Multilateration


Multilateration is a navigation and surveillance technique based on measurement of the times of arrival of energy waves having a known propagation speed. Prior to computing a solution, the time of transmission of the waves is unknown to the receiver.
A navigation system provides position information to an entity on the subject 'vehicle' ; a surveillance system provides 'vehicle' position information to an entity not on the 'vehicle'. By the reciprocity principle, any method that can be used for navigation can also be used for surveillance, and vice versa. For surveillance, a subject of interest transmits to multiple receiving stations having synchronized 'clocks'. For navigation, multiple synchronized stations transmit to a user receiver which can determine the time of transmission. To find the user's coordinates in dimensions, at least TOAs must be measured. Almost always, or .
One can view a multilateration system as measuring at least TOAs using a clock with an unknown offset from actual time. Using the measured TOAs, the user implements an algorithm that either: determines the time of transmission for the same clock and user coordinates; or ignores the TOT and forms at least time difference of arrivals, which are used to find the user coordinates. Systems that form TDOAs are also called hyperbolic systems, for reasons discussed below. A TDOA, when multiplied by the propagation speed, is the difference in the true ranges between the user and the two stations involved.
Systems have been developed for both algorithm types. In this article, the latter, TDOA or hyperbolic systems, is addressed first, as it was the first implemented. Due to the technology available at the time, those systems often determined a user/vehicle location in two dimensions. The former, TOT systems, is addressed second. It was implemented, roughly, post-1975, and usually involve satellites. Due to technology advances, TOT systems generally determine a user/vehicle location in three dimensions. However, conceptually, TDOA or TOT algorithms are not linked to the number of dimensions involved.
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 hyperbolic curve. 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, will produce a second curve, which intersects with the first. When the two curves are compared, a small number of possible user locations 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 earth-fixed radio navigation systems, where it was known as hyperbolic navigation. These systems are relatively undemanding of the user receiver, as its 'clock' can have low-performance/cost and is usually unsynchronized with station time. 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 Loran-C, which was used around the world until the system was largely shut down. The widespread use of satellite navigation systems like the Global Positioning System have made TDOA navigation systems largely redundant, and most have been decommissioned. GPS is also a hyperbolic navigation system, but also determines the TOT according to the user's clock. As a bonus, GPS also provides accurate time to users.
Pseudo range multilateration should not be confused with any of:
All of these systems are commonly used with radio navigation and surveillance systems, and in other applications. However, this terminology is not always used.

Advantages and disadvantages

AdvantagesDisadvantages
Low user equipage cost – A cooperative surveillance user only needs a transmitter. A navigation user only needs a receiver having a basic 'clock'.Station locations – Stations must nearly surround the service area
Accuracy – Avoids the ‘turn-around’ error inherent in many two-way ranging systemsStation count – Requires one more station than a system based on true ranges. Requires two more stations than a system that measures range and azimuth
Small stations – Avoids use of the large antennas needed for measuring anglesStation synchronization are required
For navigation, the number of users is unlimited Stations may require power and communications where not available
Uncooperative and undetected surveillance are possibleFor surveillance, users may mutually interfere
Viable over great distances For navigation, stations must transmit effectively simultaneously but not mutually interfere
Implementations have utilized several wave propagation phenomena: electromagnetic, air acoustics, water acoustics, and seismic--

Multilateration surveillance was used during World War I to locate the source of artillery fire using sound waves. Longer distance radio-based navigation systems became viable during World War II, with the advancement of radio technologies. The development of atomic clocks for synchronizing widely separated stations was instrumental in the development of the GPS and other GNSSs. Owing to its high accuracy at low cost of user equipage, today multilateration is the concept most often selected for new navigation and surveillance systems.

