Data processing in NMR relaxometry using the matrix pencil

The matrix pencil as a tunable filter

Despite inherent sensitivity constraints, nuclear magnetic resonance (NMR) plays an indispensable role in probing molecular structures and dynamics across scientific disciplines. Remarkably, while extensive efforts have targeted instrumental and experimental sensitivity improvements, comparatively little focus has been dedicated to sensitivity enhancement through signal analysis. Amidst this present gap, the matrix pencil method (MPM) has emerged as a versatile algorithm that offers tunable filtering and phasing capabilities. Extensive prior research has established the MPM as an adept fitting tool in signal analysis. Here, the efficacy of the MPM is investigated by precisely modeling noisy data to separate information-bearing signals from noise, thereby expanding its utility in various magnetic resonance applications. Simulated data is used to confirm the ability of the MPM to discern and separate signals from noise. Comparative analyses against standard Fourier-based filtering methods highlight the superior performance of the matrix pencil filter (MPF) in preserving signal fidelity without introducing aliasing artifacts. A variety of experimental data is then explored to demonstrate the proficiency of the MPF in characterizing signal components and correcting phase distortions. Collectively, these case studies underscore the filtering capacity of the MPM, portending its use for analytical sensitivity improvements in a wide range of NMR applications.

Bayesian NMR petrophysical characterization

Identification of reservoir rock types is necessary for the exploration and recovery of oil and gas. It involves determining the petrophysical properties of rocks such as porosity and permeability which play a significant role in developing reservoir models, estimating the volumes of oil and gas reserves, and planning production methods. Nuclear magnetic resonance (NMR) technology is a fast and accurate tool for petrophysical rock characterization. The distributions of relaxation times (T2 distributions) offer valuable insights into the distribution of pore sizes in rocks, and these distributions are closely linked to important petrophysical parameters like porosity, permeability, and bound fluid volume (BFV). This work introduces a Bayesian estimation method for analyzing NMR data. The Bayesian approach uses prior knowledge of T2 distributions in the form of the prior mean and covariance. The Bayesian approach combines prior knowledge with observed data to obtain improved estimation. We use the Bayesian estimation method where prior information regarding the rock sample type, for example shale, is available. The estimators were evaluated on decay data simulated from synthesized distributions that replicate the features of experimental T2 distributions of three types of reservoir rocks. We compared the performance of the Bayesian method with two existing methods using porosity, bound fluid volume (BFV) geometric mean (T2LM) and root mean square error (RMSE) of the estimated T2 distribution as evaluation criteria. Additional experiments were carried out using experimental T2 distributions to validate the results. The performance of the Bayesian methods was also tested using mismatched priors. The experimental results illustrate that the Bayesian estimator outperforms other estimators in estimating the T2 distribution. The Bayesian method also outperforms the ILT method in estimating derived petrophysical properties except in cases where the noise level is below 0.1 and the T2 distributions are associated with short relaxation times.

Automated component analysis in DOSY NMR using information criteria

This study introduces a model selection technique based on Bayesian information criteria for estimating the number of components in a mixture during Diffusion-Ordered Spectroscopy (DOSY) Nuclear Magnetic Resonance (NMR) data analysis. As the accuracy of this technique is dependent on the efficiency of parameter estimators, we further investigate the performance of the Weighted Least Squares (WLS) and Maximum a Posteriori (MAP) estimators. The WLS method, enhanced with meticulously tuned L2-regularization, effectively detects components when the difference in self-diffusion coefficients is more than two-fold, especially when the component with the smaller coefficient has a larger weight ratio. The MAP method, strengthened by a substantial database of prior information, exhibits outstanding precision, decreasing this threshold to 1.5 times. Both estimators provide weight ratio estimates with standard deviations of approximately around 1 percentage point, although the MAP method tends to overestimate the component with a larger self-diffusion coefficient. Deviations from the expected values can exceed 10 percentage points, often due to inaccuracies in component detection. The error estimates are determined using data resampling techniques derived from a large-scale 1000-point experiment and an additional five measurements from a single-component mixture. This approach allowed us to thoroughly examine data distribution characteristics, thereby laying a robust groundwork for future refinement efforts.

Quantitative stray-field T1 relaxometry with the matrix pencil method

The matrix pencil method (MPM) is tested as an approach to quantitatively process multiexponential low-field nuclear magnetic resonance T₁ relaxometry data. The data is obtained by measuring T₁ saturation recovery curves in the highly inhomogeneous magnetic field of a stray-field sensor. 0.9% brine solutions, doped with different concentrations of a Gd³⁺ containing contrast agent, serve as test liquids. Relaxation-times as a function of contrast-agent concentration along with the T₁ relaxation curves for combinations of multiple different test liquids are measured, and the results from processing using MPM as well as inverse Laplace transformation as a benchmark are compared. The relaxation-time resolution limits of both procedures are probed by gradually reducing the difference between the relaxation-times of two liquids measured simultaneously. The sensitivity to quantify the relative contribution of each component to the magnetization build-up curve is explored by changing their volume ratio. Furthermore, the potential to resolve systems with more than two components is tested. For the systems under test, MPM shows superior performance in separating two or three relaxation components, respectively and effectively quantifying the time constants.

