A levelling loop weighs the root, and a triangle weighs the mountain

A levelling loop reads the circulation, and the geoid has none put two observations side by side round one closed curve on a stated mountain patch. A triangle of angles reads the flux of the deflection of the vertical through its sides, which is the geoid’s total curvature inside it. A spirit-levelling loop run on the ground misses by the Jacobian of gravity and height over what it encloses — gravity crossed with the ground’s slope. The two were unrelated, and they answered the mountains’ own mass very differently: giving the terrain its full weight of rock tripled what the triangle read and moved the loop by a tenth.

That essay ended on a hope and a doubt. Two observations that respond so differently to one cause are the raw material of an inversion: a network that ran both might say how much of a range is held up by its root, which is a question normally put to gravity alone. And the triangle’s reading, a fiftieth of its own noise, might never be averaged to where it says anything.

Both halves can now be measured, and the first comes out the other way round from the hope. The triangle reads the mountain. The loop reads the root.

A root is the mountain again, upside down and far away #

A range of mountains is a load, and the ground under it does not carry the load by strength. The crust floats on a denser mantle, and where the surface stands high the light crust beneath it reaches down further, displacing mantle with something lighter. Seen from above, the range is a mass excess at the surface sitting over a mass deficit at depth, and the deficit is called its root.

The idea is a reading of a survey’s failure. In the 1850s the Great Trigonometrical Survey of India found that the latitudes of two stations south of the Himalaya, measured by the stars and carried between them by triangulation, disagreed by a little over five seconds of arc. John Henry Pratt computed in 1855 what the mountains’ attraction should have done to the plumb lines at the two stations, and got nearly sixteen. The range was pulling a third as hard as its visible rock should. George Airy proposed in the same year that the mountains were floating, carried by roots of light crust whose missing mass cancelled most of their pull — the idea now called isostasy. It was found with the plumb line: a latitude from the stars is not a latitude on the ellipsoid, and the difference between the two is the deflection of the vertical that the mountains’ mass produces.

The stated patch carries the same idea in its simplest form. Its terrain is the one the levelling-loop measurement used: mean height 2,500 metres, root-mean-square relief 600, wavelengths from 4 to 150 kilometres, sitting on a background geoid with Kaula’s spectrum between 10 and 400 — the spectrum whose curvature has a value only where the series is stopped, and whose flux through a triangle does not need one. The rock is a sheet at the datum, 2,670 kilograms a cubic metre times the terrain’s height. Its root is the same sheet turned over and placed at depth $D$, carrying a share $c$ of the rock’s mass with the opposite sign. At the datum the potential of each of the terrain’s waves is multiplied by

$$1 - c\,e^{-kD},$$

so a root cancels the terrain’s long wavelengths, whose fields reach down to it, and leaves the short ones, whose fields die away before they arrive. At $c = 0$ the mountains are carried by nothing; at $c = 1$ and $D = 0$ they are carried by a root that cancels them entirely, which is the patch with no terrain mass that the levelling-loop essay started from.

Every reading round a loop is linear in the field, so each one splits into three parts: the background geoid, which has nothing to do with the terrain here; the mountain, uncompensated; and the root, multiplied by $c$. Given the ground, the second and third are computable. The question a survey would be asked is $c$.

What the loop sees of a root #

The profile above is the mechanism, and it is worth reading slowly.

The rock’s own gravity at the ground is the terrain seen from a little below: a sheet at the datum, 2,500 metres beneath the mean ground, attracts with a field that is the terrain’s shape blurred over a few kilometres. Over the whole patch it correlates with the height of the ground at 0.99. The root’s gravity is the terrain seen from thirty kilometres below it, blurred so far that only the broad swells survive, and its correlation with the local height is −0.85 — negative because a deficit pulls the other way, and well short of one because a thirty-kilometre blur has forgotten the individual ridges.

A levelling loop is blind to exactly the part of gravity that is a function of the height under the staff, because the misclosure is gravity crossed with the slope, and gravity that rises and falls with the ground crosses its contours at no angle. So what matters is not how strong each field is but how much of it is free of the local height. Of the rock’s 54.0 milligals rms, 7.8 are free. Of the root’s 10.0, 5.3 are. The root is a fifth as strong and two thirds as visible.

Round a hundred and twenty fifty-kilometre loops the three readings respond as the profile predicts. The triangle’s excess takes 0.1112 seconds of arc from the uncompensated mountain and 0.0172 from a full root at thirty kilometres, sixteen per cent as much. A deflection observed at the loop’s centre takes 9.55 and 1.59 seconds, seventeen per cent. The loop’s misclosure takes 5.68 millimetres from the mountain and 4.80 from the root, eighty-five per cent as much.

