SB-13 – DME/DME Navigation for Aviation

Description:

Calculate an aircraft's current geographical latitude and longitude (sometimes also called a fix) from the known positions of two DME stations, and the slant range distance from both of them. Optionally, enter the height of the aircraft above each DME station, which allows to correct the slant range into horizontal distance, and which may improve the accuracy of the fix.

Disclaimer:

This program is designed for experimental navigation calculations over distances and at altitudes typical of general aviation (GA). It uses an approximate planar model of the Earth's surface, with corrections for the convergence of meridians towards the poles, rather than full spherical or ellipsoidal geometry.

This approximation introduces systematic position errors that can become substantial over long distances. The magnitude of these errors depends on the geographical configuration and generally increases with distance. Consequently, the program is not suitable for the extended navigation distances typical of high-altitude airline operations.

Operating conditions:

• Initialization and angle mode: Before first use, press 2nd D' to initialize the program. This also enforces 2nd Deg mode; do not use 2nd Rad or 2nd Grad. Press 2nd E' to calculate, not to initialize (see the BEWARE warning below).
• Station coordinates: Enter the geographical latitude and longitude of both DME stations in DDDMM.mmmmm format. Southern latitudes and western longitudes must be negative. The stations must have distinct positions; coincident stations do not provide a two-station position fix.
• Distances: Enter the two measured DME slant ranges in nautical miles (nm). The readings should refer to essentially the same aircraft position; aircraft movement between measurements is not corrected for. Negative distance inputs are treated as their absolute values.
• Optional height correction: Enter the aircraft's vertical height relative to each DME station's transmitting/receiving antenna, in feet, using SBR ( or SBR ), or SBR EE when both height differences are the same.
  These are not necessarily the aircraft's height above the terrain beneath it (AGL); if using altitudes above mean sea level (AMSL), subtract the corresponding station antenna elevation. The sign of a height difference is immaterial because the correction squares it.
  A height difference of zero leaves the measured slant range uncorrected. If height differences are not provided, the program uses the measured slant ranges as approximate horizontal distances.
• Planar approximation: The program converts corrected distances to equivalent latitude degrees and scales longitude differences by the cosine of the stations' mean latitude.
  This is an approximation intended for relatively short distances; its accuracy deteriorates over large geographical extents, near the poles, and for station pairs crossing the ±180° longitude discontinuity (related to the International Date Line, or IDL).
• Two possible positions: When the two distance circles intersect at two points, the program calculates both positions. LEFT and RIGHT are defined relative to the directed line from DME 1 to DME 2; they do not identify which position is the actual aircraft position.
  Additional position information/knowledge from the user is required to distinguish between the two. By choosing either LEFT or RIGHT, the user is giving this information.
• Rejection criteria:

  (a) If the absolute height difference (in feet, converted to nautical miles) is greater than the measured slant range (in nautical miles), which is physically impossible in reality (or a severe measurement error), the program rejects.
  The program then corrects each measured slant range for its associated height difference, obtaining a horizontal distance.
  If the horizontal distance is zero (indicating that the aircraft is directly above that station), there are no two distinct intersection points, which however are required for a valid fix.

  (b) When L is the horizontal distance between the two DME stations and d₁ and d₂ their respective corrected horizontal distances to the aircraft, the two distance circles can have five geometrical relationships:

  Condition	Geometry	Result
  L > d1 + d2	Circles too far apart to meet (externally separated)	REJECT: no intersection
  L = d1 + d2	Circles touch externally at one point	REJECT: tangency; perpendicular offset h = 0
  |d1 − d2| < L < d1 + d2	Circles cross at two distinct points	Proceed to the intersection angle quality check (see below)
  L = |d1 − d2|	Circles touch internally at one point	REJECT: tangency; perpendicular offset h = 0
  L < |d1 − d2|	One circle lies inside the other without touching (internally separated)	REJECT: no intersection

• Intersection angle quality: Two intersections can still give an unreliable fix if the circles cross at a very shallow angle, understood as the angle between the two lines from the aircraft to each of the two DME stations.

  This angle is folded into the range 0°-90°, because sin(α) = sin(180°-α); an obtuse intersection angle is equivalent to its supplementary acute angle.
  The program rejects a fix if the folded angle is less than 5°. This rejects intersections near either external or internal tangency, even when the circles cross at two distinct points.
  The error margin amplification factor is calculated as the reciprocal of the sine of this folded angle.

  Intersection angle	Error margin amplification factor	Assessment
  0° to <5°	≥11.47× (undefined at 0°)	REJECT
  5° to <15°	3.86–11.47×	Poor
  15° to <30°	2.00–3.86×	Marginal
  30° to <60°	1.15–2.00×	Good
  60° to 90°	1.00–1.15×	Excellent

Usage:

To initialize for a new pair of DME stations, press 2nd D'; this also enforces 2nd Deg mode.

Enter the geographical coordinates of both DME stations, followed by the slant range distance for each of them.

Optionally, enter the height of the aircraft above each DME station with SBR (, SBR ) or SBR EE, which allows to correct the slant range into horizontal distance, and which may improve the accuracy of the fix.

Press 2nd E' to calculate the aircraft's two possible positions. If the geometry is acceptable, you can then retrieve the two positions, either the one to the LEFT of the directed line from DME 1 to DME 2 with D; or the one to the RIGHT of the directed line from DME 1 to DME 2 with E.

After calculation, A, B, 2nd A' and 2nd B' retrieve their stored values instead of accepting new ones; C and 2nd C', as well as SBR (, SBR ) and SBR EE, always accept new entries (in order to allow for quick successive fixes).

To change station coordinates, press 2nd D' and re-enter all data; initialization clears registers 01–28, including the stored heights, and sets 2nd Deg mode.

If you press D or E before a valid result has been obtained by 2nd E', or if the calculation has been rejected, then both D and E will display a zero in both the display and in t.

Note that not only does 2nd D' reset the flag for valid output and 2nd E' in case of a rejected calculation, but so do C and 2nd C', as well as SBR (, SBR ) and SBR EE, i.e.,

whenever you re-enter a new slant range distance or a new height of the aircraft above the corresponding DME station. This prevents muddled results while allowing repeated fixes in rapid succession.

Simply press 2nd E' again.

After pressing 2nd E', the program displays the acute intersection angle (0–90°) in x and the error margin amplification factor in t.

The error margin amplification factor indicates how sensitive the calculated position is to errors in the measured distances; it is not an absolute position-error estimate.

If 2nd E' has rejected the fix, the program stops and shows a blinking "9.9999999 99" in the display (x), and t is set to zero.

The function SBR = can be used anytime after calling 2nd E' to display the intersection angle and the error margin amplification factor again, also in the case of a rejected calculation, if available (if not, zero and plus infinity will be shown as defaults).

Function keys:

Subroutines:

Output:

Good to know:

The program does not interrupt ongoing calculations. Its functions and subroutines do not use CLR or = internally, so ongoing arithmetic operations are not prematurely cleared or evaluated.

Registers used:

Glossary: