Multilateration is a navigation and surveillance technique based on the measurement of the
times of arrival (TOAs) of energy waves (radio, acoustic, seismic, etc.) having a known propagation speed. The time origin for the TOAs is arbitrary. (By the reciprocity principle, any conceptual method that can be used for navigation can also be used for surveillance, and vice versa.) For surveillance, a subject of interest – in cooperative surveillance, often a vehicle – transmits to multiple receiving stations having synchronized 'clocks'. For navigation, multiple synchronized stations transmit to a user receiver. To find the coordinates of a user in dimensions (typically, or ), at least TOAs must be measured. Multilateration systems are also called
hyperbolic systems, for reasons discussed below.
One can view a multilateration system as measuring TOAs, and then either: (a) determining the time of transmission (TOT) and user coordinates; or (b) ignoring the TOT and forming time difference of arrivals (TDOAs), which are used to find user coordinates. Systems and algorithms have been developed for both concepts. The latter is addressed first, as it was the first implemented. A TDOA, when multiplied by the propagation speed, is the difference in the true ranges between the user and the two stations involved (i.e., the unknown TOT cancels). A TOA, when multiplied by the propagation speed, is termed a pseudo range.
For surveillance, a TDOA system determines the difference in the subject of interest's distance to pairs of stations at known fixed locations. For one station pair, the distance difference results in an infinite number of possible subject locations that satisfy the TDOA. When these possible locations are plotted, they form a hyperbola. To locate the exact subject's position along that curve, multilateration relies on multiple TDOAs. For two dimensions, a second TDOA, involving a different pair of stations (typically one station is one of the first two, and one station is new), will produce a second curve, which intersects with the first. When the two curves are compared, a small number of possible user locations (typically, one or two) are revealed. Multilateration surveillance can be performed without the cooperation or even knowledge of the subject being surveilled.
TDOA multilateration was a common technique in earthfixed radio navigation systems, where it was known as hyperbolic navigation. These systems are relatively undemanding of the user receiver, as its 'clock' can have lowperformance/cost and is usually unsynchronized with station time.["The Role of the Clock in a GPS Receiver", Pratap N. Misra, GPS World, April 1996] The difference in received signal timing can even be measured visibly using an oscilloscope. This formed the basis of a number of widely used navigation systems starting in World War II with the British Gee system and several similar systems deployed over the next few decades. The introduction of the microprocessor greatly simplified operation, increasing popularity during the 1980s. The most popular TDOA hyperbolic navigation system was LoranC, which was used around the world until the system was shut down in 2010. The widespread use of satellite navigation systems like the Global Positioning System (GPS) have made TDOA systems largely redundant, and most have been decommissioned. GPS is also a hyperbolic navigation system, but also determines the TOT in order to provide accurate time.
Multilateration should not be confused with any of:

true range multilateration, which uses distance measurements from two or more sites

triangulation, which uses the measurement of angles

direction finding which does not compute a distance.
All of these systems (and combinations of them) are commonly used with radio navigation and surveillance systems.
Principle
Prior to deployment of GPS and other global navigation satellite systems (GNSSs), multilateration systems were often defined as TDOA systems – i.e., systems that form TDOAs as the first step in processing a set of measured TOAs. As result of deployment of GNSSs, two issues arose: (a) What system type are they (multilateration, trilateration, or another)? (b) What are the defining characteristic(s) of a multilateration system?

The technical answer to (a) has long been known: GPS and other GNSSs are multilateration navigation systems with moving transmitters.
[ However, because the transmitters are synchronized not only each other but a time standard, GNSS receivers are also sources of time, provided they compute TOT. This requires different solution algorithms than older TDOA algorithms. Thus, a case can also be made that GNSSs are a separate category of systems – e.g., hyperbolic navigation and timing systems or hyperbolic navigation with moving transmitters.
]

