Trámite en la Plenaria de la Cámara de Representantes

In the last two sections, we have found the gradients of the data misfit functional with respect to different model parameter types and along these gradient directions or their modifications we may iteratively find a model to reduce the data misfit to an acceptable level. However, it is unknown whether or not the model is a mathematically unique one, and more importantly, it is unknown whether or not the model is a physically plausible model. Devaney (1986) has pointed out that the uniqueness of an inverse problem is governed by the so-called "Golden Rule", which says that the number of degrees of freedom in data space should always be greater or equal to the number of degrees of freedom in model space for a "well-posed" inverse problem. In a seismic inverse problem, the number of degrees of freedom in data space is often less than the number of degrees of freedom in model space and thus the inverse problem is ill-posed. In such case, the solution of the inverse problem is intrinsically non-unique. For example, Williamson (1986) tried to use traveltimes to reconstruct the velocity field and the topography of the reflector (the full problem in his traveltime inversion in reflection seismology) and found that the resolution between the velocity field and the topography of the reflector was very poor. Actually, this full problem is an ill-posed problem because the data space has two degrees of freedom (source and receiver can be independently moved along the 1-D surface) whereas the model space has three degrees of freedom (the velocity field can vary along both x and z axes, and the topography of the reflector can vary along z axis).

So far we have two tools to tackle this intrinsic non-uniqueness of inverse problems. One tool is to use hard constraints, i.e. to obtain additional data to increase the number of degrees of freedom in data space or to simplify the model (usually

Inverse Theory 4.30 assumptions) to decrease the number of degrees of freedom in model space. The other is to use soft constraints, i.e. to use an a priori information to extrapolate the data and artificially to increase the number of degrees of freedom in data space or to use some a priori information to reduce the variability of the model parameters and to decrease the

number of degrees of freedom in model space.

For an inverse problem in reflection seismology, it is very difficult to obtain additional data (in few cases three component data or VSP data are available) or artificially to extrapolate the data to increase the number of degrees of freedom in the data space. Therefore, we are left with either simplifying the model (e.g. using models with a 2-D or even a 1-D parameter variation instead of a 3-D parameter variation, or an acoustic model instead of an elastic model) or reducing the variability of the model parameters (e.g. the gradient of density in horizontal directions may be ignored in some sedimentary regions). In 3-D seismic explorations on land (which have been simulated in section 3.1), the vertical component of displacement is usually recorded with the result that the S wave is not directly usable (m eans to neglect S w ave) and the sources and receivers are laid on a 2-D ground surface. The data space theoretically has five degrees of freedom (source and receiver can move in two dimensions and the recording can be traced along time axis, however, these five degrees of freedom are never fully realized in practice due to practical limitations). The number of degrees of freedom in the model space depends heavily on the choice of what kind of model should be obtained. A natural choice gives nine degrees of freedom in the model space (Lame parameters X(x) and |i(x), and density p(x) each has three degrees of freedom). Therefore, this problem is a very ill-posed problem (If we want this problem to be a well-posed problem, we may record three component data in 3-D space, which gives the data space with nine degrees of freedom). For a specific problem, a simplification of the model may be necessary and help to improve the condition of the ill-posedness. For example, we may regard all reflections as coming from a 2-D model, which has six degrees of freedom. A reasonable and further simplification may be achieved by imposing a fixed relation between X(x) and j i(x) , and

Inverse Theory 4.31 assuming density and elasticity as homogeneous in horizontal directions and this simplification leaves the model with two degrees of freedom (a good representation of a stable sedimentary basin without faults). The 2-D simplification of the model improves the condition of the ill-posedness but leaves the problem still being ill-posed whereas the further assumption of the fixed relation between X(x) and p.(x) and the homogeneity in horizontal directions transforms the ill-posed problem into a well-posed problem.

Once an inverse problem is posed, its condition may deteriorate further by the choices made in parameterising the data and model spaces, especially the partial realization of some degrees of freedom in the data space. For example, as we mentioned earlier, the sources and receivers are only laid out in a finite area on a 2-D surface and thus the two degrees of freedom for both the source and the receiver locations are partially realized. These partial realizations of some degrees of freedom in the data space usually leads to the so-called ill-conditionedness, i.e. some parts of the model may be better constrained (or resolved) by the data than others (In terms of singular value decomposition, the eigenvalues vary wildly). For example, Sambridge (1988) found that regions directly underneath seismic stations are better constrained than other regions in his 3D studies. Although the ill-conditionedness may arise from different causes, the effects of the ill-conditionedness are the same as those of the ill-posedness. The difficulties associated with the ill-conditionedness may be treated in the same way as those with the ill-posedness.

The most popular way to tackle the ill-posedness and the ill-conditionedness is to use a priori information to constrain the model, which is often called the regularization of the constructed model. By regularization, we mean that the constructed model should have some expected property as prescribed by the regularisation term. Simply, it is our a priori knowledge (or prejudice) about the model being constructed. This knowledge is

often inconsistent with the observed data. By minimizing the sum of the regularisation functional and the data misfit functional, a desired property can be imposed on the constructed model. However, this property may be a bias to the true model. Jackson and

Inverse Theory 4.32 Matsu'ura (1985) pointed out that this bias may not be necessarily to be avoided. From the above discussion, this bias is obviously necessary whenever the problem is ill-posed or ill-conditioned. This imposed property may take different forms. For example, Tarantola (1987) assumed that the constructed model should be as close to an a priori

model as possible whereas Constable, Parker & Constable (1987) assumed that the gradient of the constructed model should be as close to an a priori value (zero) as possible. Although the formats are quite different when different properties are imposed on the constructed model, it seems that there is no difference between them and they are simply some a priori information imposed on the current constructed model. Broadly speaking, all kind of regularisations are based on the discrepancy between some properties of the current constructed model and the a priori model.

In our hybrid inversion of reflection data, a reasonable regularization would require us to discriminate between different types of model parameters. The density and elastic contrasts should be as small as possible while the reflecting surface should be as smooth as possible or equivalently the local depth should vary smoothly. The density and elasticity of the reference medium should always be positive, possibly close to some prescribed values with given confidence intervals.

By imposing a regularization, the direction in which we update the starting model would be modified from the gradient of the data misfit functional. The modification term would be different for different types of regularisations. For example, the modification simply is the difference between the a priori model and the current model if we prescribe an a priori model as the property of the constructed model.