Two-point paraxial traveltime formula for inhomogeneousisotropic and anisotropic media: Tests of accuracy
Umair bin Waheed1, Ivan Pšenčík2, Vlastislav Červený3, Einar Iversen4, and Tariq Alkhalifah1
ABSTRACT
On several simple models of isotropic and anisotropic media,we have studied the accuracy of the two-point paraxial travel-time formula designed for the approximate calculation of thetraveltime between points S 0 and R 0 located in the vicinityof points S and R on a reference ray. The reference ray maybe situated in a 3D inhomogeneous isotropic or anisotropicmedium with or without smooth curved interfaces. The two-point paraxial traveltime formula has the form of the Taylorexpansion of the two-point traveltime with respect to spatialCartesian coordinates up to quadratic terms at points S and Ron the reference ray. The constant term and the coefficients ofthe linear and quadratic terms are determined from quantitiesobtained from ray tracing and linear dynamic ray tracing alongthe reference ray. The use of linear dynamic ray tracing allows
the evaluation of the quadratic terms in arbitrarily inhomo-geneous media and, as shown by examples, it extends the regionof accurate results around the reference ray between S and R(and even outside this interval) obtained with the linear termsonly. Although the formula may be used for very general3D models, we concentrated on simple 2D models of smoothlyinhomogeneous isotropic and anisotropic (∼8% and ∼20%anisotropy) media only. On tests, in which we estimated two-point traveltimes between a shifted source and a system of shiftedreceivers, we found that the formula may yield more accurate re-sults than the numerical solution of an eikonal-based differentialequation. The tests also indicated that the accuracy of the formuladepends primarily on the length and the curvature of the referenceray and only weakly depends on anisotropy. The greater is thecurvature of the reference ray, the narrower its vicinity, in whichthe formula yields accurate results.
INTRODUCTION
We study the accuracy of the two-point paraxial traveltime for-mula proposed by Červený et al. (2012). The formula can be usedfor the approximate determination of the two-point traveltimeTðR 0; S 0Þ between a point S 0 and another point R 0 arbitrarily chosenin the vicinities of two respective points S and R on a reference rayΩ (Figure 1). Ray Ω can be traced in a 3D laterally varying, iso-tropic, or anisotropic model of elastic medium with or withoutstructural interfaces. The two-point paraxial traveltime formula isbased on the Taylor expansion of the two-point traveltimeTðR 0; S 0Þ with respect to the differences in spatial Cartesian coor-dinates xðS 0Þ − xðSÞ and xðR 0Þ − xðRÞ of points S 0 and S andpoints R 0 and R, with an accuracy up to quadratic terms. The con-
stant term TðR; SÞ in the expansion is known from tracing the refer-ence ray. For the evaluation of the linear terms, slowness vectorspðSÞ and pðRÞ, also known from the ray tracing ofΩ, are necessary.For the evaluation of quadratic terms in the expansion, the tracingreference ray Ω is insufficient; dynamic ray tracing (DRT) or someother procedure providing quantities related to the second-order traveltime derivatives must also be performed. The DRT sys-tem used in this paper is a linear system of ordinary differentialequations of the first order, which results from the differentiationof the ray-tracing system with respect to ray parameters. Becausethe system is linear, it is possible to construct its fundamental matrixΠðR; SÞ. This matrix, specified as the identity matrix at point S, iscalled here the ray propagator matrix. It plays a basic role in theparaxial ray methods. Once matrix ΠðR; SÞ is known at point R of
Manuscript received by the Editor 25 September 2012; revised manuscript received 8 February 2013; published online 24 June 2013; corrected versionpublished online 17 September 2013.
1KAUST, Physical Sciences and Engineering Division, Saudi Arabia. E-mail: [email protected]; [email protected] of Sciences of the Czech Republic, Institute of Geophysics, Praha, Czech Republic. E-mail: [email protected] University, Faculty of Mathematics and Physics, Department of Geophysics, Praha, Czech Republic. E-mail: [email protected], Kjeller, Norway. E-mail: [email protected].
reference ray Ω, the solution of the DRT system at point R can beobtained by simple matrix multiplication for arbitrary initial condi-tions at S. Note that, beside the linear DRT, also the nonlinearversion of the DRT, based on the Riccati equation, has been usedby some authors: see, for example, Gjøystdal et al. (1984). Here,however, we consistently use the linear DRT. When we refer tothe DRT in the following text, we have the linear DRT in mind.For more details on ray tracing, linear and nonlinear dynamicray tracing, and ray propagator matrices, see Červený (2001).The DRT system and relevant ray propagator matrix can be
computed along the reference ray in various coordinate systems(Cartesian, ray-centered, etc.) We use here a two-point paraxial trav-eltime formula with the DRT performed in ray-centered coordi-nates. Although DRT is performed in ray-centered coordinates, thepositions of points S, R, S 0, and R 0 are specified in the Cartesiancoordinate system. This makes the formula practical and flexibleto use.The quadratic traveltime approximation in the vicinity of the
reference ray is usually called the paraxial traveltime approxima-tion. The vicinity in which the accuracy of the approximation issufficient is called the paraxial vicinity. The accuracy of thetwo-point paraxial traveltime formula depends on the results ofthe ray tracing and DRTalong reference rayΩ. Both, in turn, dependon the first and second (DRT) spatial derivatives of the parametersof the medium at the points of Ω. Thus, as Vanelle and Gajewski(2002) put it, “the size of the (paraxial) vicinity depends on the scaleof velocity variations in the model.” Consequently, it is not easy toestimate quantitatively the accuracy of paraxial traveltimes and thesize of the paraxial vicinity. The performed tests indicate significantdependence of the paraxial vicinity on the length of the referenceray. The longer the reference ray, the greater the size of the paraxialvicinity.One possible way of estimating the accuracy of the two-point
paraxial traveltime formula would be the use of the third-orderterms of the Taylor expansion of the traveltime. This would, how-ever, require knowledge of the third derivatives of the mediumparameters, and that, in turn, would require models with continuousthird derivatives of the model parameters. Most of the presentlyused modeling techniques employ cubic splines, which do not guar-antee continuity of the third derivatives. The use of splines of anorder higher than cubic is feasible and is expected to yield betterapproximations to two-point paraxial traveltimes. On the otherhand, such splines will lead to increased computation times, whichthus represent a tradeoff between the accuracy and efficiency.
