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Linearized multiparameter inversion is a model-driven variant of amplitude-versus-offset analysis, which seeks to separately account for the influences of several model parameters on the seismic response. Previous approaches to this class of problems have included geometric optics-based (Kirchhoff, GRT) inversion and iterative methods suitable for large linear systems. In this paper, we suggest an approach based on the mathematical nature of the normal operator of linearized inversion—it is a scaling operator in phase space—and on a very old idea from linear algebra, namely, Cramer's rule for computing the inverse of a matrix. The approximate solution of the linearized multiparameter problem so produced involves no ray theory computations. It may be sufficiently accurate for some purposes; for others, it can serve as a preconditioner to enhance the convergence of standard iterative methods.

The linearized inverse problem in reflection seismology aims at recovering model perturbations from data perturbations, assuming known reference or background model. Both reference model and model perturbations consist of material parameter fields (functions of position

With these conventions, the inverse problem may be stated as follows: given observed data

An example of this setup is linear acoustics; we will treat this example explicitly in this paper. The model

In simulating seismic data generation with models such as linear acoustics, waves typically propagate over hundreds of wavelengths, and model fields must resolve features on the wavelength scale. The normal equations thus represent millions, or billions, of equations in the same number of unknowns, ruling out the possibility of solving them by means of direct matrix methods such as Gaussian elimination. This paper presumes that the computations implementing the application of the normal operator are carried out by “wave equation” methods, that is, finite difference or finite element simulation. Thus each application is an expensive, large-scale computational procedure, a fact which places practical limits on the number of steps taken in an iterative scheme to solve the normal equations (

We present an efficient method to approximate the solution of the normal equations that requires a few applications of the normal operator (one for one-parameter inversion, two for two-parameter inversion, and six for three-parameter inversion). This method leverages the properties of the normal operator: under some conditions (background model parameter fields slowly varying on the wavelength scale, diving wave energy eliminated from data, data polarized by propagating phases), it is a special type of matrix-valued spatially varying filter [

In this work, we shall show how to separate the events corresponding to various model parameters by means of several applications of the normal operator to permuted image vectors, so that the result differs from an inversion of the data by an overall spatially dependent filter, common to all components. We have previously solved the problem of estimating and correcting for such a filter [

We review our approach to single-parameter inversion (based on constant density acoustic modeling, for instance) in Appendix

Note that data-adaptive scaling methods such as those cited in the previous paragraph differ in an essential way from scaling by an approximate diagonal of the Hessian [

It has long been recognized that the offset (or scattering angle) dependence of a reflection event encodes material parameter changes across a reflector. Approximations to this relation motivated by the analysis of reflection and transmission in layered media (the Zoeppritz equations and linearization thereof, [

Several studies have analyzed the conditioning, or error propagation properties, of multiparameter inversion [

We have used only synthetic data in the work reported here and rather simple synthetic data at that. In fact, the very few published examples of inversion (in the sense of noise-level data fit) of exploration-scale data [

Recall that the aim of this paper is to solve the normal equations

Since both input and output of the normal operator consist of

The Hessian may be treated as a matrix of spatially varying filters. The filter coefficient

Under certain conditions, the normal operator is a matrix of spatially varying filters of a special type, known as pseudodifferential, to be described below. By Beylkin [

the material parameters in the background model vary smoothly on the scale of a wavelength (since the theory is asymptotic in frequency, the technical assumption is that they are smooth, that is, infinitely differentiable, but the practical meaning is as stated here);

diving wave energy is not present in the data or has been muted or dip-filtered out;

the data has been polarized into propagating phase components.

These conditions are likely to be essential: either major reflectors in the background model or nonpolarized multiphase data lead to Hessians which produce nonphysical reflector images at some distance from their sources, hence cannot be well-approximated by their near-diagonal behavior. Since pseudodifferential operators are nearly local, in a sense to be made precise below, Hessians producing major reflector shifts cannot be approximated by them. For that matter, no near-diagonal approximation to the Hessian could be accurate in such cases, so these same limitations would seem to apply to all scaling methods.

