Wavefield extrapolation in caustic-free normal ray coordinates

KAUSTRepository

Item type Conference Paper

Authors Ma, Xuxin; Alkhalifah, Tariq Ali

Eprint version Publisher's Version/PDF

DOI 10.1190/segam2012-0104.1

Publisher Society of Exploration Geophysicists

Journal SEG Technical Program Expanded Abstracts 2012

Downloaded 6-Dec-2017 19:28:08

Link to item http://hdl.handle.net/10754/593745

Wavefield extrapolation in caustic-free normal ray coordinatesXuxin Ma and Tariq Alkhalifah, King Abdullah University of Science and Technology

SUMMARY

Normal ray coordinates are conventionally constructed fromray tracing, which inherently requires smooth velocity profiles.To use rays as coordinates, the velocities have to be smoothedfurther to avoid caustics, which is detrimental to the mappingprocess. Solving the eikonal equation numerically for a linesource at the surface provides a platform to map normal raysin complex unsmoothed velocity models and avoid caustics.We implement reverse-time migration (RTM) and downwardcontinuation in the new ray coordinate system, which allows usto obtain efficient images and avoid some of the dip limitationsof downward continuation.

INTRODUCTION

Reverse-time migration (Whitmore, 1983) and downward con-tinuation (Claerbout, 1985) are two commonly used tools forprestack depth imaging in complex geology. Conventionally,the two-way and one-way wave equations are solved on reg-ularly sampled cartesian grids. Cartesian coordinates are fa-vored because its straightforward discretization. However, itsusefulness is limited because the wavefront rarely follows x, yor z axis. A coordinate system that follows rays and wavefrontshas the benefit that wavefronts have nearly zero curvature.

Alternative coordinate frames have been proposed, for exam-ple semi-orthogonal Riemannian coordinates (Sava and Fomel,2005), which is able to follow rays with overturning angles.However, in complex geology, this method requires excessivesmoothing of the velocity model to avoid zero-division at raycaustics. Nonorthogonal Riemannian coordinates (Shragge,2008) does not require much smoothing, but the complexityof differential operators in nonorthogonal coordinate systemsincreases the cost for solving the wave equation. Another ap-proach based on vertical time instead of traveltime is free fromcaustics, however, the coordinates do not follow wavefronts instrong lateral velocity variation (Alkhalifah et al., 2001).

In this abstract, we suggest normal ray mapping that is un-conditionally stable in complex media as it uses direct eikonalsolvers to develop ray directions. This new coordinate sys-tem, which is an orthogonal curvilinear coordinate system, al-lows us to efficiently and accurately apply downward continu-ation and RTM. We show impulse responoses to point and linesources that demonstrate the potential of the approach.

THEORY

Normal ray coordinates

The position of a point in space may be determined by its carte-sian coordinates x or any other curvilinear coordinates u, forexample u = [ρ θ φ ]T for spherical coordinates. For a fan of

rays u1 and u2 are the take-off angle, u3 could be the travel-time or arclength. For normal rays u1 and u2 are the x and ycoordinates of shotpoint on the surface. In isotropic media, theray coordinates system is orthogonal because ray trajectoriesare always normal to the wavefronts.

Conventionally, ray coordinates are formed by solving the kine-matic ray tracing system

dxdτ

= v2p,

dpdτ

=−∇vv, (1)

where x is a vector defining the location of a point on the wave-front, τ is the traveltime and p = ∇τ is slowness vector, defin-ing the direction of the ray.

One common issue faced by ray coordinates transformation isthe mapping of triplications in the traveltime field that occursin the presence of a strong velocity anomaly. This poses adifficulty when interpolating a function from ray coordinatesf̃ (u) to the cartesian coordinates f (x), because of multivaluedu(x).

To avoid such caustics, we are reminded that the solution to theeikonal equation has only the first arrival time. To utilize thisfeature, rather than computing the slowness vector p from thesecond equation in (1), we can use p calculated from directlysolving the eikonal. Thus, instead of (1), ray coordinates canbe formed by solving the following system:

dxdτ

= v2p,

x(τ = 0) = x0 (2)

where x0 is the location of initial wavefront, for a fan of raysit can be a small circle around the shot point, for normal raysit is the surface z = 0.

