The ground has an indicatrix too

Every essay in this collection so far is about an instant. A datum is fitted, a projection is measured, a grid is designed, a coordinate is given a width — and underneath all of it the Earth is held still, as a shape rather than as a process.

It is not still. One essay here says so: the epoch is part of the coordinate prices the fact that the ground moves at between ten and seventy millimetres a year, and models the motion the way plate kinematics models it, as a rigid rotation of each plate about its own pole.

A rigid rotation is the one motion with no distortion in it whatsoever. So the collection’s entire treatment of a moving Earth is about the single case where there is nothing to measure.

The substitution the whole rung rests on #

A projection is a map from the sphere to a page. Its distortion is read by differencing it four ways at a point, assembling the two-by-two Jacobian, and taking the eigenvalues of its own transpose times itself — which gives the two principal scale factors, their product, and the angular deformation that this site refuses to call a projection conformal without.

The motion of the ground over an interval is also a map. It sends every place to where that place has got to, and it has four partial derivatives at a point in exactly the same sense. Feeding it to the same construction returns the same six numbers, about the Earth rather than about the paper.

That is the whole of the substitution, and it is worth being precise about what it buys. It is not an analogy. The two objects are the same kind of object — a smooth map of a surface, differentiated — and the arithmetic that reads one is not adapted to read the other, it is reused.

What comes out of it, in the units geodesy uses #

The output is a rate rather than a ratio, because the map depends on how long the interval was. Divide the departure of each principal stretch from one by the number of years and the answer is a strain rate, quoted in nanostrain a year: parts in a thousand million, per year.

The unit is not arbitrary. A plate interior sits below one nanostrain a year, a plate boundary in the hundreds, and a first-order survey’s own precision arrives somewhere between the two — so the same number that describes the physics also decides whether a network can be left alone, which is the subject of two rungs further up this ladder.

The null case, which is the check #

The temptation with a new instrument is to point it at something interesting and report what it says. The order here is the other way round, because the classical route to this quantity is a trap.

Written out in spherical components, the strain rate carries three curvature terms — a velocity times the tangent of the latitude, divided by the radius, and two relatives of it. They are there because the local east–north frame turns as the point moves across the sphere, and they are easy to write with a sign wrong. A sign wrong there produces a strain-rate field that looks entirely plausible and is a measurement of the coordinate system.

So the derivative here is taken the way this site takes every other derivative: by differencing the map in a chart. The chart is the azimuthal equidistant projection centred on the point — the tangent plane, with distances along geodesics — taken at the source point for the argument and at the image point for the value. Two charts, each centred on its own place, and nothing in between that knows what latitude is. The curvature terms are not omitted; they are absent.

And then the check. Six published plate rotation vectors, five places each, a million years:

The worst second invariant anywhere in that sweep is 6.2 × 10⁻⁸ nanostrain a year, which is the noise floor of a central difference and not a measurement. The same machinery, pointed at the stated transform boundary, returns 261. The ratio is four thousand million to one, and it is the reason every number below can be believed.

Speed and strain are unrelated, and the table says so violently #

The most useful thing the null case establishes is not that the arithmetic is right. It is that how fast the ground is moving carries no information at all about whether it is deforming.

The Pacific plate at 70.2 millimetres a year returns a strain rate indistinguishable from zero. A point in the middle of the stated transform boundary, moving at 5.3 millimetres a year in the frame the table is drawn in, returns 349. The slowest place in the table is the most strained one, by ten orders of magnitude, and no amount of looking at the arrows would say so.

This is the same distinction the collection already draws about maps, arriving in a new place. A projection can move a point a very long way from where the reader expects it and distort nothing at all — that is what a change of aspect does, and what survives a change of coordinates is the essay about it. Displacement is not deformation. Here it is the ground doing the moving.

Where the deformation actually lives #

Because the strain is a gradient, it lives entirely in the places where the velocity is changing, which are not the places where the velocity is large.

On the stated convergent boundary the peak is 71.7 nanostrain a year and it falls to half that within 213 kilometres. On the transform boundary, whose zone is stated at a quarter the width, the peak is 368 and the half-width 107. Both curves are flat at zero over most of their length, and both have their maxima where the arrow field looks least interesting.

What the three numbers separate #

A deformation gradient carries more than one kind of information, and reporting it as a single figure is what makes published strain maps hard to read. The decomposition is the standard one and this site’s machinery makes it visible.

The dilatation is the determinant minus one: the areal factor, which is the quantity an equal-area projection holds at exactly one and which here says whether the ground is gaining or losing area. On the stated convergent boundary it is −69.6 nanostrain a year, and the sign is the whole content: the ground is closing.

The maximum shear is the difference of the two principal rates, and it is what a transform boundary produces almost purely. At the stated one it is 349 nanostrain a year against a dilatation of −0.94 — the ground is being sheared and its area is very nearly conserved.

And the rotation is the antisymmetric part, which is not a strain at all.

Twelve of this collection’s essays about maps make the same separation without naming it: a projection’s Jacobian has a rotation in it too, and every invariant the site quotes is chosen precisely because the rotation cancels out of it. Doing the same for the ground is one line of the same algebra.

The refusal, and how the fields were chosen #

Everything non-rigid in this ladder is stated rather than loaded, and the reason is the coastline decision this collection made in its second phase. A real velocity field is the output of an inversion with a regularisation in it, and a reader shown a strain map computed from one cannot tell how much of the pattern is the regulariser. The published rotation vectors are a different kind of input — they are parameters, in the sense that an ellipsoid’s semi-major axis is a parameter — and this site has always used those.

A stated field also has something a real one does not: an answer written down in advance.

