# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{PP}) is derived in this paper. First, the relationship between thin-bed R

_{PP}and incidence angles is analyzed through series expansion method. For PP-wave, its reflection coefficients are even power series functions of sine incidence angles. Then, for small incidence, R

_{PP}of the thin bed is further simplified into a second-order series approximation with respect to the sine incidence angles. Simulations and accuracy analyses of the approximate formula show that approximation errors are smaller than 5% as the incidence angles smaller than 20 degrees. Based on this approximate formula, an approach is given for estimating thin-bed properties including P-wave impedance ratios and thickness. The estimation approach is applied in properties estimation of a thin bed model. Perfect performances of the model example show the future potentiality of the approximate formula in thin-bed Amplitude-Versus-Offset (AVO) analysis and inversion.

## 1. Introduction

_{2}storage at Sleipner in the Norwegian North Sea [2,13,14,15]. Therefore, seismic responses of a thin bed, which carry information of stratigraphic structure, lithology and pore fluid, are significantly important to crustal, exploration and engineering research.

_{PP}under the small-incidence assumption, and test accuracy of the approximate formula with four representative thin-bed models through numerical analysis. And then we give an approach to estimate the thin-bed properties with this approximation.

## 2. R/T Coefficients Expressed by Series Functions

_{PP}, R

_{PS}, T

_{PP}, T

_{PS}are displacement R/T coefficients of P-waves and converted S-waves, respectively;

**M**is a 4 × 4 matrix,

**n**is a 4 × 1 vector, which is presented in Appendix A in detail.

_{PP}, R

_{PS}, T

_{PP}, T

_{PS}can be calculated. When the thin-bed thickness is equal to zero, Equation (1) will be reduced to that for R/T coefficients equations of a single interface between the thin-bed roof and floor, which is consistent with the Zoeppritz equations [35].

**M**and

**n**are all expressed by series functions of sine incidence angle (sinθ). If the elements are even or odd series functions of sinθ, we mark them by ‘even’ or ‘odd’ in Equation (1) respectively as follows,

_{PP}, T

_{PP}are even power series functions of sinθ, while R

_{PS}, T

_{PS}are odd power series functions of sinθ. Therefore, R

_{PP}, R

_{PS}, T

_{PP}, T

_{PS}can be expressed by series functions of sinθ as follows,

## 3. Approximate Formula of PP-Wave Reflections

^{n}θ decreases rapidly with increasing n in Equation (3). R

_{PP}can be simplified into a second-order series approximation of sine incidence angle via a series truncation procedure as follows,

_{0}, B

_{0}, A

_{1}, B

_{1}, A

_{2}in turn from the constant term, linear term and quadratic term of sinθ in Equation (1), respectively.

_{PP}= A

_{0}, T

_{PP}= B

_{0}, R

_{PS}= 0, T

_{PS}= 0. Equation (1) is simplified as

_{P}

_{2}, z

_{P}= ρv

_{P}are P-wave impedances and their subscripts 1,2,3 indicate three layers of the single thin bed, respectively; j is an imaginary symbol.

_{0}= (z

_{P}

_{3}/z

_{P}

_{1}+ 1)cosτ + j(z

_{P}

_{2}/z

_{P}

_{1}+ z

_{P}

_{3}/z

_{P}

_{2})sinτ.

_{PP}= A

_{0}, T

_{PP}= B

_{0}, R

_{PS}= A

_{1}sinθ, T

_{PS}= B

_{1}sinθ. Equation (1) is simplified as

_{S}

_{2}, d

_{1}= ρ

_{2}/ρ

_{1}; z

_{S}= ρv

_{S}are S-wave impedances, r = v

_{P}/v

_{S}are the ratios of P-wave velocities to S-wave velocities, their subscripts 1, 2, 3 refer to three layers of the single thin bed respectively. l

_{k}= v

_{Pk}+ 1/v

_{Pk}are the ratios of the P-wave velocities in adjacent layers with k = 1, 2, a

_{1}~a

_{5}are presented in Appendix C.

_{1}= (z

_{S}

_{3}/z

_{S}

_{1}+ 1)cosq + j(z

_{S}

_{2}/z

_{S}

_{1}+ z

_{S}

_{3}/z

_{S}

_{2})sinq. a

_{6}is given in Appendix C.

