Solution of the inverse seismic problem in a layered elastic medium by means of the τ-p Radon transform

Andrey V. Baev

doi:10.1515/jiip-2017-0083

Article

Solution of the inverse seismic problem in a layered elastic medium by means of the τ-p Radon transform

Andrey V. Baev

Published/Copyright: March 13, 2018

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From the journal Journal of Inverse and Ill-posed Problems Volume 26 Issue 5

Abstract

We consider the inverse scattering problem in a layer-homogeneous elastic structure filling a half-plane. Scattering data is the seismic wave field registered on the boundary with a surface point source. We prove that velocities of longitudinal and transverse waves, and a density of the medium are recovered by a unique way. The algorithm of the inverse problem solution is based on the τ-p Radon transform. Also, we present some results of numerical modeling for waves propagation and solving of the inverse problem.

Keywords: Radon transform; layered elastic medium; longitudinal and transverse waves

MSC 2010: 81U40

Funding source: Russian Foundation for Basic Research

Award Identifier / Grant number: 17-01-00525

Funding statement: This work was supported by RFBR, project 17-01-00525, but Section 5 by Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, project AAAA-A16-116021510092-2.

A Appendix

Establish a relationship between scattering data {f1⁢(τ,p),f2⁢(τ,p)} from (4.1) and data {F2P⁢(τ,p),F1S⁢(τ,P)} from equations (4.7). We use the Laplace transform for this purpose. By definition, for ψ⁢(τ,p), equal to zero as τ<0, and a fixed p≠0, put

ψ~⁢(s)=∫-∞∞exp⁡(-s⁢τ)⁢ψ⁢(τ,p)⁢𝑑τ.

In this case, there are the obvious equalities

F~2p⁢(s)cos⁡α=f~2P⁢(s)φ~2P⁢(s),F~1S⁢(s)cos⁡β=f~1S⁢(s)φ~1S⁢(s).

Taking into account equations (4.5), (4.6), we obtain

(A.1)F~2P⁢(s)cos⁡α=A⁢sin⁡α+2⁢sin2⁡β⁢s⁢f~2⁢(s)sin⁡α⁢cos⁡2⁢β⁢s2⁢f~1⁢(s),F~1S⁢(s)cos⁡β=2⁢sin⁡β⁢f~1⁢(s)A⁢sin⁡α-cos⁡2⁢β⁢s⁢f~2⁢(s),

where A=φ0/σ1. These equations determine the functions {F2P(τ,p), F1S(τ,p)} from {f1⁢(τ,p),f2⁢(τ,p)} in a unique way.

Let us find the inverse equalities, which make it possible to obtain scattering data (4.1) from equations (4.7). Represent (A.1) in the form of a linear system with respect to the functions f~1⁢(s) and f~2⁢(s):

sin⁡α⁢cos⁡2⁢β⁢s2⁢F~2P⁢(s)⁢f~1⁢(s)-cos⁡α⁢(1-cos⁡2⁢β)⁢s⁢f~2⁢(s)=A⁢cos⁡α⁢sin⁡α,

sin⁡2⁢β⁢f~1⁢(s)+cos⁡2⁢β⁢s⁢F~1S⁢(s)⁢f~2⁢(s)=A⁢sin⁡α⁢F~1S⁢(s).

We obtain the following solution of this system:

(A.2){f~1⁢(s)=A⁢cos⁡α⁢sin⁡α⁢F~1S⁢(s)/D~⁢(s),f~2⁢(s)=A⁢sin⁡α⁢(sin⁡α⁢cos⁡2⁢β⁢s⁢F~1S⁢(s)⁢F~2P⁢(s)-cos⁡α⁢sin⁡2⁢β/s)/D~⁢(s),D~⁢(s)=sin⁡α⁢cos2⁡2⁢β⁢s2⁢F~1S⁢(s)⁢F~2P⁢(s)+cos⁡α⁢sin⁡2⁢β⁢(1-cos⁡2⁢β).

