Radio fixes and DME/DME geometry

Why are two measurements not always a good fix?

An intersection can be mathematically unique and operationally weak.

Lines and circles

A bearing defines a position line from a known station. A horizontal range defines a circle; a slant range defines a sphere before height is accounted for. A VOR radial plus suitable DME range from the associated facility gives a polar fix. Two bearings or two ranges require an intersection and may produce ambiguity without additional information.

With DME/DME, two horizontal range circles normally have two possible intersections. Prior position, additional stations and system logic distinguish the relevant solution. The laboratory chooses the northern intersection explicitly; it does not conceal the second solution.

Small crossing angle, large uncertainty

When position lines cross nearly parallel, a small measurement error can move the intersection a long way along the narrow angle. Near a right-angle crossing, the same errors constrain both directions more effectively. Geometry is independent of whether the displayed numbers look precise.

Two circles can give two answers

Place two stations 20 NM apart on an east-west line. Equal horizontal ranges of √200 = 14.142 NM intersect at two points: 10 NM north or 10 NM south of the midpoint. Both exactly satisfy the pair of numbers. Adding more decimal places does not choose between them, because the ambiguity is geometric rather than rounding error.

A prior position clearly north of the baseline, another range, a bearing or a suitable inertial solution can identify the northern candidate. A system must also avoid accepting an impossible circle pair: ranges of 2 NM and 3 NM cannot meet when the stations are 20 NM apart. The laboratory makes these conditions explicit so that a map marker is never fabricated from incompatible observations.

How can two accurate ranges make a weak position update?

Evaluate where the stations lie as seen from the aircraft.

Crossing geometry

Imagine two DME stations almost in the same direction from the aircraft. Both ranges constrain nearly the same dimension, leaving the perpendicular dimension weakly observed. Stations with a suitable angular spread provide complementary information. An aircraft nearly on the line through both stations also creates poor geometry.

For two range constraints with equal small random error, a useful qualitative amplification proxy is 1/|sin θ|, where θ is their crossing angle. It is 1 at 90° and about 5.76 at 10°. This is a teaching proxy, not an avionics protection-level calculation.

Update selection is conditional

An FMS may select or reject radio updates using facility availability, geometry, distance, masking and its installed algorithms. Receiving two signals is not proof that DME/DME navigation meets a particular specification. Check achieved performance, sensor status and approved capability, particularly when GNSS is unavailable.

Compare two station layouts

In layout A the two range constraints cross at 90° near the aircraft. A small change in either measured range moves the fix in a different direction, so both dimensions are well constrained. In layout B they cross at 10°. The same range changes can shift their intersection much farther along the weak direction. The displayed ranges can be equally precise in both layouts.

Using the transparent proxy, 1/sin 90° = 1 while 1/sin 10° = 5.76. It illustrates the amplification ratio, not a protection level or the exact covariance of a real DME/DME solution. A navigation system’s suitability assessment must also account for station quality, availability, altitude correction and its approved selection logic. Good geometry cannot rescue an invalid range, and valid ranges cannot rescue arbitrarily poor geometry.

Can you explain and apply the mechanism?

Four original questions. Work from the stated facts, then compare your reasoning with the explanation.