstabilized as defined in equation (32) involves three operations: (a) rotating the argumentk2 in the complex plane, (b) applying AWWEreal kj

and (c) rotating the resulting approximationin the k-plane. These operations, while appearing simple, cannot be applied explicitly inthe AWWE setting shown in figure 1. Fortunately, it turns out that the application ofAWWEstabilized is simply achieved by rotating the reference wavenumbers in the complexplane. More precisely, AWWE is stabilized by choosing kj = eiα/2kj . This claim is proved bythe following short argument. AWWEreal kj

is exact whenever the argument√

e−iαk2 = kj , or

equivalently, when k = eiα/2kj . This implies that AWWEstabilized is a rational approximationthat is exact for all k = eiα/2kj . Noting that AWWE is exact whenever k is equal to oneof the reference wavenumbers, we conclude that AWWEstabilized is simply a regular AWWEapproximation, but with complex reference wavenumbers kj = eiα/2kj .

The stabilization of AWWE facilitated by the rotation of kj is illustrated here for thewaveguide with wavemode distribution shown in figure 2. From the distribution of the

Subsurface imaging via fully coupled elastic wavefield extrapolation 87

Figure 5. Wavenumber distribution for the same wave guide as in figure 2, shown in the k2-plane.Wavenumbers in the marked area are those that should be mapped carefully, to avoid growingmodes in the one-way waveguide.

Figure 6. Definition of the branch-cut selection. Here figure 5 is zoomed around the origin. Therequired branch-cut rotationa angle (αbc), as can be seen from the schematic line in the figure, isabout 30◦. All of the wave modes with negative imaginary parts will lie above this line.

wavemodes in the k2-plane (see figure 5), one can see that the theoretically required branchcut is approximately α = 30◦ (see figure 6). Figures 8–11 show AWWE mapping for fourdifferent values of α for AWWEs of order 2, 6, 10, and 20, respectively. As can be seen from

88 M N Guddati and A H Heidari

Figure 7. The actual branches of AWWE approximation of order 2, 4, and 8, each plotted for 0◦,45◦ and 90◦ rotation.

figure 8, for AWWE2, all of the rotations have resulted in spurious growing modes, indicatingthat the order of AWWE is not high enough for α = 30◦. Figure 9 shows AWWE6 mapping withdifferent angles of rotation. It can be seen that AWWE6 with 90◦ rotation seems to eliminateall the growing modes. Additionally, it seems to map almost all of the real wavenumbers andmost of the complex wavenumbers properly, except for a few wavenumbers around the originand complex wavenumbers with imaginary value higher than 0.1. The mapping is clearlyimproved in figure 10(d), where AWWE10 seems to accurately map most of the wavenumbersshown in the figure. It can also be seen, from figure 11, that results of AWWE20 are very

Subsurface imaging via fully coupled elastic wavefield extrapolation 89

(a) (b)

(d)(c)

Figure 8. Exact and AWWE-approximated wavenumbers in k-plane (f = 20 Hz), using AWWE2.(a) α = 10◦, (b) α = αbc = 30◦, (c) α = 45◦ and (d) α = 90◦. Dots are the exact wavenumberswhile circles are the approximations.

close to that of AWWE10. Based on a computer-assisted parametric study, it is concludedthat AWWE of order 6–10, with a rotation of α = 90◦, can be considered a conservativelyaccurate choice for almost all the practical values of waveguide elastic properties. This is alsoverified in section 4, where AWWE8 with α = 90◦ is found effective even for heterogeneousmedia.

It is observed, both in our study and in the literature of underwater acoustics (seee.g. Yevick and Thomson (2000)) that branch-cut rotation does not completely eliminatethe negative imaginary values of the mapped wave modes, i.e., some of the propagatingwavemodes have small negative imaginary parts of the wavenumbers. These modes, althoughanalytically too slow to grow significantly for practical dimensions, could get amplified dueto numerical errors associated with discretization. Our numerical studies confirm this issue,as some of the cases with negligible imaginary value (on the order of 10−10) show growthas a function of grid size. These modes are known to propagate almost perpendicularly tothe direction of the waveguide (Yevick and Thomson 2000). Therefore, the amplification ismostly due to entrapment of these modes in the waveguide. In fact, our observations on thepattern of growth confirm this issue (not shown here). Based on this observation, it may bepossible to alleviate the grid-related growth by using absorbing boundary conditions at the

90 M N Guddati and A H Heidari

(a) (b)

(d)(c)

Figure 9. Exact and AWWE-approximated wavenumbers in k-plane (f = 20 Hz), using AWWE6.(a) α = 10◦, (b) α = αbc = 30◦, (c) α = 45◦ and (d) α = 90◦. Dots are the exact wavenumberswhile circles are the approximations.

aperture truncation points. However, it turned out that such boundary conditions were notnecessary; the growth is almost always suppressed when the grid size is equal to or less thanwhat is needed for propagation accuracy. This value is roughly one-tenth of the wave length ofthe slowest shear wave in the domain (see e.g. Marfurt (1984)). In other words, if the highestcyclic frequency to be considered is fmax and the slowest shear wave velocity is cs

min, a gridsize satisfying

�z � csmin

10fmax(33)

does not show any growth associated with the above-mentioned phenomenon. Therefore, froma practical point of view, the grid-related instability does not seem to impose any additionalconstraints on the method.

