Generalized linear inversion and the first Born theory

Generalized linear inversion and the first Born theory for acoustic media R. G. Keys and A. B. Weglein Citation: J. Math. Phys. 24, 1444 (1983); doi: 10.1063/1.525879 View online: http://dx.doi.org/10.1063/1.525879 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v24/i6 Published by the American Institute of Physics.

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Generalized linear inversion and the first Born theory for acoustic media R. G. Keys and A. B. Weglein a ) Cities Service Company, Energy Resources Group, P.O. Box 3908, Tulsa, Oklahoma 74102

(Received 3 June 1982; accepted for publication 13 January 1983) A procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional Born inversion method. The method we present is a more general Born theory which can accommodate insufficient and inaccurate data. This general method reduces to the ordinary Born procedure when the data requirements of the latter technique are satisfied. PACS numbers: 03.40.Kf, 43.40.Ph I. INTRODUCTION

The inverse problem in wave propagation is to infer the characteristics of a medium from measurements of reflected or scattered waves. Two separate approaches to the inverse problem have evolved; both were derived from linearizations of the relationship between the medium parameters and the wave field. These two approaches are known as (1) the generalized linear inversion method, and (2) the first Born approximation. In this paper, the generalized linear inverse approach will be followed to construct a multidimensional Born inversion algorithm for the acoustic medium parameter. The method of generalized linear inversion refers to that class of techniques which are designed to obtain estimates of model parameters from inexact or insufficient data. This approach was used, for example, by Backus and Gilbert' and Wiggins, Larner, and Wisecup,2 and reviewed in Parker/ Sabatier,4 and Aki and Richards. s The development of a generalized linear inversion algorithm is a two-step process. First, a linear relationship between the model parameters and the measured data is postulated. Then, using a selection criterion, a model is determined which yields the best fit to the measured data. The method of generalized linear inversion has recently been applied to the one-dimensional seismic reflection problem by Cooke and Schneider. 6 The technique they present uses a linearized relationship between the acoustic wave field and the acoustic impedance of a one-dimensional medium. An inverse operator, which in their case will typically be a generalizp,d inverse matrix, is required to solve for the variation of the parameters about some reference model. A brief summary of their procedure is given in Sec. II. The first Born approximation for the acoustic wave field is a linear expression relating the wave field to an acoustic medium. The first Born approximation is obtained by replacing the unknown wave field in a Lippmann-Schwinger integral equation with the field which would exist under the same circumstances in the reference medium. In Sec. III, we present an overview of the Born theory approach to the acoustic inverse problem. In the remaining sections of this paper, the generalized linear inversion approach will be applied to the acoustics problem to develop an inversion procedure. In Sec. IV, a linearization of the acoustic wave field with respect to the index of refraction of the medium will be derived. An opti_I

Present address: Sohio Petroleum Company, One Lincoln Centre, Dallas, TX 75240.

1444

J. Math. Phys. 24 (6), June 1983

mal solution of the linearized problem will be determined in the following section. One of the results of this paper establishes a theoretical relationship between the generalized linear inversion method and the first Born approximation. This connection provides an avenue for transferring to the first Born theories some of the results which have been established for inaccurate and insufficient data in the generalized linear inversion problem. Consequently, an optimal Born model is derived which accommodates real world considerations, such as insufficient data and measurement errors. Furthermore, this more general Born result reduces to the ordinary Born inversion when the data requirements of the earlier theory are satisfied.

II. GENERALIZED LINEAR INVERSION

The problem of one-dimensional reflection seismology 7,8 is to determine the impedance profile from reflection seismic data, In a recent paper by Cooke and Schneider,6 the generalized linear inverse approach to the one-dimensional seismic inverse problem is presented, They begin by assuming a Taylor series expansion of the acoustic wave field:

where 1

the unknown impedance profile, the reference (or guessed) impedance profile, the error in the above guess, the observed acoustic wave field, the acoustic wave field generated by the impedance profile 10,

10 1 -10 tf;(I) tf;(10)

~j

(/0)

=

the derivative operator.

