# Estimating Thermal Material Properties Using Solar Loading Lock-in Thermography
## 3 Theoretical Framework
The reference signal of a conventional lock-in amplification method is generated inherently and is available for the lock-in transform algorithm by design [8]. However, in ambient thermography, the reference signal is generated artificially. A pure sinusoidal signal \(ref_{t}(\omega)=\ell(-i\omega t)\) is used as the reference. Note that this is a set of vectors in time for each frequency considered:  
\[S_{xy}(\omega)=\sum_{t}T_{xyt}\cdot ref_{t}(\omega)\]  
The amplitude of \(S_{xy}\) is of particular interest because it depicts the amount of temperature change the surface has undergone at a particular frequency. Under simplified conditions, these amplitudes are directly related to material properties. Table 1 summarizes these relations in a table. \begin{table}
\begin{tabular}{c c} \hline
**Thermal Thickness** & \(\Re\{S_{xy}\}\) & \(\Im\{S_{xy}\}\) \\ \hline Thermally thin & 0 & \\ Thermally thick & \(\frac{p}{\sqrt{2\omega\omega\omega\rho_{x}}}\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Real and imaginary part of the temperature signal from [8] (p.120). Thermal effusivity is defined as \(b=\sqrt{\lambda\rho c_{p}}\). Substitution leads to:  
\[b=\frac{p}{\left|S_{xy}\right|\cdot\sqrt{\omega}} \tag{1}\]  
A three-step method is devised, that estimates the thermal effusivity of every pixel in the scene:  
1. Place a reference sample inside the measured frame with known material properties to get an approximation of \(p\). From Table 1 we get: \[p=\left|S_{xy}\right|\cdot\rho dc_{p}\omega\] (2) Note that this is only valid for thermally thin samples. 2. Empirically correct \(p\) to better approximate the input power density amplitude that the actual body is subjected to: \[p_{eq}=p\cdot c(\omega)\] (3) where \(c(\omega)\) is an empirical function that is determined by linear regression. 3. Approximate thermal effusivity \(b\) by inserting \(p_{eq}\) into Equation (1). This results in the final formula: \[b_{corr}=\frac{p\cdot c(\omega)}{\left|S_{xy}\right|\cdot\sqrt{\omega}}\] (4)  
The approximation of \(p\) via the reference body can only be applied to estimate \(b\) if it is used on a part of the surface that is indeed thermally thick. Additionally, approximating heat flow to be linear in relation to input power is detrimental to the accuracy if thermal radiation to the surroundings contributes to a significant part of heat transferred. Thermal radiation heat transfer is in general of the form \(p_{rad}\propto T_{obj}^{4}-T_{amb}^{4}\) and can be estimated linearly only for small amplitudes in object and ambient temperature. Simple thermal mass point simulations show that the radiative heat loss accounts for an approximatively constant part of the overall heat loss in respect with excitation strength. Linearizing radiative heat loss within naturally occurring temperatures (0 \({}^{\circ}\)C to 40 \({}^{\circ}\)C) results in a maximum error of 2.4% percent, which is less than many other approximating factors that were assumed earlier. On empirically correcting the results: One of the biggest error terms is that the real heat input density in the field experiments is unknown. Especially heat loss via convection and radiation is hard to determine directly as it is dependent on multiple significant factors that were listed above. In a first approach to measuring thermal material properties using solar loading thermography, a regression method is used to match the measured results to the literature value of a specific material (in this case brick and concrete). Other error sources include but are not limited to:  
1. Temperature distribution in thick materials is dispersive, because thermal material properties like thermal conductivity often depend on the temperature itself. 2. Heat radiation estimations should consider the temperature distribution of the half space and not estimate the ambient radiation temperature to be constant, especially at nighttime with clear skies, the sky heat sink is significant.