A weather forecast made out of orbits

An empirical thermospheric model takes a date, a place and two indices of solar and geomagnetic activity, and returns a density. Its published accuracy is about fifteen per cent, and that number has barely moved in four decades.

The reason it has not moved is worth being precise about, because it decides what can be done. The limitation is not a shortage of data: the models are fitted to decades of satellite drag, which is millions of measurements. It is that a smooth functional form driven by two ground-measured proxies cannot reproduce an atmosphere heated by extreme ultraviolet and by high-latitude particle precipitation on timescales of minutes.

A model of that kind is a climatology. It says what the density usually is under stated conditions. Adding more history makes its coefficients better determined and does not make it right on any particular afternoon.

The alternative is to stop predicting the density from proxies and start measuring it, continuously, from the thing it is being predicted for.

Every object is a thermometer #

The observation that makes assimilation possible is the one the models were built from in the first place, used in the other direction.

A tracked object’s orbit decays at a rate proportional to the density along its path divided by its ballistic coefficient. Tracking it over a few days determines that product. If the ballistic coefficient is known, the product is a density — averaged along the orbit, over the interval of the fit.

There are of order twenty thousand tracked objects in low orbit, spread across altitudes from two hundred to fifteen hundred kilometres and across every local time. Each one is a measurement of the density somewhere.

The difficulty, which is the reason this was not done for decades, is that the ballistic coefficient is generally not known. A piece of debris has an unknown shape, an unknown mass and a tumbling attitude, so tracking it determines only the product and nothing can be separated.

Two things resolve it, and they are different in kind.

Calibration objects. A small number of objects have known ballistic coefficients — spheres launched for the purpose, and satellites whose operators publish mass and attitude. Those give absolute densities.

Relative solutions. For everything else, the fitted ballistic coefficient is allowed to be whatever it is, and what the assimilation solves for is the correction to the model density that makes all the objects consistent. An object’s own coefficient is then a nuisance parameter, and the density correction is what the population has in common.

The second is how the operational systems work, and it is why they return a correction field rather than a density.

What the update replaces, and what it does not #

The distinction the hero figure draws is between two kinds of error and they behave differently in time.

A climatological error is a bias: at this place, at this time, under these indices, the model is off by some amount, and that amount persists until the conditions change. It does not average down over a forecast, because it is the same error the whole way.

An assimilated error starts at whatever the observations can determine — a few per cent — and grows as the atmosphere evolves away from the state that was observed. How fast it grows is the atmosphere’s memory: the time over which a density anomaly persists before the driving forgets it.

That memory is short. The thermosphere is driven by solar ultraviolet on timescales of a solar rotation, by geomagnetic activity on timescales of hours, and it relaxes by conduction and by the global circulation over roughly half a day to a few days depending on altitude and on how disturbed things are.

So the assimilated error rises from three per cent toward fifteen over a few days, and the advantage of having assimilated decays with it.

The practical statement is that assimilation transforms a short forecast and does nothing for a long one, which sorts the operational products into two groups.

Which predictions it changes #

The sorting is sharp and it follows from the timescales.

A conjunction assessment asks whether two objects will pass within some distance in the next few days. The screening is done three to seven days ahead and refined as the event approaches, with the decision to manoeuvre taken hours before. In that window assimilation is worth a factor of several in the along-track error, which is the dominant term in a close-approach probability.

A re-entry prediction asks when an object will come down, which for anything near the end of its life is days to weeks ahead. Over that horizon the assimilated model has relaxed to the climatology and there is nothing left to gain — and the remaining error is dominated by the future solar activity, which is a forecast of the Sun rather than of the atmosphere.

A lifetime estimate for a satellite being designed asks about decades, which is entirely a question about the solar cycle’s amplitude and nothing about the present state at all.

So assimilation is a collision-avoidance technology, and describing it as an improvement to atmospheric modelling generally is a category error: it improves one class of prediction by a large factor and leaves the others where they were.

Why the error grows as the square of the time #

The shape of every curve in the figures is the same and it is worth deriving, because it is what makes the forecast horizon the quantity that matters rather than the instantaneous accuracy.

