A grid stops fitting the ground it was laid on

A survey does not hold coordinates. It holds distances between marks, and the coordinates are what a solve produces from them — which is what a published coordinate actually is, and why a network’s specification is written in millimetres plus parts per million rather than in latitudes.

That makes the strain rate, rather than the velocity, the quantity a survey has to worry about. A distance is invariant under a rigid motion, so a plate can carry a network right round the Earth without disturbing a single measurement in it. What breaks a network is the part of the motion that is not rigid, and the previous two rungs are about exactly that part.

This rung turns the nanostrain into a number of years.

The measurement, on one baseline #

The cleanest version of the statement needs no network at all. Put two marks a hundred kilometres apart, wait, and measure the distance again.

After fifty years the four fields give:

The two rigid rows are the point of the table. The Pacific is moving at seventy millimetres a year — three and a half metres in fifty — and the two marks stay exactly a hundred kilometres apart, because they are being carried by the same rotation.

Why the rigid rows are not a rounding #

The two zeros in that table are exact rather than small, and it is worth seeing why a hundred kilometres of Pacific plate moving at seventy millimetres a year produces one.

This is the reason plate tectonics is compatible with national surveying at all. If the plates deformed as fast as they moved, no country would have been able to hold a triangulation network to a part per million for a century, and the whole apparatus of national grids would have been impossible before continuous satellite positioning. The plates are useful to a surveyor precisely to the extent that they are boring, and the interior row of the table is a measurement of how boring.

What the specification actually says #

An ordinary first-order specification is not a single number. It is a fixed part plus a proportional part: five millimetres plus one part per million of the length, or ten plus ten, or two plus a half, depending on the class of the work.

That shape matters, and it is why the years-to-failure is not simply the inverse of the strain rate. A uniform strain changes a length in proportion to the length, so the proportional part of the tolerance is a fixed number of years however long the baseline is. The fixed part is not: it protects a short baseline for a long time and a long one hardly at all.

The consequence is a clock that ticks at different speeds for the same ground:

Tightening a specification by a factor of ten shortens the life of the network by a factor of about twenty. That is not a paradox; it is the same trade the tolerance decides the model prices for a plane survey, applied along a different axis. A more accurate network goes out of date faster, and it goes out of date faster than its own accuracy improved.

The clock is the proportional part divided by the strain rate #

The four rows of that table have a closed form behind them, and extracting it says which half of a specification the lifetime actually depends on.

A tolerance of a millimetres plus b parts per million allows a baseline of length L to change by a + bL before it fails. A uniform strain rate ε changes it by εL a year. So

$$T \;=\; \frac{a + bL}{\varepsilon L} \;=\; \frac{a}{\varepsilon L} \;+\; \frac{b}{\varepsilon}.$$

The second term has no L in it. However large the network, the lifetime cannot exceed the proportional part of the tolerance divided by the strain rate — a floor set by the ground and by one of the two numbers in the specification, with the network’s own geometry contributing nothing to it.

The table is that floor almost exactly. Dividing each clock by its own proportional part gives 6.0, 6.1, 5.85 and 5.85 years per part per million on the transform field; 15.8, 15.9, 15.3 and 15.3 on the convergent one; and 1,812, 1,819, 1,782 and 1,782 in the interior. Four specifications differing by a factor of forty, and each column is constant to three per cent.

So on a four-hundred-kilometre network the fixed part of the specification does essentially nothing, and the observation that tightening by a factor of ten shortens the life by about twenty is really the observation that the two specifications differ by a factor of forty in b and the clock follows b alone. The apparent factor of twenty against ten is an artefact of quoting the fixed part first.

The two terms cross where a = bL, which for five millimetres and one part per million is a baseline of five kilometres. Below that length the fixed allowance is the larger of the two and a network’s lifetime rises steeply as it shrinks; above it, the proportional allowance dominates and the lifetime is flat in the length. That is the left-hand half of the non-monotone curve in the figure above, and it is a property of the specification rather than of the ground.

Which separates the two mechanisms the curve contains, and they are worth keeping apart because they have different remedies. The rise at the small end is the fixed millimetres protecting a short line, and it is bought by writing a specification with a larger constant in it. The rise at the large end is the network reaching outside the deforming zone onto ground that is not moving, and it is bought by putting marks somewhere else. The first is a decision about the paperwork and the second is a decision about the ground, and the minimum between them — at the zone’s own width — is where neither is available.

The floor also says what a re-observation buys. Re-observing resets the accumulated change to zero and does not change ε, a or b, so the next interval is the same length as the last one. A network in a boundary zone at a first-order specification is not slowly degrading towards a crisis; it is on a fixed six-year cycle, for as long as the specification and the plate boundary both stand.

The most fragile network is the size of the thing deforming it #

The obvious guess is that a larger network fails sooner, because a longer baseline accumulates more absolute change. It is wrong, and the shape of the answer is the finding.

At the small end the fixed five millimetres is most of the allowance, and a short baseline inside a deforming zone accumulates change slowly in absolute terms. At the large end most of the network lies outside the zone entirely, on ground that is not deforming at all, so extra length buys tolerance — a part per million of a longer line is more millimetres — without buying any extra relative motion.

Between them, at the width of the zone, is the worst case: the whole network is inside the gradient and the baselines are long enough that the fixed part no longer protects them. Fifteen point seven years.

The general statement is worth having, because it is not about tectonics. A network is most vulnerable when its own size matches the scale of the field deforming it, and a survey that is designed against a strain rate without asking about the scale of the strain has answered the wrong question.

What it looks like on the marks #

That distinction is the reason a country re-realises a datum rather than repairing one. A rigid displacement of every mark is seven parameters, and seven parameters is exactly what a datum transformation is. Publish new parameters and every old coordinate is repaired by arithmetic.

