## ABSTRACT

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.

## INTRODUCTION

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.

## TRAVELTIME FORMULA

^{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:

Note that equation 12 depends on the choice of the parameters of the reference medium because $x\u203eC$ depends on $r=\beta /\alpha $ (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.

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 $\u03f5x$, $\u03f5z$, $\delta y$, and $\gamma y$ and on the parameters of the reference medium through the ratio $r$ of S- and P-wave velocities.

## REFERENCE MOVEOUT FORMULA

## TESTS OF ACCURACY

We test equations 13 and 12 for P-SV converted waves in the isotropic model with P-wave velocity $\alpha =2.5\u2009\u2009km/s$ and varying ratio $r=\beta /\alpha $, 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 $2(cmax\u2212cmin)/(cmax+cmin)\xd7100%$, where $c$ denotes the corresponding phase velocity. The reference velocities $\alpha $ and $\beta $ are chosen equal the vertical P- and SV-wave phase velocities, which results in $\u03f5z=\gamma y=0$; see Table 1.

In the following figures, we present plots of relative traveltime errors $(T\u2212Tcal)/Tcal\xd7100%$. 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=\beta /\alpha $). 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 $x\u203e\u223c1\u22122$; 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, 23–25 is red.

#### Limestone model

In Figure ^{3}, we show the results for the weakly anisotropic limestone model. In Figure ^{3}, we can see that $x\u203eC$ 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 $x\u203eC$ 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 $vNMOex=2.296\u2009\u2009km/s$ and $vNMO=2.312\u2009\u2009km/s$, 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 $x\u203e\u223c2$. 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 18$vNMO=3.359\u2009\u2009km/s$ and the exact value $vNMOex=3.306\u2009\u2009km/s$ obtained from equation 23 is again quite small despite stronger anisotropy. For normalized offsets $x\u203e$ less than $x\u203e\u223c3$, 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 $x\u203e\u223c2$, 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 $vNMO=3.257\u2009\u2009km/s$, whereas the exact value obtained from equation 23 is $vNMOex=2.893\u2009\u2009km/s$. The reasons are stronger anisotropy and also strong positive value of the difference $\u03f5x\u2212\delta y$, 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 $x\u203e\u223c4$. 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 $\u03f5x\u2212\delta y$, 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.

## CONCLUSION

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.

## ACKNOWLEDGMENTS

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 AND MATERIALS AVAILABILITY

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

^{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 $\alpha $ and $\beta $, respectively, and their ratio $r=\beta /\alpha $. The angles of incidence and reflection of a ray of the P-S converted wave are denoted by $\theta P$ and $\theta S$. 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 $sin\theta P$ and $sin\theta S$