Editing True-range multilateration (section)

== Preliminary and final computations ==
{{expand section|date=June 2018}}

[[File:Trilateration Station Ranging Measurements.jpg|thumb|Fig. 4 2-D true-range multi-lateration (trilateration) system ranging measurements]]

The position accuracy of a true-range multilateration system—e.g., accuracy of the <math>(x,y)</math> coordinates of point '''P'''  in Fig. 1 -- depends upon two factors: (1) the range measurement accuracy, and (2) the geometric relationship of '''P''' to the system's stations '''C1''' and '''C2'''. This can be understood from Fig. 4. The two stations are shown as dots, and BLU denotes baseline units. (The measurement pattern is symmetric about both the baseline and the perpendicular bisector of the baseline, and is truncated in the figure.) As is commonly done, individual range measurement errors are taken to be independent of range, statistically independent and identically distributed. This reasonable assumption separates the effects of user-station geometry and range measurement errors on the error in the calculated <math>(x,y)</math> coordinates of '''P'''. Here, the measurement geometry is simply the angle at which two circles cross—or equivalently, the angle between lines '''P-C1''' and  '''P-C2'''. When point '''P-''' is not on a circle, the error in its position is approximately proportional to the area bounded by the nearest two blue and nearest two magenta circles.

Without redundant measurements, a true-range multilateration system can be no more accurate than the range measurements, but can be significantly less accurate if the measurement geometry is not chosen properly. Accordingly, some applications place restrictions on the location of point '''P'''. For a 2-D Cartesian (trilateration) situation, these restrictions take one of two equivalent forms:
* The allowable interior angle at '''P''' between lines '''P-C1''' and  '''P-C2''': The ideal is a right angle, which occurs at distances from the baseline of one-half or less of the baseline length; maximum allowable deviations from the ideal 90 degrees may be specified.
* The horizontal dilution of precision (HDOP), which multiplies the range error in determining the position error: For two dimensions, the ideal (minimum) HDOP is the square root of 2 (<math>\sqrt{2} \approx 1.414</math>), which occurs when the angle between '''P-C1''' and '''P-C2''' is 90 degrees; a maximum allowable HDOP value may be specified. (Here, equal HDOPs are simply the locus of points in Fig. 4 having the same crossing angle.)

[[File:2D HDOP for 2 Range Stations 2019-0118.jpg|thumb|Fig. 5 HDOP contours for a 2-D true-range multilateration (trilateration) system]]

Planning a true-range multilateration navigation or surveillance system often involves a [[Dilution of precision (navigation)|dilution of precision]] (DOP) analysis to inform decisions on the number and location of the stations and the system's service area (two dimensions) or service volume (three dimensions).<ref name="Langley">[http://www2.unb.ca/gge/Resources/gpsworld.may99.pdf "Dilution of Precision"], Richard Langeley, ''GPS World'', May 1999, pp 52–59.</ref><ref name="Lee1">[https://rosap.ntl.bts.gov/view/dot/12134 ''Accuracy Limitations of Range-Range (Spherical) Multilateration Systems''], Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Technical Note 1973-43, Oct. 11, 1973.</ref> Fig. 5 shows horizontal DOPs (HDOPs) for a 2-D, two-station true-range multilateration system.  HDOP is infinite along the baseline and its extensions, as only one of the two dimensions is actually measured. A user of such a system should be roughly broadside of the baseline and within an application-dependent range band. For example, for DME/DME navigation fixes by aircraft, the maximum HDOP permitted by the U.S. FAA is twice the minimum possible value, or 2.828,<ref name="Lilley" /> which limits the maximum usage range (which occurs along the baseline bisector) to 1.866 times the baseline length. (The plane containing two DME ground stations and an aircraft is not strictly horizontal, but usually is nearly so.) Similarly, surveyors select point '''P''' in Fig. 1 so that '''C1-C2-P''' roughly form an equilateral triangle (where HDOP = 1.633).

Errors in trilateration surveys are discussed in several documents.<ref name="Navidi">[https://web.archive.org/web/20190119230805/https://pdfs.semanticscholar.org/fdfb/dfc1de2568b71e5bdfc4348a819b72597206.pdf ''Statistical Methods in Surveying by Trilateration'']; William Navidi, William S Murphy, Jr and Willy Hereman; December 20, 1999.</ref><ref name="Provoro">[https://web.archive.org/web/20210508053225/https://apps.dtic.mil/dtic/tr/fulltext/u2/650359.pdf ''Comparison of the Accuracy of Triangulation, Trilateration and Triangulation-Trilateration'']; K.L. Provoro; Novosibirsk Institute of Engineers of Geodesy; 1960.</ref> Generally, emphasis is placed on the effects of range measurement errors, rather than on the effects of algorithm numerical errors.