A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

Converted reflected waves play an important role in the processing of multicomponent ocean-bottom measurements (Thomsen, 1999). Converted waves offer an additional information to that obtained from pure-mode reflected P-waves. Most of the existing studies are based on the Taylor series expansion of the squared traveltime of a converted wave in a transversely isotropic medium with a vertical axis of symmetry (VTI) with respect to the squared offset (see, e.g., Seriff and Sriram, 1991; Tsvankin and Thomsen, 1994; Granli et al., 1999; Thomsen, 1999; Tsvankin and Grechka, 2000; Li and Yuan, 2003; Hao and Stovas, 2016). Many details can be found in monographs of Tsvankin (2001) or Tsvankin and Grechka (2011). We concentrate on VTI media too, but we use an alternative procedure.

Following Farra and Pšenčík (2013, 2017a), Farra et al. (2016), and Pšenčík and Farra (2017), we derive a reflection moveout formula based on the combined use of the weak-anisotropy approximation and weak-anisotropy (WA) parameters. The derived approximate formula holds for P-SV and SV-P converted waves, and it offers a direct link between observed traveltimes and parameters of the medium.

For the derivation of moveout formulas for pure-mode reflected P or SV waves, Farra and Pšenčík (2013) or Farra et al. (2016) use actual rays of pure-mode P or SV waves reflected from a horizontal reflector, which coincided with one symmetry plane of the overlying anisotropic medium. For lower symmetry anisotropic media, for media with tilted symmetry elements or, for example, for converted waves, actual rays would have to be calculated by two-point ray tracing in the actual medium, which would make the procedure impractical. Fortunately, Pšenčík and Farra (2017) and Farra and Pšenčík (2017a) show that it is possible to derive simple and still sufficiently accurate moveout formulas even without knowledge of actual rays, and they extend their previous work to weakly anisotropic media of arbitrary symmetry and orientation. An important step of their procedure is the replacement of the actual ray in a studied anisotropic medium by a reference ray in a reference isotropic medium. Because the reference ray of a pure-mode reflected wave is symmetric with respect to the reflector, its construction is straightforward.

In contrast to pure-mode reflected waves, construction of a reference ray of a converted reflected wave in an isotropic medium, and, especially, the determination of the conversion point, is a more complicated task. Fortunately, there were successful attempts in the past to find the conversion point in isotropic media. Probably the first of these attempts was made by Tessmer and Behle (1988). Improved version of their formula proposed by Thomsen (1999) is used in this paper (see also Farra and Pšenčík, 2017b).

The proposed converted-wave moveout formula depends on the position of the conversion point and, through it, on the parameters of the reference isotropic medium. Most importantly, the formula depends on the parameters (four WA parameters) specifying the actual VTI medium. The formula can be used, among other applications, for the estimate of parameters directly either from the formula itself or from the expressions for the normal moveout (NMO) velocity or the quartic term derived from it. Estimated WA parameters can be used for the approximate reconstruction of P- and SV-wave phase or ray velocities. In this paper, we test its accuracy by comparing its results with results of exact and commonly used formulas.

Let us mention that the formula proposed in this paper can be generalized for the case of a dipping reflector (see Farra and Pšenčík, 2018).

The paper has the following structure: After introducing two important approximations, we use them in the derivation of the approximate converted-wave moveout formula for VTI media. From this formula, we derive expressions for basic quantities of the Taylor series expansion of the squared traveltime in terms of the squared offset, the two-way zero-offset traveltime, NMO velocity, and quartic term in terms of the WA parameters. The accuracy of the proposed formula is compared with the accuracy of the commonly used rational approximation, which is presented in the section “Reference moveout formula.” In the next section “Tests of accuracy,” the results of the proposed formula and of the rational approximation are compared with exact results. The following “Conclusions” section summarizes the main results of the paper and indicates possible extensions. Appendix  A contains description of possible ways of the determination of the conversion point in the reference isotropic medium. In Appendix  B, WA parameters used in the study are defined. Appendix  C contains derivations of expressions for the two-way zero-offset traveltime, NMO velocity, and the quartic term for the converted wave in a homogeneous VTI layer in terms of WA parameters.

