The sea is measured from a tide, and the headlands decide how much

The highest summits stay where they are, and the watershed does not found a boundary that is a height in disguise: the line where water divides sits wherever the land’s slope turns over, and an error in the heights moves it by the error divided by how sharply the slope turns. Every boundary in that essay was on land.

The boundaries that carry a state out to sea start from a line that is a height openly. A territorial sea is measured from the low-water line along the coast, and the low-water line is where a stated level of the tide meets the shore. Which level is a choice the charts make, and a shore is a slope, so the choice moves the line. This essay measures how far it moves the line, how far that moves the limit of the sea, and the one circumstance in which a few centimetres of tide move the limit by tens of kilometres.

The baseline is a height the charts choose #

The United Nations Convention on the Law of the Sea lets every state claim a territorial sea “up to a limit not exceeding 12 nautical miles, measured from baselines”, in the words of its third article, and its fifth article says what the baseline normally is: “the low-water line along the coast as marked on large-scale charts officially recognized by the coastal State.”

That sentence names no tide. Low water is not one level: the tide falls to a different height on every ebb, lower at springs than at neaps, lower in some seasons than others. A chart has to pick a level and draw the shoreline where that level meets the land, and the level it picks is its chart datum. The charts of the United States use mean lower low water, the average of the lower of each day’s two low tides. The International Hydrographic Organization recommends the lowest astronomical tide, the lowest level the tide can be predicted to reach under average weather, and many hydrographic offices chart to it or to something close to it. The two sit a few decimetres apart on most coasts and further apart where the tide is large, and older national datums add more variety still.

So the convention hands the coastal state a choice, made in its own hydrographic office for reasons of navigation rather than of law, and the line that choice draws is the one the whole territorial sea is measured from. A meridian boundary moves when its datum does found 129 metres between two readings of the 141st meridian because the treaty named a longitude without naming its datum. The fifth article names a line without naming its level, and the level is a datum of a different kind: a height, where that one was a position.

Drop over slope #

How far the low-water line moves is the difference in the datums divided by the slope of the shore. It is the same arithmetic that moved the watershed on the plateau, with the tide in place of an error in the land, and it gives answers of very different sizes on one coast. A rocky headland falling a metre in twenty moves its low-water line twenty metres for a metre of datum. A tidal flat at the head of a bay, falling a metre in a thousand, moves it a kilometre. Flats of that gentleness and gentler are common wherever a large tide works over mud and sand, and on the gentlest the low-water line moves kilometres for a metre.

That is why the choice of datum is not a detail on a shallow coast. The ground between the two low-water lines is real ground — dry at one datum’s low water, under water at the other’s — and on a coast of broad flats it runs to tens of square kilometres. Whether it matters for the sea beyond is a different question, and the answer depends on something the beach sections cannot show: which parts of the coast the limit is measured from.

The limit is measured from the nearest point of the coast #

The fourth article of the convention defines the outer limit exactly: it is “the line every point of which is at a distance from the nearest point of the baseline equal to the breadth of the territorial sea.” Draw a circle twelve nautical miles in radius about every point of the low-water line; the limit is the seaward edge of all those circles together. This is the envelope of arcs, the construction S. Whittemore Boggs put forward for the codification conference at The Hague in 1930, and it replaced an older habit of copying the coastline’s every wiggle twelve miles out.

The envelope has a property that decides everything here. A point of the limit is twelve miles from one point of the coast, its nearest, and from every other point of the coast it is further. The coast that is not nearest to any point of the limit does not affect the limit at all. On a coast of headlands and bays, the headlands stand furthest out to sea and are nearest to most of the limit, and the heads of the bays, set back from them, are often nearest to none of it.

A coast with five bays in it #

No real coast is used, because a surveyed coast arrives with its own chart datum already chosen, and the choice is the question.

The stated coast runs straight for a hundred and sixteen kilometres with five bays set into it, each five kilometres deep and shaped as half an ellipse, six, ten, sixteen, twenty-four and thirty-six kilometres wide, with four kilometres of straight shore between them. The straight shore and the mouths of the bays fall a metre in twenty. Going into each bay the beach flattens steadily, and at its head it is a tidal flat falling a metre in a thousand. Straight shore continues twelve miles beyond both ends so that the limit over the bays is not cut short.

