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GEOPHYSICS, VOL. 73, NO. 5 �SEPTEMBER-OCTOBER 2008�; P. S169–S175, 8 FIGS.10.1190/1.2956349

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restack wave-equation depth migration in elliptical coordinates

eff Shragge1 and Guojian Shan2

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ABSTRACT

We extend Riemannian wavefield extrapolation �RWE� toprestack migration using 2D elliptical-coordinate systems.The corresponding 2D elliptical extrapolation wavenumberintroduces only an isotropic slowness model stretch to thesingle-square-root operator. This enables the use of existingCartesian finite-difference extrapolators for propagatingwavefields on elliptical meshes.Apoststack migration exam-ple illustrates advantages of elliptical coordinates for imag-ing turning waves.A2D imaging test using a velocity-bench-mark data set demonstrates that the RWE prestack migrationalgorithm generates high-quality prestack migration imagesthat are more accurate than those generated by Cartesian op-erators of the equivalent accuracy. Even in situations inwhich RWE geometries are used, a high-order implementa-tion of the one-way extrapolator operator is required for ac-curate propagation and imaging. Elliptical-cylindrical andoblate-spheroidal geometries are potential extensions of theanalytical approach to 3D RWE-coordinate systems.

INTRODUCTION

Wave-equation migration techniques based on one-way extrapo-ators are often used to accurately image complex geologic struc-ures. However, most conventional downward-continuation ap-roaches cannot handle steeply propagating or turning-wave com-onents, often important for imaging areas of interest. Several novelmaging approaches address these issues through a judicious decom-osition of recorded wavefields �e.g., plane-wave migration; seehitmore, 1995�, partial and complete propagation-domain decom-

osition �e.g., Gaussian beam �Hill, 2001� and Riemannian wave-eld extrapolation �Sava and Fomel, 2005; Shragge, 2006��, or aombination thereof �e.g., plane-wave migration in tilted coordi-

Manuscript received by the Editor 24August 2007; revised manuscript rec1Stanford University, Geophysics Department, Stanford Exploration Proje

-mail: jeff@sep.stanford.edu, jshragge@gmail.com.2Formerly Stanford Exploration Project; presently Chevron Energy Techn

-mail: shan@sep.stanford.edu.2008 Society of Exploration Geophysicists.All rights reserved.

S169

ates; see Shan and Biondi, 2004�. These techniques have overcomeany, although not all, issues in the practical application of one-way

xtrapolation operators.Riemannian wavefield extrapolation �RWE� is a method for prop-

gating wavefields on generalized coordinate meshes. The centraldea of RWE is to transform the computational domain from Carte-ian to a geometry in which the extrapolation axis is oriented alonghe general wavefield propagation direction. Ideally, solving corre-ponding one-way extrapolation equations in the transform domaineads to most of the wavefield energy being propagated at angles rel-tively near the extrapolation axis, thus improving the global accura-y of wavefield extrapolation. One obvious application is generatingigh-quality Green’s functions for point sources in a dynamic coor-inate system in which a suite of rays is first traced through a velocityodel and then used as the skeleton upon which to propagate wave-elds.Although the full-domain decomposition approach naturally

dapts to propagation in a point-source ray-coordinate system, twonresolved issues make it difficult to apply RWE efficiently inrestack shot-profile migration algorithms. First, receiver wave-elds in shot-profile migration are usually broadband in plane-waveip spectrum and cannot be represented easily by a single-coordinateystem �i.e., reflections from opposing dips propagate in opposingirections�. Second, optimal meshes for source and receiver wave-elds usually do not share a common geometry. For example, a polaroordinate system is well suited for propagating source wavefields,ut elliptical meshes are more appropriate for receiver wavefields.his factor is detrimental to algorithmic efficiency, for images areenerated by correlating source and receiver wavefields: by existingn different grids, they both must be interpolated to a common Car-esian reference frame prior to imaging. The result is a significantumber of interpolations, leaving the algorithm computationally un-ttractive except in target-oriented imaging situations.