_{PP}= A

_{0}+ A

_{2}sin

^{2}θ, T

_{PP}= B

_{0}+ B

_{2}sin

^{2}θ, R

_{PS}= A

_{1}sinθ, T

_{PS}= B

_{1}sinθ. Equation (1) is simplified as

_{1}~b

_{6}are presented in Appendix C.

_{PP}, algorithmic steps of which are shown in Figure 3 detailly. The intercept and gradient of the approximate formula are dependent on frequency, which is greatly different from those of a single interface [36]. It is worthy to notice that, the approximate formula can be further simplified for ultra-thin bed cases by introducing in the approximations of sinτ ≈ τ, sinq ≈ q, cosτ ≈ 1, and cosq ≈ 1.

## 4. Accuracy Analysis

_{P}), S-wave velocity (v

_{S}), density (ρ), the normal-incident P-wave reflection coefficient of thin-bed top-interface (R

_{1}), as well as the normal-incident P-wave reflection coefficient of thin-bed bottom-interface (R

_{2}).

_{PP}in Figure 4, Figure 5, Figure 6 and Figure 7 for Models 1–4 respectively. For a better understanding of differences between thin-bed and single-interface responses, R

_{PP}of thin-bed top- and bottom- interfaces based on the Zoeppritz equations are also curved in Figure 4, Figure 5, Figure 6 and Figure 7. In the numerical analysis, the thin-bed thickness is set as 3 m and incidence frequencies are set as 20, 30, and 40 Hz, respectively. Meanwhile, we plot relative errors of R

_{PP}induced by the approximate formula under the small-incidence assumption for Models 1–4 in Figure 8. Considering small-incidence assumption of the approximate formula, the maximum incidence angle in all cases is set as 30 degrees. Due to that thin-bed R

_{PP}are complex valued, we discuss them in terms of amplitude and phase components, respectively.

_{PP}of Models 1 and 2 versus incidence angles, respectively. Models 1 and 2 are high-impedance or low-impedance thin beds with strong impedance contrasts. Destructive interference effects cause that thin-bed reflected amplitudes are obviously lower than those of top- and bottom-interfaces. Meanwhile, the thin-bed reflected phases are not constant zero or π as those of top- and bottom-interfaces, respectively. Compared with the true R

_{PP}, the approximated values show a similar change regularity that a larger incidence angle and a lower incidence frequency result in a lower reflected amplitude. For incidence angles smaller than 20 degrees, approximate thin-bed amplitudes are almost the same as the true values. When incidence angles are over 20 degrees, the approximate thin-bed amplitudes are lower than the true values and show a larger deviation from the true values at a larger incidence angle. The approximated phase is almost the same as the true phase in the angle range from 0 to 30 degrees.

_{PP}of Models 3 and 4 versus incidence angles, respectively. Models 3 and 4 are impedance transition thin beds with equal polarities, which differ from opposite polarities of Models 1–2. Constructive interference effects cause that thin-bed reflected amplitudes are higher than those of top- and bottom-interfaces. Compared with Models 1–2 cases, R

_{PP}’s amplitudes of Models 3 and 4 are less sensitive to frequency variation. The approximate Amplitude-Versus-Angle (AVA) curves coincide with the true AVA curves for incidence angles smaller than 20 degrees and deviate progressively from the true for incidence angles over 20 degrees. Comparison of Models 3 and 4 shows that the former’s phases are nearly zero while these of the latter are nearly π.

_{PP}’s relative errors caused by the approximate formula under the small-incidence assumption. In the numerical analysis, thin-bed thickness varies from λ/100 to λ/8, where λ is the P-wave wavelength in the middle layer. Approximate amplitude errors of Models 1–2 are less than 5% for incidence angles smaller than 20 degrees, and are less than 10% for incidence angles less than 25 degrees. When incidence angles are over 25 degrees, amplitude errors increase rapidly with an incidence-angle increase. Phase errors are less than 10% for incidence angles smaller than 30 degrees. For Models 3 and 4, amplitude accuracy by the approximate formula is higher than those of Models 1–2. Approximate errors of Model 3 are less than 5% for incidence angles smaller than 25 degrees and are less than 10% for incidence angles smaller than 29 degrees. For Model 4, amplitude errors versus incidence angles have similar variation regularity with those of Model 3, while, phase errors are smaller than 5% for incidence angles less than or equal to 30 degrees.

