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On inverse scattering in an elastic medium with vertical inhomogeneitiesPhilip M. Carrion
Citation: Journal of Mathematical Physics 27, 1164 (1986); doi: 10.1063/1.527160 View online: http://dx.doi.org/10.1063/1.527160 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/27/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Inversion of the elastic parameters of a layered medium J. Acoust. Soc. Am. 97, 1687 (1995); 10.1121/1.412046 An inverse scattering method to characterize inhomogeneities in elastic solids J. Appl. Phys. 62, 2771 (1987); 10.1063/1.339405 Inverse scattering in a stratified medium J. Acoust. Soc. Am. 74, 994 (1983); 10.1121/1.389846 Inverse scattering in a stratified medium J. Acoust. Soc. Am. 73, S38 (1983); 10.1121/1.2020360 Scattering in an Inhomogeneous Medium J. Acoust. Soc. Am. 29, 50 (1957); 10.1121/1.1908682
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On inverse scattering in an elastic medium with vertical inhomogeneities Philip M. Carrion Geophysics Program, Henry Krumb School of Mines, Columbia University, New York, New York 10027
(Received 13 June 1984; accepted for publication 4 December 1985)
This paper treats the nonlinear inverse scattering problem for an elastic horizontally stratified medium with vertical inhomogeneities. It is shown that the vector wave equation can be presented as a set of two scalar wave equations; the first scalar equation describes the propagation of normally incident compressional waves (it is assumed that we deal with compressional incident plane waves impinging on an elastic half space). Using any inverse scattering technique (based, for example, on the Gelfand-Levitan theorem) applied to this equation, the aco~sti~ im~dance can be uniquely recovered from the recorded impulse response. The second equatIOn IS vahd only in the region where incident compressional waves decay exponentially (evanesce~t waves) and propagate with complex angles, whereas mode-converted shear waves propagate WIth real angles. The main contribution in this region in the high-frequency approximation comes from modeconverted shear waves. We derived a new equation which describes the propagation of modeconverted shear waves in this region. Applying any inverse scattering technique to this equation, the rigidity modulus and the density can be removed separately. Knowing the acousti~ impedance, the rigidity modulus, and the density, all Lame's parameters and the densIty can be recovered separately.
I. INTRODUCTION
The interpretation of seismic data requires a solution to the inverse scattering problem. The one-dimensional inverse scattering problem in the acoustic framework has been studied over the years. A comprehensive review of one-dimensional inverse scattering techniques was given by Newton. 1
Recently several methods have been presented that treat the inverse scattering in elastic media with vertical inhomogeneities. Coen2 was probably one of the first who considered the problem in an elastic horizontally stratified medium and recovered separately the elastic parameters. His method is based on separate experiments with vertically incide~t compressional and shear waves. In that case the ~roblem IS simplified since at normal incidence compressIOnal and shear waves are uncoupled (there is no interconversion between P-waves and SV-waves at normal incidence). Aminzadeh/ Shiva and MendeV Clarke,5 and Yagle and Levy6 developed methods for estimation of all Lame's parameters and the density based on a layer-stripping technique. Recently Stickler7 presented a now approach using the "trace formula." A method for determination of material density, compressional velocity based on a solution o! a matrix Riccati equation was presented by Carazzone. Meadows and Coen9 considered exact and approximate algorithms for inversion of plane-layered isotropic and anisotropic elastic media.
All the methods mentioned above require separate experiments with incident compressional and shear waves .. The primary concern of this paper is to develop a new techmque that will provide a solution to the inverse scattering problem in elastic media with vertical inhomogeneities when only experiments with plane wave compressional waves are available. For example, in a marine environment when we try to estimate material properties of subbottom sediments the only sources available in the water are sources that generate
only compressional waves. In this paper we give an exact, formal answer giving the
circumstances under which the inverse scattering problem in elastic stratified media can be solved without using shearwave experiments.
We assume that the source that generates only compressional waves is placed in a liquid layer that covers an elastic horizontally homogeneous half space. We also assume that the time function of the source (source wavelet) is a Dirac's delta function. Incident compressional waves can be reflected, scattered, or mode converted in the elastic half space. The reflection response from the elastic half space is measured in the liquid layer in terms of negative stress (pressure) or the vertical component of particle velocity.