We consider the Cartesian coordinate system, whose x1- and x2-axes are horizontal, the x3-axis is vertical and positive downward. The coordinate system is right handed. We consider a homogeneous layer underlaid, at the depth H, by a horizontal reflector (see Figure 1). The layer is VTI. In this layer, we consider a P-SV converted wave. It propagates as a P-wave from the source S to the conversion point C at the reflector, and as an SV wave from C to the receiver R, see Figure 1 again. Without loss of generality, we can consider the profile along the x1-axis, which means that the SV wave is polarized in the vertical (x1,x3) plane. The traveltime along the ray of the converted wave from S to R via C is T=TP+TSV, where TP is the traveltime along the P-wave leg of the ray, and TSV along the SV-wave ray leg. From the geometry of the ray of the converted wave, we can derive the following expressions for the squares of traveltimes along the P- and SV-wave ray legs:
TP2(x)=xC2+H2vP2(NP),TSV2(x)=(xxC)2+H2vSV2(NSV),
(1)
where xC is the offset of the conversion point of the P-SV converted wave in the VTI medium. In equation 1, vP(NP) and vSV(NSV) denote the P- and SV-wave ray velocities in the VTI medium, respectively. They depend on vectors NP and NSV, unit vectors parallel to the P- and SV-wave ray legs of the converted wave. We call the vectors NP and NSV ray vectors. Before proceeding further, let us introduce the normalized offset x and normalized offset of the conversion point xC
x=xH,xC=xCH.
(2)
In addition, let us introduce the one-way zero-offset traveltimes T0P and T0S in the reference isotropic medium with P- and S-wave velocities α and β
T0P=Hα,T0S=Hβ.
(3)
Taking into account equations 2 and 3, equation 1 reads
TP2(x)=T0P2α21+xC2vP2(NP),TSV2(x)=T0S2β21+(xxC)2vSV2(NSV).
(4)
If xC, vP, and vSV are exact, then the traveltime T=TP+TSV, constructed from TP and TSV given in equation 4, is also exact. At this point, we shall make two important approximations, which we also did in our previous studies.
The first approximation is related to the fact that the actual ray of the converted wave is unknown. As, for example, Pšenčík and Farra (2017) or Farra and Pšenčík (2017a), we replace the actual ray by a reference ray of the converted wave in the reference isotropic medium with P- and S-wave velocities α and β, respectively. The normalized offset xC of the conversion point is thus sought in the reference isotropic medium, and it can be determined by the procedures described in Appendix  A. In this way, we replace the actual ray by the ray whose deviation from the actual ray is of the first order, and the traveltime along it represents the first-order approximation of the actual traveltime (Fermat’s principle). Because, in contrast to pure-mode reflected waves, the reference ray of a converted wave is composed of P- and S-wave ray legs, its form is affected by the ratio of P- and S-wave velocities. To minimize the deviations of the actual and reference raypaths, the ratio of the reference velocities r=β/α should be chosen close to the actual one. In the described approximation, the vectors NP and NSV, parallel to the actual ray, are replaced by vectors NP and NS parallel to the P- and S-wave legs of the reference ray. Note that we keep the same notation for NP as in the case of an actual P-wave ray, but use NS instead of NSV. Components of the ray vectors NP and NS are specified by the same formulas, as those used in the above-mentioned references. The components of the vector NP in the plane (x1,x3) are
N1P=x¯C1+x¯C2,N2P=0,N3P=11+x¯C2.
(5)
The components of the vector NS in the same plane are
N1S=x¯x¯C1+(x¯x¯C)2,N2S=0,N3S=11+(x¯x¯C)2.
(6)
The negative sign in the expression for N3S indicates upgoing character of the S-wave ray leg.
The second approximation consists in the replacement of exact squares of ray velocities vP2(NP) and vSV2(NSV) in equation 4 by their approximations vP2˜(NP) and vSV2˜(NS). A tilde above the quantities indicates that they are of the first order in the WA parameters. As Pšenčík and Farra (2017) or Farra and Pšenčík (2017a), we approximate exact squares of ray velocities by the first-order approximations of squares of phase velocities in the corresponding directions N. Using equation 24 of Pšenčík and Farra (2005) and generalization of equation 29 of Farra and Pšenčík (2013) for ϵz0, we have, in the notation of this paper
vP2˜(NP)cP2˜(NP)α2{1+2[ϵx(N1P)4+δy(N1P)2(N3P)2+ϵz(N3P)4]}
(7)
and
vSV2˜(NS)cSV2˜(NS)β2[1+2γy+2r2(ϵx+ϵzδy)(N1S)2(N3S)2].
(8)
Symbols cP and cSV denote P- and SV-wave phase velocities, r is the ratio of the reference velocities α and β, r=β/α. Symbols ϵx, ϵz, δy, and γy are the WA parameters defined in Appendix  B. It is important to note that neither equation 7 nor 8 is dependent on α or β. To prove this, it is sufficient to insert the expressions for the WA parameters given in equation B-1 to equations 7 and 8. We can also see that equations 7 and 8 would describe an isotropic medium with P- and S-wave velocities α and β if ϵx=ϵz=δy=γy=0. Let us mention that we prefer to use the above approximations rather than the approximation of the slowness v1 because we found that the use of the expressions 7 and 8 leads to more accurate (and independent of α and β) results (see Pšenčík and Farra, 2017).
Inserting equations 5 and 6 into equations 7 and 8, we obtain approximate expressions for squares of ray velocities expressed in terms of the normalized offset x and normalized conversion offset xC. The resulting expressions inserted to the traveltime formulas 4 yield
TP2(x)=T0P2(1+xC2)3PP(xC),TSV2(x)=T0S2[1+(xxC)2]3PSV(xxC).
(9)
Symbols PP(x) and PSV(x) represent polynomials
PP(x)=(1+x2)2+2ϵxx4+2δyx2+2ϵz
(10)
and
PSV(x)=(1+x2)2(1+2γy)+2r2(ϵx+ϵzδy)x2.
(11)
For the square of the total traveltime T2=(TP+TSV)2, we thus get
T2(x¯)={T0P(1+x¯C2)3/2PP1/2(x¯C)+T0S[1+(x¯x¯C)2]3/2PSV1/2(x¯x¯C)}2.
(12)
This is the final approximate converted-wave moveout formula for VTI media. Through the polynomials PP(x) and PSV(x), equation 12 depends on four WA parameters ϵx, ϵz, δy, and γy.

