# Multiparameter Determination of Thin Liquid Urea-Water Films
## 3 Evaluation Strategy
### Determination of the Additional Parameters of Temperature and Urea Concentration  
In Figure 5, the same spectra are depicted as in Figure 2 but sorted by the film concentration. It becomes clear that no region can be identified where the curves of fixed concentration overlap; instead, they show interdependencies between the other film parameters. This can be explained by the fact that the film concentration and temperature are parameters that show a nonlinear dependency to \(\alpha\) (see Equation (2)), and, thus, a new calibration approach must be pursued. Firstly, the dependencies of the film parameters are visualized in the three-dimensional \(\
u_{1}\)-\(\
u_{2}\)-\(\
u_{3}\) space. In Figure 6a, the scatter plot shows the absorbance data of the calibration data set at \(\
u_{1}\) = 4253 cm\({}^{-1}\), \(\
u_{2}\) = 4808 cm\({}^{-1}\) and \(\
u_{3}\) = 4501 cm\({}^{-1}\) with temperature-based color coding, while in Figure 6b, the same data are color coded with respect to the urea concentration. For both cases, a clear pattern can be identified, which suggests a dependency on the respective parameter to their location in the three-dimensional space. Since the visible tendencies point to a dependency of the parameter variation on the angles in the \(\
u_{1}\)-\(\
u_{2}\)-\(\
u_{3}\) space, the data are transformed from Cartesian to spherical coordinates. In spherical coordinates, the data points are specified by the radial distance \(r\), the azimuthal angle \(\varphi\) and the elevation angle \(\theta\). Figure 4: RMSE\({}_{\delta}\) of the difference between the known \(\delta\) and the calculated film thicknesses \(\delta_{calc}\) (calculated from the calibration set) over wavenumber \(\
u_{1}\). Figure 5: Absorbance spectra of all calibration measurements over the wavenumber. Spectra are color coded with respect to the film concentration of urea in water \(c\) and contain all \(T\) and \(\delta\) variations. The resulting transformation is shown in the \(\varphi\)-\(\theta\) space, while the radial distance \(r\) is neglected, since this quantity is more closely connected to \(\delta\), which was previously determined. The same transformed data set, depicted in Figure 7a, shows the data points with temperature-based color coding, while (b) is color coded with respect to the concentration. For each data set with same parameter value, a linear fit can be performed in the proposed \(\varphi\)-\(\theta\) space, leading to a set of curves that approximately intersect on one common intersection point P. It becomes clear that the gradient of the fit curve is related to the sought-for parameter value. In terms of temperature, a higher positive gradient is proportional to higher temperature values (see Figure 7a), and higher concentration values correlate with higher negative gradients (see Figure 7b). This observation suggests that a calibration can be performed by determining the coefficients of the linear fits of the set of curves and their respective intersection points \(\mathrm{P_{T}}\) and \(\mathrm{P_{C}}\). With a known intersection point, the data from a subsequent measurement can be transformed to calculate the gradient \(\mathrm{m_{T}}\) and \(\mathrm{m_{C}}\) between the intersection point \(\mathrm{P_{T}}\) and \(\mathrm{P_{C}}\) and the measurement in the \(\varphi\)-\(\theta\) space in the cases of the temperature and concentration, respectively. This way, the parameter of interest can be inferred from the resulting calibration curve, which is shown in Figure 8 (red solid line), where the predefined temperatures of the calibration data set are shown over the calculated gradients \(\mathrm{m_{T}}\) (Figure 8a) and \(\mathrm{m_{C}}\) (Figure 8b) as black dots. Similar to the film thickness calibration, the temperature and concentration calibration curves are gained by means of a linear least-squares first-order polynomial fit but in the \(\varphi\)-\(\theta\) space.