In this paper, we concentrate intentionally on simple 2D modelswithout structural interfaces because they allow us to easily estimatethe accuracy of the two-point paraxial traveltimes and to indicate theindividual effects leading to the decrease of their accuracy. In morecomplicated models, it is more difficult to separate these effects (ef-fects of inhomogeneity or anisotropy, effects of distance of points Sand R, effects of distance of points S and S 0 and of points R andR 0, etc.).The theory of two-point ray-theory traveltimes has a long history
in wave sciences, mainly in optics. It was already studied by Ham-ilton (1837), who calls the two-point traveltimes the “point charac-teristics.” Hamilton’s theory of point characteristics (also known ascharacteristic functions) was recently extended by Klimeš (2009)(see also Červený et al., 2012). Klimeš (2009) used the DRT in Car-tesian coordinates and the corresponding 6 × 6 ray-propagator ma-trix for the evaluation of two-point paraxial traveltimes TðR 0; S 0Þ. Inthis paper, we use the 4 × 4 ray-propagator matrix in ray-centeredcoordinates. Hamilton’s point characteristics also form the basis ofthe theory of optical systems, see, e.g., Luneburg (1964). Such sys-tems were introduced to seismology by Bortfeld (1989), who estab-lishes general surface-to-surface rules for rays and traveltimes ofreflected and transmitted waves in media with structural interfaces.The applicability of Bortfeld’s surface-to-surface method is consid-erably extended by Hubral et al. (1992) by using the DRT in ray-centered coordinates for layered inhomogeneous isotropic media.The method and its modifications have been used in various appli-cations, see, e.g., Schleicher et al. (1993), Červený (2001), Červenýand Moser (2007), Moser and Červený (2007). The accuracy of themethod, however, has not yet been studied. In the surface-to-surfaceapproaches, paraxial points S 0 and R 0 are situated on the surfacespassing through points S and R, respectively. This differs from thevolume-to-volume approach used in this paper, in which paraxialpoints S 0 and R 0 can be situated arbitrarily in the 3D vicinities ofpoints S and R. It is true that the two surfaces do not need to bephysical, but once they have been chosen, the surface-to-surfacemethod permits extrapolation in four coordinates only, whereasthe volume-to-volume approach permits extrapolation in six coor-dinates. Moreover, our approach permits extrapolation also alongthe ray, whereas in the surface-to-surface method the surface cannotbe oriented so that the ray is tangent or nearly tangent to the surface.Ursin (1982) proposes a traveltime formula closely related to the
one used in this paper. He develops the Taylor-type expansions forthe traveltime (parabolic formula) as well as its square (hyperbolicformula) around the reference source and receiver points. For theevaluation of the coefficients of the expansion, he uses wavefrontcurvature matrices. Ursin (1982) shows that the hyperbolic form isexact for a homogeneous isotropic medium; see also Gjøystdal et al.(1984), Schleicher et al. (1993) and Appendix A of this paper. Thehyperbolic formula of Ursin (1982) has been used and generalizedby many authors. Its accuracy, as well as the accuracy of the para-bolic formula, are tested by Gjøystdal et al. (1984). More recently,the hyperbolic formula has been used to interpolate traveltimes (Va-nelle and Gajewski, 2002) and for the determination of the geomet-rical spreading (Vanelle and Gajewski, 2003) in inhomogeneousisotropic or anisotropic media. The coefficients of the Taylor expan-sion are evaluated from traveltimes specified in known nodes of acoarse grid. The traveltimes are computed using an eikonal solver(not ray tracing and the DRT). Gjøystdal et al. (1984) make the firstattempts to use the DRT in ray-centered coordinates for evaluating
Figure 1. The traveltime between points S 0 and R 0 is estimatedfrom the traveltime between points S and R on the reference ray(black solid curve; dashed curve is shown only for illustration,no ray connecting S 0 and R 0 is necessary), along which the resultsof ray tracing and dynamic ray tracing in ray-centered coordinatesare known.
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the coefficients of the Taylor expansion of squared traveltime in 3Dlaterally varying layered isotropic models. They, however, use anonlinear form of DRT (Riccati equation), which does not allowconstruction of the ray propagator matrix. The ray propagator ma-trix is introduced in the context of two-point paraxial traveltimes byČervený et al. (1984), where the two-point paraxial traveltime for-mula for isotropic inhomogeneous media is presented for the firsttime. Mispel et al. (2003) study the behavior of the parabolic andhyperbolic approximations in a typical seismic reflection imagingcontext, where the source and receiver points are fixed and the scat-tering point is allowed to vary.In this paper, we use the parabolic formula of Červený et al.
(2012). For computation in isotropic media, we use the standardray tracer for computing reference rays with linear dynamic ray trac-ing along them. Along the reference rays, we compute standardray-theory traveltimes. The ray-theory traveltimes obtained by raytracing are used in the two-point paraxial formula and also for test-ing its accuracy. For computations in anisotropic media, we usefirst-order ray tracing and DRT (FORT and FODRT), see Pšenčíkand Farra (2005, 2007). This is a technique that allows for approxi-mate, but relatively accurate, computations in inhomogeneousweakly and moderately anisotropic media. We test the accuracyof the two-point paraxial traveltime formula by comparing the trav-eltimes between S 0 and R 0 with the traveltimes obtained with thestandard ray tracer in the isotropic case and with the FORT in theanisotropic case.In the following section, we present the two-point paraxial trav-
eltime formula derived by Červený et al. (2012) and briefly describethe quantities required to evaluate it. We then present numerical ex-amples, which we use to illustrate the applicability of the formula inisotropic and anisotropic models. We also show its application to anexperiment, in which we estimate the traveltimes between a shiftedsource and arbitrarily shifted systems of receivers. For this purpose,we use the traveltimes, ray-tracing, and DRT quantities obtainedalong the reference rays connecting the source and receivers beforethe shift. Results are shown for isotropic as well as anisotropic mod-els. We end the paper with a short section containing concludingremarks. Appendix A is devoted to the analysis of the two-pointparaxial traveltime formula in homogeneous media. Appendix Bis devoted to the Shanks transform.We use the componental and matrix notation. In the componental
notation, the upper-case indices (I; J; K; : : : ) take the values 1 and2, and the lower-case indices (i; j; k; : : : ) the values 1, 2, or 3. TheEinstein summation convention is used. In the matrix notation, thematrices and vectors are denoted by bold upright symbols.
TWO-POINT PARAXIAL TRAVELTIME FORMULA
Let us consider a single reference rayΩ with points S and R on it,situated in a 3D laterally inhomogeneous, isotropic, or anisotropicmedium of arbitrary symmetry. Let us denote τ the traveltime var-iable alongΩ, increasing from τ0 at S to τ at R. The medium may ormay not contain curved structural interfaces. Let us try to estimatethe traveltime between points S 0 and R 0 arbitrarily chosen in theparaxial vicinities of points S and R, respectively. If traveltimeTðR; SÞ between S and R and the results of ray tracing and DRTalong the reference ray are available, we can estimate TðR 0; S 0Þfrom the two-point paraxial traveltime formula proposed by Čer-vený et al. (2012),
TðR 0; S 0Þ ¼ TðR; SÞ þ δxRi piðRÞ − δxSi piðSÞ
þ 1
2δxRi ½fRMiðP2Q−1
2 ÞMNfRNj þ ΦijðRÞ�δxRj
þ 1
2δxSi ½fSMiðQ−1
2 Q1ÞMNfSNj − ΦijðSÞ�δxSj
− δxSi fSMiðQ−1
2 ÞMNfRNjδx
Rj : (1)
The symbols Q1 ¼ Q1ðR; SÞ, Q2 ¼ Q2ðR; SÞ, and P2 ¼P2ðR; SÞ in equation 1 represent the 2 × 2 submatrices of the4 × 4 ray propagator matrix ΠðR; SÞ in ray-centered coordinates,
The ray propagator matrix ΠðR; SÞ is determined by solving theDRT system,
dΠðτ; τ0Þ∕dτ ¼ SðτÞΠðτ; τ0Þ; (3)
along ray Ω from S to R, with initial conditions at τ ¼ τ0,
Πðτ0; τ0Þ ¼ I: (4)
Matrix SðτÞ in differential equation 3 is the 4 × 4 DRT system ma-trix, matrix I in equation 4 is the 4 × 4 identity matrix. For moredetails, see Červený et al. (2012).The other symbols in equation 1 are defined as follows
Symbols xiðSÞ and xiðRÞ denote the Cartesian coordinates of pointsS and R on reference ray Ω; xiðS 0Þ and xiðR 0Þ denote the Cartesiancoordinates of points S 0 and R 0 situated in the paraxial vicinities ofS and R, respectively, see Figure 1. Symbols pi, Ui, and ηi are thecomponents of slowness vector p, of ray-velocity vector U and ofthe time derivative of p along the ray, ηðτÞ ¼ dpðτÞ∕dτ, respec-tively, all determined while tracing the reference ray. The upper in-dices S and R indicate that the corresponding quantities areconsidered at point S or R.Symbols fSMi and f
RMi ðM ¼ 1; 2Þ in formula 1 represent the Car-
tesian components of vectors f1 and f2 perpendicular to referenceray Ω at S and R, respectively. Vectors f1 and f2 are determinedfrom equations
f1 ¼ C−1ðe2 × UÞ; f2 ¼ C−1ðU × e1Þ: (7)
Here, U is the ray-velocity vector along reference ray Ω and C is therelevant phase velocity. Vectors eI (I = 1, 2, 3) can be obtained bysolving, along Ω, the vectorial, ordinary differential equation
deI∕dτ ¼ −ðeI · ηÞp∕ðp · pÞ: (8)
Vectors eI are situated in the plane tangent to the wavefront andform a right-handed orthonormal triplet with vector e3 ¼ Cp. It issufficient to solve equation 8 for only one of the vectors e1, e2. The
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other can be determined from the condition of orthonormality ofvectors ei. Note that vectors f1 and f2 need not necessarily be unitand orthogonal in anisotropic media.Once reference ray Ω and the above-mentioned quantities calcu-
lated along it are available, the two-point paraxial traveltimes be-tween points S 0 and R 0, arbitrarily chosen in the vicinities of Sand R, can be calculated without a problem. Let us mention thatformula 1 fails to work properly if the variation of the model param-eters in the vicinity of the reference ray is too strong, or if matrixQ2
is singular at point R. The latter problem occurs when there is acaustic at point R or when point R is too close to S, see the testslater. This is physically understandable as we cannot extrapolate thetraveltime to large distances from a short segment of the refer-ence ray.In the next section, in one example, we also test the formula for
T2ðR 0; S 0Þ. It has the following form (Červený et al., 2012):
The meaning of the quantities appearing in formula 9 is the same asin formula 1. The formula for T2ðR 0; S 0Þ was obtained by squaringformula 1 and neglecting the terms of higher order than two in δx.