A deformation is not seven parameters. It is a field, with as many degrees of freedom as there are marks, and no transformation of the whole sheet puts it back — which is precisely the finding where a fit leaves residuals makes about an old triangulation network, arriving here from the other direction. There the strain was accumulated survey error; here it is the ground.

The interior, which is where most people live #

The two interesting rows of the clock are the ends. The row that decides most national practice is the middle one.

That last observation is not a complaint about the figure. A strain map has to normalise to something, so the shape of the field is legible and the size of it is not — which puts a strain map in exactly the position an indicatrix field drawn at a size somebody chose is in on the other side of this collection. Two maps of the same construction, one showing 0.0075 and one showing 349, are the same picture.

The two things a country can do, and both are visible in the arithmetic #

The first is to re-realise: recompute every coordinate at a new epoch and publish the whole set again. Australia did it in 1994 and again in 2020, and the second move was 1.8 metres. The clock above says how often that has to happen, and for a plate interior at a working specification the answer is once a century or so — which is why most of the world does not think about it.

The second is to hold the plate, which is what the previous rung’s frames are for: publish in a frame in which the plate’s own rotation is zero, so the coordinates stay put and the epoch stops mattering for anything except the residual. This works exactly as well as the plate is rigid, and the plate-interior row of the table is a measurement of how well that is: 0.0075 nanostrain a year, eighteen centuries to a first-order tolerance.

Neither works in the third case.

New Zealand and Japan both do exactly that: a national datum with a velocity field attached, so that a coordinate is a position and a rule for where it will be. That is a genuine change in what a coordinate is, and what a coordinate refers to has to be read with it in mind.

What the clock is checked against #

A number of years is easy to produce and hard to trust, because everything in it — the field, the geometry of the marks, the shape of the tolerance — is a choice. Two controls make it a measurement.

The first is the rigid case, and it is a refusal rather than an agreement: the clock is required to return never for a plate whose motion is a published rotation vector, at any speed, over any network. A clock that returned a finite answer there would be measuring the arithmetic. It returns infinity, because the baseline change comes back at 10⁻⁸ parts per million a year and the tolerance is a positive number.

The second is the scaling. The years-to-failure has to be inversely proportional to the strain rate with the geometry held fixed, because that is what the definition says, and the transform and convergent rows differ by a factor of 2.6 in both directions: 349 nanostrain a year against 71.7 gives 4.9 in rate, and the clocks are 6.1 and 15.9 years, a factor of 2.6 — the difference being that the two zones have different widths, so the network sees different fractions of each. Holding the field and varying only the specification recovers the proportionality exactly.

Between them those two say the clock is reading the ground rather than the model of it, which is the standing requirement on every number this collection prints.

What the clock does not measure #

The years computed here are the interval before a network fails its own internal specification, and that is one of three clocks a mapping agency runs.

The network’s clock is this one: when do the marks stop agreeing with each other.

The frame’s clock is when the published coordinates stop agreeing with the international frame, which is a rigid motion and is therefore much faster — Australia crosses an ordinary tolerance in under two years, as the epoch rung measures, while its network’s own internal geometry is good for centuries.

And the map’s clock is when the features drawn on the sheet stop agreeing with the ground, which is a different thing again: a road resurveyed after an earthquake moves relative to a fence that was not.

The three run at different speeds and a country needs all three answers. What this rung supplies is the slowest of them, and the reason it matters is that it is the only one a new realisation cannot reset — a datum can be republished, a frame can be re-fitted, and a network that has genuinely deformed has to be re-observed.

Where the model stops #

The clock computed here has three simplifications in it, and each of them makes the answer optimistic.

The fields are smooth. A real boundary zone concentrates most of its motion on faults, and a fault that slips four metres in a minute breaks a network in one event rather than over six years. The rate is the long-run average and the failures are not distributed like the average.

The marks are assumed stable. A survey mark that is moving relative to the ground it is set in — frost heave, subsidence, a wall settling — contributes to the same measurement and is not tectonics, and a residual has more than one explanation is the essay about that confusion.

And the tolerance is treated as a hard edge. A specification is a probabilistic statement about a measurement, so a network does not pass on the sixth year and fail on the seventh; the probability of a detectable discrepancy rises smoothly, and the year quoted is where the systematic part reaches the whole allowance.

The generalisation #

Strip out the geodesy and what is left is a statement about any built thing that references a moving frame.

The lifetime of a reference system is its own tolerance divided by the rate at which the thing it references deforms, and neither term is a property of the system alone. Tightening the tolerance shortens the lifetime; the rate is set by where the system was built. A specification with no date on it is therefore incomplete in the same way a coordinate with no epoch is incomplete, and for the same reason.

The pleasant part is that the number is small enough to matter and large enough to plan against. Six years is a survey contract. Eighteen centuries is not anybody’s problem.

Who found it, and when #

Repeated triangulation across the San Andreas fault, from surveys begun in the 1850s and re-observed after 1906, is the first measurement of this kind: Harry Fielding Reid read the ground’s accumulated strain out of two sets of angles taken half a century apart. The instrument was a theodolite and the answer was a strain rate.

The modern version is continuous, and the change it made was to the timescale rather than to the argument. A network that once had to wait fifty years for a second epoch now produces one every day, which is why the deformation models attached to national datums are possible at all.

The specification shape — a fixed part plus a proportional part — is older than either, and comes from the error budget of a measured distance: an instrument has a constant error and a scale error, and a tolerance written to match it inherits both.

Where the ladder goes next #

Every number in this essay is a distance on the ground. Every number anybody actually has is a difference of coordinates on a map, and a map has a scale factor that varies. The last rung of this ladder is what that does to a strain rate, and the answer is that a ground which is not deforming at all reports one.