The strain a map adds to the ground's

Every number in the three rungs below this one is a quantity on the sphere. Every number anybody actually has is a difference of grid coordinates, because that is what a station’s position is delivered as and what a spreadsheet of them contains.

Those are not the same thing, and the gap between them is this collection’s entire subject arriving inside somebody else’s measurement.

The composition, in one line #

A strain rate is the gradient of a displacement. On the page the displacement is measured in page units, so the derivative taken there is

$$\mathbf{F}_{\text{page}} = \mathbf{J}(P')\,\mathbf{F}_{\text{ground}}\,\mathbf{J}(P)^{-1},$$

where J is the projection’s own Jacobian and P and P′ are the place and where it has got to. That is the same product a reprojection composes, with a deformation of the ground where a second projection would be.

Two things follow immediately. If the projection’s Jacobian were the same at both ends, the two Js would cancel and the page would report the ground exactly. And if the ground’s own deformation gradient is the identity — a rigid motion, no strain at all — what is left is J(P′)J(P)⁻¹, which is not the identity unless the map is doing the same thing at both places.

So the map’s contribution is the change in the projection’s distortion along the displacement. It is a directional derivative of a first-order quantity, which makes it second-order information about the map arriving inside a first-order measurement of the ground — and Tissot stops at the first derivative is the essay about why nothing in the site’s usual apparatus reports it.

Holding the ground rigid, which took two attempts #

Isolating the map’s half needs a motion with exactly zero ground strain, and the obvious construction is wrong.

Displacing every probe point by the same easting and northing in its own local frame is not a rigid motion of a sphere. The meridians converge, so a common bearing at every place is a deformation, and the first version of this measurement was reporting a ground strain it had manufactured — a control that was not controlling anything. The symptom was mild and entirely plausible: a projection centred on the very point being measured reported three nanostrain a year where it should have reported nothing.

What holds the gradient at exactly the identity is the rotation that carries the point to its image, applied to the probes as well: a rigid rotation of the whole sphere, which the first rung has already shown deforms nothing to eight decimal places. With that substitution the centred control returns 0.0066 nanostrain a year against Mercator’s 10.93, a factor of 1,664, and every number below is the map talking.

The library, and a structure nobody put there #

Reading the second column is more instructive than reading the bars:

Every equal-area projection reports its invented strain as pure shear, and every conformal one reports it as pure dilatation. Not approximately — the equal-area rows are zero to the last digit the difference can resolve.

That is not a coincidence and it is not an artefact. An equal-area map holds its areal factor at one everywhere, so the determinant of J is constant, so the determinant of J(P′)J(P)⁻¹ is exactly one and the invented deformation preserves area by construction. A conformal map holds the ratio of the principal scales at one everywhere, so its invented deformation is a similarity and has no shear in it. Each family’s defining property survives into a quantity nobody was thinking about when the property was chosen.

And the two are almost exactly equally wrong: Mercator at 10.93 against the Lambert cylindrical at 10.89. Choosing between them chooses what kind of nonsense to publish, not how much.

The plate carrée is the odd row, and its oddity has a one-line cause: its meridional scale is constant, so only the parallel scale varies and the invented deformation is uniaxial — one principal rate equal to the dilatation and the other exactly zero.

The null direction #

The mechanism is a directional derivative, so it has a direction in which it gives nothing.

Mercator’s scale factor is a function of latitude alone, so ground that moves along a parallel arrives somewhere the map is doing exactly the same thing, and there is nothing for the composition to report. Ground that moves north gets the whole of it.

This has an uncomfortable practical edge. A transform boundary running east–west and a transform boundary running north–south, with identical physics, produce different invented strain on the same sheet — so the pattern of a published strain map can be partly a picture of how the tectonics happen to be oriented relative to the projection’s own symmetry.

Where it lives on the sheet #

Nowhere on that axis is the number large compared with a plate boundary, which is 349. Everywhere on it the number is large compared with a plate interior, which is 0.0075 — by three orders of magnitude at sixty degrees north.

