# Estimating Thermal Material Properties Using Solar Loading Lock-in Thermography
## 3 Theoretical Framework  
Solar loading lock-in thermography uses no artificial heat source, like heat lamps, but only naturally occurring heat sources. These consist of the following: Heat transfer within the structure (conduction). Heat transfer with the surrounding air, which is free to move (convection) and heat transfer via radiation (solar irradiation, radiative heat loss). The fourth heat source, evaporative cooling, is neglected in this work. Conduction within a static material is a linear process \(\sim\Delta T\), governed by the heat equation. Natural convection is a non-linear process that is commonly linearized by defining an \"overall heat transfer coefficient\" [6], convection in a natural setting, where ambient temperatures, humidity and air speeds are always changing is very hard to accurately represent in a single coefficient. Thermal radiation is a major non-linear process \(\sim T^{4}\) that may be linearized for a single temperature region. The accuracy of thermal radiation linearization is discussed later in more detail. To estimate thermal material properties the value of the heat input density amplitude (irradiance) of a periodic excitation is needed. The heat input is also assumed to be periodic and sinusoidal. Estimating the heat input that occurs in a specific frequency content of the total excitation is inaccurate. Especially for overcast days, and frequencies higher than 1/24 h direct measurements are necessary. One way of measuring the heat input is a measuring solar irradiance (e.g., via a pyranometer). Another way of determining heat input, presented in this work, is done by placing thermally known reference samples into the measurement setup and using the thermal response of the reference samples to infer the preceding irradiation amplitude in frequency domain. Subsequently using formulas from lock-in thermography to determine the heat flow amplitude, which in turn arise from the heat equation using various assumptions and simplifications. The thermal imaging long wave infrared (LWIR) camera is set up in a fixed manner to capture a set of thermal images that are regularly spaced in time. These images represent a 2D temperature field that is assumed to accurately depict the surface temperature of the captured structure. Thermal imaging cameras can be calibrated and adjusted to radiation to measure more accurate results [7]. A two-point calibration with a calibration radiator is performed within the approximate temperature range to get temperature estimates as close to the real temperature as possible. Other effects, like emissivity/reflectivity can be corrected as well to further improve absolute measuring accuracy for a given surface. These kinds of corrections were not performed in the scope of this work but are hypothesized to further increase measurement accuracy. A time series matrix is arranged from the temperature field \(T_{xy}\), which is called \(T_{xyt}\) with the third dimension being time. The time series for each pixel is then transformed into frequency domain by multiplying with the complex frequency \(\omega\) in time. This is similar to using a lock-in amplifier. The main difference is that the reference oscillator is generated inside the Fast Fourier Transform (FFT) and never subject to any feedback from the signal itself. Therefore, two main conditions have to be met to ensure the error being as low as possible. 1. For the fundamental frequency and its respective harmonics amplitudes to result in a single bin, the measurement time shall be an exact multiple of the fundamental interval (in this work 24 h). 2. The local start time of the measurement is captured because it defines the reference Phase value, each calculated phase is related to. The resulting matrix \(S_{xy}(\omega)\) contains the complex coefficients describing either amplitude/phase or real/imaginary (often called \(0^{\circ}/90^{\circ}\)) components for each pixel of the image [8].