The chain the satellite does not have

The reduction chain is four corrections, and each one supplies something the instrument could not know. The slope reduction supplies the vertical, because a tape stretched between two marks does not know which way is down. The height reduction supplies the distance from the centre of the Earth, because the tape does not know how high it is. The scale factor supplies the position, because the grid’s own scale depends on where the line is and the tape does not know that either.

What a tape measures takes the four apart and finds their sizes are not in the order anybody expects. What none of the essays on this ladder has done is check the chain against anything, because there has been nothing to check it against.

There is now, and it comes from an instrument of a completely different kind.

What a satellite observation actually is #

What comes back from a satellite baseline is a vector between two markers in a geocentric frame: three numbers that already contain the positions. The grid distance follows by projecting the two endpoints and subtracting, and that route has no scale factor in it, no height reduction and no slope correction. There is nothing to get in the wrong order, which is the whole subject of setting out runs the chain backwards.

So the two routes have almost nothing in common. One is four corrections applied to a length; the other is a projection formula evaluated twice. If the chain is right they must agree, and where they disagree, one of the steps is wrong or missing.

They agree to 0.17 millimetres over 76.7 kilometres — two parts in a thousand million.

The grid does not know the height #

Raise both marks four kilometres and the grid distance between them does not move — not approximately, but to the last bit of a double-precision number, because the projection formula never sees the height. The chord a tape would read grows by 48 metres.

That is the sharpest statement of what the chain is for. Everything the chain does between those two lines is the work of getting from the first quantity to the second, and every correction in it exists to remove a fact about the instrument’s position that the coordinate system does not care about. A satellite observation supplies the position directly, so the corrections have nothing to remove.

It also explains a practical asymmetry that is otherwise puzzling. A grid coordinate can be published for a mark on a mountain and one at sea level with no note about height, and the pair of coordinates gives the right grid distance for both — while a tape measurement between the same two marks needs the heights to three metres to reach a part per million, which is why the ground is not the grid and why the height that goes into the chain has to be the height above the ellipsoid rather than the one above the sea.

A chord is not a distance #

The satellite route delivers a chord — a straight line through the rock — and the chain has to know that. The shortfall against the arc is 0.128 millimetres over five kilometres, 128 over fifty, and 8.18 metres over two hundred, following the cube of the length exactly: the fitted exponent is 3.000 against a predicted three, and the coefficient is 1/(24R²).

For the short lines a tape can measure, that term is invisible: at one kilometre it is a micron. For the lines a satellite baseline routinely spans it is the difference between a chain that closes and one that does not, and it is one of the two places where the two routes could disagree without either being wrong.

What was computed, and how #

The two routes are built from different libraries and share nothing but the two marks.

The satellite route. Convert both marks to geocentric Cartesian coordinates at their ellipsoidal heights, take the difference for the chord; separately, project both marks onto the grid with the ellipsoidal transverse Mercator and take the plane distance.

The tape route. From the chord and the two heights, solve for the angle at the centre of the Earth by the cosine rule, multiply by the Gaussian radius to get the arc on the ellipsoid, and apply the line’s own scale factor.

There is a third quantity that neither route needs and that both can be checked against: the ellipsoidal geodesic between the two marks, by Vincenty’s iteration. It comes out at 76,772.185 metres against the spherical reduction’s 76,772.185 — the two agree to 0.2 millimetres, on a line where the ellipsoid’s flattening might have been expected to matter more than that.

What the textbook chain leaves on the table #

The reduction as it is taught takes the height difference off the chord by Pythagoras and then divides by 1 + h/R. Both steps are plane arithmetic on a curved Earth, and putting the two routes side by side prices them:

• over 13 kilometres, the textbook chain is out by 1.9 millimetres;
• over 77 kilometres, by 462 millimetres;
• over 215 kilometres, by 10.2 metres.

The exact one-step reduction — the cosine rule in the triangle whose apex is the centre of the Earth — agrees with the projection route to 0.36, 0.17 and 5.5 millimetres over the same three lines.

This site’s own machinery is on the wrong side of that comparison, and the note belongs here rather than in a correction. The reduction chain in what a tape measures does the plane arithmetic, in reverse, on lines of 13 and 77 kilometres. At 13 kilometres it is right to two millimetres, which is inside the tolerance every figure on that essay is drawn against; at 77 it would not be. The formulae are correct for the lines they were written about and are not a formula for a two-hundred-kilometre line — which is exactly the shape of statement the tolerance decides the model is about, arriving this time about the site’s own arithmetic.

The chain has not become unnecessary #

It would be a poor reading of this to conclude that the chain is obsolete. Three reasons, each of which is a working fact rather than a caution.

A total station is still the instrument for a short line. Over a few hundred metres a tape or an electronic distance meter is more accurate than a satellite baseline, cheaper, and works under a canopy or indoors where nothing can see the sky.

The chain runs backwards more often than forwards. Setting out a design coordinate on the ground means converting a grid distance into something to observe, and that is the chain in reverse — with the extra difficulty that reversing a chain reverses the order as well as the operations.

And the two routes measure different things when the marks move. A satellite baseline is an observation now; a published grid coordinate is the result of a computation somebody did once, on a network, at an epoch. Re-observing a control mark perfectly and finding a hundred metres of disagreement is not a failure of either route.

The height that goes into the chain is not the one on the map #

The height the chain wants is the height above the ellipsoid, and the height a levelling run produces is the height above the geoid. The two differ by the geoid separation, which is 48 metres in northern England and reaches a hundred in places, and using one where the other is wanted is an error of N/R — about 7.5 parts per million here, or half a metre on the 77-kilometre line above.

The satellite route sidesteps that too, and for the same reason as everything else: the ellipsoidal height is what the observation contains, and the geoid never enters. It is the levelling instrument that measures the other one, which is why the two heights have to be reconciled at all and why a levelled height is not a distance.