In summary, the complete procedure to construct ray coordi-nates is

1 solve eikonal for first arrival traveltimes τ(x)2 compute slowness vector p(x) = ∇τ(x)3 solve ray tracing system (2) for x(u)

A cartesian space function f (x) can then be interpolated to raycoordinates by simple substitution f̃ (u) = f (x(u)).

In order to interpolate from the ray coordinates to the carte-sian coordinates, the inverse function to x(u), i.e. u(x) needsto be evalutated at every grid. This can be achieved by trac-ing rays backward starting from every location x in the space,with the sign of p reversed in system (2) and terminate ray trac-ing when x reaches the surface z = 0. The interpolation fromray coordinates to cartesian coordinates is done by substitutionf (x) = f̃ (u(x)).

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-0104.1SEG Las Vegas 2012 Annual Meeting Page 1

Dow

nloa

ded

01/1

8/16

to 1

09.1

71.1

37.2

10. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Application to the acoustic wave equation

The acoustic wave equation, casted as a set of first order linearpartial differential equations, is given by

ρ∂wi

∂ t=− ∂ p

∂xi(i = 1,2,3),

∂ p∂ t

=−k3∑

i=1

∂wi

∂xi, (3)

where p is the stress and w is the particle velocity, ρ is themass density and k is the bulk modulus.

In an orthogonal curvilinear coordinate system u, the gradientand divergence operators are (Riley et al., 2006)

∂φ

∂xi=

1hi

∂φ

∂ui(i = 1,2,3),

3∑i=1

∂φi

∂xi=

1∏3i=1 hi

3∑i=1

∂

∂ui

∏j 6=i

h jφi

, (4)

where hi = |∂x/∂ui| is a scale factor such that ei = h−1i ∂x/∂ui

is a unit vector everywhere tangential to coordinate ui.

Substituting Equations (4) into Equation (3) yields the ray-coordinate acoustic wave equation

ρ∂wi

∂ t=− 1

hi

∂ p∂ui

(i = 1,2,3),

∂ p∂ t

=− k∏3i=1 hi

3∑i=1

∂

∂ui

3∏j 6=i

h jwi

. (5)

The second-order scalar wave equation is often used for imag-ing purposes

1v2

∂ 2 p∂ t2 =

3∑i=1

∂ 2 p∂x2

i, (6)

where v =√

k/ρ is P-wave velocity.

By combining the two equations in (4), the Laplacian operatorin the orthogonal curvilinear coordinate u is

3∑i=1

∂ 2φ

∂x2i=

1∏3i=1 hi

3∑i=1

∂

∂ui

(∏j 6=i h j

hi

∂φ

∂ui

)

≈3∑

i=1

1h2

i

∂ 2φ

∂u2i. (7)

The approximation in the second line is justified by the factthat odd-order terms in a second-order evolution PDE are dif-fusive rather than dispersive (Courant and Hilbert, 1989). Thusby dropping the first order terms in this equation the solutionretains kinematically correct. Substitute (7) into Equation (6)gives the ray-coordinate two-way wave equation.

1v2

∂ 2 p∂ t2 =

3∑i=1

1h2

i

∂ 2 p∂u2

i. (8)

In the special case when velocity is homogeneous, the nor-mal ray coordinate u becomes cartesian coordinate x, hi =

|∂x/∂xi| = 1 (i = 1,2,3), it is easy to verify that equation (8)reduces to Equation (6).

For example, in two-dimensional space, rays initiated froma line source may be specified by its components shotpointu1 = s and traveltime u3 = τ , in which the scale factors arecomputed using

hs =

∣∣∣∣∂x∂ s

∣∣∣∣=√(

∂x∂ s

)2+

(∂ z∂ s

)2,

hτ =

∣∣∣∣∂x∂τ

∣∣∣∣=√(

∂x∂τ

)2+

(∂ z∂τ

)2, (9)

for the corresponding wave equation:

1v2

∂ 2 p∂ t2 =

1h2

s

∂ 2 p∂ s2 +

1h2

τ

∂ 2 p∂τ2 . (10)

Downward continuation equation

The dispersion relation related to equation (8) is obtained byFourier transforming it in space and time as follows

ω2

v2 =

3∑i=1

k2i

h2i. (11)

For downward continuation, we can extract the vertical wavenum-ber

k3 =±ωh3

v

√√√√1−2∑

i=1

v2k2i

ω2h2i, (12)

with + for upgoing waves and − for downgoing waves. In thetwo-dimensional space, the ω− k downward continuation is

kτ =±ωhτ

v

√1− v2k2

sω2h2

s, (13)

and ω − x downward continuation with, for example, a 15◦

approximation

kτ =±ωhτ

v

(1+

v2

2ω2h2s

∂ 2

∂x2

). (14)

When hs = hτ = 1 this equation becomes the cartesian domain15◦ paraxial wave equation (Claerbout, 1985).