The main goal of this paper is to specify one coordinate systemhat accurately propagates high-angle and turning-wave compo-ents of source and receiver wavefields. We demonstrate that an el-iptical-coordinate system forms a natural computational grid for

February 2008; published online 26August 2008.ford, California, U.S.A.

ompany, San Ramon, California, U.S.A.

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restack shot-profile migration and has useful geometric propertieshat facilitate numerical implementation. An elliptical-coordinateystem originates on the horizontal acquisition surface and stepsutward as a series of ellipses. Thus, the coordinate system expandsn a radial manner appropriate for computing accurate point-sourcereen’s functions while allowing the dipping plane-wave compo-ents in the receiver wavefield to propagate at large angles to eitheride of the acquisition surface.

One consequence of using a 2D elliptical-coordinate system ishat the corresponding extrapolation operator must be modified.owever, we show that elliptical geometry introduces only an iso-

ropic velocity model stretch. Existing high-order implicit Cartesiannite-difference extrapolators with accuracy as much as 80° from

he extrapolation axis �Lee and Suh, 1985� can be used to propagateavefields, readily and accurately imaging wide-angle and turningaves at a cost competitive with that of Cartesian downward contin-ation.

This paper begins with a discussion of why elliptical meshes are aatural coordinate-system choice for shot-profile prestack depth mi-ration �PSDM�. We develop an extrapolation wavenumber appro-riate for wavefield propagation on a 2D elliptical-coordinate sys-em. Then we present poststack and prestack migration exampleshat illustrate the scheme’s ability to image steep structure usingurning waves. The paper concludes with a discussion of advantagesf analytic-coordinate systems relative to more dynamically gener-ted meshes and two potential extensions of the analytical approacho 3D RWE-coordinate systems.

ELLIPTICAL-COORDINATE EXTRAPOLATION

Generating an effective RWE-coordinate system for prestack mi-ration requires appropriately linking mesh geometry with wave-eld-propagation kinematics. Figure 1a illustrates this concept for

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igure 1. Elliptical-coordinate system. �a� Constant-velocity imag-ng experiment with punctual source and receiver wavefieldsdashed lines� from locations marked S and R, respectively. The cor-esponding image is an elliptical isochron surface derived by cross-orrelating source and receiver wavefields �solid line�. �b� Grid-rota-ion angles for the elliptical-coordinate system.

n idealized shot-profile imaging experiment through a medium ofonstant wavefield slowness s. Here, we specify source and receiveravefields �S and R� as impulses at source position and time �xs,� s

0� and at receiver position and time �xr,� r � � �, where � is anrbitrary time lag. Wavefields expand outward as spherical wave-ronts �dashed lines� according to

S�xs,x;t� � � �t � s�x � xs�� and,

R�xr,x;t� � � �t � � � s�x � xr�� , �1�

here �x� is the Euclidean norm of the vector x and s is wavefieldlowness. An image I �x� can be generated by applying a correlationmaging condition at t � 0 �Claerbout, 1985�,

I�x� � � �� � s��x � xr� � �x � xs��� , �2�

hich is the equation of an ellipse �solid line�.This suggests a natural correspondence between an elliptical-co-

rdinate system and prestack-migration isochrons for a constant-ve-ocity model. One can observe this in Figure 1 by how well the isoch-on image conforms to the underlying coordinate mesh. In this ex-mple, we intentionally did not fix locations of elliptical mesh focielative to xs and xr. Adjusting these points will alter the ellipticalesh and how well it conforms to isochrons. These parameters rep-

esent two degrees of freedom that allow us to match mesh geometryo the bulk-propagation direction.

elmholtz equation in elliptical coordinates

Propagating a wavefield � in generalized coordinates requiresncoding the mesh geometry directly into one-way extrapolationquations �Sava and Fomel, 2005; Shragge, 2006�. Hence, derivingn elliptical-coordinate extrapolation operator requires introducinglliptical geometry into the Laplacian operator �2 of the Helmholtzquation:

�2� � �2s2� � 0. �3�

e begin with the definition of the analytic transformation betweenhe elliptical- and Cartesian-coordinate systems �Morse and Fesh-ach, 1953�:

