Schlumberger Limited is a multinational company supplying oilfield and nformation services to a worldwide energy market. These services include both exploration and production tools ranging through seismic and remote sensing, well-logging and reservoir optimization. The problem described in this report is related to well-logging via Nuclear Magnetic Resonance (NMR), a relatively new and developing tool with potential to reveal a range of reservoir properties including porosity and saturation, as well as physical properties of the petroleum deposit. In order to recover this information from NMR spectra the company must have an effective, efficient and robust algorithm to perform inversion from the dataset to the unknown probability distribution on magnetic relaxation times. This ill-posed problem is encountered in diverse areas of magnetic imaging and there does not appear to be an ‘off-the-shelf’ solution which the company can apply to its problem. Company scientists have developed a sophisticated algorithm which performs well on some simple test datasets, but they are interested in knowing if there are simpler approaches which could work effectively, or if some limited but useful properties of the density are accessible with a totally different approach. Our report is organised as follows. In Section 1.2 we present a careful and complete description of the problem and the work already done by the company. In Section 1.3 we discuss Truncated Singular Value Regularisation and Tikhonov Regularisation and show how some ‘off-the-shelf’ Matlab code may be used to good effect on the test datasets provided by the company. In Section 1.4 we show that one can incorporate higher order regularisation into the company’s existing algorithm, answering one specific question raised at the beginning of the workshop. Finally, in Section 1.5 we record our unsuccessful attempt to establish an iterative algorithm for the positively constrained inversion. Finally in the last section we review our conclusions and make suggestions for future work
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