A field declared at 25 and −10 nanostrain a year comes back at 25.03 and −9.995. That is a check on the two charts, on the Runge–Kutta step that advances the points, and on the eigenvalue routine, all at once, and it is available only because nobody had to go and measure the field.

The velocity picture is the one that misleads #

It is worth drawing the thing this rung is arguing against, because it is what nearly every published account of a moving Earth shows.

An arrow field is a velocity, and a velocity is not a strain. What a reader can actually see in one is the difference between neighbouring arrows, which is the gradient, taken by eye, at whatever spacing the author chose to draw. Halving the spacing halves the visible difference and changes nothing about the ground.

Why a tensor rather than a number #

A strain rate is reported here as six numbers, and geophysics usually reports one — the second invariant — so it is worth saying what the other five are for.

The two principal rates say what kind of deformation it is. Equal and positive is a uniform spreading; equal and opposite is a pure shear; one zero is a uniaxial stretch. Those are three different physical situations with the same second invariant.

The azimuth says where it points, which is the quantity a fault map is compared against and the only one that can be checked against geology rather than against another geodetic solution.

And the rotation is not a strain at all, so a summary that includes it is reporting a rigid motion as a deformation.

The single number survives because a map has to be drawn and a map draws a scalar. That is the same pressure the indicatrix field is under on the other side of this collection, and it has the same answer: draw the scalar, and print the tensor beside it.

Where the model stops #

Three limits, each of them the kind that would change a number rather than the argument.

The motion is horizontal. Real ground moves vertically too, and the third coordinate moves too is the essay about that half; the strain tensor here is the two-by-two horizontal one, which is what a horizontal network measures and what a map cares about.

The interval is a million years, chosen so the deformation is large enough to difference cleanly and small enough that the linearisation holds. Over a hundred million years the strains here would compound rather than add, and the rate would stop being the useful quantity.

And the fields are smooth. A real plate boundary has faults in it, which are discontinuities rather than gradients, and a discontinuity has no derivative at all. What a geodesist does about that — model the fault and take the strain of what is left — is a modelling choice this site does not make, and stating it is why the fields here are declared with a half-width.

The generalisation #

The move made here works for any smooth map of a surface to itself, and the surface does not have to be the Earth.

That matters because it puts the site’s own subject in an unexpectedly general position. The apparatus built to measure what a page does to the sphere turns out to measure what the sphere does to itself, and — as the anchor that carves a map to a stated density found last time this happened — the apparatus does not care where its argument came from. A projection, a cartogram, and a year of tectonics are three maps with derivatives.

The fourth rung of this ladder is where the generality becomes a problem rather than a convenience, because a strain rate is computed from coordinates and every coordinate anybody has is on a map.

Who found it, and when #

The velocity gradient tensor and its decomposition into strain and spin are nineteenth-century continuum mechanics, and were applied to the Earth’s crust from geodetic observations long before the plates were accepted: Harry Fielding Reid’s elastic rebound account of the 1906 San Francisco earthquake rests on triangulation from the 1850s onwards, read as a strain field.

The plate-kinematic half is younger. Rigid rotation about an Euler pole was put to the sea floor in the mid-1960s, and the rigidity is the hypothesis rather than an approximation — it is what makes plate tectonics a theory with content, because a rigid plate has three degrees of freedom and a deforming one has infinitely many.

What is new here is nothing about the geophysics. It is that the strain of the ground and the distortion of a map are being computed by one piece of arithmetic, on a site that had been using it on maps alone for two hundred and forty-six essays.

Two tensors, one page #

The arithmetic being shared has a consequence nobody in either field has to think about until the two objects are drawn together, which is exactly what a published strain map does.

A crustal strain field is drawn on a projected sheet. The ellipses in every such figure are plotted in page coordinates, at page positions, with page orientations — and the page has a deformation of its own, computed by the same machinery, which is the whole subject of this site.

So what a reader sees is a composition. The ground’s strain tensor, pushed through the projection’s Jacobian, becomes something else: the principal directions of the composite are not the principal directions of the ground’s strain, and the magnitudes are not its magnitudes. Two deformations applied in sequence do not commute, and their principal axes do not add.

Whether the picture is readable depends on which projection was used, and the rule is clean.

A conformal sheet keeps the directions and scales the magnitudes. Its own Jacobian is a rotation times a scalar, so it takes a strain ellipse to a similar strain ellipse: the principal axes point the same way relative to the ground, the ratio between them is preserved, and only the overall size changes with position. A reader comparing the orientation of strain in two places is right, and a reader comparing its magnitude in two places is wrong by the ratio of the scale factors.

An equal-area sheet keeps neither. Its Jacobian has unequal principal stretches, so it rotates the strain axes by an amount that varies across the sheet and changes the ratio between them. The direction of maximum extension read off such a map is not the direction on the ground, and nothing on the page says by how much.

Which gives the same answer as this collection reached for a contour map, arriving from a different anchor: a field whose direction is the quantity of interest wants a conformal sheet, and a field whose total is the quantity of interest wants an equal-area one. Strain direction is a direction, so the choice is settled.

The same reasoning applies to any tensor field drawn on a map — stress, anisotropy, a diffusion tensor, a fabric orientation. None of them is a scalar, all of them are plotted as oriented ellipses, and every one of them inherits the sheet’s own deformation in the same way.

And the composite is computable rather than merely a caution, since both tensors come from the same arithmetic. That is the practical value of the substitution this rung is built on: the correction a strain map needs is one matrix multiplication with the projection’s own Jacobian, which the plotting code already has.

Where the ladder goes next #

The strain rate is a gradient, and a gradient does not care what was added to the field before it was taken. That has a consequence about reference frames which is sharp enough to be its own rung: the same place reports four different speeds in four defensible frames, and one strain rate.