There is no authoritative answer to (b). However, a reasonable twopart answer is (1) a system whose use only involves measurement of TOAs or (if the propagation speed is accounted for) only pseudo ranges; and (2) a system whose station clocks must be synchronized. This definition includes GNSSs as well as TDOA systems.
Multilateration is commonly used in civil and military applications to either (a) locate a vehicle (aircraft, ship or car/truck/bus) by measuring the TOAs of a signal from the vehicle at multiple stations having known coordinates and accurate, synchronized 'clocks' (surveillance application) or (b) enable the vehicle to locate itself relative to multiple transmitters at known locations (stations) and having synchronized clocks using their signal TOAs (navigation application). When the stations are fixed to the earth and do not provide time, the measured TOAs are almost always used to form one fewer TDOAs.
For vehicles, surveillance or navigation stations (including required associated infrastructure) are often provided by government agencies. However, privatelyfunded entities have also been (and are) station/system providers – e.g., wireless phone systems. Multilateration is also used by the scientific and military communities for noncooperative surveillance.
Surveillance application: locating a transmitter from multiple receiver sites
If a pulse is emitted from a vehicle, it will generally arrive at slightly different times at spatially separated receiver sites, the different TOAs being due to the different distances of each receiver from the vehicle. However, for given locations of any two receivers, a set of emitter locations would give the same time difference (TDOA). Given two receiver locations and a known TDOA, the locus of possible emitter locations is one half of a twosheeted
hyperboloid.
In simple terms, with two receivers at known locations, an emitter can be located onto one hyperboloid.[In other words, given two receivers at known locations, one can derive a threedimensional surface (characterized as one half of a hyperboloid) for which all points on said surface will have the same differential distance from said receivers, i.e., a signal transmitted from any point on the surface will have the same TDOA at the receivers as a signal transmitted from any other point on the surface. ]
Therefore, in practice, the TDOA corresponding to a (moving) transmitter is measured, a corresponding hyperbolic surface is derived, and the transmitter is said to be "located" somewhere on that surface. Note that the receivers do not need to know the absolute time at which the pulse was transmitted – only the time difference is needed. However, to form a useful TDOA from two measured TOAs, the receiver clocks must be synchronized with each other.
Consider now a third receiver at a third location which also has a synchronized clock. This would provide a third independent TOA measurement and a second TDOA (there is a third TDOA, but this is dependent on the first two TDOAs and does not provide additional information). The emitter is located on the curve determined by the two intersecting hyperboloids. A fourth receiver is needed for another independent TOA and TDOA. This will give an additional hyperboloid, the intersection of the curve with this hyperboloid gives one or two solutions, the emitter is then located at one of the two solutions.
With four synchronized receivers there are 3 independent TDOAs, and three independent parameters are needed for a point in three dimensional space. (And for most constellations, three independent TDOAs will still give two points in 3D space).
With additional receivers enhanced accuracy can be obtained. (Specifically, for GPS and other GNSSs, the atmosphere does influence the traveling time of the signal and more satellites does give a more accurate location).
For an overdetermined constellation (more than 4 satellites/TOAs) a least squares method can be used for 'reducing' the errors. Extended Kalman filters are used for improving the individual signal timings.
Averaging over longer times can also improve accuracy.
The accuracy also improves if the receivers are placed in a configuration that minimizes the error of the estimate of the position.
The emitter may, or may not, cooperate in the multilateration surveillance process. Thus, multilation surveillance is used with noncooperating 'users' for military and scientific purposes as well as with cooperating users.
Navigation application: locating a receiver from multiple transmitter sites
Multilateration can also be used by a single receiver to locate itself, by measuring signals emitted from synchronized transmitters at known locations (termed stations). At least three emitters are needed for twodimensional navigation; at least four emitters are needed for threedimensional navigation. For expository purposes, the emitters may be regarded as each broadcasting pulses at exactly the same time on separate frequencies (to avoid interference). In this situation, the receiver measures the TOAs of the pulses. In TDOA systems, the TOAs are differenced and multiplied by the speed of propagation to create true range differences. In GNSS systems, different algorithms are used that implicitly form TDOAs but also compute TOT.
In operational systems, several methods have been implemented to avoid selfinterference. A historic example is the British Decca system, developed during World War II. Decca used the phasedifference of three transmitters. LoranC, introduced in the late 1950s, used time offset transmissions. A current example is the Global Positioning System (GPS). All GPS satellites broadcast on the same carrier frequency, which is modulated by different pseudorandom codes.
Another complication occurs when the transmitters are moving along known trajectories, which is the situation for GNSSs. This requires that the satellites broadcast their trajectory information.