_{PP}are significantly different from the top- or bottom-interface R

_{PP}calculated by the Zoeppritz equations. The second-order series approximation has high accuracy for incidence angles less than 20 degrees and deviates progressively from the true values over 20 degrees. Correspondingly, the relative errors are less than 5% for incidence smaller than 20 degrees.

## 5. Application Example

_{PP}’s A

_{0}, A

_{2}of 120 thin-bed models. Figure 9 shows the R

_{1}and R

_{2}distributions of 120 thin-bed models, which reveal that those 120 models include most of the cases for thin-bed R

_{1}and R

_{2}ranging from −0.2 to 0.2. Meanwhile, those 120 models cover all types of thin-bed models, i.e., high-impedance thin beds, low-impedance thin beds, high-to-low impedance transition thin beds and low-to-high impedance transition thin beds, of which each type includes 30 models.

_{PP}’s A

_{0}, A

_{2}of those 120 thin-beds at different R

_{1}, R

_{2}and thicknesses. Taking the cases of λ/10, λ/20, λ/40, and λ/60 for examples, the contour maps of A

_{0}and A

_{2}of those 120 thin-bed models are shown in Figure 10, Figure 11, Figure 12 and Figure 13 respectively. Considering that the coefficients A

_{0}and A

_{2}are also complex valued, we discuss them in terms of amplitude and phase components, respectively. The expressions of amplitude and phase of A

_{0}and A

_{2}are shown in Appendix D in detail.

_{0}’s amplitude and phase of those 120 models versus R

_{1}and R

_{2}, respectively. For the λ/10 case, amplitude contours are similar to ellipses with the major axis along the line R

_{2}= −R

_{1}and minor axis along the line R

_{2}= R

_{1}. The amplitude increases with increasing R

_{1}and R

_{2}. The amplitude growth rate of thin-bed models with equal polarities is larger than opposite-polarity cases, especially for a thinner bed. For the cases of thin beds with higher R

_{1}, R

_{2}and thinner thickness, the amplitude contours in the direction of major axis are approximately parallel to the line R

_{2}= −R

_{1}. A

_{0}’s phases show that a positive R

_{2}results in a negative phase, while a negative R

_{2}results in a positive phase. For the same R

_{2}, a positive R

_{1}results in a relatively smaller phase than that of a negative R

_{1}case.

_{2}’s amplitude and phase of those 120 models versus R

_{1}and R

_{2}, respectively. Compared with A

_{0}, A

_{2}has a relatively higher amplitude and a completely different phase variation. For A

_{2}, a positive R

_{2}results in a positive phase, while a negative R

_{2}results in a negative phase.

_{1}and R

_{2}range from −0.2 to 0.2, thin-bed properties can be estimated based on the templates of A

_{0}and A

_{2}. Once the amplitude and phase values of A

_{0}and A

_{2}are obtained from thin-bed AVA data, we can exhibit the corresponding contours in the templates and seek the intersections of those contours. The intersection of the four contour curves, including those of amplitude and phase of A

_{0}and A

_{2}, indicates thin-bed R

_{1}, R

_{2}and thickness. The P-wave impedance ratios at the top- and bottom-interfaces of the thin bed can be obtained from R

_{1}, R

_{2}as follows,

_{0}are 0.1006 and 0.9355 respectively, amplitude and phase of A

_{2}are 0.2364 and −2.2545 respectively. We obtain the corresponding contour lines in the amplitude and phase templates at different thicknesses. Taking the cases of λ/10, λ/20, λ/40, and λ/60 for examples, we exhibit the contours lines of amplitude and phase of A

_{0}and A

_{2}at fixed thicknesses, which are shown in Figure 14.