We propose a two-step process: The first step is the computation of the acoustic impedance from the wave equation that describes the experiment with compressional waves at normal incidence. The second step involves the derivation of a new equation that governs the propagation of evanescent compressional waves at complex angles of propagation and mode-converted shear waves at real angles of propagation. Under the assumptions of geometrical optics, evanescent waves propagating with complex angles decay very fast. Therefore in the region of rapidly decaying compressional waves that propagate with real angles (we certainly assume that such a region exists), only mode-converted shear waves occur. After solving the inverse scattering problem for this region using the derived equation, the rigidity modulus and the density can be computed separately. This allows for estimation of Lame's parameters as well as the density.
II. DECOMPOSITION OF THE VECTOR ELASTIC WAVE EQUATION
Let us consider the elastic wave equation, which can be written as
1164 J. Math. Phys. 27 (4), April 1986 0022-2488/86/041164-05$02.50 ® 1986 American Institute of Physics 1164
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(2.1 )
where uj is the 3-D displacement vector,p is the density, an~ Tij is a nine component stress tensor. In the case of a medlum with vertical inhomogeneities, Eq. (2.1) can be reduced to
J.L(z)V2u + [...t(z) + J.L(z) ]VV· u + a...t(z) (V ·u)ez az
aJ.L(z) [ au ] a2u +-- 2-+ez XVXu =p(z) --2 '
az az at (2.2)
wherep,...t,J.L are defined forO <z < 00 and take values inR;...t andJ.L are Lame's parameters; and ez is a unit vector directed along the z axis.
Let us consider the cylindrical coordinate system and introduce a Hankel transform of order a:
ju(kr,m) = 1"0 j(r,m)Jer(krr)rdr = Her(j), (2.3)
whereJu (.) is a Bessel function of order a and kr is the radial wave number. Applying a Laplace transform to (2.1) and setting s = im, where s is the argument of the Laplace transform, the following equation can be obtained after application of the Hankel transform of order zero (a = 0) and after projection of (2.1) onto the z axis:
~ + -- - p2 Uz + -In [...t(z) + '2J.L(z)] ~ a 2A (1 ) a aA
ar a 2 (z) az az
+ mp [aA.(z) + (...t(z) +J.L(Z)~]HI(Ur) (A.(z) + 2#(z») az az
+ (mp)2 [...t(z) +J.L(z)] Uz
=0, (2.4) [...t(z) + 2J.L(z)]
where
uz=Ho(u z ) (2.5)
and U r is the radial component of the displacement (Carrion and Hassanzadeh 10). In Eq. (2.4), P is the ray parameter or horizontal slowness,
p = kJm, (2.6)
and a (z) is the velocity of compressional waves as a function ofz,
a(z) = ~[...t(z) + '2J.L(z) ]lp(z) . (2.7)
Equation (2.4) is the plane wave decomposition of the elastic wave equation (2.2) for the vertical component of the displacement; it describes the propagation of plane waves in an elastic medium with vertical inhomogeneities. Setting p = 0 in Eq. (2.4), we obtain the following equation:
a~ ~ A a ~ --+ -2- Uz + -In[...t(z) + 2J.L(z)] - = o. az2 a (z) az az
Let us introduce a travel-time coordinate q:
q(z) = ("" --!!L. Jo act)
(2.8)
(2.9)
Using the definition of the travel-time coordinate q(z), Eq. (2.8) can be presented as follows:
1165 J. Math. Phys., Vol. 27, No.4, April 1986
[~ alnI(q)~+ 2]A ( )=0 2 + m Uz m,q ,
aq aq aq (2.10)
where
I(q) = p(q)a(q)
is the acoustic impedance. Equation (2.10) is called the "reflectivity equation" since the second term in brackets on the left-hand side of this equation is proportional to the reflection coefficient in the single scattering approximation. This equation has been studied by several authors (see, for example, Ware and Aki,lI Gray, 12 and Carrion et al. 13
).
In order to compute the acoustic impedance I(q) by inversion of the differential operator in (2.10) several approaches can be used. Some of them will be discussed in the next paragraph.
