A loop reads a root's depth from its own pattern, and a triangle reads none of it

A levelling loop weighs the root, and a triangle weighs the mountain asked how much of a mountain range’s mass is carried by a root of light crust beneath it. On a stated range with the root as a sheet thirty kilometres down, a levelling loop’s misclosure moved 85 per cent as much for a full root as for the range’s own rock, and a triangle’s spherical excess 16 per cent. A root’s gravity is a smoothed copy of the ground’s, upside down and far away, and a loop, which reads gravity crossed with the slope of the ground, sees the smooth part that a triangle does not. The essay counted what it took to weigh the root: 2,742 loops to pin its share to a tenth, or about a million triangles.

It held the root’s depth fixed at thirty kilometres, and it ended on the number it had not asked for. The loop’s response to a root peaks near ten kilometres of depth and falls slowly, while the triangle’s falls from the start, so the ratio of what the two read of the same root is a function of the depth — about two at five kilometres, five and a half at thirty, seven at sixty. Depth is the other thing isostasy has always been asked for. Hayford’s depth of compensation, a little over a hundred kilometres, was a depth, and so is the thirty kilometres of crust Airy’s hypothesis is usually given. If the ratio of two readings could say it, a network that ran both would answer a question normally put to gravity.

The ratio of two readings is a ratio of one reading and noise #

A ratio is a gauge of depth only if both of its terms are measured, and the earlier essay had already found the reason the triangle’s term is not. A triangle’s excess — the geoid’s curvature summed over what it encloses, which a triangle reads the geoid’s curvature, and that one converges showed is a well-defined reading — is read from three angles each good to a second, and so carries an observing error of 1.7 seconds, and the background geoid adds very little to that. Its response to a root carrying all of the terrain’s mass is three hundredths of that noise at ten kilometres, a hundredth at thirty and two thousandths at sixty. At the share the earlier essay assumed, six tenths, every one of those is smaller again.

The loop is different in kind. Its noise is larger in absolute terms — the background geoid’s own misclosure, 21 millimetres round a 50 km loop, and a millimetre per root kilometre of levelling, together 25 millimetres — but its response to a full root is more than a third of that noise for roots between five and fifteen kilometres deep, a fifth at thirty, and still a twentieth at sixty. Loops of 25 and 100 kilometres a side follow the same curve within a few per cent: a bigger loop sees more of the root and more of the background in about the same proportion. A triangle is the opposite case: how big a triangle it takes found its excess growing with its area against an angle noise that stays fixed, so a bigger triangle always reads better.

So the ratio the earlier essay drew is real as a ratio of responses and useless as a ratio of readings. Its denominator, the triangle’s response, sits under the triangle’s noise by a factor of thirty for a root at ten kilometres and five hundred for one at sixty. Averaged over a million triangles it would be measured; averaged over the hundreds a survey actually observes — Struve’s arc had 141, which how many triangles it takes priced — it is noise. The question becomes whether the loops alone carry the depth.

The depth is in the pattern from loop to loop #

A root at depth $D$ cancels each wave of the terrain’s potential by a factor $1 - c\,e^{-kD}$ at the surface, with $k$ the wave’s wavenumber. A shallow root cancels short wavelengths as well as long ones; a deep one cancels only the long ones, because the short ones’ fields have died away before they reach it. The misclosure a root adds to each loop is therefore not the same pattern at every depth scaled up or down. It changes shape: a shallow root’s contribution follows the ground’s short wavelengths, a deep root’s follows only its broad swells.

That is information the ratio did not need. Over many loops placed across a range, the root’s contribution at one depth is a different set of numbers from its contribution at another, and the survey’s misclosures, less the background and the mountain’s own part, should match one of those sets better than the others. The measurement here does exactly that. A survey is simulated as the background plus the mountain plus $c$ times the root at the true depth plus observing noise, at loops placed at random over the range. The share $c$ is then fitted by weighted least squares at each of fourteen candidate depths from 5 to 100 kilometres, and the depth chosen is the candidate whose best fit leaves the least misfit.

On a survey with no noise and no background the procedure is exact: the true depth fits perfectly and every other depth leaves a misfit, so the root’s pattern at one depth is not a scaled copy of its pattern at any other. With the depth stated correctly, the share comes back unbiased, 0.59 for 0.6 at 1,600 loops, as the earlier essay’s estimate did. What is unknown is how much of the depth survives the noise.