Note that equation 12 depends on the choice of the parameters of the reference medium because xC depends on r=β/α (see equation A-3 or equations A-4 and A-5). Also note that it would be possible to expand the square roots and terms in the denominators to get rid of them. The cost would be a significant loss in accuracy. Compare the results of traveltime approximations I and II in Pšenčík and Farra (2017), where a similar expansion was made.

In an isotropic medium with P- and S-wave velocities α and β, equation 12 reduces to
T2(x)={T0P(1+xC2)1/2+T0S[1+(xxC)2]1/2}2.
(13)
Because we are considering a VTI layer with a horizontal reflector, equation 12 can be rewritten into the form of the Taylor series expansion of the squared traveltime T2 with respect to the squared offset x2
T2(x)=T2(0)+vNMO2x2+A4x4+..
(14)
For dipping reflector, equation 14 would also contain odd terms. From equation 14, it is possible to find the expressions for the two-way zero-offset traveltime T(0), the NMO velocity vNMO, and the quartic term A4 in terms of WA parameters.
We start with the two-way zero-offset traveltime T(0). Although the corresponding amplitude of the converted wave is zero, T(0) is of practical use. If we take equation C-2 and insert into it expressions for T0P, T0S, ϵz, and γy, we obtain
T(0)=H(A331/2+A551/2).
(15)
The two-way zero-offset traveltime T(0) is exact (parameters A33 and A55 represent squares of vertical P and SV ray velocities in the VTI medium), and it is thus independent of the choice of the reference velocities α and β.

