The geoid model stops at a degree

Eleven essays on this ladder treat the geoid as a surface that is there. Height is measured above it; a deflection is its slope; the line a height is measured along curves because it is perpendicular to it; a mountain is not a buried sphere because the mass that produces it is shaped.

Every geoid anybody uses is a series stopped at a degree. EGM96 stops at 360, EGM2008 at 2190, a national model at whatever its own data supported. So an orthometric height derived from one is a height above a surface that is missing everything shorter than the model’s own resolution, and the size of what is missing is not printed beside the height.

One empirical statement, and the rest is arithmetic #

Everything here follows from a single input, so it is worth isolating it.

Kaula’s rule, from 1963: the normalised spherical-harmonic coefficients of the Earth’s gravity field at degree n have an RMS of about 10⁻⁵/n². It is an empirical summary of the spectrum’s fall, not a theorem, and the whole of this essay is that statement plus arithmetic.

From it: a degree-n geoid contribution has amplitude R × 10⁻⁵ × √(2n+1) / n², so the degree variance goes as 1/n³, the tail above degree N sums to 1/N², and

$$\text{omitted height} \approx R \times 10^{-5} / N = 63.7 / N \ \text{metres}$$

The fitted exponent over the working range is −0.998, which is the check that the arithmetic is doing what the algebra says.

Those two bold numbers are worth carrying. Eighteen centimetres is the omission error of the model most textbooks are written around; three centimetres is the omission of the best global model there is. Neither is quoted with a height, and both are larger than the formal uncertainty usually attached to one.

The slope loses far more, and does not converge #

Now ask the same question of the geoid’s slope, which is the deflection of the vertical and is what a plumb line actually feels.

A degree-n undulation has a half-wavelength of about 20,000/n kilometres, so its slope is its amplitude divided by that length: the slope spectrum carries an extra factor of n, its degree variance goes as 1/n rather than 1/n³, and the tail of 1/n is a logarithm.

That has a consequence sharper than a large number:

The omitted height settles by the second row and does not move again. The omitted slope keeps growing, and the square of it is exactly linear in the logarithm of the cutoff, at R² = 1.000000.

So there is no such thing as “the omitted deflection of a degree-360 model” under Kaula’s rule. Any figure quoted for it is a statement about where somebody stopped summing, and a longer sum gives a larger answer for ever.

What the degree means on the ground #

A degree is an abstraction until it is a distance, and the conversion is one division: degree n resolves a half-wavelength of about 20,000/n kilometres.

So EGM2008 describes the geoid down to features nine kilometres across and says nothing about anything smaller. A valley four kilometres wide, a granite intrusion two kilometres across, the mass of a mountain range’s individual ridges: all of it is in the omission.

That is the practical reading of the whole essay. The geoid drawn in this ladder’s opening figure is a smooth curve, and it is smooth because every published version of it is band-limited. The real equipotential surface is not smooth at that scale, and the difference is the eighteen centimetres — concentrated, by the spectrum’s own shape, in exactly the short wavelengths a survey walks across.

The refusal, and what it rules out #

An omission that falls with the degree is only informative if it could have failed to.

The same code, given a flat spectrum — equal power at every degree, which is what a surface with no preferred scale would have — returns an omission that barely moves as the expansion is extended: the sum above degree 36 and the sum above degree 360 differ by under ten per cent, because each new degree adds as much as the last. The Earth’s, over the same range, falls by a factor of ten.

So the fall is a property of the Earth’s own spectrum rather than of the arithmetic, and the specific thing it is a property of is the 1/n² in Kaula’s rule. A body whose gravity field were dominated by short wavelengths — a rubble pile, a body with no isostatic compensation — would gain almost nothing from a longer expansion, and its geoid would be unrepresentable by any model of a practical degree.

That connects this ladder to a body that is not an ellipsoid, where the same question is asked of the shape rather than of the field: the reason a spherical harmonic expansion is the right tool for the Earth is that the Earth’s spectrum falls, and the reason it is the wrong tool elsewhere is that some bodies’ do not.

