A meridian boundary moves when its datum does

The first rung of this anchor sets three real boundary segments beside each other and finds that one of them has no ambiguity at all. Along the 141st meridian, the geodesic, the rhumb line, the parallel-analogue and a straight line on a conic sheet with that meridian as its axis are the same curve — exactly, to the last digit the arithmetic carries. That panel is the control every other number in the anchor is measured against.

It is also, on that boundary, a false reassurance. The Anglo-Russian Convention of 1825 fixes the line at the meridian line of the 141st degree of west longitude, and a longitude is a number about a model. The convention names no model, because in 1825 there was none to name.

A hundred and twenty-nine metres is a thousand times the fifty-eight-millimetre scallop the rung below measures on a chorded parallel. The description with no geometric ambiguity has the largest datum ambiguity in the anchor, and the two facts have nothing to do with each other.

What moves and what does not #

A datum transformation moves a ground point in three dimensions and only one of the three is a boundary question.

The figure carries a second result that is worth more than the first. The east component varies by five centimetres over nine and a third degrees of latitude — from −129.050 metres at Mount Saint Elias to −128.997 at the Arctic coast. That is not a wedge, it is a ribbon: the two lines are parallel to within a part in two thousand over a thousand kilometres.

Five centimetres over a thousand kilometres is worth putting in the units the rung is about. It means the strip’s western and eastern edges are parallel to a part in two and a half thousand, so treating the disputed ground as a rectangle 129 metres by 1,039 kilometres is right to four significant figures — and the area figure quoted above is that rectangle. On a transformation with rotations in it that shortcut would be wrong, and the rotation has two sign conventions is the essay about how easily a datum’s rotations are got wrong.

That constancy is a property of this particular transformation rather than a general fact, and it is worth saying why it holds here. NAD27 to WGS84 is published as three translations with no rotations and no scale, so the shift in space is a fixed vector; converting that vector into east and north components at a point involves only the local orientation of the ellipsoid, and along a meridian the east direction turns very slowly. A transformation with rotations in it — the one for Germany’s Potsdam datum has a rotation of 2.5 arcseconds — produces a shift that does vary along a line, and the ribbon becomes a wedge.

The largest term in the transformation contributes nothing #

A datum transformation of this kind is three translations, and the natural expectation is that the largest of them does most of the work. It does none of it.

That exact zero is the reason this rung has a figure of the parts rather than a sentence about them. Three published numbers of 8, 160 and 176 metres go into a boundary movement of 129, and the arithmetic that turns the one set into the other is entirely about the direction each translation points relative to the local east at a particular place. At a boundary on the equator the tz term would still contribute nothing; at a boundary running east–west it would contribute everything, because there the north component is the one that moves the line.

So the size of a datum shift and the size of its effect on a boundary are different quantities, and which of the seven parameters matters depends on the boundary’s own direction. A published transformation says nothing about that until it is resolved at a place.

The bar at the bottom of the figure is the term the seven parameters do not account for: 0.36 metres, which is the change of ellipsoid itself. NAD27 is on Clarke 1866 and WGS84 is on its own, and the two have different shapes as well as different placements. It is the smallest term here and it is the one that would survive if every translation were zero.

The size is not a property of one boundary #

One transformation at one place could be an accident of Alaska. Four, each evaluated where it is used, is a statement about the size of the effect.

That figure is the second version. The first asked every datum the question at the 141st meridian and reported 665 metres for OSGB36, which is a number about extrapolating a transformation fitted to Britain across a quarter of the world — a fact about the arithmetic and about nothing else. A datum is a fit to a region and its published parameters say nothing outside it, which is exactly the failure where a fit leaves residuals is about, met here in a place where the answer was going into a table.

The corrected reading has a shape worth noticing. The four numbers cluster because the datums they compare are all mid-twentieth-century regional fits to the same planet, made with similar data and similar methods, and the systematic difference between such a fit and a global one is the offset between the local geoid and the global ellipsoid — which is a few tens of metres of separation turned into a horizontal shift by the geometry of the transformation. There is no reason for that quantity to depend much on which country is doing the fitting, and it does not.

