


 Anisotropic tomography with rock physics constraints  

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For a regularized problem, an Lcurve analysis is often useful to determine the damping parameter
in equation 7 and investigate the posterior distribution of the inversion (Hansen and O'Leary, 1993). Typical Lcurve has two distinct parts: one vertical part where the solution is dominated by the data fitting and one horizontal part where the solution is dominated by the model styling. The corner of the Lcurve corresponds to a good balance between minimization of both fitting goals. In Figure 5, we obtain the Lcurve for each inversion scheme for the shale (sandy shale) model in loglog scale by varying
from
to
. The model residual is defined in the preconditioning space. Therefore, the shape of the Lcurve also depends on the covariance matrix. A good estimation of the covariance and a proper
will place the solution right at the corner of the Lcurve, as in the case of using the full covariance. The "7curve" shape in the loglog scale (which is still an "Lcurve" in absolute scale) shows a relatively poor estimation of the covariance matrix, hence indicating difficulties in finding a proper damping parameter
.
Finally, to move beyond the deterministic inversion which produces only one solution, we perform the nullspace analysis following the work flow proposed by Osypov et al. (2008). The effective nullspace of an operator can be sampled by an iterative Lanczos eigendecomposition method. The right panel on Figure 1 shows the nullspace projection (darker dots) overlaid on the approximate prior distribution (smaller dots) when the full covariance scheme is used. It is obvious that the full covariance matrix representation produces a good estimation to the true prior (by the similarity of the cloud shape on the left panel and the right panel). Also, the nullspace projection suggests that higher uncertainty in anisotropic parameters for higher velocities, which often means greater depth, is embedded in rock physics knowledge. The reduced volume of the cloud shows the value of information that the data bring into the inversion.


Lcurve
Figure 5. Lcurve for the shale (sandy shale) model inversion.







 Anisotropic tomography with rock physics constraints  

Next: Conclusions
Up: Li et al.: RP
Previous: Numerical tests
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