TESTS
We test the two-point paraxial traveltime formula 1 in models ofhomogeneous and inhomogeneous and isotropic and anisotropicmedia. Standard ray tracing and DRT along the reference ray fromS to R are used in isotropic media, even in cases when analytic sol-utions are known. In anisotropic media, we use the FORT approach(Pšenčík and Farra, 2005, 2007) instead of standard ray tracing foranisotropic media, for which formula 1 was designed. Along first-order rays, we perform first-order DRT in ray-centered coordinates.Quantities obtained in this way are used in formula 1.As an illustration of the accuracy of the two-point paraxial trav-
eltime formula, we compare its results TðR 0; S 0Þ with TexðR 0; S 0Þ,where TexðR 0; S 0Þ is obtained from standard two-point ray tracing.We consider 2D models, mostly of vertically inhomogeneous me-dia. Note that in 2D models of vertically inhomogeneous media,vectors eIðτÞ and fIðτÞ can be determined analytically, without solv-ing equation 8.We consider first the case of S 0 ≡ S with the point source fixed at
S. In this case, equations 1 and 9 simplify considerably becauseδxSi ¼ 0, and several terms on the right-hand side of equations 1and 9 vanish. Points R 0 are situated at the nodes of a rectangulargrid covering the studied region. The separation of horizontal andvertical grid lines is 0.1 km. In addition to studying the completeformula 1 for the two-point paraxial traveltime, we also study theeffects of the linear and quadratic terms in δx in formula 1. Moreprecisely, we study formula 1 with the quadratic terms ignored,yielding the two-point traveltime T linðR 0; SÞ (“lin” indicates thatonly linear terms are retained) and with the linear terms ignored,yielding the two-point traveltime TquadðR 0; SÞ (“quad” indicates thatonly quadratic terms are retained). This separation of “linear” and
“quadratic” terms clearly shows where they play an important role.The white curve (line) in the plots indicates the reference ray be-tween point S and point R. If not specified otherwise, S ≡ ð0; 0Þand R ≡ ð2.5; 2.5Þ.
Isotropic models
We first investigate the accuracy of formula 1 for TðR 0; SÞ in ahomogeneous isotropic medium. In certain situations in thiscase, it is possible to compare the approximate formulas withthe exact, see Appendix A. Here we show the comparison ofresults of the approximate two-point paraxial traveltime formulaand of the two-point traveltimes based on standard ray formulas.We emphasize again that we are studying the accuracy of thetwo-point paraxial traveltime formula 1 in all figures. Only inone figure, we briefly discuss the accuracy of T2ðR 0; SÞ. Forsupplementary test results in isotropic media, see Gjøystdal et al.(1984).In Figure 2, we can see the traveltime differences
TðR 0; SÞ − TexðR 0; SÞ measured in seconds, where TðR 0; SÞ standsfor T linðR 0; SÞ in the upper plot, for TquadðR 0; SÞ in the middle plotand for the complete two-point paraxial traveltime, given by for-mula 1, in the bottom plot. Note that TðR; SÞ ¼ 1.7678 s. As shownin Appendix A, see equations A-2 and A-4, formula 1 for TðR 0; SÞreduces in a homogeneous medium to the formula for T linðR 0; SÞand yields the exact traveltime along the reference ray. This canbe seen in the upper plot of Figure 2. Perpendicular to the ray,T linðR 0; SÞ is constant, equal to the two-point traveltime TðR; SÞat point R on the reference ray. The exact traveltime is, therefore,always larger than T linðR 0; SÞ in the vicinity of the reference ray.Thus, we can see only negative or zero (on the reference ray) trav-eltime differences in the upper plot. As the curvature of the wave-front (with the exact two-point traveltime on it) decreases withincreasing distance from point S, the region of small traveltimedifferences (and thus the higher accuracy of T linðR 0; SÞ) broadens.In accordance with the conclusions of Appendix A, the accuracy ofTquadðR 0; SÞ is very high along the normal to the reference ray atpoint R. The strong gradient of the traveltime differences parallelto the reference ray is caused by the linear term missing inTquadðR 0; SÞ. It is interesting to see the combined effects of the lin-ear and quadratic terms in the bottom plot of Figure 2. The bottomplot also shows that the two-point paraxial traveltime formula 1increasingly underestimates the exact traveltime with increasingperpendicular distance from the reference ray between S and R.On the contrary, the approximate traveltimes are increasingly over-estimated in the direction perpendicular to the reference ray beyondpoint R (this part of the reference ray is not shown). Figure 3 showsthe same as the bottom plot of Figure 2, but in the form of isolines. Itprovides a better quantitative estimate of the accuracy of formula 1.The reference ray between S and R is shown as the bold black curve.We can see that the region of high accuracy of TðR 0; SÞ forms a kindof cross at point R with the part perpendicular to the reference rayslightly curved.In Figure 4, we can see the same as in Figure 2, but for the iso-
tropic model with P-wave velocity of 2 km∕s at z ¼ 0 km and witha constant vertical gradient of 0.9 s−1. TðR; SÞ ¼ 1.1594 s in thiscase. We can see that the velocity gradient has distorted consider-ably the distribution of the traveltime differences as well as thetrajectory of the reference ray. Note that the color scales for thedifferences of T linðR 0; SÞ, TquadðR 0; SÞ, and TðR 0; SÞ from
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TexðR 0; SÞ are the same in Figures 2 and 4. In the upper plot ofFigure 4, we can see that, except for the vicinity of point R, thereference ray is no longer a place of the highest accuracy ofT linðR 0; SÞ. In fact, the region of the highest accuracy ofT linðR 0; SÞ around R splits in the direction to and away from pointS. The traveltime differences T linðR 0; SÞ − TexðR 0; SÞ are no longeronly negative or zero, they are now also positive. Interesting also isthe behavior of TquadðR 0; SÞ. The curve, along which TquadðR 0; SÞ ¼TexðR 0; SÞ is still perpendicular to the reference ray at R, but withincreasing distance from it, it is strongly curved. The strong gradientof the traveltime differences TquadðR 0; SÞ − TexðR 0; SÞ along thereference ray remains. The map of the complete traveltimedifferences TðR 0; SÞ − TexðR 0; SÞ in the bottom plot differs fromits counterpart in Figure 2. The traveltime differences are not zeroalong the reference ray as they are in homogeneous media. Thegreatest distinction is a relatively large traveltime difference inthe vicinity of point S. The symmetric picture from the bottom plotof Figure 2 is completely distorted due to the velocity gradient. Thiscan also be seen in Figure 5, which shows the same as the bottomplot of Figure 4, but in the form of isolines.Next, we study the effects of the length of the reference ray be-
tween the points S and R, specifically of the size of TðR; SÞ, on theaccuracy and size of the region of applicability of formula 1. Weconsider an isotropic model with a P-wave velocity of 2 km∕s atz ¼ 0 km and a constant vertical gradient of 0.7 s−1 in all plots.Figure 6 shows the traveltime differences TðR 0; SÞ − TexðR 0; SÞfor points R whose distance from point S is successively increasing.Point S is situated at (0, 0), point R is situated successively at
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Figure 3. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the isotropic homogeneous model. TexðR 0; SÞ — the stan-dard ray theory traveltime. The plot corresponds to the bottom plotof Figure 2, where TðR 0; SÞ is the complete two-point paraxial trav-eltime determined from formula 1. Points S and R are situated at thebeginning and end of the reference ray — black curve. Point S issituated at (0, 0). Point R is situated at (2.5, 2.5) and points R 0 in thegrid covering the studied region.
