Multilateration
Multilateration is a technique for determining a stationary or moving "vehicle's" position based on measurement of the "times of arrival" (TOAs) of virtually any type (physical phenomenon) of energy wave having a known waveform and propagation speed when traveling either "from" (navigation) or "to" (surveillance) multiple system stations. The stations are at known locations and have synchronized "clocks". Systems have been developed for waves that follow straight-line and curved earth trajectories. Multilateration is: abbreviated MLAT; more completely termed pseudo-range multilateration; and is also termed hyperbolic positioning. Prior to computing a solution, the common "time of transmission  (TOT) of the waves is unknown to the receiver(s), either on the vehicle (one receiver, navigation) or at the stations (multiple receivers, surveillance). Consequently, also unknown is the wave times of flight" (TOFs) the ranges of the vehicle from the stations multiplied by the wave propagation speed. Each "pseudo-range" is the corresponding TOA multiplied by the propagation speed with the same arbitrary constant added (representing the unknown TOT). In navigation applications, the vehicle is often termed the "user"; in surveillance applications, the vehicle may be termed the "target". For an exact solution, the ranges must not change during the  period the signals are received (between first and last to arrive at a receiver). Thus, for navigation, an exact solution requires a stationary vehicle; however, multilateration is often applied to the navigation of moving vehicles whose speed is much less than the wave propagation speed. If formula_1 is the number of physical dimensions being considered (thus, vehicle coordinates sought) and formula_2 is the number of signals received (thus, TOAs), it's required that formula_3. Then, the fundamental set of formula_2 measurement equations is Processing is usually required to extract the TOAs (or their differences) from the received signals, and an  algorithm is usually required to solve these equations. An algorithm either: (a) determines numerical values for the time of transmission (TOT) (for the receiver(s) "clock") and formula_1 vehicle coordinates; or (b) ignores the TOT and forms formula_10 (at least formula_1) time difference of arrivals (TDOAs), which are used to find the formula_1 vehicle coordinates. Almost always, formula_13 (e.g., a plane or the surface of a sphere) or formula_14 (e.g., the real world). Systems that form TDOAs are also called "hyperbolic" systems, for reasons discussed below. A "navigation" system provides vehicle position (and perhaps other) information to an entity "on" the  vehicle (e.g., aircraft pilot or GPS user). A "surveillance" system provides vehicle position to an entity "not on" the vehicle (e.g., air traffic controller or cell phone provider). By the reciprocity principle, any method that can be used for navigation can also be used for surveillance, and vice versa (the same information is involved). Systems have been developed for both TOT and TDOA algorithms. In this article, TDOA algorithms are addressed first, as they were implemented first (circa 1916). Due to the technology available at the time, TDOA systems often determined a vehicle location in two dimensions. TOT systems are  addressed second. They were implemented, roughly, post-1975 and usually involve satellites. Due to technology advances, TOT algorithms generally determine a user/vehicle location in three dimensions. However, conceptually, TDOA or TOT algorithms are not linked to the number of dimensions involved. Pseudo-range multilateration should not be confused with any of: All of these systems (and combinations of them) are commonly employed. However, the terminology employed herein is not always used. Background. Multilateration definition. Prior to deployment of GPS and other global navigation satellite systems (GNSSs), 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 received signals. However, as result of deployment of GNSSs (which determine TOT), two issues arose: (a) What system type are GNSSs (pseudo-range multilateration, true-range multilateration, or a new system type)? (b) What are the defining characteristic(s) of a pseudo-range multilateration system? (There are no deployed multilateration surveillance systems that determine TOT, but they have been analyzed.) Applications overview. Pseudo-range multilateration systems have been developed with virtually every wave phenomena -- electromagnetic (various frequencies and waveforms), acoustic (audible or ultrasound, in water or air), seismic, etc. The multilateration  technique was apparently first used during World War I to locate the source of artillery fire using audible sound waves (TDOA surveillance). Multilateration surveillance is related to passive towed array sonar target localization (but not identification), which was also first used during World War I. Longer distance radio-based navigation systems became viable during World War II, with the advancement of radio technologies. For about 1950-2000, 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. 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 development of atomic clocks for synchronizing widely separated stations was instrumental in the development of the GPS and other GNSSs. The widespread use of satellite navigation systems like the Global Positioning System (GPS) have made Earth-fixed TDOA navigation  systems largely redundant, and most have been decommissioned. 