�x1

x2� � �a cosh � 2 cos � 1

a sinh � 2 sin � 1� , �4�

here �x1,x2� are underlying Cartesian-coordinate variables, �� 1,� 2�re RWE elliptical coordinates defined on intervals � 1 � �0,�� and2 � �0,��, and a is a stretch parameter controlling coordinate-sys-

em breadth. �The definition in equation 4 is analogous to definitionsf cylindrical or spherical coordinates.� As illustrated in Figure 1,ines of constant � 3 represent ellipses, whereas those of constant � 1

orm hyperbolas. Figure 1 also shows our definition of the angle��� between the elliptical- and Cartesian-coordinate systems. Ad-itional information on elliptical coordinates is presented in Appen-ix A.

The metric tensor gij � ��xk/�� i� ��xk/�� j� with an implicitum over index k describing the elliptical-coordinate system geome-ry is

�g � � �A2 0 � , �5�

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here A � a�sinh2 � 2 � sin2 � 1. The metric tensor determinant gA4 is required to specify the associated �gij, or the inverse of gij�

nd weighted �mij � �ggij� metric tensors:

�gij� � �A�2 0

0 A�2� and �mij� � �1 0

0 1� . �6�

nterestingly, the weighted metric tensor mij in equation 6 is an iden-ity matrix, indicating the transformation in equation 4 causes spaceo contract or dilate isotropically.

Inserting tensor components in equation 6 into the standard ex-ression for the Laplacian operator �Morse and Feshbach, 1953�,

�2 �1

�g�

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�� j� �

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eads to the elliptical-coordinate Laplacian operator,

�2 �1

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�� 22 �

� 2

�� 12� . �8�

he elliptical-coordinate Helmholtz equation is derived by introduc-ng equation 8 into equation 3 and rearranging terms to yield

� � 2

�� 22 �

� 2

�� 12�� � A2�2s2� � 0. �9�

efining an effective-slowness field seff � As, we can rewrite thequation 9 as

� � 2

�� 22 �

� 2

�� 12�� � �2seff

2 � � 0. �10�

quation 10 is this paper’s most important result:An elliptical-coor-inate system introduces only an isotropic-slowness model stretchn the Helmholtz equation.

ispersion relation in elliptical coordinates

Deriving an elliptical-coordinate dispersion relation from equa-ion 10 proceeds in the usual manner. In the following 2D develop-

ent, we use the convention that � 2 and � 1 are the extrapolation di-ection and orthogonal coordinate, respectively. Replacing partialifferential operators with their Fourier-domain duals �Claerbout,985� gives

k� 1

2 � k� 2

2 � �2seff2 . �11�

solating k� 2wavenumber contributions leads to a recursive wave-

eld-extrapolation operator for stepping outward in concentric el-ipses in the � 2 direction:

� �� 2 � � 2,k� 1,�� � � �� 2,k� 1

,��e�i� 2k� 2

� � �� 2,k� 1,��e�i� 2��2sef f

2�k� 1

2,

�12�

here � 2 is the extrapolation step size and � determines whether aavefield is propagating causally �i.e., source wavefield� or acaus-

lly �i.e., receiver wavefield�.The dispersion relation in equation 12 generally will not be an ex-

ct expression because s varies spatially. This situation is similar to

eff

hat in Cartesian-wavefield extrapolation in laterally varying media,nd equation 12 can be implemented easily with existing Cartesianxtrapolation schemes �e.g., finite differences, phase-shift plus inter-olation� and using an effective-slowness model seff � As.An additional question worth addressing is to which angle is prop-

gation in elliptical coordinates accurate? Because geometric effectsf the elliptical coordinate can be incorporated into an effective-lowness model seff, the local angular accuracy for finite-differenceropagation is equivalent to that of a Cartesian-domain implementa-ion. Globally, however, the maximum propagation angle for a givenxtrapolation accuracy depends on the orientation of the local ex-rapolation axis. Figure 1b illustrates how the angle of the extrapola-ion axis � � � �� � changes locally in an elliptical-coordinate sys-em. In the following examples, our �80° finite-difference propaga-ors �Lee and Suh, 1985� have a maximum extrapolation angle equalo � �� � � 80°.

