Cartographic Projection
every map is wrong, and by exactly how much
Concept

Bottleneck — where it appears

The weakest link in a chain, which decides what the whole chain can do. A crossing joined by a sequence of diversion airfields is legal at the allowance its widest gap demands and at no smaller one, so the binding pair is a bottleneck and the rest of the route is slack.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

In a wind every circle moves upwind, and the route moves with them. New York to London with the sixty-minute allowance at a stated one-engine speed of 745 km/h. In still air every airfield's circle has a radius of 745 km, drawn faint and dashed. In a northerly of 150 km/h the places from which each airfield can be reached in sixty minutes form a circle of the same radius moved 150 km upwind, drawn solid. The dashed route is the shortest the still-air rule allows, 5,698 km; from one point of it the quickest diversion in this wind takes 68.7 minutes. The solid route stays inside the moved circles, 5,765 km, and its longest diversion takes 60.8 minutes — the sixty the moved circles stand for, to within what the Earth's curvature does to a wind that keeps one bearing.

A diversion allowance in a wind is the same circle, moved upwind

A diversion rule written in minutes is turned into a distance by assuming still air. In a uniform wind the places from which an airfield can be reached in sixty minutes are not a stretched region but the still-air circle moved upwind by the wind's run. A 150 km/h northerly makes the still-air route across the North Atlantic need a diversion of 68.7 minutes, a southerly makes it 54.0 — and the least radius at which the ocean can be crossed barely moves.

paths · Paths A fast stretch over one half of a gap, with nothing changed across the jet. A jet of 100 km/h blowing along the Goose Bay–Narsarsuaq gap, carrying a streak that reaches 300 km/h over a stretch about 1,000 km long, centred 500 km from the gap's midpoint on the Goose Bay side. The dotted bands are where the wind has slowed to 150, 100 and 50 km/h, and they are drawn from the jet's own cross-section — nothing about it changes across the flow. The solid outlines are the places within sixty minutes of each airfield. The least allowance at which New York is joined to London is 665.6 km of still-air radius against the plain jet's 624.7 — a move of +40.9 km, pinned between Goose Bay–Iqaluit.

A streak along the jet moves the floor while the crossing is flown

The diversion floor answers only to the difference between the wind on a gap's two halves, and every case so far produced that difference by moving a jet sideways. A jet carrying a streak — a thousand kilometres of core markedly faster than the rest — produces the same difference with nothing changed across the flow at all, and moves the floor by up to 43 km where the same jet at one speed along its whole length moves it by ten metres. Plotted against the half-integral difference the two geometries fall on one line: the floor cannot tell them apart. And because a streak drifts with the flow, the allowance an operator must satisfy is 30 km higher than the one the departure analysis reported.

paths · Paths The floor against where the streak is, and the stretch of it a crossing sweeps. The diversion floor for the New York–London chain with a streak 200 km/h faster than a 100 km/h jet core, against where along the jet the streak's centre lies, computed exactly on a grid of positions 125 km apart and read between them. With the streak far off the floor is the plain jet's, 624.7 km. With it over the Goose Bay half it rises by up to 41.1; over the Narsarsuaq half it falls by up to 44.8. The bar is the stretch a streak starting at -1400 km sweeps in a four-hour crossing at 250 km/h; the floor to be met is the worst on it, 666.0 km. The shaded band either side is a forecast position error of ±200 km at one standard deviation.

A forecast's error in where the streak is becomes a margin

A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.

paths · Paths Nineteen forecasts of one crossing, and the floor each says must be met. One crossing, with the forecast's error in where the streak starts at 150 km and in its drift speed at 50 km/h. The line is the worst floor over a four-hour crossing against where the streak starts, at the assumed drift of 250 km/h. Dots: nineteen ensemble members, each drawn with its own start and drift speed and placed at the worst floor its own crossing window meets. Open circle: the truth, 660.3 km. Square: the ensemble's centre, 640.3 km. Dashed: the centre's floor plus the fixed margin that covers 95 per cent of crossings at these errors, 653.3 km. Solid: the highest member, 663.2 km — the plan the ensemble gives with no margin added.

An ensemble carries its own margin, if its spread is honest

A fixed margin of about eight kilometres for every hundred of forecast error covers the diversion floor in nineteen crossings of twenty, but it needs the error's size and it is spent on every crossing alike. Nineteen forecasts of the same crossing need neither. If the truth is one more draw from the spread they come from, the highest of them covers the floor met in nineteen crossings in twenty by rank alone — whatever the error, the streak or the drift — and it holds a fifth to two fifths less in hand than the fixed margin, because it spends nothing on the crossings the streak cannot reach. What it cannot do is notice that its own spread is too small: at half the true spread it covers 88 in a hundred.

paths · Paths

Named alongside it

The objects these essays reach for when they reach for this one.

ToleranceVerificationConventionGreat circleEstimatorFlow fieldMarginReach setCoverageDegeneracyDiversionEtops

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