Fault Offsets

Tobias Stephan

2026-11-02

This tutorial demonstrates how you can derive displacement components from fault slip.

The offset along a fault can be factorized into several components (Figure 1).

Figure 1: Graphic illustration of displacement components along a fault. $\mathbb{G}$ is the geographic reference frame with D = Down, E = East, N = North.

Get different components with trigonometry

Fault orientation (dip angle, dip direction), shortening direction, and horizontal throw

Knowing the horizontal throw (e.g. from plate motion parameters), the remaining components of the displacements along a given fault are as follows. The angle $\delta$ between the direction of horizontal displacement (here assumed parallel to $e_3$) and the fault strike is

$$\delta = |e_3 - (\text{dip direction} + 90^{\circ})|$$

Slip components in the horizontal plane:

$$\begin{aligned} f_\text{strike slip} &= |\cos{\delta} \cdot f_\text{horizontal throw}|\\ f_\text{heave} &= \sqrt{f_\text{horizontal throw}^2 - f_\text{strike slip}^2} \end{aligned}$$

Slip components in the vertical plane perpendicular to the strike of the fault:

$$\begin{aligned} f_\text{dip slip} &= \frac{f_\text{heave}}{\cos{(\text{dip})}}\\ f_\text{vertical throw} &= \sqrt{f_\text{dip slip}^2 - f_\text{heave}^2} \end{aligned}$$

Slip components in the fault plane:

$$\begin{aligned} f_\text{net slip} &= \sqrt{f_\text{strike slip}^2 + f_\text{dip slip}^2}\\ \text{rake} &= \arctan{\left(\frac{f_\text{dip slip}}{f_\text{strike slip}}\right)} \end{aligned}$$

Thus, the rake angle describes the ratio between the dip slip and the strike slip component.

Knowing the vertical throw (e.g. from thermochronology or petrology), the (assumed) fault dip, and the direction and amount of horizontal offset, the strike of the fault is as follows:

$$\begin{aligned} f_\text{heave} &= \frac{f_\text{vertical throw}}{\tan{(\text{dip})}}\\ \delta &= \arcsin{\left(\frac{f_\text{heave}}{f_\text{horizontal throw}}\right)}\\ \text{strike} &= |e_3 - \delta| \end{aligned}$$

Knowing the vertical throw (e.g. from thermochronology or petrology), the fault strike (e.g. from geomorphology), and the direction and amount of horizontal offset, the dip of the fault is as follows:

$$\begin{aligned} \delta &= |e_\text{3} - \text{strike}|\\ f_\text{heave} &= f_\text{horizontal throw} \cdot \sin{\delta}\\ \text{dip} &= \arctan{\left(\frac{f_\text{vertical throw}}{f_\text{heave}}\right)} \end{aligned}$$

Knowing the vertical throw (e.g. from thermochronology or petrology) and the fault’s dip and rake, the heave, the strike slip, the horizontal throw, and the net slip are as follows:

$$\begin{aligned} f_\text{dip slip} &= \frac{f_\text{vertical throw}}{\sin{(\text{dip})}}\\ f_\text{heave} &= \frac{f_\text{vertical throw}}{\tan{(\text{dip})}}\\ f_\text{strike slip} &= \frac{f_\text{dip slip}}{\tan{(\text{rake})}}\\ f_\text{horizontal throw} &= \sqrt{f_\text{strike slip}^2 + f_\text{heave}^2}\\ f_\text{net slip} &= \frac{f_\text{dip slip}}{\sin{(\text{rake})}} \end{aligned}$$

Fault displacement tensors

Each fault component is a vector describing its direction and length. For instance, the vector of the strike slip is:

$$\vec{f}_\text{strike slip} = \begin{pmatrix} \lVert f_\text{strike slip}\rVert \\ 0 \\ 0 \end{pmatrix}$$

Net slip vector

The net slip vector is the sum of the strike slip, heave, and vertical throw vectors:

$$\vec{f}_\text{net} = \begin{pmatrix} \lVert f_\text{strike slip}\rVert \\ \lVert f_\text{heave}\rVert \\ \lVert f_\text{vertical throw}\rVert \end{pmatrix}$$

fault_displacements(strikeslip = 2, verticalthrow = -5, heave = 3)

Principal displacement tensor

The eigenvalues of $\mathsf{F}_{ij}$ are the magnitudes of the heave, strike slip, and vertical throw components, and the corresponding eigenvectors $\{ \vec{f}_1, \vec{f}_2, \vec{f}_3 \}$ are mutually orthogonal unit vectors along these directions. In this principal frame the tensor is diagonal, i.e. the principal displacement tensor $\mathsf{F}_\mathbb{F}$:

$$\mathsf{F}_{\mathbb{F}} = \begin{bmatrix} \lVert f_\text{heave}\rVert & 0 & 0\\ 0 & \lVert f_\text{strike slip}\rVert & 0\\ 0 & 0 & \lVert f_\text{vertical throw}\rVert\end{bmatrix}$$

The tensor $\mathsf{F}$ can also be defined by the magnitudes of the fault displacements:

Fu <- displacement_tensor(s = 2, v = -5, h = 3)
print(Fu)

     [,1] [,2] [,3]
[1,]    3    0    0
[2,]    0    2    0
[3,]    0    0   -5
attr(,"class")
[1] "matrix"  "array"   "ftensor"

Orientation tensor

The fault orientation tensor is defined by the fault plane’s orientation (dip direction and dip angle) and the fault’s slip (direction and magnitude):

$$\mathsf{F}_{ij} = \begin{bmatrix}f_{11} & f_{12} & f_{13}\\ f_{21} & f_{22} & f_{23}\\ f_{31} & f_{32} & f_{33}\end{bmatrix}$$

Fg <- displacement_tensor(s = 2, v = -5, h = 3, dip_direction = 45)
print(Fg)

        [,1]      [,2] [,3]
[1,] 2.12132  1.414214    0
[2,] 2.12132 -1.414214    0
[3,] 0.00000  0.000000   -5
attr(,"class")
[1] "matrix"  "array"   "ftensor"

From principal displacement tensor to orientation tensor

Translate the origin of $\mathsf{F}_\mathbb{F}$ to the point of measurement and rotate it into the fault orientation, giving $\mathsf{F}_\mathbb{FG}$:

displacement_tensor_decomposition(Fg, dip_direction = 45)

$displacements
          dip    delta     rake verticalthrow horizontalthrow heave  dipslip
[1,] 59.03624 56.30993 71.06818            -5        3.605551     3 5.830952
     strikeslip  netslip
[1,]          2 6.164414

$fault
Fault object (n = 1):
dip_direction           dip       azimuth        plunge         sense
     45.00000      59.03624     258.69007     -54.20424      -1.00000

$strain_tensor
     [,1] [,2] [,3]
[1,]  3.0 -1.5 -1.5
[2,] -1.0  1.0 -1.0
[3,]  2.5  2.5 15.0

$volumetric_strain
[1] 19

$shear_strain
[1] 5.196152