Principle

Prior to deployment of GPS and other global navigation satellite systems, pseudo-range multilateration systems were often defined as TDOA systems – i.e., systems that formed TDOAs as the first step in processing a set of measured TOAs. As result of deployment of GNSSs, two issues arose: What system type are GNSSs ? What are the defining characteristic of a pseudo-range multilateration system?
Multilateration is commonly used in civil and military applications to either locate a vehicle by measuring the TOAs of a signal from the vehicle at multiple stations having known coordinates and synchronized 'clocks' or enable the vehicle to locate itself relative to multiple transmitters at known locations and having synchronized clocks based on measurements of signal TOAs. 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 are often provided by government agencies. However, privately funded entities have also been station/system providers –- e.g., wireless phone providers. Multilateration is also used by the scientific and military communities for non-cooperative surveillance.

TDOA algorithm principle / surveillance

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. Given two receiver locations and a known TDOA, the locus of possible emitter locations is one half of a two-sheeted hyperboloid.
In simple terms, with two receivers at known locations, an emitter can be located onto one hyperboloid. 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. 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..
With additional receivers enhanced accuracy can be obtained..
For an over-determined constellation a least squares method can be used for 'reducing' the errors. 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, multilateration surveillance is used with non-cooperating 'users' for military and scientific purposes as well as with cooperating users.

TDOA algorithm principle / navigation

Multilateration can also be used by a single receiver to locate itself, by measuring signals emitted from synchronized transmitters at known locations. At least three emitters are needed for two-dimensional navigation ; at least four emitters are needed for three-dimensional navigation. For expository purposes, the emitters may be regarded as each broadcasting pulses at exactly the same time on separate frequencies. In this situation, the receiver measures the TOAs of the pulses. In TDOA systems, the TOAs are immediately differenced and multiplied by the speed of propagation to create range differences.
In operational systems, several methods have been implemented to avoid self-interference. A historic example is the British Decca system, developed during World War II. Decca used the phase-difference of three transmitters. Later, Omega used the same principle. Loran-C, introduced in the late 1950s, used time offset transmissions.

TOT algorithm principle

The TOT concept is illustrated in Figure 2 for the surveillance function and a planar scenario. Aircraft A, at coordinates, broadcasts a pulse sequence at time. The broadcast is received at stations, and at times, and, respectively. Based on the three measured TOAs, the processing algorithm computes an estimate of the TOT, from which the range between the aircraft and the stations can be calculated. The aircraft coordinates are then found.
When the algorithm computes the correct TOT, the three computed ranges have a common point of intersection which is the aircraft location. If the algorithm's computed TOT is after the actual TOT, the computed ranges do not have a common point of intersection. Similarly, if the algorithm's computed TOT is after the actual TOT, the three computed ranges do not have a common point of intersection. It is clear that an iterative TOT algorithm can be found. In fact, GPS was developed using iterative TOT algorithms. Closed-form TOT algorithms were developed later.
TOT algorithms became important with the development of GPS. GLONASS and Galileo employ similar concepts. The primary complicating factor for all GNSSs is that the stations move continuously relative to the earth. Thus, in order to compute its own position, a user’s navigation receiver must know the satellites’ locations at the time the information is broadcast in the receiver’s time scale. To accomplish this: satellite trajectories and TOTs in the satellites’ time scales are included in broadcast messages; and user receivers find the difference between their TOT and the satellite broadcast TOT. GPS satellite clocks are synchronized to UTC as well as with each other. This enables GPS receivers to provide UTC time in addition to their position.

Measurement geometry and related factors

Rectangular/Cartesian coordinates

Consider an emitter at an unknown location vector
which we wish to locate. The source is within range of receivers at known locations
The subscript refers to any one of the receivers:
The distance 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, which makes its distance to the emitter

Spherical coordinates

Low-frequency radio waves follow the curvature of the earth rather than straight lines. In this situation, equation is not valid. Loran-C and Omega are examples of systems that utilize spherical ranges. When a spherical model for the earth is satisfactory, the simplest expression for the central angle between vehicle and station is
Here, latitudes are denoted by and longitudes are denoted by. Alternative, better numerically behaved equivalent expressions, can be found in great-circle navigation.
The distance from the vehicle to station is along a great circle will then be
Here, is the assumed radius of the earth and is expressed in radians.

