A levelling loop reads the circulation, and the geoid has none

A triangle reads the geoid’s curvature, and that one converges found what a surveyor’s triangle actually measures of the surface heights are referred to. Not the curvature at a point, which the geoid does not have, but the total curvature over the triangle — and to first order that is the flux of the deflection of the vertical outward through the triangle’s three sides, divided by the Earth’s radius. A fifty-kilometre triangle reads about 0.038 seconds of arc of it, a number that settles as the geoid model is refined, and forty-six times smaller than the noise of three one-second angles.

It ended on a symmetry. A closed curve has two line integrals of a vector field round it: the component across the curve, which is the flux, and the component along it, which is the circulation. A triangle observed with a theodolite takes the first. A levelling network is built out of closed loops, and a loop carries a height along the curve, which looks like the second. Whether the geoid’s share of a triangle’s excess and its share of a levelling loop’s misclosure are correlated, independent, or one number seen twice was left as the question a triangle alone could not ask.

The answer has three parts, and each is measured below on one stated patch of ground. The geoid contributes nothing whatever to the circulation. A levelling loop run on the ground does misclose, by an amount that is not a property of the geoid at all but of gravity crossed with the terrain. And the two readings round one loop are uncorrelated — as unrelated as two integrals of one field can be.

Two integrals round one loop #

The patch is stated so that every number can be traced. Its geoid is a sum of undulations with the degree variances of Kaula’s rule — the spectrum the triangle’s reading was computed from — between wavelengths of 10 and 400 kilometres, continued upward from the geoid as a harmonic potential. Its ground is a mountain terrain, mean height 2,500 metres and root-mean-square relief 600, with wavelengths from 4 to 150 kilometres. Its normal gravity is the field about latitude 45°, increasing northward by 0.81 milligals a kilometre and decreasing upward by 0.3086 milligals a metre.

On it, a triangle fifty kilometres on a side. Along each side the deflection of the vertical has a component across the side and a component along it, and each can be integrated from one corner to the next. The first figure gives both for all three sides. Across the sides the three integrals are −0.689, −0.538 and +0.499 metres, and they sum to −0.727 metres of flux, which divided by the Earth’s radius is −0.0235 seconds of arc of excess: the geoid’s share of this triangle’s reading, of the size the triangle essay found for triangles in general.

Along the sides the three integrals are −0.235, +0.275 and −0.040 metres, individually as large as the ones across, and they sum to zero. Not to something small: to zero, with the digits that survive being the quadrature’s.

A slope carried round a loop comes back to itself #

The reason is one line. The deflection of the vertical is, to first order, the negative gradient of the geoid’s height: $\boldsymbol\varepsilon = -\nabla N$. Its component along a path, integrated, is the change in $N$ from one end to the other,

$$\int_A^B \boldsymbol\varepsilon\cdot\mathbf t\,ds \;=\; N(A) - N(B),$$

and round a closed loop the two ends are the same place. Carried round any loop, however large, whatever the geoid looks like inside it, the deflection returns the height it started with. This is what astronomic levelling does — it recovers the geoid’s shape along a line by integrating the deflections observed at stations along it, which how far the plumb line bends describes from the station’s side — and the statement is that a loop of astronomic levelling, reduced to the geoid, closes identically.

The figure puts a number on “identically”. Across a hundred and twenty triangles dropped at random on the patch, the flux through the sides has an rms of 1.147 metres. The circulation along them has an rms of 2.6 × 10⁻⁵ metres when each side is integrated in 160 steps, and 1.5 × 10⁻⁶ when it is integrated in 640 — seventeen times smaller for a step four times shorter, which is how the error of the rule used falls and not how any physical quantity would. What is left is arithmetic. The circulation is zero.

So the first part of the answer is already a complete one: the geoid’s flux and its circulation are not the same number seen twice. One carries the whole of what the triangle reads; the other carries nothing, round any loop at all. A direction carried round a loop found the other thing a loop can carry — a bearing, turned on return by the curvature enclosed — and that turn belongs to the surface’s curvature as surely as a height carried round a loop does not belong to its slope. A quantity that is the gradient of something cannot be turned by going round.

What a staff reads is gravity crossed with the ground #

That would end the matter if a levelling loop integrated the geoid’s slope. It does not. A spirit-level and a pair of staves measure, at each set-up, the height of the ground at the forward staff above the ground at the back one, along the local vertical — and a levelled height is not a distance is the reminder that such increments do not add up to anything that closes. The quantity that closes is the geopotential number, the sum of each increment multiplied by the gravity where it was measured, because that sum is a difference of potential and a potential has one value at each place:

$$\oint g\,dn \;=\; 0 \qquad\text{so}\qquad \oint dn \;=\; -\oint \frac{g - \gamma_0}{\gamma_0}\,dn.$$

The raw increments round a loop miss by the second integral. With $g$ the gravity at the ground and $H$ the ground’s height, and Green’s theorem turning a line integral round a loop into an integral over what it encloses, the misclosure is

$$\oint dn \;=\; -\frac{1}{\gamma_0}\iint (\nabla g \times \nabla H)\,dA,$$

the Jacobian of gravity and height over the loop — the rate at which gravity changes across the ground’s slope rather than along it.

