This paper describes a method to compute tapered areas of a probability density function by means of integral transforms of its noisy Laplace transform. This problem is directly relevant to Nuclear Magnetic Resonance data for fluid and rock characterization. We describe integral transforms that are directly applicable to the measured magnetisation data, to estimate the tapered areas. In petrophysics, these tapered areas are useful in estimation of fluid saturations and bound and free fluid volumes. Since integral transforms are linear, uncertainty in the tapered areas can be computed as a function of signal-to-noise (SNR) in the data. Performance of these transforms is demonstrated on simulated data and compared to results from the traditional inverse Laplace transform.

In this paper we describe a new approach for the estimation of the porosity and its uncertainty from Nuclear Magnetic Resonance relaxation measurements in porous media. The new approach is based on the Fourier transform of the measured data in √t domain. This approach was found to work reasonably well and had a smaller bias and variance in comparison to traditional methods of computing the porosity.

Interior sampling and exterior sampling (or enclosure) signal-subspace-based imaging methodologies for extended scatterers derived in previous work are reformulated and reinterpreted in terms of the concepts of angles and distances between subspaces. The insight gained from this reformulation renders a broader, more encompassing inversion methodology based on a (pseudo) cross-coherence matrix associated to the singular vectors of the scattering or response matrix and the singular vectors intrinsic to a given, hypothesized support region for the scatterers (under a known background Green's function associated to a known embedding medium where the scatterers reside). A number of new imaging functionals based on that cross-coherence matrix are proposed and numerically shown to perform well in both imaging and shape reconstruction problems. The proposed approaches do not require for their implementation the estimation of a cutoff in the singular value spectrum separating signal from noise subspaces, which is a common computational difficulty in signal subspace methods. In the shape reconstruction context it is also shown how to combine the signal subspace approach with the level set method.

This paper develops, within a general framework that is applicable to rather arbitrary electromagnetic and acoustic remote sensing systems, a theory of time-reversal "MUltiple Signal Classification" (MUSIC)-based imaging of extended (nonpoint-like) scatterers (targets). The general analysis applies to arbitrary remote sensing geometry and sheds light onto how the singular system of the scattering matrix relates to the geometrical and propagation characteristics of the entire transmitter-target-receiver system and how to use this effect for imaging. All the developments are derived within exact scattering theory which includes multiple scattering effects. The derived time-reversal MUSIC methods include both interior sampling, as well as exterior sampling (or enclosure) approaches. For presentation simplicity, particular attention is given to the time-harmonic case where the informational wave modes employed for target interrogation are purely spatial, but the corresponding generalization to broadband fields is also given. This paper includes computer simulations illustrating the derived theory and algorithms.