Range (km)
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Figure 2. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the isotropic homogeneous model. TexðR 0; SÞ — standardray theory traveltime. Here, (a) TðR 0; SÞ ¼ T linðR 0; SÞ — quadraticterms suppressed; (b) TðR 0; SÞ ¼ TquadðR 0; SÞ — linear terms sup-pressed; (c) TðR 0; SÞ — the complete two-point paraxial traveltimedetermined from formula 1. Points S and R are situated at the be-ginning and end of the reference ray — white curve. Point S issituated at (0, 0). Point R is situated at (2.5, 2.5) and points R 0 in thegrid covering the studied region.
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(0.1, 0.1), (0.5, 0.5), (1, 1), (3, 3), (4, 4), and (5, 5) and the corre-sponding two-point traveltimes TðR; SÞ are 0.0695, 0.3225, 0.6041,1.4221, 1.72, and 1.9717 s, respectively. Point S 0 is again chosen ascoinciding with S, and points R 0 are situated at the grid points of arectangular grid covering the studied region. For better comparison,we use the same color scale in all the plots. The reference ray isshown again as a white curve connecting points S and R. In theupper left corner of Figure 6, we can see the confirmation of thetheoretical observation that formula 1 is less accurate if the refer-ence ray between S and R is very short (small TðR; SÞ). Point R issituated at (0.1, 0.1) in this case. The region, in which formula 1works relatively well, say with an accuracy well below 0.2 s, is verynarrow. It underestimates the exact traveltimes in the direction ofpropagation (approximately along the diagonal of the plot). Onthe contrary, the traveltimes determined from formula 1 overesti-mate the exact traveltimes outside the narrow diagonal region.For R situated at (0.5, 0.5) and (1, 1), the region of accuracy under0.2 s broadens and for R in the middle of the studied region, spe-cifically at (3, 3), the approximately determined traveltimes differby less than 0.2 s from the exact ones in the whole studied region.With further increase of distance of R from S, for R situated at (4, 4)and (5, 5) the accuracy around S slightly decreases in the way ob-served in Figure 4, but remains high in the rest of the studiedregion.In Figure 7, we compare the accuracy of equations 1 and 9. In
the left column, we present the traveltime differences jTðR 0; SÞ−TexðR 0; SÞj calculated for TðR 0; SÞ obtained from formula 1.In the right column, we present the traveltime differences for
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Figure 5. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the isotropic model with constant vertical gradient of0.9 s−1. TexðR 0; SÞ — the standard ray theory traveltime. Theplot corresponds to the bottom plot of Figure 4, where TðR 0; SÞis the complete two-point paraxial traveltime determined from for-mula 1. Points S and R are situated at the beginning and end of thereference ray — black curve. Point S is situated at (0, 0). Point R issituated at (2.5, 2.5) and points R 0 in the grid covering the studiedregion.
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Figure 4. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the isotropic model with constant vertical gradient of0.9 s−1. TexðR 0; SÞ — the standard ray theory traveltime. Here,(a) TðR 0; SÞ ¼ T linðR 0; SÞ — quadratic terms suppressed;(b) TðR 0; SÞ ¼ TquadðR 0; SÞ — linear terms suppressed;(c) TðR 0; SÞ — the complete two-point paraxial traveltime deter-mined from formula 1. Points S and R are situated at the beginningand end of the reference ray — white curve. Point S is situated at(0, 0). Point R is situated at (2.5, 2.5) and points R 0 in the grid cover-ing the studied region.
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TðR 0; SÞ obtained as the square root of T2ðR 0; SÞ obtained fromformula 9. The comparisons of the traveltime differences are madefor the isotropic homogeneous model (Figure 7a and 7d) and iso-tropic models with constant vertical gradients of 0.5 (Figure 7band 7e) and 0.9 s−1 (Figure 7c and 7f). Again, the color scaleis the same for all plots. As shown in Appendix A, equation 9is exact in isotropic homogeneous media. This is clearly seenin Figure 7d. In inhomogeneous media, however, the accuracyof equation 9 rapidly decreases and becomes comparable or evenlower than the accuracy of formula 1, see Figure 7d and 7f. Let usemphasize that all the above plots were obtained with the use of asingle reference ray only.To illustrate how formula 1 works in more complicated models,
we test it in an isotropic anticline model; see Figure 8a, whichshows the P-wave velocity distribution in the considered model.In this model, we perform a similar test as in Figures 2 and 4.The bottom plot of Figure 8 shows the traveltime differencesTðR 0; SÞ − TexðR 0; SÞ as in the bottom plots of Figures 2 and 4.Point S is situated outside the plot, at (−1; 0);point R is located at (2.5, 2.5). Both points areconnected by the reference ray (white curve).Note that TðR; SÞ ¼ 1.733 s. Despite the morecomplicated structure, the performance of for-mula 1 seems to be even better than in the 1Dmodel of Figure 4 (note the different colorscales). This indicates that the main factor reduc-ing the accuracy of formula 1 is the curvature ofthe reference ray. The curvature of the referenceray in Figure 8 is smaller than in Figure 4.So far, we have studied formula 1 with S 0 ≡ S,
i.e., with S corresponding to a point source. InFigure 9, we test the performance of formula 1for both points, S and R, shifted. Because, in thisexperiment, points S 0 and R 0 differ from points Sand R, all terms on the right-hand side of for-mula 1 are involved in the procedure. The testis made in the isotropic model with a P-wavevelocity of 2 km∕s at z ¼ 0 km and with a con-stant vertical gradient of 0.7 s−1. Figure 9 showsthe traveltime differences TðR 0; S 0Þ − TexðR 0; S 0Þ,where TexðR 0; S 0Þ represents the standard ray-theory traveltime and TðR 0; S 0Þ is the two-pointparaxial traveltime obtained from formula 1along the reference ray (denoted by the whitecurve) between S (outside the plot) and R. PointS situated at (0;− 0.5) in the upper plot and at(−0.5; 0) in the bottom plot is shifted to pointS 0 (0, 0). Points R 0 are again situated at the nodesof a rectangular grid covering the model. Travel-times TðR; SÞ are 1.5 s in Figure 9a and 1.3725 sin Figure 9b. We can see that the effects of theshift of point S are only small. When comparingthe plots with Figure 4c (where a higher gradientis used), effectively we can observe little changewhen S is shifted vertically (top). A slightlybroader region of lower accuracy around pointS 0 and a faster decrease of the accuracy awayfrom point R can be observed for the horizontalshift of point S (bottom).
