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

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

Andrey V. Baev

Veröffentlicht/Copyright: 13. März 2018

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Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 26 Heft 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.

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Received: 2017-09-12

Accepted: 2018-03-04

Published Online: 2018-03-13

Published in Print: 2018-10-01

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

Schlagwörter für diesen Artikel

Radon transform; layered elastic medium; longitudinal and transverse waves