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 -- e.g., surveillance of flying (alternative to radar) and taxiing (alternative to visual) aircraft. Multilateration is commonly used in civil and military applications to either (a) locate a vehicle (aircraft, ship, car/truck/bus or wireless phone carrier) by measuring the TOAs of a signal from the vehicle at multiple stations having known coordinates and synchronized "clocks" (surveillance application) or (b) enable the vehicle to locate itself relative  to multiple transmitters (stations) at known locations and having synchronized clocks based on measurements of 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, privately funded entities have also been (and are) station/system providers e.g., wireless phone providers. Multilateration is also used by the scientific and military communities for non-cooperative surveillance. Advantages and disadvantages. The following table summarizes the advantages and  disadvantages of pseudo-range multilateration, particularly relative to true-range measurements. The advantages of systems employing pseudo-ranges largely benefit the vehicle/user/target. The disadvantages largely burden the system provider. Principle. Coordinate frame selection. Virtually always, the coordinate frame is selected based on the wave trajectories. Thus, two- or three-dimensional Cartesian frames are selected most often, based on straight-line (or line-of-sight) wave propagation. However, polar (also termed circular/spherical) frames are sometimes used, to agree with earth-surface wave propagation paths. Given the frame type, the origin and axes orientation can be selected, e.g., based  on the station locations. Some solution algorithms require the stations to be in specific locations (e.g., the origin). This can be accomplished by coordinate frame translation and rotation. TDOA formation. Given formula_2 received signals, TDOA systems form formula_16 differences of TOA pairs (see "Calculating TDOAs or TOAs from received signals" below). All received signals must be a member of at least one TDOA pair, but otherwise the differences used are arbitrary (any two of the several sets of TDOAs can be related by an invertible linear transformation). Thus, when forming a TDOA, the order of the two TOAs involved is  not important. Some operational TDOA systems (e.g., Loran-C) designate one station as the "master" and form their TDOAs as the difference of the master's TOA and the formula_16 "secondary" stations' TOAs. When formula_18, there are formula_19 possible TDOA combinations, each corresponding to a station being the de facto master. When formula_20, there are formula_21 possible TDOA sets, formula_22 of which do not have a de facto master. When formula_23, there are formula_24 possible TDOA sets, formula_25 of which do not have a de facto master. TDOA 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 (TDOA). 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 (see Figure 1). 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 over-determined constellation (more than 4 satellites/TOAs) 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 (e.g., in civil transportation). TDOA principle / navigation. Multilateration can also be used by a single  receiver to locate itself, by measuring signals emitted from synchronized transmitters at known locations (stations). At least three emitters are needed for two-dimensional navigation (e.g., the Earth's surface); at least four emitters are needed for three-dimensional navigation. Although not true for real systems, for expository purposes, the emitters may be regarded as each broadcasting narrow pulses (ideally, impulses) at exactly the same time on separate frequencies (to avoid interference). In this situation, the receiver measures the TOAs of the pulses. In actual TDOA systems, the received signals are cross-correlated with an undelayed replica to extract the  pseudo delay, then differenced with the same calculation for another station and multiplied by the speed of propagation to create range differences. 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 elaborated on this principle. For Loran-C, introduced in the late 1950s, all transmitters broadcast pulses on the same frequency with different, small time delays. GNSSs continuously transmitting on the same carrier frequency modulated by different pseudo random codes (GPS, Galileo, revised GLONASS). TOT principle. The  TOT concept is illustrated in Figure 2 for the surveillance function and a planar scenario (formula_13). Aircraft A, at coordinates formula_27, broadcasts a pulse sequence at time formula_28. The broadcast is received at stations formula_29, formula_30 and formula_31 at times formula_32, formula_33 and formula_34 respectively. Based on the three measured TOAs, the processing algorithm computes an estimate of the TOT formula_28, from which the range between the aircraft and the stations can be calculated. The aircraft coordinates formula_27 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 (the solid-line circles in Figure 2). If the computed TOT is after the actual TOT, the computed ranges do not have a common point of intersection (dashed-line circles in Figure 2). 