What are , and what are not , Inverse Laplace Transforms

Time-domain NMR, in one and higher dimensionalities, makes routine use of inversion algorithms to generate results called “T2-distributions” or joint distributions in two (or higher) dimensions of other NMR parameters, T1, diffusivity D, pore size a, etc. These are frequently referred to as “Inverse Laplace Transforms” although the standard inversion of the Laplace Transform long-established in many textbooks of mathematical physics does not perform (and cannot perform) the calculation of such distributions. The operations performed in the estimation of a “T2-distribution” are the estimation of solutions to a Fredholm Integral Equation (of the First Kind), a different and more general object whose discretization results in a standard problem in linear algebra, albeit suffering from well-known problems of ill-conditioning and computational limits for large problem sizes. The Fredholm Integral Equation is not restricted to exponential kernels; the same solution algorithms can be used with kernels of completely different form. On the other hand, (true) Inverse Laplace Transforms, treated analytically, can be of real utility in solving the diffusion problems highly relevant in the subject of NMR in porous media.