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

A streak along the jet moves the floor while the crossing is flown found that a stretch of jet core running faster than the rest moves the diversion floor for the New York–London chain by up to forty-three kilometres, depending on which half of the Goose Bay–Narsarsuaq gap its fast air lies over. Because a streak drifts with the flow, the floor an operator must satisfy is the worst over the hours the aircraft is over water, and in the case measured that was thirty kilometres above what the departure analysis reported.

That measurement put the streak where it was. An operator has it where a forecast says it is, and a forecast places the along-axis structure of a jet — where a streak begins and ends — much less well than it places the jet’s core, because the streak is carried by the flow rather than steered by the geometry. The earlier essay ended on the precision this would need: the floor moves from nothing to its maximum over about five hundred kilometres of streak position, so an error of a hundred kilometres might be worth a fifth of the effect and an error of three hundred most of it.

That was an estimate from the shape of one curve. What an operator needs is a margin: the number of kilometres to add to a floor computed from a forecast so that the floor actually met is covered, say, nineteen times in twenty. And the question behind it is whether reading the streak from a forecast is worth doing at all, against the alternative of ignoring the streak and carrying enough allowance for the worst it could do.

One table of exact floors #

Each floor is the same computation the earlier essays ran: the least still-air radius at which a chain of diversion circles, each blown downwind by the field, keeps the route within reach of an airfield, found by a shortest-path search over the northern field. That search takes a few seconds, and a trial that needed a fresh one for every crossing would need hours. But the floor is a smooth function of one number, the streak’s position along the jet, so it is computed exactly at 35 positions 125 kilometres apart and read between them by cubic interpolation. At positions half-way between grid points the interpolated floor agrees with a fresh search to within 0.8 kilometres.

The shape is the one the earlier essay measured. With the streak far off, the floor is the plain jet’s. As the streak’s fast air moves over the Goose Bay half of the gap the floor rises, by up to 41 kilometres; as it moves on over the Narsarsuaq half it falls, by up to 45, because the fast air is then on the half where a tailwind helps rather than hurts. Beyond, the floor returns to the plain jet’s. A crossing of four hours with the streak drifting at 250 km/h sweeps a thousand kilometres of this curve, and the floor that has to be met is its highest point over that stretch.

Three ways to plan #

Every trial draws a true starting position for the streak, uniformly over the 3,100 kilometres of the jet from which a four-hour drift can bring it over the gap, and a forecast position that differs from it by a normal error of stated size. The floor actually met is the worst over the true crossing window. Against it, three plans:

The forecast’s own window. The worst over the crossing window the forecast predicts: the right procedure if the forecast were perfect, and what the earlier essay recommended.

The departure analysis. The floor at the forecast’s position at departure, which is what a single analysis valid at one time gives.

No streak at all. The plain jet’s floor, as though the streak were not there.

Each plan then needs a margin, the amount to add so that the planned floor plus margin covers the floor met in 95 per cent of crossings. With the margin added, each plan also holds something in hand on average — the planned floor plus margin, less the floor met — which is the allowance the operator carries beyond what the crossing needed. Twenty thousand seeded crossings at each forecast error.

The margin grows in proportion to the error #

The window plan’s margin is nothing with a perfect forecast, since the plan is then the truth, and grows almost exactly in proportion to the error: 4.2 kilometres at 50, 7.9 at 100, 11.5 at 150, 15.0 at 200, 22.1 at 300 and 28.6 at 400. That is about eight kilometres of margin for every hundred of error, and it is close to the curve’s own average slope where the streak matters: the floor rises by forty kilometres over roughly five hundred of streak position. A forecast misplacing the streak by a given distance misreads the window’s worst by about that slope times the distance, more where the curve is steep and less near its crest, and the ninety-fifth percentile of those misreadings is the margin.

The rate is the whole of the practical answer. A forecast that places a streak’s along-axis structure to a hundred kilometres makes the window plan cost eight kilometres of allowance. One that places it to four hundred makes it cost twenty-nine, and the plan ignoring the streak costs forty-one. At about six hundred kilometres of error the window plan’s margin reaches the flat one, and reading the forecast has bought nothing a flat allowance would not.