Mixed domain extrapolators may be derived in a similar fash-ion. For example, the split-step Fourier migration, kz can befound using a first-order Taylor expansion about reference slow-ness s0 = 1/v0, and thus, the reference scale factors hs0 andhτ0

kτ ≈±hτ0

√ω2s2

0−k2

s

h2s0

+ωhτ0∆s+ωs0∆hτ +k2

s hτ0

ωs0h3s0

∆hs, (15)

which upon setting hs = hτ = 1 and ∆hs = ∆hτ = 0 reducesto the cartesian-coordinate split-step Fourier migration (Stoffaet al., 1990). In Equation (15) the first term is ω − k domain

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-0104.1SEG Las Vegas 2012 Annual Meeting Page 2

Dow

nloa

ded

01/1

8/16

to 1

09.1

71.1

37.2

10. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

phase shift and the other three terms apply ω−x domain phaseshift. The ∆hs term is still has mixed domain dependence.One simplification could be setting ks = 0 and thus ignore thisterm (Sava and Fomel, 2005). This could however introduceerror to the downward continued wavefield when lateral varia-tion of hs is large.

EXAMPLES

The presence of a negative anomaly in the velocity causesmulti-pathing, and thus triplication of rays and wavefronts.An example of such triplication is illustrated in Figure 1(a)and 1(c), which are computed by the kinematic ray tracing sys-tem (1). This could pose a potential difficulty for interpolatingfrom ray coordinates u to cartesian coordinates x, due to mul-tivalued u(x) at caustics. By computing rays using eikonal-based ray tracing system (2), such triplications are effectivelyavoided, because eikonal contains only the first arrival. Ourapprouch guarantees single valued mapping relations x(u) andu(x). An example of such first arrival rays are plotted in Fig-ures 1(b) and 1(d), in which ray caustics below 1500m depthare eliminated.

(a) (b)

(c) (d)

Figure 1: A velocity model overlayed by normal rays (aand b) and a fan of rays (c and d). The velocity includesa −750m/sec anomaly with a homogeneous background of2250m/sec. Rays in (a) and (c) are computed using Equa-tion (1). Rays in (b) and (d) are computed using Equation (2).Ray caustics below z = 1500m are removed in (b) and (d).

As an example of wavefield extrapolation in ray coordinates,we extrapolate a zero-offset section backward in time and indepth, in both cartesian coordinates and normal ray coordi-nates, and compare the results. The zero-offset data has hor-izontal events at 0.44sec, 0.84sec and 1.24sec. The velocitymodel has a background linear gradient and a negative lensanomaly, as shown in Figure 2(a). Figure 2(b) is the same ve-locity interpolated to normal ray coordinates, which has hori-

zontal axis of shotpoint s and vertical axis of traveltime τ . Thewavefield could also be extrapolated in ray coordinates initi-ated from a point, in which case the horizontal axis is take-offangle.

(a)

(b)

Figure 2: (a) Velocity model overlayed by normal ray co-ordinates. The background velocity is 2100 + 0.025z +0.14x m/sec and the lens has a −750m/sec velocity pertur-bation. (b) The velocity interpolated to the normal ray coordi-nates.

Figure 3(b) is the image obtained from poststack reverse-timemigration in the cartesian domain, using the two-way waveequation (6). In the normal ray coordinates, reverse-time mi-gration is achieved by solving Equation (10). This result isplotted in Figure 3(c). As expected, the reflectors in ray coordi-nates are flat because the surface τ = const is always tangentialto first arrival wavefronts. To check consistency of migrationin the two domains, the image obtained in normal ray coordi-nates is interpolated to cartesian coordinates, in Figure 3(d).The agreement between Figure 3(b) and 3(d) verifies the cor-rectness of the ray-coordinates two-way wave equation (10).

