Multilateration (
MLAT) is a surveillance technique based on the measurement of the
difference in distance to two stations at known locations by broadcast signals at known times. Unlike measurements of absolute distance or angle, measuring the difference in distance between two stations results in an infinite number of locations that satisfy the measurement. When these possible locations are plotted, they form a
hyperbola. To locate the exact location along that curve, multilateration relies on multiple measurements: a second measurement taken to a different pair of stations will produce a second curve, which intersects with the first. When the two curves are compared, a small number of possible locations are revealed, producing a "fix".
Multilateration is a common technique in radio navigation systems, where it is known as hyperbolic navigation. These systems are relatively easy to construct as there is no need for a common clock, and the difference in the signal timing can 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 introduced over the next few decades. The introduction of the microprocessor greatly simplified operation, greatly increasing popularity during the 1980s. The most popular hyperbolic navigation system was LORANC, which was used around the world until the system was shut down in 2010. Other systems continue to be used, but the widespread use of satellite navigation systems like GPS have made these systems largely redundant.
Multilateration should not be confused with any of;

trilateration, which uses distances or absolute measurements of timeofflight from three or more sites

triangulation, which uses the measurement of angles

direction finding which does not compute a distance.
All of these systems are also commonly used with radio navigation systems.
Principle
Multilateration is commonly used in civil and military applications to accurately locate an aircraft, vehicle or stationary emitter by measuring the "time difference of arrival" (TDOA) of a signal from the emitter at three or more synchronized receiver sites (surveillance application) or the signals from three or more synchronized emitters at one receiver location (navigation application).
Surveillance application: locating a transmitter from multiple receiver sites
If a pulse is emitted from a platform, it will generally arrive at slightly different times at two spatially separated receiver sites, the TDOA being due to the different distances of each receiver from the platform. In fact, for given locations of the two receivers, a whole set of emitter locations would give the same measurement of 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 a 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 any two 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 (measured by 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.
Consider now a third receiver at a third location. This would provide one extra independent TDOA measurement (there is a third TDOA, but this is dependent on the first two TDOA and does not provide extra information) and the emitter is located on the curve determined by the two intersecting hyperboloids. A fourth receiver is needed for another independent TDOA, this will give an extra hyperboloid, the intersection of the curve with this hyperboloid gives one or two solutions, the emitter is then located at the one or at the one of the two solutions.
With four receivers there are 3 independent TDOA, three independent parameters are needed for a point in three dimensional space. (And for most constellations, three independent TDOA will still give two points in 3D space).
With additional receivers enhanced accuracy can be obtained. (Specifically for GPS 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) a least squares 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 emitting platform may, or may not, cooperate in the multilateration surveillance processes.
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 three or more synchronized transmitters at known locations. 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 TDOAs of the pulses, which are converted to range differences.
However, operational systems are more complex. These methods have been implemented:

(a) Pulses are broadcast by different emitters on the same frequency, with known delays between transmission times.

(b) Continuous signals are broadcast on different frequencies, and their measured phase differences are converted to range differences.

