A more accurate estimate of T2 distribution from direct analysis of NMR measurements.

In the past decade, low-field NMR relaxation and diffusion measurements in grossly inhomogeneous fields have been used to characterize pore size distribution of porous media. Estimation of these distributions from the measured magnetization data plays a central role in the inference of insitu petro-physical and fluid properties such as porosity, permeability, and hydrocarbon viscosity. In general, inversion of the relaxation and/or diffusion distribution from NMR data is a non-unique and ill-conditioned problem. It is often solved in the literature by finding the smoothest relaxation distribution that fits the measured data by use of regularization. In this paper, estimation of these distributions is further constrained by linear functionals of the measurement that can be directly estimated from the measured data. These linear functionals include Mellin, Fourier-Mellin, and exponential Haar transforms that provide moments, porosity, and tapered areas of the distribution, respectively. The addition of these linear constraints provides more accurate estimates of the distribution in terms of a reduction in bias and variance in the estimates. The resulting distribution is also more stable in that it is less sensitive to regularization. Benchmarking of this algorithm on simulated data sets shows a reduction of artefacts often seen in the distributions and, in some cases, there is an increase of resolution in the features of the T(2) distribution. This algorithm can be applied to data obtained from a variety of pulse sequences including CPMG, inversion and saturation recovery and diffusion editing, as well as pulse sequences often deployed down-hole.

Estimation of petrophysical and fluid properties using integral transforms in nuclear magnetic resonance.

In the past decade, low-field NMR relaxation and diffusion measurements in grossly inhomogeneous fields have been used to characterize properties of porous media, e.g., porosity and permeability. Pulse sequences such as CPMG, inversion and saturation recovery as well as diffusion editing have been used to estimate distribution functions of relaxation times and diffusion. Linear functionals of these distribution functions have been used to predict petro-physical and fluid properties like permeability, viscosity, fluid typing, etc. This paper describes an analysis method using integral transforms to directly compute linear functionals of the distributions of relaxation times and diffusion without first computing the distributions from the measured magnetization data. Different linear functionals of the distribution function can be obtained by choosing appropriate kernels in the integral transforms. There are two significant advantages of this approach over the traditional algorithm involving inversion of the distribution function from the measured data. First, it is a direct linear transform of the data. Thus, in contrast to the traditional analysis which involves inversion of an ill-conditioned, non-linear problem, the estimates from this new method are more accurate. Second, the uncertainty in the linear functional can be obtained in a straight-forward manner as a function of the signal-to-noise ratio (SNR) in the measured data. We demonstrate the performance of this method on simulated data.

Continuous moment estimation of CPMG data using Mellin transform.

This paper provides a theoretical basis to directly estimate moments of transverse relaxation time T(2) from measured CPMG data in grossly inhomogeneous fields. These moments are obtained from Mellin transformation of the measured CPMG data. These moments are useful in computing petro-physical and fluid properties of hydrocarbons in porous media. Compared to the conventional method of estimating moments, the moments obtained from this method are more accurate and have a smaller variance. This method can also be used in other applications of NMR in inhomogeneous fields in characterizing fluids and porous media such as in the determination of hydrocarbon composition, estimation of model parameters describing relationship between fluid composition and measured NMR data, computation of error-bars on estimated parameters, as well as estimation of parameters and σ(lnT(2)) often used to characterize rocks. We demonstrate the performance of the method on simulated data.

Mellin transform of CPMG data.

This paper describes a new method for computing moments of the transverse relaxation time T(2) from measured CPMG data. This new method is based on Mellin transform of the measured data and its time-derivatives. The Mellin transform can also be used to compute the cumulant generating function of lnT(2). The moments of relaxation time T(2) and lnT(2) are related to petro-physical and fluid properties of hydrocarbons in porous media. The performance of the new algorithm is demonstrated on simulated data and compared to results from the traditional inverse Laplace transform. Analytical expressions are also derived for uncertainties in these moments in terms of the signal-to-noise ratio of the data.

The contrast-source stress-velocity integral-equation formulation of three-dimensional time-domain elastodynamic scattering problems: a structured approach using tensor partitioning.

The contrast-source stress-velocity integral-equation formulation of three-dimensional time-domain elastodynamic scattering problems is discussed. A novel feature of the formulation is a tensor partitioning of the relevant dynamic stress and the contrast source volume density of deformation rate. The partitioning highlights several features about the structure of the formulation. These can advantageously be incorporated in a computational implementation of the method. An application to the case of a scatterer composed of isotropic material and embedded in an isotropic elastic background medium shows that the corresponding newly introduced constitutive coefficients are more natural as a characterization of the media than the traditional Lame coefficients.

The diagonalized contrast source inversion approach for elastic wave inversion.

This study focuses on the inverse scattering of objects embedded in a homogeneous elastic background. The medium is probed by ultrasonic sources, and the scattered fields are observed along a receiver array. The goal is to retrieve the shape, location, and constitutive parameters of the objects through an inversion procedure. The problem is formulated using a vector integral equation. As is well-known, this inverse scattering problem is nonlinear and ill-posed. In a realistic configuration, this nonlinear inverse scattering problem involves a large number of unknowns, hence the application of full nonlinear inversion approaches such as Gauss-Newton or nonlinear gradient methods might not be feasible, even with present-day computer power. Hence, in this study we use the so-called diagonalized contrast source inversion (DCSI) method in which the nonlinear problem is approximately transformed into a number of linear problems. We will show that, by using a three-step procedure, the nonlinear inverse problem can be handled at the cost of solving three constrained linear inverse problems. The robustness and efficiency of this approach is illustrated using a number of synthetic examples.

A multiplicative regularization approach for deblurring problems.

In this work, an iterative inversion algorithm for deblurring and deconvolution is considered. The algorithm is based on the conjugate gradient scheme and uses the so-called weighted L2-norm regularizer to obtain a reliable solution. The regularizer is included as a multiplicative constraint. In this way, the appropriate regularization parameter will be controlled by the optimization process itself. In fact, the misfit in the error in the space of the blurring operator is the regularization parameter. Then, no a priori knowledge on the blurred data or image is needed. If noise is present, the misfit in the error consisting of the blurring operator will remain at a large value during the optimization process; therefore, the weight of the regularization factor will be more significant. Hence, the noise will, at all times, be suppressed in the reconstruction process. Although one may argue that, by including the regularization factor as a multiplicative constraint, the linearity of the problem has been lost, careful analysis shows that, under certain restrictions, no new local minima are introduced. Numerical testing shows that the proposed algorithm works effectively and efficiently in various practical applications.

Inversion of guided-wave dispersion data with application to borehole acoustics.

The problem of inferring unknown geometry and material parameters of a waveguide model from noisy samples of the associated modal dispersion curves is considered. In a significant reduction of the complexity of a common inversion methodology, the inner of two nested iterations is eliminated: The approach described does not employ explicit fitting of the data to computed dispersion curves. Instead, the unknown parameters are adjusted to minimize a cost function derived directly from the determinant of the boundary condition system matrix. This results in an efficient inversion scheme that, in the case of noise-free data, yields exact results. Multimode data can be simultaneously processed without extra complications. Furthermore, the inversion scheme can accommodate an arbitrary number of unknown parameters, provided that the data have sufficient sensitivity to these parameters. As an important application, we consider the sonic guidance condition for a fluid-filled borehole in an elastic, homogeneous, and isotropic rock formation for numerical forward and inverse dispersion analysis. We investigate numerically the parametric inversion with errors in the model parameters and the influence of bandwidth and noise, and examine the cases of multifrequency and multimode data, using simulated flexural and Stoneley dispersion data.