## 2 Materials and Methods
### The Spectral Induced Polarization Method  
The induced polarization (IP) imaging method, also known as complex resistivity (or by its inverse the complex conductivity, CC) imaging, or electrical impedance tomography, is an extension of the electrical resistivity tomography (ERT) method. It is based on four-electrode measurements, where two electrodes are used to inject current (current dipole), and another pair of electrodes is used to measure the resulting voltage (potential dipole). Modern devices can simultaneously measure tens of potential dipoles for a given current dipole, permitting data collection with high spatial resolution and reduced acquisition times [36]. IP data can be collected in the time domain (TDIP) or the frequency domain (FD). Most commercial instruments work in TDIP, using direct current, with measurements of the transfer resistance during the current injection. At the same time, the integral chargeability is computed from voltage measurements collected after shutting off the current to record the decay of the secondary voltage [22]. In FDIP, an alternating current is injected into the ground to measure the resulting electrical impedance in terms of its magnitude (i.e., transfer resistances) and the phase shift between the periodic voltage and current signals. FDIP measurements can be conducted at different frequencies using the so-called spectral IP (SIP) method to gain information about the frequency dependence of the electrical properties. Recent studies have proposed using TDIP measurements to record the full waveform to gain information about the frequency dependence [37; 38]. Nevertheless, we will only investigate SIP signatures collected from FDIP in this study. Further details of the IP method can be found in the books by [PERSON] and [PERSON] or [PERSON], as well as the studies by [PERSON] and [PERSON] and [PERSON] [22; 39; 40; 41].  
IP imaging measurements (TDIP or FDIP) can be inverted to solve the spatial (i.e., lateral and vertical) changes of the complex resistivity in the subsurface. The complex conductivity (\(\sigma^{*}\))--or its inverse, the complex resistivity (\(\rho\), with \(\sigma^{*}=1/\rho\))--can be expressed in terms of its real (or in-phase, \(\sigma^{\prime}\)) and imaginary (or quadrature, \(\sigma^{\prime\prime}\)) components, or by its magnitude (\(\,|\,\sigma\,|\,\)) and phase (\(\rho\)), such as  
\[\sigma^{*}=\sigma^{\prime}+i\sigma^{\prime\prime}=|\sigma|e^{i\varphi} \tag{1}\]  
where \(i=\sqrt{-1}\). The real part of the complex conductivity (\(\sigma^{\prime}\)) is related to the ohmic conduction, while the imaginary part (\(\sigma^{\prime\prime}\)), is related to the polarization.  
We use the Cole-Cole type model as proposed by [PERSON] et al. to describe the frequency dependence of the complex resistivity in SIP measurements. The Pelton model (PM) can be expressed in terms of the electrical resistivity by [20]:  
\[\rho^{*}=\rho_{0}\bigg{[}1-M\bigg{(}1-\frac{1}{1+\left(i\omega\tau\right)^{c} }\bigg{)}\bigg{]} \tag{2}\]  
where \(M\) is the chargeability (describing the amplitude of the polarization), \(\rho_{0}\) the low-frequency limit of the resistivity, c is the dispersion coefficient and \(\tau=\frac{1}{2\pi/f_{c}}\frac{1}{(1-M)^{\frac{1}{2}}}\) is the relaxation time, with \(f_{c}\) being the characteristic frequency at which the maximum of the imaginary part of the complex resistivity is observed. The ratio of the chargeability and the \(\rho_{0}\) gives the normalized chargeability:  
\[\text{M}_{n}=\frac{M}{\rho_{0}} \tag{3}\]We obtained the fit of the PM to our measured spectra (field and laboratory data) using routines developed by [PERSON], which are part of the Python geophysical imaging and modeling library pyGIMLi [42].