Advantages and disadvantages
Measurement geometry and related factors
Rectangular/Cartesian coordinates
Consider an emitter (E in Figure 2) at an unknown location vector
 $\backslash vec\; E\; =\; (x,\; \backslash ,\; y,\; \backslash ,\; z)$
which we wish to locate. The source is within range of N + 1 receivers at known locations
 $\backslash vec\; P\_0,\; \backslash ,\; \backslash vec\; P\_1,\; \backslash ,\; \backslash ldots,\; \backslash ,\; \backslash vec\; P\_m,\; \backslash ,\; \backslash ldots,\; \backslash ,\; \backslash vec\; P\_N.$
The subscript
m refers to any one of the receivers:
 $\backslash vec\; P\_m\; =\; (x\_m,\; \backslash ,\; y\_m,\; \backslash ,\; z\_m)$
 0 ≤ m ≤ N
The distance ( R_{m}) from the emitter to one of the receivers in terms of the coordinates is
For some solution algorithms, the math is made easier by placing the origin at one of the receivers (P_{0}), which makes its distance to the emitter
Spherical coordinates
Lowfrequency radio waves follow the curvature of the earth (great circle paths) rather than straight lines. In this situation, equation is not valid. LoranC
[ The Development of LoranC Navigation and Timing, Gifford Hefley, U.S. National Bureau of Standards, Oct. 1972.] and Omega
[ Omega Navigation System Course Book, Peter B. Morris et al, TASC, July 1994.] are examples of systems that utilize spherical ranges. When a spherical model for the earth is satisfactory, the simplest expression for the
central angle (sometimes termed the
geocentric angle) between vehicle
v and station
m is
 $$
\cos\sigma_{vm}=\sin\varphi_v\sin\varphi_m+\cos\varphi_v\cos\varphi_m\cos\lambda_{vm}.
Here: latitudes are denoted by
φ; longitudes are denoted by
λ; and
λ_{ vm} =
λ_{ v} −
λ_{ m}. Alternative, better numerically behaved equivalent expressions, can be found in greatcircle navigation.
The distance R_{ m} from the vehicle to station m is along a great circle will then be
 $$
R_m=R_E \sigma_{vm}
Here
R_{E} is the assumed radius of the earth and
$\backslash sigma\_\{vm\}$ is expressed in radians.
Time of transmission (user clock offset or bias)
Multilateration systems measure TOAs in order to compute the user's location in dimensions. It's possible to also compute the transmitter(s) time of transmission (TOT)
in the clock time scale used to measure the TOAs. Prior to GNSSs, there was little value to determining the TOT as known to the receiver. Moreover, computing resources were quite limited. Consequently, in those systems (e.g., LoranC, Omega, Decca), receivers treated the TOT as a nuisance parameter and eliminated it by forming TDOA differences (hence were termed TDOA or rangedifference systems). This simplified solution algorithms. Even if the TOT (in 'receiver time') was needed (e.g., to calculate vehicle velocity), TOT could be found from one TOA, the location of the associated station, and the computed vehicle location.
With the advent of GPS, TOT as known to the user receiver provides useful information (and computing power had increased significantly). GPS satellite clocks are synchronized not only with each other but also with Coordinated Universal Time (UTC) (with a published offset) and their locations are known relative to UTC. However, user receivers are usually not synchronized with UTC. Thus, algorithms used for satellite navigation solve for the receiver position and its clock TOT simultaneously. The receiver clock is then adjusted so its TOT matches the satellite TOT (which is known in UTC by the GPS message). The adjustment value is called the clock offset or bias. Thus, GNSS receivers are a source of time as well as position information. Computing the TOT is a practical difference between GNSSs and earlier TDOA multilateration systems, but is not a fundamental difference. To first order, the user position estimation errors are identical.[
]
TOA adjustments
Multilateration system governing equations – which are based on 'distance' equals 'propagation speed' times 'time of flight' – assume that the energy wave propagation speed is constant along all signal paths. This is equivalent to assuming that the propagation medium is homogeneous. However, that is not always sufficiently true; some paths may involve unacceptable propagation delays due to inhomogeneities in the medium. Accordingly, to improve solution accuracy, some systems adjust (generally, move to an earlier time) measured TOAs to account for known propagation delays. Thus, GNSS augmentation systems – e.g., Wide Area Augmentation System (WAAS) and European Geostationary Navigation Overlay Service (EGNOS) – provide TOA adjustments in real time to account for the ionosphere. Similarly, U.S. Government agencies used to provide adjustments to LoranC measurements to account for variations in the conductivity of soil.
Calculating the time difference in a TDOA system
The basic measurements are the TOAs of multiple signals (
$T\_i$) at the vehicle (navigation) or at the stations (surveillance). The distance
$R\_m$ in equation is the wave speed (
$v$) times transit time, which is unknown, as the time of transmission is not known. A TDOA multilateration system calculates the time differences (
$\backslash tau\_m$) of a wavefront touching each receiver. The TDOA equation for receivers
m and 0 is
The quantity $v\; T\_i$ is often termed a pseudo range. It differs from the true range between the vehicle and station $i$ by an offset or bias which is the same for every station. Differencing two pseudo range yields the difference of two true ranges.
Figure 3a is a simulation of a pulse waveform recorded by receivers $P\_0$ and $P\_1$. The spacing between $E$, $P\_1$ and $P\_0$ is such that the pulse takes 5 time units longer to reach $P\_1$ than $P\_0$. The units of time in Figure 3 are arbitrary. The following table gives approximate time scale units for recording different types of waves.