In the following, we use formula 1 in an experiment performedearlier by Alkhalifah and Fomel (2010), see the sketch in Figure 10.Alkhalifah and Fomel (2010) use two-point traveltimes calculatedfrom one point source (S) to a system of receivers (R; in Figure 10represented by two points R at selected nodes) distributed on arectangular grid to estimate the traveltimes between the shiftedsource (S 0) and shifted receivers (R 0). In Figure 10, source S isshifted horizontally by one grid interval and vertically by two gridintervals. The described procedure is important, for example,in Kirchhoff modeling, migration or in velocity estimation ap-proaches. To calculate the two-point traveltimes, Alkhalifah andFomel (2010) use the approach based on the numerical solutionof the eikonal equation. Here, we use the two-point paraxial trav-eltime formula 1. We can proceed in several ways. We can use asingle reference ray between S and one of the points R situatedin the middle of the grid and then use formula 1 to estimateTðR 0; S 0Þ. The results of such an experiment would not differ muchfrom the tests in Figure 9.
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Figure 6. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in seconds) in the isotropicmodel with constant vertical gradient of 0.7 s−1. Here, TðR 0; SÞ is the two-point paraxialtraveltime determined from formula 1 and TexðR 0; SÞ is the standard ray theory trav-eltime. Points S and R are situated at the beginning and end of the reference ray— white curve. (a) S is situated at (0, 0), point R at (0.1, 0.1), (b) (0.5, 0.5),(c) (1, 1), (d) (3, 3), (e) (4, 4), and (f) (5, 5). Points R 0 are situated in the grid coveringthe studied region.
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To generate the plots in Figure 11, we have used a different andmore accurate approach. We calculated the reference rays betweensource S and every receiver R, and solved the DRT equations alongthem. Using formula 1, we then recalculated the original two-pointtraveltime table TðR; SÞ to yield a new table TðR 0; S 0Þ. The shifts ofthe source and of the receivers may be the same or different (inprinciple, each of the receivers R could be shifted in a differentway). The original traveltime table TðR; SÞ may be used to generatean arbitrary number of new tables TðR 0; S 0Þ without need of newray tracing and DRT.Figure 11a shows the absolute values of the traveltime difference
jTðR 0; S 0Þ − TexðR 0; S 0Þj in seconds, calculated by formula 1 forthe model and configuration used by Alkhalifah and Fomel(2010). The model is isotropic with a P-wave velocity of 2 km∕sat z ¼ 0 km and a constant vertical gradient of 0.7 s−1. Source Sand all receivers R are shifted by 0.2 km in the positive z direction.In both plots of Figure 11, the shifted source S 0 is at (0, 0) (the
position of the original source S is thus outside the frames ofthe plots). Comparison with Figure 2b of Alkhalifah and Fomel(2010) shows that the traveltime differences of the two-point parax-ial traveltime formula are: (1) in most of the studied regionby nearly one order lower (less than 0.001 s) than those of theeikonal-based approach, the highest traveltime differences (in theright upper corner) being around 0.003 s, and (2) varying negligiblyin the whole studied region. The superiority of our approach isnatural because Alkhalifah and Fomel (2010) use the first-order ex-pansion, while here we have used the second-order expansion.Figure 11b shows a map of the absolute traveltime differences cor-responding to a vertical shift of the receivers different from thesource. The source is shifted by 0.2 km and the receivers by0.4 km. We can observe generally larger differences than inFigure 11a. The largest differences are concentrated in the closevicinity of the shifted source S 0 (0, 0). In a substantial part of themodel, however, the traveltime differences do not exceed 0.005 s.
In Figure 12a, we show the traveltimedifferences jTðR 0; S 0Þ − TexðR 0; S 0Þj (in sec-onds) in the isotropic model with a constant ver-tical gradient of 0.7 s−1 and horizontal gradientof 0.5 s−1. The source and receivers are shiftedvertically by 0.8 km. The shifted source S 0 islocated at (0, 0). Despite the considerably largeshift, the traveltime differences are reasonablysmall, not exceeding 0.025 s. For formulas likeformula 1, which have the form of the Taylorexpansion formula, Alkhalifah and Fomel (2010)suggest using the Shanks transform (Bender andOrszag, 1978). The Shanks transform can en-hance the accuracy of such formulas if appliedwith care. Figure 12b shows the top map afterthe application of the Shanks transform, seeAppendix B. We can see that the accuracy hasincreased dramatically. The traveltime differencein the whole studied region does not exceed0.005 s.
Anisotropic models
Previous tests were made on models of homo-geneous or inhomogeneous isotropic media.Now, we show several tests on anisotropic mod-els. We start with the model of a transverselyisotropic medium (∼8% anisotropy) with thehorizontal axis of symmetry (HTI) varying lin-early with depth, see the schematic picture inFigure 13, which shows how the axis of sym-metry varies with depth. We call the model the“twisted crystal model 8%.” We consider tworealizations of the model. In one, which we callthe “homogeneous model,” density-normalizedelastic moduli, measured in ðkm∕sÞ2, are con-stant throughout the model; only the axis of sym-metry rotates with depth. In the other model,called the “inhomogeneous model,” the density-normalized elastic moduli vary with depth in ad-dition to the variation of the axis of symmetry.The moduli are specified as follows:
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Figure 7. Traveltime differences jTðR 0; SÞ − TexðR 0; SÞj (in seconds) for TðR 0; SÞ de-termined from formula 1 (left column) and for TðR 0; SÞ determined as the square root ofT2ðR 0; SÞ calculated using formula 9. (a and d) The isotropic homogeneous model, (band e) isotropic model with constant vertical gradient of 0.5 s−1, and (c and f) 0.9 s−1 areconsidered. TexðR 0; SÞ — the standard ray theory traveltime. Points S and R are situatedat the beginning and end of the reference ray — white curve. Point S is situated at (0, 0),point R at (2.5, 2.5) and points R 0 in the grid covering the studied region.
at z ¼ 5 km. The Thomsen parameters (referenced to the axis ofsymmetry) are constant throughout: ϵ ¼ 0.0866 and δ ¼ 0.0816.
In the homogeneous model, the matrix for z ¼ 0 km is usedthroughout the model; in the inhomogeneous model, the density-normalized elastic moduli between z ¼ 0 and 5 km are obtainedby linear interpolation from the above values. In both cases, the axisof symmetry makes an angle of −45° with the x-axis at z ¼ 0 km
and 0° at z ¼ 5 km. We consider S 0 ≡ S at (0, 0) and point R at (2.5,2.5). This again leads to considerable simplification of equation 1because δxSi ¼ 0, and several terms on the right-hand side of equa-tion 1 vanish.In Figure 14, we can see that the behavior of the traveltime
differences TðR 0; SÞ − TexðR 0; SÞ in the homogeneous twistedcrystal model 8% resembles very much their behavior in the homo-geneous isotropic model, see Figure 2. In this case, TðR; SÞ ¼0.917 s. The greatest difference between Figures 2 and 11 arethe traveltime differences. For T linðR 0; SÞ and TquadðR 0; SÞ, thedifferences in Figure 14 are about half of the differences in Figure 2.The differences of the complete two-point paraxial traveltime areeven smaller. The explanation is simple. The average velocity in thehomogeneous twisted crystal model is approximately 4 km∕s, while
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Figure 8. (a) Isotropic anticline model and (b) traveltime differencesTðR 0; SÞ − TexðR 0; SÞ (in seconds) in this model. Here, TðR 0; SÞ isthe two-point paraxial traveltime determined from formula 1 andTexðR 0; SÞ is the standard ray theory traveltime. Points S ≡ S 0 andR are situated at the beginning and end of the reference ray — whitecurve. Point S is situated at (−1; 0) and, point R at (2.5, 2.5). PointsR 0 are situated in the grid covering the studied region.
