A velocity needs a frame and a strain rate does not

A velocity is a difference of two coordinates divided by a difference of two dates, and a coordinate without its system is not a location. So a velocity without its system is not a velocity, and the system has one more freedom in it than the coordinate did: two frames can agree about the shape of the Earth, about where its centre is and about how large it is, and still disagree about which way it is turning.

There is no measurement that settles the disagreement. What settles it is a convention, and the convention is chosen for convenience.

Four defensible answers to one question #

Take a single place in the middle of the stated transform boundary and ask how fast the ground there is moving. Four frames, all of them in use, all of them arrived at by subtracting some rigid rotation from the same field:

The speeds span a factor of 10.4. The bearings span 161 degrees, which is to say that two of these frames have the ground going in roughly opposite directions. Every row is correct.

What survives, and how exactly #

The strain rate at that place is 348.934 nanostrain a year, and it is 348.934 in all four. Not approximately: the spread across the four frames is 6.8 × 10⁻⁹ nanostrain a year, which is two parts in a hundred thousand million of the value, and is the noise floor of the finite difference rather than a disagreement.

The principal azimuth survives too, at 89.977326° in every frame, to every digit computed.

The mechanism is one line. The strain is a gradient of the velocity field, and the thing a frame change adds is a rigid rotation of the whole Earth — whose gradient is the antisymmetric part and contributes nothing to the symmetric one. Adding a constant to a function does not change its derivative; adding a rotation to a velocity field does not change its strain.

That is exactly the shape of argument what survives a change of coordinates makes about maps: the scale along a meridian depends on which graticule was drawn, the two principal scales do not. Here the moving quantity is a velocity and the surviving one is its symmetric gradient, and the reason is the same reason.

The place that is not moving at all #

The sharpest version of the disagreement is not the fast case, it is the zero.

In the stated plate-interior field — a Eurasian rotation with a half-millimetre ripple on it — a place in central Europe moves at 0.00 millimetres a year in the Eurasia-fixed frame, 25.0 in no-net-rotation, 5.8 with Africa held, and 76.0 with the Pacific held. Its strain rate is 0.00748 nanostrain a year in every one of them.

A national mapping agency publishes coordinates in a plate-fixed frame precisely so that this number is zero, and the zero is what makes a national grid usable for decades at a time. It is worth being clear about what that zero is: it is not a statement that the ground is not moving. It is a statement that a rotation has been subtracted, and the residual — which is the part that will eventually break the grid — is the 0.00748.

The picture every frame agrees about #

Drawing the strain field instead of the velocity field makes the agreement visible rather than tabular, because there is only one picture to draw.

The three velocity panels and this one carry the same information about the ground, and only this one carries it without a convention attached. A reader handed the velocity panels has to be told which is which; a reader handed this one does not have to be told anything.

What a frame change is made of #

It is worth writing down what is actually being subtracted, because the smallness of it is surprising.

A frame is fixed to a plate by subtracting that plate’s rotation vector — three numbers, in milliarcseconds a year about the three Cartesian axes, of order one. The Pacific’s is the largest in the published model at 2.4 milliarcseconds a year in total, which is 1.2 × 10⁻⁸ radians a year: a full turn every five hundred million years.

Multiplied by the Earth’s radius that is seventy millimetres a year at the equator of that rotation, which is why the Pacific-fixed column in the table above is the loud one. Divided into a gradient it is nothing at all, because a rotation has no gradient to give: its velocity varies from place to place only in the way a rigid body’s does, and a rigid body’s variation is exactly the part the symmetric gradient throws away.

That is the whole asymmetry in one line. The same three numbers are enormous in the velocity and absent in the strain, and which of the two a study reports decides whether those three numbers are part of its result.

Adding an arbitrary rotation, on purpose #

The invariance can be tested rather than argued. Take the field, add a rigid rotation of the whole Earth of a stated size, and read the two quantities off again.

No frame in use was consulted for that figure. Any rotation at all does the same thing, because the property being demonstrated is a property of gradients rather than a property of geodesy.

Where the invariance stops, which is the refusal #

An invariance that held against everything would be a statement about the instrument’s resolution rather than about the field. So the check has a second half: something that is not a rigid rotation has to break it.

The stated graded field is a uniform extension with a ramp on it. Its strain rate at one place is 25.3 nanostrain a year and four degrees to the east it is 40.3 — a difference of 59 per cent, from a field whose velocities differ by very little. A frame change cannot produce that, and nothing that a frame change does can hide it.

The trend in that second row is worth naming rather than smoothing: the relative spread of the strain rate grows as the strain gets smaller, from 1.9 × 10⁻¹¹ where the shear is 349 nanostrain a year to 1.1 × 10⁻⁷ where it is 0.036. That is a fixed absolute noise floor divided by a shrinking number, which is what a finite difference always does, and it is the same shape as the departure a finite indicatrix has from its limit.

What no-net-rotation actually is #

The frame most often used for global work is defined by a condition rather than by a place: the integral of the velocity field’s rotation over the whole Earth’s surface is required to vanish.

That is a reasonable convention and it is not a measurement. It depends on a model of what every plate is doing, so a new plate model gives a new no-net-rotation frame, and the velocities of every station on Earth change by the difference — while every strain rate stays where it was. A published station velocity therefore has a citation attached to it in a way that a published strain rate does not.

It is also worth noticing what the condition does not fix. The Earth’s rotation about its own spin axis is enormous and is removed long before any of this; what remains is the residual turning of the crust relative to the deep interior, and there is no reason for it to be zero. Requiring it to be zero is a choice of origin, in exactly the sense that a grid has an origin that is not there.

The catalogue, seen from here #

The previous rung’s catalogue of stated fields can be read again in this light, and it says something it could not say before.