The figure draws the integrand over the loop. It is patchy and of both signs, largest where a ridge runs across a gradient of gravity, and zero wherever gravity changes only up and down the slope. Summed over the triangle it is −22.5 millimetres; summed as levelled increments round the triangle’s sides it is also −22.5. The two computations share nothing but the fields: one walks the perimeter in 2,400 steps with gravity taken at each step’s mean height, and the other differences gravity and height at about twenty-five thousand points inside. They agree to within a thousandth.

That is what a levelling loop reads, and none of it is the geoid’s curvature. It is not even a property of gravity alone: take the mountains away and it vanishes, however strong and however irregular the gravity field.

The largest change in gravity is invisible to the loop #

The Jacobian has a feature that decides what it sees. Any part of gravity that depends on height and nothing else has its gradient parallel to the ground’s, and the cross product of two parallel vectors is zero. Such a part contributes nothing to the misclosure, whatever its size.

The free-air gradient is the extreme case. Gravity falls by 0.3086 milligals for every metre climbed, and over the patch’s relief — from about 300 to about 4,600 metres — that is a change of 1,333 milligals, by far the largest variation of gravity anywhere on it. Its contribution to the misclosures of a hundred and twenty loops is 6 × 10⁻¹⁰ millimetres, which is the precision of the arithmetic. A gravity that is a function of height is a gravity whose level surfaces are parallel to each other along every contour of the ground, and the staff never feels it.

The two parts that do contribute are the ones that vary sideways. Normal gravity increases northward at 0.81 milligals a kilometre, so a loop fifty kilometres across has 40.5 milligals of it, and that gradient crossed with the east–west slopes of the ground misses by 18.1 millimetres rms. This is exactly the effect a levelled height is not a distance computed for a rectangle a hundred kilometres long at a thousand metres, where it came to 82 millimetres: the orthometric correction. The anomalous field — the gravity of the geoid’s own undulations, about 20 milligals rms at the ground here — adds 22.0 millimetres more, and the two together give 30.8, adding roughly as independent parts do.

So the geoid does reach a levelling loop, but not through the curvature a triangle reads. It reaches it through its gravity, and only through the part of that gravity which changes across the ground’s contours.

The star and the staff agree on the ground #

There is one more loop a surveyor can run, and it closes the circle back to the deflection. Astronomic levelling on the geoid closes identically; astronomic levelling on the ground does not. A theodolite at a station on a mountain observes the direction of the plumb line there, and the deflection it reads is the horizontal gradient of the potential at that station’s height, not the gradient of the geoid.

The two differ in one term. Along the ground the potential changes by its horizontal gradient times the step plus its vertical gradient times the change of height, and the vertical gradient is minus gravity:

$$dW \;=\; \nabla_{\!h} W\cdot d\mathbf l \;-\; g\,dH.$$

Round a loop $\oint dW = 0$, so the circulation of the surface deflection is $\oint g\,dH/g$ — the same integral of gravity against height that the staff misses by, reached from the other end.

Computed both ways round the same hundred and twenty loops, the two misclosures sit on the diagonal. They differ by 0.012 millimetres rms, four parts in ten thousand of the misclosures themselves, which is the size of the terms a first-order account drops. A staff and a star, run round one loop on the ground, miss by one number.

That identity is what the reduction of astronomic observations to the geoid is for. How far the plumb line bends measured how the plumb line curves between the ground and the geoid under a mountain; carried round a loop, the correction that curvature demands is this Jacobian, and once it is made the astronomic loop closes as the geoid’s gradient requires.

The two readings are unrelated #

The question the triangle left is now well posed. Round one loop there are two readings: what a triangle of angles reads of the geoid, which is the flux, and what a levelling loop misses by, which is the Jacobian. Are they related?

Across the hundred and twenty loops the correlation is −0.008. Between their sizes, whichever way they point, it is −0.12, which for a hundred and twenty pairs is within the scatter chance allows. The two readings are uncorrelated, and there is a reason they must be rather than merely happen to be.

The flux is linear in the geoid and does not involve the terrain at all. The misclosure is a product of gravity and the terrain’s slope. On this patch gravity does not depend on the terrain, so turn the terrain upside down — every hill a valley of the same shape — and every misclosure changes sign while every triangle’s reading stays exactly as it was. Averaged over terrains, the product of the two has to vanish. They are not independent: a patch whose gravity field is violent will tend to produce large values of both. But neither says anything about the sign or the size of the other.

The mountains move one reading and not the other #

The patch so far has had no mountains in its gravity: the terrain was carried by a root that cancels its attraction, and gravity came only from the geoid’s undulations. Real mountains are partly compensated and partly not, and their own rock has a gravity field of its own.

