Transformation to the principal stress coordinate system

We have seen above that calculations of normal and shear stresses on a plane and stresses in a borehole are simple in the principal stress coordinate system. It is therefore advantageous to transform fault plane normals and borehole coordinates to the principal system before carrying out the calculations. We shall also see that the transformation of a vector from the geographic North, East, down (NEd) coordinate system, $\ensuremath{\mbox{\boldmath ${n}$}}^{NEd}$, to the principal stress system, $\ensuremath{\mbox{\boldmath ${n}$}}^P$, is very simple in the case where the principal stress axes are represented in NEd coordinates. We write the transformation

$$\ensuremath{\mbox{\boldmath${n}$}}^P = \ensuremath{\mbox{\boldmat... ...ath{\mbox{\boldmath${d}$}}\end{pmatrix}\ensuremath{\mbox{\boldmath${n}$}}^{NEd}$$ (29)

where $\ensuremath{\mbox{\boldmath ${\hat{\sigma_i}}$}}$ are unit vectors in the principal stress directions, in the NEd coordinate system, and $\ensuremath{\mbox{\boldmath ${N}$}} = (1,0,0)$, $\ensuremath{\mbox{\boldmath ${E}$}} = (0,1,0)$ and $\ensuremath{\mbox{\boldmath ${d}$}} = (0,0,1)$ are the basis vectors in the NEd system. Writing $\sigma_{iN}$, $\sigma_{iE}$ and $\sigma_{id}$ for the three components of ${\hat{\sigma_i}}$, Equation 29 simplifies to

$$\ensuremath{\mbox{\boldmath${n}$}}^P = \begin{pmatrix}\ensuremath... ...${\hat{\sigma_3}}$}}\cdot\ensuremath{\mbox{\boldmath${n}$}}^{NEd} \end{pmatrix}$$ (30)

Naturally, transforming vectors from the principal system to the NEd system is just as easy, we simply use the transpose of the transformation matrix above. This transformation matrix will be frequently used in the stress tensor inversion scheme for earthquake focal mechanisms described below. The stress tensor is there known in the principal system and we need to transform the planes of the earthquake focal mechanisms to the principal system.