Inverse methods assumptions

The generalized inverse method represents another formulation of multilinear least-squares analysis. All the usual assumptions involved with least squares apply. 

A straightforward application of an inverse method clearly shows that no useful information can be directly retrieved from borehole paleothermometry at low accumulation sites like Vostok (Rommelaere, 1997). To overcome this problem, Salamatin et al. (1998 and references therein) developed an inverse procedure based on the assumption that the inferable components of the surface temperature at Vostok can be expressed as... 

Constraining In the case of partial melting the unknowns are (1) the chemistry of the source, (2) partial melting the bulk partition coefficient for each element considered and (3) the degree of using an partial melting for each sample. There are too many unknowns for a direct solution inversion method and so a number of initial assumptions have to be made. 

If the above-mentioned assumptions are reasonably satisfied, the stress inversion methods are capable to determine four parameters of the stress tenson three angles defining the directions of the principal stress directions, Ci, <72 nd <73, and shape ratio R. The methods are unable to recover the remaining two parameters of the stress tensor. Therefore, the stress tensor is usually searched with the normalized maximum compressive stress... 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

In the course of investigation of reactivity of the mesoionic compound 44 (Scheme 2) the question arose if this bicyclic system participates in Diels-Alder reactions as an electron-rich or an electron-poor component <1999T13703>. The energy level of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) orbitals were calculated by PM3 method. Comparison of these values with those of two different dienophiles (dimethyl acetylenedicarboxylate (DMAD) and 1,1-diethylamino-l-propyne) suggested that a faster cycloaddition can be expected with the electron-rich ynamine, that is, the Diels-Alder reaction of inverse electron demand is preferred. The experimental results seemed to support this assumption. 

A preference of hemodynamic response is mainly due to better contrast-to-noise ratio and the ease of comparability with fMRI BOLD response which is inversely related to changes in Hb, over measurement of neuronal activity. However, measurement of the fast neuronal response can potentially benefit NIR methods by addressing the inherent pitfalls of MBLL assumptions in slow response measurements. Fast response, aside from being a direct measurement of neuronal activity, also has the potential to provide better spatial resolution. 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

With respect to the spreading calibration, several methods have been suggested e.g. (6-1 ) Numerous techniques have been proposed for solving the inverse filtering problem represented by Equation 1, with different degrees of success e.g. (it,15-19) Only references (M, (l8) and (I9) make no assumptions on the shape of g(t,x). 

Biraud s method starts by considering the centermost value of <9 + (0) as known, because it is at co = 0 that the inverse filter gives its most reliable estimate. Furthermore, this assumption guarantees that the total flux in the restoration is the same as that measured. Thus we find... 

As for the non-uniqueness of the solution, there is no method that can bypass this inherent problem. In inverse problems, one of the common practices to overcome the stability and non-uniqueness criteria is to make assumptions about the nature of the unknown function so that the finite amount of data in observations is sufficient to determine that function. This can be achieved by converting the ill-posed problem to a properly posed one by stabilization or regularization methods. In the case of groundwater pollution source identification, most of the time we have additional information such as potential release sites and chemical fingerprints of the plume that can help us in the task at hand. 

From Table X it is evident that fcc0 decreases with increasing granule size, and is inversely proportional to the granule size in the range of 0.55 mm. to 1.82 mm. The assumption that breakdown time is dependent on an increase in surface area is not confirmed. In Table XI, measurements of the surface area by the B.E.T. method are given for... 

The above formulas may become inapplicable for systems with adsorption processes or/and coupled chemical steps in solution whose characteristic times are comparable with the inverse frequency within the impedance measurement interval. In this case the charge-transfer resistance, Rct, must be replaced by a complex charge-transfer impedance, Zct. Another restriction of this treatment is its assumption of the uniform polarization of the m s interface which requires to ensure a highly symmetrical configuration of the system. Refs. [i] Sluyters-Rehbach M, Sluyters JH (1970) Sine wave methods in the study of electrode processes. In Bard A/ (ed) Electroanalytical chemistry, vol. 4. Marcel Dekker, New York, p 1 [ii] Bard A], Faulkner LR (2001) Electrochemical methods, 2nd edn. Wiley, New York [iii] Retter U, Lohse H (2005) Electrochemical impedance spectroscopy. In Scholz F (ed) Electroanalytical methods. Springer, Berlin, pp 149-166 [iv] Bar-soukov E, Macdonald JR (ed) (2005) Impedance spectroscopy. Wiley, Hoboken... 

An important feature of PLS is that it takes into account errors in both the concentration estimates and the spectra. A method such as PCR assumes that the concentration estimates are error free. Much traditional statistics rests on this assumption, that all errors are of the variables (spectra). If in medicine it is decided to determine the concentration of a compound in the urine of patients as a function of age, it is assumed that age can be estimated exactly, the statistical variation being in the concentration of a compound and the nature of the urine sample. Yet in chemistry there are often significant errors in sample preparation, for example accuracy of weighings and dilutions, and so the independent variable in itself also contains errors. Classical and inverse calibration force the user to choose which variable contains the error, whereas PLS assumes that it is equally distributed in both the x and c blocks. 

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