Expressions for the NMO velocity and the quartic term of the Taylor series expansion of the squared traveltime of the converted wave can be obtained from equations given in Appendix  C. The two quantities depend on four WA parameters ϵx, ϵz, δy, and γy and on the parameters of the reference medium through the ratio r of S- and P-wave velocities.

In the following, we use a special choice of reference velocities α and β, which leads to the simplification of the formulas. We choose
α2=A33,β2=A55,
(16)
which yields
ϵz=0,γy=0.
(17)
With the specification in equation 16, the following expressions for the NMO velocity and the quartic term depend on only two WA parameters ϵx and δy.
Inserting equations A-5, C-1, C-2, C-7, and C-8 to equation C-5, we obtain for the NMO velocity the expression
vNMO2=(αβ)1[12δy(r1)+ϵxr(r+1)].
(18)
Equation 18 corresponds to equation 27 of Tsvankin and Grechka (2000) or equation 5.130 of Tsvankin (2001). Mentioned references use Thomsen’s (1986) δT instead of δy used in equation 18 and defined in equation B-1.
In an isotropic layer, equation 18 reduces to
vNMO2=(αβ)1.
(19)
The approximate formula for the quartic term is obtained from equation C-13 in combination with equations A-5, C-1, C-2, C-7, C-8, C-14, and C-15
A4=14T2(0)(αβ)2r1{(1r)2+4(1r)1+r[ϵx2r1(1r)(rδy+ϵxδy)]41+r[3δy2+3r1(ϵxδy)2+(rδy+ϵxδy)2r(1+r)]}.
(20)
In an isotropic layer with P- and S-wave velocities α and β, respectively, the approximate expression in equation 20 reduces to
A4=14T2(0)(αβ)2r1(1r)2.
(21)
The series expansion of equation 12 contains, in addition to the terms shown in equation 14, also higher order terms, which we do not present here. Use of the expansion 14 up to the quartic term instead of equation 12, thus leads to the reduction of accuracy of the moveout formula 12. The accuracy of equation 14 is usually enhanced by adding a denominator to the quartic term, the so-called rational approximation. See the following section.
In the following, we also compare the accuracy of the moveout formulas 12 and 13 with the rational approximation of moveout formula proposed by Tsvankin and Thomsen (1994). It was later modified and generalized by, for example, Thomsen (1999) and Li and Yuan (2003). Hao and Stovas (2016) propose the generalized moveout approximation that includes all types of rational approximations as special cases. For comparison with equation 12 proposed in this paper, we use the formula
T2(x)=T2(0)+(vNMOex)2x2+A4exx41+Bx2.
(22)
The symbol T2(0) in it is the exact two-way zero-offset traveltime 15, vNMOex is the exact NMO velocity
(vNMOex)2=αTβT[1+2δT(rT1)+ϵTrT(1+rT)];
(23)
see Thomsen (1999), equation 23, and Hao and Stovas (2016), equation 26. Let us note that equation 18 derived from the proposed moveout formula represents the first-order weak-anisotropy approximation of the exact NMO velocity equation 23. The symbol A4ex in equation 22 represents the exact quartic term
A4ex=14T2(0)(vNMOex)4rT1(1rT2+2ϵT)2[1+rT+2δT+2rT1(ϵTδT)]2.
(24)
From comparison of equation 24 with equation 20, one can find that equation 20 represents the first-order weak-anisotropy approximation of the exact quartic term 24. In equations 23 and 24, ϵT and δT are Thomsen’s (1986) parameters and rT is the ratio of vertical velocities βT and αT, rT=βT/αT=A55/A33. The factor B in equation 22 is given by
B=A4exA11(vNMOex)2(vNMOex)2A11,
(25)
which follows, for example, from equations of Appendix A of Thomsen (1999). The parameter A11 can be expressed through the Thomsen’s (1986) parameter, ϵT, A11=αT2(1+2ϵT).