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 (transmitters on satellites) 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 (which is used to measure the TOAs). To accomplish this: (1) satellite trajectories and TOTs in the satellites’ time scales are included in broadcast messages; and (2) user receivers find the difference between their TOT and the satellite broadcast TOT (termed the clock bias or offset). GPS satellite clocks are synchronized to UTC (to within a published offset of a few seconds), 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 (E in Figure 3) at an unknown location vector which we wish to locate (surveillance problem). The source is within range of formula_38 receivers at known locations The subscript formula_40 refers to any one of the receivers: The distance (formula_43) 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"), which makes its distance to the  emitter Spherical coordinates. Low-frequency radio waves follow the curvature of the Earth (great-circle paths) rather than straight lines. In this situation, equation is not valid. Loran-C and Omega are examples of systems that use spherical ranges. When a spherical model for the Earth is satisfactory, the simplest expression for the central angle (sometimes termed the "geocentric angle") formula_44 between vehicle formula_45 and station is where latitudes are denoted by formula_47, and longitudes are denoted by formula_48. Alternative, better numerically behaved equivalent expressions can be found in great-circle navigation. The distance formula_43 from the vehicle to station  is along a great circle will then be where formula_51 is the assumed radius of the Earth, and formula_44 is expressed in radians. Time of transmission (user clock offset or bias). Prior to GNSSs, there was little value to determining the TOT (as known to the receiver) 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 (e.g., Loran-C, Omega, Decca), receivers treated the TOT as a nuisance parameter and eliminated it by forming TDOA differences (hence were termed  TDOA or range-difference 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 and subsequently other satellite navigation systems: (1) TOT as known to the user receiver provides necessary and useful information; and (2) 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. Thus, algorithms used for satellite navigation solve for the receiver position and its clock offset (equivalent to TOT) simultaneously. The receiver clock is then adjusted so its TOT matches the satellite TOT (which is known by the GPS message). 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 (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 Loran-C measurements to account for soil conductivity variations. Calculating TDOAs or TOAs from received signals. Assume a surveillance system calculates the time differences (formula_53 for formula_54) of wavefronts touching each receiver. The TDOA equation for receivers formula_40 and formula_56 is (where the wave propagation speed is formula_57 and the true vehicle-receiver ranges are formula_58 and formula_43) The quantity formula_60 is often termed a pseudo-range. It differs from the true range between the vehicle and station formula_40 by an offset, or bias, which is the same for every signal. Differencing two pseudo -ranges yields the difference of the same two true-ranges. Figure 4a (first two plots) show a simulation of a pulse waveform recorded by receivers formula_62 and formula_63. The spacing between formula_64, formula_63 and formula_62 is such that the pulse takes 5 time units longer to reach formula_63 than formula_62. The units of time in Figure 4 are arbitrary. The following table gives approximate time scale units for recording different types of waves: The red curve in Figure 4a (third plot) is the cross-correlation function formula_69. 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 formula_70 value needed for equation . Figure 4b shows 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 formula_71. 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 (recall the same math works for moving receiver and 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, narrow-band waveforms because of the relation between phase formula_72, frequency formula_73 and time formula_74: 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. Navigation systems employ similar, but slightly more complex, methods than surveillance systems to obtain delay differences. The major change is DTOA navigation systems cross-correlate each received signal with a stored replica of the transmitted signal (rather than another received signal). The result  yields the received signal time delay plus the user clock's bias (pseudo-range scaled by formula_76). Differencing the results of two such calculations yields the delay difference sought (formula_53 in equation ). TOT navigation systems perform similar calculations as TDOA navigation systems. However, the final step, subtracting the results of one cross-correlation from another, is not performed. Thus, the result is formula_2 received signal time delays plus the user clock's bias (formula_79 in equation ). Solution algorithms. General algorithm behavior. Generally, using a direct (non-iterative) algorithm, formula_80 measurement equations can be reduced to a single scalar nonlinear "solution  equation" having one unknown variable (somewhat analogous to Gauss–Jordan elimination for linear equations) e.g., a quadratic polynomial in one vehicle Cartesian coordinate. The vehicle position and TOT then readily follow in sequence. When formula_80, the measurement equations generally have two solution sets (but sometimes four), only one of which is "correct" (yields the true TOT and vehicle position in the absence of measurement errors). The "incorrect" solution(s) to the solution equation