Inverse models/modeling error analysis

As described in more detail below, the inverse approach generally leads to underdetermined mathematical systems that are much harder to solve than the systems encountered in forward models. Error and resolution analysis are two issues of particular importance when solving underdetermined inverse problems. First, the solved-for physical and biogeochemical parameters depend directly on the tracer data, and errors in the data propagate into errors in the solution. Second, owing to the incompleteness of information in underdetermined systems, the unknowns are usually not fully resolved. Instead, only specific linear combinations of unknowns may be well constrained by the data, while individual unknowns or other combinations of unknowns may remain poorly determined. Both, error and resolution analysis are essential for a quality assessment of the solution of underdetermined systems. 

The 50.31 MHz 13C NMR spectra of the chlorinated alkanes were recorded on a Varian XL-200 NMR spectrometer. The temperature for all measurements was 50 ° C. It was necessary to record 10 scans at each sampling point as the reduction proceeded. A delay of 30 s was employed between each scan. In order to verify the quantitative nature of the NMR data, carbon-13 Tj data were recorded for all materials using the standard 1800 - r -90 ° inversion-recovery sequence. Relaxation data were obtained on (n-Bu)3SnH, (n-Bu)3SnCl, DCP, TCH, pentane, and heptane under the same solvent and temperature conditions used in the reduction experiments. In addition, relaxation measurements were carried out on partially reduced (70%) samples of DCP and TCH in order to obtain T data on 2-chloropentane, 2,4-dichloroheptane, 2,6-dichloroheptane, 4-chloroheptane, and 2-chloroheptane. The results of these measurements are presented in Table II. In the NMR analysis of the chloroalkane reductions, we measured the intensity of carbon nuclei with T values such that a delay time of 30 s represents at least 3 Tj. The only exception to this is heptane where the shortest T[ is 12.3 s (delay = 2.5 ). However, the error generated would be less than 10%, and, in addition, heptane concentration can also be obtained by product difference measurements in the TCH reduction. Measurements of the nuclear Overhauser enhancement (NOE) for carbon nuclei in the model compounds indicate uniform and full enhancements for those nuclei used in the quantitative measurements. Table II also contains the chemical... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

There are two primary difficulties in directly inverting equation (12.7). First, the system is usually underdetermined, which means there are more variables (e.g., wavelengths) than equations (e.g., number of calibration samples). Thus, direct inversion does not always yield a unique solution. Second, even if a pseudoinverse exists and results in a unique solution, the solution tends to be unstable because all measurements contain noise and error. That is, small variations in c or S can lead to large variations in b. Underdetermined and unstable models can be avoided by using data reduction methods, such as factor analysis, which reduce the dimensionality of the spectral data and much of the underlying noise within each spectrum. 

The die-away curves were analyzed by an unweighted least squares method. Parameter variances were estimated from the inverse design matrix defined by a Taylor expansion [40] of the non-linear model and the estimated error variance. Unweighted analysis was selected because measurement errors in the mass spectrometer are constants, approximating 0.003 atom percent excess. 

In analysis, the response of a system is sought if the system model and excitatimi are known. This is sometimes termed a forward problem. In this interpretation, the inverse problem of finding a system model given the excitation and response is called system identification. System identification in the time domain involves the determination of unspecified parameters of an assumed system model. This can always be formulated as an optimization problem. In the present context, suppose the differential model of hysteresis is adopted and a set of measured excitation-response data from cyclic performance tests of an inelastic structure is given. How can the loop parameters of hysteresis be estimated from the measured data For each choice of the parameters, the response of the degrading structure subjected to the given excitation can be obtained by numerical simulation. The calculated response data can then be compared to the measured data to see if there are large errors. Obviously, the assumed loop parameters... 

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