III. COMPUTATION OF THE ACOUSTIC IMPEDANCE
Suppose that I(q) is smooth enough so I(q)EC 2. Then Eq. (2.10) can be transformed to the Schrodinger equation by the following substitutions:
G(q) =I-I/2(q) (3.1)
and
t/J(m,q) = u(m,q)G -I(q).
This yields
[:q: + m2] t/J(m,q) = S(q)t/J(m,q),
where the scattering potential S is
a2
Seq) = G(q) -2 G -I(q). aq
(3.2)
(3.3 )
(3.4 )
Now the Gel'fand-Levitan treatment can be applied as long as the boundary conditions are specified. Suppose that the boundary conditions are taken in terms of the observed data at the plane z = 0:
d(t,r) = u(t,r,z = 0) is available for any re[O,oo).
A Laplace-Hankel transform applied to the data yields
u(m,p) = Ho[d(t,r)]. (3.5)
The boundary condition in the form of (3.5) with plane wave decomposed elastic wave equation (2.4) provides us with all ingredients needed for the inversion procedure. Setting p = 0 in Eq. (3.5) and using Eq. (3.3) the acoustic impedance I(q) can be directly recovered using the Gel'fand-Levitan treatment. Ifwe require that I(q)EC I, we can avoid the use of the Schrodinger operator and the acoustic impedance can be recovered from the reflectivity equation (2.10) using the algorithm proposed by Carrol and Santosa.14
Since the analyticity of the wave field in the lower half space is equivalent to the causality in the time domain, the principles of causality of the wave field can be used to construct different algorithms in the time domain. Burridgel5
extended the Gel'fand-Levitan theorem to the time variable wave fields and developed time domain algorithms for the recovery of the acoustic impedance using time domain Schrodinger operators.
An inverse Fourier transform of equation (1.10) yields
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[a 2 alnJ(q) a a 2
] aq2 + aq aq - at 2 Uz (t, q) = O. (3.6)
This equation is equivalent to the following system of partial differential equations (PDE's):
a J(q) ~ P(t,q) at aq
a J(q) ~ aq at
=0, (3.7)
V(t,q)
where V(t,q) = (a lat)u z (t, q) is the vertical component of particle velocity. Setting the initial conditions in the form
P(t, q) = V(t, q) = 0, t < 0,
and the boundary conditions
V(t,O) = d(t, q),
P(t,O) = b(t),
(3.8)
(3.9)
the acoustic impedance can be computed on one of the characteristics of a set of hyperbolic PDE's:
J(q) = P(q, q)IV(q, q). (3.10)
The downward continuation algorithm similar to (3.7)(3.10) has been discussed by Santosa and Schwetlick/6
Bube and Burridge, 17 and Foster and Carrion. IS The discrete implementation of these schemes is directly related to the Cholescky factorization of ToepIitz operators.
One of the advantages of such schemes is that they are computationally efficient and fast in comparison with algorithms based on a solution ofintegral equations. It also should be mentioned that in (3.10) the acoustic impedance or its derivatives can have a finite set of points where they have jump discontinuities.
IV. DERIVATION OF AN EQUATION FOR MODECONVERTED SHEAR WAVES
Let us consider formula (2.3) for u = O. The inverse Hankel transform of order zero can be presented as
(4.1 )
Since k, = UJp = UJ sin; la, where ;is the angle ofpropagation, (4.1) can be rewritten as
2
1;1
I(UJ,r) =~ 1 (UJ,;)Jo(UJ,;) sinscos;d;, (4.2) a 0
where; 1= 1T/2 - ioo (seeBathI9). Formula (4.2) is a com
plete description of the point-source reflection response. The upper limit of integration in (4.2) is complex because the angle of propagation; can take on values which correspond to supercritical incidence (postcritical incidence). Complex values of the angular spectrum correspond to complex values of vertical number kz and describe exponentially decay evanescent waves. It is important to investigate in which cases the contribution of the evanescent waves can be neglected. Since for evanescent waves the amplitudes behave as
exp{ -\kz\z),
it is obvious that for z> \ kz \- I the contribution of the evanescent waves is negligibly small.
