A vertical rate needs a height system

The four rungs below this one measure a two-by-two matrix, and the reason is a line of code rather than an argument: chartAt returns two numbers, because a tangent chart on a sphere has two axes.

That was recorded as a shortfall on the day the anchor was built, in the plainest terms available. Vertical deformation is measured by the same instruments and is the larger signal in a subsiding basin. This rung pays it, and paying it turns out to require two separate things — a third basis vector, and a decision about what the vertical rate is measured against.

Six derivatives, not nine #

A three-dimensional velocity field has nine partial derivatives. A geodetic network measures six of them, and which six is decided by the fact that a network lives on a surface.

Marks on the ground can be differenced horizontally, so a network forms:

• the four horizontal derivatives of the horizontal velocity, which are the two-by-two tensor the four rungs below measure;
• the two horizontal derivatives of the vertical velocity, which together are the tilt — a rate of rotation of the surface about a horizontal axis, with a magnitude and an azimuth.

The three it cannot form all differentiate with respect to height, and there is no second mark above a mark. ∂u/∂z, ∂v/∂z and ∂w/∂z are not small, not noisy and not badly conditioned. They are absent.

That distinction has a cost attached and it is not rhetorical. A strain rate computed from a horizontal network and reported as the strain rate of the ground is a two-dimensional projection of a three-dimensional object, and the projection is onto the surface the marks happen to sit on. In a compacting basin the vertical shortening is the dominant term and the horizontal extension is its Poisson consequence, so the quantity being reported is the smaller half of a mechanism whose larger half is not in the data at all.

The tilt peaks where the subsidence does not #

The stated field is a Gaussian bowl, chosen because it is smooth and because its gradient has a property worth demonstrating: a Gaussian has zero derivative at its own centre, so the tilt vanishes exactly where the vertical rate is greatest.

The prediction is worth stating because it is a check rather than an observation. For w = w₀exp(−(r/L)²), the derivative dw/dr is largest at r = L/√2 = 84.85 km for L = 120 km. The measurement on the 401-sample profile gives 86 km, which is within the two-kilometre sample spacing.

The consequence is operational. A GNSS station at the centre of a subsiding basin sees the maximum of one quantity and the minimum of the other, and a levelling line run across the centre — the obvious line to run — crosses the tilt’s null. The two instruments disagree about where the interesting part of the basin is, and both are right about what they measure.

The vertical is the larger signal, measured #

The shortfall’s claim was that the vertical is larger. It is larger by a factor of a hundred and sixty-five against a plate interior, and the comparison is fair in the only sense available: both quantities are dimensionless rates per year, both are what one geodetic network measures, and both are quoted in nano-somethings for exactly that reason — the same unit discipline a coordinate is a number with a width imposes on a static position.

Against a plate boundary the comparison runs the other way — the transform boundary reaches 349 nanostrain a year, which is nearly three times the basin’s tilt. So the honest statement is not that the vertical always dominates but that it dominates wherever the horizontal signal is small, which is most of the continental interior and all of the places where somebody is deciding whether a datum still fits.

What a vertical rate is measured against #

Here is the half of the rung that has nothing to do with adding an axis, and it is the half that makes a vertical rate a different kind of quantity from a horizontal one.

A horizontal velocity needs a frame — that is rung two, and the answer there is that the frame changes every velocity and no strain rate. A vertical velocity needs a surface, and the two surfaces on offer are not related by a rigid motion.

A GNSS station reports a rate of change of ellipsoidal height, because that is what the geometry gives. A levelling network reports a rate of change of orthometric height, because that is what a level and a staff give. The two are related by a definition,

$$h = H + N,$$

so their rates are related by the same definition differentiated:

$$\dot{h} = \dot{H} + \dot{N}.$$

Ṅ is not zero whenever mass moves. The geoid is an equipotential surface of the actual gravity field, the actual gravity field has the mass in it, and a basin that subsides because water was pumped out of it has lost the water.

