Abstract
An analysis of one and two-dimensional data sets using the so-called inverse Laplace transforms (ILT) and a discrete approach called Anahess is presented. Actually, the ILT is in mathematical terms an ill posed numerical inversion of Fredholm integrals with smooth kernels, meaning that several and very different solutions may fit the experimental data equally well. This inversion is referred to as an ILT in this book. The ILT and Anahess are fundamentally different methods for analysing the data, and the two methods are evaluated against each other on synthetic data sets. These data sets will represent the NMR data typically found from analysing real systems such as cheese or fluid saturated porous rock core plugs. Singular components and distribution are represented in the synthetic data sets. While the ILT algorithm fits the experimental data to a large set of fixed points in a grid, the Anahess algorithm minimizes the number of points, which are not fixed due to user defined grid as for the ILT algorithm. From the comparison, it will become obvious that both approaches have their benefits and drawbacks. Ultimately, user interpretation is required to decide which particular model fits the system better, either the Anahess algorithm, which fits a limited number of discrete components that are continuously variable, or the ILT algorithm, which assigns a larger set of components from a fixed grid.
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Appendix
Appendix
In the following a Matlab program is provided that perform a weighted linear fit on a data set.
When running the program above with a weighting of each data point as the intensity to a power of 4, I4, the fit will look like the one shown in Fig. 5.1.
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Sørland, G.H. (2014). Analysis of Dynamic NMR Data. In: Dynamic Pulsed-Field-Gradient NMR. Springer Series in Chemical Physics, vol 110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44500-6_5
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