Which is why a deflection has to be measured from mass #

That divergence is not a defect of Kaula’s rule; it is the reason the ladder’s earlier rungs took the route they took.

How far the plumb line bends computes a deflection from a mass distribution — a buried body with a stated density and depth — rather than from a spectrum, and gets a number. A deflection is the slope of a mass makes the same move. At the time that looked like a modelling preference. It is forced: the spectral route does not converge, so a spectrum cannot bound a deflection and only a physical model of the near masses can.

The same statement is why a deflection is dominated by local topography. The omitted signal above any model degree is concentrated in the shortest wavelengths present, and the shortest wavelengths in the Earth’s gravity field are the hill the instrument is standing on.

What each degree omits, as a share #

Put as fractions of the whole, with the sum bounded at degree 36,000 — a half-wavelength of about half a kilometre, which is where a gravity field stops being a smooth surface and starts being terrain:

A model to degree 360 has 99.6 per cent of the geoid’s height and a third of its slope. The best model there is has 99.93 per cent of the height and slightly under half the slope.

That asymmetry is the rung’s central number, and it explains something the ladder had recorded without explaining. A global geoid model is excellent for converting an ellipsoidal height to an orthometric one — the height signal is nearly all there — and poor for predicting which way a plumb line will hang. Those two are the same surface, and the second is the derivative of the first.

What a height difference actually carries #

The 17.7 centimetres above is a point RMS, and nobody uses a geoid at a point. A height is transferred from a benchmark, so the operative quantity is the omitted signal’s difference between two places — and two nearby places share most of it.

The interpolation between those two limits is the spectrum’s own correlation function, Σcₙ Pₙ(cos ψ) ÷ Σcₙ, computed from the same Kaula coefficients:

Over a kilometre the omission costs 19 mm rather than 177, because both ends of the baseline sit under nearly the same missing bump. By fifty kilometres the two ends are independent, and the difference slightly overshoots √2 times the point value — 274 against 250 — because a band-limited signal’s correlation goes negative at separations near its own wavelength, which is a real feature and not a numerical wobble.

And with EGM2008’s degree 2,190 the same baselines cost 13.9, 40.5 and 39.9 mm: the point value falls by six but the short-baseline value falls by less than a third, because the omitted signal is now short enough that even a one-kilometre baseline sees most of it.

Commission and omission are different errors #

One distinction that a stated geoid accuracy usually blurs.

Omission is what the model does not contain: everything above its highest degree. It is what this essay measures, it is a property of the Earth and the degree, and no amount of better data reduces it without raising the degree.

Commission is the error in the coefficients the model does contain: the uncertainty on each one, propagated. It falls with better data, it is what a published accuracy usually reports, and it is generally the smaller of the two — for EGM2008 the commission error is quoted at a few centimetres and the omission above degree 2,190 is the 2.9 centimetres computed here.

They add in quadrature and behave oppositely under improvement: a decade of new satellite data shrinks the commission and leaves the omission exactly where it was. Which is why the degree is the number to quote — it fixes the half of the error that data cannot touch.

The same pairing runs through the seven parameters’ own uncertainty, where the fitted values have a formal error and the model’s own inadequacy is a separate, larger thing that no number of common points removes.

What this changes about quoting a height #

Three readings, and the first is the one that costs nothing.

Quote the model’s degree with the height. It is one word and it fixes the half of the error that no amount of data removes. 145.32 m orthometric says nothing about which surface; 145.32 m above EGM2008 says the surface is missing everything below nine kilometres, which is 3 cm of height and half the slope. The degree is metadata the height has never carried and the ladder has been saying so about coordinates for eleven rungs.

Use the difference, not the point value, when the height is transferred. A survey referencing a benchmark ten kilometres away carries about a tenth of the model’s quoted omission, not all of it. Quoting the point value there overstates the error by an order of magnitude, which is its own kind of wrong.

And do not take a deflection from a global model. Half of it is missing at any degree, the missing half is unbounded in the spectrum, and the part that matters is the terrain within a few kilometres — which is a local computation with local data, and is what every national geoid programme actually does.