That is the honest summary of the whole rung, and it is worth stating before the profile that demonstrates it. The map’s contribution is negligible where the ground is loud and dominant where the ground is quiet.

The addition, checked #

The composition is a product of matrices and the strains are their logarithms, so to first order the two contributions add. That is a prediction with three independently computed quantities in it.

Ground −0.438, map −1.766, predicted −2.204, measured off the page −2.128. The 3.6 per cent gap is the commutator the linearisation throws away, and it shrinks as the interval does — which is the check that it is a truncation rather than a mistake.

The profile, which is where it bites #

Eight hundred and eighty-four per cent, at the place a reader is looking hardest.

The interesting question about a strain map is not where the maximum is — everybody knows where the plate boundary is — it is how far out the deformation reaches, because that is what decides how much of a country needs a deformation model rather than a datum. That question is asked exactly where the ground’s signal has fallen to a few nanostrain a year, which is exactly where the sheet’s own contribution is most of the answer.

Why a scale factor is not the repair #

The obvious objection is that surveyors already correct for this. Every national grid publishes a line scale factor, and the scale factor of a line is the essay about computing one properly; dividing a grid distance by it recovers the ground distance. If distances are corrected, why is the gradient of them wrong?

Because a scale factor repairs a length and the strain is a ratio of two lengths at different places. Applying the same factor to both ends of a baseline cancels out of the ratio entirely; applying the correct — different — factor at each end is precisely the composition above, and doing it point by point is doing the whole calculation properly rather than correcting it afterwards.

There is a sharper version. The scale factor is a scalar, and what has to be undone is a matrix: at sixty degrees north on Mercator the invented deformation is a pure dilatation, which a scalar could in principle absorb, and on the Lambert cylindrical it is a pure shear, which no scalar can touch at all. A pipeline that corrects distances with a factor and then differences the corrected coordinates has removed the equal-area case’s contribution not at all, and it is the same size as the conformal one.

This is the same distinction an area on the grid is not an area on the ground draws for a second-order quantity, one derivative further out: a correction that is right for the thing it was derived for is not automatically right for anything computed from it.

What to do about it, and it is cheap #

The repair is one line and this rung’s own control is the proof that it works: compute the gradient in a chart centred on the point, rather than in whatever grid the coordinates arrived in.

That costs a projection evaluation per sample and removes the whole effect down to the noise floor — 0.0066 against 10.93. It is the same move a local model has an order makes for a different quantity, and the same one the plane a file is stored in recommends for a different operation. A chart per point sounds extravagant and is four multiplications.

What is not cheap is noticing that it was necessary, because every intermediate result in the wrong calculation is a plausible number in the right units.

Why the control has to be a rotation #

The whole rung rests on being able to hold the ground still, and there is exactly one way to do it on a sphere — which is worth stating because the obvious alternatives all fail.

A common velocity fails, as above: a common bearing at every place is a deformation, because the meridians converge.

Holding every point fixed is not a motion, so there is no displacement for the map’s Jacobian to be evaluated at two ends of; the quantity being measured is a change along a displacement, and with no displacement there is nothing.

A common displacement in a single tangent plane fails for the same reason as the first: parallel transport of one vector to every point of a sphere is impossible, which is the hairy-ball theorem two rungs along in the impossibility field.

What is left is a rigid rotation, which moves everything, deforms nothing, and has a displacement at every point. That it is the only available control is a fact about the sphere rather than about the experiment, and it is the same fact that makes plate kinematics use rotations in the first place.

Where the model stops #

The addition is first-order and the interval used here is a million years, which is long enough that the dropped commutator is measurable at four per cent. Over a decade — which is what a real study has — the term is smaller by five orders of magnitude and the decomposition is exact for any purpose.