restack-migration algorithm

The expression in equation 12 can be extended to prestack migra-ion. An initial step is defining foci locations of the elliptical-coordi-ate system. Unfortunately, choosing the optimal answer, relative tohe acquisition geometry �i.e., the source and farthest-offset loca-ions�, is not straightforward. For example, situating foci too closeogether pulls wavefields toward the focus because the local extrapo-ation axis angle � becomes steep rapidly �see Figure 1b�. In con-rast, placing foci too distant from each other leads to a near-rectilin-ar coordinate system that affords little improvement over Cartesianxtrapolation. We determined heuristically that optimal foci loca-ions are an additional 10%-20% �of the aperture length� beyond theource point and farthest receiver offset.

The remaining six prestack-migration algorithmic steps follow.

� Specify a shot-specific elliptical-coordinate system for sourcelocation si, and interpolate the Cartesian velocity model to thismesh.

� Generate the shot-specific image I ��,si� in the elliptical-coor-dinate system at step � 2 from the source S and receiver R wave-fields:

I��,si� � ��

R�S*��,�,si�R��,�,si�� . �13�

� Propagate source and receiver wavefields �for all frequencies�by a step � 2:

S�� 2 � � 2,k� 1,�,si� � S�� 2,k� 1

,�,si�e�i� 2k� 2,

R�� 2 � � 2,k� 1,�,si� � R�� 2,k� 1

,�,si�e�i� 2k� 2.

�14�

� Repeat steps 2-3 until reaching the end of the elliptical-coordi-nate mesh.

� Interpolate the single-shot elliptical-coordinate image I ��,si�to Cartesian coordinates, and update the global Cartesian imageI �x�.

� Repeat steps 1–5 for all shot locations.

2D MIGRATION TESTS

This section presents 2D test results for poststack turning-wavend prestack-velocity-benchmark �Billette and Brandsberg-Dahl,

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005� data sets. We propagate all wavefields with one-way extrapo-ators described in Lee and Suh �1985� on an elliptical-coordinateystem defined by equation 4, assuming effective-slowness fieldseff � As. Imaging results are generated with poststack and shot-pro-le migration algorithms using the recursive extrapolation relations

n equations 12 and 14, respectively. Image volumes in elliptical co-rdinates subsequently are transformed back to the Cartesian do-ain using sinc interpolation. The extra computational cost of gen-

rating RWE migration results, relative to those in Cartesian imag-ng, is roughly two additional interpolations per shot: one for theartesian velocity model to the elliptical mesh and another for the el-

iptical image to the Cartesian grid.

oststack migration example

The first elliptical-coordinate migration example uses the post-tack data set shown in Figure 2. The data were generated from andapted Sigsbee model �Figure 3a�, using exploding reflector �two-ay time-domain finite-difference� modeling from all salt-body

dges �Sava, 2006�. The imaging test involved only turning compo-ents of the wavefield in Figure 2. Figure 3a also presents the experi-ental geometry for the coordinate system, with foci situated at90 km �not shown� and 5 km. Because propagation directions of

he wavefield are known, we could choose foci locations that en-ured the grid generally conformed to the turning-wave propagationirection. �We recognize this example is a special case in which theip field is oriented largely in one direction.� Figure 3b shows the ef-ective-slowness model seff in the transformed coordinate system,arameterized by extrapolation step and surface-takeoff-positionxes.