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Figure 9. Traveltime differences TðR 0; S 0Þ − TexðR 0; S 0Þ (in sec-onds) in the isotropic model with constant vertical gradient of0.7 s−1. TðR 0; S 0Þ is the two-point paraxial traveltime determinedfrom formula 1 and TexðR 0; S 0Þ is the standard ray theory travel-time. Points S and R are situated at the beginning and end ofthe reference ray — white curve. Point S is situated (a) at(0;−0.5) and (b) at (−0.5; 0). Point R is situated at (2.5, 2.5) in bothplots. Point S 0 is in both plots at (0, 0). Points R 0 are situated at thenodes of the grid covering the studied region.
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in the homogeneous isotropic model it is 2 km∕s. Due to this,TðR; SÞ in Figure 2 is 1.7678 s and in Figure 14, it is 0.917 s.For higher velocities, the traveltime differences are smaller.Figure 15 shows, as in the isotropic case, the same as the bottomplot in Figure 14, but in the form of isolines. We can see that theregion of high accuracy of TðR 0; SÞ again forms a kind of cross atpoint R, similarly as in isotropic media.Figure 16 shows the same as Figure 14, but for the inhomo-
geneous twisted crystal model 8%. In this case, TðR; SÞ ¼0.8069 s. The resemblance to Figure 4 is not as strong as in thecase of the homogeneous models. This is mostly caused by the dif-ferent gradients in Figures 4 and 16. The fact that the gradient usedto generate Figure 16 is weaker is indicated by the small curvatureof the reference ray connecting points S and R. Because the averagevelocity in the model used to generate Figure 16 is approxi-mately 5 km∕s, the differences of the two-point paraxial travel-time TðR 0; SÞ in Figure 16 are again substantially smaller than inFigure 4. The bottom plot of Figure 16 in the form of isolines can beseen in Figure 17.Figure 18 shows the traveltime differences jTðR 0; SÞ − TexðR 0; SÞj
in seconds for a similar experiment as in Figure 12, now for the inho-mogeneous twisted crystal model 8%. The source and receivers areshifted vertically by 0.8 km. The shifted source S 0 is located at (0, 0).Despite the considerably large shift, the traveltime differences shownin the upper plot are small, not exceeding 0.005 s. In a large part ofthe studied region, they do not exceed 0.003 s. Figure 18b shows, asin Figure 12, the traveltime differences after application of the Shankstransform. As in the isotropic case, we can see a dramatic reduction inthe traveltime differences. They do not exceed 0.002 s in the wholeregion; mostly they are even smaller. In contrast to Figure 18a, thetraveltime differences vary only negligibly.Figure 19 shows the same as Figure 18, but for the model with the
stronger anisotropy. We call the model the “twisted crystal model
20%” because we deal again with an HTI model whose axis of sym-metry rotates with depth, and its anisotropy is 20%. The density-normalized elastic moduli are specified as follows:
Figure 10. Use of the two-point paraxial traveltime formula to es-timate two-point traveltime between the shifted source S 0 and theshifted receivers R 0. The source S is shifted horizontally (by onegrid interval) and vertically (by two grid intervals) to the new posi-tion S 0. Every receiver R (the receivers are situated in all grid points,here represented by two selected grid points) is shifted vertically (bytwo grid intervals) to the new position R 0. Two-point traveltimesalong reference rays (black solid curve), from source S to receiversR are assumed to be known. Dashed curves are shown only forillustration, no rays connecting S 0 and R 0 are necessary.
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Figure 11. Application of the two-point paraxial traveltime formulato the estimation of the traveltimes between shifted source S 0 andthe shifted system of receivers R 0. The plots show the traveltimedifferences jTðR 0; S 0Þ − TexðR 0; S 0Þj (in seconds) in the isotropicmodel with constant vertical gradient of 0.7 s−1. Shifted sourceS 0 at (0, 0) and shifted receivers R 0 in the grid covering the studiedregion. (a) Source S and receivers R shifted vertically by 0.2 km; cf.Figure 2b of Alkhalifah and Fomel (2010). (b) Source S shifted ver-tically by 0.2 km, receivers R shifted vertically by 0.4 km.
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at z ¼ 5 km. The Thomsen parameters (referenced to the axis ofsymmetry) are constant throughout: ϵ ¼ 0.197 and δ ¼ 0.2. Theaxis of symmetry makes an angle of 00 at z ¼ 0 km and −450at z ¼ 5 km. The moduli between z ¼ 0 and 5 km are again ob-tained by linear interpolation.Figure 19a shows the traveltime differences for the twisted crystal
model 20% and the same configuration as in Figure 18. The bottomplot shows the differences after the Shanks transform. We cansee that traveltime differences in both plots are smaller than inFigure 18. This is because of the smaller vertical gradient in themodel used to generate the plots in Figure 19. Due to it, rays inFigure 19 are less curved than in Figure 18 and formula 1 worksbetter. The more the reference ray is curved, the narrower the vicin-ity of the reference ray, in which formula 1 yields accurate results.
Figure 13. Schematic picture of rotating horizontal axis of sym-metry in the twisted crystal model 8%. The axis of symmetry makesan angle of −45° and 0° with the x-axis at z ¼ 0 and z ¼ 5 km,respectively.
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Figure 14. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the homogeneous twisted crystal model 8%. TexðR 0; SÞ— FORT traveltime. Here, (a) TðR 0; SÞ ¼ T linðR 0; SÞ — quadraticterms suppressed; (b) TðR 0; SÞ ¼ TquadðR 0; SÞ — linear terms sup-pressed; (c) TðR 0; SÞ — the complete two-point paraxial traveltimedetermined from formula 1. Points S and R are situated at the be-ginning and end of the reference ray — white curve. Point S issituated at (0, 0). Point R is situated at (2.5, 2.5) and points R 0 in thegrid covering the studied region.
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Figure 12. Application of the two-point paraxial traveltime formulato the estimation of the traveltimes between shifted source S 0 andthe shifted system of receivers R 0. The plots show traveltimedifferences jTðR 0; S 0Þ − TexðR 0; S 0Þj (in seconds) in the isotropicmodel with constant vertical gradient of 0.7 s−1 and horizontal gra-dient of 0.5 s−1. Shifted source S 0 at (0, 0) and shifted receivers R 0in the grid covering the studied region. (a) The source and receiversshifted vertically by 0.8 km. (b) The same as above after the Shankstransform.