We test equations 13 and 12 for P-SV converted waves in the isotropic model with P-wave velocity α=2.5  km/s and varying ratio r=β/α, in the limestone model, whose P- and SV-wave anisotropy are approximately 8% and 5%, respectively, and the Mesa Verde mud shale and the hard shale models with P-wave anisotropy of approximately 6% and 25%, respectively, and an SV-wave anisotropy of approximately 12%. The anisotropy percentage is defined as 2(cmaxcmin)/(cmax+cmin)×100%, where c denotes the corresponding phase velocity. The reference velocities α and β are chosen equal the vertical P- and SV-wave phase velocities, which results in ϵz=γy=0; see Table 1.

In the following figures, we present plots of relative traveltime errors (TTcal)/Tcal×100%. Here, T is the traveltime calculated from equation 12, 13, or 22, and Tcal is the traveltime calculated using the package ANRAY (Gajewski and Pšenčík, 1990), which we take as an exact reference.

Isotropic model

In Figure 2, we show results of tests for the isotropic model. The dashed black curve in Figure 2 shows that the use of the numerical solution of equation A-3 in equation 13 leads to exact results. It is because equation A-3 is an equation for the exact determination of the conversion point in an isotropic medium. Use of the approximate expression A-4 instead of the solution of equation A-3 introduces certain errors (the colored solid curves corresponding to varying ratios r=β/α). These errors however, do not exceed 0.5%. Figure 2 shows comparison of the dashed black curve for r=0.4 (it corresponds to the solid black curve in Figure 2) with colored curves obtained for varying values of r from the reference formula in equation 22, which represents the so-called rational approximation (Tsvankin and Thomsen, 1994). We can see that this formula yields satisfactory results (comparable with formula 13) only to the normalized offsets of x12; the higher the ratio r, the larger the offset to which the formula in equation 22 yields satisfactory results. For larger offsets, the accuracy of the reference formula rapidly decreases, whereas the results of formula in equation 13 remain satisfactorily accurate.

Anisotropic models

Figures 3, 4, and 5 contain results of tests for the limestone, the Mesa Verde mud shale and the hard shale models, respectively. Each figure consists of two plots. The (a) plots contain two curves, both obtained from equation 12. The black curve is obtained with the conversion point determined from the approximate expression A-4, and the red curve is obtained by numerically solving the quartic equation A-3. In the plots (b) , we compare the performance of equation 12 with the performance of equation 22. The curve corresponding to equation 12 with the conversion point determined from the approximate expression A-4 is black, and the curve corresponding to the reference formula 22 with equations 15, 2325 is red.

Limestone model

In Figure 3, we show the results for the weakly anisotropic limestone model. In Figure 3, we can see that xC determined by solving numerically quartic equation A-3 leads to relative traveltime errors less than 0.1% for normalized offsets from 0 to 8. The use of the normalized conversion offset xC determined from the approximate equation A-4 leads to slightly larger errors, but they are still less than 0.2% for normalized offsets between 0 and 8. These relative traveltime errors are comparable with relative errors of the first-order formula for pure-mode reflected P waves, but they are less than errors of the first-order formula for pure-mode reflected SV waves. Compare Figure 3 of this paper with Figures 1 and 3 of Farra and Pšenčík (2013). It is also of interest to compare exact and approximate values of NMO velocity in the limestone model. From equations 23 and 18, we get vNMOex=2.296  km/s and vNMO=2.312  km/s, respectively. Thus, equation 18 yields quite accurate results. In Figure 3, we compare the black curve corresponding to the black curve in Figure 3, obtained from formula 12, with the red curve obtained from the reference formula in equation 22. The latter formula yields very accurate results up to the normalized offset x2. Then, its accuracy strongly decreases. The accuracy of the formula in equation 12 remains high.

