Four satellites in view, and which four

Five satellites can keep every point on the Earth in sight of at least one — that is a statement about coverage, about whether some satellite is above every horizon at every moment. Navigation asks for more. A receiver that wants its position from satellite signals needs four satellites in view at once, and a constellation built for navigation is designed so that four or more are above every place at every time.

The question this essay asks is the one that follows: given that four are in view, how good is the position? The answer turns out to depend less on how many satellites are visible than on where they are in the sky, and it is this dependence — not the count — that sets how many satellites a navigation constellation actually carries.

Why four #

A navigation satellite broadcasts its position and the time by its clock. The receiver notes when each signal arrives, by its own clock, and multiplies the delay by the speed of light to get a range. With three ranges from three satellites at known positions, the receiver’s position is the intersection of three spheres — two points, one of them far out in space and easily discarded.

But the receiver’s clock is not accurate. A satellite carries an atomic clock, corrected for its motion and its height in the Earth’s gravity, and steered to a common time scale; a receiver carries a quartz oscillator that drifts by microseconds a second. A microsecond of clock error is three hundred metres of range, the same for every satellite. So each measured quantity is not a range but a pseudorange — the true range plus the speed of light times the receiver’s clock offset — and the unknowns are four: three coordinates and the offset. Four unknowns need four measurements.

The receiver’s clock offset is therefore a fourth coordinate of the solution, and the satellites are, in effect, determining the receiver’s position in space and time together. A navigation receiver is also a clock set to atomic time, accurate to tens of nanoseconds, which is how most of the world’s networks keep time.

From ranges to positions #

The solution is a least-squares fit. Linearised about an estimate of the receiver’s position, each pseudorange gives an equation in four small corrections: three components of a position shift and one of a clock shift. The coefficients of the position shift are the components of the unit vector from the receiver toward the satellite, with a minus sign — moving toward a satellite shortens its range — and the coefficient of the clock shift is one. Stacked for all the satellites, the coefficients form the geometry matrix $H$, one row per satellite and four columns.

If every pseudorange has an independent error of standard deviation $\sigma$, the errors of the fitted position and clock are given by the matrix $(H^{\mathsf T}H)^{-1}$ multiplied by $\sigma^2$. The square root of the sum of its three spatial diagonal terms is the position dilution of precision, PDOP:

$$\sigma_{\rm position} = \mathrm{PDOP}\times\sigma_{\rm range}.$$

PDOP depends only on the directions to the satellites. It is a pure property of the geometry, the same for every receiver at the same place and moment, and it multiplies whatever error the ranges carry.

The first figure shows its range. When the four satellites are spread — one overhead, three around the horizon — each constrains the receiver along a different direction, and the dilution is about two: the position error is twice the range error. When the four are bunched in one patch of sky, they all constrain the receiver along nearly the same direction, and a shift perpendicular to that direction changes all four ranges by nearly the same amount, which the clock offset can absorb. The dilution is sixty.

A day at one place #

A real constellation’s satellites move, and the geometry changes from minute to minute.

The spikes are the geometry’s signature. At most moments, six or seven satellites are visible and spread well enough that the dilution is two. Every few hours a satellite sets just as another is about to rise, and for a few minutes the remaining five happen to cluster; the dilution climbs to five or nine, and a receiver measuring ranges to a metre fixes its position to five or nine metres. Those minutes are predictable from the orbits — the dilution is computed for every place and time in advance — and surveyors and pilots plan around them.

The constellation drawn here is an idealised version of the American system: twenty-four satellites in six planes, half a day per orbit, arranged in a regular pattern. The real system has more satellites, placed irregularly within the planes precisely to remove the worst of these moments.

Up is always worse #

The asymmetry is one of the most consistent facts about satellite navigation. A horizontal shift of the receiver lengthens the ranges to satellites on one side and shortens those on the other; no adjustment of the clock offset can mimic that, because the clock changes every range by the same amount. A vertical shift, by contrast, shortens every range a little — every satellite is above — and a slight change of the clock offset can absorb most of it. Height and clock are partly degenerate, and the degeneracy is broken only by the satellites near the horizon, whose ranges change little with height.

The practical consequence is familiar to anyone who has watched a receiver’s altitude reading wander while its horizontal position stays steady: vertical errors are typically one and a half to two times horizontal ones. It is not a defect of any receiver. It is the geometry of measuring a position from one side only.