LoranC and Decca mentioned at earlier (recall the same math works for moving receiver & multiple known transmitters), use spacing larger than 1 wavelength and include equipment, such as a
Phase detector, to count the number of cycles that pass by as the emitter moves. This only works for continuous, narrowband waveforms because of the relation between phase (
$\backslash theta$), frequency (
ƒ) and time (
T)
 $\backslash theta\; =\; 2\; \backslash pi\; f\; \backslash cdot\; T.$
The phase detector will see variations in frequency as measured
phase noise, which will be an uncertainty that propagates into the calculated location. If the phase noise is large enough, the phase detector can become unstable.
Solution algorithms
Overview
First, assume that the station locations are known as a function of time. For TDOA systems, the stations are fixed to the earth and their locations are surveyed. Satellite systems follow welldefined orbits and broadcast orbital information. Equation is the hyperboloid described in the previous section, where 4 receivers (0 ≤ m ≤ 3) lead to 3 nonlinear equations in 3 unknown values (x,y,z). The system must then solve for the unknown user (often, vehicle) location in real time. Civilian air traffic control multilateration systems use the Mode C SSR transponder message to find an aircraft's altitude. Three or more receivers at known locations are used to find the other two dimensions—either (x,y) for an airport application, or latitude and longitude for larger areas.
S. Bancroft was apparently the first to publish an algebraic solution to the problem of locating a user (vehicle) and the common TOT using only TOA measurements involving four transmitters.[ "An Algebraic Solution of the GPS Equations", Stephen Bancroft, IEEE Transactions on Aerospace and Electronic Systems, Volume: AES21, Issue: 7 (Jan. 1985), pp 56–59.] Bancroft's algorithm, as do many, reduces the problem to the solution of a quadratic equation, and yields the three Cartesian coordinates of the receiver as well as the common time of signal transmissions. Other, comparable solutions and extensions were subsequently developed.["Trilateration and extension to global positioning system navigation", B.T. Fang, Journal of Guidance, Control, and Dynamics, vol. 9 (1986), pp 715–717.][“A direct solution to GPStype navigation equations”, L.O. Krause, IEEE Transactions on Aerospace and Electronic Systems, AES23, 2 (1987), pp 225–232.][Analytical GPS Navigation Solution, Alfred Kleusberg, University of Stuttgart Research Compendium, 1994.][ A Synthesizable VHDL Model of the Exact Solution for Threedimensional Hyperbolic Positioning System, Ralph Bucher and D. Misra, VLSI Design, Volume 15 (2002), Issue 2, pp 507–520.][ EarthReferenced Aircraft Navigation and Surveillance Analysis, Michael Geyer, U.S. DOT John A. Volpe National Transportation Systems Center, June 2016.] The latter reference provides the solution for locating an aircraft with known altitude using TOA measurements at 3 receivers.
When there are more TOA measurements than unknown quantities – including the common time/range offset – e.g., 5 or more GPS satellite TOAs, the iterative Gauss–Newton algorithm for solving nonlinear least squares (NLLS) problems is generally preferred.["Closedform Algorithms in Mobile Positioning: Myths and Misconceptions", Niilo Sirola, Proceedings of the 7th Workshop on Positioning, Navigation and Communication 2010 (WPNC'10), March 11, 2010.] An overdetermined situation eliminates the possibility of ambiguous and/or extraneous solutions that can occur when only the minimum required number of measurements are available. Another important advantage of the Gauss–Newton method over some closedform algorithms is that it treats measurement errors linearly, which is often their nature, thereby reducing the effect measurement errors by averaging. The Gauss–Newton method may also be used with the minimum number of measurements. Since it is iterative, the Gauss–Newton method requires an initial solution estimate.
Multilateration systems employing sphericalrange measurements (e.g., LoranC, Decca) utilized a variety of solution algorithms based on either spherical trigonometry["Explicit (Noniterative) Loran Solution", Sheldon Razin, Navigation: Journal of the Institute of Navigation, Vol. 14, No. 3, Fall 1967, pp. 265–269.] or the Gauss–Newton NLLS iterative method.
Threedimensional Cartesian solutions
For Cartesian coordinates, when four TOAs are available and the TOT is needed, Bancroft's
[ or another closedform (direct) algorithm are one option, even if the stations are moving. When the four stations are stationary and the TOT is not needed, extension of Fang's algorithm (based on DTOAs) to three dimensions is a second option.][ A third option, and likely the most utilized in practice, is the iterative Gauss–Newton NLLS method.][
]
Most closedform algorithms reduce finding the user vehicle location from measured TOAs to the solution of a quadratic equation. One solution of the quadratic yields the user's location. The other solution is either ambiguous or extraneous – both can occur (which one depends upon the dimensions and the user location). Generally, eliminating the incorrect solution is not difficult for a human, but may require vehicle motion and/or information from another system. An alternative method used in some multilateration systems is to employ the Gauss–Newton NLLS method and require a redundant TOA when first establishing surveillance of a vehicle. Thereafter, only the minimum number of TOAs is required.
Satellite navigation systems such as GPS are the most prominent examples of 3D multilateration.["Existence and uniqueness of GPS solutions", J.S. Abel and J.W. Chaffee, IEEE Transactions on Aerospace and Electronic Systems, vol. 26, no. 6, pp. 748–53, Sept. 1991.]["Comments on "Existence and uniqueness of GPS solutions" by J.S. Abel and J.W. Chaffee", B.T. Fang, IEEE Transactions on Aerospace and Electronic Systems, vol. 28, no. 4, Oct. 1992.] Wide Area Multilateration (WAM), a 3D aircraft surveillance system, employs a combination of three or more TOA measurements and an aircraft altitude report.