_{1}and R

_{2}are estimated through the coordinates of the intersect point “p”, as R

_{1}= 0.087 and R

_{2}= −0.085. The estimated R

_{1}and R

_{2}are very close to those of Model 5 (R

_{1}= 0.085, R

_{2}= −0.085) respectively. Meanwhile, the estimated P-wave impedance ratios at the top- and bottom-interfaces are 1.190 and 0.843, of which the relative errors are 0.32% and 0.08%, respectively. The good estimated results verify the availability of the estimation approach for thin-bed properties based on the approximate formula.

## 6. Discussion

_{PP}’s approximate formula of a single thin bed. When the target thin bed is embedded in a finely layered background, the reflection problems become much more complex. The study shows that interlayer structure and lithologies of finely layered reservoirs can be determined by long-wavelength approximation [37,38,39,40]. Therefore, we will introduce long-wavelength approximation in our further AVA analysis and inversion of finely layered reservoirs.

_{0}, A

_{2}from 120 thin-bed models with R

_{1}and R

_{2}between −0.2 and 0.2. Meanwhile, the S-wave velocity and density of those 120 thin-bed models are determined by P-wave velocity through Castagna’s Relationship [41] and Gardner’s Equation [42], respectively. We will develop the database of templates gradually for a wider application in future thin-bed properties analysis.

## 7. Conclusions

_{PP}can reveal the thin-bed true reflections for small incidence exactly. Relative errors are less than 5% for incidence angles smaller than 20 degrees. The approximate formula is derived with no weak impedance contrast hypothesis, so it is valid for thin-bed models with strong impedance contrasts. The proposed approach to estimate thin-bed properties is established by thin-bed models with R

_{1}and R

_{2}between −0.2 and 0.2. The templates of amplitude and phase of A

_{0}, A

_{2}can be utilized to estimate P-wave impedance ratios and thin-bed thickness, which verify the availability of the approximate formula.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**M**is a 4 × 4 matrix with 4 column vectors,

**M**≡ [

**m**

_{1}

**m**

_{2}

**m**

_{3}

**m**

_{4}], defined as:

**n**is a 4 × 1 vector defined as:

**A**is a 4 × 4 matrix with elements a

_{ij}defined as:

_{P}and v

_{S}are P-wave and S-wave velocities, respectively, ρ is density, δ is S-wave transmitted or reflected angle, and their subscripts 1, 2, 3 refer to three layers of the thin bed respectively; θ is P-wave incidence angle, θ

_{2}and θ

_{3}are P-wave transmitted or reflected angle in the target thin bed and its floor respectively; h is the thickness of the middle layer, ω is angular frequency, and j is the imaginary symbol.

## Appendix B

_{P}

_{2}, q = ωh/v

_{S}

_{2}.

## Appendix C

_{1}~a

_{6}and b

_{1}~b

_{6}represent as follows:

## Appendix D

_{0}, A

_{2}are complex valued, their real, imaginary, amplitude and phase components are as follows. For A

_{0}, there are:

_{2}, there are:

