Three-dimensional data calibration

Figure 8.39 Three-dimensional scatter plot of the first three PCA scores obtained from a set of original calibration data. The calibration samples selected by the cluster analysis method are marked with an x. ...

Contaminant Transport Modeling. A major difficulty in the calibration of any two-dimensional contaminant transport model is relating the two-dimensional simulated plume to the real three-dimensional plume. A model based on Equation 4 can simulate two dimensions in cross section or areal view. An areal view was selected for the problem considered here. Use of a two-dimensional areal view model implies that the contaminant is uniformly spread out through the entire saturated thickness of the aquifer. However, in the field the aldicarb plume is only around 10 feet (3m) thick while the aquifer is around 70 feet (21 m) thick. Moreover, the concentration data were collected from wells having 3 ft (0.91 m) well screens and hence are representative of only a small fraction of the total aquifer thickness. It was decided to calibrate the model to concentrations representative of the center of the plume vertically. That is, the model was calibrated to maximum measured concentrations in each well nest. As a result, the loading rate to the model is inflated over probable field values. The model assumes the load to the model is distributed over the full aquifer thickness, when in the field the zone of maximum concentration is probably no more than 3 feet thick. Therefore, the probable loading rate in the field is roughly 3/70 or 4% of that used to calibrate the model. 

In chemometrics we are very often dealing not with individual signals, but with sets of signals. Sets of signals are used for calibration purposes, to solve supervised and unsupervised classification problems, in mixture analysis etc. All these chemometrical techniques require uniform data presentation. The data must be organized in a matrix, i.e. for the different objects the same variables have to be considered or, in other words, each object is characterized by a signal (e.g. a spectrum). Only if all the objects are represented in the same parameter space, it is possible to apply chemometrics techniques to compare them. Consider for instance two samples of mineral water. If for one of them, the calcium and sulphate concentrations are measured, but for the second one, the pH values and the PAH s concentrations are available, there is no way of comparing these two samples. This comparison can only be done in the case, when for both samples the same sets of measurements are performed, e.g. for both samples, the pH values, and the calcium and sulphate concentrations are determined. Only in that case, each sample can be represented as a point in the three-dimensional parameter space and their mutual distances can be considered measures of similarity. 

This report presents our approach for calculation of the Task. Our numerical code THAMES is the three-dimensional finite element simulator of fully coupled processes. First, we defined the input data for THAMES from the supplied properties of FEBEX bentonite. After calibrations of some parameters such as thermal vapour diffusivity, the analysis that treats fully coupled thermal, hydraulic and mechanical processes was carried out. 

Up to this point, regression has been restricted to two blocks of two-way data Y and X. In chemical analysis, however, a growing number of problems can be cast as three-way regression analysis. Consider the calibration of chemical constituents on the basis of their fluorescence excitation/emission spectrum or of gas chromatography/mass spectrometry (GC/MS) data. For each sample, two-dimensional measurements are available that constitute a three-way data array, X. This data array has to be related to sample concentrations of one, vector y, or several analytes, matrix Y. Cases can be imagined where even the matrix Y constitutes a three-way data array. 

The temperature calibration is then made by fitting a three-dimensional third-degree polynomial to the RGB and HSI data and a two-dimensional third-degree polynomial to the POD data. This was done using a standard... 

Although colorimetric methods were the earliest to be used for pesticide analysis [203], competitive spectroscopic methodologies for the determination of these pollutants were not developed until the last decade. The spectroscopic determination of several pesticides in mixtures has been the major hindrance, especially when their analytical characteristics are similar and their signals overlap as a result. Multivariate calibration has proved effective with a view to developing models for qualitative and quantitative prediction from spectroscopic data. Thus, partial least squares (PLS) and principal component regression (PCR) have been used as calibration models for the spectrofluorimetric determination of three pesticides (carbendazim, fuberidazole, and thiabendazole) [204]. A three-dimensional excitation-emission matrix fluorescence method has also been used for this purpose (Table 18.3) [205]. 

Matrices (arrays) can be multidimensional three-dimensional matrices are also called tensors. Analysis of tensors is frequently called 3-way analysis. Typical example is the data from a hyphenated technique, e.g., gas chromatography-mass spectrometry (GC-MS) data one direction (way) is the mass spectrum, second direction is the chromatographic separation (time, scan), and the third direction is the samples (of different origin, repetitions, calibration series, etc.). The 3-way analyses can easily be generalized into n-way analysis including more directions. 

In this section we examine the effect of PS concentration on the fluorescence of miscible PS/PVME blends cast from toluene. We then show how fluorescence data for miscible blends may be used to establish a "calibration curve for M, the probability of eventual non-radiative or radiative decay of the excitation by monomer fluorescence. Finally, we present a spatially periodic lattice model that leads to reasonable predictions for Wx for the three dimensional energy migration process. 

Once a three-dimensional seismic facies model has been generated and calibrated with well data, the next step consists in translating the facies model more efficiently to the simulation domain. The challenge faced is to minimize the manual work and to preserve as much as possible the detailed representation carried by the three-dimensional seismic facies model in order to improve the seismic to simulation workflow. 

The reason for this effect has to be attributed to a better and adequate ratio between sample size and array dimensionality. For a significant clustering of the patterns, with an array of six sensors a sample size of at least 18 is required [149, 184]. As a consequence, the discrimination based on only 12 measurements has poor statistical relevance. Most of the applications with sensor arrays found in the literature do not consider this fact frequently discriminations with 12-32 sensors in an array and with a sample size of three to four are described. All of them are of limited feasibility with concurrent poor validation, especially in terms of reproducibility and predictive ability. In other words, if there are not enough calibration measurements one can separate data in a predetermined way, but will fail to verify the result using independent test samples. 

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