Twodimensional solutions
For finding a user's location in a two dimensional (2D) geometry, one can adapt methods used for 3D geometry. For example, assuming that three TOAs are available, Bancroft's["Solving Passive Multilateration Equations Using Bancroft’s Algorithm", Michael Geyer and Anastasios Daskalakis, Digital Avionics Systems Conference (DASC); Seattle, WA; Oct. 31Nov. 6, 1998.] or another 3D/4TOA algorithm can be adapted. Additionally, there are specialized TDOA algorithms for twodimensions and stations at fixed locations — notable are the methods published by Fang["Simple Solutions for Hyperbolic and Related Position Fixes", Bertrand T. Fang, IEEE Transactions on Aerospace and Electronic Systems, September 1990, pp 748–753] (for a Cartesian plane) and Razin[ and Stuifbergen]["Intersection of Hyperbolae on the Earth", Nicholas H.J. Stuifbergen, University of New Brunswick Report, Fredericton, New Brunswick, Canada; December 1980.] (for a spherical earth). A comparison of 2D Cartesian algorithms for airport surface surveillance has been performed.["Localization algorithms for multilateration (MLAT) systems in airport surface surveillance", Ivan A. MantillaGaviria, Mauro Leonardi,·Gaspare Galati and Juan V. BalbastreTejedor, SpringerVerlag, London, 2014 ] However, as in the 3D situation, likely the most utilized algorithms are based on Gauss–Newton NLLS.[
]
When necessitated by the combination of vehiclestation distance (e.g., hundreds of miles or more) and required solution accuracy, the ellipsoidal shape of the earth must be considered if ground waves are used (e.g., LoranC and Omega). This has been accomplished using the Gauss–Newton NLLS[ Minimum Performance Standards (MPS) Automatic Coordinate Conversion Systems, Report of RTCM Special Committee No. 75, Radio Technical Commission for Marine Services, Washington, D.C, 1984] method in conjunction with ellipsoid algorithms by Andoyer,["Formule donnant la longueur de la géodésique, joignant 2 points de l'ellipsoide donnes par leurs coordonnées geographiques", Marie Henri Andoyer, Bulletin Geodsique, No. 34 (1932), pages 77–81] Vincenty["Direct and Inverse Solutions of Geodesics on the Ellipsoids with Applications of Nested Equations", Thaddeus Vincenty, Survey Review, XXIII, Number 176 (April 1975)] and Sodano.["General noniterative solution of the inverse and direct geodetic problems", Emanuel M. Sodano, Bulletin Géodésique, vol 75 (1965), pp 69–89]
Examples of 2D multilateration are Shortwave radio long distance communications through the Earth's atmosphere, acoustic wave propagation in the SOFAR channel of the oceans and the LoranC radionavigation system.
Solution with limited computational resources
Improving accuracy with a large number of receivers can be a problem for devices with small embedded processors because of the time required to solve several simultaneous, nonlinear equations (, & ). The TDOA problem can be turned into a system of linear equations when there are three or more receivers, which can reduce the computation time. Starting with equation , solve for R_{ m}, square both sides, collect terms and divide all terms by $v\; \backslash tau\_m$:
Removing the 2 R_{0} term will eliminate all the square root terms. That is done by subtracting the TDOA equation of receiver m = 1 from each of the others (2 ≤ m ≤ N)
Focus for a moment on equation . Square R_{ m}, group similar terms and use equation to replace some of the terms with R_{0}.
Combine equations and , and write as a set of linear equations of the unknown emitter location x, y, z'
Use equation to generate the four constants $A\_m,B\_m,C\_m,D\_m$ from measured distances and time for each receiver 2 ≤ m ≤ N. This will be a set of N − 1 inhomogeneous linear equations.
There are many robust linear algebra methods that can solve for the values of ( x, y, z), such as Gaussian elimination. Chapter 15 in Numerical Recipes[ Numerical Recipes official website] describes several methods to solve linear equations and estimate the uncertainty of the resulting values.
Accuracy
For trilateration or multilateration, calculation is done based on distances, which requires either the time of arrival or the frequency and the wave count of a received transmission. For triangulation or multiangulation, calculation is done based on angles. For electronic systems, this requires the phases of received transmission plus the wave count.
For lateration compared to angulation, the numerical problems compare, but the technical problem is more challenging with angular measurements, as angles require two measures per position when using optical or electronic means for measuring phase differences instead of counting wave cycles.
Trilateration in general is calculating with triangles of known distances/sizes, mathematically a very sound system. In a triangle, the can be derived if one knows the length of all sides, (see congruence), but the length of the sides cannot be derived based on all of the angles, not without knowing the length of at least one of the sides (a baseline) (see similarity).
In 3D, when four or more angles are in play, locations can be calculated from n + 1 = 4 measured angles plus one known baseline or from just n + 1 = 4 measured sides.
Multilateration is often more accurate for locating an object than trilateration or triangulation, as (a) it is inherently difficult or expensive to accurately measure the true range (distance) between a moving vehicle and a station, and (b) accurate angle measurements require large antennas which are costly and difficult to site. Multilateration serves for several aspects:

overdetermination of an nvariable quadratic problem with ( n + 1) + m quadratic equations

Stochastics errors prohibiting a Determinism approach to solving the equations

Cluster analysis needs to segregate members of various clusters contributing to various Model theory of solving, i.e. fixed locations, Oscillation locations and moving locations
Accuracy of multilateration is a function of several variables, including:

The antenna or sensor geometry of the receiver(s) and Transmitter for Electronics or Optics transmission.

The timing accuracy of the receiver system, i.e. thermal stability of the clocking .

The accuracy of frequency synchronisation of the transmitter with the receiver oscillators.

Phase synchronisation of the transmitted signal with the received signal, as propagation effects as e.g. diffraction or reflection changes the phase of the signal thus indication deviation from line of sight, i.e. multipath reflections.

The bandwidth of the emitted pulse(s) and thus the risetime of the pulses with pulse coded signals in transmission.

Inaccuracies in the locations of the transmitters or receivers when used as a known location
The accuracy can be calculated by using the Cramér–Rao bound and taking account of the above factors in its formulation. Additionally, a configuration of the sensors that minimizes a metric obtained from the Cramér–Rao bound can be chosen so as to optimize the actual position estimation of the target in a region of interest.[
]
Planning a multilateration system often involves a 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). In a DOP analysis, the TOA measurement errors are taken to be statistically independent and identically distributed. This reasonable assumption separates the effects of userstation geometry and TOA measurement errors on the error in the calculated user position.[ Accuracy Limitations of Hyperbolic Multilateration Systems, Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Technical Note 197311, March 22, 1973][ Improved Satellite Constellations for CONUS ATC Coverage, Harry B. Lee and Andrew E. Wade, Massachusetts Institute of Technology, Lincoln Laboratory, Project Report ATC23, May 1, 1974]
Station synchronization
Multilateration requires that the spatially separated stations – either transmitters (navigation) or receivers (surveillance) – have synchronized 'clocks'. Except for GNSSs, station 'clocks' could, and often have been, devices that measured elapsed time from a common arbitrary origin (zero time) and returned to zero (reset) frequently. Multiple methods have been used for station synchronization. Typically, method selection is based on the distance between stations. In approximate order of increasing distance, methods have included:

