## 1 Introduction  
Rainfall-induced landslides are major geologic disasters that cause significant losses to people's lives and properties. The real-time assessment of landslides is crucial for disaster prevention and mitigation. Various techniques have been developed for landslide assessment. Traditional methods rely on geological surveys [1; 2; 3]. However, observations at discrete sites are restrictive in temporal and spatial scales and cannot accurately monitor rainfall-induced landslides. In recent decades, remote sensing technology has been extensively applied for landslide assessment. Synthetic Aperture Radar (SAR) with numerous operational systems onboard satellites [4; 5; 6] and aircraft [7; 8] has become an effective and valuable tool for mapping and quantifying geomorphological changes. It achieves high spatial resolution and provides all-weather, all-time observations, making it suitable for assessment rainfall-induced landslides. However, due to high costs, remote sensing technology is not widely used in many mountainous areas [9; 10]. More importantly, almost all of these technologies are restricted to surface observations and cannot monitor subsurface movements of landslides, particularly changes in soil moisture during the formation of the slide face. As a high-efficiency and low-cost nondestructive detection technology, the geophysical direct resistivity (DC) method offers a new approach for assessing rainfall-induced landslides [11; 12]. It involves injecting an electrical current into the ground through a pair of electrodes and measuring the resulting potential differences with other pairs of electrodes. The data collected are used to create a subsurface resistivity distribution, which can indicate different geological and hydrological conditions [13; 14; 15]. Rainfall generally increases the water content of underground soil, leading to changes in soil resistivity. Studying these resistivity changes is useful for tracking the sliding surface and the movement of groundwater [16; 17]. This information aids in the risk analysis of natural hazards caused by landslides. Landslides generally occur in mountainous areas; traditional ground geophysical surveys often require the establishment of a manual survey grid, which is a laborious field task with low efficiency and high cost. In mountainous and forested areas with poor line of sight conditions, few national control points that are far from the work area make it nearly impossible to determine the coordinates of electric observation points using traditional theodolites [18; 19; 20]. Additionally, when the terrain is highly undulating and complex, interpreting apparent resistivity data can lead to erroneous results. Many scholars have conducted research on how to mitigate the effects of undulating terrain. A common approach is to use numerical simulations to apply topographic corrections to observational data by accounting for the pure topographic response [21; 22]. Topographic correction methods have a certain effect on mitigating the impact of terrain. However, they can only completely eliminate the influence of topography when the subsurface electrical properties are homogeneous. To overcome this issue, inversion with topography is essential. The arbitrary dipole-dipole configuration offers significant advantages in terms of flexibility, resolution, efficiency, and cost-effectiveness. Its ability to adapt to complex terrains and specific monitoring needs makes it a valuable tool for a wide range of geophysical applications, particularly in the assessment and prevention of geological hazards like rainfall-induced landslides [23]. In this paper, arbitrarily placed electrodes are combined with unstructured finite element methods to achieve the incomplete Gauss-Newton inversion of arbitrary dipole-dipole apparent resistivity observation data under arbitrary terrain conditions (flat or undulating). This strategy enables the effective handling of measurement challenges posed by complex terrain, thereby enhancing the accuracy of landslide surface detection.