Mesa Verde mud shale model

Figure 4 shows relative traveltime errors of equation 12 applied to the Mesa Verde mud shale model. In Figure 4, the errors are slightly larger than in Figure 3 (the S-wave anisotropy is stronger), but they do not exceed 0.5% for the normalized offsets between 0 and 8. These errors are substantially smaller than the errors of the first-order formula for the pure-mode reflected SV wave, compare Figure 4 with Figure 4 of Farra and Pšenčík (2013). The difference between the approximate value of the NMO velocity obtained from equation 18vNMO=3.359  km/s and the exact value vNMOex=3.306  km/s obtained from equation 23 is again quite small despite stronger anisotropy. For normalized offsets x less than x3, the reference formula in equation 22 represented by the red curve in Figure 4, yields better results than the black curve obtained from formula in equation 12 with the conversion point determined from the approximate expression A-4. For larger offsets, however, its accuracy rapidly decreases. The formula in equation 12, although also more inaccurate (but with errors not exceeding 0.5%) than in Figure 3, yields more satisfactory results.

Hard shale model

Because of the stronger P-wave anisotropy of the hard shale, relative traveltime errors of equation 12 in Figure 5 are larger. They nearly reach 2% around the normalized offset x2, see Figure 5. For the remaining offsets, the errors are, however, smaller, and they tend to zero for increasing offsets. In this case, the accuracy of equation 12 is higher than the accuracy of the first-order formula for the pure-mode reflected SV wave. This follows from comparison of Figure 5 with Figure 5 of Farra and Pšenčík (2013). The accuracy of the approximate NMO velocity formula in equation 18 is, however, lower than in the previous models. Equation 18 yields vNMO=3.257  km/s, whereas the exact value obtained from equation 23 is vNMOex=2.893  km/s. The reasons are stronger anisotropy and also strong positive value of the difference ϵxδy, which was not the case in the previous models. In Figure 5, we can see that the reference formula in equation 22 yields much better results (errors of less than 1%) than formula in equation 12 for the normalized offsets less than x4. For larger offsets, the accuracy of formula in equation 22 decreases, whereas the formula in equation 12 yields results that converge to zero error with the increasing offset. The increased errors of formula in equation 12 for small and intermediate offsets are obviously caused by strong anisotropy, the strong positive difference ϵxδy, and, as in our previous studies, by the deviation of ray N and phase n vectors (also related to the strength of the anisotropy). The phase vector n is a unit vector in the direction of the slowness vector.

We derived an approximate, explicit, and relatively simple reflection-moveout formula for a converted wave in a weakly or moderately anisotropic homogeneous VTI layer. The formula relates, in a simple and transparent way, traveltimes calculated along a reference ray of the converted wave to the parameters of the medium represented by WA parameters. Along a profile, the formula depends on four WA parameters and the ratio r of S- and P-wave velocities of the reference medium. The ratio r controls the form of the reference ray. The reference ray approximates the actual one well when the ratio r does not differ significantly from the actual velocity ratio.

Although the derivations and tests were performed for the P-SV converted wave, due to its kinematic reciprocity, the formula holds also for the SV-P converted wave.

Performed tests indicate that the accuracy of the moveout formula is close to the accuracy of formulas derived in a similar way earlier for pure-mode reflected P or SV waves. As in similar previous studies, the moveout formula yields satisfactory results for all offsets without necessity to trace rays in actual anisotropic medium to finite offsets as in some moveout approximations to increase their accuracy.

The tests also show that the moveout formula can be used not only for weakly but also for moderately anisotropic media. In isotropic media, the proposed moveout formula yields results of high accuracy; when numerical solution of quartic equation is used for the determination of the conversion point, the moveout formula yields exact results.

The procedure described in this paper can be extended to waves reflected at the bottom of a stack of VTI layers of moderate anisotropy. As in this paper, two basic approximations will be used. The approximate traveltimes will be calculated along the reference rays of reflected waves in a reference medium composed of isotropic layers. For the evaluation of traveltimes, the weak-anisotropy approximation of the ray velocity will be used. Another possible extension of the presented moveout formula is to the so-called dip-constrained TI medium, whose symmetry axis is perpendicular to the reflector. Using the concept of a common S-wave, the moveout formula for the converted wave could probably be generalized for anisotropic media of arbitrary symmetry and orientation.