Time of transmission (user clock offset or bias)

Prior to GNSSs, there was little value to determining the TOT or its equivalent in the navigation context, the offset between the receiver and transmitter clocks. Moreover, when those systems were developed, computing resources were quite limited. Consequently, in those systems, receivers treated the TOT as a nuisance parameter and eliminated it by forming TDOA differences. This simplified solution algorithms. Even if the TOT was needed, TOT could be found from one TOA, the location of the associated station, and the computed vehicle location.
With the advent of GPS and subsequently other satellite navigation systems: TOT as known to the user receiver provides necessary and useful information; and computing power had increased significantly. GPS satellite clocks are synchronized not only with each other but also with Coordinated Universal Time and their locations are known relative to UTC. Thus, algorithms used for satellite navigation solve for the receiver position and its clock offset simultaneously. The receiver clock is then adjusted so its TOT matches the satellite TOT. By finding the clock offset, 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 and equal along all signal paths. This is equivalent to assuming that the propagation medium is homogeneous. However, that is not always sufficiently accurate; some paths may involve additional propagation delays due to inhomogeneities in the medium. Accordingly, to improve solution accuracy, some systems adjust measured TOAs to account for such propagation delays. Thus, space-based GNSS augmentation systems – e.g., Wide Area Augmentation System and European Geostationary Navigation Overlay Service – provide TOA adjustments in real time to account for the ionosphere. Similarly, U.S. Government agencies used to provide adjustments to Loran-C measurements to account for soil conductivity variations.

Calculating the time difference in a TDOA system

The basic measurements are the TOAs of multiple signals at the vehicle or at the stations. The distance in equation is the wave speed times transit time, which is unknown, as the time of transmission is not known. A TDOA multilateration system calculates the time differences of a wavefront touching each receiver. The TDOA equation for receivers m and 0 is
The quantity is often termed a pseudo range. It differs from the true range between the vehicle and station by an offset or bias which is the same for every station. Differencing two pseudo range yields the difference of the two true ranges.
Figure 4a is a simulation of a pulse waveform recorded by receivers and. The spacing between, and is such that the pulse takes 5 time units longer to reach than. The units of time in Figure 4 are arbitrary. The following table gives approximate time scale units for recording different types of waves.
Type of waveMaterialTime units
AcousticAir1 millisecond
AcousticWater1/2 millisecond
AcousticRock1/10 millisecond
ElectromagneticVacuum, air1 nanosecond

The red curve in Figure 4a is the cross-correlation function. The cross correlation function slides one curve in time across the other and returns a peak value when the curve shapes match. The peak at time = 5 is a measure of the time shift between the recorded waveforms, which is also the value needed for Equation.
Figure 4b is the same type of simulation for a wide-band waveform from the emitter. The time shift is 5 time units because the geometry and wave speed is the same as the Figure 4a example. Again, the peak in the cross correlation occurs at.
Figure 4c is an example of a continuous, narrow-band waveform from the emitter. The cross correlation function shows an important factor when choosing the receiver geometry. There is a peak at Time = 5 plus every increment of the waveform period. To get one solution for the measured time difference, the largest space between any two receivers must be closer than one wavelength of the emitter signal. Some systems, such as the LORAN C and Decca mentioned at earlier, 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, narrow-band waveforms because of the relation between phase, frequency and time
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

