Gravity is worth four loops a station, and the background still hides a deep root

A loop reads a root’s depth from its own pattern, and a triangle reads none of it asked whether a mountain range’s compensating root could be found at its depth as well as weighed. On a stated range with relief from four to a hundred and fifty kilometres in wavelength, the root modelled as Airy’s mirror of the terrain at a stated depth carrying a stated share of its mass, levelling loops did carry the depth: the pattern of their misclosures from loop to loop changes as the root deepens, and 1,600 loops of fifty kilometres placed a ten-kilometre root within a factor of 1.5 of its depth three times in four and a thirty-kilometre root about half the time. A sixty-kilometre root they did not place, even from 6,400 loops, and when the depth was fitted rather than stated they read it as a light shallow root, carrying a quarter of the mass where it carried six tenths.

That essay ended on gravity, which every measurement on this ground since a levelling loop reads the circulation, and the geoid has none had left unobserved — an odd omission in one sense, since a levelled height is not a length until gravity has been observed along the line, which a height that is not a length set out, so a levelling survey usually carries gravity already. A loop’s misclosure reads gravity crossed with the slope of the ground. A gravimeter on the ground reads the vertical attraction itself, at every wavelength the root still reaches, and it is also far cheaper: a gravity station is one reading at one place, where a loop is a hundred and fifty kilometres of levelling. The question was whether a few hundred stations separate the depth from the share where the loops cannot, whether loops add anything to them, and whether the two together reach a root at sixty kilometres.

The answers are yes, somewhat, and no — and the no comes with a reason that is not about either reading.

A station sees the root better than a loop does #

The model is the earlier essay’s, with a gravimeter set at the centre of each of the 160 placements where a loop and a triangle were. All three readings come from one field: the background geoid — the part of the Earth’s field with wavelengths from ten to four hundred kilometres that the range does not explain, drawn with a degree spectrum of the kind the global models give it — plus the range’s own rock, plus the root. A gravity station reads the downward attraction of all three, in milligals, with an observing error of a tenth of a milligal, which a modern relative gravimeter on a carefully levelled station does better than.

Each reading’s response to a root carrying all of the terrain’s mass is set against the reading’s own noise, and the figure is the earlier one with gravity drawn on it. The station’s noise is almost all background: 28.6 milligals of the geoid band’s attraction, against which the observing error is invisible. Its response to a full root is half again its noise at five kilometres, slightly more than its noise at ten, 38 per cent of it at thirty and 9 per cent at sixty. The loop’s, on the same placements, are 37, 38, 20 and 5.5 per cent.

So a gravity station sees the root three to four times as clearly as a loop when it is shallow, twice as clearly at thirty kilometres, and about half again as clearly at sixty. The advantage shrinks with depth because of what each reading weights. Gravity is the root’s attraction, which falls off with depth as each wavelength’s share decays by $e^{-kD}$ and weights the long waves the deep root still reaches; a loop reads the slope of the geoid round a closed path, which weights the shorter waves a deep root has lost. The deeper the root, the more both are reading the same long-wave part of it — and that part, as below, is where the background is.

A station does the work of four loops, then two #

The depth is estimated as before. At every candidate depth from five to a hundred kilometres the root’s share is fitted by weighted least squares, the depth is chosen where the fit is best, and three hundred simulated surveys show how the choice spreads. A survey draws its readings at random placements with each reading’s background taken from a different placement, as the earlier essay did, because the background varies on scales the placements average across.

For a ten-kilometre root a gravity station is worth four loops almost exactly. A hundred stations place the depth within a factor of 1.5 in 55 surveys of a hundred, as 400 loops do in 50; 400 stations, 83, between 1,600 loops’ 75 and 6,400’s 97; and 1,600 stations, 99, as 6,400 loops do. For a thirty-kilometre root the exchange rate falls to about two: 400 stations place it in 46 surveys of a hundred against 400 loops’ 38, and 1,600 stations in 73, between 1,600 loops’ 54 and 6,400’s 89. The middle half of the depths 1,600 stations choose for a thirty-kilometre root is 25 to 40 kilometres, as tight as 6,400 loops’.

For a sixty-kilometre root neither reading does anything. A hundred stations place it within a factor of 1.5 in 15 surveys of a hundred, 400 in 14, 1,600 in 18 — the share of candidates that a factor of 1.5 either side of sixty happens to cover, chosen nearly at random. The loops were at 18 per cent from 1,600 and 32 from 6,400. Gravity is a better reading, by the margin the response figure promised, everywhere the loops could already read something, and no better where they could not.

