When fault-slip P and T axes average to nothing

Fault-slip analysis has an unusual property among the methods in this series. The others tell you when they are unhappy. Fit a great circle to a cluster of poles instead of a girdle and the fit is visibly bad. Ask for a mean direction from a bimodal population and the confidence cone blows out. Kinematic axes do not behave that way. Give the moment tensor one fault with one striation and one sense and it returns a P axis and a T axis, cleanly, to a tenth of a degree. Give it four hundred faults from four hundred kilometres and unrelated tectonic phases and it returns a P axis and a T axis, just as cleanly.

So the interesting question is not how to compute the axes. It is how to tell a real answer from an arithmetic one. This walks through both on the same dataset, because the dataset happens to contain one of each.

The data

Köhler and colleagues published a structural survey of northern Bavaria, 1,660 records from 48 outcrops, openly licensed through PANGAEA under CC BY 4.0. Of those, 470 are fault planes carrying both a striation and a determined sense of movement, which is what kinematic analysis needs and what field datasets usually lack.

Those 470 are not evenly spread. Seven outcrops supply about two thirds of them, because those are the outcrops with good exposure and a long afternoon. Pool everything and you have weighted the regional answer by where the fieldwork was comfortable. So the file used here holds one measurement per station, 34 of them, each chosen as the measurement whose own axes sit closest to the average of everything else at that station. One station, one vote.

The mix of movement senses says the same thing more bluntly. Sixteen normal, nine reverse, five dextral and four sinistral. A region under one stress state for one interval does not usually produce that spread.

Computing the axes

Each fault becomes a P, B and T triple through the moment tensor: build the tensor from the fault pole and the slip vector, take its eigenvectors, and the largest and smallest give T and P. This is the Marrett and Allmendinger construction, and it is worth being precise about what it produces. P and T are the axes of maximum shortening and maximum extension for that fault. They are geometry, not stress. A P axis is not σ₁, and treating it as one is the most common way to overstate a fault-slip result.

The regional answer, and why to distrust it

Averaging the 34 P axes gives a mean shortening direction of 056/73, and the eigenvector version of the same question puts it at 050/83. Steeply plunging shortening, which means horizontal extension: a clean extensional result, the kind of number that ends up in a table.

Before it does, plot the individual axes and look at them. That is the step the calculation cannot do for you, and the one worth building a habit around. The 34 P axes and 34 T axes go on a net of their own and the orientation tensor describes their shape.

They have almost no shape. The three eigenvalues of the P axes come out at 0.44, 0.33 and 0.22, where a perfectly random set of axes gives 0.33, 0.33 and 0.33. Woodcock's K is 0.69, on the girdle side of the boundary at 1 rather than the cluster side. Vollmer's random index is 0.67, so two thirds of the fabric is scatter. The confidence cone on the mean is 21.6 degrees wide, which is less a direction than a hemisphere with an opinion. The T axes tell the same story.

So 056/73 is the average of a distribution barely distinguishable from noise. It is not wrong arithmetic. It is a real mean of a population that has no mean worth reporting, and nothing about the number itself would have told you.

A second check makes the same point from a different direction. Go back to the full archive, average all 470 raw measurements rather than one per station, and the shortening axis moves to 004/26. That is not a small revision of the first result, it is a different structural claim. Two defensible reductions of one dataset disagreeing that completely is diagnostic. When a population has real structure, how you weight it changes the answer by a few degrees.

The same method, one outcrop

Kirchleus is the largest single exposure in the survey, with 74 usable fault-slip pairs from one outcrop. Run exactly the same workflow on it alone.

Now the axes have a shape. The leading eigenvalue rises from 0.44 to 0.61, the Vollmer point index more than triples from 0.11 to 0.38, and Woodcock's K goes from 0.69 to 3.39, across the boundary from girdle to firm cluster. The confidence cone halves, from 21.6 degrees to 12.3. The principal shortening axis sits at 013/04, horizontal and pointing very nearly due north.

One caveat about reading those panels, since it is visible in the figure below and would otherwise look like a contradiction. Both classify as uniform. That label is a three-way vote between the point, girdle and random indices, and the random index is three times the smallest eigenvalue, so it stays above 0.5 until that eigenvalue drops below 0.167. At Kirchleus it is 0.1671. The label loses by a thousandth. Read the continuous indices rather than the bucket they fall into, which is good practice anywhere the boundary is a convention rather than a measurement.

Checking it against the fault geometry

A shortening axis derived from slip vectors can be tested against something it did not use, namely which faults moved which way.

At Kirchleus the sinistral faults strike about 040 and the dextral faults about 150. Both sets are near vertical, both carry horizontal striations, and the acute angle between them is bisected by an axis pointing roughly 005. That is a conjugate strike-slip pair, and its bisector is an independent estimate of the shortening direction. It lands within about ten degrees of the 013/04 the tensor produced. Separately, the 26 reverse faults strike near 110 with shallow dips, close to perpendicular to that same axis, which is where thrusts belong under north-south shortening.

Three lines of evidence, two of them independent of the tensor arithmetic, all pointing north-south. That is what a result looks like. The regional number had none of this available to it, because there was no coherent geometry to check against.

What is left over

Eleven normal faults at Kirchleus do not belong to the conjugate pattern. They dip more steeply, their striations plunge steeply rather than lying flat, and no amount of north-south shortening produces them. They are real measurements from the same outcrop recording something else, most likely a separate episode.

This is the ordinary condition of fault-slip data rather than a flaw in this particular outcrop. Faults are reused. A surface that slipped as a thrust can slip again as a normal fault under later extension, and the two striations sit on the same plane. Separating them is its own literature, and the honest first step is noticing that separation is needed. A population that will not cluster is the notification.

Where this could still be wrong

Kinematic axes are not stress axes. P and T describe the shortening and extension accommodated by slip on a surface, which depends on the orientations of the faults that happened to be available as much as on the stress that drove them. Twiss and Unruh set out the distinction more carefully than a paragraph can. Reporting P as a compression direction is defensible; reporting it as σ₁ is a stronger claim than fault-slip data alone support.

The 45-degree assumption inside the moment tensor deserves the same scepticism. P and T are placed at 45 degrees to the fault plane in the slip direction, exact for an isotropic material failing fresh and approximate for reactivated surfaces. Since reactivation is what the leftover normal faults suggest, the Kirchleus axes are well constrained rather than exact.

A striation also has to lie in the plane it was measured on. Many pairs in the source archive miss by a few degrees, which is rounding and field reality rather than error, so the published files here have each striation projected into its fault plane first. Fifteen pairs at Kirchleus missed by more than ten degrees and were dropped, since at that magnitude the record is wrong rather than imprecise.

Data and sources

Two files, both importable into Stereogram Pro in one pass. Set the plane convention to right-hand rule. Every figure and number above can be reproduced from them.