## References

- Backus, G.E. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res.
**1962**, 67, 4427–4440. [Google Scholar] [CrossRef] - Ghaderi, A.; Landrø, M. Estimation of thickness and velocity changes of injected carbon dioxide layers from prestack time-lapse seismic data. Geophysics
**2009**, 74, O17–O28. [Google Scholar] [CrossRef] - Zeng, H.; Marfurt, K.J. Recent progress in analysis of seismically thin beds. Interpretation
**2015**, 3, SS15–SS22. [Google Scholar] [CrossRef] - Chang, C.H.; Shih, R.H. Characteristics of reflectivity strength on a thin bed. Terr. Atmos. Atmos. Ocean Sci.
**1996**, 7, 269–276. [Google Scholar] [CrossRef] - Jones, L.E.A.; Wang, H.F. Ultrasonic velocities in Cretaceous shales from the Williston Basin. Geophysics
**1981**, 46, 288–297. [Google Scholar] [CrossRef] - Ye, F. Sensitivity of Seismic Reflections to Variations in Anisotropy in the Bakken Formation, Williston Basin, North Dakota. Master’s Thesis, University of Texas at Austin, Austin, TX, USA, 2010. [Google Scholar]
- Sayers, C.M.; Dasgupta, S. Elastic anisotropy of the upper and middle Bakken. In Second International Workshop on Rock Physics; EAGE Publication: Houten, The Netherlands, 2013. [Google Scholar]
- Chung, H.; Lawton, D.C. Some properties of thin beds. In Seg Technical Program Expanded Abstracts; Society of Exploration Geophysicists: Tulsa, OK, USA, 1991; pp. 224–227. [Google Scholar]
- Chopra, S.; Castagna, J.P.; Xu, Y. Thin-bed reflectivity inversion and some applications. First Break
**2014**, 31, 27–34. [Google Scholar] [CrossRef] - Zhang, Q.F.; Wang, Y.Q.; Wang, T.Q. Seismic prediction technique of thin interbed channel sand in SongLiao Basin. Lithol. Reserv.
**2007**, 19, 92–95. [Google Scholar] - Huifeng, L.; Xiangtong, Y.; Jiangyu, L.; Pengyao, Z.; Dengfeng, R.; Sheng, Y. Lesson learned from an unsuccessful multi-stage fracturing case and an improved design in Tarim Basin, China. Int. Petrol. Technol. Conf.
**2016**, 26, 1923–1927. [Google Scholar] - Widess, M.B. How thin is a thin bed? Geophysics
**1973**, 38, 1176–1180. [Google Scholar] [CrossRef] - Rubino, J.G.; Velis, D. Seismic characterization of thin beds containing patchy carbon dioxide-brine distributions: A study based on numerical simulations. Geophysics
**2011**, 76, R57–R67. [Google Scholar] [CrossRef] - Williams, G.; Chadwick, A. Quantitative seismic analysis of a thin layer of CO
_{2}in the Sleipner injection plume. Geophysics**2012**, 77, R245–R256. [Google Scholar] [CrossRef] - Cowton, L.R.; Neufeld, J.A.; White, N.J.; Bickle, M.J.; White, J.C.; Chadwick, R.A. An inverse method for estimating thickness and volume with time of a thin CO
_{2}-filled layer at the Sleipner Field, North Sea. J. Geophys. Res.-Sol. Earth**2016**, 121, 5068–5085. [Google Scholar] [CrossRef] - Almoghrabi, H.; Lange, J. Layers and bright spots. Geophysics
**1986**, 51, 699–709. [Google Scholar] [CrossRef] - Juhlin, C.; Young, R. Implications of thin layers for amplitude variation with offset (AVO) studies. Geophysics
**1993**, 58, 1200–1204. [Google Scholar] [CrossRef] - Guo, Z.; Li, X. Cracked thin layered reservoir analysis using offset-dependent spectrum characteristics. In Seg Technical Program Expanded Abstracts; Society of Exploration Geophysicists: Tulsa, OK, USA, 2010; pp. 478–482. [Google Scholar]
- Brekhovoskikh, L.M. Waves in Layered Media; Academic Process: Tamil Nadu, India, 1960. [Google Scholar]
- Liu, Y.; Schmitt, D.R. Amplitude and AVO responses of a single thin bed. Geophysics