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CONCLUSIONS
The above tests of the two-point paraxial traveltime formulashow its potential to estimate, approximately, two-point traveltimesin a rather broad vicinity of a single reference ray, along whichquantities obtained during ray tracing and linear dynamic ray trac-ing between points S and R are available. The formula is applicableto very general 3D smoothly laterally varying isotropic or aniso-tropic structures with or without smooth curved interfaces.Anisotropy of arbitrary type and strength may be considered. Thepositions of points S and R may be varied arbitrarily. In the numeri-cal tests, we concentrated on studying the two-point paraxial trav-eltimes of P-waves propagating in 2D inhomogeneous isotropic andanisotropic media. The tests were made for point S fixed and Rvarying as well as for S and R varying.Our tests demonstrated the crucial role of dynamic ray tracing in
the two-point paraxial traveltime computations. Although the linearpart of the two-point paraxial traveltime formula (with respect to thespatial coordinates) yields accurate estimates of the two-point trav-eltime only in the very close vicinity of the reference ray aroundpoint R, addition of the quadratic terms, whose coefficients are cal-culated from dynamic ray tracing, broadens the region of high ac-curacy of the estimated two-point traveltimes considerably. Theaccuracy increases not only in the region between S and R, but alsobeyond both points. The performed tests indicate that the accuracyof the two-point paraxial traveltime formula in inhomogeneous me-dia is primarily sensitive to the length and the curvature of the refer-ence ray. The accuracy depends only weakly on anisotropy.The tests performed have also shown that the accuracy of the
formula for T2ðR 0; SÞ, which is exact in a homogeneous isotropic
medium, decreases with increasing inhomogeneity of the mediumto become comparably or even less accurate than the formula forTðR 0; SÞ. This observation, however, may be model-dependent anddeserves further investigation.
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Figure 15. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the homogeneous twisted crystal model 8%. TexðR 0; SÞ— FORT traveltime. The plot corresponds to the bottom plot ofFigure 14, where TðR 0; SÞ is the complete two-point paraxial trav-eltime determined from formula 1. Points S and R are situated at thebeginning and end of the reference ray — black curve. Point S issituated at (0, 0). Point R is situated at (2.5, 2.5) and points R 0 in thegrid covering the studied region.
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Figure 16. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the inhomogeneous twisted crystal model 8%. TexðR 0; SÞ— FORT traveltime. Here, (a) TðR 0; SÞ ¼ T linðR 0; SÞ — quadraticterms suppressed; (b) TðR 0; SÞ ¼ TquadðR 0; SÞ — linear terms sup-pressed; (c) TðR 0; SÞ — the complete two-point paraxial traveltimedetermined from formula 1. Points S and R are situated at the begin-ning and end of the reference ray — white curve. Point S is situatedat (0, 0). Point R is situated at (2.5, 2.5) and points R 0 in the gridcovering the studied region.
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As shown in numerical examples, the two-point paraxial travel-time formula can be used for very efficient estimation of the trav-eltime at an arbitrary point situated in a rather broad vicinity of thereference ray between points S and R. Because the ray tracing anddynamic ray tracing quantities are available at any point of the refer-ence ray between S and R, we can, without any additional expenses,consider any such point as point R and apply the two-point paraxialtraveltime formula at its position.The applicability of the two-point paraxial traveltime formula
also has some limitations. The formula is not expected to workproperly in strongly varying media and in media in which multi-pathing may occur. Its accuracy decreases, as illustrated in this pa-per, with decreasing time TðR; SÞ between points S and R on thereference ray.The properties of the two-point paraxial traveltime formula were
mostly illustrated by numerical comparisons. Certain conclusionsabout the properties of the formula can also be deduced from itsform and behavior in homogeneous media. We have shown that,in homogeneous isotropic or anisotropic media, the formula yieldsexact results along the reference ray, inside and outside intervalspecified by points S and R. In homogeneous or weakly inhomo-geneous isotropic or anisotropic media, it yields highly accurateresults in the direction perpendicular to the reference ray at R. This
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Figure 17. Traveltime differences TðR 0; SÞ − TexðR 0; SÞ (in sec-onds) in the inhomogeneous twisted crystal model 8%.TexðR 0; SÞ — FORT traveltime. The plot corresponds to the bot-tom plot of Figure 16, where TðR 0; SÞ is the complete two-pointparaxial traveltime determined from formula 1. Points S and Rare situated at the beginning and end of the reference ray — blackcurve. Point S is situated at (0, 0). Point R is situated at (2.5, 2.5)and points R 0 in the grid covering the studied region.
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Figure 18. Application of the two-point paraxial traveltime formulato the estimation of traveltimes between shifted source S 0 and thesystem of receivers R 0. The plots show the traveltime differencesjTðR 0; S 0Þ − TexðR 0; S 0Þj (in seconds) in the inhomogeneoustwisted crystal model 8%. Shifted source S 0 at (0, 0) and shiftedreceivers R 0 in the grid covering the studied region. (a) The sourceand receivers shifted vertically by 0.8 km. (b) The same as aboveafter the Shanks transform.
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Figure 19. Application of the two-point paraxial traveltime formulato the estimation of traveltimes between shifted source S 0 and thesystem of receivers R 0. The plots show the traveltime differencesjTðR 0; S 0Þ − TexðR 0; S 0Þj (in seconds) in the inhomogeneoustwisted crystal model 20%. Shifted source S 0 at (0, 0) and shiftedreceivers R 0 in the grid covering the studied region. (a) The sourceshifted vertically by 0.8 km and receivers horizontally by 0.4 km.(b) The same as above after the Shanks transform.
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can be observed as cross-like regions of high accuracy in the vicin-ity of point R in several figures.There are many possible applications of the two-point paraxial
traveltime formula in seismology and seismic exploration. Theseapplications include fast and flexible two-point traveltime calcula-tions between sources and receivers, whose positions are specifiedin Cartesian coordinates and which are situated close to a knownreference ray. A few possible applications were discussed in ourprevious publications. For example, they include typical situationsin reflection surveys, in which source S and receiver R are situatedon a measurement surface (the earth’s surface, ocean bottom). If weknow the reference ray of a reflected wave from S to R and thecorresponding traveltime TðR; SÞ, the proposed formula can be usedto calculate traveltime TðR 0; S 0Þ for any points S 0 and R 0 situated ina vicinity of S and R without ray tracing between S 0 and R 0. Thisapplies to common source, common offset, common midpoint,common reflection surface configuration, etc. The formula may alsofind useful applications in microseismic monitoring, location of mi-croearthquakes, computation of Fresnel volumes and Fresnel zones,etc. In this paper, we have presented another application, which mayplay an important role in Kirchhoff modeling, migration or velocityestimation. Specifically, we presented the estimation of the two-point traveltimes TðR 0; S 0Þ between the shifted source S 0 and theshifted system of receivers R 0. If TðR; SÞ and the ray-tracing anddynamic ray-tracing quantities calculated between S and R areknown, it is an easy task, which can be repeated without additionalray tracing and dynamic ray tracing as many times as necessary.Comparison with results of the numerical solution of the eiko-nal-based partial differential equation indicates that the two-pointparaxial traveltime formula may provide results of higher accuracy.As shown in the paper, the shifts of points S and R may be quitelarge in comparison with their distance. Points S and R maybe shifted, without any problem, in different ways; the mediummay be 3D inhomogeneous isotropic or anisotropic, with or withoutsmooth curved interfaces. In this paper, we have concentrated onmedia without interfaces. Preliminary tests in media with interfacesindicate again the high accuracy of the two-point paraxial traveltimeformula.Many published papers have been devoted to the theory of the
two-point paraxial traveltimes, for more details refer to the Intro-duction, but only a few of them also to the study of their accuracy.The most extensive tests were performed, however, only with iso-tropic media and used the nonlinear DRT based on the Ricatti equa-tion. The approach tested in this paper is applicable to isotropic andanisotropic media and uses the linear DRT, which allows the use ofthe ray propagator matrix concept. This makes it much easier tocompute the quadratic terms of the two-point traveltime formula,as compared to using DRT based on the Riccatti equation.