One survey’s misfit against the depth tried #

One survey shows the shape of the answer. With 1,600 loops of 50 kilometres and the root carrying six tenths of the terrain’s mass, the misfit as a function of the depth tried is a well, and its width is the precision. For a root at ten kilometres the well is narrow: depths from 10 to 15 kilometres are within one unit of the best, and a depth of 5 or of 25 is three to six units worse. For a root at thirty it is broad and shallow, from 25 to 60 kilometres within one unit. For a root at sixty there is no well: every depth from 25 to 100 kilometres is within one unit of the best, and the survey chose 80.

The difference follows from the response curve. A shallow root’s contribution to the misclosures is large — a third of the noise per loop — and its shape changes quickly with depth, because the short wavelengths it still cancels are the ones that switch off first as it deepens. A deep root’s contribution is a twentieth of the noise per loop, and what is left of it is the terrain’s longest swells, which every deep root cancels in nearly the same proportions. Twice as deep is mostly fainter rather than differently shaped, and a fainter copy of one pattern is what a smaller share would also produce.

The resemblance can be put as a number. Across the 160 placements, the misclosures a root adds at ten kilometres and at fifteen correlate at 0.993; at thirty and forty, 0.994; at forty and sixty, 0.986; at thirty and sixty, 0.963. Two patterns that correlate at r differ, once each is scaled to fit the other, by a remainder of $\sqrt{1 - r^2}$ of their size, and that remainder is all a survey has to tell the two depths apart with. Between forty and sixty kilometres it is a sixth of the sixty-kilometre pattern, which is itself a twentieth of a loop’s noise: under a hundredth of the noise per loop, which takes something like ten thousand loops to see once. Between ten and fifteen kilometres it is an eighth of a pattern a third the noise’s size, about a twenty-fifth of the noise per loop, which a few hundred loops begin to see. The two counts are the shape of everything that follows.

Three hundred surveys #

One survey’s well could be a lucky draw, so the measurement repeats it three hundred times, each with its own placements, backgrounds and observing errors. For a root at ten kilometres the chosen depths cluster: their middle half runs from 7.5 to 12.5 kilometres, and three surveys in four choose a depth within a factor of 1.5 of the truth. The errors that remain lean shallow — nearly a fifth of the surveys choose five kilometres — because the shallowest roots differ from one another least.

For a root at thirty kilometres the middle half runs from 15 to 40, and a little over half the surveys land within a factor of 1.5. That is a measurement of depth, and a coarse one — what the answer is a set asked every estimate in this subject to admit: it would distinguish thirty kilometres from a hundred, and Airy’s thirty from Hayford’s hundred and more, but not twenty-five from forty.

For a root at sixty kilometres the choice is nearly uninformative, and its shape says why. The chosen depths pile up at the two ends of the candidates, 77 surveys at five kilometres and 87 at a hundred, with the rest spread thinly between. A faint deep root and a faint shallow root carrying less mass leave patterns the noise cannot separate, so the fit runs to whichever end the noise favours.

More loops narrow a middling root and not a deep one #

The obvious remedy is more loops, and it works where the response is large enough to work with. For a root at ten kilometres the share of surveys within a factor of 1.5 rises from 28 per cent at a hundred loops to 97 at 6,400. For thirty kilometres it rises from 25 to 89. For sixty it barely moves: 16 per cent at a hundred loops, 18 at 1,600, 32 at 6,400.

Those counts should be read against what a range offers. The earlier essay found that two hundred loops in one range, sharing one background, were worth about two hundred and twenty independent ones, because a misclosure is gravity’s short wavelengths crossed with the ground’s slopes and those decorrelate within a loop or two. But the stated patch is a thousand kilometres across, and an equilateral loop of fifty kilometres encloses about 1,080 square kilometres, so the patch holds fewer than a thousand such loops without overlapping them. The counts of 1,600 and 6,400 are more loops than the range has room for. On that evidence a root’s depth is readable to a factor of 1.5 from levelling alone only when it is shallow — ten kilometres or so — and is bracketed rather than read at thirty.

Several sizes of loop, and triangles too #

The earlier essay’s closing question had a second half: whether a survey that observed loops of several sizes could read the depth with one kind of reading alone. The idea is sound in outline. A large loop and a small one weight the terrain’s wavelengths differently, so their responses to a root might change with depth in different ways, and comparing them would be a ratio of two measured terms.

The measurement says the sizes add almost nothing. 1,600 loops split evenly among sides of 25, 50 and 100 kilometres choose a depth within a factor of 1.5 in 59 per cent of surveys; 1,600 loops all of 100 kilometres do exactly as well, and all of 50 kilometres slightly worse, 54. Loops of 25 kilometres do worst, 43 per cent, because their response is the smallest share of their noise. What carries the depth is the pattern across placements, and every size of loop carries roughly the same pattern, so mixing sizes is worth no more than choosing the best single size.