Figure 4 presents results for migration in selected RWE ellipticaloordinates. Figure 4a shows the monochromatic poststack migra-ion result with the elliptical-coordinate system overlain. Poststack

igration in elliptical coordinates successfully propagates turningaves, which arrive at normal incidence to the salt flank as expected

or exploding-reflector modeling. Figure 4b shows the elliptical-co-rdinate version of the image in Figure 4a at successive extrapola-ion steps and illustrates that wavefield energy propagates at fairlyteep angles relative to the extrapolation axis. Figure 4c and d shows

igure 2. Poststack turning-wave data. Data set generated by two-ay time-domain, finite-difference modeling from all salt-body

dges of the velocity model in Figure 3a.

roadband image results in Cartesian- and elliptical-coordinate sys-ems, respectively. Salt flanks beneath the salt nose are positionedccurately, demonstrating the potential for imaging turning waves inlliptical coordinates with wide-angle extrapolation operators.

restack migration example

We performed a prestack elliptical-coordinate migration test us-ng a velocity benchmark model �Figure 5a�. Full data offsets wereot used for each shot. Rather, we used a 10-km initial migration ap-rture to enable more accurate propagation of turning waves withinur propagation domain. Foci for each migrated shot were located andditional 15% �of the 10-km acquisition aperture� beyond the shot-oint and farthest receiver offset. Figure 5b shows the effective-lowness model seff in the elliptical-coordinate system. The ellipticalesh is parameterized by extrapolation step and surface-takeoff-po-

ition axes. Importantly, the steep salt-body structure to the right be-omes relatively low angle under this coordinate transformation andhould be better imaged.

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ξigure 3. Poststack turning-wave model. �a� Velocity model used toenerate turning-wave data in Figure 2 with coordinate system over-ain. �b� Effective-slowness model seff in elliptical coordinates.

igure 4. Poststack turning-wave migration result. �a� Monochro-atic Cartesian image with overlain elliptical coordinates. �b�onochromatic elliptical-coordinate image. �c� Broadband Carte-

ian image. �d� Broadband elliptical-coordinate image.

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Figure 6 presents prestack migration results for the elliptical-co-rdinate system. Figure 6a shows the RWE shot-profile migrationesult in elliptical coordinates; Figure 6b presents the correspondingartesian image generated by finite-difference operators of equiva-

ent accuracy. �Slightly different source wavelets were used in thewo migrations, leading to a phase rotation between the two images.�he salt body to the left is well imaged in most areas in both images,lthough it is improved in, for example, the circled location. Thealt-body flanks to the right �circled�, illuminated largely by turning,nd prismatic waves are better imaged in the elliptical system.

DISCUSSION

One question naturally arises when using RWE propagation in arestack-migration algorithm: How does one obtain the optimalrade-off between using �1� low-order extrapolators on a more dy-amic coordinate system �e.g., ray coordinates� and �2� high-orderxtrapolators on a mesh less conformal to the wavefield-propagationirection? Based on our experience, we argue that a parametric coor-inate system �such as a tilted Cartesian or an elliptical mesh� offershe advantage of developing analytic extrapolation operators thateadily lend themselves to high-order finite-difference schemes.

Although coordinate systems based on ray tracing conform betterothewavefield-propagationdirection, numerically generated mesh-s do not lend themselves as easily to high-order extrapolators be-ause of the greater number and spatial variability of correspondingixed-domain coefficients. In addition, analytic coordinates allow

he user to specify a coordinate system adequate for propagatingource and receiver wavefields, rather than optimizing for one or thether. One caveat, however, is that higher-order extrapolators usual-y are required for analytic coordinate systems because, althoughhey are more optimal for global propagation, they are less confor-

al to the local extrapolation direction.A second question worth addressing is how the analytic coordi-

ate approach can be extended to 3D prestack shot-profile migra-

Distance (km)

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Dep

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b)

igure 5. Prestack migration test in elliptical coordinates. �a� Bench-ark synthetic velocity model with the elliptical-coordinate system

verlain. �b� Effective-slowness model in the transformed elliptical-oordinate system.

ion. Although a complete study of 3D possibilities is beyond thecope of this paper, two potential coordinate systems for wavefieldxtrapolation in the � 3 direction are elliptical-cylindrical and oblate-pheroidal coordinates �discussed inAppendix B�.

