## 2 ULMA-Based borehole PEC Signal Model  
Consider a ULMA-based borehole PEC system equipped with a single transmitter and \(M\) receivers. The receivers, each composed of a coaxial coil wound around a soft magnetic core in a cylindrically layered medium, are separated by \(\Delta z\). The electrical and geometrical parameters of each layer in the multilayered borehole PEC system are denoted as \(\epsilon_{j}\), \(\mu_{j}\), \(\sigma_{j}\), and \(r_{j}\) (\(j=1,\,2,\ldots\,J\)) (see Figure 1). The magnetic core is considered as the innermost layer. The transmitter and \(M\) receivers, with \(N_{\text{T}}\) and \(N_{\text{R}}\) turns, respectively, are located in the second layer. Note that all receivers have the same number of turns. When \(M\) is large, both \(N_{\text{R}}\) and \(N_{\text{T}}\) can be configured sufficiently small to diminish the TRD of each single receiver. The individual TRDs can then be ignored and approximated as a single value. Moreover, if the diameters of all coils are assumed sufficiently small, the source region can be considered as the second layer alone. The induced electromagnetic force (EMF) of the receiving coils is then related only to the vertical component of the magnetic field of the first layer (the magnetic core). The remaining layers (such as liquid mud, casing, and formation) are regarded as source-free regions. When measuring the thickness of the metal pipes, we further assumed that the electrical parameters of all layers and the inner radius of the metal pipes were fixed. Considering one transmitter and \(M\) receivers with inter-element spacing \(\Delta z\), the induced EMF in the \(m\)-th receiver can be calculated as [21]  
\[U_{m}(t,z_{m},d_{m})=-\frac{\xi\chi}{t_{\text{of}}}\sum_{s=1}^{S}\sum_{q=1}^{Q }\sum_{p=1}^{P}g_{s,q,p}(t,d_{m})v_{p}(z_{m}) \tag{1}\]  
where \(g_{s,\,q,\,p}(t,d_{m})=D_{s}[exp[(-\text{sln}2t_{\text{of}})/t]-1]A_{q}(B_{q}+ 1)A_{p}x_{1}C_{1}I_{0}(x_{1}r)/2s\), \(v_{p}(z_{m})=\cos\left[\lambda_{0}z_{m}(B_{p}+1)/2\right]\), \(\xi=\mu_{1}N_{\text{R}}N_{\text{T}}t_{\text{T}}/\pi\), and \(\chi=\pi r_{1}{}^{2}\lambda_{0}/2\). In Equation (1), \(t\) and \(t_{\text{of}}\) denote the observation and turn-off times, respectively, \(z_{m}\) and \(d_{m}\) are the TRD and thickness of the metal casing with respect to the observation position of the \(m\)-th receiver, respectively, and \(S\) and \(D_{s}\) denote the stage and the integral coefficient of the Gaver-Stehfest inverse Laplace transform, respectively. \(P\) and \(Q\) represent the number of stages of the two Legendre polynomials with quadrature coefficient \(A\) and zero-point \(B\). \(C_{1}\) denotes the reflection coefficient in the innermost layer. \(x_{j}\) and \(\lambda\) are introduced variables satisfying \(x_{j}^{2}=\lambda^{2}-k_{j}^{2}\). \(I_{0}(\cdot)\) is a modified zero-order Bessel function of the first kind. Considering the monotonically decreasing characteristic of modified Bessel functions, we approximated the upper limit of integration of \(\lambda\) as \(\lambda_{0}\). Note that \(g_{s,q,p}(t,d_{m})\) is related to \(s\), \(q\), and \(p\) whereas \(v_{p}(z_{m})\) is only related to \(p\). Accordingly, Equation (1) can be concisely expressed as  
\[U_{m}(t,z_{m},d_{m})=-\frac{\xi\chi}{t_{\text{of}}}\sum_{p=1}^{P}g_{p}(t,d_{m} )v_{p}(z_{m})=-\frac{\xi\chi}{t_{\text{of}}}\textbf{v}(z_{m})\textbf{g}^{T}( t,d_{m}) \tag{2}\]  
Figure 1: ULMA-based borehole PEC system. where \(\mathbf{g}(t,d_{m})=[g_{1}(t,d_{m}),\ldots,g_{p}(t,d_{m}),\ldots,g_{p}(t,d_{m})]\) with elements \(g_{p}(t,d_{m})=\sum_{s=1}^{S}\sum_{q=1}^{Q}g_{s,q,p}\) (\(t\), \(d_{m}\)) and \(\mathbf{v}(z_{m})=[v_{1}(z_{m}),\)\(\ldots,\)\(v_{p}(z_{m}),\)\(\ldots,\)\(v_{p}(z_{m})]\). As the receiver array is much shorter than the distance changes between two neighboring thicknesses, the thicknesses of the metal casing with respect to the observation position of each receiving coil are identical along the borehole axis, namely, \(d_{1}=d_{2}=\ldots=d_{M}=d_{0}\).