Adding 1,600 triangles to 1,600 loops moves the share from 54 to 57 per cent, which is within what three hundred surveys can resolve. The triangle’s contribution to a depth estimate is, as its response curve predicted, nothing that can be told from nothing.

Fitting the depth costs the share its accuracy when the root is deep #

The earlier essay’s share was read with the depth assumed. A survey that does not know the depth has to fit both, and the two trade against each other: a root that is deeper and carries more mass leaves nearly the same faint pattern as one that is shallower and carries less. Fitting the depth therefore puts a cost on the share as well, the way a datum shift hid inside a projection’s parameters in the datum hides inside the projection’s parameters, and the cost depends on how deep the truth is.

Down to forty kilometres the cost is small. The median share read with the depth fitted is within three hundredths of the median read with it stated, at ten, twenty, thirty and forty kilometres. At sixty kilometres it collapses: stated correctly, the share comes back as 0.55 for a true 0.6, with its middle half from 0.29 to 0.87; fitted, its median is 0.25. Half the surveys of a deep root choose a shallow one, and a shallow root that leaves the same faint pattern must carry less mass. A survey that fitted both numbers to a range compensated at sixty kilometres would report a range mostly uncompensated, carried by a thin root near the surface.

That is the measurable form of a confusion the subject has carried since Pratt and Airy argued in 1855: whether a range is held up by roots of light crust or by lighter rock beneath it, and at what depth. Read by levelling alone, a deep compensation and a weak shallow one are the same answer, and the depth has to come from somewhere else.

What each number was held to #

Only the true depth may fit a noiseless survey exactly. With no background and no observing error, the share fitted at the true depth must leave no misfit, and every other candidate must leave some. The true depth fits to zero; the best other candidate leaves 4.7 square millimetres over 160 loops.

With the depth stated, the share must come back unbiased. At 1,600 loops and a root at thirty kilometres, the median share over three hundred surveys must be within three hundredths of the truth. It is 0.59 for 0.6.

The finding must be there to fail. The triangle’s response to a root at thirty kilometres must be under a fiftieth of its noise and the loop’s over a sixth of its noise; they are 1.0 and 20.1 per cent of it.

Independent backgrounds. Each simulated loop takes its background from a different placement than its mountain and its root, as the earlier essay’s independent estimate did. A real range shares one background across its loops, which that essay measured to be worth slightly more than the same number of independent loops; the counts here are of independent loops, and a range’s are of its own.

Where the range stops being the Earth #

A sheet, not a crust. The root is a sheet of mass at one depth. A real root has thickness, and its mass is spread over tens of kilometres of depth. That probably smooths the difference between depths further and makes the depth harder to read than it is here, though nothing here measures it.

Airy only. The compensation here is a mirror of the terrain at depth. Pratt’s alternative, lighter columns of rock beneath the range down to a fixed depth, puts the compensating mass at every depth to that one, and a loop would respond to it with a different curve. Telling the two apart is the older form of this question.

One range. The stated terrain, 600 metres of rms relief on a 2,500 metre plateau, with wavelengths from 4 to 150 kilometres. A range whose relief is concentrated at long wavelengths gives a root less to change shape with, and every depth here would be harder to read.

The share held at six tenths. Every surface in this essay is drawn with the root carrying 60 per cent of the terrain’s mass. A root carrying less is fainter at every depth, and the depth is harder to read in proportion.

A deflection is the slope of a mass gives the closed form for a single buried body, which is where the trade between depth and mass can be seen in one line: a body’s deflection profile widens with its depth and grows with its mass, and a profile measured over too short a baseline cannot tell which. The loops here are that profile sampled at random over a range.

Still open: whether gravity breaks the tie #

Levelling leaves a deep root and a weak shallow one nearly indistinguishable, because both cancel the terrain’s longest swells by similar amounts. Gravity measured on the ground does not depend on the terrain’s slopes the way a misclosure does: it reads the root’s field directly, at every wavelength the root still reaches. The measurements of a levelling loop reads the circulation, and the geoid has none treated gravity as unobserved, and every essay on this ground since has done the same.

Whether a few hundred gravity stations, added to a few hundred loops, separate the depth from the share where the loops alone cannot; whether the loops then add anything to what gravity says; and whether the combination reads a root at sixty kilometres, which neither levelling nor triangles reach, are questions a survey that observes no gravity cannot ask.