The triangle and the station agree because they are one reading: a triangle’s share of the geoid’s curvature is the flux of the deflection through its sides, so whatever a root does to the deflection at each point it does to the triangle as an average. Both are first derivatives of the potential at the datum, and a root thirty kilometres down has a weak first derivative under a fifty-kilometre loop. The loop is a different kind of reading. It does not ask how strongly gravity varies but in which direction, against the ground, and a root’s field varies in directions the ground does not.

It is worth stating what this does to the levelling-loop essay’s own result. There, the uncompensated mountains moved the loop’s anomalous misclosure from 22.0 millimetres rms to 24.1, a tenth, and the conclusion was that the loop was nearly blind to mass. It is nearly blind to the rock. Add a root at thirty kilometres and the total comes to 23.1, a different figure made of different parts, and the parts are what a survey would be trying to tell apart.

The deeper the root, the more of what remains is the loop’s #

The depth decides how much of the root reaches the surface at all, and every reading weakens as it falls: at a hundred kilometres the root is a few per cent of the mountain for all three. But they weaken at different rates, and the loop’s curve has a shape the other two do not.

A root five kilometres down is close enough to the rock to undo most of it: the triangle and the station feel it at seven tenths of the mountain’s strength, and the loop at more than one and a half times. From there the triangle and the station fall away steadily — half at ten kilometres, a sixth at thirty, a tenth at forty. The loop holds up first. A root at ten kilometres moves the loop by more than half again what the mountain does, because at that depth its field has been blurred just enough to cross the contours and not yet enough to have faded. Only from fifteen kilometres down does the loop’s response fall, and at every depth it stays at least twice the triangle’s: five and a half times at thirty kilometres, six at forty, seven at sixty.

That is the inversion of the hope the levelling-loop essay ended on. It expected one quantity that was “mostly the compensated part of the field and one that is mostly the uncompensated part”. There are such quantities, and the loop is the one that is mostly compensation.

A triangle cannot tell a root from lighter rock #

A root that cancels a sixth of what the mountain gives a triangle has a second difficulty besides its size, and it is the one that would stop a survey even with unlimited triangles.

Plotted loop by loop, the triangle’s response to the root is very nearly a fixed negative multiple of its response to the mountain: the correlation is −0.85. A root that removes the same fraction of every triangle’s mountain reading is indistinguishable, to a triangle, from rock that is lighter by that fraction. And the rock’s density is not a known constant. The 2,670 kilograms a cubic metre used here is the conventional crustal value; real mountains range by a tenth either side of it, which is the same order as the effect being sought.

So a survey that does not know the density must estimate it alongside the root, and two regressors that point nearly the same way share the information between them. The variance of the root’s estimate is multiplied by $1/(1 - r^2)$: by 3.7 for the triangle. The loop’s two responses correlate at −0.67, a pattern of their own, and the same penalty for the loop is 1.8. The station’s deflection, which reads what the triangle reads, pays 4.1.

This is the ambiguity Pratt and Airy argued about without the language for it. Pratt’s own account of the Himalayan deficit was not a root at depth but lighter rock under the mountains, the deficit spread through a column; Airy’s was a root. For a measurement that sees only the first derivative of the potential at the datum, the two readings are nearly the same, and a century of geodesy since has been spent finding observations that separate them.

What it costs to weigh a root #

A response is only a measurement once it is read against what else moves the same reading. For each of the three, the noise has two parts: the observer’s own, and everything in the field that the survey cannot compute — here the background geoid, whose undulations owe nothing to the terrain. The sensitivity is the root’s part at $c = 1$. Pinning $c$ to a tenth from independent observations takes $100 \times (\text{noise}/\text{sensitivity})^2$ of them.

The triangle. Three one-second angles carry 1.73 seconds of noise against a root’s 0.0172: a hundredth. How big a triangle it takes found three such angles too coarse to read even the sphere’s own curvature to a per cent in a triangle smaller than 281 kilometres on a side, and the sphere gives a fifty-kilometre triangle 5.49 seconds of excess — three hundred times what a root gives it. Pinning $c$ to a tenth takes a million fifty-kilometre triangles, and 3.7 million if the density is unknown. How many triangles it takes found that averaging can at best divide the noise by the square root of the count, and here the count is larger than every triangle ever observed. Nothing about a better theodolite rescues it: at a tenth of a second the noise is still ten times the root.

The loop. A fifty-kilometre loop is 150 kilometres of levelling, at a millimetre per square root of a kilometre twelve millimetres of noise, and the background field adds twenty-two more. Against a root’s 4.8 millimetres that is a fifth, and pinning $c$ takes 2,742 loops — 5,027 with the density unknown. That is roughly the levelling of a large country many times over. It is three hundred and seventy times fewer than the triangles need.

The station. A deflection observed to a second of arc against a root’s 1.59 seconds and a background of 3.6 is the best of the three per observation, at 545 stations. A modern astronomic position to three tenths of a second barely improves it, to 509, because the noise is the geoid’s background and not the instrument. With the density unknown, 2,251.