It is easy to see that as a result of the inverse Laplace transform we get integral equations of the second kind. Indeed, D~⁢(s)≠0 for s→+∞ since

lims→+∞⁡s⁢F~2P⁢(s)=F2P⁢(+0,p)=1,lims→+∞⁡s⁢F~1S⁢(s)=F1S⁢(+0,p)=1.

A similar consideration yields the useful formula

R⁢(p)≡f2⁢(+0,p)f1⁢(+0,p)=sin⁡(α-2⁢β)cos⁡α,

which allows us to determine a1 and b1 if the ratio R⁢(p) is known for two different values p≠0.

Suppose that the reflection coefficients

(A.3)Kja≡a⁢(Zj+1,p)-a⁢(Zj,p)a⁢(Zj+1,p)+a⁢(Zj,p),Kjb≡b⁢(Zj+1,p)-b⁢(Zj,p)b⁢(Zj+1,p)+b⁢(Zj,p),

are small for any j=1,…,N; this often occurs in applications. In this case, the so-called Born (linearized) approximation, as a rule, is used. In the framework of this approximation, according to [18], we have

s⁢F~2P⁢(s)=1+2⁢∑j=1NKja⁢e-2⁢s⁢Zj+𝒪⁢(ε2),s⁢F~1S⁢(s)=1+2⁢∑j=1NKjb⁢e-2⁢s⁢Yj+𝒪⁢(ε2),

where ε=max1≤j≤N⁡{|Kja|,|Kjb|} is a small parameter. In a similar way, we can linearize formula (A.2). However, the result is very complicated, so it is easier to solve system (A.2) for specific values of α and β.

Let us demonstrate the transition to the Born approximation as α=2⁢β. We have

sin⁡α⁢s2⁢F~2P⁢(s)⁢f~1⁢(s)-(1-cos⁡α)⁢s⁢f~2=A⁢sin⁡α,

sin⁡α⁢f~1⁢(s)+cos⁡α⁢s⁢F~1S⁢(s)⁢f~2⁢(s)=A⁢sin⁡α⁢F~1S⁢(s),

whence

f~1⁢(s)=A⁢F~1S⁢(s)/D~1⁢(s),f~2⁢(s)=A⁢sin⁡α⁢(s⁢F~1S⁢(s)⁢F~2P⁢(s)-1s)/D~1⁢(s),

where D~1⁢(s)=s2⁢cos⁡α⁢F~1S⁢(s)⁢F~2P⁢(s)+1-cos⁡α.

In the linearized approximation, the following equalities hold:

s⁢F~1S⁢(s)⁢F~2P⁢(s)=F~1S⁢(s)+F~2P⁢(s)-1s,

1/D~1⁢(s)=1+cos⁡α⁢(2-s⁢F~1S⁢(s)-s⁢F~2P⁢(s)),

which finally give

(A.4){f~1⁢(s)=A⁢(2⁢cos⁡α/s+(1-cos⁡α)⁢F~1S⁢(s)-cos⁡α⁢F~2P⁢(s)),f~2⁢(s)=A⁢sin⁡α⁢(F~1S⁢(s)+F~2P⁢(s)-2s).

Let us return to the τ-p representation and obtain an approximation uniform with respect to τ with an accuracy of 𝒪⁢(ε2):

(A.5){f1⁢(τ,p)=A⁢(θ⁢(τ)-2⁢∑j=1N(Kja⁢cos⁡α⁢θ⁢(τ-2⁢Zj)-Kjb⁢(1-cos⁡α)⁢θ⁢(τ-2⁢Yj))),f2⁢(τ,p)=2⁢A⁢sin⁡α⁢∑j=1N(Kja⁢θ⁢(τ-2⁢Zj)+Kjb⁢θ⁢(τ-2⁢Yj)).

We use solutions (A.5) for the numerical simulation in Section 5.

