## 1 Introduction  
Sill emplacement in the shallow upper crust is an important process of the volcanic plumbing system with great significance for the evolution of our planet [1, 2]. Magma propagation is controlled by its interaction with the mechanical behavior of the host rock, such as deformation and fracture [3, 4]. At shallow crustal depths, host rock deformation associated with sill emplacement is generally expressed as overburden uplifting [3]. For mafic sills, which are often characterized by a high aspect ratio (length vs. thickness), the associated host rock deformation is dominated by elastic bending [4, 5, 6]. Fractures can also occur at the periphery of a sill when the sill reaches a critical length [4, 7]. Previous studies suggested that these inclined fractures at the sill tips were caused by tensile stress [4, 8], while more recent research indicates that these fractures are more likely the result of shear failures [3, 9, 10]. These inclined fractures provide pathways for inclined sheets and promote the formation of saucer-shaped sills, which represent fundamental geometries for magma intrusions in shallow sedimentary sequences [11, 12]. In other words, shear failure greatly controls the deflection of flat sills to inclined sheets [9].  
Recent studies such as those from Sierra Negra volcano and Alcedo volcano, Galapagos, showed that shear fractures at the tips of mafic sills can also extend to the surface [13, 14].  
These faults can delimit the uplift area of the host rock and affect the pathway of ascending magma (Figure 1a) [14]. This so-called trapdoor faulting mechanism is considered an end-member type of host rock deformation associated with mafic sill intrusion [15]. Similar faulting has also been observed in analog experiments, in which reverse faults are observed at the terminations of the inclined sheets of saucer-shaped sills (Figure 1b) [16]. As saucer-shaped sills often consist of dolerites in nature [17; 18], it is likely that reverse faulting at the tips of mafic sills is also a common mechanism of host rock deformation in contrast to the elastic bending model. Note that these reverse faults will be hereafter referred to as peripheral faults.  
However, the current understanding of the structure of these peripheral faults is mostly derived from analog modeling, field observations, and theoretical modeling based on field observations [13; 16; 19]. Note that these faults are often not well exposed in outcrops and forward and inverse modeling from field observations largely depends on the initial model, which is inherently subject to non-unique solutions [8; 20]. For example, at Sierra Negra, the fault is often modeled from surface deformation data such as Global Positioning System (GPS), tilt, and Interferometric Synthetic Aperture Radar (InSAR) data [14; 19]. These methods have relatively weak constraints on the fault dip angle. As a matter of fact, a shallower outward dip angle (e.g., 70\({}^{\circ}\)) or a subvertical inward value for the fault in the model will yield similar results [21]. Analog experiments can reproduce the growth of mafic sills and peripheral faults, yet a study from natural examples may help us better understand the geometry of peripheral faults [16].  
A typical example from the Tarim Basin was selected as a case study, where typical saucer-shaped sills and peripheral faults were observed [18; 22]. The seismic reflection data were first interpreted to decipher the structure of peripheral faults. Due to the limitations of the vertical resolution of the seismic reflection data, there were still some errors in determining the dip of the peripheral faults based on seismic data. Therefore, trishear kinematic modeling and finite element limit analysis were used to constrain the detailed geometry and dip angle of the peripheral faults at the sill tips from a kinematic and mechanical perspective, respectively.