Modeling inverse

The inverse model of a plant provides a eontrol veetor u(kT) for a given output veetor y(lcT) as shown in Figure 10.27. 

So, for example, with the ship model shown in Figure 10.26, the inverse model eould be trained with time, forward veloeity, lateral veloeity and yaw-rate as input data and rudder angle and engine speed as output data. 

In situ CO titration experiments have also been conducted on multicomposition systems, that is, inverse model catalyst. Schoiswohl et al. [68] in their studies compared the CO titration reaction on three surfaces clean Rh(l 1 1) surface, Rh (111) surface covered with large 2D V309 islands (mean size >50 nm), and Rh(l 11) surface covered with small 2D V309 islands (meansize<15 nm). Prior to CO titration, the three surfaces were exposed to 10-7 mbar 02 to form a (2 x l)-0 phase at room temperature. In situ STM was used to follow the titration reaction in the presence of 10 x-10 7 m liar CO. CO titration on the clean Rh(l 1 1) surface or the Rh(l 1 1) surface with large V309 islands exhibits similar reaction kinetics. Figure 3.19 shows... 

Within the inverse model catalyst approach, the y/7-V309-Rh(l 11) nanostructures have been used to visualize surface processes in the STM with atomic-level precision [104]. The promoting effect of the V-oxide boundary regions on the oxidation of CO on Rh(l 1 1) has been established by STM and XPS by comparing the reaction on two differently prepared y/7-V309-Rh(l 11) inverse catalyst surfaces, which consist of large and small two-dimensional oxide islands and bare Rh areas in between [105]. A reduction of the V-oxide islands at their perimeter by CO has been observed, which has been suggested to be the reason for the promotion of the CO oxidation near the metal-oxide phase boundary. 

Parkhurst, D.L. Appelo, C.A.J. 1999. User s guide to PHREEQC (Version 2)—A computer program for speciation, batch-reaction, one-dimensional transport, and inverse modeling. 

The Rothamsted Carbon Model (RothC) uses a five pool structure, decomposable plant material (DPM), resistant plant materials (RPM), microbial biomass, humified organic matter, and inert organic matter to assess carbon turnover (Coleman and Jenkinson 1996 Guo et al. 2007). The first four pools decompose by first-order kinetics. The decay rate constants are modified by temperature, soil moisture, and indirectly by clay content. RothC does not include a plant growth sub-module, and therefore NHC inputs must be known, estimated, or calculated by inverse modeling. Skjemstad et al. (2004) tested an approach for populating the different pools based on measured values. 

Discussing practical aspects of the inverse model such as stability, resolution, and statistical assessment would require considerably more development of the subject than presented here. For these important topics, the reader can consult the references quoted above and their enclosed bibliography. 

The simplest inverse model consists in finding liquid and solid phase proportions assuming a melt and source composition. This case is depicted in Figure 9.1 and may be modeled quantitatively with no extra assumption. Equation (9.2.1) expresses the fact that, in the m-dimensional composition space, the source composition must be the centroid of melt and residual mineral compositions, each being weighted by the... 

Table 9.2. Melt and mineral fractions (%) in the molten source assumed to calculate concentrations in 5 melts used as an example for inverse modeling of partial melting.

Table 9.3. Synthetic example of batch-melting inverse modeling.

Feigenson, M. D. Carr, M. J. (1993). The source of Central American lavas inferences from geochemical inverse modeling. Contrib. Mineral. Petrol., 113, 226-34. 

Table 8.3 Global distributions of CH4 emissions (Tg CH4year ) calculated using inverse modelling. In Scenario A rice contributes 50-80Tg year and in B 15-30Tg year the net contribution of natural wetlands and ricelands is constant...

Heimann M, Kaminski T. 1999. Inverse modelling approaches to infer surface trace gas fluxes from observed atmospheric mixing ratios. In Bouwman AF, ed. Approaches to Scaling of Trace Gas Fluxes in Ecosystems. Amsterdam Elsevier. 

Hein R, Crutzen PJ, Heimann M. 1997. An inverse modeling approach to investigate the global atmospheric methane cycle. Global Bio geochemical Cycles 11 43-76. 

Houweling S, Kaminski T, Dentener F, Lelieveld I, Heimann M. 1999. Inverse modeling of methane sources and sinks using the adjoint of a global transport model. Journal of Geophysical Research-Atmospheres 104 26137-26160. 

Where C is a matrix of component concentrations or sample properties, S is a matrix of basis vectors (pure component spectra, or spectral profiles reflecting a pure sample property), and E and Ec are model residuals. The direct model expresses the analyzer responses (X) as a function of concentrations, whereas the inverse model expresses concentrations as a function of the analyzer responses. Because the former is more in line with the Beer-Lambert Law (absorbance = concentration x absorptivity), it is given the label direct . 

Strategies for direct versus inverse modeling methods... 

In Section 12.3.2, the fundamental differences between direct and inverse modeling methods were discussed. As will be discussed here, this distinction is not just a convenient means for classifying quantitative regression methods, but has profound implications regarding calibration strategy and supporting infrastructure. 

The use ofdiese classical experimental designs requires that the variables be set to prefetermined levels. Therefore, additional effon must be made to account for wiables that are not controllable. One chemometric approach is to allow these variables to vary naturally and to collect enough data to adequately modd their effect. This is the difference between the so-called natural and controllei calibration experiments (Martens and Nxs, 1989). When it is possible to mr iu e the variables, this can be done to verify that an adequate range has covered. Inverse models as discussed in Chapter 5 can then be used to implic y model their effect. (See also Appendix A.)... 

To see how the inverse and classical modeling approaches differ, the inverse model n resented by Equation A.2 is examined. 

In maiK- applications, the spectral residuals will not behave in the ideal manner as picted here. Some nonrandom behavior may be tolerable depending on the performance requirements of the model. If the residuals are lai e and fire model performance is not acceptable, an inverse model approach sudias PLS or PCR (Section 5.3) can be considered. 

The maiber of samples that need to be measured in the calibration phase is smiS as compared to inverse modeling (Section 5.3). The minimum number of mistures to measure is equal to the number of analytes in the system. 

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