Let us now project Eq. (2.2) onto the z axis:
1166 J. Math. Phys .• Vol. 27. No.4, April 1986
p(z)V2u + [(A(Z) + p(z») ~ + ~A(Z)]V.u az az
+ 2 ap(z) au = (z) a2u . (4.3)
az az P at 2
Let us now introduce two functions (potentials) qJ and X which generate the displacement via
u = VqJ + VX(O,X,O) (4.4)
(see Richards and Frasier20).
The scalar potential qJ can be presented as
qJ(r, z) = A (r, z) exp(iUJn), (4.5)
where A (r, z) is the amplitude and n describes the phase of the scalar potential. In the high-frequency approximation n satisfies the eikonal equation:
[Vn]2 = a- 2 (z), (4.6)
where a (z) is the local velocity for compressional waves. The amplitude A (r, z) satisfies the transport equation:
AV2n + 2VA·Vn = O. (4.7a)
Usually the eikonal equation (4.6) can be solved using ray tracing and then the transport equation (4.7a) can be reduced to the ordinary differential equation along the rays that are characteristics of the eikonal equation. In our derivation we are not interested in finding the amplitudes of the scalar and the vector potentials. They can be found from the continuity of traction and displacement across boundaries between each pair of adjacent layers.
Let us assume that for the incident compressional field
n = pr + z cosjla(z). (4.7b)
It is easy to see that ( 4. 7b) satisfies the eikonal equation. It is also easy to see that the transmitted compressional wave can be described in the approximation of geometrical optics by the phase of the scalar potential
n =pr + r cos S df: (4.8) t Jo act) ~,
which certainly also satisfies the eikonal equation (4.6). In the eikonal approximation the amplitUde of transmitted Pwaves can be estimated from a balance of energy flux toward and away from the interfaces between layers.
Since there are only two types of speeds for body waves in an elastic medium--compressional and shear-the eikonal equation for shear waves can be written as
(V~)2 =/3 -2(Z), (4.9)
where ~ is the phase of the vector potential X. Suppose that a compressional wave was converted to shear wave at depth z and propagates as a shear wave. Then the reflected mode converted shear wave can be presented by the vector potentialX:
x(r,z) =pr+[ coss dt. z pes)
( 4.10)
Let us consider now the second term on the left-hand side of equation (4.3). It is quite obvious that only gradients of the scalar potential contribute to this term. Let us also consider the "postcritical" region for compressional waves, where
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pa(z) > 1, for all values of z. (4.11 )
Then the scalar potentiallp for transmitted compressional waves in the "postcritical" region can be expressed as follows:
lpt (r, z) = At (r, z) eXP(iwpr - w f I :~~~ I ds ) , ( 4.12)
whereAt (r, z) is the amplitUde ofthe transmitted compressional wave. Since the integrand in the exponential function is positive if condition (4.11) holds, then (4.12) describes compressional evanescent waves that exponentially decay with z. In the approximation of geometrical optics (w- 00 ),
lpt-o. This means that in the region described by (4.11), Eq. (4.3) can be approximately presented as
,u(z)V2u + 2 a,u(z) au = _ p(z)w2u. (4.13) az az
Applying a Hankel transform to Eq. (4.13) yields
a 2uz + 2 a In,u (z) au 2 (1 2) A 0 -- -+w ----p u= , ar az az P2(Z)
(4.14 )
where p 2 = ,ul p is the local speed of shear waves. Let us introduce a variable h similar to (2.9):
h(z) = r ~1-p2pl(S) dS. Jo pes)
(4.15 )
Using this coordinate whose physical meaning is the vertical travel time, Eq. (4.16) can be rewritten as
(~22 + ~ In [(cos ;-)K(h),u(h)] ~ + ( 2)U(W,h) = 0,
( 4.16)
where cos;- = ~ 1 - plp 1 and K(h) is the quantity which we call the "shear impedance,"
K(h) =,u(h)p(h).
Equation (4.16) has an interesting property. This equation is similar to the acoustic impedance in the travel-time coordinate (2.10), which describes the propagation of the compressional waves. This remarkable property will allow us to obtain an algorithm that recovers the modulus and the density separately.