Where the numbers here are weakest #

Three limits, stated because the whole essay rests on one empirical rule.

Kaula’s rule is a summary, not the spectrum. The real degree variances depart from 10⁻⁵/n² by a factor of two either way at some degrees, and systematically at the very low ones — degree 2 is dominated by the flattening and is an order of magnitude above the rule. Every number here is therefore an order-of-magnitude statement with one significant figure of authority, and the shapes — the 1/N fall, the logarithmic divergence, the correlation function — are the parts that do not depend on the rule’s constant.

The cutoff at degree 36,000 is a choice. It is where a half-wavelength reaches about half a kilometre, which is roughly where a gravity field stops being usefully described as a surface at all. Every “share of the slope omitted” in this essay moves if that choice moves, and the essay says so rather than picking a number and presenting it as the answer.

And the correlation function assumes the spectrum is isotropic. Σcₙ Pₙ(cos ψ) is the covariance of a field with no preferred direction, and the Earth’s gravity field has plenty — a baseline along a mountain range and one across it do not see the same correlation. Recorded as a shortfall: the anisotropic version needs a real regional model rather than a rule.

What a model should publish beside its degree #

The degree is printed on every geoid model’s name, and the finding here is that the same degree means two very different things depending on what the model is used for — 99.6 per cent of the height and 33 per cent of the slope, from one truncation. That gap is the argument for a small addition to what a model ships.

Publish the omission per operation, not per model. A table of four or five rows, computed from the same spectrum the truncation is already stated in:

• the omission in a point height, which is the number everybody quotes;
• the omission in a height difference, at several separations — a kilometre, ten, fifty, a hundred — since it rises from a small fraction of the point value towards $\sqrt2$ times it and passes that mark;
• the omission in a deflection of the vertical, with the honest note that it does not converge and so is a lower bound rather than a budget;
• and the separation at which the difference’s omission stops being smaller than the point value’s, which is the only number in the list that tells a user whether their job is in the safe regime.

None of that is new computation. Every entry follows from the degree and the same empirical spectrum the model’s own truncation error was derived from; it is a page of arithmetic against a model that took years of data to build.

And it would replace the thing users currently do, which is to take the point-value figure and apply it to whatever operation they are performing. That is right for a height, conservative by an order of magnitude for a short height difference, and optimistic by a factor of three for a slope. One published number is being asked to describe three quantities that differ from each other by more than the improvements between successive generations of model.

There is one objection to publishing such a table and it is worth answering. A producer might say that a user who needs the height-difference figure can derive it, since the spectrum is public and the arithmetic is in this essay. That is true and it is the same argument that leaves every ratio in the collection uncorrected: a derivation available to a specialist is not available to the engineer who downloads a grid and reads the number in its documentation, and the number in the documentation is the one that ends up in the specification.

The general shape is the one this ladder keeps finding. A quantity is measured carefully, published as a single figure, and used for operations that figure was never about — and the fix is almost never more measurement. It is publishing the shape of what was already measured.

What this rung establishes #

Every geoid is truncated, and the omission is R × 10⁻⁵ ÷ N — 17.7 cm at degree 360 and 2.9 cm at 2,190, with a fitted exponent of −0.998 against the −1 the algebra requires.

The slope’s omission does not converge. Its spectrum falls as 1/n, the tail is a logarithm, and the square of the omitted slope is linear in the logarithm of the cutoff at R² = 1.000000 — so no covariance model built on Kaula’s rule alone can bound a deflection, which is why this ladder’s deflection essays compute from a mass instead.

A truncated model keeps almost all the height and about half the slope. 99.6 per cent against 33 per cent at degree 360, which is the same surface measured two ways and is why a global geoid is excellent for one job and unusable for the other.

And a height difference carries far less than the point value. 19 mm over a kilometre against 177 at a point, rising to √2 times the point value and slightly past it by fifty kilometres — which is a correlation function rather than an error budget, and no published geoid quotes one. Every rung of this ladder has found the same shape of gap: a quantity that is measured carefully and then reported without the thing that says what it means. Here it is a degree, and it is already printed on the model’s own name.