(c) Continuous signals are broadcast on the same carrier frequency, but each emitter modulates the carrier with a unique, known code. Correlation processing is used to obtain TDOAs.
The multilateration technique is used by several navigation systems. A historic example is the British DECCA system, developed during World War II. Decca used the phasedifference of three transmitters (method (b)). LORANC, introduced in the late 1950s, uses method (a). A current example is the Global Positioning System, or GPS. All GPS satellites broadcast on the same carrier frequency, which is modulated by pseudorandom codes (method (c)).
TDOA geometry
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
 $\backslash vec\; P\_0,\; \backslash ,\; \backslash vec\; P\_1,\; \backslash ,\; \backslash cdots,\; \backslash ,\; \backslash vec\; P\_m,\; \backslash ,\; \backslash cdots,\; \backslash ,\; \backslash vec\; P\_N.$
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 rather than a straight line. In this situation, equation is not valid. LORANC and Omega are primary examples of systems that utilize spherical (vice slant) 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
 $$
\cos\sigma_{vm}=\sin\phi_v\sin\phi_m+\cos\phi_v\cos\phi_m\cos\lambda_{vm}.
Here: latitudes are denoted by φ; longitudes are denoted by λ; and λ
_{vm} = λ
_{v} − λ
_{m}. Alternative, better numerically behaved equivalent expressions, can be found in Greatcircle navigation.
The distance R_{m} from the vehicle to station m is along a great circle will then be
 $$
R_{m}=R_{E} \sigma_{vm}
Here
R_{E} is the assumed radius of the earth and σ
_{vm} is expressed in radians.
Measuring the time difference in a TDOA system
The distance
$R\_m$ in equation is the wave speed (
$v$) times transit time (
$T\_m$). A TDOA multilateration system measures the time difference (
$\backslash tau\_m$) of a wavefront touching each receiver. The TDOA equation for receivers m and 0 is
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 (f) 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
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 emitter location in real time. Civilian air traffic control multilateration systems use the Mode C SSR transponder return to find the 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 first published the algebraic solution to the problem of locating a receiver using TDOA measurements involving 4 transmitters.[ An Algebraic Solution of the GPS Equations, Stephen Bancroft, IEEE Transactions on Aerospace and Electronic Systems, Volume: AES21, Issue: 1 (Jan. 1989), Pages 5659.] Bancroft's algorithm 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 transmission. Other, comparable solutions and extensions were subsequently developed.[ 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, Pages 507520.][ Aircraft Navigation and Surveillance Analysis for a Spherical Earth, Michael Geyer, John A. Volpe National Transportation Systems Center (U.S.), Oct. 2014, Section 7.4.] The latter reference provides the solution for locating an aircraft with known altitude using TDOA measurements at 3 receivers.
When there are more measurements equations than unknown quantities (overdetermined situation), the iterative GaussNewton 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. The GaussNewton method may also be used with the minimum number of measurements — e.g., when an ellipsoidal model for earth must be employed. Since it is iterative, the GaussNewton method requires an initial solution estimate (which can be generated using a spherical earth model).
Multilateration systems employing sphericalrange measurements utilize a combination of solution algorithms based on spherical trigonometry[Explicit (Noniterative) Loran Solution, Sheldon Razin, Navigation: Journal of the Institute of Navigation, Vol. 14, No. 3, Fall 1967, pp. 265–269.] and the GaussNewton NLLS method.
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 3 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
$v\; \backslash tau\_m$:
 . \\
\end{align}
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)
 \\
& 0 = v \tau_{1}  2 R_{0}  \frac {(R_{0}^2  R_{1}^2)} {v \tau_{1}} \\
\hline \\
& 0 = v \tau_{m}  v \tau_{1} + \frac {(R_{0}^2  R_{m}^2)} {v \tau_{m}}
 \frac {(R_{0}^2  R_{1}^2)} {v \tau_{1}}.
\end{align}
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 N1 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.
Twodimensional solution
For finding the emitter location in a two dimensional geometry, one can generally adapt the methods used for the 3D geometry. Additionally, there are specialized algorithms for twodimensions—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 748753] (for a Cartesian plane) and Razin
[ (for spherical earth).
]
When necessitated by the combination of vehiclestation distance (e.g., hundreds of miles or more) and required solution accuracy (e.g., less than 0.3% of the vehiclestations distance), the ellipsoidal shape of the earth must be considered. This has been accomplished using the GaussNewton LLS 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 7781] 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 6989]
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 Loran C navigation system.
Accuracy
For trilateration or multilateration, calculation is done based on distances, which requires the frequency and the wave count of a received transmission. For triangulation or multiangulation, calculation is done based on angles, which 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, in general, far more accurate for locating an object than sparse approaches such as trilateration, where with planar problems just three distances are known and computed. 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 oscillators.

The accuracy of frequency synchronisation of the transmitter oscillators 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.[
]
Example applications

Sound ranging – Using sound to locate 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 – British aircraft location technique from World War II, using accurate reference transmitters

LORANC – Navigation system using TDOA of signals from multiple synchronised transmitters

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 (by either the phone itself, or the phone network)

Reduced Vertical Separation Minima (RVSM) monitoring using Secondary Surveillance Radar – Mode C/S transponder replies to calculate the position of an aircraft. Application to RVSM was first demonstrated by Roke Manor Research Limited in 1989.

Wide area multilateration (WAM)  Surveillance system for airborne aircraft that measures the TDOA 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 TDOA of emissions from the aircraft transponder (on 1090 MHz); ASDEX is U.S. FAA terminology, equivalent systems are in operational service in several countries.
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 based on the Time Difference Of Arrival (TDOA) principle

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 distance (e.g. timeofflight) measurement on coincident signals 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).
Notes