Two metres overhead and thirty at the horizon

Of the four media a tracking signal crosses on its way between an antenna and a spacecraft, the lowest is the most stubborn. The ionosphere and the solar plasma are dispersive, so measuring at two frequencies removes them. The gravitational delay is calculable. The neutral atmosphere is neither: it delays every frequency equally, and its delay depends on the pressure, temperature and water vapour along the path, none of which the tracking measures.

The standard treatment splits the problem in two. The delay straight up — the zenith delay — is about two and a half metres of range, most of it from the dry air and proportional to the surface pressure, the rest from water vapour, variable and hard to predict. The delay along the actual line of sight to the spacecraft is the zenith delay multiplied by a function of elevation: the mapping function. The zenith delay can be measured by a barometer and a radiometer or fitted from the data. The mapping function has to be assumed.

That assumption is harmless for most of the sky and the dominant error near the horizon, and the horizon is where tracking cannot avoid looking.

Why the cosecant fails #

If the atmosphere were a flat slab, a line of sight tilted to elevation e would cross it along a path longer than the vertical by one over the sine of the elevation, and the delay would scale the same way. That is the rule that works for most of the sky, and it is the rule that the four-media calculation used. It predicts an infinite delay at the horizon, where a line of sight would travel through an infinite slab.

The atmosphere is not a slab; it is a spherical shell. A line of sight at a low elevation does not travel horizontally forever: the Earth curves away beneath it, and within a few hundred kilometres the line has climbed out of the dense lower atmosphere through the top of the shell. The path through the atmosphere is long but finite, and the delay at the horizon is large but finite — a few tens of metres for a radio signal. The flat rule overestimates it by a growing margin below about fifteen degrees, by thirty per cent at three degrees, and by an infinite amount at zero.

The same curvature is what lets light at the horizon bend round the Earth instead of escaping in a straight line. The curvature correction is purely geometric and is easy to compute for an atmosphere of known structure: integrate the refractivity along a straight ray through spherical shells. Refraction also bends the ray, lengthening the path a little more, which adds centimetres below five degrees and is ignored in the figures. What is not easy is knowing the structure, and the mapping function’s job is to encode it.

A function written for one atmosphere #

The conventional mapping functions are continued fractions in the sine of the elevation, with three coefficients chosen so that the fraction reproduces the delay through a typical atmosphere. The form, introduced by John Marini in 1972, is chosen because it behaves like the cosecant at high elevation and bends over correctly near the horizon; its coefficients depend on the atmosphere’s scale height, which depends on its temperature. Modern versions take the coefficients from numerical weather models, day by day and site by site.

The error has a characteristic shape. Above thirty degrees every mapping function is right to a millimetre, whatever the atmosphere, because at high elevation the delay is just the zenith delay divided by the sine and the details of the profile hardly matter. Below fifteen degrees the error rises steeply, roughly as the cube of the cosecant, and its size depends on how different the day’s atmosphere is from the one the coefficients assumed. A scale height wrong by eight per cent — a temperature difference of about twenty degrees through the column — puts an error of decimetres at five degrees.

The practical conclusion is that the zenith delay and the mapping function are different kinds of uncertainty. The zenith delay is a single number per station per hour, measured or fitted, and its error scales with the cosecant like the delay itself. The mapping function’s error is a shape, concentrated in the lowest elevations, and it cannot be fitted out with a single number because it changes the relation between elevations. Modern practice therefore computes the mapping function from a numerical weather model of the actual atmosphere above the station at the time of the observation, which reduces the shape error several-fold, and still treats the lowest elevations with suspicion.

The atmosphere is not the same in every direction #

A mapping function depends only on elevation, which assumes the atmosphere is horizontally uniform: the same profile in every direction from the station. It is not. The air thins systematically towards the poles, weather fronts tilt the layers, and a coastal station sees moist air over the sea and dry air over the land. The delay looking north is then different from the delay looking south at the same elevation.

The gradient term grows faster with decreasing elevation than the delay itself, roughly as the square of the cosecant, because a line of sight at low elevation samples the atmosphere over a long horizontal distance and so integrates the gradient along it. A gradient invisible at the zenith is ten centimetres at five degrees. And because its sign depends on the azimuth, it does not average away as a spacecraft is tracked through a pass: the spacecraft rises in one direction and sets in another, and the gradient adds to the range at one end and subtracts at the other, producing an error that looks like a slope in time — which is exactly how an error in the spacecraft’s velocity would look.