V. SEPARATE RECOVERY OF ALL LAME'S PARAMETERS AND THE DENSITY
Equation (4.16) is equivalent to the following set of PDE's in the time domain:
aW(t,h) + K(h),u (h) cos;- aV(t,h) = 0, (5.1) at ah
aW(t,h) + K(h),u(h) cos;- aV(t,h) = 0, (5.2) ah at
where W(t,h) is an auxiliary function that satisfies the boundary condition
W(t,h = 0) =,u(O)I5(t)/2,
and the initial condition
W(t,h) = 0, for all t <0.
1167 J. Math- Phys., Vol. 27, No.4, April 1986
(5.3)
(5.4)
Since V(t,h = 0) represents the observed data (boundary condition for the vertical component of particle velocity) the Gopinath-Sondhi-type integral operator can be written as follows:
1 f+h, j(hjlt} + - V(lt -rl )j(r,h;) dr = 1,
2 -h,
i=I,2, r<lh;l, (5.5)
which simply means that the (5.5) should be solved twice for any two experiments with plane compressional waves such that condition (4.11) is satisfied. Then we can estimate the quantity B = K(h),u(h)cos;- from
d Soh, B; = - j (hjlt) dt, i = 1,2.
dh; 0 (5.6)
This means that the rigidity modulus and the density can be recovered separately by solving two equations (5.6) with two unknowns. Knowing the acoustic impedance (Sec. III) all Lame's parameters can be estimates. We should mention that the procedure [(5.5) and (5.6)] is similar to one proposed by Coen21 for an acoustic medium using a Gel'fandLevitan treatment.
VI. REMARKS
Sometimes it is important to calculate the parameters of a medium (acoustic or elastic) as functions of depth. As it was shown we calculate all parameters as function of travel times (for compressional and shear waves). For this reason, in order to rescale the computed parameters an integral equation similar to the one described by Howard22 should be solved. Carrion23 proposed a numerical recursive scheme for rescaling the computed parameters and recovering them directly as functions of depth.
It was shown (Carrion24 et al. and Santosa and Symes25
) that the accuracy of methods for joint reconstruction of the velocity and the density of an acoustic medium depends on the difference of the angles of incidence. It is important that the angular difference be large, otherwise the reconstruction problem fails. This means that the difference between two rays chosen for the determination of the rigidity modulus and the density should not be small.
VII. DISCUSSION
The inverse problem for an acoustic or an elastic medium can be treated as a special case of the generalized Riemann boundary value problem, which was extensively studied in recent years (see Chudnovsky26). In order to solve the Riemann problem the boundary conditions along with the incident wave field should be specified.
We demonstrated the possibility of recovering all Lame's parameters and the density using experiments with compressional plane waves only. (Until now, mathematically rigorous justifications have been given only when experiments with incident shear waves are available.) Although Eq. (4.14) is approximate (it was derived in the approximation of geometrical optics), actually it can be used for a much wider range offrequencies. For example, in exploration geophysics typical frequencies can be considered as satisfying
Philip M. Carrion 1167
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the "high frequency" approximation and the impact of the evanescent waves will be very small.
It is important to mention that although we require that condition ( 4.11) holds, our formulation is valid when, below the depthz = Ikz I-I, there are so-called low-velocity zones, where possibly pa (z) is less than 1. In general, this can cause the "tunneling effect" when an evanescent wave propagates with real angles. However, below a certain depth (z> Ikz I-I), which is called the critical depth, the influence of the low-velocity zones is negligibly small since an evanescent wave arriving to this depth is characterized by negligibly small amplitude. In order to use Eq. (4.16) fortherecovery of the rigidity modulus and the density in the region of compressional evanescent waves we should choose those values of Snell's parameter (ray parameter) p that satisfy pP(z) < 1 for all values of z. Let us consider the following example: Suppose compressional plane waves impinge on the ocean bottom. Taking p. = lIao, where ao is the average sound speed in the water, we are assured that for any depth below the sea bottom, p·a(z) > 1. However, we also should be sure that p·P(z) < 1. This means that we can recover only those shear velocities that satisfy P(z) < lip·. If we want to recover higher shear velocities (if they exist) an additional approach will be discussed elsewhere.
ACKNOWLEDGMENTS
I would like to thank an anonymous reviewer whose suggestions and comments made the paper clearer.
1168 J. Math. Phys., Vol. 27, No.4, April 1986
This work was supported by the National Science Foundation No. EAR-850 59 22.
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