do not correspond to the vehicle position and TOT and are either ambiguous (yield other vehicle positions which have the same measurements) or extraneous (do not provide vehicle  positions which have the same measurements, but are the result of mathematical manipulations). Without redundant measurements (i.e., formula_80), all valid algorithms yield the same "correct" solution set (but perhaps one or more different sets of "incorrect" solutions). Of course, statistically larger measurement errors result in statistically larger errors in the correct computed vehicle coordinates and TOT. With redundant measurements (i.e., formula_83), a loss function or cost function (also called an error function) is minimized (a quadratic loss function is common). With redundant measurements in the absence of measurement errors, the measurement equations usually have a unique solution. If measurement errors  are present, different algorithms yield different "correct" solutions; some are statistically better than others. Algorithm selection considerations. 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 (as do iterative algorithms) or is it not? Direct (closed-form) 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 (closed form ) algorithms have multiple solutions, which is detrimental to their automation. A third factor is: Does the algorithm function well with both the minimum number (formula_84) TOA measurements and with additional (redundant) measurements? Direct algorithms can be further categorized based on energy wave propagation path—either straight-line or curved. The latter is applicable to low-frequency radio waves, which follow the earth's surface; the former applies to higher frequency (say, greater than one megahertz) and to shorter ranges (hundreds of miles). This taxonomy has five categories: four for direct algorithms and one for iterative algorithms (which can be used  with either formula_84 or more measurements and either propagation path type). However, it appears that algorithms in only three of these categories have been implemented. When redundant measurements are available for either wave propagation 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 TOA systems, the satellites follow well-defined  orbits and broadcast orbital information. (For navigation, the user receiver's clock must be synchronized with the transmitter clocks; this requires that the TOT be found.) Equation is the hyperboloid described in the previous section, where 4 receivers (0 ≤ m ≤ 3) lead to 3 non-linear equations in 3 unknown Cartesian coordinates (x,y,z). The system must then solve for the unknown user (often, vehicle) location in real time. (A variation: 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/longitude for off-airport applications.) Steven Bancroft was apparently the first to publish a closed-form solution to the problem of locating a user (e.g., vehicle) in three dimensions and the common TOT using four or more 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 signal TOT. 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 (fourth-order) polynomial. Multilateration systems and studies employing spherical-range measurements (e.g., Loran-C, Decca, Omega) utilized a variety of solution algorithms based on either iterative methods or spherical trigonometry. Three-dimensional Cartesian algorithms. For Cartesian coordinates, when four TOAs are available and the TOT is needed, Bancroft's or another closed-form (direct) algorithm are options, 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 an option. Another 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 (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 3-D multilateration. Wide Area Multilateration (WAM), a 3-D aircraft surveillance system, employs a combination of three or more TOA measurements and an aircraft altitude report. Two-dimensional Cartesian algorithms. For finding a user's location  in a two dimensional (2-D) 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 algorithms. Razin developed a closed-form algorithm 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 (e.g., hundreds of miles or more) 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 (respectively), 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 (say, formula_86, numbered formula_87) 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 formula_43, square both sides, collect terms and divide all terms by formula_89: Removing the formula_90 term will eliminate all the square root terms. That is done by subtracting the TDOA equation of receiver formula_91 from each of the others (formula_92) Focus for a moment on equation . Square formula_58, group similar terms and use equation  to replace some of the terms with formula_58. Combine equations and , and write as a set of linear equations (for formula_95) of the unknown emitter location formula_96 Use equation to generate the four constants formula_97 from measured  distances and time for each receiver formula_95. This will be a set of formula_99 inhomogeneous linear equations. There are many robust linear algebra methods that can solve for formula_100, 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 'vehicle'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) a transient ‘acquisition’ (surveillance) or ‘cold start’ (navigation) mode, whereby the user’s location is found from the current measurements only; and (b) a steady-state ‘track’ (surveillance) or ‘warm start’ (navigation) mode, whereby the user’s previously computed location is updated