Clockless stations wired to single clock (navigation and surveillance) – Clockless stations are hard wired to a central location having the single system clock. Wire lengths are generally equal, but that may not be necessary in all applications. This method has been used for locating artillery fire (stations are microphones).

Test target (surveillance) – A test target is installed at a fixed, known location that's visible to all receivers. The test target transmits as an ordinary user would, and its position is calculated from the TOAs. Its known position is used to adjust the receiver clocks. ASDEX uses this method.

Clockless stations radiolinked to single clock (navigation and surveillance) – Clockless stations are radiolinked (e.g., microwave) to a central location having the single system clock. Link delays are equalized. This method is used by some WAM systems.

Fixed transmitter delays (navigation) – One transmitter is designated the master; the others are secondaries. The master transmits first. Each secondary transmits a fixed (short) time after receiving the master's transmission. LoranC originally used this method.

Satellite time transfer (navigation and surveillance) – There are multiple methods for transferring time from a reference site to a remote station via satellite. The simplest is to synchronize the stations to GPS time.
[ One Way GPS Time Transfer, National Institute of Standards and Technology] Some WAM systems use this method.

Atomic clocks (navigation and surveillance) – Each station has one or more synchronized atomic clocks. GNSSs use this method and Omega did. LoranC switched to it. Even atomic clocks drift, and a monitoring and correction system may also be required.
Service area or volume
While the performance of all navigation and surveillance systems depends upon the user's location relative to the stations, multilateration systems are more sensitive to the user's location than most. For example, consider a hypothetical twostation surveillance system that monitors the location of a railroad locomotive along a straight stretch of track (a one dimension situation). The locomotive carries a transmitter and the track is straight in both directions beyond the stretch that's monitored. Such a system would work well when a locomotive is between the two surveillance system's stations. But it would not work at all when a locomotive is not between the stations. Then the multilateration system would always indicate that the locomotive was at the nearer station.
In two or three dimensions, a multilateration system's accuracy is not twovalued, as it is in a onedimensional situation. A dilution of precision (DOP) analysis is often employed to quantify the effect of userstation geometry on accuracy.[ "Dilution of Precision", Richard Langeley, GPS World, May 1999, pp 52–59.] In general, multilateration accuracy is quite good in the perimeter enclosing the stations but degrades rapidly for locations outside it. Fundamentally, inside the perimeter, the measurements amplify the effect of vehicle motion, while outside of the perimeter they attenuate it. Figure 4 illustrates the service area of twodimensional system having three stations forming an equilateral triangle (BLU denotes baseline unit).
Outside the stations' perimeter, a multilateration system should typically only be used near the center of the closest baseline connecting two stations (two dimensional situation) or near the center of the closest plane containing three stations (three dimensional situation). Moreover, it should only be used in locations that are a fraction of an average baseline length (e.g., less than 20%) from the closest baseline or plane. This is the situation for an earthbased GPS user. While outside the perimeter of the visible satellites, a user is typically close to the center of the nearest plane containing three satellites and between 5% to 10% of a baseline length from that plane.
Having more than the minimum number of stations mitigates the worst situations (e.g., no solution), but not dramatically. The major value of an additional station is fault tolerance. To provide the required userstation geometry, LoranC stations were often placed at perimeter locations that some would consider 'remote' – e.g., to provide navigation service to the North Atlantic, there were stations at: Faroe Islands (Denmark), Jan Mayen Island (Norway) and Angissq (Greenland).
Example applications

GPS (U.S.), GLONASS (Russia), Galileo (E.U.) – Global navigation satellite systems. Two separate complicating factors relative to TDOA systems are: (1) the transmitter stations (satellites) are moving; and (2) Receivers compute TOT, requiring a more complex algorithm but providing accurate time to users.

Sound ranging – Using sound to locate the source of artillery fire.

Electronic targets – Using the Mach wave of a bullet passing a sensor array to determine the point of arrival of the bullet on a firing range target.