Automated optimization of spatial resolution for single-sided NMR

Single-sided NMR instruments utilize inhomogeneous magnetic fields with strong gradients to nondestructively probe physical properties of materials. The sensitive region of this type of magnet is often a thin slice above the magnet's surface; measuring planar samples with high spatial resolution requires coplanarity between the sensitive region of the magnet and the sample region of interest. We developed an algorithmic approach to position flat samples coplanar with the magnet's sensitive region. The efficient and objective positioning process utilizes an adjustable stage that offers control over three degrees of freedom, and the optimal position for each sample is found with a quadtree algorithm. We show this algorithm is effective for positioning samples with various relaxation behaviors. We report resolution values that describe position optimization, acquisition constraints, and final spatial resolution for each sample. Measurements after optimized positioning had appropriate spatial resolution to distinguish physical regions of layered samples with different physical properties, namely, relaxation behavior. Our algorithmic positioning process can be implemented for planar samples in research and industrial settings to enhance spatial resolution of single-sided NMR measurements.

Quadratic spacing of the effective gradient area for spatially encoded diffusion NMR

Diffusion NMR experiments rely on the measurement of signal attenuation as a function of the area of diffusion-encoding pulsed magnetic-field gradients. In conventional experiments, arbitrary series of gradient values can be used, and different gradient spacing strategies have different advantages. Ultrafast diffusion NMR relies on the spatial parallelisation of effective gradient area values to collect full 2D diffusion data sets in a single scan. Until recently, only linear spacing was available. We have shown that quadratic spacing can be achieved using a tailored frequency swept pulse. Here we describe the design of the pulse and validate it with numerical spin simulations, that make it possible to check the effect of the quadratic spacing pulse at different stages of the pulse sequence. We also show that quadratic spacing makes it possible to use a recently reported analysis method for diffusion NMR, the Matrix Pencil Method. We describe the results obtained with the MPM and those obtained with the direct exponential curve resolution algorithm (DECRA), which also requires quadratic gradient spacing. Overall, these developments open new opportunities for applications of spatially encoded diffusion experiments, such as ultrafast DOSY NMR and ultrafast Laplace NMR.

Ultrafast methods for relaxation and diffusion

Relaxation and diffusion NMR measurements offer an approach to studying rotational and translational motion of molecules non-invasively, and they also provide chemical resolution complementary to NMR spectra. Multidimensional experiments enable the correlation of relaxation and diffusion parameters as well as the observation of molecular exchange phenomena through relaxation or diffusion contrast. This review describes how to accelerate multidimensional relaxation and diffusion measurements significantly through spatial encoding. This so-called ultrafast Laplace NMR approach shortens the experiment time to a fraction and makes even single-scan experiments possible. Single-scan experiments, in turn, significantly facilitate the use of nuclear spin hyperpolarization methods to boost sensitivity. The ultrafast Laplace NMR method is also applicable with low-field, mobile NMR instruments, and it can be exploited in many disciplines. For example, it has been used in studies of the dynamics of fluids in porous materials, identification of intra- and extracellular metabolites in cancer cells, and elucidation of aggregation phenomena in atmospheric surfactant solutions.

Rapid parameter determination of discrete damped sinusoidal oscillations

We present different computational approaches for the rapid extraction of the signal parameters of discretely sampled damped sinusoidal signals. We compare time- and frequency-domain-based computational approaches in terms of their accuracy and precision and computational time required in estimating the frequencies of such signals, and observe a general trade-off between precision and speed. Our motivation is precise and rapid analysis of damped sinusoidal signals as these become relevant in view of the recent experimental developments in cavity-enhanced polarimetry and ellipsometry, where the relevant time scales and frequencies are typically within the ∼1 - 10 µs and ∼1 - 100 MHz ranges, respectively. In such experimental efforts, single-shot analysis with high accuracy and precision becomes important when developing experiments that study dynamical effects and/or when developing portable instrumentations. Our results suggest that online, running-fashion, microsecond-resolved analysis of polarimetric/ellipsometric measurements with fractional uncertainties at the 10^-6 levels, is possible, and using a proof-of-principle experimental demonstration we show that using a frequency-based analysis approach we can monitor and analyze signals at kHz rates and accurately detect signal changes at microsecond time-scales.