The combined factor above is the shape this makes in practice. Scale factor and height factor pull in opposite directions and cancel at one elevation, so there is a height at which the ground and the grid agree exactly — and construction sites that want a tape reading to equal a computed distance exploit it, which is what a grid scaled to the ground is.

Two observations, one network #

There is one more difference between the two routes and it is not arithmetic. A tape measurement of a line is complete in itself; a satellite baseline is complete in itself as well, but the coordinates it is converted into are not, because they depend on the frame the observation is expressed in and on the date.

That is the reason a survey based on satellite baselines still needs the chain’s vocabulary even where it does not need the chain. Every quantity in a reduction is a statement about a model — a datum, an ellipsoid, a grid, a scale factor — and the satellite route replaces four of those statements with one: project the endpoints, with the projection the grid declares. The declaration does not go away, and a coordinate without its system is not a location whichever instrument produced it.

Where the model stops #

The comparison assumes both marks are known exactly, which is the boundary this field agreed to when it opened: every threshold here survives being handed perfect observations, and an argument about how errors accumulate belongs to a different subject.

It also assumes the two routes refer to the same datum at the same epoch. They frequently do not — a satellite baseline arrives in a global frame and the grid is on a national datum realised decades ago — and the transformation between them is hundreds of metres before any of this arithmetic starts. Nothing in this essay is about that step; everything here happens after it.

And the exact reduction used above is spherical: it takes a Gaussian radius at the mid-latitude of the line. The Vincenty comparison shows that costing 0.2 millimetres over 77 kilometres, which is why it is used, and it would not survive a line ten times longer or one running east–west near the equator where the two radii of curvature differ most.

What each route costs to compute #

A last practical difference, and it runs the other way from everything above.

The tape route is four multiplications once the corrections are known, and the corrections are lookups: a slope angle, a height, a scale factor read off a table or a short series. It was designed to be done by hand and it can be.

The satellite route needs the full ellipsoidal transverse Mercator at both endpoints — the Krüger series, four coefficients, hyperbolic functions — and then a subtraction. It is trivial for a machine and was impossible in the field before there were machines, which is the historical reason the chain has the shape it has: the chain is not merely the arithmetic a tape needs, it is the arithmetic a tape needs that a surveyor could do on a hillside.

That is worth keeping in view when the chain looks like an accumulation of fiddly corrections. Each of them replaced a computation nobody could do in the field with one somebody could, and the cost of that trade is the two millimetres over 13 kilometres and the 462 over 77 that the exact route now makes visible.

The generalisation #

The pattern is worth naming because it recurs wherever instruments of different kinds measure the same thing. A measurement chain exists to supply what one instrument cannot observe, and a second instrument that observes it directly does not need a shorter chain — it needs no chain at all. The corrections do not become smaller; they become inapplicable.

The consequence is the one this essay is built on. Two instruments with disjoint chains give an independent route to the same number, and independence is what makes a check possible. A chain checked against itself — by re-deriving a step, or by running it backwards — cannot find a missing step, because a missing step is missing in both directions.

Who found it, and when #

The four-step reduction is nineteenth-century practice, refined through a century of national triangulation, and it was never checkable in the way described here: the only instruments available all measured lengths and all needed the same corrections.

Satellite geodesy changed that in the 1980s, and the first effect was not a check but a shock — networks adjusted from satellite baselines disagreed with published triangulation coordinates by metres, which is the disagreement that is definitional rather than erroneous. The check in this essay is only possible once both routes are computed rather than observed, which is to say once the arithmetic of both is available in one place.

How many independent routes there actually are #

The check rests on independence, and independence is easy to claim and hard to have. It is worth counting what is genuinely available, because the number is small and it is shrinking.

Two receivers are not two routes. Two satellite baselines observed over the same lines share the constellation, the broadcast or precise ephemerides, the ionospheric and tropospheric models, the antenna phase-centre corrections and usually the processing software. They will agree closely, and the agreement establishes repeatability rather than correctness — a fault in any shared model displaces both identically and is invisible in the comparison.

Independence lives between classes of instrument, not between instances of one. A taped or electro-optically measured distance and a satellite baseline share nothing: one needs the height above the ellipsoid, the atmosphere along a horizontal path and a scale reduction; the other needs orbits, clocks and a vertical atmosphere. That is why the check in this essay is possible, and it is why it needs both traditions present.

The count of classes is two, and it has been two for forty years. Levelling and gravimetry give independent routes to heights, not to horizontal lengths; a laser tracker or an interferometer is the same class as a tape for this purpose. So the whole checking capacity of the subject rests on the coexistence of terrestrial and satellite measurement.

Which is the uncomfortable consequence. As triangulation networks stop being observed and the instruments and the practitioners who can use them become rare, the number of independent routes to a length falls towards one — and a single route cannot be checked, only repeated. The chain does not become unnecessary; the ability to test it does.

It is also a reason to be careful about what a modern agreement between two providers establishes. Two commercial correction services, two post-processing packages, two national reference frames — each pair looks like a comparison and each shares most of its model stack, so the agreement is evidence about implementation and almost none about the physics in between.

That is an argument for keeping the arithmetic of the classical chain alive even where nobody swings a tape, which is what this essay’s own machinery is: both routes computed rather than observed, in one place, so that the comparison survives the instruments.

Where the ladder goes next #

This rung uses an independent route to check a chain of corrections and finds one of the corrections in this collection’s own machinery to be a short-line formula. What it does not do is ask what the grid costs a job when the marks are far apart in a different sense — across a zone boundary, where two grids meet and a line has two right answers. That is where two zones meet, and it is the case where the projection rather than the reduction decides the number.