# A New Calibration Method for Broadband Complex Resistivity Measurement System of Rocks and Ores
## 1 Introduction  
The complex resistivity (CR) method, also known as spectral-induced polarization (SIP), is a geophysical technique used to describe the polarization characteristics of rocks. This is done by testing the impedance spectra of various samples under the influence of an external electric field [1]. Complex resistivity data consist of two components: the real part, which reflects the conductivity of charge flow within the sample, and the imaginary part,which is associated with the accumulation of polarized charges on the internal and surface boundaries of the sample [2]. Due to the strong induced polarization (IP) effects of metallic ones under an external electric field, the CR method is commonly used in mineral exploration and development [3,4]. Typically, measurement frequencies for this method are below 1 kHz [3,5]. As measurement techniques and theoretical understanding have improved, the method has been widely applied in various fields. These include engineering geophysics [6,7], hydrogeological surveys [8,9], biomedical engineering [10], and plant disease and pest control [11,12]. Measurement frequencies have expanded from the traditional kHz range to hundreds of kHz. Although the physical mechanisms behind CR-induced polarization are not fully understood, several theories have gained traction. Electrochemical polarization (or double-layer polarization) occurs at the interface between solid particles and pore fluids. This is attributed to reversible electrochemical diffusion processes, mainly at low frequencies [13,14,15]. Interfacial polarization (or Maxwell-Wagner polarization) dominates at higher frequencies. It describes the dielectric properties of rocks [2]. Despite the incomplete understanding of polarization mechanisms, observed polarization effects still reveal key relationships between the physical properties, material composition, and structural characteristics of samples. In some cases, these effects can be weak. For example, phase shifts in non-mineralized sedimentary samples may only be a few millivolts, which makes them difficult to detect with traditional measurement systems [16,17]. To capture the actual polarization characteristics of a sample, both the testing instrument and measurement apparatus must meet stringent requirements. At high frequencies (>1 kHz), studies have shown that the distributed capacitance in the instrument's acquisition channels can lead to significant phase errors [18,19,20]. Therefore, calibration of high-frequency data is essential before use. In addition to internal circuit design and material selection, the design of the measurement apparatus also plays a crucial role [21]. Polarization effects occur at low frequencies with measurement electrodes [22,23], electromagnetic coupling interference exists between measurement wires [24,25], and contact impedance from structural design [26] can affect measurement accuracy. Even when using the same apparatus for different sample types (e.g., rock samples vs. solutions), varying error responses can occur. This makes error analysis for the CR measurement system essential. Given these testing challenges, increasing attention has been given to developing high-precision data calibration methods. [PERSON] et al. [27] designed an impedance analyzer and analyzed the effects of the instrument's acquisition channel distributed capacitance, and the material and positioning of the measurement electrodes on test data using sampling resistors below 1 k\(\Omega\). This led to accurate CR measurements across the mHz to kHz frequency range. Building on this work, [PERSON] et al. [28] improved calibration accuracy by using reverse current and potential electrodes to precisely estimate electrode impedance. [PERSON] and [PERSON] [29] developed an error estimation model using the error term \(\omega\)C\({}_{\text{in}}\)Z\({}_{\text{x}}\) to approximate the impact of distributed capacitance in the acquisition channels.