The station’s lead is the history in a figure. Compensation was found at a handful of stations because the Himalaya is not this patch: its root is enormous and its long wavelengths, which the patch does not have, are exactly the ones a root keeps. When John Fillmore Hayford set out half a century later to measure the depth of compensation across the United States, he used deflections at several hundred astronomic stations, and found a depth a little over a hundred kilometres — a count of the order the arithmetic here asks for.

The loop’s place in the ranking is the finding that stands. Per observation it is five times worse than a station and three hundred and seventy times better than a triangle, and unlike either it pays only 1.8 for not knowing the rock. A levelling network that ignores a root does not merely carry an error: it carries one whose pattern differs from the mountain’s, and could in principle be read.

The arithmetic is checked by doing it #

The budget assumes the observations are independent. Loops in one range are not obviously so: they share a background whose wavelengths reach 400 kilometres, in a patch a thousand across.

Both assumptions are tested by simulating the survey. Misclosures are built from each loop’s background, rock and root at $c = 0.6$, with levelling noise added, and $c$ is recovered by least squares on the root’s pattern. With backgrounds drawn independently, the estimates centre on 0.60 at every count from a hundred upward and spread as the budget says: 0.480 from a hundred loops against 0.463 predicted, 0.115 from sixteen hundred against 0.116. At twenty-five loops the spread is an eighth wider than predicted, which is the ratio estimator’s own skew when its denominator is small.

Then the same two hundred loops are kept in one range and the background redrawn twenty-four times beneath the same mountains. The twenty-four estimates spread by 0.31, against 0.33 for two hundred independent loops: the range is worth about 220. A levelling misclosure is the background’s short wavelengths crossed with the ground’s slopes, and those decorrelate within a loop or two, so a range’s loops behave as if they were independent even though its geoid is one surface. The one draw that lands far from the truth — an estimate of −0.34 among twenty-four — is what a single range risks, and its size is the size the budget said it would be.

What each number was held to #

A root carrying nothing must change nothing. At $c = 0$ the patch must read exactly as the uncompensated one; it does, to rounding.

A root at no depth must cancel the rock. At $c = 1$ and a depth of a tenth of a metre every reading must return to the patch with no terrain mass, to a thousandth; it does.

The split must be linear. A half-compensated patch must read as the mountain plus half the root. The triangle does so exactly. The loop divides each levelled step by gravity, so it is linear only to first order, and agrees to 0.02 millimetres, a five-hundredth of its misclosures’ rms.

The finding must be there to fail. At thirty kilometres the loop must respond to the root at least half as strongly as to the mountain, and the triangle at most a quarter as strongly. They read 85 and 16 per cent.

And the estimate must behave as the arithmetic says, centred on the true share within three standard errors and spread within a fifth of the predicted spread, from a hundred and from four hundred independent loops. It does.

Where the sheet stops being a mountain #

The patch is stated, not surveyed. The terrain is a random surface with stated relief and wavelengths, the background geoid is independent of it, and every count above belongs to this patch. In a real range much of the short-wavelength geoid is the terrain, so the background a survey cannot compute is smaller than Kaula’s whole spectrum and every count would fall; how far is a property of the range.

Mass as sheets. The rock is a thin layer at the datum and the root a thin layer at depth, so the rock’s attraction at its own summit and the root’s finite thickness are both missing; how far the plumb line bends is the measurement of what a column of rock does to the plumb line on its way down, and a sheet has no column. Pratt’s alternative — a deficit spread through the column under the mountains rather than concentrated at a depth — is a different filter on the same terrain and is not drawn.

One loop size. Every reading is round a fifty-kilometre triangle. The loop’s advantage comes from the root’s field crossing the ground’s contours, and a larger loop encloses more of the broad swells a deep root follows; how the ranking changes with the loop’s size is not measured.

Gravity is not observed. A national levelling network observes gravity along its lines, and with gravity in hand a misclosure is computable and carries nothing a gravimeter has not already said; a height that is not a length sets out what the heights mean once it has. Everything here is about what the geometry of the three readings could say on its own.

Still open: whether the ratio of two readings reads the root’s depth #

The three curves against depth are not proportional. The loop’s response peaks near ten kilometres and falls slowly; the triangle’s falls from the start. So the ratio of what a loop and a triangle read of the same root is a function of the root’s depth — about two at five kilometres, five and a half at thirty, seven at sixty — and depth is the other number isostasy has always been asked for: Hayford’s hundred kilometres was a depth, not a share.

Whether that ratio is steep enough, against the noise both readings carry, to say at what depth a range is compensated rather than only how much, and whether a survey that observed loops of several sizes could do it with one kind of reading alone, are questions one loop size and one stated root cannot answer. A deflection is the slope of a mass gives the closed form for a single buried body, which is where such a ratio could be checked exactly before it is trusted on a range.