Stress Inversion

A stress inversion finds the stress state that best explains a set of faults and the directions they slipped. It works out the principal stress axes and the stress ratio, with their confidence from a bootstrap, and adds the result as a new stress tensor. The faults come from a CSV file, or from fault planes and slickenline lineations measured in VRGS.

Geomechanics module

Stress inversion needs the Geomechanics module on your licence.

From a CSV file

One row per fault, with its dip direction, dip and rake, and optionally a weight:

dip_direction,dip,rake,weight
120,60,-85,1
305,55,-95,1
210,70,10,0.5

• Columns are found by their header, ignoring case: the dip direction is a header containing direction or azimuth, or dipdir or dd, or failing those a header containing strike, read by the right-hand rule (dip direction = strike + 90°); the dip a header starting dip; the rake a header containing rake; the weight a header containing weight. A file with no header row is read as dip direction, dip and rake, in that order.
• Dip is 0–90°.
• Rake follows Aki & Richards: the angle in the fault plane from the strike (the dip direction less 90°) to the slip, positive upward. It gives the motion of the hanging wall: −90 normal, +90 reverse, 0 left-lateral, 180 right-lateral.
• Rows that cannot be read are skipped, and the messages panel counts them.

To run it, right-click Stress Tensors (or any tensor) in the Collections tree, choose Invert from Fault Slip (CSV)..., pick the file and choose the method.

From measurements

A fault measured in VRGS is a plane (a Fault, or any plane type) and the slickenline lineations measured on it. Invert Stress from Selected Faults... on the Orientation menu pairs them up:

1. Give each lineation, or its plane, a Sense attribute (text) recording the hanging wall's motion: N or normal, R, reverse or thrust, D, dextral, right or RL, S, sinistral, left or LL. A lineation's own sense wins over its plane's. Slip Sense is read too.
2. Select the planes and the lineations, or the groups that hold them, anywhere in the tree.
3. Choose Invert Stress from Selected Faults..., and in the dialog the method and how a lineation finds its plane:

Field	Description
Method	Least angular misfit (Angelier 1979) (default) or Linear least squares (Michael 1984). See Method.
Lineation to plane, at most (m)	A lineation goes with the nearest plane within this distance that it lies in. 0 for any distance. Default 5 m.
Off its plane, at most (deg)	How far a lineation may lie out of its plane; it is projected into the plane. Default 15°.

Each lineation becomes a fault: its plane's normal and the lineation, pointed by the sense — down the dip for normal, up it for reverse, against the strike for dextral, along it for sinistral. The messages panel says how many lineations were paired, and why the others were not: no plane near enough, none it lies in, no sense, or a sense the lineation cannot show (a normal or reverse sense on a lineation within 10° of the strike, a dextral or sinistral one within 10° of the dip, or a plane within 5° of horizontal). At least four faults are needed.

The result

The messages panel gives the method, the number of faults, the plunge and trend of s1, s2 and s3, the stress ratio φ with its 95% interval, how far the bootstrap axes of s1 and s3 spread, and the mean misfit:

Stress inversion of N faults (method): s1 plunge/trend, s2 …, s3 … (plunge/trend), phi x (95% a-b); 95% of B bootstrap s1 within d deg, s3 within d deg; mean misfit m deg

A new tensor, Inverted (N faults), is added to Stress Tensors and selected. It is not made active: check it, then use Set as Active if you want the stress analyses to use it. Its page has an Inversion group: the method, the faults, φ and its interval, the 95% cones and the mean misfit.

On the stereonet, with Stress Axes on and the tensor chosen, the bootstrap axes of σ1 and σ3 are drawn as faint red and blue dots about the axis markers, and the key gives their count and the 95% cones. A tight cloud is a well-constrained axis; a cloud smeared along a girdle says the faults cannot tell those axes apart.

Axes and ratio only

An inversion of slip directions resolves the orientation of the stress and its shape, not its size. The new tensor takes the magnitudes of σ1 and σ3, the failure parameters (friction coefficient, pore pressure and cohesion) and the depth model from the active tensor, and sets σ2 = σ3 + φ (σ1 − σ3). Give it real magnitudes before you use it for slip tendency, fracture susceptibility or the Mohr diagram.

Method

Assumptions. Each fault slipped in the direction of the resolved shear stress on it (the Wallace–Bott hypothesis), and the stress was uniform across the faults.

Least angular misfit (Angelier 1979). The reduced stress tensor has four parameters: the direction of s1, the turn of s3 about it, and φ. The search finds the four that minimise

$$\sum_i w_i \,(1 - \cos\alpha_i),$$

where α_i is the angle between fault i's slip and the slip the tensor drives on it. It starts from a global grid — s1 about every 5°, s3 every 10°, φ every 0.1 — and from Michael's solution, and refines the best few by a pattern search to a hundredth of a degree. It makes no assumption about how large the shear stress is on each fault, so exact slips give the exact tensor. It is slower than Michael's method; the classic inverse of Vollmer's Orient is the same idea.

Linear least squares (Michael 1984). If, in addition, the shear stress has about the same magnitude on every fault, the slip direction s on a fault with normal n is a linear function of the deviatoric stress tensor σ:

$$\boldsymbol\sigma\mathbf n - (\mathbf n\cdot\boldsymbol\sigma\mathbf n)\,\mathbf n = \tau\,\mathbf s .$$

With the trace of σ set to zero and the shear magnitude τ fixed, each fault gives three equations in the five unknowns of σ. The stacked equations, weighted by each fault's weight, are solved by least squares. The equal-shear assumption is not exactly true, so even perfect data from faults of every orientation come back a few degrees off — about 2–3° in VRGS's tests, where the angular misfit search is within a few hundredths of a degree.

Outputs.

• The principal axes: s1 the most compressive, s2, s3.
• The stress ratio φ = (σ2 − σ3) / (σ1 − σ3), from 0 to 1.
• Each fault's misfit: the angle between its observed slip and the slip the result drives on it. The mean misfit says how well one stress explains the faults.

Confidence. The faults are resampled with replacement, with a fixed seed, and inverted again: 500 times for Michael's method, 200 for the angular misfit (each refined from the best solution). The result reports the angle about s1, and about s3, within which 95% of the resampled axes lie, and the 2.5–97.5% interval of φ.

Limitations

• One uniform stress is fitted to all the faults. Faults from more than one deformation event need separating first; a large mean misfit is the warning sign.
• Measurements are paired by position: a lineation measured far from its plane needs a larger distance, and closely spaced faults a smaller one.
• Angelier's (1990) direct inversion is not offered; the angular misfit search does what it is usually wanted for.

References

1. Aki, K., & Richards, P. G. (1980). Quantitative Seismology. Freeman, San Francisco.
2. Angelier, J. (1979). Determination of the mean principal directions of stresses for a given fault population. Tectonophysics, 56, T17–T26.
3. Bott, M. H. P. (1959). The mechanics of oblique slip faulting. Geological Magazine, 96, 109–117.
4. Michael, A. J. (1984). Determination of stress from slip data: faults and folds. Journal of Geophysical Research, 89(B13), 11517–11526.
5. Wallace, R. E. (1951). Geometry of shearing stress and relation to faulting. Journal of Geology, 59, 118–130.

See also

• Stress Tensors — the tensors and the active one.
• Mohr Diagram
• Stereonet user guide — the Stress Axes overlay.
• Geomechanics