The two rigid rotations at the bottom of that chart are the extreme case of the whole argument: their velocities are the largest in the table and are entirely a frame choice, in the sense that a frame exists in which each of them is exactly zero everywhere. Their strain rates are zero in every frame, including the ones where they are moving fastest.

Why the invariant is the one to publish #

The practical consequence is a recommendation the geodetic community reached independently and for these reasons.

A velocity map is the wrong product to compare between two studies, because two studies in different frames produce different pictures of identical ground. A strain-rate map is the right one: it is what the observations constrain, and two groups who disagree about which rotation to subtract will still agree about it.

The trap is that a velocity map is much easier to draw and much easier to read, so it is the one that gets published — and a reader shown one has been shown a quantity with a convention baked into it and no way to see the convention. This is the same failure the indicatrix drawn at a size somebody chose has on the other side of the collection: a picture whose content is partly an unstated choice.

What a published velocity is worth #

The practical consequence is a rule about citation rather than about geodesy, and it is sharp.

A station velocity cannot be compared with another study’s without both frames. Two papers reporting the same station at 5.3 and 54.6 millimetres a year may agree perfectly about the ground and disagree only about which rotation they subtracted, and nothing in the two numbers says so.

A strain rate can. Two studies reporting 349 nanostrain a year at the same place agree about the ground, whatever frames they used and whether or not either states one.

So the reproducible product of a geodetic campaign is the gradient, and the headline product is the velocity field, because a velocity field is a picture and a strain rate is a tensor. That mismatch between what reproduces and what publishes is not unique to geodesy and is worth naming when it appears: the quantity that survives is rarely the quantity that illustrates.

Where the model stops #

The invariance proved here is exact for a rigid rotation and only for a rigid rotation. Three things that look like frame changes are not.

A change of scale — a frame whose metre is a different length — multiplies every coordinate and does change the dilatation, by the scale rate itself. Frames do carry a scale rate, at the level of a part per thousand million a year, and it lands directly on the areal part of every strain rate computed in them.

A change of origin with a velocity in it — a frame whose centre is drifting — is a translation rather than a rotation, and a translation of a sphere is not a rigid motion of its surface: a common bearing at every place is not a common motion, because the meridians converge. That distinction is not a subtlety here; it is a defect this ladder’s own machinery had to be repaired for, and the repair is in the third rung’s control.

And a change of epoch is not a frame change at all. It is a different question, with a different answer, and it is what the epoch is part of the coordinate prices.

Which invariant is protected against what #

The caveat about a frame’s scale rate deserves more than a mention, because at the sizes involved it decides whether the invariance is a curiosity or a working guarantee — and the answer depends on which strain quantity is being reported.

A frame carrying a scale rate multiplies every coordinate by 1 + σt, which adds an isotropic term σ to the strain-rate tensor: both principal rates rise by σ and their difference does not move. So

• the dilatation — the trace, the areal rate — changes by 2σ, exactly;
• the maximum shear — the difference of the principal rates — does not change at all;
• and the principal azimuth does not change either, since adding a multiple of the identity cannot rotate the eigenvectors.

The quantity this rung reports is therefore protected twice over. The 348.934 nanostrain a year is a maximum shear, and it is invariant under a rigid rotation because a rotation contributes only to the antisymmetric part, and invariant under a scale rate because a scale contributes only to the isotropic part. Two different freedoms, two different halves of the tensor, and the shear lives in neither.

The dilatation has only the first protection, and the sizes make that matter. At the essay’s own quoted level for a frame’s scale rate — a part per thousand million a year, which is one nanostrain a year — the boundary field’s 349 nanostrain a year would be perturbed by 0.3 per cent and the plate interior’s 0.00748 would be perturbed by more than a hundred times itself.

So the invariance is not uniform across the catalogue. On a plate boundary every strain quantity is frame-independent to well inside any plausible scale rate. In a plate interior, which is where most national networks are and where the residual strain is the number a mapping agency cares about, the areal part of the strain rate is dominated by the frame’s scale realisation and the shear part is not.

That has a practical form. A study reporting a plate interior’s deformation should report the shear rather than the dilatation, or should report the dilatation with the frame’s scale rate stated beside it — and the second is not a convention anybody follows, because a scale rate is a property of a frame’s realisation rather than of the study, and it lives in a different document.

It also explains a choice this rung makes without justifying it. Every figure here reports the maximum shear or the second invariant, and neither is the areal rate. That was the right choice and the reason is now available: the areal rate is the one quantity in the tensor that a frame can move.

The generalisation #

The pattern is older and wider than geodesy, and stating it plainly is worth more than the tectonics.

A quantity defined by a difference inherits every freedom of the thing it differences. A quantity defined by a gradient loses the ones that are constant. Position has an origin freedom, so a displacement has none; velocity has a rotation freedom, so its symmetric gradient has none. Each derivative discards exactly the group it is blind to, and what is left is what two observers with different conventions can argue about.

This collection has now made that argument three times about three different objects — the invariants that survive a change of graticule, the difference of two coordinates and its covariance, and the velocity field here — and it is the same argument each time.

Who found it, and when #

Frame dependence became a practical problem when space geodesy started producing velocities good to a millimetre a year in the 1980s, and two groups could then disagree about a station’s motion while agreeing about every observation that went into it. The no-net-rotation condition dates from the plate models of that decade and is a direct response.

The mathematics is much older. That the symmetric part of a velocity gradient is invariant under superposed rigid motion is the principle of material frame indifference, which continuum mechanics has held since the nineteenth century and which is the reason a constitutive law can be written down at all.

Where the ladder goes next #

If the strain rate is the invariant, it is also the quantity a survey has to care about — because what a survey holds is the distance between two marks, and a distance is another thing a rigid rotation cannot touch. The next rung turns the nanostrain into a number of years.