Gradients are now estimated alongside the zenith delay, as two extra numbers per station per few hours, one for the north–south tilt and one for the east–west. The estimate needs low-elevation observations spread round the compass, and a station tracking one spacecraft through one pass has little azimuth spread to offer, which is part of why tracking stations increasingly take their troposphere from a separate instrument rather than from the tracking data.

Two delays with different habits #

The zenith delay is not one quantity but two, and the reason for splitting it is that they behave differently. The hydrostatic part comes from the dry air, and it is almost exactly proportional to the pressure at the surface: about 2.3 millimetres of delay for every hectopascal, so a standard atmosphere of a thousand hectopascals gives about 2.3 metres. A barometer at the station gives it to a millimetre, and it changes only as weather systems pass, by a few centimetres over days.

The wet part comes from water vapour, and it is small, unruly and local. A dry winter day over a desert station may contribute a few centimetres; a humid summer day over a coastal one, forty. Its relation to anything measured at the ground is poor, because the water is concentrated in the lowest two kilometres in patches that drift with the wind, and the line of sight to a spacecraft at low elevation passes through tens of kilometres of that patchwork, and bends through layers whose structure varies from day to day. The wet delay is roughly six and a half times the depth of liquid water the vapour along the path would make if condensed, which is how radiometers and weather models express it.

The two parts also have different mapping functions, because they have different scale heights. The dry air thins over eight kilometres, the water vapour over about two, so the wet layer is thinner and its curvature correction smaller: near the horizon a wet delay maps up more steeply than a dry one of the same zenith value. A single mapping function applied to the total would err by the difference, which is why every modern treatment carries two.

A delay and its rate #

Deep-space navigation relies more on Doppler than on range, and Doppler measures not the delay but its rate of change. A troposphere that delayed every signal by the same amount would leave Doppler untouched; what matters is how fast the delay changes as the station’s rotation carries the line of sight through the atmosphere. That rate is the zenith delay times the derivative of the mapping function, times the rate at which the elevation is changing — and both of the latter are largest near the horizon, where the mapping function is steepest and a rising or setting spacecraft’s elevation changes fastest.

So the Doppler error from the troposphere is concentrated at the ends of a pass even more strongly than the range error is. A position measured from a frequency recovers the spacecraft’s declination from how the Doppler signature varies over the day, and that variation is carried mostly by the low-elevation ends of each pass — the same ends where the atmosphere’s rate error is largest. A mapping-function error that is a fraction of a per cent of the delay becomes a velocity error that the orbit fit can mistake for a small error in where the spacecraft is pointing from.

A pass from rise to set #

Follow a single tracking pass to see how this plays out. A spacecraft at a declination of twenty degrees, tracked from a station at thirty-five degrees north, rises in the east-north-east, climbs to a maximum elevation of about seventy-five degrees, and sets in the west-north-west some ten hours later. At rise and set, at five degrees elevation, the tropospheric delay is about twenty-five metres; at culmination it is two and a half. Over the pass the delay falls by more than twenty metres and rises again.

With a mapping function in error by the cold-day amount drawn above, the range at five degrees is wrong by more than a decimetre and at fifteen degrees by a centimetre, while at culmination it is right to a millimetre. Add a gradient of a millimetre, pointing north, and the rising and setting ends — both looking somewhat northward — acquire errors of several more centimetres with the same sign. A fit of the spacecraft’s state to that pass sees a systematic signal in its residuals, symmetric and concentrated at the ends, and it will absorb part of it into whichever of the spacecraft’s parameters has the most similar shape.

That is the general hazard of a correction that is right where the data are plentiful and wrong where they are scarce but influential: the fit will not report the error as a large residual. It will report it as a small bias in the answer.

Why the low elevations cannot be dropped #

All of this suggests an obvious remedy: ignore the observations near the horizon. Above twenty degrees the mapping function is good to a millimetre and the gradient to a few. Below it the errors are centimetres. A cutoff would remove them.

The remedy has a cost, and it is not only the lost data. The station’s own position is one of the unknowns, and its height and the zenith delay affect every range in nearly the same way: raising the antenna shortens every range by the height times the sine of the elevation, while a larger zenith delay lengthens every range by the delay times the mapping function. For observations near the zenith both are nearly constant, and a fit cannot tell them apart. They differ only near the horizon, where the sine falls towards zero and the mapping function rises steeply.

