The effect of topography on the state of stress in the crust: Application to the site of the Cajon Pass Scientific Drilling Project

Abstract

This paper describes a generalized approximate method for assessing the effect of topography on the state of stress at depth in the crust and applies it to the site of the Cajon Pass Scientific Drilling Project. While previous studies have addressed idealized topography with regular shapes, we develop a three‐dimensional (3‐D) method for arbitrary surface topography. The analysis leads to four sets of elastic boundary value problems that are solved by convolutional methods based on the Green's functions of Boussinesq's problem and Cerruti's problem. The effect of topography on the state of stress can be divided into two parts by the linear superposition theorem. The first part is the effect on gravitational stress in the crust. In this case the effect of the topography can be expressed by the summation of the zero‐order vertical load and the first‐order horizontal shear at the base of the relief. The second part is the effect of topography on tectonic stress. In this case the effect of topography involves only the first‐order horizontal shear. The total effect of the topography on the state of stress in the crust can be obtained by adding the effect on gravitational stress with the effect on tectonic stress. The stresses caused by a simple case of 3‐D topography are calculated with this approach to serve as illustration. The identity of this study with previous studies (McTigue and Mei, 1981; Savage et al., 1985) for the same two‐dimensional case demonstrates the correctness of this method. In applying this technique to the Cajon Pass Scientific Drilling Project, we considered stresses induced by the topography associated with the San Gabriel and the San Bernadino mountains as a function of depth at the site. The regional topography induces stresses of several megapascals at the drilling site which decrease rapidly with depth. The induced maximum horizontal compressive stress is approximately in the N‐S direction. If we assume that in the absence of topography there is little right‐lateral shear stress acting on the San Andreas fault at depth (a weak fault assumption), the regional topography tends to cause a slight increase of right‐lateral shear on the San Andreas fault at a shallow depth in the crust. On the other hand, if we assume the strong fault model, the topography has no effect on the sense of shear on the San Andreas fault at all. Therefore it is clear that the small amount of left‐lateral shear stress on the San Andreas fault observed in the upper 3.5 km at the Cajon Pass site is not associated with the topographic effect.