An Upgrade in Linearized Modeling of Wave Propagation
Quantitative migration or linearized inversion techniques rely on a linearization in the wave equation: the accuracy of the result depends on the accuracy of the linearization. The kinematics of events modeled using such a linearization depends on the ref
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Summary. Quantitative migration or linearized inversion techniques rely on a linearization in the wave equation: the accuracy of the result depends on the accuracy of the linearization. The kinematics of events modeled using such a linearization depends on the reference velocity model. In this paper we show that, for acoustic wave propagation, the amplitudes of the reflected events modeled by such a linearization technique are drastically influenced by the choice of the parameters that describe the model perturbations. We propose a choice for these parameters that makes the linearization error small. This is demonstrated by means of analytic calculations for plane waves propagating in a two-layer model as well as by comparison with a finite difference modeling.
1 Introduction Migration/inversion techniques (see for instance [1, 2]) have met considerable interest : those techniques aim at a quantitative characterization of heterogeneities within the subsurface. That is why they are so called "true amplitude migration" . Some of them (known as ray Born migration/inversion) rely on a linearization in the wave equation. Such a linearization depends of course on how accurate the reference model is. But it also depends on how we represent, in terms of physical parameters, the model perturbations. The goal of this paper is to clarify this dependency. Then we will address the question: can we choose a representation of model perturbations that makes the linearization error small?
2 Linearized modeling For the propagation of acoustic and isotropic waves, a model m is characterized by two parameter distributions, the density p(x) and the bulk modulus h:(x) for instance. The pressure field P(x, w) is the solution of the wave equation (written here in the frequency domain) [W
2
1 . 1 ] h:(x) - dIV(p(X) grad) P(x,w)
= S(x,w),
(1)
G. C. Cohen et al. (eds.), Mathematical and Numerical Aspects of Wave Propagation WAVES 2003 © Springer-Verlag Berlin Heidelberg 2003
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Florence Delprat-Jannaud et al.
where S(x, w) is the source, and x denotes a point in the subsurface with coordinates x = (h, z) where h is the horizontal location and z is the depth (for a 2D problem). From equation (1), the relationship between the seismic response P( x, w) and parameters (p, "") appears to be non linear. As a consequence, inverting equation (1) to compute, from some recorded pressure P, the model parameters (p, "") is far from trivial, and it is common to use a linear approximation of (1). To perform a linearization, we split the model m into two parts:
m=mo+om,
(2)
where mo is the reference model supposed to be "close" to model m so that the perturbation model Om is "small". The linearized wavefield is defined as Po + OP where
Po is the solution of (1) when the acoustic parameters are set to those describing model mo, oP = J(mo)om, J(mo) being the Jacobian operator evaluated at model mo· Of course, the goal is to make the non linear forward map and its linearization close each other. The attemps [3] that have been made to derive err
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