A fractional error $\varepsilon$ in the density produces a fractional error $\varepsilon$ in the drag acceleration. An acceleration error integrates once into a velocity error and twice into a position error, so after a time $t$ the displacement error is of order $\varepsilon\, a_{\rm drag} t^2$.

For a low orbit the relevant displacement is along-track rather than radial, and the reason is the one the first essay on this subject is about: drag removes energy, energy loss lowers the orbit, and a lower orbit goes round faster. So a drag error is a period error, and a period error accumulates in phase — the object is where the model says it will be, and not when.

That gives the observed pattern. A fifteen per cent density error is metres after an hour, hundreds of metres after a day, and tens of kilometres after a week.

Because the growth is quadratic, halving the density error halves the position error at every horizon — it does not delay the growth, it scales it. That is why the vertical distance between the curves is constant on the logarithmic axes and the ratio is the whole story.

It also means the crossing point is not where the curves meet but where they meet a threshold. A conjunction becomes actionable when the along-track error is comparable to the miss distance, which is of order a kilometre, and reading off where each curve crosses a kilometre is the practical content of the figure.

The same method as a weather forecast, and the differences #

The analogy to numerical weather prediction is exact in structure and instructive in where it breaks.

A weather forecast runs a physical model forward, compares its state with observations, and adjusts the state to be consistent with both — weighted by their respective uncertainties. That is the assimilation step, and it is repeated every few hours.

Three things differ here and each makes the problem harder.

The observations are indirect and integrated. A weather station measures a temperature at a point. A tracked object measures a density averaged along an orbit over a fitting interval, which is a line integral through a rotating atmosphere over hours. Inverting that for a local density is an under-determined problem, and the assimilation has to carry the integration operator rather than compare point to point.

The model is empirical. A weather model integrates the fluid equations; a thermospheric model is a fit. Assimilating into a fit means adjusting its output rather than its state, which cannot propagate an increment forward physically. The physics-based thermospheric models exist and are used, and they do not outperform the empirical ones for operational prediction — because their inputs carry the same uncertainty and their extra physics buys resolution rather than accuracy.

And the driving is not observed where it acts. A weather model’s boundary conditions are measured. The thermosphere’s are the solar extreme ultraviolet flux and the high-latitude energy input, neither of which is measured at the cadence or the spatial resolution the response demands.

The third is the binding one, and it is why the assimilation’s advantage decays rather than persisting: the model has no way to propagate the observed state correctly, so the information is lost at the rate the atmosphere changes rather than being carried forward.

What is actually measured, and by whom #

The operational picture has changed in the last few years and the change is commercial rather than scientific.

Historically the tracked-object catalogue was maintained by a small number of government radar and optical networks, and the orbit determinations were the raw material for anything of this kind. The precision was adequate for cataloguing and marginal for density work.

Three things have shifted.

Commercial tracking networks now produce independent orbit determinations at comparable or better precision, and publish them.

Large constellations in low orbit carry precise navigation receivers and are tracked continuously by their operators to centimetres, with known masses and commanded attitudes. Thousands of such objects distributed in altitude and local time are a far better sampling of the thermosphere than the historical record ever was, and their ballistic coefficients are known rather than fitted.

And the operators manoeuvre, frequently — often to avoid each other, which is a conjunction screening’s own product — and that is a difficulty rather than a help: a manoeuvre breaks the orbit fit, and the density information in the interval around it is lost.

The assimilation systems that exist use whichever of these they can get. The constellation data are the most valuable and the least available, and whether they become a scientific resource is a question about arrangements.

What a storm does, and why it is the hard case #

The atmosphere’s behaviour under geomagnetic disturbance is where every part of this argument is stressed at once, and it is worth walking through because it is the case the systems exist for.

A geomagnetic storm deposits energy at high latitudes by particle precipitation and by resistive heating in the auroral electrojets. The energy input rises within minutes and can exceed the solar ultraviolet input over the polar regions. The atmosphere responds by expanding, and the expansion propagates equatorward as a travelling disturbance at hundreds of metres a second.

Four things then happen that the assimilation has to cope with.

