Permittivity Measurement in Multi-Phase Heterogeneous Concrete Using Evidential Regression Deep Network and High-Frequency Electromagnetic Waves

Permittivity measurements of concrete materials benefit from the application of high-frequency electromagnetic waves (HF-EMWs), but they still face the problem of being aleatory and exhibit epistemic uncertainty, originating from multi-phase heterogeneous materials and the limited knowledge of HF-EMW propagation. This limitation restricts the precision of non-destructive testing. This study proposes an evidential regression deep network for conducting permittivity measurements with uncertainty quantification. This method first proposes a finite-difference time-domain (FDTD) model with multi-phase heterogeneous concrete materials to simulate HF-EMW propagation in a concrete sample or structure, obtaining the HF-EMW echo that contains aleatory uncertainties owing to the limited knowledge of wave propagation. A U-net-based model is then proposed to denoise an HF-EMW, where the difference between a couple of observed and denoised HF-EMWs characterizes aleatory uncertainty owing to measurement noise. Finally, a Dempster–Shafer theory-based (DST-based) evidential regression network is proposed to compute permittivity, incorporating the quantification of two types of uncertainty using a Gaussian random fuzzy number (GRFN): a type of fuzzy set that has the characteristics of a Gaussian fuzzy number and a Gaussian random variable. An experiment with 1500 samples indicates that the proposed method measures permittivity with a mean square error of 7.50% and a permittivity uncertainty value of 74.70% in four types of concrete materials. Additionally, the proposed method can quantify the uncertainty in permittivity measurements using a GRFN-based belief measurement interval.