The invented strain computed here is for a rigid ground, which is the clean control and not the general case. When the ground is deforming, the map’s contribution depends on the actual displacement at each point, so it is not a fixed field that could be tabulated and subtracted once.

And the projections here are the site’s library in their normal aspects. A national grid is a transverse Mercator on a narrow zone, where the scale factor varies with distance from the central meridian rather than with latitude — so the null direction turns through ninety degrees and the numbers are smaller, because a zone is six degrees wide and a hemisphere is not.

The generalisation #

The pattern this ladder ends on is one the collection has now met from four directions.

A derivative taken in the wrong chart reports the chart. A mean of scattered positions moves because an average is taken on the page; a centroid belongs to a plane for the same reason; a simplification tolerance is a promise about the picture rather than the ground; and a strain rate differenced from eastings is partly a measurement of the eastings. Each is an operation that is not equivariant under the map, performed after the map instead of before it.

What makes this instance the sharpest is the size of the thing being measured. A centroid moved by a kilometre is visibly wrong. A strain rate of three nanostrain a year where the truth is 0.3 is a perfectly reasonable-looking number, in the right units, on the right sheet, and there is nothing in it to notice.

Who found it, and when #

Geodesists computing strain from triangulation have worked in local plane coordinates since the nineteenth century, and the reduction to a plane is one of the standard steps: the observations are reduced to the ellipsoid, then to the grid, and every textbook says so. The step is a correction with a name, applied because the alternative is arithmetic on a sphere.

What is new here is nothing about the practice. It is that the correction and this site’s subject are the same object — the correction is the projection’s own distortion, differenced — so the machinery that has audited fourteen projections for two hundred and forty-six essays audits this without being adapted.

Why a plausible number is the worst kind of error #

The remark that a strain rate of three nanostrain where the truth is 0.3 has nothing in it to notice deserves its own statement, because it identifies which errors survive and which do not.

An implausible number is caught by everybody. A centroid a kilometre out of place, a distance twice what it should be, a negative area — these are found immediately, by whoever looks at the output, because a person carrying any intuition about the quantity rejects them on sight. They are the cheapest errors to have.

A plausible number is caught by nobody. It has the right sign, the right order of magnitude, the right units, and it sits comfortably among the other numbers in the table. The only way to discover it is to compute the quantity a second way, and there is no reason to do that for a value that looks fine.

And a factor of ten in nanostrain is squarely in the plausible band, because nobody has an intuition for nanostrain. Crustal strain rates span orders of magnitude between a stable craton and an active margin, so almost any value is defensible for almost any place, and the contaminated figure is indistinguishable from a real one.

Which is the argument for a control rather than a review. No amount of care reading the output finds this, because the output is not wrong-looking. What finds it is running the machinery on a case whose answer is known — a rigid rotation, which must give exactly zero strain — and seeing whether it does.

It is also why the control had to be a rotation rather than a small strain. A control whose right answer is a small number tests whether the machinery gets close; a control whose right answer is exactly zero tests whether it is measuring the thing at all, and only the second can distinguish a pipeline that adds the page’s own contribution from one that does not.

That is this collection’s standing habit and it earns its keep here. A check that could not fail proves nothing; a check on an input the machinery must refuse is the only thing that separates a correct pipeline from one producing plausible numbers in the right units.

The same reasoning picks the controls elsewhere on the site: an identity request that must cost nothing, a uniform density that must produce no distortion, a rigid motion that must produce no strain. In each case the value being asserted is zero, and zero is the only value an instrument cannot arrive at by being approximately right.

A control of that kind also survives being ported: it is a statement about the machinery rather than about the data, so it keeps working when the inputs change.

Where the ladder goes next #

This anchor has taken the collection’s instrument and pointed it at the ground. Four rungs in, the thing it has found is that the ground and the page contribute to one measurement in the same units and cannot be told apart without a control. What is missing is the vertical, where the same argument runs against a level surface rather than a sheet, and where the field the strain is measured against is itself the thing being mapped.