We thank Y. Sripanich, A. Stovas and an anonymous referee for useful comments. We are grateful to the Research Project 16-05237S of the Grant Agency of the Czech Republic and the project “Seismic waves in complex 3-D structures” (SW3D) for support.

Data associated with this research are available and can be obtained by contacting the corresponding author.

DETERMINATION OF THE CONVERSION POINT IN THE REFERENCE ISOTROPIC MEDIUM

Let us consider a homogeneous isotropic layer underlaid by a horizontal reflector. On the surface of the layer, we consider source S and receiver R. These two points are connected by the ray of a P-S converted wave with the conversion point C at the reflector (see Figure 1). In the following, we follow closely the derivation of Thomsen (1999), but we use slightly different notation corresponding to the notation, which we used in our previous studies. We denote the P- and S-wave velocities by α and β, respectively, and their ratio r=β/α. The angles of incidence and reflection of a ray of the P-S converted wave are denoted by θP and θS. By x, we denote the source-receiver offset and by xC, the offset of the conversion point. The depth of the layer is H. Elementary trigonometry considerations based on the sketch in Figure 1 yield for sinθP and sinθS
sinθP=xCxC2+H2,sinθS=xxC(xxC)2+H2.
(A-1)
Combination of equation A-1 with the Snell law, sinθP/α=sinθS/β, leads to the equation:
(xxC)2(xC2+H2)r2=xC2[(xxC)2+H2].
(A-2)
Equation A-2 is equivalent to equation 14 of Thomsen (1999). Normalizing it by H4, using notation introduced in equation 2 and rearranging equation A-2 to the form of a polynomial equation, we get
xC42xxC3+(1+x2)xC22xxC1r2+x21r2=0.
(A-3)
This is a quartic polynomial equation for the normalized offset of the conversion point xC. For the pure-mode reflected wave (r=1), equation A-3 yields the expected xC=x/2. Equation A-3 can be solved analytically using, for example, the so-called Ferrari procedure. It can also be solved numerically. Tessmer and Behle (1988) derive an approximate explicit formula for the determination of the offset of the conversion point. The formula is improved by Thomsen (1999). Taking into account the normalization specified in equation 2, we use here the approximate formula of Thomsen (1999) in the form
x¯Cx¯(C0+C2x¯21+C3x¯2),
(A-4)
where
C0=11+r,C2=r21r(1+r)3,C3=121r(1+r)2.
(A-5)
From equations A-4 and A-5, we can see that the normalized offset xC of the conversion point depends on the parameters of the reference medium only through the ratio r of the S- and P-wave velocities. For a detailed study of the accuracy of the expression A-4, see Thomsen (1999).

WA PARAMETERS USED IN THE STUDY

WA parameters ϵx, ϵz, δy, and γy used in equations 7, 8, 10, and 11 of the main text are defined as
ϵx=A11α22α2,ϵz=A33α22α2,δy=A13+2A55α2α2,γy=A55β22β2,
(B-1)
where A11, A13, A33, and A55 denote density-normalized elastic moduli in the Voigt notation, and α and β are the P- and S-wave velocities in the reference isotropic medium.