4. Results

The accuracy of the stabilized elastic AWWE was studied semi-analytically in the previoussection by examining the mapping properties of the AWWE in the k-plane. To assess theaccuracy of numerical implementation of the elastic AWWE as formulated in equation (26),

Subsurface imaging via fully coupled elastic wavefield extrapolation 91

(a) (b)

(d)(c)

Figure 10. Exact and AWWE-approximated wavenumbers in k-plane (f = 20 Hz), usingAWWE10. (a) α = 10◦, (b) α = αbc = 30◦, (c) α = 45◦ and (d) α = 90◦. Dots are theexact wavenumbers while circles are the approximations.

we obtained its impulse response, which is shown in figure 12. The material properties ofthe domain are described in the caption. The impulse response is calculated using stabilizedAWWE4. By comparison with the exact impulse response, overlaid on the figure, it can beseen that both shear and pressure wave fronts are captured accurately for almost the entirerange of the dip angle (entire semi-circle). The head wave is also captured, which indicatesproper coupling between the modes.

A flat reflector in a homogeneous domain is studied next. Figure 13 shows the layoutof the reflector, along with the tensile source that is used for the exploding reflector model(Loewenthal et al 1976) to obtain the surface trace. Figure 14 shows both components of thesurface trace (u and v) obtained from the forward model, with the details of the forward modeldescribed in the caption. The surface trace is transformed into frequency domain and usedfor elastic AWWE migration. The resulting image using AWWE4 is shown in figure 15. Arotated AWWE with α = 90◦ is used for stability. For the sake of accuracy of propagation,and to prevent grid-related instability, the depth step is chosen as 2.5 m.

It can be seen from figure 15 that the reflector is accurately imaged and the downwardcontinuation is stable. A very low amplitude noise can be seen in the bottom of the imagewhich is believed to be related to wave modes with small negative imaginary values (see thediscussion at the end of section 3). However, as discussed in the previous section, these modes

92 M N Guddati and A H Heidari

(a) (b)

(d)(c)

Figure 11. Exact and AWWE-approximated wavenumbers in k-plane (f = 20 Hz), usingAWWE20. (a) α = 10◦, (b) α = αbc = 30◦, (c) α = 45◦ and (d) α = 90◦. Dots are theexact wavenumbers while circles are the approximations.

are suppressed by the right choice of grid size satisfying the requirement in equation (33). Itshould be noted that since the image is obtained at t = 0, the top of the pulse will be locatedon the actual position of the reflector, as is the case with this image.

A dipping reflector in a homogeneous medium, as shown in figure 16, is studied next. Thereflector has a dip of approximately 50◦. A vertical excitation in the form of an initial conditionon v is used in the forward (exploding reflector) model. The components of the surface traceare shown in figure 17. AWWE8, with a rotation of α = 90◦ for stabilization, is used to imagethe surface trace. The resulting image, obtained on a grid of (dx = 5 m) × (dz = 1 m), isshown in figure 18. Aside from a small amount of dispersion, which originates from boththe surface trace and the relatively large horizontal grid size, the image is located right on thetarget.

Finally, in order to assess the performance of the elastic AWWE migration forheterogeneous media, we apply the method to the model shown in figure 19. The modelconsists of two different materials (properties are described in the figure caption), with flatand dipping interfaces. The exploding reflector model with vertical excitation is used toobtain the surface trace shown in figure 20. As marked on the figure, both pressure and shearwave fronts originated from the dipping interface can be seen in the surface trace. However,

Subsurface imaging via fully coupled elastic wavefield extrapolation 93

Figure 12. Impulse response of AWWE6 in a homogenous elastic medium, using a smoothedimpulse on the surface trace. Material properties are cp = 4000 m s−1, cs = 1950 m s−1,ν = 0.35, ρ = 2048 kg m−3. The dotted lines are exact pressure and shear impulse responses.

tensile source for exploding reflector

Figure 13. The reflector layout for a single flat reflector in a homogeneous elastic medium.The material properties are as follows: cp = 4000 m s−1, cs = 1950 m s−1, ν = 0.35, ρ =2048 kg m−3. The inset shows the tensile source used on the reflector for the exploding reflectormodel to obtain the surface trace.

the flat reflectors, due to vertical excitation, only result in a pressure wave front. Figure 21shows the image obtained with AWWE8 and 90◦ rotation. The downward continuation iscompletely stable in this example, which confirms the conservative choice of the rotationangle and the order of AWWE. Relatively large horizontal grid spacing of dx = 8 m is usedin this model for computational efficiency. Naturally, this choice results in some dispersionon the dipping interface as can be seen in figure 18. Downward continuation step of dz = 1 mis chosen in order to satisfy the requirements for accuracy and grid-related stability. Thisnumerical experiment confirms the applicability of the proposed stabilized elastic-AWWEimaging method to heterogeneous media.