What was computed, and how #

The geoid rate comes from a closed form rather than a model. The potential at the centre of a thin disc of surface density σ and radius R is 2πGσ**R, so a disc whose surface density is changing at σ̇ changes the geoid there at

$$\dot{N} = \frac{2\pi G \dot{\sigma} R}{\gamma}.$$

For the pumped basin — 55 millimetres a year of equivalent water thickness over a disc 120 kilometres in radius — that is 0.282 millimetres a year. Against 18 millimetres a year of subsidence it is 1.57 per cent, which sounds negligible and is not, for a reason of units rather than of size.

Nothing compares a subsidence rate against a geoid rate. What gets compared is a height difference over an interval. Thirty years of that geoid change is 8.5 millimetres, and a first-order levelling loop closes at about 4 millimetres times the square root of its length in kilometres. So the geoid’s own motion is comparable to the closure of the network being used to measure the subsidence, it is systematic rather than random, and it is in the same direction everywhere in the basin — which is the shape of error a network is least able to detect, as the blunder the network cannot see sets out for a different systematic.

The controlled pair is the point of the calculation. Compaction and pumping produce the same subsidence by construction, 18 millimetres a year at the centre of the same bowl. They differ only in whether mass left. The geoid rate is exactly zero for the first and not for the second, so the disagreement between a GNSS network and a levelling network is not an error in either — it is a measurement of the mass, made by the difference of two instruments neither of which was pointed at it.

The dome, which is the same argument at a thousand kilometres #

The basin is 120 kilometres across and its tilt is a local quantity. A postglacial dome is ten times wider, its rate is half as large, and the two facts pull the tilt in opposite directions — which makes it the useful second case rather than a repetition.

A tilt is a rate over a distance, so widening a feature by ten and halving its rate divides the tilt by twenty. The dome’s tilt is therefore a few nanoradians a year where the basin’s is a hundred and twenty-six, and that single ratio explains why the two are studied with different instruments: a basin is a levelling problem and a dome is a GNSS problem, and neither choice is about which is the larger motion.

It also decides which of the two height systems matters. A local subsidence bowl is small enough that the geoid is essentially flat across it, so the whole geoid effect is the 0.28 millimetres a year computed above and it is a nearly uniform offset — the shape of the subsidence survives. A dome is large enough that the geoid’s own change varies across it, so the two height systems disagree about the shape as well as the rate, and a national vertical datum spanning the dome is being deformed by the disagreement rather than merely shifted by it. That is the mechanism behind a grid stopping fitting the ground it was laid on, one rung down, running in the vertical rather than the horizontal.

The geoid term grows with the width of the load #

The 1.57 per cent computed for the basin is quoted as a number about that basin, and the expression it comes from says which parameter it is sensitive to — which is the one that separates the basin from the dome.

Writing the ratio out,

$$\frac{\dot N}{\dot w} = \frac{2\pi G \rho\, R}{\gamma}\cdot\frac{\dot t}{\dot w},$$

with R the radius of the load and ṫ the rate of change of its equivalent thickness. Everything but R is either a constant or a property of the mechanism, so for a given pair of rates the geoid’s contamination of a vertical rate is proportional to how wide the deforming region is.

That single dependence explains the difference between the two cases in this rung better than the numbers do. The basin is 120 kilometres across and its geoid term is 1.6 per cent. A feature ten times wider, moving at the same rates, would have a geoid term ten times larger — sixteen per cent — and would therefore have two height systems disagreeing not by a rounding but by a sixth of the signal.

It also bounds the effect, which is worth having because an unbounded one would make the whole comparison hopeless. Setting the ratio to one gives R = 7,640 kilometres for water: a surface load would have to be the size of a hemisphere before its geoid rate matched its own subsidence rate. So on Earth the geoid term is always a fraction of the vertical motion, never a comparable quantity, and the question is only what fraction.

Two readings follow, and they are the practical content of the whole section.

A local network can ignore it and a national one cannot. At a hundred kilometres the term is under two per cent and is inside the levelling network’s own noise over a decade; at a thousand it is a sixth of the signal and is systematic across the whole network, which is the regime a national vertical datum spanning a rebounding region is in.

