Lawrence Berkeley Laboratory
Email:
vasco@ccs.lbl.gov
poster/oral: oral
| Non-uniqueness is an important aspect of geophysical inverse problems. Techniques for assessing the non-uniqueness associated with linear inverse problems are well developed. However, many of the current techniques face computational difficulties when the inverse problem is large. The first part of the talk will examine an efficient method for computing resolution and covariance matrices for large sparse linear inverse problems. The method will be illustrated using an application from whole Earth tomography. In particular, I shall examine elements of the resolution matrix associated with the inversion of body wave arrival times. In the upper and mid mantle the well resolved cells lie beneath Eurasia and a narrow zone surrounding the mantle beneath the Pacific basin. In the lowest mantle the resolution decreases significantly. An examination of the averaging kernels associated with selected locations near the base of the mantle reveals the significant lateral and vertical averaging inherent in estimates of compressional velocity. In well sampled regions of the outer core we are able to resolve compressional velocity variations in our 6 degree by 6 degree cells. However, velocity variations in the inner core are poorly constrained at the scale of our blocks. In well sampled regions of the mantle and outer core model parameter standard deviations remain below 0.25 percent. For much of the inner core the standard deviations are quite large, exceeding 0.5 percent. In contrast to the linear case, techniques for non-linear inverse problems are not as well developed. Typically, non-linear inverse problems are linearized and a conventional linear assessment is conducted. In the second part of this talk I shall discuss alternative approaches for quantifying the non-uniqueness of non-linear inverse problems. Such methods are based upon algebraic constructs such as polynomial ideals and topological ideas such as homotopy. |