ACKNOWLEDGMENTS
We would like to express our gratitude to Joachim Mispel, threeanonymous referees and the associate editor, Claudia Vanelle, fortheir stimulating comments. We are grateful to Seismic WaveAnalysis Group (SWAG) of KAUST, Saudi Arabia, project “Seis-mic waves in complex 3-D structures” (SW3D), to ResearchProjects 210/11/0117 and 210/10/0736 of the Grant Agency ofthe Czech Republic and to Research Project MSM0021620860of the Ministry of Education of the Czech Republic for support.E. Iversen acknowledges support from the Research Council of
Norway and from the European Community’s FP7 ConsortiumProject AIM “Advanced Industrial Microseismic Monitoring,”Grant Agreement no. 230669.
APPENDIX A
TWO-POINT PARAXIAL TRAVELTIME T�R 0;S 0�IN A HOMOGENEOUS MEDIUM
Formula 1 for the two-point paraxial traveltime TðR 0; S 0Þ simpli-fies considerably in homogeneous anisotropic or isotropic media.Rays in homogeneous media are straight lines, therefore, numericalray tracing is not necessary. In this case, the 4 × 4 ray propagatormatrix ΠðR; SÞ can be determined analytically so that neitherthe dynamic ray tracing is required. Moreover, only one of the four2 × 2 submatrices of ΠðR; SÞ is nonzero, specifically Q2ðR; SÞ.Exact expressions for TðR 0; S 0Þ in homogeneous media can be
determined from simple geometrical considerations. This offers asimple way of estimating the accuracy of the approximate formula 1in this special case. In addition, these exact expressions are usefultools for the correct understanding of phenomena present in thefigures of this paper.In a homogeneous anisotropic medium, the two-point paraxial
traveltime formula 1 reduces to
TðR 0; S 0Þ ¼ TðR; SÞ þ piðδxRi − δxSi Þ
þ 1
2fMiðQ−1
2 ÞMNfNjðδxRi − δxSi ÞðδxRj − δxSj Þ:(A-1)
Here we have used the same notation as in formula 1. Because slow-ness vector p and vectors fM are constant along the reference ray,i.e., they are the same at points S and R, we do not use argumentsand superscripts S and R for them.Let us consider points S 0 and R 0 situated on the reference ray, i.e.,
let us consider vectors δxS and δxR parallel to the reference ray.Because vectors fM are perpendicular to the reference ray, the quad-ratic term vanishes and equation A-1 reduces to
If we take into account the obvious relation TðR; SÞ ¼ð∂τ∕∂xiÞ½xiðRÞ − xiðSÞ� ¼ pi½xiðRÞ − xiðSÞ� (remember: in aniso-tropic media, slowness vector p need not be parallel to vectorxðRÞ − xðSÞ), we have
Symbol TexactðR 0; S 0Þ in equation A-3 denotes the exacttraveltime between the points specified in the argument. Insertingequation A-3 into equation A-2 yields
TðR 0; S 0Þ ¼ TðR; SÞ þ TexactðR 0; S 0Þ − TðR; SÞ¼ TexactðR 0; S 0Þ: (A-4)
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We can see that the two-point paraxial traveltime formula 1 yieldsexact results along reference rays in homogeneous isotropic oranisotropic media also for points situated inside and outside theinterval SR. It should be emphasized that the two-point paraxialtraveltime formula 1 is exact along a ray only in a homogeneousmedium (when the ray is a straight line). In inhomogeneous media,formula 1 is always only approximate.Another situation, in which formula d1 yields exact results is for
δxS ¼ δxR in homogeneous isotropic or anisotropic media. This fol-lows immediately from equation A-1 and it is understandable be-cause this case represents a parallel shift of the whole reference raybetween points S and R into the new position between S 0 and R 0.Formula A-1 further simplifies in homogeneous isotropic media.
In this case, the expression for Q2ðR; SÞ simplifies considerably,
½Q2ðR; SÞ�MN ¼ V2TðR; SÞδMN; (A-5)
see Červený (2001, equation 4.8.3). In equation A-5, V denotes theP- or S-wave velocity in a homogeneous isotropic medium.Inserting equation A-5 into equation A-1 yields
TðR 0; S 0Þ ¼ TðR; SÞ þ piðδxRi − δxSi Þ
þ 1
2V2TðR; SÞ ðδij − v2pipjÞ
× ðδxRi − δxSi ÞðδxRj − δxSj Þ: (A-6)
Here, we have used the fact that vectors fM are unit and mutuallyorthogonal in isotropic media, and the relation fMifNj ¼ δij−V2pipj. For vectors δxS and δxR perpendicular to the referenceray, i.e., for piðδxRi − δxSi Þ ¼ 0, equation A-6 reduces to
TðR 0; S 0Þ ¼ TðR; SÞ þ 1
2V2TðR; SÞ ðδxRi − δxSi ÞðδxRi − δxSi Þ:
(A-7)
This is an approximate, but highly accurate equation. This can beproved by comparing it with the exact expression TexactðR 0; S 0Þ,which can be obtained from simple geometric considerations. Theexact expression reads
For small V−2T−2ðR; SÞðδxRi − δxSi ÞðδxRi − δxSi Þ, we can expandequation A-8 into a Taylor series whose leading terms are
TexactðR 0; S 0Þ ¼ TðR; SÞ
þ 1
2V2TðR; SÞ ðδxRi − δxSi Þ
× ðδxRi − δxSi Þþ · · · : (A-9)
Thus, the approximate expression A-7 coincides with the expansionof the exact expression A-8 to the quadratic terms in δxRi − δxSi .Because the next nonzero term in the Taylor expansion of expres-sion A-8 is of the order ðδxRi − δxSi Þ4, the error of the approximateexpression A-7 is of this order. Because this term is negative, equa-tion A-7 yields approximate traveltimes, which are always larger
than exact ones. Expression A-7 approximates expression A-8 wellif the term V−2T−2ðR; SÞðδxRi − δxSi ÞðδxRi − δxSi Þ is small. Thiscondition explains unsatisfactory performance of formula 1 forTðR; SÞ → 0; see, for example, Figure 6a.For completeness, let us also describe the properties of the ex-
pression for the square of the two-point paraxial traveltime for ahomogeneous isotropic medium, T2ðR 0; S 0Þ. If we neglect termsof order higher than two in ðδxR − δxSÞ in the squared expres-sion A-6, we get,
Equations A-10 and A-13 are identical, which implies that equa-tion A-10 is exact. We can also see that the problems with theaccuracy of the two-point paraxial traveltime formula forTðR; SÞ → 0, observed in equation A-7, disappeared in the expres-sion for T2ðR 0; S 0Þ. See also Ursin (1982), Gjøystdal et al. (1984),Schleicher et al. (1993), who prove it for equations of different formfrom ours.
APPENDIX B
SHANKS TRANSFORM
The Shanks transform is a useful way of improving the conver-gence rate of a series. Let us consider a Taylor series expansion offunction TðxÞ,
TðxÞ ¼ C0 þ C1xþ C2x2þ · · · : (B-1)
Here, C0; C1, and C2 are coefficients of the expansion. We can iso-late and remove the most transient behavior of expansion B-1 byfirst defining the following parameters:
A0 ¼ C0; A1 ¼ C0 þ C1x; A2 ¼ C0 þ C1xþ C2x2.
(B-2)
The Shanks transform representation (Bender and Orszag, 1978) ofB-1 is then given by
TðxÞ ≈ A0A2 − A21
A0 − 2A1 þ A2
. (B-3)
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This transformation creates a new series which often convergesmore rapidly than the old series B-1.Numerical computations using the Shanks transform must be
handled with caution because computer processors are limited inprecision to the numbers they can resolve. Thus, when x isextremely small, the denominator A0 − 2A1 þ A2 of B-3 is domi-nated by this round-off error and its accuracy is likely to be lowerthan the original series B-1.
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