The correlation is not the same in every direction

The geoid model stops at a degree measured what a truncated geoid leaves out, and the number that matters is the omission in a height difference rather than at a point — because nobody uses a geoid at a point. A height is carried from a benchmark — along a line, not down a normal —, and what a survey needs to know is how much of the omitted signal the two ends share.

That calculation used a correlation function of the form $\sum c_n P_n(\cos\psi)$: a covariance depending on the angular distance between the two places and on nothing else. The essay recorded a shortfall against it, and the objection is short.

The omitted signal is the short-wavelength part of the gravity field, the short-wavelength part is topography and its compensation, and topography has grain. A baseline running along a mountain range does not see the same correlation as one running across it.

The smallest model that says so #

An anisotropic covariance can be built in many ways and most of them stop being covariance functions — a symmetric positive-definite kernel is not something one can perturb casually.

The model here keeps the isotropic function and changes its argument. The correlation is evaluated not at the angular distance $\psi$ but at an effective distance from an elliptical metric,

$$\psi_{\text{eff}} = \psi\sqrt{\frac{\cos^2\alpha}{\rho} + \rho\sin^2\alpha},$$

with $\alpha$ the azimuth from the grain and $\rho$ the anisotropy ratio. Along the grain the effective distance is shorter, so the correlation is higher and the two ends share more; across it the effective distance is longer and they share less.

At $\rho = 1$ the factor is one at every azimuth and the model is the isotropic one exactly. That is the refusal, it is checked, and it holds to machine precision.

The size, at a modest anisotropy #

At $\rho = 2$ — which is modest for terrain, and is well below what a linear mountain belt would give — the same 20 km baseline omits 165 millimetres along the grain and 241 across it.

Seventy-six millimetres of difference, on a quantity a survey is trying to control at the centimetre. And the isotropic model reports 203, understating the worse direction by 18.4 per cent and overstating the better one by 23.

That is the shortfall’s answer in one line. The isotropic quotation is not a compromise between the two directions in any useful sense; it is a number that is wrong in both, by amounts that depend on the azimuth of the work.

It matters most where the model is used #

This is the part that decides whether the correction is worth having, and it goes the awkward way.

Over a long baseline the two ends are uncorrelated whatever direction the line takes, so the omitted difference is $\sqrt{2}$ times the point value in every direction and the anisotropy is invisible. Over a short one the two ends share most of the signal, how much they share is exactly what the correlation function decides, and the direction decides it.

A geoid model is used to transfer a height from a benchmark a few kilometres away. That is the short end. The direction-dependence is largest at precisely the separations at which the model is doing its work, and it vanishes at the separations where nobody needs it.

What the anisotropy is a model of #

It is worth naming the physical thing the ratio stands for, because a stated parameter with no referent is not a model.

The omitted part of a geoid model is the signal above its maximum degree, which at degree 360 means wavelengths under about 110 kilometres. At those wavelengths the gravity field is dominated by topography and by the mass deficiency compensating it — a mountain range and its root — and topography at that scale is emphatically directional.

A linear range has a correlation length along its axis that is the length of the range and one across it that is the width. Ratios of five and ten are ordinary for real belts; the ratio of two used in the figures here is conservative and was chosen to be so.

The same argument applies to a coastline, a rift, a fault-bounded basin and a glacial trough. Almost every landform that produces short-wavelength gravity signal is elongated, and the exceptions — a shield, an abyssal plain — are exactly the places where the omission is small anyway.

What a practitioner does with this #

The number that comes out is a modelling uncertainty rather than a correction, because the anisotropy ratio for a given region is not something this collection can supply.

Quote a range rather than a value. An omission error stated as a single number is a claim that the correlation is isotropic, and that claim is testable and usually false. Stating the isotropic value with a factor either side of it is more honest and costs nothing.

Expect the azimuth to matter for short lines, which is where a levelled height stops being a distance matters too. Two benchmarks the same distance away, in different directions, do not carry a height equally well, and the difference is tens of millimetres at ordinary separations.