The departure analysis loses to ignoring the streak #

The dashed line is the surprise. Planning to the floor at the forecast’s departure position needs a margin of 39 kilometres with a perfect forecast — almost as much as ignoring the streak entirely — and at an error of six hundred kilometres, 51, more than ignoring it.

The distribution of shortfalls shows why. The departure analysis reads the floor at one moment, and in about half of all crossings the streak is short of the gap at departure and over its Goose Bay half three hours later. The analysis is then short by nearly the whole of the streak’s effect — thirty or forty kilometres — and a 95-per-cent margin has to cover that. Worse, in the crossings where the streak starts over the Narsarsuaq half, the analysis reads a floor below the plain jet’s, and a plan built on it is lower than a plan that ignored the streak.

So the only use of the streak that a single analysis supports is a wrong one. A plan that reads one moment of a moving field and a plan that ignores the field need about the same margin, and the first holds more allowance in hand on average — 20.6 kilometres against 18.0 — because its planned floors scatter both ways. Reading the streak from a forecast is worth something only when it is read as a window, which is what the streak essay argued from the timing alone; the trials say it is not a refinement of the departure analysis but a replacement for it.

What the margin holds in hand #

A margin sized for the worst twentieth of crossings is carried in all of them, and what it costs in the others is the average allowance held beyond the floor met. For the plan that ignores the streak that is 18.0 kilometres: most crossings do not meet the streak’s full effect, and all of them carry the allowance for it. For the window plan it is nothing with a perfect forecast and grows with the error — 7.8 kilometres at 100, 14.8 at 200 — until between 200 and 300 kilometres of error it passes the flat plan’s.

That is a sharper break-even than the margin gave. Covering the bad crossings, reading the forecast is better than ignoring the streak up to an error of about six hundred kilometres; averaged over all crossings, it is better only up to about two hundred and fifty. Past that, the window plan’s margin is doing the work the flat allowance does, and doing it less evenly.

An error in the drift speed is the same kind of error #

A forecast can place a streak correctly and move it at the wrong speed. The window plan assumes the streak drifts at 250 km/h, and an error in that speed shifts where the streak is at the end of the crossing by the error times four hours. An error of 50 km/h, one standard deviation, is two hundred kilometres at the far end. With the position wrong by a hundred kilometres as well, the margin rises from 8.3 kilometres with the drift speed right to 11.4 with it wrong by 50 km/h, and to 13.8 at 100 km/h, in the twelve thousand crossings the figure runs at each.

The two errors enter the same way because the floor depends on one number — where the streak is at each moment of the crossing — and both errors move it. A forecast’s skill for this purpose is the error in that position over the crossing window, not the error at the analysis time, and a forecast that is accurate at analysis and wrong in its advection has the same cost as one that starts wrong.

The flat allowance is the streak’s whole effect #

The plan that ignores the streak needs a margin of 41.4 kilometres, the whole of the most the streak can add. That is not a coincidence: a plan that does not know where the streak is must allow for it being in the worst place, and nineteen crossings in twenty include enough of those to make the margin the maximum. It is the natural reference for everything else, because it is what an operator carries who has no forecast of the streak at all.

Against it, the window plan with a forecast good to a hundred kilometres needs a fifth of that margin and holds less than half as much in hand. With a forecast good to three hundred it needs half the margin and holds more in hand. Somewhere in that range the value of reading the streak turns over, and the whole of the difference is the forecast’s error in where the streak’s fast air is along the jet.

Why the answer is a margin and not a better number #

It would be natural to want the forecast error folded into a better estimate of the floor — an expected floor, say, averaged over where the streak might be. That would be the wrong quantity. The rule is a wall: the rule that keeps a route near land is pinned at both ends found a crossing refused below one radius and free above another, each wall set by a single feature of the ocean, and the floor here is the lower wall with a wind on it. A plan that meets the floor in an average crossing is on the wrong side of the wall in half of them.

Nor does re-planning in the air rescue a plan made to the wrong floor. A crossing is a chain of decisions found re-planning worth nothing when the forecast turns out right and worth something only when it turns out wrong — and the diversion allowance is fixed before departure, so the forecast’s error has to be paid for in the plan.

