The highest summits stay where they are, and the watershed does not

A river boundary is at least a line anybody can find. A river boundary goes where the river goes, or stays where it was measures how much land it moves when the river migrates, but nobody standing on its bank doubts where the river is today. The other great natural boundary is the opposite kind of object. A watershed — the line from which rain runs one way on one side and the other way on the other — does not move on any timescale a treaty cares about, and it cannot be seen. It is a property of where every drop of water would go, and that is decided by the shape of the land for kilometres around.

Mountain boundaries are very often written in terms of both at once, because a range of mountains usually is both: its highest ground is where the water parts. This essay is about what happens where it is not, and about a second difference between the two lines that has nothing to do with where they lie. One of them can be found to within a few metres from land known only roughly. The other cannot be found to within a kilometre.

A phrase that names two lines #

The boundary treaty of 1881 between Argentina and Chile put the frontier, as far south as the fifty-second parallel, on the Cordillera of the Andes, running — in the usual English rendering of its first article — along the highest crests of the Cordillera which divide the waters. Along the high Andes of the north those words describe one line. The highest ground is a continuous crest, rain falling west of it reaches the Pacific and rain falling east of it reaches the Atlantic, and nobody had any reason to ask which half of the phrase was doing the work.

In Patagonia the phrase came apart. Rivers rising on the plateau east of the high peaks turned out to flow west, through gaps in the range, to the Pacific, so the water divided on low rolling country well east of any summit. When the commissions went to mark the line in the 1890s, the Argentine expert, Francisco Moreno, held that the treaty meant the chain of highest peaks, and the Chilean, Diego Barros Arana, that it meant the continental water divide. Both readings are in the sentence. The ground between them ran to tens of thousands of square kilometres.

The two governments took the question to arbitration by the British Crown. A technical commission under Sir Thomas Holdich surveyed the disputed country, and the award of Edward VII in 1902 followed neither line throughout; it drew a boundary between them that shared the disputed land. That did not end it. In 1994 an arbitral tribunal was still deciding, for the country around Laguna del Desierto, where the award’s water-parting actually ran on the ground, and a stretch across the Southern Patagonian Ice Field was agreed on paper in 1998 and has waited for detailed mapping since.

The history is usually told as a dispute about intentions: what the negotiators of 1881 meant. The measurements below ask something the history does not. Supposing the treaty had named only the watershed, and both governments had agreed to it without reservation, could the line have been found?

A range with one gorge in it #

No real mountain range is used, because a surveyed elevation model is somebody’s survey, with its own errors folded in, and the question here is about the errors.

The stated range runs north to south. Its crest stands three thousand metres high about thirty-five kilometres from the western edge of the ground, wandering a kilometre and a half either way along its eighty-kilometre length. West of the crest the land falls evenly to the western edge, which is the western ocean. East of it the land drops steeply for six kilometres onto a plateau at about eleven hundred metres, and the plateau falls gently, twelve metres in every kilometre, to the eastern edge, which is the other ocean.

Over forty kilometres of the range’s length the plateau carries a broad low rise, and a gorge cuts clean through the range at the middle of that stretch. The rise sits up to eighteen kilometres east of the summits. Its top is rounded over three kilometres rather than coming to an edge, which is what the top of a rise on an eroded plateau looks like, and it is no higher than the rest of the plateau — the whole of its effect is that the land between it and the mountains tilts back towards them.

The section at forty-eight kilometres north is where the two lines part, and it shows why. Rain falling just east of the summits runs down the mountainside into the trough at its foot, and rain falling on the tilted plateau between the trough and the rise runs back into the same trough. The trough runs north and south along the foot of the range, falling towards the gorge, and the gorge carries its water out to the west. So everything from the summits to the top of the rise drains to the western ocean, and the water divides on the rise, eleven and a half kilometres from the highest ground, at a height nineteen hundred metres lower. Twenty-two kilometres further north there is no rise, the plateau falls away east from the foot of the range, and the water divides on the summits exactly as the northern Andes would lead anybody to expect.

How a watershed is found #

A watershed is not traced; it is what is left between two regions once every piece of ground has been assigned the ocean its water reaches. That assignment is a computation on the heights, and it has two steps, because neither step alone gets it right.

The first fills the hollows. Water poured into a closed hollow does not stop there; it rises until it reaches the lowest point of the hollow’s rim and spills out, so for the purpose of where it ends up, the hollow might as well already be full. The standard method, a priority flood, starts at the two ocean edges and works inward in order of height, always taking the lowest ground not yet reached, and raises every cell it reaches to at least the height it was reached at. When it finishes, every hollow has been filled to its pass. The height of the pass between two basins is the same quantity arriving in a different subject: which way something settles is decided by the lowest point on the rim, not by the depth.