Pseudodifferential operators are distinguished from other types of spatially varying filters by strong constraints on the filter coefficients

Condition (

By convention, filter coefficients obeying the growth rules described here are called

Equation (

the product (in other words, composition)

consequently, if

That is,

The theory cited above also showed that the normal operator is “partly invertible”, that is, at each point in the subsurface, the Hessian scales spatial Fourier components with a certain range of dips by a

Multiparameter scattering results in a matrix Hessian, mixing influences between various parameters. In some cases, the Hessian is a matrix of pseudodifferential operators. For example, the Hessian for acoustic scattering or for polarized elastic scattering (

To this end, recall the definition of the

The significance of the adjugate is this: when the matrix

Note also that while

For inversion of

Equation (

The reader will recall that

The algorithm explained in Appendix

Having constructed

Equations (

In fact, it is possible to compute

Equation (

The situation is more complicated for

As a final note, we can show explicitly for

As a first application to the two parameter inversion, we construct a variable density acoustics model perturbation consisting of a thin oscillatory velocity layer and a thin oscillatory density layer in a different place (see Figures

vp, velocity perturbation.

dn, density perturbation.

We simulated reflection data for this model using the IWAVE software developed by The Rice Inversion project in linearized (Born) modeling mode [

All (reverse-time) migrations were also carried out with IWAVE [

Migrating the Born data shows how migration mixes the effects of the two models in the two components of the migrated images (Figure

Migrated images mixing the contributions from density and velocity, and effecting a phase space scaling.

Applying the scheme outlined above, we form

The application of the adjugate separates the velocity and density contributions. This result is a phase space scaling of the true model.

Scaling of the migrated images by

The final step corrects the amplitudes of

The approximate inverse. The contributions from velocity and density are separated, and the amplitudes are corrected.

Data misfit versus target data. The inverted model fits 70% of the data.

Data misfit

Target Data

We have presented a method to approximate the solution of the multiparameter linearized inverse problem an extension of Cramer's rule for matrices of pseudodifferential operators. The method consists of two steps:

reducing the multiparameter problem to a one parameter problem, which yields an amplitude scaling of the solution;

correcting the amplitudes of the result from the previous step to approximate the solution.

The application of the work flow to a layered variable density acoustics example shows how the effects of different material parameters, indistinguishable in the migrated images, are separated in the first step. The amplitude correction step successfully yields an approximate solution to the linearized inverse problem.

The work flow presented above applies without modification to any model. Results of application to models more complex than the layered model described above will be presented elsewhere. We point out that other aspects of the physics of seismic data generation will need to be accommodated if this (or any other) algorithm is to extract accurate results from field data. For example, as shown already by Minkoff and Symes [

We have extended the Pseudodifferential Operator Algorithm (Appendix

The explicit derivation analogous to

We end by reminding the reader that the approach to approximate inversion pursued here relies on the validity of the pseudodifferential characterization of the Hessian. As mentioned in the theory section, two of the three conditions (background model smooth on the wavelength scale, polarized data) are essential for

Bao and Symes [

This discussion is restricted to 2D, so we may write

Thus, writing

Notice that

Fourier transform theory identifies

Sampling the field

Choosing

Equation (

Compute

For each

Initialize

compute

accumulate

A straightforward discretization of (

The dependence on dip is captured in the angle variable

In this appendix, we review the method developed in [_{mig} and the remigrated image

The scaling factor

We enforce the continuity of

We use limited memory BFGS (lBFGS) to minimize the objective function (

This work was supported in part by the National Science Foundation under grant DMS 0620821, and by the sponsors of The Rice Inversion Project. The authors are grateful to an anonymous reviewer and to Guest Editor Sergey Fomel for thoughtful critiques.