References

[1] A. S. Alekseev and A. G. Megrabov, Direct and inverse scattering problems of plane waves in inhomogeneous layers (in Russian), Mat. Problemy Geofiz. (1972), no. 3, 8–36. Search in Google Scholar

[2] A. V. Baev, A method of solving the inverse scattering problem for the wave equation, Comp. Math. Math. Phys. 28 (1988), no. 1, 15–21. 10.1016/0041-5553(88)90211-XSearch in Google Scholar

[3] A. V. Baev, On the solution of the inverse problem of scattering of a plane wave by a layered-inhomogeneous medium, Sov. Phys. Dokl. 32 (1988), no. 1, 26–28. Search in Google Scholar

[4] A. V. Baev, On the solution of the inverse boundary value problem for the wave equation with a discontinuous coefficient, Comp. Math. Math. Phys. 28 (1988), no. 6, 11–21. 10.1016/0041-5553(88)90037-7Search in Google Scholar

[5] A. V. Baev, On t-local solvability of inverse scattering problems in two dimensional layered media, Comp. Math. Math. Phys. 55 (2015), no. 6, 1033–1050. 10.1134/S0965542515060032Search in Google Scholar

[6] A. V. Baev, Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media, Comp. Math. Math. Phys. 56 (2016), no. 12, 15–21. 10.1134/S0965542516120034Search in Google Scholar

[7] A. V. Baev and N. V. Kutsenko, Solving the inverse generalized problem of vertical seismic profiling, Comput. Math. Model. 15 (2004), no. 1, 1–18. 10.1023/B:COMI.0000011680.61982.4fSearch in Google Scholar

[8] A. V. Baev and N. V. Kutsenko, Identification of a dissipation coefficient by a variational method, Comp. Math. Math. Phys. 46 (2006), no. 1, 1796–1807. 10.1134/S0965542506100150Search in Google Scholar

[9] E. N. Bessonova, V. M. Fishman, V. Z. Ryaboy and G. A. Sitnikova, The tau method for inversion of travel times 1. Deep seismic sounding data, Geophys. J. R. Astron. Soc. 36 (1974), 377–398. 10.1111/j.1365-246X.1974.tb03646.xSearch in Google Scholar

[10] A. S. Blagoveshchenskii, On local method for solving of nonstationary inverse problem for inhomogeneous string (in Russian), Trudy MIAN V. A. Steklova 115 (1971), 28–38. Search in Google Scholar

[11] T. M. Brocher and R. A. Phinney, Inversion of slant stacks using finite-length record sections, J. Geophys. Res. 86 (1981), 7065–7072. 10.1029/JB086iB08p07065Search in Google Scholar

[12] C. H. Chapman, A new method for computing synthetic seismograms, Geophys. J. R. Astron. Soc. 54 (1978), 481–518. 10.1111/j.1365-246X.1978.tb05491.xSearch in Google Scholar

[13] C. H. Chapman, Generalized radon transform and slant stacks, Geophys. J. R. Astron. Soc. 66 (1981), 445–453. 10.1111/j.1365-246X.1981.tb05966.xSearch in Google Scholar

[14] J. F. Claerbout, Fundamentals of Geophysical Data Processing, McGraw-Hill, New York, 1976. Search in Google Scholar

[15] J. F. Claerbout and A. G. Jonson, Extrapolation of time dependent waveforms along their path of propagation, Geophys. J. R. Astron. Soc. 26 (1971), no. 1–4, 285–295. 10.1111/j.1365-246X.1971.tb03402.xSearch in Google Scholar

[16] R. W. Clayton and G. A. McMechan, Inversion of refraction data by wavefield continuation, Geophys. 46 (1981), 860–868. 10.1190/1.1441224Search in Google Scholar

[17] A. A. Kaufman, A. L. Levshin and K. L. Larner, Acoustic and Elastic Wave Fields in Geophysics. II, Elsevier, New York, 2002. Search in Google Scholar