Decca Navigator System – A system used from the end of World War II to the year 2000, employing the phasedifference of multiple transmitters to locate on the intersection of hyperboloids

Omega Navigation System – A worldwide system similar to Decca, shut down in 1997

Gee (navigation) – British aircraft location system used during World War II

LoranC – Navigation system using TDOA of signals from multiple synchronised transmitters, shut down in the U.S. and Europe

Passive ESM multilateration systems, including Kopáč, Ramona, Tamara, VERA and possibly Kolchuga – location of a transmitter using multiple receivers

Mobile phone tracking – using multiple base stations to estimate phone location, either by the phone itself (termed downlink multilateration), or by the phone network (termed uplink multilateration)

Reduced Vertical Separation Minima (RVSM) monitoring to determine the accuracy of Mode C/S aircraft transponder altitude information. Application of multilateration to RVSM was first demonstrated by Roke Manor Research Limited in 1989.

Wide area multilateration (WAM) – Surveillance system for airborne aircraft that measures the TOAs of emissions from the aircraft transponder (on 1090 MHz); in operational service in several countries

Airport Surface Detection Equipment, Model X (ASDEX) – Surveillance system for aircraft and other vehicles on the surface of an airport; includes a multilateration subsystem that measures the TOAs of emissions from the aircraft transponder (on 1090 MHz); ASDEX is U.S. FAA terminology, equivalent systems are in operational service in several countries.

Flight Test "Truth" – Locata Corporation offers a groundbased local positioning system that augments GPS and is used by NASA and the U.S. military

Seismic Event Location – Events (e.g., earthquakes) are monitored by measuring TOAs at different locations and employing multilateration algorithms
["A ClosedForm Solution for Earthquake Location in a Homogeneous HalfSpace Based on the Bancroft GPS Location Algorithm", Demian Gomez, Charles Langston & Bob Smalley, Bulletin of the Seismological Society of America, January 2015.]

Towed array sonar / SURTASS / SOFAR (SOund Fixing And Ranging) – Employed operationally by the U.S. Navy (and likely similar systems by other navies). The purpose is determining the distance and direction of a sound source from listening.

MILS and SMILS Missile Impact Location Systems – Acoustic systems deployed to determine the 'splash down' points in the South Atlantic of Polaris, Poseidon and Trident missiles that were testfired from Cape Canaveral, FL.

Atlantic Undersea Test and Evaluation Center (AUTEC) – U.S. Navy facility that measures trajectories of undersea boats and weapons using acoustics
Simplification
For applications where no need for absolute coordinates determination is assessed, the implementing of a more simple solution is advantageous. Compared to multilateration as the concept of crisp locating, the other option is fuzzy locating, where just one distance delivers the relation between detector and detected object. This most simple approach is unilateration. However, such unilateration approach never delivers the angular position with reference to the detector. Many solutions are available today.
Some of these vendors offer a position estimate based on combining several laterations. This approach is often not stable, when the wireless ambience is affected by metal or water masses. Other vendors offer room discrimination with a roomwise excitation, one vendor offers a position discrimination with a contiguity excitation.
See also

Ranging

Rangefinder

Hyperbolic navigation – General term for multiple navigation systems that measure at least one TOA more than the physical dimensions involved

FDOA – Frequency difference of arrival. Analogous to TDOA using differential Doppler measurements.

Triangulation – Location by angular measurement on lines of bearing that intersect

Trilateration – Location by multiple distances – e.g. timeofflight measurements from multiple transmitters

Mobile phone tracking – used in GSM networks

Multidimensional scaling

Radiolocation

Radio navigation

Realtime locating – International standard for asset and staff tracking using wireless hardware and realtime software

Real time location system – General techniques for asset and staff tracking using wireless hardware and realtime software

Greatcircle navigation – Provides the basic mathematics for addressing spherical ranges

Nonlinear least squares  Form of leastsquares analysis when nonlinear equations are involved; used for multilateration when (a) there are more rangedifference measurements than unknown variables, and/or (b) the measurement equations are too complex to be inverted (e.g., those for an ellipsoidal earth), and/or (c) tabular data must be utilized (e.g., conductivity of the earth over which radio wave propagated).

Coordinated Universal Time (UTC)  Time standard provided by GPS receivers (with published offset)

Clock synchronization  Methods for synchronizing clocks at remote stations

Atomic clock – Sometimes used to synchronize multiple widely separated stations

Dilution of precision – Analytic technique often used with multilateration systems
Notes