## 1 Introduction  
In the domain of material and microelectronic device characterizations, many groups and studies have emerged with the goal to estimate both the thermal properties and resistance of these thin multilayers. One of the most common procedures is based on the 3\(\omega\) technique [1], which has been extensively used since it offers absolute measurements of the heat flux and the temperature and is also well suited for characterizations at low temperatures. At high temperatures, contactless photothermal methods, such as the thermoreflectance [2; 3] and infrared radiometry [4; 5; 6] techniques, have been implemented and allow measuring the relative change in temperature and heat flux. Nevertheless, calibration is a complex task within those experimental configurations. In this community, very few studies are based on field characterizations. The notion of fields introduces the requirement to propose new methods for which the computation time is reasonable, passing from only one time vector, to find the inverse of a matrix. On the other hand, within the research groups involved in quantitative infrared thermography, many authors have developed a technique for qualitative in-depth defect detection by using flash thermography and thermal signal reconstruction (TSR) [7; 8] or the lock-in technique [9]. Few authors, e.g., [PERSON] et al. [10] and [PERSON] et al. [11], have linked the TSR method with quantitative estimation of the defect depth. The same kind of studies have been performed by [PERSON] et al. [12] with a technique based on pulsed phase transformation (PPT) and thermographic signal reconstruction similar to [13]. As depicted in [14] one alternative to these qualitative methods is the use of parametrized 1D simulations with the quadrupole method as presented in the book of [PERSON] et al. [15]. Some works were performed by coupling infrared thermography (IRT) and the quadrupole approach, such as the one of [PERSON] et al. [16; 17; 18]. In fact, with these analytical methods, it becomes easy to calculate a multilayer sample with several thermal properties as well as thermal resistance. Then, by using nonlinear regression methods as proposed by [PERSON] et al. [14], the parameters of the model can be adjusted by least square minimization between the model output and the experimental data. This approach is difficult when the initial values are poorly known and could be time consuming when addressing a large amount of data, such as temperature fields, with IRT. In the publication of [PERSON] et al. [19], artificial delaminations were manufactured by introducing PTFE (Teflon) inlays in carbon-fiber-reinforced plastics, and they attempted to reduce the inversion by using singular value decomposition (SVD) [20; 21; 22; 23]. Nevertheless, in this challenging problem, the complete inverse processing of a multilayer analytical model is difficult due to the lack of sensitivity of some parameters (layer thickness, depth of thermal resistance, etc.) and the expansive computational iterative processing. For these reasons, the proposed strategy is to use a simple multilayer problem where only one resistive layer is estimated. Moreover, to simplify the inverse processing often based on iterative methods, an asymptotic development method is proposed. The main novelty comes from the calibrating method between the analytical calculation of a theoretical base and the estimated parameters of the asymptotic method. Then with the knowledge of this calibration curve, the estimated parameters based on the measured one can be turn to quantitative ones.  
Finally, in this work, a well-known flash setup is presented. A method is developed to model the heat transfer within the multilayer sample by using the quadrupole formulation. Then, in the inverse processing section, the autoregressive asymptotic development method is presented as well as the calibration procedure. Finally, the results and discussion section presents the characterization of a standard sample as a validation of the method.