# Non-Destructive Infrared Evaluation of Thermo-Physical Parameters in Bamboo Specimens
## 3 Inverse Problem Formulation  
In this paper, we analyze the effects of bamboo preservation on bamboo's thermo-physical parameters. Consequently, we focus on the inverse problem of retrieving \(k\), and \(\rho c_{p}\) of a bamboo sample heated with an electric resistor. From Equation (2), one can also think of estimating \(h\) in which case we would cope with an _inverse problem_ in which not only are the thermophysical properties of the material unknown but the boundary conditions as well. This problem, however, is not easily solved because of strong correlation between parameters and ill-conditioning. In Section 4, we show some disadvantages of including \(h\) in the estimation process. Given \(p=\left[k,\rho c_{p}\right]\) as vector of unknown parameters in Equation (1), we intend to estimate these quantities through nonlinear least-squares regression using the surface temperature evolution \(T_{cam}(R,it_{s})\) on the cylinder's convex side measured with an infrared camera, where \(t_{s}=\dfrac{1}{30}\) is the sampling time, \(i\in\{0,1,2,3,4,5,6,7,...,m\}\), and \(T_{s}(it_{s})\) is determined as:  
\[\dfrac{1}{a_{0}}\sum_{j=0}^{j=1}b_{j}T_{res}(t_{s}(i-j))+a_{1}T_{s}(t_{s}(i-1)), \tag{7}\]  
where \(T_{res}(it_{s})\) is the known temperature of a tubular resistor placed at \(r=0<r_{0}\). Coefficients \((a_{j},b_{j})\) represent the heat losses in the experimental platform that cannot be included explicitly in the heat diffusion equation. Section 5 explains the estimation of these parameters. Figure 2 shows the methodology used here to estimate these parameters [25]. The datasets are available to the interested readers upon request to the authors. Measuring an estimate's accuracy is difficult, above all when there is little to no clue about the true values. Nonetheless, we can measure the consistency and robustness of a solution with respect to unexplained variability in the data through the asymptotical standard parameter error \(\sigma_{p}\) defined in Equation (8). This variability may be a consequence of something simple such as instrumental noise, or parameter correlation, or something more complex like the lack of completeness in the model that disregards some facts about the true nature of the problem [26]:  
Figure 2: Methodology to address the Inverse Heat Conduction Problem (IHCP) presented above. Assuming \(a_{i},b_{i},T_{res}\), and \(h\) as known. \(T(R,t,p)\) is fitted to \(T_{cam}\) by varying \(p\) using the Levenberg–Maquardt (LM) algorithm. The algorithm stops and yields \(p_{opt}\) when one of these criteria is met (i) \(max|J^{T}W(T_{cam}-T(p))|<\epsilon\) (ii) \(max|\dfrac{\Delta p_{j}}{p_{i}}|<\epsilon\) (iii) \(\dfrac{\chi^{2}}{m-n+1}<\epsilon\). The LM algorithm is highly sensible to \(p_{init}\) especially when the HKCP is nonlinear in the parameters, thus it must be carefully chosen. \[\sigma_{p}=\sqrt{diag\left(\left[J^{T}W\right]^{-1}\right)}, \tag{8}\]  
where \(J\) is the sensitivity matrix, \(W\) is diagonal matrix whose diagonal is \(diag(W)=\sigma_{T}^{2}=\dfrac{1}{m-n+1}(T-\hat{T}(p_{fit}))^{T}(T-\hat{T}(p_{fit}))\), and \(n\) is the parameter vector length. There are three important facts to highlight about this metric. First, very noisy data increase the value of \(\sigma_{p}\). Second, highly correlated parameters certainly increase \(\sigma_{p}\), meaning that any change in one parameter results in changes in the other parameters. In our case, the convective coefficient \(h\) is a difficult parameter to control and know beforehand since it varies depending on the materials surface and the environmental conditions. Third, ill-conditioning of an IHCP can be noticed through \(\sigma_{p}\), as it depends on \(J\), small eigenvalues of \(J\) causes the standard error to rise drastically [26]. Additionally, sensitivity coefficients allow us to characterize the IHCP as either linear or nonlinear in the parameters. Non-linearity in the parameters makes any IHCP sensitive to measurement errors [27].