Together they place a middling root, and not a deep one #

The earlier essay’s last question was whether loops add anything once gravity is observed. For a thirty-kilometre root they do: 400 stations and 1,600 loops together place it in 70 surveys of a hundred, against 46 for the stations alone and 54 for the loops alone, about what 1,600 stations do on their own. Four hundred stations and 1,600 loops is a reasonable description of a regional survey that has both, and it reads a middling root’s depth to within a factor of 1.5 seven times in ten.

That figure flatters the combination by a stated amount. Each reading’s background is drawn independently, so a loop and a station near each other carry independent errors here, where on the real Earth they share one background field, read once as a slope and once as an attraction. Two readings of one error are worth less than two readings of two, so a real survey would get somewhat less from adding loops to gravity than the 70 per cent shows. The direction is safe: the combination does no better than that.

At sixty kilometres every design fails, the best — 1,600 of each — placing the root in 30 surveys of a hundred. The middle half of its chosen depths runs from 40 to a hundred kilometres. It has learned that the root is deep, and nothing more.

What hides the deep root is the background #

A sixty-kilometre root still has a pattern of its own. Its attraction is the terrain’s, upside down, smoothed by $e^{-kD}$ and so concentrated in the range’s longest swells — but no root at another depth with another share draws exactly that pattern, which the earlier essay checked: with no noise, only the true depth fits exactly. So something in the noise is hiding it, and the two candidates are the observing error and the background.

Take the background away, keep the tenth of a milligal of observing error, and gravity places a sixty-kilometre root within a factor of 1.5 in every one of three hundred surveys of 1,600 stations, at the right share. It places it from a hundred stations too, and with the observing error raised tenfold to a full milligal a hundred stations still place it in 99 surveys of a hundred. The reading is not the limit. A root at sixty kilometres carrying all of the terrain’s mass attracts the stations by 2.6 milligals rms, and this one carries six tenths of it, against a background of 28.6; the background is the whole of the difficulty.

Which part of the background? Cut it to its waves shorter than a hundred kilometres and its rms falls to 22 milligals; the sixty-kilometre root is placed in 32 surveys of a hundred. Cut it to its waves longer than a hundred and its rms is 17; the root is placed in 34. Either half alone is enough to hide the root, and neither half is the half that matters, because the deep root’s signal lies across the boundary: it is the terrain’s longest swells, around a hundred kilometres in wavelength and longer, filtered by depth, and the boundary between the halves runs through the middle of it, each half with power on its own side. A thirty-kilometre root, which still has signal at shorter wavelengths where the background thins out, is placed nine times in ten against either half and seven in ten against the whole.

The background is not noise in the instrument’s sense, and it cannot be modelled away by stopping the field at some degree either: the geoid model stops at a degree found what truncation leaves behind, and here what is left behind is exactly the band the root shares. It is real mass — density in the crust and mantle that has nothing to do with this range — and it attracts a gravimeter exactly as the root does. No improvement in the gravimeter reduces it, and no regional gravity model subtracts it without subtracting the root’s long waves along with it, since a model built from satellite tracking sees the total field at those wavelengths and cannot say which part of it sits under the range. What separates them is a statement about the background from somewhere else: a seismic Moho, which fixes the depth of the density contrast directly; or a model of the background’s own sources. That is the sense in which gravity does not break the tie. It reads more than the loops do, and the tie at depth was never theirs.

Fitting the depth: the deep root grows heavy #

The loops read a deep root, once its depth was fitted, as a light shallow one: a quarter of the mass at sixty kilometres where it carried six tenths. Gravity errs the other way. With the depth stated correctly the share it reads stays near the truth — 0.58 at ten kilometres, 0.55 at thirty, falling to 0.40 at sixty as the background eats a weak signal and pulls the estimate toward nothing. With the depth fitted, it reads 0.59 at ten and twenty, 0.65 at thirty, 0.94 at forty, and 1.38 at sixty: a root more than twice as heavy as the true one, which no Airy root can be, since a root carrying more than the terrain’s own mass would make the range float higher than it stands.

The mechanism is the trade between depth and mass that a deflection is the slope of a mass wrote in one line for a single body: a deeper body spreads its signal wider and must be heavier to make the same anomaly. Against a background, a fit can buy a better match to the background’s long waves by putting the root deeper and making it heavier, because a deep heavy root makes long waves and nothing else. Gravity weights the long waves, so its fit takes that trade; a loop weights the short waves, so its fit took the opposite one, a shallow root light enough to stay out of the long waves altogether. Both are the background shaping the answer, from opposite sides of the spectrum, and the true depth sits between their two errors.