So the forecast error has to become an allowance for the bad cases, and the question is only which bad cases and how many. Nineteen in twenty is a stated choice, not a regulation, and the whole comparison would move if it were ninety-nine in a hundred: every margin would grow, and the flat allowance — already the whole of the streak’s effect — would grow least, since it cannot exceed the most the streak can do. A stricter coverage favours ignoring the streak, and a looser one favours reading it.

That is the same shape a week of twilights learns what a sight is worth found for celestial fixes: knowing the error’s size is worth something only where the decision is sensitive to it, and a steady error that nobody knows the size of is often cheaper to allow for than to estimate. Here the decision is sensitive — the floor moves by forty kilometres over a few hundred of streak position — so the forecast is worth reading, up to the point where its own error is as wide as the curve.

The flow field that moves the floor is the same one the quickest route is not the shortest steered a single aircraft through, and the diversion circles are that aircraft’s reach under it. A forecast error in the field is an error in every reach set at once, and it enters the floor through the one gap that binds.

What an operator can take from this #

Read the streak as a window or not at all. The departure analysis needs nearly the margin of ignoring the streak at every forecast error, and holds more allowance in hand than ignoring it at every one.

The margin is about eight kilometres per hundred of position error. That is a property of this streak on this gap — the floor’s slope against position — and a stronger or shorter streak would change it in proportion, as a jet moves the floor that a uniform wind cannot found the floor’s movement linear in the wind’s excess.

The break-even is a statement about the forecast, not the flight. Reading the streak pays while the forecast places its along-axis structure to better than about two hundred and fifty kilometres over the crossing window. This essay makes no claim about any forecast system’s actual error; that is a verification statistic a forecast centre could compute for streak ends, and nothing here assumes one has.

The allowance was already a margin. A diversion allowance in a wind is the same circle, moved upwind is the reminder that the rule’s stated radius is itself set with room to spare. The margins here are room the rule does not know it is spending, and the comparison between plans is the useful part even for an operator whose own allowance already covers them.

What each number was checked against #

The table must be the floor. At six positions half-way between grid points, the interpolated floor agrees with a fresh shortest-path search to within 0.8 kilometres.

Far from the gap the floor must be the plain jet’s. The first grid position, 2,650 kilometres short of the gap’s midpoint, gives the plain jet’s floor to within half a kilometre.

A perfect forecast must need no margin on the window plan. With a forecast error of zero the window plan’s planned floor is the floor met in every crossing, and its margin is zero exactly.

Every plan is scored on the same crossings. One seeded stream of true positions and forecast errors per error size, shared by the three plans, so the comparisons between them carry no sampling difference of their own.

Where the model stops #

One streak, one gap. A streak 200 km/h faster than a 100 km/h core and a thousand kilometres long, on the one gap that pins this chain. The margin per hundred kilometres scales with the floor’s slope, which scales with the streak’s excess speed; its break-even does not obviously scale with anything and would have to be measured again.

A uniform prior on where the streak starts. Every starting position from which the drift can reach the gap is taken as equally likely. A real forecast week has a climatology, and a prior concentrated where the floor is steep would raise every margin.

Normal errors, independent of position. A real forecast’s along-axis error may grow with the streak’s speed or its distance from the analysis, and may be biased — a streak that is consistently forecast too slow is a drift-speed bias, not a scatter, and a margin sized for scatter would not cover it.

The floor is the rule’s, not the aircraft’s. Everything is in kilometres of still-air radius, the quantity the rule states. What an operator does with a margin — carry more fuel, pick a different track, delay — is outside it.

Still open: whether the streak can be forecast as a window #

Every plan here reads the streak as a point that moves: a position at departure, advected at a stated speed. A forecast is richer than that. It gives the wind field at every forecast hour, and the floor at each hour can be computed from the field forecast for that hour directly, without assuming anything about how the streak moves. That is the window plan done properly, and its error is then the forecast’s error at each hour rather than an initial error carried forward.

Whether the forecast’s error at the end of a four-hour crossing is larger or smaller than the carried-forward error assumed here, and whether an ensemble — many forecasts from slightly different starts — gives a spread of window floors whose ninety-fifth percentile is a better margin than any fixed rate per hundred kilometres, are questions a single streak with a stated error cannot ask.