# Non-Destructive Infrared Evaluation of Thermo-Physical Parameters in Bamboo Specimens
## 3 Inverse Problem Formulation
Figure 3a displays the relative sensitive coefficients for our inverse heat conduction scenario. Here, we show that, as \(h\) increases, not only does \(J_{h}\) change, implying that it depends on \(h\) and so the IHCP is nonlinear with respect to \(h\), but \(h\) variation also causes \(J_{k}\) and \(J_{\rho c_{p}}\) to vary, thus demonstrating correlation between the parameters. Likewise, the variation of \(k\) and \(\rho c_{p}\) also introduces an effect on the other relative sensitivity coefficients. Figure 3a also shows that the volumetric heat capacity \(\rho c_{p}\) strongly affects the transitory state of the heat conduction. In contrast, \(h\) and \(k\) most influence the temperature behavior at the steady state. In estimating parameters through temperature measurements, the sensor position is also fundamental; Figure 3b shows that \(J_{r}\) varies with position. For instance, at \(r=R\), where we have located our sensor (with no possibility of relocation since we intend to perform a non-destructive test), temperature response is lagged and attenuated, making it harder to estimate parameters by using these temperature measurements. As our IHCP also exhibits correlation among parameters and nonlinearity, the problem of estimating thermo-physical properties of materials, in this scenario, is challenging. An additional fact to point out is that a substantial difference in the range of the parameters to estimate makes the LM algorithm very sensitive to noise, preventing it from converging, as volumetric heat capacity and thermal conductivity take values proportional to \(10^{6}\) and \(10^{-1}\) for bamboo, respectively, we decided to optimize the volumetric heat capacity in the range of 2-10, but the model  
Figure 3: Relative sensitivity coefficients. (**a**) \(J_{k}\), \(J_{h}\) and \(J_{\rho c_{p}}\); and (**b**) \(J_{r}\) are the first derivatives of the temperature with respect to \(k\), \(h\), \(\rho c_{p}\), and \(r\) (position within the cylinder). internally multiplies it by \(10^{6}\). This approach increases the probability of convergence and produces better Jacobean matrices. Estimation of Thermo-Physical Properties of Bamboo Using Synthetic Data with Additive White Gaussian Noise  
In order to gain insight into the estimation of the thermo-physical properties of a material and its surrounding medium from transient temperature measurements, we generate synthetic data by solving the heat diffusion equation explained in Section 2 through the Crank-Nicholson method for a known set of parameters \(p_{t}=(k,\rho c_{p},h)\). Even though this experiment uses data from the solution of the direct model, it allows us to observe how the LM regression algorithm performs under different conditions such as noisy data and random initialization. The noisy data was generated by adding additive white Gaussian noise (with standard deviations \(\sigma\) of 0.1) to the data. Random initialization, on the other hand, was performed by randomly choosing parameters in the set of initial random guesses \(S=\{p_{init}=(k,\rho c_{p},h)\mid 0.4p_{t,1}<k<1.6p_{t,1},\ 0.4p_{t,3}<\rho c_{p}<1.6p_{t,3},\ 0.4p_{t,2}<h<1.6p_{t,2}\}\). It is necessary to take into account that as \(k\), \(\rho c_{p}\), and \(h\) form fractions in the heat diffusion model, a scaled version of them will always allow a perfect curve-fitting even if the parameters are not the ones we used to calculate the numerical solution. That is why it is preferable to have an initial guess whose components (or individual parameters) do not have a scaling factor in common. For instance, if \(p_{t}\) is the actual value toward which the LM algorithm must converge, \(p_{init}\) cannot be \(1.2p_{t}\). Instead, we should multiply every individual component by a different factor i.e., \(p_{init}=(1.2p_{t,1},0.9p_{t,1},0.6p_{t,1})\). These kinds of initial guesses prevent us from obtaining, at least in the first iteration, a valid solution but far from the actual one.