There are multiple categories of multilateration algorithms, and some categories have multiple members. Perhaps the first factor that governs algorithm selection: Is an initial estimate of the user's position required or is it not? Direct algorithms estimate the user's position using only the measured TOAs and do not require an initial position estimate. A related factor governing algorithm selection: Is the algorithm readily automated, or conversely, is human interaction needed/expected? Most direct algorithms can have ambiguous solutions, which is detrimental to their automation. A third factor is: Does the algorithm function well with both the minimum number and with redundant measurements?
Direct algorithms can be further categorized based on:
This taxonomy has five categories: four for direct algorithms and one for iterative algorithms. However, it appears that algorithms in only three of these categories have been implemented. When redundant measurements are available for either path, iterative algorithms have been strongly favored over closed-form algorithms. Often, real-time systems employ iterative algorithms while off-line studies utilize closed-form algorithms.
All multilateration algorithms assume that the station locations are known at the time each wave is transmitted. For TDOA systems, the stations are fixed to the earth and their locations are surveyed. For TOT systems, the satellites follow well-defined orbits and broadcast orbital information. Equation is the hyperboloid described in the previous section, where 4 receivers lead to 3 non-linear equations in 3 unknown Cartesian coordinates. The system must then solve for the unknown user location in real time.
Steven Bancroft was apparently the first to publish a closed-form solution to the problem of locating a user in three dimensions and the common TOT using only four TOA measurements. Bancroft's algorithm, as do many, reduces the problem to the solution of a quadratic algebraic equation; its solution yields the three Cartesian coordinates of the receiver as well as the common time of signal transmissions. Other, comparable solutions were subsequently developed. Notably, all closed-form solutions were found a decade or more after the GPS program was initiated using iterative methods.
The solution for the position of an aircraft having a known altitude using 3 TOA measurements requires solving a quartic polynomial.
Multilateration systems and studies employing spherical-range measurements utilized a variety of solution algorithms based on either iterative methods or spherical trigonometry.

Three-dimensional Cartesian solutions

For Cartesian coordinates, when four TOAs are available and the TOT is needed, Bancroft's or another closed-form 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 to three dimensions is a second option. A third option, and likely the most utilized in practice, is the iterative Gauss–Newton Nonlinear Least-Squares method.
Most closed-form 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. 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 3-D multilateration. Wide Area Multilateration, a 3-D aircraft surveillance system, employs a combination of three or more TOA measurements and an aircraft altitude report.

Two-dimensional Cartesian solutions

For finding a user's location in a two dimensional Cartesian geometry, one can adapt one of the many methods developed for 3-D geometry, most motivated by GPS—for example, Bancroft's or Krause's. Additionally, there are specialized TDOA algorithms for two-dimensions and stations at fixed locations — notable is Fang's method.
A comparison of 2-D Cartesian algorithms for airport surface surveillance has been performed. However, as in the 3-D situation, it's likely the most utilized algorithms are based on Gauss–Newton NLLS.
Examples of 2-D Cartesian multilateration systems are those used at major airports in many nations to surveil aircraft on the surface or at very low altitudes.

Two-dimensional spherical solutions

Razin developed a closed-form solution for a spherical earth. Williams and Last extended Razin's solution to an osculating sphere earth model.
When necessitated by the combination of vehicle-station distance and required solution accuracy, the ellipsoidal shape of the earth must be considered. This has been accomplished using the Gauss–Newton NLLS method in conjunction with ellipsoid algorithms by Andoyer, Vincenty and Sodano.
Examples of 2-D 'spherical' multilateration navigation systems that accounted for the ellipsoidal shape of the earth are the Loran-C and Omega radionavigation systems, both of which were operated by groups of nations. Their Russian counterparts, CHAYKA and Alpha, are understood to operate similarly.

Cartesian solution with limited computational resources

Consider a three-dimensional Cartesian scenario. 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, non-linear 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, square both sides, collect terms and divide all terms by :
Removing the term will eliminate all the square root terms. That is done by subtracting the TDOA equation of receiver from each of the others
Focus for a moment on equation. Square, group similar terms and use equation to replace some of the terms with.
Combine equations and, and write as a set of linear equations of the unknown emitter location
Use equation to generate the four constants from measured distances and time for each receiver. This will be a set of inhomogeneous linear equations''.
There are many robust linear algebra methods that can solve for, such as Gaussian elimination. Chapter 15 in Numerical Recipes describes several methods to solve linear equations and estimate the uncertainty of the resulting values.