The elliptical-cylindrical coordinate system �Figure 7� extendshe 2D elliptical mesh invariantly in the second lateral dimension �� 2

ow� and leads to a fairly basic expression for the extrapolationavenumber k� 3

�see Appendix B�. This coordinate system shoulde well suited for narrow-azimuth geometries in which the ellipticalurfaces �i.e., � 1� are oriented in the inline direction, leaving the cy-indrical coordinate �i.e., � 2� to extend in the crossline direction. Oneisadvantage is that waves cannot be overturned in the crossline di-ection. However, the degree to which this is problematic largely de-ends on the velocity model and reflector geometry.

Oblate-spheroidal coordinates �Figure 8� incorporate a morepherical geometry and would enable steep-angle and turning-waveropagation at all azimuths. This coordinate system is likely welluited for wide-azimuth data sets containing arrivals from a richwath of azimuths. One drawback, however, is that this geometryeads to a more complicated and difficult-to-implement expressionor extrapolation wavenumber k� 3

�seeAppendix B�.Overall, the applicability of imaging in 3D analytic coordinate

ystems remains an open research topic that will be addressed by theuthors in a future paper.

Distance (km)

Distance (km)

Dep

th(k

m)

Dep

th(k

m)

a)

b)

igure 6. Synthetic migration results. �a� Elliptical-coordinate mi-ration result using finite-difference propagators. �b� Cartesian mi-ration result generated by finite-difference extrapolators of accura-y equivalent to those in �a�.

igure 7. Four extrapolation steps of an elliptical-cylindrical-coor-inate system.

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CONCLUSIONS

This paper extends RWE to 2D prestack shot-profile migration.e choose an elliptical-coordinate system that generally conforms

o the wave-propagation direction and enables wide-angle extrapo-ation of source and receiver wavefields. Poststack migration resultsf a turning-wave data set validate the approach, whereas 2Drestack imaging results show that the RWE-migration algorithmenerates images more accurate than corresponding Cartesian re-ults. The cost difference between the elliptical and Cartesian imag-ng algorithms is only two additional interpolations per migratedhot profile. Finally, we argue that parametric-coordinate systemsre a good trade-off between the competing constraints of meshesonforming to the wavefield propagation direction and coordinate-ystem simplicity because one can readily develop analytic wave-umbers and more accurate high-order extrapolation implementa-ions.

ACKNOWLEDGMENTS

We thank Biondo Biondi, Ben Witten, Brad Artman, and Kenarner for enlightening discussions. We thank SEP sponsors for

heir continuing support, and we acknowledge BP for the syntheticelocity data set.

APPENDIX A

ELLIPTICAL-COORDINATE SYSTEMS

Similar to cylindrical or spherical coordinate systems, relation-hips between Cartesian and elliptical geometries in equation 4 can-ot be derived per se. Exploring constant coordinate surfaces, how-ver, provides additional insight into some characteristics of the el-iptical-coordinate system. As illustrated by the following trigono-

etric identities, curves of constant � 1 represent hyperbolas:

x12

a2 cos2 � 1�

x22

a2 sin � 1� cosh2 � 2 � sinh2 � 2 � 1.

�A-1�

urves of constant � form ellipses:

igure 8. Three extrapolation steps of an oblate-spheroidal-coordi-ate system.

2

x12

a2 cosh2 � 2�

x22

a2 sinh � 2� cos2 � 2 � sin2 � 2 � 1.

�A-2�

hus, outward extrapolation in the � 2 direction would step a wave-eld through a family of elliptical surfaces defined by equation A-2.

Equation A-1 also can be used to derive an expression that de-nes the local extrapolation axis angle � �� � relative to vertical refer-nce. Taking the total derivative of equation A-1,

2x1dx1

a2 cosh2 � 1�

2x2dx2

a2 sinh2 � 1� 0, �A-3�

nd further manipulating the result yields the local extrapolationxis angle � �� �:

tan � �dx1

dx2�

x2 cos2 � 1

x1 sin2 � 1� tanh � 2 cot � 1. �A-4�

igure 1a illustrates these angles for the elliptical-coordinate sys-em.

