Inversion

It is important to notice that a point $(\omega, k_{s}, k_{h})$ in data space fixes the angle $\theta$ of the incident plane wave as well. Because of the Bragg condition, the $\omega$ dependence in model space is transformed into incidence-angle dependence, according to

 

$$\cos \theta = {{ ({\omega \over v_{i}})^{2} - ({\omega \ove... ...over { 2 {\omega \over v_{i}} \sqrt{ k_{x}^{2}+ k_{z}^{2}} }}.$$ (4)

For each point in the (k_x, k_z) domain, that is for each sinusoidal component in the space domain, the $\theta$ dependence is a vector ${\bf d} (\theta)$. This represents the Phase versus Angle (PvA) and Amplitude versus Angle (AvA) of the sinusoidal component of the medium (Figure 4),

 

$${\bf d}(\theta) = {\bf G}(\theta) \; {\bf m},$$ (5)

with ${\bf m} = [\Delta Ip / Ip \; , \; \Delta Is / Is \; , \; \Delta \rho / \rho].$

The matrix G describes the reflection angle dependence, the wavelet spectrum, the propagation effects and the directivity of sources and receivers. Figure (5) illustrates the reflection coefficient of a sinusoidal medium due to unitary parameter perturbations for P-P and P-SV reflections.

 

PvA-AvA
Figure 4 Interpretation of ${\bf d} (\theta)$ for a given (k_x, k_z).

 

unitary Figure 5 Theoretical AvA of a sinusoidal medium for a unitary change of parameters.

The inversion is a least-squares fit of the measured PvA and AvA, which is the result of the mapping process, with the theoretical PvA and AvA in the complex plane. We use the SVD decomposition of the relation data-parameters ${\bf G} = {\bf U} {\bf \Lambda} {\bf V}^{T}$ (Lines and Treitel 1984). The matrix ${\bf U}$ contains the eigenvectors in data space. ${\bf \Lambda}$ is the diagonal matrix of singular values. They represent the energy in the data due to a unitary variation along eigenvectors axis in the model space. ${\bf V}$ contains the eigenvectors in model space: they form an orthogonal basis in the model space and are linear combinations of the elastic parameters perturbations $\Delta Ip / Ip$, $\Delta Is / Is$ and $\Delta \rho / \rho$.

The reflections of each sinusoidal component are inverted one by one: given (k_x, k_z) and the corresponding migrated data ${\bf d}_{m}(\theta)$,

 

$${\bf m} \simeq {\bf V} \; {\bf \Lambda}^{-1} {\bf U}^{T} {\bf d}_{m}.$$ (6)

The data ${\bf d}_{m}(\theta)$, plotted in the complex plane, form a line (Figure 4). The inclination of the line gives the phase of the sinusoidal component. The absolute value of the data is the measured AvA of the sinusoidal medium.

Because of the finite range of sources, receivers, frequency components and observation angles, only a part of the entire model spectrum can be recovered. Non-observable components of the spectrum (null space) are not inverted. Sharp transitions of the angular coverage in the (k_x, k_z) domain are tapered before transformation to the x-z panels.

The ill-conditioning of the inverse elastic problem can lead, because of the presence of numerical inaccuracies and interpolation errors, to interference between parameters estimates. De Nicolao et al. (1993) have found that with realistic signal-to-noise-ratios, in this linearized approach, P-P reflections allow an accurate estimation of P-impedance perturbation only. S-impedance perturbation can be recovered with multicomponent data, namely P-SV reflections. Density perturbations are practically unrecoverable. We have tested these results on synthetic examples, which are described below.