Análisis sobre la obligatoriedad de la consulta previa en el asunto de la referencia

As we have learned from the three sections above, an unbiased estimation of model parameters in a linear inverse problem can be obtained by an inversion algorithm with a varying reference model. If a nonlinear inverse problem can be linearized, the linear inverse theory can be directly applied. Observable data and model parameters in reflection seismology are not normally related in a linear relationship. For nonlinear inverse problems such as seismic traveltime inversions and seismic waveform inversions, a local linearization can be employed and an unbiased estimation of the model parameters may be obtained iteratively. In this section, we will first formulate the seismic inverse problem by a Bayesian approach and generate a common formulation for seismic inversions, and then derive the Frechet derivative and gradient of elastic seismic inversion in a subspace fashion.

4.4.1 Formulation of nonlinear seismic inversion

Two types of observed data are currently used in seismic inverse problems. The first is seismic wave traveltimes and the second is seismic waveforms. Cary & Chapman (1988) have demonstrated that waveforms provide more information than traveltime alone in matching data with synthetics. However, a waveform misfit functional is a highly non-linear functional (Macdonald et al., 1987; see also section 4.2) and it is almost impossible to obtain an acceptable model in an economic way [The misfit functional is highly oscillatory due to the nature of seismic waves. If the starting model is very close to the true model, then we have a good chance to arrive at the global minimum. For real problems, this assumption can never be true. Furthermore if the starting model is already close enough to the true model, there would be no need to do an inversion.]. From the data residual analysis in section 4.2, we can make a best use of information contained in the seismic recordings by breaking an inversion into several stages through which the solution may gradually come to approach the true model instead of attempting to achieve an elegant solution in one big step. In each stage, the observed datasets may take

Inverse Theory 4.26

different forms.

A new dataset generated by reorganizing seismic recordings may have different error statistical distributions from the original recordings. For example, if a stochastic variable has a Gaussian distribution, the square of the variable with its mean subtracted would have a y} distribution with one degree of freedom. Therefore, the statistical characteristics of the observed data and the a priori model may be quite different in different stages. In each stage, the statistical errors associated with the observed data and the a priori model may also have different distributions. Furthermore, the error distribution for the new dataset may sometimes be difficult to be represented by a simple formula. In this case, a bold simplification is usually made, which causes the inversion formula to be quite different from what it should be and so make it difficult to be aware of use of Bayes' rule.

As a simple example, we may divide a seismic waveform inversion into two stages. In the first stage, the dataset is composed of the delay times of different seismic wave phases. In the second stage, the dataset consists of the delay times and the seismic waveforms. In this way, we hope to resolve the large scale structure in the first stage, and to achieve a better convergent rate and resolve smaller scale structure in the second stage. The statistical distribution of errors in the two stages are obviously different. We will see this later.

A seismic model may be sampled at finite spatial gridpoints (which will lose all the information above Nyquist wavenumber, cf Fourier Sampling Theory, e.g. Kanasewich (1981)) and discretely represented by a vector m of a 3-D function in a spatial domain T)

(4.42) m = [ m(x); x e *D],

where m(x) can represent various quantities such as density and elastic moduli, density and velocities or reflectivity. The seismic wavefield may be represented as

Inverse Theory 4.27

(4.43) d = f(m ),

where the operator f denotes the transformation from a model m to amplitude-time data d.

Although the error distributions in the two stages may be different, the Bayesian approach often gives the same general formulation, i.e. minimisation of a scalar functional F(d0bs, mpn0r, mest),

(4.44) F(d0bs> m prior> m est) = 2 [^(dobs» dpre) + ^(mprior» m est) ]

where

\

4.4.2. The gradients o f the data misfit functional &( m)

As stated in the section above, the data misfit functional depends on the probabilistic nature of the observed data set. For a two-stage inversion, the probabilistic characters of the observed data sets in the two stages are different. In the first stage, the observed data set is the traveltime of a seismic phase and errors associated with the picking of traveltimes are often considered as having Jeffreys' distribution. For the sake of numerical convenience, a Gaussian distribution is often assumed. In the second stage of inversion, the observed data set is the original field recordings. Gaussian statistics give a reasonable good description of the discrepancy between observed and synthetic seismograms due e.g. to noise.

With the assumption of Gaussian errors in the observed dataset, the data misfit may be transformed into its dual space by multiplying the inverse of the data covariance (see section 4.2),

(4.45) 8d(xr,t; xs) = C^(d(xr,t; xs)0bs - d(xr,t; xs)cai)

The dual of model perturbation can then be obtained by projecting the dual of data misfit into the dual space of model space using a local linearization, i.e.

Inverse Theory 4.28

(4.46) 5m = FT5d.

where F is the Frechet derivative of the data misfit functional with respect to model parameters. Obviously, the Frechet derivative would be different at different stages as well as at different iterations. Explicitly, we can give expressions for perturbations in the dual spaces of density contrasts, elastic modulus contrasts, local depths, density and elastic modulus as follows,

t2 (4.47) pc (x) = Z Z Jd t Fp (x,t)Cj15d(xr,t; xs), s r ti t2 * c(x )= S S J dt Fi (x.t)C i15d(xr,t; xs). s r ti c t2 h (x )= I I J dt Fh(x,t)C j ’5d(xr,t; xs), s r tj t2 P = I I j d t F0(t)Cä15d(xr,t; xs), s r ti K t2 Ä. = Z I j dt Fx (t)C ^5d(xr,t; xs), s r tj

By transforming the above expressions back into the model space, finally we obtain local gradients of misfit functional with respect to different model parameters as follows,

(4.48) Ypc(x) = CPcp 0(x)

\ ( x ) = CXc Xc(x)