Every point of the charted low-water line is moved seaward, along the shore’s own normal, by one metre divided by its slope, and the limit is found again. At the mouth of a bay, where the shore turns a corner into the land, the moved line is not one point but an arc about the corner, and that arc is part of what the limit is measured from; leaving it out moves the limit over a narrow bay by seventeen metres where the true answer is twenty. The limit is then compared, at every point along the coast, with the limit measured from the charted datum.

Along most of the coast the two curves have nothing to do with each other. The shore moves by up to a kilometre in every bay, and the limit twelve miles out moves by twenty metres, the headlands’ figure, because the headlands are what it is measured from. Over four of the five bays the kilometre of tidal flat is invisible from the sea.

Over the fifth, the widest, the limit moves with the flat. The difference between the twenty-four-kilometre bay, whose flat changes nothing, and the thirty-six-kilometre bay, whose flat moves the limit by the whole kilometre, is not one of degree.

A bay’s own shore takes over at one width #

Why a wide bay behaves differently has a closed answer. Over the middle of a bay of width w and depth d, the nearest land to the limit is either the corner of the bay’s mouth, half the width away along the coast, or the head of the bay, set back by d. A limit of breadth R measured from the corner reaches $\sqrt{R^2 - w^2/4}$ out from the line of the mouth; measured from the head it reaches R − d. The head is the nearer land once the first is the smaller, which is once the bay is wider than $2\sqrt{2Rd - d^2}$. For a twelve-mile limit and a five-kilometre bay, that is 28.1 kilometres.

The measured transition is not a single step at 28.1 kilometres, and the reason is worth having. Between the corner of the mouth and the head of the bay lie the bay’s shoulders — shore partway up its sides, set back less than the head and nearer the middle than the corners — and as the bay widens they take over first, from about twenty-five kilometres. Their beaches are partway between a headland’s and a flat’s, so the limit over the bay moves partway: two hundred metres, then five hundred and seventy, then seven hundred and sixty. By 28.1 kilometres the head itself is nearest and the limit moves the flat’s whole kilometre. Below about twenty-four kilometres none of the bay’s inside is nearest to anything, and the limit over it does not care what the tide does there.

Two further rules of the convention act in the same direction. Its tenth article closes off a bay behind a straight line across its mouth when the bay is deep enough — when its area is at least that of a semicircle on the line — and the two narrowest bays here pass that test, so for them the baseline is the closing line, their flats are internal waters, and the tide inside them could not move the sea if it tried. The thirty-six-kilometre bay fails it badly: it would have to be eighteen kilometres deep. And its seventh article allows straight baselines along a deeply indented coast, which again replaces the insides of bays with lines between headlands. The shallow wide bay, the one kind that the tide moves the sea from, is exactly the kind those rules leave alone.

Fifty-six square kilometres of shore, eleven of sea #

The coast as a whole says the same thing in areas.

Lowering the datum a metre uncovers 56.1 square kilometres of shore along the stated coast, most of it in the bays, and moves 11.1 square kilometres of sea into the territorial sea at its outer edge. The sea gained is a fifth of the shore uncovered. It is also bounded on both sides by the two things the limit could have followed: at least the coast’s length times the headlands’ shift, 2.32 square kilometres, and at most the coast’s length times the flats’ shift, 116. The stated coast sits near the bottom of that range, and nearly four fifths of what it does gain comes from the one bay wide enough for its flat to be nearest.

On a straight coast of one slope the limit moves exactly as far as the shore does, everywhere, and the sea gained is the coast’s length times that shift to the last decimal. That is the case in which the shore and the sea agree, and it is the only one: every headland on a real coast is a place where the limit stops listening to the shore behind it. The envelope reading of the limit was also checked against a plain count of every quarter-kilometre cell of sea within twelve miles of the charted low-water line, and the two areas agree to 0.02 per cent.

A bank that dries #

The fourth article measures from the nearest point of the baseline, and the thirteenth says what else can be baseline. A low-tide elevation is “a naturally formed area of land which is surrounded by and above water at low tide but submerged at high tide”, and where one lies within the breadth of the territorial sea from the mainland, “the low-water line on that elevation may be used as the baseline.” A sandbank that dries at low water, anywhere within twelve miles of the coast, is a place the limit can be measured from.

Whether it dries at low water depends on which low water. A bank whose crest stands half a metre below one chart’s datum is under water at that chart’s low water and is no low-tide elevation. On a chart whose datum is a metre lower, its crest stands half a metre above low water and it is one.