The second step routes the water. On the filled surface each cell sends its water to whichever of its eight neighbours lies most steeply below it, and across a filled hollow, where no neighbour is lower, towards the cell the flood reached it from, which leads to the hollow’s outlet. Following those steps from every cell ends at one edge or the other, and that is the cell’s ocean.

The routing step is not optional. A flood left to assign oceans as it goes gives each cell the ocean it happened to reach that cell from, and on an even mountainside both oceans can be reached without climbing — straight down the slope, or sideways along it at constant height. The flood then spreads along contours, and a mountainside whose water plainly runs down into the valley at its foot is handed to whichever ocean arrived along the contour first. That is a wrong answer given with complete confidence, and it moved the edge of the westward drainage on this range by several kilometres along stretches where the true answer is that it does not move at all.

Three checks hold the method to account. On a plane tilted west, every cell between the two edges drains west, and on a plane tilted east every one drains east. On the stated range with the rise removed, the water divides on the summits. And at the middle of the stretch, the westward drainage reaches the top of the rise — 52.98 kilometres from the western edge on the stated terrain, 53.00 by the count of cells, which is inside one cell’s width.

Eighteen kilometres, and three hundred and thirty-one square kilometres #

On the stated range the two lines part for twenty-four kilometres of its length, from twenty-eight to fifty-two kilometres north, and the ground between them is 331 square kilometres. All of it is the same kind of ground: east of the highest summits, and draining to the western ocean. A boundary on the summits gives it to the eastern state; a boundary on the watershed gives it to the western.

How much ground that is depends on how far out the rise sits, and not in proportion.

With no rise at all the two lines are the same line, and the 5.3 square kilometres the count finds between them is the width of the cells the land is drawn on: the summit line falls between cell centres, on average sixty-six metres from the nearest, and eighty kilometres of that is a little over five square kilometres. It is the count’s floor, and every other number in the figure should be read against it.

A rise six kilometres out does almost nothing, for a reason the section makes visible. The range’s eastern slope reaches the plateau six kilometres from the summits, so a rise closer than that is under the mountainside, and there is no tilted plateau between it and the mountains to drain back towards them. Only near the middle of the stretch, where the rise stands a little further out, does it clear the slope, and the two lines part over a few kilometres of the range for twenty-nine square kilometres between.

Once the rise clears the mountainside, the dispute grows steadily: 199 square kilometres at twelve kilometres out, 331 at eighteen, 458 at twenty-four. The widest parting follows the rise exactly, because on this range the water divides on the top of the rise wherever it divides off the summits.

That is the dispute as the treaty’s two criteria define it, on land known perfectly. The award of 1902 had to settle it on land that was not.

The summits are pinned and the watershed is not #

Every elevation model carries error, and an error of several metres to a few tens of metres in a height is ordinary in a global one. So the stated range is redrawn with a smooth random error added to every height — a surface made of forty waves between three and twelve kilometres long in random directions, scaled to a stated root-mean-square height — and both lines are found again. The same measure is used for both: the ground each line sweeps across when the error is added, divided by the length of range it was counted along, is how far that line moved on average.

The two lines come out different by a factor of hundreds. Under ten metres of error the watershed on the plateau moves 1.24 kilometres, and the highest summits move three metres — which is to say that in almost every row of cells the highest cell is still the highest cell, and the few that change move by one cell. Even at forty metres the summits move eleven metres, still far below the 250-metre cells the land is drawn on. The count cannot see a summit move by less than a cell, and it does not need to: in eight redrawings, 23 of 2,560 rows moved their highest cell at all, each by a single cell, while the watershed on the plateau moved five cells’ width on average.

Between those extremes, the watershed where it runs along the summits moves more than the summits do, up to 175 metres, and that comes from the ends of the stretch. There, the rise is just clearing the mountainside, and a few metres of error decide whether the land at the foot of the range tilts back towards the mountains or away from them, so the point at which the watershed leaves the summits slides north and south along the range.

The reason the summits stay put is arithmetic about slopes. A summit is where the land stops rising and starts falling, and on this range it does so abruptly: the western slope climbs eighty-six metres in a kilometre and the eastern falls three hundred and seventeen. An error can move the highest point only by adding a slope steeper than that. The error surface at ten metres has an east–west slope of about seven metres a kilometre, a tenth of the gentler of the two, and at forty metres about thirty. A peak is a place where the land’s own slope reverses by hundreds of metres a kilometre in a single step, and no error of this size competes with that.

On the plateau the water divides where the land’s slope reverses by twelve metres a kilometre, and not in a step: over three kilometres. That is the same size as the error’s slope, and an error of that size moves the top of the rise a long way.

Seen on the ground rather than as an average, the error does not blur the watershed evenly. It shifts it in patches, kilometres long, as one wave of the error tilts one stretch of the rise east and the next tilts the stretch beyond it west. The shaded band is every cell that went to one ocean in some redrawings and to the other in others — ground whose country, under a boundary on the watershed, this error does not settle.