**2003**, 68, 1161–1168. [Google Scholar] [CrossRef] - Rubino, J.G.; Velis, D. Thin-bed prestack spectral inversion. Geophysics
**2009**, 74, R49–R57. [Google Scholar] [CrossRef] - Yang, C.; Wang, Y.; Lu, J. Weak impedance difference approximations of thin-bed PP-wave reflection responses. J. Geophys. Eng.
**2017**, 14, 1010–1019. [Google Scholar] [CrossRef] - Yang, C.; Wang, Y.; Wang, Y.H. Reflection and transmission coefficients of a thin bed. Geophysics
**2016**, 81, N31–N39. [Google Scholar] [CrossRef] - Chung, H.M.; Lawton, D.C. Amplitude responses of thin beds: Sinusoidal approximation versus Ricker approximation. Geophysics
**1995**, 60, 223–230. [Google Scholar] [CrossRef] - Chung, H.M.; Lawton, D.C. Frequency characteristics of seismic reflections from thin beds. Can. J. Explor. Geophys.
**1995**, 31, 32–37. [Google Scholar] - Zeng, H.; Backus, M.M. Interpretive advantages of 90°-phase wavelets, Part I: Modeling. Geophysics
**2005**, 70, C7–C15. [Google Scholar] [CrossRef] - Zeng, H.; Backus, M.M. Interpretive advantages of 90°-phase wavelets, Part II: Seismic applications. Geophysics
**2005**, 70, C17–C24. [Google Scholar] [CrossRef] - Thomson, W. Transmission of elastic waves through a stratified solid medium. J. Appl. Phys.
**1950**, 21, 89–93. [Google Scholar] [CrossRef] - Haskell, N. The dispersion of surface waves in multilayered media. Bull. Seismol. Soc. Am.
**1953**, 43, 17–34. [Google Scholar] - Restrepo, D.; Gόmez, J.D.; Jaramillo, J.D. SH wave number green’s function for a layered, elastic half-space. Part i: Theory and dynamic canyon response by the discrete wave number boundary element method. Pure Appl. Geophys.
**2014**, 171, 2185–2198. [Google Scholar] [CrossRef] - Kumar, S.; Pal, P.C.; Majhi, S. Reflection and transmission of plane SH-waves through an anisotropic magnetoelastic layer sandwiched between two semi-infinite inhomogeneous viscoelastic half-spaces. Pure Appl. Geophys.
**2015**, 172, 2621–2634. [Google Scholar] [CrossRef] - Sahu, S.A.; Paswan, B.; Chattopadhyay, A. Reflection and transmission of plane waves through isotropic medium sandwiched between two highly anisotropic half-space. Wave Random Complex
**2015**, 26, 1–26. [Google Scholar] [CrossRef] - Paswan, B.; Sahu, S.A.; Chattopadhyay, A. Reflection and transmission of plane wave through fluid layer of finite width sandwiched between two monoclinic elastic half-spaces. Acta Mech.
**2016**, 227, 3687–3701. [Google Scholar] [CrossRef] - Singh, P.; Chattopadhyay, A.; Srivastava, A.; Singh, A.K. Reflection and transmission of p-waves in an intermediate layer laying between two semi-infinite media. Pure Appl. Geophys.
**2018**, 175, 4305–4319. [Google Scholar] [CrossRef] - Zoeppritz, K. On the reflection and penetration of seismic waves through unstable layers. Göttinger Nachrichten
**1919**, 1, 66–84. [Google Scholar] - Castagna, J.P.; Swan, H.W. Principles of AVO cross plotting. Lead. Edge
**1997**, 1, 66–84. [Google Scholar] - An, Y.; Lu, J. Calculation of AVA responses for finely layered reservoirs. Math. Probl. Eng.
**2018**, 1–11. [Google Scholar] [CrossRef] - Roganov, Y.; Stovas, A. Low-frequency wave propagation in periodically layered media. Geophys. Prospect.
**2012**, 60, 825–837. [Google Scholar] [CrossRef] - Stovas, A.; Landrø, M.; Avseth, P. AVO attribute inversion for finely layered reservoirs. Geophysics
**2006**, 71, C25–C36. [Google Scholar] [CrossRef] - Stovas, A.; Landrø, M. Uncertainty in discrimination between net-to-gross and water saturation for fine-layered reservoirs. In SEG Technical Program Expanded Abstracts; Society of Exploration Geophysicists: Tulsa, OK, USA, 2006; pp. 1698–1702. [Google Scholar]
- Castagna, J.P.; Batzle, M.L.; Eastwood, R.L. Relationship between compressional-wave and shear-wave velocities in clastic silicate rocks. Geophysics
**1985**, 50, 571–581. [Google Scholar] [CrossRef] - Gardner, G.H.F.; Gardner, L.W.; Gregory, A.R. Formation velocity and density—The diagnostic basics for stratigraphic traps. Geophysics
**1974**, 39, 770–780. [Google Scholar] [CrossRef]