The strip is real ground and it is on both sides #

A hundred and thirty-four square kilometres of Alaska–Yukon border country is a real quantity and it is worth being precise about who it belongs to, which is: nobody is arguing.

The boundary is not the meridian. It is the line the International Boundary Commission surveyed between 1907 and 1913, monumented, and maintains — the same principle the rung below invokes, that monuments control over descriptions. So the strip in the hero figure is not disputed territory; it is the difference between two ways of saying where a line is that was settled by putting marks in the ground.

What the strip is, is a measurement of how much the words would move the line if anybody ever went back to them. And there are boundaries where somebody does. A maritime boundary defined by coordinates has no monuments — there is nothing to put a mark on — so the description is all there is, and a stated longitude on an unstated datum is the whole of the definition. Two grids over the same ground prices the same problem for a national grid replaced by its successor: the ground has not moved, every published coordinate has, and everything referred to the old numbers is now referred to a surface nobody uses.

The practical rule that follows is short and is not always followed. A boundary description that names a coordinate must name the datum, and a boundary description written before there were datums has to be given one by a later agreement — which is a decision about a hundred and thirty metres of territory, taken by people choosing a technical convention.

There is a further wrinkle that makes the rule harder than it sounds. Naming a datum is necessary and is not sufficient, because a datum is realised by a set of marked points and the realisation is revised: NAD83 has had four adjustments and the coordinates of a given mark differ between them by up to a metre. A published coordinate is a result is the essay about that distinction — the datum is a definition, the realisation is a solve, and a coordinate belongs to a realisation rather than to a datum. A boundary agreement that names the datum and not the realisation has narrowed a hundred and thirty metres to one, which is a great improvement and is not zero.

What a hundred metres of longitude is made of #

A hundred metres is a strange size for a discrepancy between two descriptions of the same planet, and it is worth saying where it comes from, because the answer is not measurement error.

A regional datum is an ellipsoid chosen and placed so that it fits the geoid over one country as closely as possible. A global datum is one placed so that it fits the whole geoid. The two placements differ by whatever the local geoid does relative to the global mean — a few tens of metres of separation in most inhabited places — and that vertical difference is converted into a horizontal one by the geometry of the fit, because moving an ellipsoid up under one region tilts it relative to every other.

So the hundred metres is not error in any useful sense. Both datums are correct statements about the surface they define; they simply define different surfaces, and a longitude on one is a different ground point from the same longitude on the other. What a coordinate refers to is the essay about that distinction and this is its consequence for territory.

The number’s stability across four countries follows from the same account. Every one of these datums was fitted in the mid-twentieth century by broadly similar methods to broadly similar geoid separations, so the horizontal consequence lands in the same range wherever the work was done. A datum fitted to a region with an unusually large geoid anomaly would be an outlier, and none of the four is.

Why this is larger than everything else in the anchor #

Three quantities have now been measured on real boundary segments, and putting them in one column is the useful summary of the anchor so far.

The scallop a chorded parallel leaves, at the spacing anybody uses: fifty-eight millimetres. The spread between readings of “a straight line” over seven hundred and eighty kilometres: 7.5 kilometres. The movement of a stated longitude between two datums: 129 metres.

Those are three orders of magnitude apart and they are not ordered the way the descriptions’ precision would suggest. The vaguest description — “a straight line” — produces the largest number; the most specific one, a stated meridian, produces the middle one; and the geometric operation a survey actually performs contributes the smallest.

The order also inverts the amount of work each ambiguity took to find. The 7.5 kilometres needed three curves and a separation measure. The fifty-eight millimetres needed a closed form and a seven-doubling ladder. The hundred and twenty-nine metres needed a published transformation and one call — it has been sitting in every table of datum parameters ever printed, in a column headed “shift”, and nobody in this collection had asked what a shift does to a boundary.

The reason the middle number is not the smallest is the one this rung exists for. A description that names a coordinate has moved the ambiguity out of the geometry and into the reference frame, where it is invisible: nothing about the sentence “the meridian of the 141st degree of west longitude” suggests that it depends on an unstated choice, and the choice is worth more than the entire chord geometry of a two-thousand-kilometre parallel.

The boundaries where the number is still live #

Three classes of boundary have no monuments, and on all three the description is the whole of the definition.