[18] A. N. Kolmogorov, Sur l’interpolation et l’extrapolation des suites stationnaires, C. R. Acad. Sci. Paris 208 (1939), 2043–2045. Search in Google Scholar

[19] M. E. Kupps, A. J. Harding and J. A. Orcutt, A comparison of tau-transform methods, Geophys. 55 (1990), no. 9, 1202–1215. 10.1190/1.1442936Search in Google Scholar

[20] M. M. Lavrent’ev, K. G. Reznitskaia and V. G. Iakhno, One-Dimensional Inverse Problems of Mathematical Physics (in Russian), Nauka, Novosibirsk, 1982. Search in Google Scholar

[21] G. A. McMechan, Migration by extrapolation of time-dependent boundary value, Geophys. Prosp. 31 (1983), 413–420. 10.1111/j.1365-2478.1983.tb01060.xSearch in Google Scholar

[22] G. A. McMechan, P-x imaging by localized slant stacks of T-x data, Geophys. J. R. Astr. Soc. 72 (1983), 213–221. 10.1111/j.1365-246X.1983.tb02813.xSearch in Google Scholar

[23] A. G. Megrabov, Method for restoring of density and velocity in an inhomogeneous layer as functions of the depth from a set of plane waves reflected from the layer at different angles (in Russian), Mat. Problemy Geofiz. 5 (1974), no. 2, 78–107. Search in Google Scholar

[24] B. Milkereit, W. D. Mooney and W. M. Kohler, Inversion of seismic refraction data in planar dipping structure, Geophys. J. Intern. 82 (1985), no. 1, 81–103. 10.1111/j.1365-246X.1985.tb05129.xSearch in Google Scholar

[25] W. Nowacki, Theory of Elasticity (in Polish), PWN, Warsawa, 1970. Search in Google Scholar

[26] I. Psencik, V. Cerveny and L. Klimes, Seismic Waves in Laterally Inhomogeneous Media. Part I, Birkhäuser, Berlin, 1996. 10.1007/978-3-0348-9213-1Search in Google Scholar

[27] V. G. Romanov, Inverse Problems of Mathematical Physics, VSP, Netherlands, 1984. Search in Google Scholar

[28] V. G. Romanov, Determining the parameters of a stratified piecewise constant medium for the unknown shape of an impulse source, Sib. Math. J. 48 (2007), no. 6, 1074–1084. 10.1007/s11202-007-0109-ySearch in Google Scholar

[29] V. G. Romanov, On the problem of determining the parameters of a layered elastic medium and an impulse source, Sib. Math. J. 49 (2008), no. 5, 919–943. 10.1007/s11202-008-0090-0Search in Google Scholar

[30] P. S. Schultz, Velocity estimation and downward continuation by wave front synthesis, Geophys. 43 (1978), 691–714. 10.1190/1.1440847Search in Google Scholar

[31] A. N. Tikhonov, On uniqueness of solution of the electroprospecting problem (in Russian), Dokl. Akad. Nauk SSSR 69 (1949), no. 6, 797–800. Search in Google Scholar

[32] A. N. Tikhonov, Mathematical basis of the theory of electromagnetic soundings, Comp. Math. Math. Phys. 5 (1965), no. 3, 207–211. 10.1016/0041-5553(65)90157-6Search in Google Scholar

[33] V. G. Yakhno, Inverse Problem for Differential Equations of Elasticity (in Russian), Nauka, Novosibirsk, 1990. Search in Google Scholar

[34] B. Zhou and S. A. Greenhalgh, Linear and parabolic τ-transforms revisited, Geophys. 59 (1994), no. 7, 1133–1149. 10.1190/1.1443669Search in Google Scholar

Received: 2017-09-12

Accepted: 2018-03-04

Published Online: 2018-03-13

Published in Print: 2018-10-01

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Articles in the same Issue

https://doi.org/10.1515/jiip-2017-0083

Keywords for this article

Radon transform; layered elastic medium; longitudinal and transverse waves