Iterative algorithms

The defining characteristic and major disadvantage of iterative methods is that a 'reasonably accurate' initial estimate of the user's location is required. If the initial estimate is not sufficiently close to the solution, the method may not converge or may converge to an ambiguous or extraneous solution. However, iterative methods have several advantages:
Many real-time multilateration systems provide a rapid sequence of user's position solutions -- e.g., GPS receivers typically provide solutions at 1 sec intervals. Almost always, such systems implement: a transient ‘acquisition’ or ‘cold start’ mode, whereby the user’s location is found from the current measurements only; and a steady-state ‘track’ or ‘warm start’ mode, whereby the user’s previously computed location is updated based current measurements. Often the two modes employ different algorithms and/or have different measurement requirements, with being more demanding. The iterative Gauss-Newton algorithm is often used for and may be used for both modes.
When there are more TOA measurements than the unknown quantities – e.g., 5 or more GPS satellite TOAs – the iterative Gauss–Newton algorithm for solving non-linear least squares problems is often preferred. Except for pathological station locations, an over-determined situation eliminates possible ambiguous and/or extraneous solutions that can occur when only the minimum number of TOA measurements are available. Another important advantage of the Gauss–Newton method over some closed-form 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.
While the Gauss-Newton NLLS iterative algorithm is widely used in operational systems, the Nelder-Mead iterative method is also available. Example code for the latter, for both TOA and TDOA systems, are available.

Accuracy

Multilateration is often more accurate for locating an object than true range multilateration or multiangulation, as it is inherently difficult and/or expensive to accurately measure the true range between a moving vehicle and a station, particularly over large distances, and accurate angle measurements require large antennas which are costly and difficult to site.
Accuracy of a multilateration system is a function of several factors, including:
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.
Concerning the first issue, planning a multilateration system often involves a dilution of precision analysis to inform decisions on the number and location of the stations and the system's service area or volume. In a DOP analysis, the TOA measurement errors are assumed to be statistically independent and identically distributed. This reasonable assumption separates the effects of user-station geometry and TOA measurement errors on the error in the calculated user position.