A clock that counts as a satellite #

The height–clock degeneracy suggests its own remedy. If the receiver’s clock were already known — if it were an atomic clock synchronised to the system’s time, or if the time were held steady from an earlier fix — the fourth unknown would disappear, the clock column of the geometry matrix would drop out, and three satellites would suffice.

The improvement is not only in the count. With the clock fixed, a vertical shift of the receiver can no longer be absorbed by an adjustment of the clock, and the vertical dilution of precision falls sharply — for a typical sky, by about a factor of two. Receivers on aircraft and ships that carry good oscillators exploit this by “coasting” the clock through short gaps in the satellite coverage, and timing receivers at fixed sites, whose positions are known exactly, run the problem the other way and solve only for time.

A clock is a measuring instrument for more than time, and in navigation it is literally worth a satellite. The receiver’s oscillator stands in for the fourth range; the better the oscillator, the fewer satellites the receiver needs and the better its height.

A hole in the northern sky #

The constellation’s orbits are inclined at 55 degrees, so no satellite’s ground track goes beyond 55 degrees of latitude, the most northerly point of every orbit. An observer at 52 degrees north therefore sees satellites in every direction except one: toward the northern sky beyond a certain elevation, where a satellite would have to be over a latitude no orbit reaches. The result is a hole in the sky, centred somewhat north of the zenith, that no satellite ever crosses.

The hole matters for the geometry because it removes a direction. A receiver at high latitude sees satellites to the south, east, west and overhead, and only low in the north; its constraint from the north is weak, and its north–south position is somewhat worse than its east–west. At the equator the hole is absent, and at the poles every satellite is low, which gives excellent horizontal geometry and poor vertical. The dilution of precision therefore has a latitude dependence set by the inclination of the orbits, and systems designed for high-latitude users choose higher inclinations or add satellites in highly elliptical orbits that spend most of their time over the north.

Corrections from a fixed point in the sky #

The ranges themselves can be improved as well as the geometry. A network of reference stations at known positions measures the errors in every satellite’s range — from orbit and clock errors and from the ionosphere — and broadcasts corrections that receivers apply before solving. In several regions those corrections are broadcast from geostationary satellites, which hang at a fixed point in each region’s sky and add a ranging signal of their own.

A geostationary satellite contributes little to the geometry at mid-latitudes — it sits low in the southern sky, close to the direction of other satellites — but its corrections reduce every range error by a factor of several, and the position error with it. The two improvements multiply: better ranges times better geometry. For aviation, where both the integrity of the position and its vertical precision are certified, the combination is what allows approaches to runways with no ground-based landing aids.

How many satellites the worst place needs #

This is the essay’s central result, and it explains the design of every navigation constellation. A constellation good enough for the median user is small: eighteen satellites give a typical dilution of under four, and twenty-four give two. What forces more satellites is the tail — the places and moments where the geometry happens to fail. With eighteen, nearly four per cent of place-moments have fewer than four satellites in view and no fix at all. With twenty-four in a regular pattern, a few still do. With thirty, none do, and the worst in a hundred has a dilution of under four.

The extra satellites are bought for the worst minute at the worst place, not for the typical user, and they buy it by filling gaps in the geometry rather than by adding to the count. That is why the operational constellations carry thirty or more satellites, not twenty-four, and why their satellites are placed irregularly within each plane — an irregular arrangement removes the specific moments when a regular one bunches.

The same reasoning, applied with different weights, sets the size of every multi-satellite system whose usefulness depends on geometry rather than coverage: the number of stations needed to triangulate an earthquake, the number of radio telescopes in a very-long-baseline array, the number of receivers in a network that times lightning strikes. In each, a minimum number gives a solution and many more are needed for the solution to be uniformly good.

A street as an elevation mask #

The figures so far assume the receiver can see every satellite more than ten degrees above its horizon. In a city, a valley or a forest, it cannot.

The mask is cruel because it removes the satellites the geometry most needs. The satellites near the horizon are the ones whose directions differ most from each other and from the satellite overhead; they are what spread the geometry out. A receiver in a street sees only the satellites near the zenith, which are bunched by definition, and its dilution of precision is that of the second panel of the first figure. Its vertical error suffers most, because the low satellites were also what broke the degeneracy between height and clock.