The density rises by tens of per cent globally within hours, which is far outside what a climatology driven by a three-hourly ground index can track.

The response is not uniform: it is largest at high latitudes and arrives at the equator hours later, so a correction derived from objects at one inclination does not apply to another.

The composition changes as the circulation brings heavier species upward, which alters the mean molecular weight and therefore the scale height — so the vertical structure the model assumes is wrong at the same time as its amplitude.

And the observations degrade, because the orbit fits that supply the assimilation are themselves being perturbed by the thing being measured.

The result is that the assimilated model’s advantage is smallest exactly during the events that make it necessary, and quantifying that is the main open question in the operational literature. The figures here show the effect as a shorter memory and a larger observation error, which is the right direction and understates the problem.

Where the approach stops #

It cannot forecast the driver. Everything assimilation does is about the atmosphere’s present state. The future depends on the Sun, and the useful horizon for solar activity is hours for a flare’s effect and a few days for a coronal-hole stream. Beyond that the forecast reverts to climatology whatever has been assimilated.

The observations are where the objects are. The tracked population is concentrated in particular altitude bands and particular inclinations, so the assimilation is well constrained at eight hundred kilometres and poorly constrained at two hundred and at two thousand. That is a sampling problem no amount of processing fixes.

And a correction field is not a model. An assimilation that adjusts a climatology’s output rather than its state produces a correction valid where and when it was derived. Extrapolating it to a different altitude or a different local time is an interpolation with no physics behind it, and the operational systems limit how far they extrapolate for exactly that reason.

What the correction field looks like #

It is worth saying what an assimilation system actually produces, because “an improved density model” is vaguer than the output.

The output is a set of multiplicative corrections to a background model, on a grid in altitude, latitude and local time, valid for a stated interval. A typical system solves for a few tens of coefficients — an overall scaling, a low-order spatial variation, and a couple of terms describing the high-latitude enhancement — rather than for a full three-dimensional field, because the observations cannot support more.

The corrections are not small. Over a solar rotation the overall scaling wanders by tens of per cent relative to the background, and during a storm it jumps by more. That wandering is the climatological error, exhibited.

Two features of the solutions are worth noticing because they are diagnostics rather than outputs.

The corrections are coherent across objects. Twenty thousand independent measurements agreeing on a correction is what makes it a density and not a set of ballistic-coefficient errors, and the coherence is checked before the field is published.

And they are persistent across days. A correction derived today is a better predictor of tomorrow’s than the background is, which is the same statement as the atmosphere having a memory — and measuring how fast that persistence decays is how the memory in the figures is determined.

The assimilation therefore measures its own useful lifetime, which is an unusually tidy property for an operational system to have.

The general structure #

There is a lesson here that generalises past the thermosphere and is worth stating because it decides where effort should go.

A forecast has two error sources: not knowing the present state, and not knowing how it will evolve. Assimilation attacks the first. Whether that is worth doing depends entirely on which of the two dominates at the horizon that matters.

When the state is forgotten faster than the forecast runs, improving the state buys nothing. The atmosphere’s memory is a day or two; a fortnight’s forecast is a statement about the driving, and the best possible measurement of today’s density is irrelevant to it.

The same arithmetic decides where to spend effort in an ephemeris, where the question is whether the error is in the initial conditions or in the force model, and in every chaotic system, where the horizon past which a trajectory cannot be followed is set by how fast the system forgets rather than by how well it was measured.

The diagnostic is the same in all three: plot the error against the horizon with and without the improvement, and see where the curves meet. If they meet inside the horizon that matters, the improvement is real; if they meet before it, the effort belongs elsewhere.

Still open: what the next question is #

Drag has been treated as an unwanted force throughout these essays — something to be predicted, corrected for and survived. It can also be used.

A vehicle arriving at a planet on a hyperbola can be captured into orbit by a single pass through the atmosphere, spending no propellant at all, if it aims at exactly the right periapsis. That is the corridor problem run deliberately, and the corridor is narrow by exactly the amount the density is uncertain by. It has been done slowly, over months of repeated dips. The single-pass version has never been flown, and the reason is the factor of two these essays have been about.