And expect it to be worst in mountains, which is where the anisotropy is largest and where orthometric heights are hardest anyway — the deflection of the vertical, the plumb line’s bend and the geoid’s own roughness all peak in the same places, and they are not independent problems.

The sweep, and why it is a sweep #

Every figure here varies the anisotropy ratio rather than fixing it, and that is a deliberate refusal to invent a number.

The alternative would be to pick a ratio — from a published regional covariance, from a topographic autocorrelation, from a plausible-sounding argument about mountain belts — and quote a single corrected figure. That would be a more satisfying essay and a worse one, because the ratio would then be doing the arguing and it would be somebody’s estimate for somewhere else.

What a sweep gives instead is a sensitivity: how much the answer moves per unit of anisotropy, over the whole range a practitioner might be in. At $\rho = 1.25$ the isotropic model understates by 6.1 per cent; at 1.5 by 11.0; at 2 by 18.4; at 4 by 31.8; at 6 by 34.8.

A reader with a regional model reads their own number off that curve. A reader without one reads the shape, which is the honest thing to take away: the effect is roughly linear in the ratio at small anisotropies and saturates above about four, so the first fifty per cent of grain costs more than the next two hundred.

Why Kaula’s rule cannot know #

The isotropic model is not a careless choice; it follows directly from what the rule supplies.

Kaula’s rule gives a degree variance — how much signal there is at each spherical-harmonic degree — and a degree variance is a sum over all the orders within a degree. Summing over orders is exactly averaging over direction, so a degree-variance model has thrown away every directional distinction before it starts.

Recovering one needs the individual coefficients, or an order-dependent model, or a regional covariance fitted to local data. Each of those is a bigger object than a rule of thumb, and each is available for some regions and not others.

So the isotropy is not an approximation somebody made; it is the only thing a degree-variance model can say, and using one is choosing to say nothing about direction.

What would settle it #

A regional covariance fitted to gravity observations, which is standard practice in physical geodesy and is what least-squares collocation runs on.

Such a model can be anisotropic, is fitted rather than assumed, and gives the local correlation lengths along and across whatever grain the region has. What it needs is data — a network of gravity observations dense enough to resolve the short wavelengths in question — and that is exactly the data whose absence created the omission error in the first place.

That circularity is real and is not fatal. A region with enough gravity data to fit an anisotropic covariance has less omission error to worry about, and a region with little data has more of both. The practical consequence is that the uncertainty on the uncertainty is largest where the uncertainty is largest.

An error budget with an azimuth in it #

The awkwardness of this result is what it does to the format an error budget comes in.

A budget is a list of variances added in quadrature, and every entry is a scalar. The geoid’s omission appears as one line — so many millimetres, for a baseline of so many kilometres, at a model of such a degree.

Making it directional turns that line into a function of azimuth, and a budget has no column for that. The practical responses are all unsatisfying: quote the worst direction, which is conservative by up to eighteen per cent; quote the isotropic value, which is wrong in both directions; or quote a range, which the format does not accommodate.

The same problem appears one anchor over for the radius of curvature, where a directional quantity has to be summarised by a scalar and every choice of scalar is wrong in some direction. The two are the same failure of the format rather than of the geodesy, and the honest remedy in both cases is to carry the direction rather than to average it away.

Where this sits in the ladder #

Ten rungs of this anchor have taken a height apart: what it is above, why it is not a distance, what the plumb line does, where the geoid comes from and what a truncated model leaves out.

The last two are a pair. The ninth measured the size of the omission and its dependence on the model’s degree; this one measures its dependence on the direction of the work, which the ninth’s machinery could not express. Neither is a correction anybody applies; both are numbers that belong in an error budget and are not in one.

The refusal #

Two checks, and the first is the one that makes the model legitimate.

At $\rho = 1$ the answer must be the isotropic one, at every azimuth, to machine precision. It is: the spread across azimuth is one to within $10^{-12}$, and the value matches the previous rung’s function exactly. So the anisotropic model contains the isotropic one rather than replacing it.