That disagreement is itself a reading. A survey with both loops and gravity, fitted separately, gets two depths and two shares that bracket the truth when the root is deep and agree when it is shallow; the 1,600-of-each survey reads 1.18 for the share at sixty kilometres, between the two. Where the separate fits disagree, the honest report is that the depth is not known, which is the report every survey of a deep root here should be making.

Gravity under the levelling, for a century #

Gravity was the instrument isostasy was first argued from. Pratt and Airy in the 1850s were explaining why the Himalaya pulled the plumb line less than their visible mass should have, which is a deflection of the kind how far the plumb line bends measured, but by the end of the century the case was being made from pendulum gravity, and John Hayford’s depth of compensation, about 113 kilometres, was fitted from deflections across the United States in 1909, with William Bowie extending the work to gravity. Veikko Heiskanen’s tables of the 1920s and 1930s made Airy’s model the working one, and the standard way to estimate the depth of compensation since the 1970s has been the spectral one this essay’s fit is a version of: compare gravity with topography wavelength by wavelength, and read the depth from where the two stop moving together.

The admittance and coherence methods that do it — associated with Dan McKenzie, Anthony Watts and Donald Forsyth among others — run into the problem measured here, and it has a name in that literature: loads not visible in the topography. Forsyth’s method of 1985 added a second load, buried and unseen, to explain the long waves the visible relief could not, and much of the argument since has been about how badly unmodelled loads bias the depth and the strength that come out. The background field here is such a load at its simplest — mass with no relation to the range — and the bias it puts on the share, heavy from gravity and light from the loops, is the kind that literature has spent forty years measuring.

The loops add one small thing to that story. A levelling loop weighs the root, and a triangle weighs the mountain found that a loop reads gravity through the geoid’s slope, so it weights the spectrum differently from a gravimeter, and the opposite signs of the two biases are what that difference in weighting produces. Two readings biased in opposite directions by the same unseen load are a check that one reading biased in one direction is not.

What the stations were held to #

With no background, gravity must place a deep root. Three hundred surveys of 1,600 stations with the background removed must place a sixty-kilometre root within a factor of 1.5 in more than 95 surveys of a hundred. They place it in all of them.

A station must be worth at least a loop. At thirty kilometres, 400 stations must place the root at least as often as 400 loops. They place it in 46 surveys of a hundred against 38.

The finding has to be there to fail. With the whole background, 1,600 stations must place a sixty-kilometre root in fewer than half of surveys. They place it in 18.

One range, one set of placements. Every station is at the centre of one of the 160 placements the loops and triangles used, and reads the same background, mountain and root, so the comparisons are between readings and not between ranges.

Where the stations stop being a survey #

Backgrounds drawn independently. Each reading’s background comes from another placement, as the loops’ did. On the Earth neighbouring stations share a background that varies over tens to hundreds of kilometres, which a dense survey could partly model as a smooth surface — and not an isotropic one, as the correlation is not the same in every direction found of the field’s own covariance; and a station and a loop near each other share one background, so a survey with both learns less from the second reading than the designs here credit it with.

A sheet at one depth. The root is a sheet, and a mountain is not a buried sphere found how much the shape of a compensating mass changes what a reading makes of it. A root with thickness, or Pratt’s compensation by lighter columns, has a different attraction at every wavelength, and the depth it would be read at is a different question.

Terrain known exactly. Every reading here has the range’s own attraction removed exactly, which is the terrain correction a gravimetric survey makes from a digital elevation model. The model’s errors, and the density assumed for the rock, add to the background at the shortest wavelengths, which hurts the shallow root most and the deep root hardly at all.

One background. The geoid band here is drawn once, with one spectrum. A region whose background is weaker in the band where the root’s signal lies — or one where a seismic survey has fixed the Moho — would read the deep root better, and the arithmetic here says by how much: in proportion to the background, since without it a hundred stations do.

Still open: what a known Moho buys #

The deep root is hidden by mass that the range does not explain, and nothing measured at the surface tells that mass from the root at the wavelengths they share. A seismic survey does: refraction and receiver-function studies fix the depth of the crust’s base directly, with an uncertainty of a few kilometres, at a handful of places along a profile.

Whether a few seismic depths, fed to the gravity fit as stated depths at their own places, pin the share well enough to read the background back out of the gravity; whether the loops then add anything beside them; and whether a depth known at five points constrains a root that varies in depth across the range, are questions a survey of gravity and levelling alone cannot ask.