Everything else in this essay has been continuous: a centimetre more of datum moves the shore a little further and the limit a little further. The bank is not. While its crest is under water it contributes nothing, and the moment its crest is dry its whole low-water line is baseline, and a circle of twelve miles about it joins the envelope. At ten kilometres offshore that circle reaches eleven kilometres beyond the coast’s own limit, and the limit bulges out round it.

In the stated case the bulge is 298 square kilometres when the bank is dry over a disc two hundred metres across, the smallest the five-centimetre steps of the figure can show. Widening the dry disc tenfold, to two kilometres across, adds 42 square kilometres more, and widening it to six kilometres adds 143 — less than half of what the first dry sand added. A bank’s contribution to the territorial sea is decided almost entirely by whether its crest clears the chosen low water, and hardly at all by how much of it does. Which chart datum a state uses decides, on a coast with banks, whether hundreds of square kilometres are territorial sea.

The bank that adds most adds nothing a step further out #

The thirteenth article’s second half is the reason the jump has a second threshold, in distance rather than in height. A low-tide elevation “wholly situated at a distance exceeding the breadth of the territorial sea from the mainland or an island” has no territorial sea of its own. A bank can extend the sea only if some part of it is inside the sea already.

The closer to the coast a bank lies, the less it adds, because its circle of twelve miles mostly overlaps the coast’s own. The further out, the more of its circle lies beyond the coast’s limit — until its nearest edge passes twelve miles from the coast, at which point it adds nothing at all. The largest contribution a drying bank can make is made by the bank at the last position where it still counts: 926 square kilometres for a bank dry over a kilometre’s radius, with its near edge a hair inside the limit, and zero a hundred metres further out.

That makes the outermost banks the most sensitive to everything at once. Their distance from the coast is measured from the low-water line, which moves with the datum; their being dry at all depends on the datum; and a small error in charting either their position or their crest decides between the largest bulge and none. A boundary commission on land could at least walk to its monuments, and the line a commission can actually run is drawn between things it can stand on; a bank whose crest clears low water by a few centimetres is a baseline point nobody can stand on for more than an hour a day.

What the datum decides #

The shape of the answer is the useful part. A river boundary goes where the river goes, or stays where it was found the same pairing in time rather than in height — land that changes country a little at a time under accretion, and a whole loop at once under avulsion. A choice of tidal datum moves a coast’s shoreline smoothly and by a lot, by the drop over the slope, wherever the shore is gentle. It moves the territorial sea smoothly and by little, because the sea is measured from the headlands and the headlands are steep, except over bays wide enough that their own heads are the nearest land. And it moves the sea by a great deal and not smoothly at all wherever a bank’s crest lies between the two datums and inside the limit.

An equidistance line belongs to a surface measured a maritime boundary between two states from four basepoints and found forty kilometres between readings of “equidistant”. Every basepoint in a real equidistance line is a point on a low-water line, and many of the most consequential are exactly the rocks and drying banks this essay ends on, so the datum question sits underneath that one. A tripoint defined three times found three answers to where three boundaries meet under three conventions, and one sentence, and the ground between its readings the ground between readings of a single sentence. The fifth article is a sentence with a height inside it, and the height’s convention is left to a hydrographic office.

It is also the counterpart of the level a country measures its heights from. Every country’s zero is a different surface finds each national height datum pinned to mean sea level at one tide gauge, and eight European zeros spread over 462 millimetres; a chart datum is the same kind of object, a tidal surface chosen for a purpose, taken downwards instead of up. Height above what separates a height above a mathematical surface from a height above the surface water settles on, and a chart datum is a third answer to that question: a surface defined by where the tide itself stops.

Still open: straight baselines, where the coast is replaced by chosen points #

Everything here measured the sea from the coast as it is. The seventh article lets a state with a deeply indented coast, or a fringe of islands along it, measure instead from straight lines joining appropriate points, provided they do not depart appreciably from the general direction of the coast. The low-water line then matters only at the points chosen, and the tide inside the lines stops mattering at all.

That replaces a question about a datum with a question about a selection: which headlands, rocks and islands are joined, how far the lines may stray from the general direction of a coast that has no single direction, and how much sea each choice of points adds over the envelope of arcs from the coast itself. It is a boundary drawn by picking points, and a coast whose shape is fixed cannot ask how much the picking is worth.