APPENDIX B

ONE-WAY EXTRAPOLATION IN 3DCOORDINATE SYSTEMS

This appendix develops the dispersion relationship for extrapo-ating waves for two 3D elliptical-coordinate systems: elliptical cy-indrical and oblate spheroidal. Readers interested in additional in-ormation are directed to Shragge �2006�.

lliptical-cylindrical coordinates

The analytic transformation between the elliptical-cylindrical-nd Cartesian-coordinate systems is

x1

x2

x3� � a cosh � 3 cos � 1

� 2

a sinh � 3 sin � 1� , �B-1�

here �x1,x2,x3� are the underlying Cartesian-coordinate variables;� 1,� 2,� 3� are the RWE elliptical-cylindrical coordinates defined onntervals � 1 � �0,2��, � 2 � ���,��, and � 3 � �0,��; and a is atretch parameter controlling the breadth of the coordinate system.

The metric tensor gij � ��xk/�� i���xk/�� j� describing the geome-ry of the elliptical-coordinate system is given by

�gij� � A2 0 0

0 1 0

0 0 A2 � , �B-2�

here A � a�sinh2� 3 � sin2� 1. The determinant of the metric ten-or is g � A4. The associated �inverse� metric tensor is given by

�gij� � A�2 0 0

0 1 0

0 0 A�2 � , �B-3�

nd weighted metric tensor �mij � �ggij� is given by

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�mij� � 1 0 0

0 A2 0

0 0 1� . �B-4�

he corresponding extrapolation wavenumber is generated by usingensors gij and mij in the general wavenumber expression for a 3Donorthogonal coordinate system �Shragge, 2006�. Although the el-iptical-coordinate system varies spatially, local curvature parame-ers ni � ��mij/�� j� remain constant: n1 � n2 � n3 � 0. Thus, in-erting values of gij, mij, and nj leads to the following extrapolationavenumber k� 3

for stepping outward in concentric ellipses:

k� 3� ��A2s2�2 � k� 1

2 � Ak� 2

2 . �B-5�

he wavenumber for 2D extrapolation in elliptical coordinates re-uces to

k� 3k� 2

�0 � ��A2s2�2 � k� 1

2 . �B-6�

blate-spheroidal coordinates

The analytic transformation between the oblate-spheroidal- andartesian-coordinate systems is

x1

x2

x3� � a cosh � 3 cos � 1 cos � 2

a cosh � 3 cos � 1 sin � 2

a sinh � 3 sin � 1� , �B-7�

here coordinates are defined on intervals � 1 � �0,2��,2 � �0,2��, and � 3 � �0,��, and a is a stretch parameter controlling

he breadth of the coordinate system. The metric tensor gij describinghe geometry of oblate-spheroidal coordinates is

�gij� � A2 0 0

0 B2 0

0 0 A2 � , �B-8�

here A � a�sinh2� 3 � sin2� 1 and B � a cosh � 3 cos � 1. The met-ic tensor determinant is given by g � A4B2, the associated �in-erse� metric tensor by

�gij� � A�2 0 0

0 B�2 0

0 0 A�2 � , �B-9�

nd the weighted metric tensor by

�mij� � B 0 0

0A2

B0

0 0 B� . �B-10�

e generate the corresponding extrapolation wavenumber by input-ing tensors gij and mij into the generalized wavenumber expressionor 3D nonorthogonal coordinate systems �Shragge, 2006�. Unliken elliptical-cylindrical coordinates, however, the oblate-spheroidalystem has nonstationary ni coefficients: n1 � a cosh � 3 sin � 1, n2

0, and n3 � a sinh � 3. The resulting extrapolation wavenumber is

k� 3�

i tanh � 3

2

��A2s2�2 � k� 1

2 ��sinh2 � 3 � sin2 � 1

cosh � 3 cos � 1

k� 2

2 � ik� 1tan � 1 �

1

4tanh2 � 3.

�B-11�

he wavenumber for 2D extrapolation in elliptical coordinates re-uces to

k� 3k� 2

�0

� ��A2s2�2 � k� 1

2 � ik� 1tan � 1 �

1

4tanh2 � 3.

�B-12�

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