**Figure 1.**A thin-bed model with two horizontal interfaces. P is incident P-wave, PS is reflected or transmitted S-wave, PP is reflected or transmitted P-wave; h is thin-bed thickness; θ is incidence or reflected angle of P-wave, θ

_{3}is P-wave transmitted angle, δ

_{1}and δ

_{3}are S-wave reflected and transmitted angles respectively; v

_{P}and v

_{S}are P-wave and S-wave velocities respectively, ρ is density, and their subscripts 1, 2, 3 refer to three layers of the thin bed respectively.

**Figure 2.**Flowchart of the second-order series approximation simplified from the thin-bed exact R/T coefficient equations.

**Figure 4.**Comparison of Model 1’s R

_{PP}calculated by true value and the second-order series approximation, and top-interface and bottom-interface R

_{PP}calculated by the Zoeppritz equations, (

**a**) amplitude, and (

**b**) phase.

**Figure 5.**Comparison of Model 2’s R

_{PP}calculated by true value and the second-order series approximation, and top-interface and bottom-interface R

_{PP}calculated by the Zoeppritz equations, (

**a**) amplitude, and (

**b**) phase.

**Figure 6.**Comparison of Model 3’s R

_{PP}calculated by true value and the second-order series approximation, and top-interface and bottom-interface R

_{PP}calculated by the Zoeppritz equations, (

**a**) amplitude, and (

**b**) phase.

**Figure 7.**Comparison of Model 4’s R

_{PP}calculated by true value and the second-order series approximation, and top-interface and bottom-interface R

_{PP}calculated by the Zoeppritz equations, (

**a**) amplitude, and (

**b**) phase.

**Figure 8.**Relative errors of R

_{PP}induced by the second-order series approximation, (

**a**) amplitude of Model 1, (

**b**) phase of Model 1, (

**c**) amplitude of Model 2, (

**d**) phase of Model 2, (

**e**) amplitude of Model 3, (

**f**) phase of Model 3, (

**g**) amplitude of Model 4, and (

**h**) phase of Model 4.

**Table 1.**Rock properties of thin-bed models, units of velocities and densities are m/s and g/cm

^{3}, respectively.

Layer No. | v_{P} | v_{S} | ρ | R_{1} | R_{2} | |
---|---|---|---|---|---|---|

Model 1 | 1 | 3000 | 1414 | 2.29 | / | / |

2 | 3800 | 2103 | 2.43 | 0.1468 | –0.1468 | |

3 | 3000 | 1414 | 2.29 | / | / | |

Model 2 | 1 | 3000 | 1414 | 2.29 | / | / |

2 | 2400 | 897 | 2.17 | −0.1376 | 0.1376 | |

3 | 3000 | 1414 | 2.29 | / | / | |

Model 3 | 1 | 3000 | 1414 | 2.29 | / | / |

2 | 3200 | 1586 | 2.33 | 0.0409 | 0.0388 | |

3 | 3400 | 1759 | 2.37 | / | / | |

Model 4 | 1 | 3400 | 1759 | 2.37 | / | / |

2 | 3200 | 1586 | 2.33 | −0.0388 | –0.0409 | |

3 | 3000 | 1414 | 2.29 | / | / |

**Table 2.**Rock properties, A

_{0}and A

_{2}of the testing thin-bed model, units of velocities and densities are m/s and g/cm

^{3}, respectively.

Layer No. | v_{P} | v_{S} | ρ | h | R_{1} | R_{2} | |
---|---|---|---|---|---|---|---|

Model 5 | 1 | 3000 | 1414 | 2.29 | / | / | / |

2 | 3440 | 1793 | 2.37 | λ/10 | 0.0854 | −0.0854 | |

3 | 3000 | 1414 | 2.29 | / | / | / | |

A_{0} | Amplitude | 0.1006 | A_{2} | Amplitude | 0.2364 | ||

Phase | 0.9355 | Phase | −2.2545 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, C.; Wang, Y.; Lu, J.; Chen, B.; Shi, L. A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections. *Appl. Sci.* **2019**, *9*, 709.
https://doi.org/10.3390/app9040709

**AMA Style**

Yang C, Wang Y, Lu J, Chen B, Shi L. A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections. *Applied Sciences*. 2019; 9(4):709.
https://doi.org/10.3390/app9040709

**Chicago/Turabian Style**

Yang, Chun, Yun Wang, Jun Lu, Benchi Chen, and Lei Shi. 2019. "A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections" *Applied Sciences* 9, no. 4: 709.
https://doi.org/10.3390/app9040709