Similar to the horizontal events, we computed the impulse re-sponse of reverse-time migration in cartesian and normal raycoordinates. The migration images are plotted in Figures 4(b)to 4(d). Again, the two migration images are consistent.

Downward continuation in normal ray coordinates is achievedby solving the corresponding one-way wave equation. In thisparticular example, we use the split-step Fourier migration de-

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-0104.1SEG Las Vegas 2012 Annual Meeting Page 3

Dow

nloa

ded

01/1

8/16

to 1

09.1

71.1

37.2

10. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

(a) (b)

(c) (d)

Figure 3: Poststack reverse-time migration. (a) zero-offsetdata featuring three horizontal events at 0.44sec, 0.84sec and1.24sec. Migration images obtained in (b) cartesian coordi-nates and (c) normal ray coordinates. (d) shows the image (b)interpolated to cartesian coordinates.

(a) (b)

(c) (d)

Figure 4: Impulse response of poststack reverse-time migra-tion. (a) zero-offset data featuring three spikes at 0.44sec,0.84sec and 1.24sec. Migration images obtained in (b) carte-sian coordinates and (c) normal ray coordinates. (d) shows theimage (b) interpolated to cartesian coordinates.

scribed by Equation (15). The migration images of the samezero-offset data are plotted in Figure 5(a) for cartesian coordi-nates and in Figure 5(b) for the ray coordinates result interpo-lated to cartesian coordinates. The impulse responses of thesetwo migration operators are plotted in Figures 6(a) and 6(b).

(a) (b)

Figure 5: Poststack split-step Fourier migration of data in Fig-ure 3(a), showing (a) cartesian coordinates migration image (b)normal ray coordinates migration image interpolated to carte-sian coordinates.

(a) (b)

Figure 6: Impulse response of poststack split-step Fourier mi-gration, showing (a) cartesian coordinates migration image (b)normal ray coordinates migration image interpolated to carte-sian coordinates.

In both Figures 5(b) and 6(b), the discrepancies observed nearx= 1km of the deep reflector is an indication of the error due toks = 0 assumption, as addressed previously with Equation (15).

CONCLUSION

We introduced an approach to construct normal ray coordi-nates based on first arrival traveltimes. Interpolation betweenthe cartesian coordinates and the ray coordinates are free fromcaustics. We derived a two-way acoustic equation and one-waydownward continuation equations for normal ray coordinates.Using these equation, zero-offset sections are migrated in bothcartesian and normal ray coordinates using reverse-time mi-gration and downward continuation. The migration imagesobtained by reverse-time migration in the cartesian and nor-mal ray coordinates are consistent. Downward continuation innormal ray coordinates less accurate because of ignoring lat-eral variation of hs.

ACKNOWLEDGEMENTS

We are grateful to KAUST for supporting this research. Wethank Anton Duchkov for helpful discussions.

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-0104.1SEG Las Vegas 2012 Annual Meeting Page 4

Dow

nloa

ded

01/1

8/16

to 1

09.1

71.1

37.2

10. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

http://dx.doi.org/10.1190/segam2012-0104.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Alkhalifah, T., S. Fomel, and B. Biondi, 2001, The space-time domain: theory and modelling for anisotropic media: Geophysical Journal International, 144, 105-113.

Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.

Courant, R., and D. Hilbert, 1989, Methods of mathematical physics: John Wiley & Sons.

Riley, K., M. Hobson, and S. Bence, 2006, Mathematical methods for physics and engineering: Cambridge University Press.

Sava, P., and S. Fomel, 2005, Riemannian wavefield extrapolation: Geophysics, 70, no. 3, T45-T56.

Shragge, J. C., 2008, Riemannian wavefield extrapolation: Nonorthogonal coordinate systems: Geophysics, 73, no. 2, T11-T21.

Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, 1990, Split-step Fourier migration: Geophysics, 55, 410-421.

Whitmore, N. D., 1983, Iterative depth migration by backward time propagation: SEG Technical Program Expanded Abstracts, 2, 382-385.

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-0104.1SEG Las Vegas 2012 Annual Meeting Page 5

Dow

nloa

ded

01/1

8/16

to 1

09.1

71.1

37.2

10. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/