The network’s answer is decided before it is measured makes the same point about a survey: how well a quantity can be known is fixed by geometry before a single observation is taken. Here the geometry is the land’s. Which of the two lines can be found from an imperfect model of the land was settled by the shape of the land, not by the model.

What pins a watershed is how sharply the slope turns over #

The error chart suggests that low ground is the problem. It is not, and the stated range can separate the explanations: keep the rise at the same height, with the same twelve-metre grade on both sides, and change only how sharply its top turns over — rounded across half a kilometre, one, two, three or five.

The watershed moves 0.27 kilometres on the sharpest top and 1.69 on the broadest, six times as far, on land that is identical except within a few kilometres of the line itself. The height of the land has nothing to do with it and neither do the grades.

The mechanism is the one that pinned the summits, at a smaller scale. A smooth top is where the land’s slope passes through zero. An error that adds a slope t moves that point to where the land’s own slope is −t, and on a top whose slope turns from +g to −g across a width r, that lies roughly t r / g away. With an error slope of seven metres a kilometre on a top turning over twelve metres a kilometre across three kilometres, the estimate is about 1.8 kilometres, and 1.24 is measured. At half a kilometre the estimate is 0.3 and 0.27 is measured. The estimate overstates the broad tops, because on a top five kilometres wide the error’s shorter waves no longer tilt the whole top one way, but the order of it holds everywhere.

So the measure of how well a watershed can be known is the curvature of the land along it, and the summits are simply the extreme case: a crest is a top with its slope turning over in a single step, the sharpest there is. A boundary on a knife-edge ridge and a boundary on a peat bog can both be described as watersheds, and one of them is a fact about the land to within metres while the other is a fact about an elevation model.

The ground in doubt, set against the ground in dispute #

Put the two quantities side by side and the treaty’s difficulty changes shape.

The ground a stated error leaves unsettled grows in proportion to the error, as the displacement does: about eleven square kilometres for every metre. At ten metres it is a third of the ground in dispute between the two criteria. At forty it is larger than the whole of it.

That reverses the order in which the question is usually asked. The argument in Patagonia was about which of two lines the treaty meant, as though each line, once chosen, could be run. But one of the two lines was a question the surveying of the 1890s could hardly answer on a broad plateau: to choose the watershed was to choose a line whose position depended on levelling across country with almost no relief, where the difference between draining west and draining east is a few metres in tens of kilometres. The summits could be seen from a valley and fixed by triangulation from a distance. The line a commission can actually run is the one it can see, and on a plateau the watershed is not one of those.

None of this says the award should have followed the summits. It says that the two readings of the treaty differ in two ways at once — in where they put the boundary, and in how well the boundary each puts can be known — and that the second difference falls entirely on one side.

How this sits with the other boundaries priced here #

The sensitivity has a familiar shape. A meridian boundary moves when its datum does finds the 141st meridian 129 metres apart on two datums, and an equidistance line belongs to a surface finds a median line nearly forty kilometres apart on different surfaces. Both of those are conventions a treaty could fix by naming them. An elevation error is not a convention and cannot be named away; it can only be reduced by measuring the land better, and on a plateau it has to be reduced by a factor of several hundred before the watershed is known as well as a summit is known from a rough model.

One sentence, and the ground between its readings prices the ambiguity of a sentence and a tripoint defined three times the ambiguity of three sentences meeting. This one prices an ambiguity of a different order: a sentence that is perfectly clear about what it names, where the thing named is only as definite as the heights it is computed from. The water divides on this plateau at a line that is real — every drop of rain goes one way or the other — and a map can put it anywhere within a kilometre.

Whether a height is the right quantity to route water by at all is its own question. Water runs from higher to lower potential rather than from greater to lesser length above an ellipsoid, and a height that is not a length shows a lake that runs downhill in a height system that reports lengths. On this plateau the difference is about half a millimetre in a kilometre against a grade of twelve metres, so it does not decide anything here. On a lake plateau with no grade at all, it could.

Still open: a coast whose line is a height #

Every boundary here is a line on land. The boundaries that extend a state out to sea start from a line that is on land and water at once: the low-water line along the coast, from which the territorial sea is measured twelve nautical miles outward.

That line is a height too. It is where a stated tidal level meets the beach, and which tidal level — mean low water, the lowest astronomical tide, a chart datum of some country’s choosing — is a convention the law of the sea leaves largely to the coastal state. A beach is a slope, so a difference in the height of the datum becomes a difference in the position of the line, divided by the slope, in exactly the arithmetic that moved the watershed on the plateau. How far that moves the outer limit of a territorial sea, and how much sea it adds or takes away along a coast of stated shape, is what a watershed cannot ask.