Inverse Laplace

Exponential decay often occurs in measurements of diffusion and spin-relaxation and both properties are sensitive probes of the electronic and molecular structure and of the dynamics. Such experiments and analysis of the decay as a spectrum of 7i or D, etc., are an analog of the one-dimensional Fourier spectroscopy in that the signal is measured as a function of one variable. The recent development of an efficient algorithm for two-dimensional Laplace inversion enables the two-dimensional spectroscopy using decaying functions to be made. These experiments are analogous to two-dimensional Fourier spectroscopy. 

The general strategies to solve this problem have been discussed extensively in the literature on mathematics [47]. Numerical Recipes [48] and other NMR literature [30, 31, 49] are a good introduction. Even though there are well-established algorithms for performing a numerical Laplace inversion [29-31], its use is not necessarily trivial and requires considerable experience. It is thus useful to understand the essential mathematics involved in the analysis as a better guide to its... 

The ID Laplace inversion can be approximated by a discretized matrix form ... 

Novel Two Dimensional NMR of Diffusion and Relaxation for Material Characterization 171 2.7.3.2 Fast Laplace Inversion - FLI... 

The 2D Laplace inversion, such as Eq. (2.7.1), can in fact be cast into the ID form of Eq. (2.7.11). However, the size of the kernel matrix will be huge. For example, a Ti-T2 experiment may acquire 30 % points and 8192 echoes for each X assuming that 100 points for Ti and T2 are used, respectively. Thus, the kernel will be a matrix of (30 8192) 10 000 with 2.5 109 elements. SVD of such a matrix is not practical on current desktop computers. Thus the ID algorithm cannot be used directly. 

With this tensor structure of the kernel, 2D Laplace inversion can be performed in two steps along each dimension separately [50]. Even though such procedure is applicable when the signal-to-noise ratio is good, the resulting spectrum, however, tends to be noisy [50]. Furthermore, it is not dear how the regularization parameters should be chosen. 

The Laplace inversion (LI) is the key mathematical tool of the DDIF experiment. The ability to convert the measured multi-exponential decay into a distribution of decay times is crucial to the DDIF pore size distribution application. However, unlike other mathematical operations, the Laplace inversion is an ill-conditioned problem in that its solution is not unique, and is fairly sensitive to the noise in the input data. In this light, significant research effort has been devoted to optimizing the transform and understanding its boundaries [17, 53, 54],... 

Here K is the kernel matrix determining the linear operator in the inversion, A is the resulting spectrum vector and Es is the input data. The matrix element of K for Laplace inversion is Ky = exp(—ti/xy) where t [ and t,- are the lists of the values for tD and decay time constant t, respectively. The inclusion of the last term a 11 A 2 penalizes extremely large spectral values and thus suppresses undesired spikes in the DDIF spectrum. 

Secondly, although stable solutions covering the entire temporal range of interest are attainable, the spectra may not be well resolved that is, for a given dataset and noise, a limit exists on the smallest resolvable structure (or separation of structures) in the Laplace inversion spectrum [54]. Estimates can be made on this resolution parameter based on a singular-value decomposition analysis of K and the signal-to-noise ratio of the data [56], It is important to keep in mind the concept of the spectral resolution in order to interpret the LI results, such as DDIF, properly. 

An example of DDIF data on a Berea rock sample is shown in Figure 3.7.1 illustrating the decay data (A), the pore size distribution after Laplace inversion... 

In order to make the meaning of this theorem more precise, let us consider the Laplace inverse of (Q (z) ky, which is defined by a formula similar to (65) ... 

Often the data cannot be satisfactorily fit to multiple specific exponentials because the sample contains a natural distribution of exponentials centered on the specific exponential time decays utilized in the fitting procedure. When specific exponentials cannot adequately describe the data, Laplace inversion routines are... 

Figure 10.7 Carr-Purcell-Meiboom Gill (CPMC) sequence and NMR signal. (A) CPMC pulse sequence, (B) CPMC decay signal, and (C) Laplace inversion of the CPMC decay signal.

On Laplace inversion and then inserting the rate kernel into the Noyes expression for the rate coefficient [eqn. (191)], the rate coefficient is seen to be exactly that of the Collins and Kimball [4] analysis [eqn. (25)]. It is a considerable achievement. What is apparent is the relative ease of incorporating the dynamics of the hard sphere motion. The competitive effect comes through naturally and only the detailed static structure of the solvent is more difficult to incorporate. Using the more sophisticated Gaussian approximation to the reactant propagators, eqn. (304), Pagistas and Kapral calculated the rate kernel for the reversible reaction [37]. These have already been shown in Fig. 40 (p. 219) and are discussed in the next section. 

The above equations are valid for any experiment in a cell with finite planar geometry. For example, they apply to the experiment described in Sect. 4.7 in fact, eqns. (116) can be derived from eqns. (123) and (124) by setting i(t) equal to the constant i and performing a Laplace inversion. The Laplace inversion is difficult in this derivation and the interested reader is referred to ref. 79 for guidance. 

The Laplace inversion of eqn. (123) or eqn. (124) when the surface concentrations are constant is accomplished rather easily and leads to... 

For g3, Kirkwood s superposition approximation is assumed. Next a double derivative of Eq. (115) is taken and Laplace inversion is performed. Now equating the terms in this expression and in Eq. (119) (after setting q = 0) of the order of (1/z3) the following expression for the binary time constant is obtained ... 

By comparison with Eq. (55), the Laplace inversion of Eq. (63) leads to the Mittag-Leffler shape [105]... 

Since, as noted above, the success in determining G(D) is actually not in the choice of a computer program for Laplace inversion but reducing the noise level in measured g( (t,q). Thus, it is crucial that the solution is cleaned (i.e. "dust-freed") very thoroughly before it is subjected to laser light scattering measurements. For example, in studies conducted by the author, efforts were made to ensure that the relative difference between the measured and calculated baselines did not exceed 0.1%. The error analysis related to the above problem can be found elsewhere [42,43]. 

Once the scaling relation of Eq. (39) is known, the molar mass distribution can, at least in principle, be obtained from a Laplace inversion of the multi-exponential decay function as defined in Eq. (40). At this point, the differences between PCS and TDFRS stem mainly from the different statistical weights and from the uniform noise level in heterodyne TDFRS, which does not suffer from the diverging baseline noise of homodyne PCS caused by the square root in Eq. (38). 

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