The three steps involved in this generalized linear inverse approach are (1) the linearization of the Taylor series expansion, realized by truncating the series after the second term, (2) the discretization of the problem, which results in a finite order matrix for atf;(lo)/al and finite dimensional vectors for tf;(I), tf;(/o), and 1 - 10 , and (3) a modified least-square error, generalized inverse technique, used to invert atf;lal and solve for 1 - /0 '

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© 1983 American Institute of PhysiCS

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1444

Knowing 10 , and calculating ~I provides a linear approximation for the impedance profile: I = (I - 10 ) + Io'Z~I + 10 , The technique presented by Cooke and Schneider can be applied iteratively by replacing 10 with the new estimate for I and repeating steps 1-3. The generalized linear inversion algorithm that will be derived later in this paper differs from the procedure developed by Cooke and Schneider in the following three ways. First, the proposed algorithm is a multidimensional technique. Second, the model parameter which will be determined is the index of refraction rather than the acoustic impedance. Third, the analysis is performed for the continuous case. There are no discretizations introduced into the problem; as a consequence, the optimum index of refraction is the solution of an integral equation. III. BORN INVERSION THEORY

Within a given model, Born inversion consists of, first, linearizing an integral equation between the model parameters and the wave field; and second, solving the resulting Fredholm I integral equation for the model parameters. Linearization in this problem is accomplished by replacing, in the integral equation, the actual wave field with the corresponding wave field in a reference model. This linearization is equivalent to truncating the Born series9 after the second term. In this paper, the expression "ordinary Born theory" refers to first Born approximations derived from general reference models, which are not necessarily constant background models. We now present a brief review ofthe multidimensional Born theory as it pertains to the acoustic wave equation. The pressure field p(x,t ) in a constant density acoustic medium satisfies (

V2 -

1

2

a at

--2 -2

)

p(x,t) = f(x,t),

c(x) wheref(x,t )is the source function, and c(x) is the local acoustic velocity at the spatial coordinate x. Let P (x,w) and F (x,w) denote the temporal Fourier transform ofp(x,t ) and F (x,t ), respectively. Then P(x,w) =

and F(x,w) =

L"" p(x,t )e

J: "

iwr

f(x,t )e

dt,

iwr

dt.

P (x,w) satisfies the differential equation

( V2

+ (

2 ) P(x,w) = F(x,w).

c(x) 2 A Lippmann-Schwinger integral equation for P 9 which incorporates boundary conditions can be written as P(x,w) = Po(x,w) -

J

W2

Go(x,S,w) - - 2 a(s)P(s,w)d S, cots)

where Po is the solution of ( V2 1445

+ ~) Po(x,w) = co(x)

F(x,w),

J. Math. Phys., Vol. 24, No.6, June 1983

and co(x) and a(x) are the reference velocity and variation in the index of refraction, respectively; Co and a are related to the acoustic velocity c by the equation 1

1

(1 +a(x)). C(X)2 co(x) Go(x,s,w) is the Green's function for the reference velocity problem -=--2

( V2

+ ~) Go(x,s,w) =

o(x - s). co(x) The Born approximation consists of replacing P (x,w) by Po(x,w) in the right-hand side of the Lippmann-Schwinger integral equation. The resulting equation is P(x,w) = Po(x,w) -

f Go(x,s,w) ~ cots)

a(s)po(s,w)d s·

Born inversion involves solving this Fredholm I integral equation for a(x). For the multidimensional case, this problem has been addressed by a number of authors. 10--13 IV. LINEARIZATION OF THE WAVE FIELD

The first step in developing linear inversion procedure is to find an approximation for the wave field which is linear with respect to the model parameter. It is convenient to choose the square of the index of refraction as the model parameter. This parameter, which we will denote by mix), is related to the acoustic velocity by the equation mix) = V6/c(x)2,

where Vo is a chosen constant reference velocity. The wave field associated with the model mix) satisfies the equation [V2

+ k 2m (x)]If(x,k,m) =

for

F(x,k)

xEfl,

(1)

where k is the wave number (k = wlVo) and Fis the source term. We will assume that fl is a bounded domain. Linearization of the wave field is accomplished by expanding the wave field in a generalized Taylor series about some reference model mo(x). The objective of the section is to derive a linear expression of the form IfL(m) = If(mol

+ 1f'(moHm -

mol·

(2)

In the above equation, If is the wave field; and Ifdm) is the linearized approximation to If(m) which becomes exact as m approaches mo. If'(mol is the directional derivative of If with respect to the model parameter. In this section, a closed form expression for the directional derivative will be found. In addition, the connection between generalized linear inversion and the first Born theory will be made. We begin by defining the set M of model parameters to be the space of complex-valued functions on fl which are bounded almost everywhere, i.e., M = L "'(fl). The norm for this space is the essential supremum, given by Ilmll""

=

inf[B:lm(xll 0 such that

X

(x,~,k,mo)(m - mo)(~)(ifJ -

ifJ

')(~)d ~ 12 d

IG (x,~,k,mo)12Im - mol2 d ~ dx

{}

1IifJ-ifJ'12d~.

From condition (3), and the fact that mEN (mo,r),

i

ITm(ifJ)-Tm(ifJ'Wdx

[}

4 2 2