Fold axis three ways: pi diagram, lineation, hinges

There is a nice piece of redundancy built into a folded metamorphic terrane. Fit a great circle through the poles of the folded surface and its pole is the fold axis, which is the π-diagram. Average the mineral lineation and you get the fold axis too, but only if the lineation happens to lie parallel to the hinge. Or skip the inference entirely and average the hinges of minor folds that somebody measured in outcrop.

Three measurements of different things. If they agree, the fabric is telling a simple story and you can say what it is. If they disagree, the disagreement is the result. Either way it is worth checking, and it takes about ten minutes.

The dataset, and one trap in it

These are Archean rocks of the Superior Province, mapped by Hemstad, Southwick and Ojakangas for the Minnesota Geological Survey at 1:50,000 and digitised by the National Park Service. The extract holds 1,237 foliation planes, 796 mineral lineations and 78 minor fold axes across the park, along with the unit and age each was mapped in.

It also holds 40 glacial striations, and those are the trap. They are linear features sitting in the same map layer as the tectonic lineations, with a trend and no plunge because they are scratches on a horizontal rock surface. They record Pleistocene ice flow, not Archean deformation. Average them in with the mineral lineation and you have quietly mixed two datasets separated by two and a half billion years. They import as their own feature type here, so leaving them out is a decision you make rather than one you forget.

The foliation is folded

Plot poles to the 1,237 foliations, contour them, and the distribution is a girdle: a broad band running across the net rather than a single maximum. That is the signature of a surface that has been folded, and the orientation tensor puts a number on it. Woodcock's K comes out at 0.58, comfortably on the girdle side of the boundary at 1, with a fabric strength that says the girdle is well developed rather than a smear.

Fitting the great circle gives a π-circle of 164/62 and an axis of 074/28. Worth noting what the alternative would have given: taking a simple mean of the same poles returns a mean foliation plane of 271/62, which is a real number and a misleading one. It is the average attitude of a surface that is folded, so no outcrop need look like it. The girdle describes the rock. The mean plane only summarises it.

The lineation is not

The 796 mineral lineations behave in the opposite way. They form a cluster, tight enough that the mean direction is pinned down to better than two degrees, at 076/33.

One thing to read carefully in that panel. The confidence cone on the mean is 1.6°, but the angular radius containing roughly two thirds of the individual lineations is closer to 25°. Both are true. The first says the mean is well determined, which it is with 796 measurements; the second says the fabric itself is genuinely dispersed, which it also is. Report only the first and you will mislead somebody, probably yourself.

And the hinges themselves

The third estimate needs no inference at all. Seventy-eight minor fold axes were measured directly in outcrop, most of them logged as crests of refolded folds. Their mean is 076/35, with a wider confidence cone than the lineation mean because there are ten times fewer of them.

So do they agree?

Drop any two of the three results into the angle calculator and it tells you directly.

The axis is not the same everywhere

Before making too much of that agreement, it is worth asking what the three numbers are averaging. All of them are single values fitted to the whole park, and a single value is only a fair summary if the thing it describes is constant. Here it is not.

Put the lineations in a three-dimensional box with position on the floor and plunge running up the vertical axis, and the shape of the problem becomes visible. The plunge is not spread evenly through the box: the steep population bunches toward one end and thins away across the rest. Being able to orbit the box matters here, because a single static view of a point cloud is always partly a projection artefact. Sliding a section slab along the horizontal axis makes the same point a slice at a time.

Split the park into thirds by longitude and both independent methods shallow together, which is the test that matters. If only one of them moved, the gradient would more likely be an artefact of the method than a property of the rocks.

So the park-wide plunge of 33° is a number no third of the park actually shows. The trend is genuinely constant at around 075°, and that part of the earlier result stands. The plunge is a regional average across a gradient, and it should be quoted that way. The foliation flattens eastward in step with it, from a median dip near 80° to under 50°, which is the kind of thing you would expect if the section exposes a different structural level from one end of the park to the other.

What the agreement means

The lineation is parallel to the fold hinge. In the older terminology that makes it a b-lineation, formed in the deformation that folded the foliation, rather than a stretching lineation oblique to the hinges. The π-axis agreeing with hinges measured directly in outcrop also says the folding is close to cylindrical across the whole park. Two fold generations with strongly different axes would leave a diffuse pole girdle and a fold-axis population that refused to cluster, and neither is what the data show.

Grouping by rock unit adds a second-order result worth having. The axis is the same everywhere, within a few degrees, in the schist-rich migmatite, the granite-rich migmatite and the Quetico biotite schist. What changes between them is how strongly folded each unit is: the migmatites give clear girdles, while the Quetico schist sits on the cluster side of the boundary, its foliation more nearly planar over the area sampled. That is a difference in fold intensity, not in kinematics, and it is the kind of thing that is invisible until you group the plot.

Where this could still be wrong

The π-method assumes the folding is cylindrical. It does not test that. A conical fold scatters poles along a small circle, and a great-circle fit will happily return an axis for one anyway, quietly wrong. On a girdle as broad as this one it is worth spending thirty seconds on a conical fit to see whether it describes the poles better.

Woodcock's K is not a strain ratio either. It describes the shape of a distribution of directions, so applied to poles of a folded surface it is measuring the geometry of the folding. It is not the Flinn parameter and should not be read as prolate against oblate strain, however similar the diagrams look.

And as with everything else in this series, these are attitudes selected for portrayal on a published map rather than sampled at random from outcrop. Three independent estimates agreeing to within seven degrees is strong evidence that they record a real structure. The dispersion figures still describe the mapped population and not the rock mass.

Data and sources

The extract is a single CSV of 2,154 records covering foliation, mineral lineation, minor fold axes, glacial striations and flow banding, with strike, dip, trend, plunge, position, unit, lithology and age. It imports into Stereogram Pro in one pass, and every figure and table above can be reproduced from it.