based current measurements (rendering moot the major disadvantage of iterative methods). Often the two modes employ different algorithms and/or  have different measurement requirements, with (a) being more demanding. The iterative Gauss-Newton algorithm is often used for (b) and may be used for both modes. When there are more TOA measurements than the formula_84 unknown quantities – e.g., 5 or more GPS satellite TOAs – the iterative Gauss–Newton algorithm for solving non-linear least squares (NLLS) 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 (e.g., ASDE-X), 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 (a) it is inherently difficult and/or expensive to accurately measure the  true range (distance) between a moving vehicle and a station, particularly over large distances, and (b) 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 (user-station geometry), 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 volume (three dimensions). 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 (navigation) or receivers (surveillance) – have synchronized 'clocks'. There are two distinct synchronization  requirements: (1) maintain synchronization accuracy continuously over the life expectancy of the system equipment involved (e.g., 25 years); and (2) for surveillance, accurately measure the time interval between TOAs for each 'vehicle' transmission. Requirement (1) is transparent to the user, but is an important system design consideration. To maintain synchronization, station clocks must be synchronized or reset regularly (e.g., every half-day for GPS, every few minutes for ASDE-X). Often the system accuracy is monitored continuously by "users" at known locations - e.g., GPS has five monitor sites. 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: Service area or volume. Sensitivity of accuracy to vehicle-station geometry. 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 formula_104. 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 formula_105 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 formula_106 away from the origin, in the absence of measurement errors, the TDOA would be formula_107 (where formula_57 is the known wave propagation speed). Thus, (ignoring the scale-factor formula_57) 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 (i.e., formula_110 or formula_111) 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 formula_106 which occurred during that interval—i.e., formula_113. Conversely, outside the perimeter, consecutive TDOAs will typically attenuate but not cancel associated vehicle movement—i.e., formula_114. 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. Dilution Of Precision (DOP). When analyzing a 2D or 3D multilateration system, dilution of precision (DOP) is usually employed  to quantify the effect of user-station geometry on position-determination accuracy. The basic DOP metric is The symbol formula_116 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 (PR) measurement error. (When redundant stations are involved, it is possible to  have 0 < ?DOP < 1.) HDOP is usually employed (? = H, and XXX = horizontal position) when interest is focused on a vehicle position on a plane. Pseudo-range errors are assumed to add to the measured TOAs, be Gaussian-distributed, have zero mean (average value) and have the same standard deviation formula_117 regardless of vehicle location or the station involved. Labeling the orthogonal axes in the plane as formula_118 and formula_119, the horizontal position error is characterized statistically as Mathematically, each DOP "flavor" is a different sensitivity ("derivative") of a solution quantity (e.g., horizontal position) standard deviation with  respect to the pseudo-range error standard deviation. (Roughly, DOP corresponds to the condition formula_121.) 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 that a linearized least squares algorithm is used. (It is also the smallest variance for any algorithm.) Specifically, HDOP is the sensitivity ("derivative") of the user's horizontal position standard deviation (i.e., its sensitivity) 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 (station separation formula_122). The inner circle is more "conservative" and corresponds to a "cold start" (no knowledge of vehicle's initial position). 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 (and would be the same value for true range measurements). Beginning with HDOP = 1.25, the contours shown follow a factor-of-2 progression. Their roughly equal spacing (outside of the three V-shaped areas between the baseline extensions) 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. (HDOP is mathematically undefined at the stations; hence multiple DOP contours can terminate on a station.) 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 (two dimensional planar situation) or near the center of the closest plane containing three stations (three dimensional situation). Additionally, a multilateration system should only be employed for user locations that are  a fraction of an average baseline length (e.g., less than 25%) from the closest baseline or plane. For example: When more than the required minimum number of stations are available (often the case for a GPS user), HDOP can be improved (reduced). However, limitations on use of the system outside the polygonal station perimeter largely remain. Of course, the processing system (e.g., GPS receiver) must be able to utilize the additional TOAs. This is not an issue today, but has been a limitation in the past.