TWO-WAY ZERO OFFSET TRAVELTIME T(0), NMO VELOCITY vNMO and quartic term A4

From equations 9 to 11 specified for the zero-offset case, we get
TP(0)=T0P(1+2ϵz)1/2andTSV(0)=T0S(1+2γy)1/2.
(C-1)
From equation C-1, we get the expressions for the two-way zero-offset traveltime T(0):
T(0)=T0P(1+2ϵz)1/2+T0S(1+2γy)1/2.
(C-2)
Note that TP(0), TSV(0), and T(0) do not depend on the choice of the parameters of the reference medium. It can be proved by substituting equations 3 and B-1 to equations C-1 and C-2.
Inverse square of the NMO velocity vNMO of the P-SV converted wave is given by the standard expression
vNMO2=dT2dx2|x=0=d(TP+TSV)2dx2|x=0=2T(0)(dTPdx2+dTSVdx2)|x=0.
(C-3)
Let us introduce NMO velocities vNMOP and vNMOSV related to the P- and SV-wave ray legs of the converted wave in the following way:
(vNMOP)2=dTP2dxC2|xC=0,(vNMOSV)2=dTSV2d(xxC)2|xxC=0.
(C-4)
Using the definitions C-4, equation C-3 can be rewritten in the form
vNMO2=T(0)[C02(vNMOP)2TP(0)+(1C0)2(vNMOSV)2TSV(0)].
(C-5)
In the derivation of equation C-5, we used the relations:
dxC2dx2|x=0=C02andd(xxC)2dx2|x=0=(1C0)2
(C-6)
resulting from the differentiation of equation A-4 and the specification of the result for the zero offset x=0.
To evaluate equation C-5, we need to specify expressions for the NMO velocities vNMOP and vNMOSV. Differentiation of formulas in equation 9 with respect to xC2 and (xxC)2, respectively, yields expressions
(vNMOP)2=α21+6ϵz2δy(1+2ϵz)2
(C-7)
and
(vNMOSV)2=β21+2γy2r2(ϵx+ϵzδy)(1+2γy)2.
(C-8)
Note that NMO velocities vNMOP and vNMOSV do not depend on the parameters of the reference medium. The NMO velocity vNMO in equation C-5, however, depends on them through the factor C0 (see equation A-5), which depends on the ratio r=β/α.
The derivation of the quartic term A4 is similar, but slightly more involved. The quartic term A4 is given by the expression
A4=12ddx2(dT2dx2)|x=0.
(C-9)
We have
ddx2(dT2dx2)=12(TP1dTP2dx2+TSV1dTSV2dx2)2+(TP+TSV)[TP1ddx2(dTP2dx2)+TSV1ddx2(dTSV2dx2)12TP3(dTP2dx2)212TSV3(dTSV2dx2)2].
(C-10)
Taking into account equations 2, 3, A-4, C-7 to equation C-9, we can express the term A4 in the following way:
A4=14[C02(vNMOP)2TP(0)+(1C0)2(vNMOSV)2TSV(0)]2+12T(0)[TP1(0)ddx2(dTP2dx2)|x=0+TSV1(0)ddx2(dTSV2dx2)|x=0C042(vNMOP)4TP3(0)(1C0)42(vNMOSV)4TSV3(0)].
(C-11)
In a way similar to the introduction of NMO velocities vNMOP and vNMOSV in equation C-4, we introduce quartic terms A4P and A4SV:
A4P=12ddxC2(dTP2dxC2)|xC=0,A4SV=12dd(xxC)2(dTSV2d(xxC)2)|xxC=0.
(C-12)
Using equations C-5 and C-12, and with the help of equation A-4, we can rewrite equation C-11 into the form
A4=14vNMO4T2(0)+12T(0)[4C0C2(vNMOP)2TP(0)H2C042(vNMOP)4TP3(0)+2C04A4PTP(0)4(1C0)C2(vNMOSV)2TSV(0)H2(1C0)42(vNMOSV)4TSV3(0)+2(1C0)4A4SVTSV(0)].
(C-13)
To express the quartic term A4 in equation C-13 in terms of parameters of the medium, we need to evaluate A4P and A4SV in equation C-12. From equations 9 to 11, we get
A4P=2α4T0P2(1+2ϵz)3[(1+2ϵz)(δyϵxϵz)+2(δy2ϵz)2]
(C-14)
and
A4SV=2β4T0S2(1+2γy)3[r2(1+2γy)(ϵx+ϵzδy)+2r4(ϵx+ϵzδy)2].
(C-15)
Inserting equations C-14 and C-15 into equation C-13, we get the first-order expression for the quartic term A4 of the converted wave. Note that the terms A4P and A4SV in equations C-14 and C-15 are independent of the choice of the reference medium, but the term A4 in equation C-13 depends on it through the terms C0 and C2, which depend on the ratio r=β/α.
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