Forward stress simulation in a homogeneous stress field


This algorithm calculates the theoretical movements expected on faults under a homogeneous stress field. We assume the movement vectors be parallel to the maximum resolved shear stress acting on a surface, as derived from the Wallace-Bott equation. Fault orientations can be provided by the user or they can be simulated by the program.


Program input

The program accepts a user-provided text file that stores fault orientations as input (see below for data format). Otherwise, the program creates random simulated fault orientations. The user enters additional parameters at the start of the program execution:

  • stress field characteristics (required)
  • locational information (optional)
  • time information (optional)
  • fault data source (required)
  • output file name (required)

We inspect them in greater detail below.

Stress field characteristics

Required parameters.
They are:
  • Sigma1 (i.e., the most compressive stress axis) and sigma 3 trends, plunges and magnitudes (positive: compression, negative: extension). Their values follow the geological convention, e.g., angle from geographical North, measured clockwise, for the trend.
  • Phi value is the stress ratio (sigma 2 - sigma 3)/(sigma 1 - sigma 3) that varies between 0 and 1.

Locational information

When present (or simulated), the used coordinate system can be planar (x-y-z) or spherical (lat-long-z). When simulating faults, also a randomly chosen location can be attributed to these faults, within a chosen spatial domain.

Time information

Optional parameters. A random, uniformly-distributed time tag can be attributed to simulated faults. The user defines the time ranges in float format (e.g., between 0.0 and 1000.0).

Fault data source

Required parameters. The user can use its own data set or let the program simulate a chosen number of randomly oriented faults.
If the user provides an input file, it must have a column format, with commas as field separators. There can be zero, one or more header lines (they will not be considered). The field format in this user-defined file depends on whether locational and/or time data is available. Their required order is the following:

  • fault id (required): integer that uniquely identifies the individual fault datum;
  • x-latitude, (optional): x or latitude coordinate;
  • y-longitude (optional): y or longitude coordinate;
  • z-depth (optional): z coordinate, positive downwards;
  • time (optional): float value representing a time;
  • fault strike, right-hand-rule (required): numeric value, 0°-360°, representing the fault strike according the right-hand rule. Its value follows the geological convention: angle from geographical North, measured clockwise.
  • fault dip angle (required): numeric value, 0°-90°, for the fault dip angle.

Name of output files

Required. It is the prefix of the two output files created by the program. See next paragraph.

Program output

The algorithm creates two textual output files, respectively containing the resulting data and their metadata. The data file stores the result in a tabular format. First line is the field header, and the following lines contain the resulting data. The data file can be imported in GIS, in order to create a new point layer.

Fields in output file

  • Id_rec: numeric id of the fault
  • X or Lat (optional): X (Cartesian coordinates) or Latitude (spherical coordinates) value
  • Y or Lon (optional): Y (Cartesian coordinates) or Longitude (spherical coordinates) value
  • Z (optional): Z value
  • Time (optional): time value
  • Strike_rhr: strike, right-hand rule
  • Dip_angle: dip angle
  • Th_rake: theoretical rake (Aki and Richards, 1980 convention, see figure below)
  • Th_trend_slick: trend of the maximum resolved shear stress
  • Th_plunge_slick: plunge of the maximum resolved shear stress (positive values: downwards movements; negative values: upwards movements)
  • Sigma_Magn: magnitude of the traction vector acting on the fault plane
  • SigmaN_Magn: magnitude of the normal component of the traction vector
  • SigmaT_Magn: magnitude of the tangential component of the traction vector
  • Mod_SlipTendency: "normalized slip tendency", equal to the shear stress magnitude divided by the traction stress vector magnitude (see Alberti, 2010). Its range is from 0 for a traction vector normal to the fault, to 1, for a traction vector parallel to the fault
  • Deformation_Index: "deformation index" in Alberti (2010). Its value is equal to (|t|-tau)/t - where t is the magnitude of the traction vector (negative when compressive) and tau is the shear stress magnitude. Its range is from -1 to 1: compressed surfaces have a value of -1, sheared surfaces of 0, and surfaces under tension of 1.

The figure below illustrates the Aki and Richards (1980) convention for rake angle. Reverse movements have positive rake values, while normal movements have negative rakes.

Case study

We simulated a test data set with 100 randomly-oriented faults under a stress field corresponding to a transcurrent stress regime, with N-S horizontal sigma 1 and E-W horizontal sigma 3. Used stress parameters are:

  • sigma 1 trend and plunge: 0 0
  • sigma 3 trend and plunge: 90 0
  • sigma 1 magnitude: 1
  • sigma 3 magnitude: -1
  • stress ratio: 0.5

We show the result in the stereonet below (created using the program Multiple Inverse Method software Package by Yamaji et al.). Red colors indicate reverse faults, blue ones the normal faults.

As we can see in the tangent lineation diagram (figure below, created with the program Stereonet for Fault-Striation Analysis by Yamajj), the tangent lineations departs from the sigma 1 axis (compressive) and they point towards the sigma 3 axis (extensional).

We inverted the fault data set of simulated fault movements with the program Multiple Inverse Method software Package by Yamaji et alii. The inverted stress field corresponds well with the simulated one (left: inverted sigma 1, right: inverted sigma 3).

Related article

Aki, K., Richards, P.G., 1980. Quantitative seismology. Theory and Methods. Vol. I, W.H. Freeman and Company, San Francisco, 557pp.

Alberti, M., 2010. Analysis of kinematic correlations in faults and focal mechanisms with GIS and Fortran programs. Computers & Geosciences 36, 186-194.


Mauro Alberti