Giving the mountains their full mass — a layer of rock of the standard crustal density, supported by nothing beneath — multiplies the triangle’s reading by three, from 0.0371 to 0.1124 seconds of arc, because a mountain’s attraction raises the geoid over it and the geoid’s curvature follows the mountains. The levelling misclosure’s anomalous part moves by ten per cent, from 22.0 to 24.1 millimetres.

The asymmetry is the Jacobian again. The mountains’ own gravity is nearly a function of the mountains’ height: largest over the peaks and least in the valleys. A gravity that follows the ground crosses its contours at almost no angle, so it adds almost nothing to the loop however large it is — what it adds comes only from the way a mountain’s field smooths with distance above it, which differs from wavelength to wavelength. A deflection is the slope of a mass is the one-mountain version of the first half of this: mass changes the geoid’s slope directly. The loop is blind to the part of the mass that the ground’s own shape describes.

One of the two readings can be observed #

A reading that exists and a reading anyone can take are different things, and here they part completely.

Precise levelling is good to about a millimetre over a kilometre of line, with its error growing as the square root of the distance, so a loop fifty kilometres on a side, 150 kilometres round, carries about twelve millimetres of noise. The misclosure it reads on the mountain patch is 30.8 millimetres rms, two and a half times that. From loops of twenty-five kilometres upward the reading stands above the noise; at ten it is level with it; at two hundred it has fallen back a little, because the terrain here has no wavelengths longer than 150 kilometres and the loop begins to average them away.

The triangle’s reading never comes close. Against the noise of three one-second angles it is a two-hundred-and-fiftieth at ten kilometres and a fourteenth at two hundred.

This is why levelling networks carry gravity and triangulation networks never carried the geoid. A levelling party in mountains that did not observe gravity along its lines would find its loops misclosing by several times their noise, systematically, and the misclosures would be real: they would be the Jacobian of a field the party had not measured. Every national levelling system reduces with observed gravity for exactly that reason, and a height that is not a length sets out what the heights mean once it has. A triangulation network that ignored the geoid’s curvature lost a fiftieth of its angle noise and could not have noticed.

The patch and what was checked on it #

A gradient must have no circulation. Round every one of the hundred and twenty loops the deflection’s circulation must sit below a ten-thousandth of the flux’s rms; it sits at 2.6 × 10⁻⁵ metres against 1.147, and falls seventeenfold with a step four times shorter.

Gravity that depends on height alone must contribute nothing to a loop. The free-air gradient changes gravity by 1,333 milligals over the relief and must contribute less than a ten-thousandth of the misclosure; it contributes 6 × 10⁻¹⁰ millimetres, the precision of the arithmetic, once each levelled step is reduced with gravity at its mean height — as a levelling computation does.

The line integral must equal the area integral. Round the stated loop the levelled increments give −22.5 millimetres and the Jacobian over the enclosed ground gives −22.5, agreeing to within a thousandth.

Spirit and astronomic levelling on the ground must miss by one number, to a five-hundredth of the misclosure. They differ by 0.012 millimetres against 30.8.

And the finding must be there to fail. The two readings must be uncorrelated, below 0.15 in magnitude, and the mountains’ mass must more than double the triangle’s reading while moving the loop’s by less than fifteen per cent. It triples one and moves the other by ten.

Where the patch is a model #

The fields are stated, not surveyed. The geoid’s undulations are Kaula’s spectrum over one band of wavelengths, the terrain is a stated random surface, and the two are independent unless the terrain’s mass is added. Every size above — the 30.8 millimetres, the ratio to levelling noise, the factor of three — belongs to this patch, and a real mountain range would give its own. The statements that do not depend on the patch are the three identities: the geoid’s circulation is zero, the levelling misclosure is the Jacobian, and the free-air gradient drops out.

The plane replaces the sphere. A loop of a few hundred kilometres on a plane misses the sphere’s own curvature, which enters the triangle’s excess but not either misclosure; the triangle’s reading here is only the geoid’s share.

First order throughout. The deflection is the gradient of the geoid only to first order in the geoid’s height, and the astronomic and spirit misclosures agree only to that order, which is the four parts in ten thousand by which they differ.

The terrain’s mass as a sheet. Rock supported by nothing is placed as a thin layer at the datum rather than as a body with height; a real mountain’s attraction at its own summit is not what a sheet predicts, and a partial root would put the answer between the two cases drawn.

Still open: whether the two loops together say where the mass is #

The triangle and the loop read two integrals of fields that the same mass produces: the triangle the geoid’s flux, which the mountains’ own mass tripled, and the loop gravity crossed with the ground, which the same mass barely touched. Two observations that respond so differently to one cause are the raw material of an inversion. A network that ran both round the same loops would hold one quantity that is mostly the compensated part of the field and one that is mostly the uncompensated part.

Whether that difference is large enough, in any real network, to say how much of a mountain range is carried by its root — a question normally answered from gravity alone, with a density assumed — and whether the triangle’s reading, a fiftieth of its own noise in a single triangle, could ever be averaged up to where it says anything, are questions one stated patch cannot settle. How many triangles it takes priced the averaging, and the triangle essay found it cannot help: the geoid’s share of a network’s excess is the flux through its outline, however many triangles fill it.