94 M N Guddati and A H Heidari

Figure 14. The components of the surface trace for the flat reflector in a homogeneous elasticmedium. The surface trace is obtained by forward finite-difference solution of the full elastic waveequation for the exploding reflector model. The grid parameters are dx = 2.5 m, dz = 2.5 m, dt =0.25 ms.

Figure 15. The image resulting from elastic AWWE imaging, using rotated (α = 90◦) AWWE4. Itcan be seen that the image is accurately on the target, both in terms of position and length. Imagingis performed on the (dx = 8) × (dz = 5) grid. Frequencies up to 40 Hz have been considered inthis image.

5. Summary and conclusions

We have developed a new elastic migration technique based on wavefield extrapolationusing arbitrarily wide-angle wave equations (AWWEs), which are high-order elastic one-waywave equations amenable to imaging applications. AWWE extrapolation is achieved by anintegrated procedure that combines the concepts of wavefield extrapolation and the derivationof AWWE. The procedure involves (a) discretization of the full elastic wave equation indepth, (b) simulation of one-way propagation by attaching a halfspace to each depth layer,(c) effectively representing the halfspace using special finite elements with complex lengthsand (d) discretization in the horizontal direction using regular finite elements. While theentire formulation is presented in terms of finite element discretization, the implementationcan be viewed as equivalent to the finite difference method commonly employed in standardwave-based migration methods.

Subsurface imaging via fully coupled elastic wavefield extrapolation 95

Figure 16. The reflector layout for a single dipping reflector in a homogeneous elastic medium.The material properties are cp = 4000 m s−1, cs = 1950 m s−1, ν = 0.35, ρ = 2048 kg m−3.Vertical excitation on the reflector is used for the exploding reflector model to obtain the surfacetrace.

Figure 17. The components of the surface trace for the dipping reflector in a homogeneous elasticmedium. The surface trace is obtained by forward finite-difference solution of the full elastic waveequation for the exploding reflector model. The grid parameters are dx = 5 m, dz = 5 m, dt =0.25 ms.

Due to the improper mapping of complex wavenumbers by real-coefficient AWWEs,similar to other real-coefficient elastic one-way wave equations, AWWE extrapolation resultsin spurious growing modes that transform into numerical instabilities. Starting with the ideaof rotating the branch cut proposed by Milinazzo et al (1997), we show that AWWE can bestabilized by simply rotating its parameters (reference wavenumbers) in the complex plane.This stabilization is verified for homogeneous media analytically, by studying the AWWEmapping of wave modes, as well as numerically, by applying to various migration problems.We have also shown, through numerical examples, that conservative choice of branch-cutrotation angle and AWWE order stabilizes the extrapolation in heterogeneous media as well.

96 M N Guddati and A H Heidari

Figure 18. The image resulting from elastic AWWE imaging, using rotated (α = 90◦) AWWE8.It can be seen that the image is accurately on the target, both in terms of position and length.Imaging is performed on the (dx = 5 m) × (dz = 1 m) grid. Frequencies up to 50 Hz have beenswept for this image.

Figure 19. Velocity field for the heterogeneous model. Material I has a pressure wave velocity of,cp = 2500 m s−1, while in Material II cp = 4000 m s−1. Other properties, which are constant inthe medium, are ν = 0.35, ρ = 2048 kg m−3. Vertical excitation at the interfaces between the twomaterials is used for the exploding reflector model.

The accuracy of the proposed elastic AWWE migration is illustrated by verifying itsimpulse response, and by applying it to imaging flat and dipping events in homogeneous andheterogeneous media.

While the current study illustrates the effectiveness of elastic AWWE migration, it appearsthat several improvements and extensions would be desirable. They include: optimizing theAWWE layers and the branch-cut rotation to achieve stability and accuracy with minimumcomputational cost, implementation of prestack elastic AWWE migration and application ofAWWE wavefield extrapolation for material property mapping. These extensions are subjectsof ongoing and future studies.

Subsurface imaging via fully coupled elastic wavefield extrapolation 97

Figure 20. Components of the surface trace obtained form the heterogeneous model shown infigure 19. Both pressure and shear wave events from the inclined surface in the model can be seenin the figure. Events A and B are the first arrivals of pressure and shear waves, while events C andD are their corresponding second arrivals. Note that since the excitation is only vertical, flat eventsdo not result in any shear wave fronts.

Figure 21. The image of the model in figure 19, using AWWE8 with 90◦ rotation, and a grid of(dx = 8 m)× (dz = 1 m). The inclined interface shows slight dispersion, which is due to both theforward model (see figure 20) and relatively large horizontal grid spacing. The conservative choiceof AWWE order (8) and 90◦ rotation, as discussed in section 3, stabilizes AWWE extrapolation inthis heterogeneous media.

Acknowledgments

This material is based upon work supported by the National Science Foundation under grantno. CMS-0100188. Any opinions, findings and conclusions or recommendations expressed inthis material are those of the authors and do not necessarily reflect the views of the National

98 M N Guddati and A H Heidari

Science Foundation. The authors wish to thank Professor William Symes (Editor-in-Chief)for his suggestions for improving the presentation.

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