The trade is sharp. A cutoff high enough to remove the mapping function’s error makes the station’s height and the atmosphere’s delay almost perfectly correlated, so that an error in either appears in the other, and the height uncertainty grows several-fold. A cutoff low enough to separate them admits the mapping function’s error. The working compromise in geodesy is a cutoff of five to ten degrees, with low-elevation observations down-weighted rather than discarded and the mapping function taken from weather data, and the residual error of the compromise is the dominant error in station heights measured by satellite navigation and interferometry.

For deep-space navigation the same trade appears in a different guise. A spacecraft’s declination is measured from how its Doppler signal varies as the Earth’s rotation carries the station under it, and that variation is largest when the spacecraft is low in the sky. Tracking passes begin and end near the horizon whether the navigators like it or not, and the data from those ends carry both the geometry the navigation needs and the troposphere’s largest errors.

Fifty years of mapping functions #

The mapping function has a history that tracks the precision of the measurements it served. In the late 1960s, when ranges to spacecraft were good to tens of metres, Helen Hopfield modelled the atmosphere as two layers with refractivity falling as a fourth power of height and integrated through them; the cosecant would have done almost as well. Jaakko Saastamoinen’s formula of 1972 gave the hydrostatic zenith delay from surface pressure to a millimetre, a result still used unchanged. Marini’s continued fraction of the same year fixed the form every later function has kept.

As interferometry pushed station positions to the centimetre in the 1980s, the coefficients became the issue. Arthur Niell’s mapping functions of 1996 tied them to latitude and season, removing most of the systematic error for an average day at a given site. And from the 2000s the coefficients have been computed from global numerical weather models every six hours for every station — the Vienna mapping functions — which capture the actual temperature and humidity profile above each site rather than a climatology.

Each step reduced the low-elevation error by a factor of a few, and each was driven by a measurement that had become precise enough to see the previous step’s error. The pattern is the same as for every correction in tracking: it is invisible until the data are good enough to be limited by it, and then it is the whole problem.

What the instruments add #

The troposphere’s wet part can be measured directly. A water-vapour radiometer pointed along the line of sight measures the microwave emission of the water molecules and converts it into a wet delay to a few millimetres, independently of any model. The deep-space network uses them at its antennas for the most demanding measurements — the radio-science experiments that find oceans in Doppler residuals need the troposphere’s delay rate removed to a part in ten to the fourteen — and they have turned the wet troposphere from the dominant error into a minor one for those experiments.

The dry part is measured by the surface pressure, which fixes the dry zenith delay to a millimetre, but its mapping function still depends on the temperature profile, and that comes from weather models. Satellite navigation receivers networked across the globe provide a third input: each one estimates its own zenith delay and gradients every few minutes from dozens of satellites spread over the sky, and the resulting maps of water vapour are now assimilated into weather forecasts. The error source of one discipline has become the observation of another.

The same problem in starlight #

The radio problem has an exact optical counterpart. Starlight passing through the atmosphere is delayed and bent, and astrometry from the ground has to correct positions for refraction that grows towards the horizon, with a formula that is the cotangent of the elevation at high altitude and fails near the horizon for the same reason the cosecant fails here: the atmosphere is curved and its profile is uncertain. The optical correction is also dispersive, so a star’s image near the horizon is spread into a short spectrum, while the radio delay at frequencies below about twenty gigahertz is not. Both disciplines arrive at the same practice: observe high where precision matters and accept low elevations only when the geometry requires them, with the atmosphere’s structure taken from outside the measurement.

Still open: the atmosphere as an unknown rather than a correction #

The mapping function was invented as a correction, a known function applied to remove a nuisance. At the millimetre level it has become an unknown in its own right, estimated from weather models and from the tracking data together, with gradients and time variations of its own. Whether the next factor of ten in tracking and geodetic precision comes from better instruments or from treating the atmosphere’s three-dimensional structure as part of the solution — ray-traced through weather-model fields for every observation rather than summarised by a function of elevation — is a choice being made now. Ray tracing removes most of the mapping function’s error and inherits the weather model’s, and how good those models are at the scale of a single line of sight through the lowest kilometre of the atmosphere is the question on which the choice depends.