Maritime boundaries. There is nothing to monument. A delimitation agreement lists coordinates, and every one of them refers to a datum that the agreement may or may not name. Agreements from before about 1990 frequently do not, and re-expressing their coordinates in a modern frame is a decision worth the numbers in this rung — over a boundary hundreds of kilometres long, an unnamed datum is tens of square kilometres of seabed.

Boundaries in inaccessible country. Parts of the 141st meridian itself run through terrain where the monuments are tens of kilometres apart, and between them the boundary is the description. The datum question there is live in exactly the intervals the survey could not reach.

Boundaries under ice. A line described by coordinates across an ice sheet has nothing to attach a mark to that will stay where it is put. A velocity needs a frame is about the same problem for ground that deforms, and ice is the extreme case: the description is all there is, and the ground under it is moving.

In every one of the three the practical rule is the same and it is not always followed: state the datum, and state the epoch. A coordinate without both is a description of an unnamed surface at an unnamed date.

And a longitude has a second origin nobody states either #

There is one more unstated convention in “141 degrees west”, and it is a different kind of thing from the datum.

A longitude is measured from a prime meridian, and the prime meridian is a choice. The 1825 convention was drawn when several were in use — Paris, Ferro, Cadiz, Washington and Greenwich among them — and a boundary at 141 degrees west of one is not the same line as 141 degrees west of another.

The offsets are large: Paris is 2°20′14″ east of Greenwich, which at 65 degrees north is a hundred and ten kilometres of ground.

That is not a subtle ambiguity and it was not one in practice, because a treaty naming a longitude was understood to name a specific published meridian and the parties knew which. What it does illustrate is that a coordinate carries at least three conventions before it names any ground — a prime meridian, an ellipsoid, and a placement of that ellipsoid relative to the Earth — and what a coordinate refers to is the essay in this collection that lists them.

The modern versions of the same convention are also not zero. The International Reference Meridian used by satellite positioning sits about 102 metres east of the transit circle at Greenwich, for reasons entirely about the deflection of the vertical at that spot, which is the same order as the datum shift measured here and arrives from a different direction.

The datum a boundary was actually given #

The three ambiguities in “the meridian of the 141st degree of west longitude” — which prime meridian, which ellipsoid, and where that ellipsoid sits — were all resolved for this boundary, and none of them was resolved by the treaty.

They were resolved by the survey. Between 1907 and 1913 the International Boundary Commission determined the meridian astronomically and monumented it, and from that moment the boundary was the monuments. Every subsequent description is a description of where the monuments are, so the datum question stopped being about the treaty’s words and became about how to express the marks’ positions — which is a question with no territory attached to it at all.

That is the general pattern and it is worth stating as one. A boundary described by a coordinate is ambiguous until it is surveyed, and the survey both resolves the ambiguity and makes it moot: whatever datum the surveyors were implicitly working in becomes the boundary’s datum by the act of putting marks in the ground. The chain the satellite does not have is the essay about what a modern position is missing that a classical one carried, and this is the same observation with the sign reversed — the classical survey carried its datum in the marks, and the marks are still there.

Where the pattern fails is where there are no marks, and that is the whole of maritime delimitation.

What a description that names no coordinate at all does #

Every segment in this anchor so far names two monuments or a coordinate curve, and each turns out to leave something undecided — a curve between the monuments, or a surface for the coordinate.

There is a third kind of description in wide use and it names neither. A maritime boundary is very often defined as the line equidistant from the two coasts, which is not a coordinate, not a monument, and not a curve anybody can draw without computing it. It looks immune to both of the ambiguities priced so far: no reading to choose, and no datum in the words.

It is not, and the reason is that “equidistant” is a statement about distances, and a distance is a statement about a surface. The surface enters twice over: once because the basepoints on the two coasts are coordinates and therefore carry the datum ambiguity this rung has just priced, and once again because the distance between two coordinates depends on whether it is computed on an ellipsoid, on a sphere, or with a ruler on a chart.

The second of those has no counterpart in anything measured so far. Every segment in this anchor has been defined by points, and a point is a point whatever surface it is expressed on. An equidistance line is defined by an operation, and an operation has to be carried out somewhere.