Station synchronization

Multilateration requires that spatially separated stations – either transmitters or receivers – have synchronized 'clocks'. There are two distinct synchronization requirements: maintain synchronization accuracy continuously over the life expectancy of the system equipment involved ; and accurately measure the time interval between TOAs for each transmission. Requirement is transparent to the user, but is an important system design consideration. To maintain synchronization, station clocks must be synchronized or reset regularly. Station clocks must be accurate enough to satisfy requirement between resets. Often the system accuracy is monitored continuously by "users" at known locations.
Multiple methods have been used for station synchronization. Typically, the method is selected based on the distance between stations. In approximate order of increasing distance, methods have included:
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-station geometry than are most systems. To illustrate, consider a hypothetical two-station surveillance system that monitors the location of a railroad locomotive along a straight stretch of track -- a one dimensional situation. The locomotive carries a transmitter and the track is straight in both directions beyond the stretch that's monitored. For convenience, let the system origin be mid-way between the stations; then occurs at the origin.
Such a system would work well when a locomotive is between the two stations. When in motion, a locomotive moves directly toward one station and directly away from the other. If a locomotive is distance away from the origin, in the absence of measurement errors, the TDOA would be . Thus the amount of displacement is doubled in the TDOA. If true ranges were measured instead of pseudo ranges, the measurement difference would be identical.
However, this one-dimensional pseudo range system would not work at all when a locomotive is not between the two stations. In either extension region, if a locomotive moves between two transmissions, necessarily away from both stations, the TDOA would not change. In the absence of errors, the changes in the two TOAs would perfectly cancel in forming the TDOA. In the extension regions, the system would always indicate that a locomotive was at the nearer station, regardless of its actual position. In contrast, a system that measures true ranges would function in the extension regions exactly as it does when the locomotive is between the stations. This one-dimensional system provides an extreme example of a multilateration system's service area.
In a multi-dimensional situation, the measurement extremes of a one-dimensional scenario rarely occur. When it's within the perimeter enclosing the stations, a vehicle usually moves partially away from some stations and partially toward other stations. It is highly unlikely to move directly toward any one station and simultaneously directly away from another; moreover, it cannot move directly toward or away from all stations at the same time. Simply put, inside the stations' perimeter, consecutive TDOAs will typically amplify but not double vehicle movement which occurred during that interval -- i.e.,. Conversely, outside the perimeter, consecutive TDOAs will typically attenuate but not cancel associated vehicle movement -- i.e.,. The amount of amplification or attenuation will depend upon the vehicle's location. The system's performance, averaged over all directions, varies continuously as a function of user location.
When analyzing a 2-D or 3-D multilateration system, dilution of precision is usually employed to quantify the effect of user-station geometry on position-determination accuracy. The basic DOP metric is
The symbol conveys the notion that there are multiple 'flavors' of DOP -- the choice depends upon the number of spatial dimensions involved and whether the error for the TOT solution is included in the metric. The same distance units must be used in the numerator and denominator of this fraction -- e.g., meters. ?DOP is a dimensionless factor that is usually greater than one, but is independent of the pseudo range measurement error. HDOP is usually employed when interest is focused on a vehicle's position on a plane.
Pseudo range errors are assumed to add to the measured TOAs, have zero mean and have the same standard deviation, regardless of vehicle's location or the station involved. Labeling the orthogonal axes in the plane as and, the horizontal position error is characterized statistically as:
Mathematically, each DOP 'flavor' is a different derivative of a solution quantity's standard deviation with respect to the pseudo range error standard deviation. That is, ?DOP is the rate of change of the standard deviation of a solution quantity from its correct value due to measurement errors -- assuming certain conditions are satisfied, which is typically the case. Specifically, HDOP is the derivative of the user's horizontal position standard deviation to the pseudo range error standard deviation.
For three stations, multilateration accuracy is quite good within almost the entire triangle enclosing the stations -- say, 1 < HDOP < 1.5 and is close to the HDOP for true ranging measurements using the same stations. However, a multilateration system's HDOP degrades rapidly for locations outside the station perimeter. Figure 5 illustrates the approximate service area of two-dimensional multilateration system having three stations forming an equilateral triangle. The stations are M-U-V. BLU denotes baseline unit. The inner circle is more 'conservative' and corresponds to a 'cold start'. The outer circle is more typical, and corresponds to starting from a known location. The axes are normalized by the separation between stations.
Figure 6 shows the HDOP contours for the same multilateration system. The minimum HDOP, 1.155, occurs at the center of the triangle formed by the stations. Beginning with HDOP = 1.25, the contours shown follow a factor-of-2 progression. Their roughly equal spacing is consistent with the rapid growth of the horizontal position error with distance from the stations. The system's HDOP behavior is qualitatively different in the three V-shaped areas between the baseline extensions. HDOP is infinite along the baseline extensions, and is significantly larger in these area. A three-station system should not be used between the baseline extensions.
For locations outside the stations' perimeter, a multilateration system should typically be used only near the center of the closest baseline connecting two stations or near the center of the closest plane containing three stations. Additionally, a multilateration system should only be employed for user locations that are a fraction of an average baseline length from the closest baseline or plane. For example:
When more than the required minimum number of stations are available, HDOP can be improved. However, limitations on use of the system outside the polygonal station perimeter largely remain. Of course, the processing system must be able to utilize the additional TOAs. This is not an issue today, but has been a limitation in the past.

Example applications

For applications where there is no need for absolute coordinates determination, implementing a simpler 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. The simplest approach is Unilateration. However, the unilateration approach never delivers the angular position with reference to the detector.
Many products are available. Some 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 room-wise excitation; one vendor offers a position discrimination with a contiguity excitation.