And the term is a measurement of the mass. Since it is proportional to R and to ṫ, the disagreement between a GNSS network and a levelling network over the same ground is a reading of how much mass left, obtained from two instruments that were both pointed at heights. That is the same argument the compaction–pumping pair makes, generalised: the difference of two height systems is a gravimeter nobody installed.

Where the model stops #

The disc is not a mantle. The rebound case comes out at 1.8 per cent, and the published figure for Fennoscandia is around six: the ratio of geoid rate to uplift rate is close to 0.06 there. The gap is the model’s, and it is understood — the disc represents a surface mass and the actual mechanism moves mantle material at depth, which contributes to the geoid without contributing surface load. The closed form is right about the sign, right about the order, and short by a factor of three, and it is used here because it can be checked from the formula rather than believed.

The tilt is not the full vertical story. A tilt is two of the three missing pieces of information about the vertical field; the third, ∂w/∂z, is the compaction rate itself and is what a borehole extensometer measures. That instrument is not a geodetic network and its data is not a coordinate, which is why it sits outside this collection.

The bowl is stated, not fitted. Real subsidence fields are not Gaussian, and the tilt maximum’s position is a property of the shape rather than a general result. What is general is that a smooth maximum has zero gradient, so the tilt and the rate always peak in different places; where the second one peaks depends on the field.

And nothing here is a rheology. The relation between the horizontal extension on the flank of a basin and its vertical shortening is a material property, and this rung states two independent fields rather than deriving one from the other. Making them consistent would require a constitutive model, which is a different subject — and one this collection has already declined once, in a mountain is not a buried sphere, for the same reason.

The generalisation #

This anchor’s whole method has been to take the instrument the collection built for measuring maps and point it at the ground. Four rungs did that in two dimensions and the answer was clean: the same six numbers, about the Earth rather than about the page.

The third dimension breaks the symmetry, and it breaks it in a way that is characteristic of the whole site. A horizontal position is a coordinate and a height is a measurement against a surface, so the two halves of a three-dimensional coordinate are not the same kind of object — which is exactly what height above what? establishes for a static coordinate, arriving here as a statement about a rate.

The general pattern is the one what survives a change of coordinates states: a quantity is a measurement when it survives the conventions used to express it. A horizontal strain rate survives a change of frame. A vertical rate does not survive a change of height system, so a vertical rate quoted without one is not yet a number.

Who found it, and when #

Levelling networks have measured tilt since the nineteenth century, and the separation of geometric height change from geoid change is the central problem of every modern vertical reference frame — the International Height Reference System exists to state it. The relation ḣ = Ḣ + Ṅ is a definition and is in every geodesy textbook.

What this rung adds is nothing about the geodesy. It is that the collection’s own instrument, built to audit projections, had a shortfall recorded against it in exactly this place, and paying the shortfall required admitting that the two halves of the answer are different kinds of quantity — six observable derivatives rather than nine, and a rate that is not a rate until a surface is named.

What this rung does not repair #

The shortfall recorded against rung 4 asked for the vertical, and the vertical is here. Two things it named are still not here and it is worth saying which.

The first is a three-dimensional strain tensor. Nothing above is one: what is here is a six-element observable block with a shape rather than a symmetric three-by-three, and the missing column is not a limitation of the code. A network on a surface cannot produce it, so no amount of library work would.

The second is the coupling between the horizontal and vertical halves. The basin’s horizontal extension and its vertical shortening are one mechanism, and this rung states them as two independent fields whose numbers happen to be reasonable. Making them one field requires a material model, and the moment a material model enters, every number stops being a closed form and starts being a consequence of assumed parameters — which is the trade this collection has refused everywhere it has come up.

Where the ladder goes next #

Five rungs have taken one instrument round the ground: its horizontal gradient, its frame-independence, the survey clock it drives, the map’s contribution to it, and now the vertical and the surface that vertical is measured against. What is missing is the time axis — every field here is steady, and the interesting ones are not. A coseismic step, an aseismic transient, and a seasonal signal are three different objects with the same units, and a network sampling once a year cannot tell them apart.