## 3 Model Inversion and Application Case
### Application Case  
Three-dimensional resistivity tomography technology for constrained spaces was used to investigate the foundation of an ancient pagoda in Longyou County, Zhejiang Province. The construction site is shown in Figure 8. The original construction date of the pagoda is unknown, but it was rebuilt during the third year of the Northern Song Dynasty's Jiayou period (1058 AD). The existing structure is the Northern Song pagoda body, with a height of 27.31 m and an octagonal shape. This ancient pagoda is an important heritage site in the history of Chinese Buddhist architecture and one of the cultural heritage  
Figure 7: Sliced images of the inversion results for three electrode arrays: (**a,b**) the horizontal slice at \(Z=-2.5\) m and the vertical slice at Y = 0 m of the “\(\square\)”-shaped electrode array; (**c,d**) the horizontal slice at \(Z=-2.5\) m and the vertical slice at Y = 0 m of the “U”-shaped electrode array; (**e,f**) the horizontal slice at \(Z=-2.5\) m and the vertical slice at Y = 0 m of the “L”-shaped electrode array. sites in the Jiangnan region. Its architectural style and design have had a significant impact on the development of Chinese Buddhist and pagoda architecture, earning it the reputation of being the \"first ancient pagoda in Jiangnan.\" The primary objective of this survey was to determine the scale of the pagoda's foundation, providing a reference for future preservation efforts. The field survey employed a DZD-6 multifunctional DC resistivity instrument produced by the Chongqing Geological Instrument Factory. The electrode array used a \"\(\square\)\"-shaped electrode array, as shown in Figure 9a, with the planar coordinate point (0,0) approximately at the center of the pagoda's foundation. Due to observation constraints, it was not feasible to place electrodes at infinity around the pagoda, so the Wenner-alpha electrode array was used for the observation setup. The electrode spacing was 1 m. In actual measurements, since the instrument's default setting assumes straight survey lines and equal point spacing, the collected resistivity data are inaccurate, and the geometric factor needs to be recalculated according to the real corrdinates of the electrodes to obtain the real apparent resistivity distribution underground. Abnormal data were manually removed due to the complexity in the relative positioning of certain power electrode pairs and measurement electrode pairs caused by the circular layout of the cables, which resulted in very large geometric factor values. In the field, 436 data points were measured, and after discarding the unusable data, 362 valid data points remained, as shown in Figure 9b. Figure 8: Construction site. (**a**) Present situation of the ancient pagoda. (**b**) Survey site layout (taken from north to south). Figure 9: Electrode deployment and interval distribution of measured resistivity data. (**a**) Design of a “\(\square\)”-shaped electrode array for the survey aera. (**b**) Distribution of intervals before and after correction of measured resistivity data. Figure 10 illustrates the variation in measured apparent resistivity before and after the correction process, which involved the removal of outliers. As depicted in the figure, the inherent complexity of surface conditions and electrode placements during power supply measurements in field surveys resulted in anomalies in the measured resistivity data, including negative values. By applying the correction outlined in Equation (21) and taking the absolute value, we obtained valid and accurate measurements of resistivity. The corrected apparent resistivity data were converted into potential differences, followed by a three-dimensional inversion of the potential data, iterated five times in total. Figure 11a presents the convergence curve of the inversion's average mean square error across iterations, revealing a rapid descent in the error curve with increasing iteration count, dropping from 120% to 40%. However, the final error remains at a relatively high level.