There is a counterweight, and it is the reason navigation receivers apply a mask even in open country. A signal from a satellite near the horizon passes through far more atmosphere than one from overhead, and the delay the air adds grows from two metres overhead to thirty at the horizon. The low satellites give the best geometry and the worst ranges. A mask of five to fifteen degrees is a compromise between the two, and the best choice depends on how well the atmospheric delay can be modelled — which is why precise receivers, which model it carefully, use lower masks than ordinary ones.

More than one constellation #

There are now four global navigation systems — American, Russian, European and Chinese — with about thirty satellites each, and a modern receiver uses all of them. The effect on the figures is to move every curve to the right: a hundred and twenty satellites, spread across many more orbital planes, leave no place-moment with poor geometry in open country and make even a city street usable.

The multiple systems also make the geometry robust in a way a single system cannot. Each has its own orbital planes, inclinations and altitudes, so their satellites bunch at different times and places; a moment of poor geometry in one is almost never a moment of poor geometry in all. The worst-case tail that forced each single system to carry thirty satellites is, in combination, filled by the others. That redundancy is the reason the dilution of precision, once a number pilots checked before an approach, has become invisible to most users.

What remains is the vertical. More satellites above the horizon still constrain the receiver from one side, and the height–clock degeneracy is weakened, not removed. Aircraft landing on satellite guidance, which need vertical precision of a few metres, add ground stations that broadcast corrections and, at airports, satellites whose positions are monitored — measures that improve the ranges rather than the geometry.

What the figures assume #

The constellation is a regular Walker pattern. Real constellations place their satellites unevenly and replace them at different times, and the dilution-of-precision statistics of a real constellation are better in the tail than a regular pattern’s at the same number of satellites, because the irregularity is chosen to remove its worst moments.

Every range has the same independent error. In reality ranges to low satellites are noisier, because of the atmosphere and of multipath reflections from the ground, and receivers weight them accordingly. Weighting changes the dilution of precision into a weighted version that depends on the error model.

The Earth is a sphere and the orbits are circles. Neither matters for the geometry at the level drawn; both matter for the operational system, which uses precise orbits, an ellipsoidal Earth and a detailed model of every perturbation.

The grid is coarse. The global statistics are sampled every 12 minutes at places 10 degrees apart in latitude and 30 in longitude. The worst place and moment found on that grid is not the worst that exists; the true tail is somewhat worse, and the gaps for the eighteen- and twenty-four-satellite constellations somewhat larger.

The geometry is the measurement #

The dilution of precision is a small piece of linear algebra, and it carries a general lesson about any measurement made by combining distances or directions to known points. The precision of the combination is not the precision of the parts. It is the precision of the parts multiplied by a factor that depends only on how the known points are arranged around the thing being located, and that factor can be large enough to overwhelm any improvement in the parts.

For satellite navigation the lesson was learned early and built into the system: the satellites’ orbits are chosen for geometry as much as for coverage, the number carried is set by the geometry’s worst moments, and every receiver reports a dilution of precision alongside its position. The same factor appears in the ground-track coverage problem as a statement about which stations see which satellites, and in astrometry as the parallax factor that makes some times of year better than others for measuring a star’s distance. In each case the arrangement of the measurements is a quantity in its own right, and it can be designed.

The factor also explains a habit of navigation receivers that otherwise looks like waste: they track every satellite in view, not the best four. Adding a fifth, sixth or tenth satellite never makes the dilution of precision worse — each added row of the geometry matrix can only add information — and it usually makes it better, by filling in whichever direction the first four left weak. A receiver that chose the four best satellites would get a position nearly as good most of the time and a much worse one exactly when the best four happen to bunch, which is when it most needs the others. Using all of them turns the geometry’s worst moments into its ordinary ones.

Still open: navigation from a planet with no constellation #

Every result here assumes a constellation built for the purpose. The Moon and Mars have none. Spacecraft there are navigated from Earth by the Doppler shift and delay of radio signals, a single direction of measurement from a single point, whose geometry is as poor as the second panel of the first figure and is rescued only by watching the spacecraft move along its orbit for days.

Plans for lunar navigation services propose a handful of satellites in high, eccentric orbits around the Moon, enough to give a receiver on the lunar surface four ranges for part of each orbit. With so few satellites the dilution of precision is the central design parameter, and the regions of the Moon with the worst geometry — near the poles, where the most interesting landing sites lie, or in craters whose rims raise the elevation mask — are exactly the regions where the service is most needed. The trade drawn here for a constellation of thirty will be made again for a constellation of four, where the count is barely enough and the geometry is everything.