And the effect must vanish at long baselines, because two uncorrelated points cannot care about direction. It does: the spread falls from 1.91 at one kilometre to 1.04 at two hundred.

Together those say the measure returns zero when it should, twice, for two different reasons.

Where the model stops #

The elliptical metric is the simplest anisotropy available and is not the one real terrain has. Real grain is not a single direction over a region — it curves, branches and changes scale — so a covariance with one preferred azimuth is a local model at best.

Nor is the ratio derived from anything. It is a stated parameter and the figures sweep it, which is the honest form for a quantity this collection cannot measure: what is demonstrated is the size of the effect per unit of anisotropy, and a practitioner with a regional model supplies the rest.

What is also omitted is the deflection of the vertical, whose spectrum does not converge at all — the same question cannot be asked of a slope, and adding anisotropy to a divergent sum does not make it converge.

Two benchmarks, one distance, two answers #

The result stated as a field problem, which is where a surveyor meets it.

A job needs a height at a new point. Two benchmarks are available, both eight kilometres away, one to the north and one to the east, and both have published orthometric heights of the same quality. The obvious rule is to take the nearer, and they are the same distance.

At an anisotropy of two, with the grain running east–west, the eastern benchmark carries the height with about 30 per cent less omission error than the northern one. Nothing about the benchmarks differs; nothing about the instruments differs; the difference is entirely in how much of the omitted gravity signal the two ends of each line share.

A surveyor cannot know that without a regional covariance. What they can know is that the two are not equivalent, and that a rule preferring the shorter line is answering a question about distance when the question is about correlation.

Who found it, and when #

Anisotropic covariance models are old in physical geodesy and are standard in the collocation literature from the 1970s onward; the isotropic Kaula-type model is used precisely where nothing better is available, which is most global work.

What is unusual is stating the consequence for an omission error quotation. The omission is normally reported as a single global figure per model degree — a few centimetres at degree 2190, tens at degree 360 — and that figure is a point value, isotropically derived, quoted without a direction and used for baselines.

Each of those three steps loses something, and this rung is about the third.

The isotropic model is not conservative #

One more consequence, and it is the one that would matter to somebody writing a specification.

An approximation that errs on the safe side is tolerable, and a reader might hope the isotropic model does. It does not. It sits between the two directions by construction — it is the average of a quantity that varies with azimuth — so it is optimistic in one direction and pessimistic in the other, in equal measure.

At $\rho = 2$ it understates the across-grain omission by 18.4 per cent and overstates the along-grain one by 23. Half of all baselines are worse than the quoted figure and half are better, and which half a particular job is in is not knowable from the quotation.

That is the worst arrangement for a specification, which needs a bound rather than a central value. A specification written against the isotropic figure is met by about half the work it governs, and the half it fails is not identifiable in advance.

What a specification should quote instead #

The complaint is that a central value is the wrong statistic for a specification, so it is worth saying what the right one is, since it comes out of the same sweep.

Quote the worst azimuth. The sweep already evaluates the omission over the full range of directions; taking the maximum rather than the mean costs nothing and produces a figure that every baseline meets rather than half of them. At $\rho = 2$ that is the along-grain extreme, about 23 per cent above the isotropic number.

And quote the range beside it, because the spread is the thing a user needs in order to decide whether direction is worth modelling for their particular network. A job whose baselines all run one way can use the figure for that azimuth and do much better than the bound; a job with baselines in every direction has to take the bound.

Neither number is new work. Both are readings of a curve the model already produces, and the only change is which point on it gets printed. That is the smallest possible remedy for a defect that currently makes a specification unmeetable by half the work it governs — and it is the same remedy the whole collection keeps arriving at, which is to publish the shape of an answer rather than one point of it.

Where the ladder goes next #

Ten rungs have priced a height and the last two have priced what is missing from the model that supplies it. The unasked question is temporal: a geoid model is a snapshot of a field that changes with water, ice and mass redistribution, and an orthometric height computed from one has an epoch as surely as a horizontal coordinate does.