# 3D In Situ Stress Estimation by Inverse Analysis of Tectonic Strains
## 1 Introduction  
The magnitude, orientation and distribution of in situ stress are of fundamental importance for not only design and stability assessment in underground rock engineering [1, 2] but also the assessment of the excavation-induced rock behaviors, such as the deformation, disturbed zone, damage, permeability variation, etc. [3, 4, 5]. Field measurement to obtain in situ stress directly is costly and time-consuming. In engineering practice, only a limited number of points in a region are chosen to perform in situ stress measurement [6, 7]. Theoretically, the in situ stress determined at the limited number of points only reflects local stress states near the testing points, and practically, the in situ stress in the region that includes the testing points are estimated by forward geomechanical modeling and inverse analysis [8, 9, 10, 11, 12]. The traditional stress-based and displacement-based approaches are always used in simulating an in situ stress field, however, the selection of the boundary mode is still the key problem for both approaches. In the stress- or displacement-based approach, there are about 14 variables for only the linear boundary mode, and there will be more variables in quadric or other complicated boundary conditions. Nevertheless, the resulting high dimensionality of the optimization process will make it extremely difficult to solve the problem in an efficient way. Furthermore, more complicated expression for stress or displacement boundary mode does not necessarily lead to a better assessment for in situ stress field and it is not taken into account for the connection between the normal and shear stress or displacement on boundaries. To date, there are at least two challenges in determining in situ stress by forward modeling and inverse analysis. The first challenge is the ambiguity and uncertainty of assumptions in the distribution of either tectonic stress or tectonic displacement at theboundary in forward modeling [13; 14; 15; 16]. In forward modeling, either a stress or displacement boundary condition needs to be assumed to obtain the in situ stress. However, for either of the boundary conditions, an assumption of linear distribution or quadratic distribution leads to the different final results of the engineering-scale in situ stress estimation. It is unclear which assumption leads to a more accurate estimation of the engineering-scale in situ stress. The second challenge is the high dimensionality of input and output variables involved in the inverse analysis [17]. Here, inverse analysis is mainly an optimization process with computational efforts growing exponentially with dimensions [18]. In the aforementioned stress- or displacement-based approach, there are about 10 variables for only linear boundary mode, and there will be more variables in quadric or other complicated boundary conditions, whereas there are usually multiple local in situ stress states as multi-objectives, and the resulting high dimensionality of the optimization process will make it extremely difficult to solve the problem in an efficient and accurate way. It is well known that, from a geological point of view, the in situ stress field should be an unsteady stress field and changes with time, however, from the point view of engineering construction, the in situ stress field can be considered as a relatively steady stress field. A large number of practical projects indicate that the region of interest in the project is mostly controlled by a big tectonic stress field caused by modern crustal movement and all kinds of tectonic activities [19; 20], and except for the individual place largely influenced by local stress, the directions of principal stresses in most regions are roughly consistent. Therefore, it can be considered that tectonic movement is relatively uniform on the boundaries of the region of interest and the tectonic stress field can be simulated by optimizing the regional tectonic strain of the boundary.