Abstract:
A method of locating a terrestrial emitter of electromagnetic radiation in the midst of a plurality of emitters in a satellite in orbit about the earth which utilizes a location estimation and location probability determination process with respect to each possible emitter site and its corresponding error region and then using both feedback and feed forward interaction between location and phase ambiguity resolution processes to generate resolved phase from emitter location, update emitter location or some or all of the emitters, and subsequently utilizing the probabilities thus determined to produce a single estimate of the desired emitter&#39;s location.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This is a continuation of application Ser. No. 11/806,179 filed May 30, 2007 for Method for Single Satellite Geolocation of Emitters Using an Ambiguous Interferometer array. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    This invention relates generally to the location of objects from space via a satellite and, more particularly, to the accurate geolocation of possibly non-cooperative emitters from a single low earth orbit (LEO) satellite utilizing the minimum number of antennas required to derive ambiguous emitter direction of arrival. 
         [0004]    2. Description of Related Art 
         [0005]    Geolocation as used here refers to the determination of the emitter in both earth-center inertial (ECI) coordinates, and geodetic latitude and longitude. Using satellites to locate ground emitters is an important civilian and military task. Examples are emergency rescue of downed aircraft, determination of inadvertent sources of communications interference and location of malicious jammers disrupting military satellite usage. The transmitters may or may not aid or cooperate in their location. 
         [0006]    Cooperative emitter location can take many forms. A common method is to transmit a stable CW narrow band signal. These signals may be continuous, or sent in burst; however, the transmitters always endeavor to be on during ranging. This allows a single LEO satellite to locate the emitter using the signal frequency&#39;s Doppler shifts. To provide location to a useable accuracy, a typical transmitter requirement is frequency stability of one part in 10 −8  for twenty minutes over a wide range of adverse climatic conditions. Thus the transmitter is specifically designed and built to facilitate its location. 
         [0007]    Non cooperative transmitters are not necessarily attempting to avoid geolocation. That is just not their reason for operating. Hence they are not required to have the frequency stability needed for delta-Doppler shift location, nor are they required to transmit for an extended period. However, rapid, accurate location of these non cooperative emitters becomes important when they interfere with satellite communication. 
         [0008]    This interference is not necessarily malicious. The Ku band and C band of the electromagnetic spectrum are crowded, and inadvertent interference is commonplace. By contrast, X band activity is generally military, and interference is typically hostile. But whether inadvertent or hostile, the geolocation method used to find non-cooperative interferers must quickly produce emitter latitude and longitude or its equivalent to an accuracy eventually providing unique identification. Multiple satellite solutions have heretofore been favored for this. Using multiple satellites means Doppler shift techniques can still be used, but now it is the shift measured simultaneously between satellite pairs. So the emitter frequency stability requirement is not nearly as stringent as when locating using dwell-to-dwell measurements. 
         [0009]    Thus simultaneous intercept of the interfering signals is a strong point of multiple satellite solutions. But it is also a problem since the emitter must lie simultaneously within the field of view of all the satellites, and be simultaneously detected by them. 
         [0010]    To minimize the impact of these problems, U.S. Pat. No. 6,417,799, “Method of Locating an Interfering Transmitter for a Satellite Telecommunications System,” Aubain et al., discloses a known two satellite solution. They reduce the number of satellites required to two by using combinations of three measurements: (a) signal time difference of arrival (TDOA) between the two satellites; (b) frequency Doppler difference (FDOP) between the two satellites; and, (c) signal angle of arrival (AOA) using an interferometer on a single satellite. The implementation Aubain et al uses in U.S. &#39;799 to illustrate their method is locating the jammer of a geosynchronous orbit (GEO) telecommunications satellite. A second special “detection” satellite is in a LEO orbit with an interferometer mounted on it. Aubain assumes a linear interferometer as shown in  FIG. 1  of subject applicant&#39;s set of drawings where a linear interferometer measures phase (φ)  104  and from this obtains the (θ) angle of arrival (AOA)  100  between the baseline vector ({right arrow over (d)})  101  and normal to the signal wavefront, or direction of arrival (DOA) unit vector ({right arrow over (μ)})  102 . 
         [0011]    Thus the linear interferometer determines a cone  103  that the emitter DOA vector lies on. The cone intersects the earth giving a line of position (LOP) for the emitter. The intersection of this cone with a tangent plane at the emitter is a conic section, usually a parabola. This parabolic LOP has a thickness or uncertainty due to the interferometer phase measurement error (ε)  106 . The AOA error is reduced by extending the baseline length  101  between the antenna phase centers. It is also reduced at higher emitter frequencies, or shorter signal wavelengths (λ)  107 . 
         [0012]    Aubain et al. uses the interferometer in two ways. First, frequency and time measurements are performed and the resulting TDOA and FDOP LOP have multiple points of intersection. The AOA parabola then provides an additional measurement picking out the correct intercept. In this application the extent of the AOA error or, equivalently, the thickness of the parabolic LOP is not critical because the TDOA and FDOP lines of position determine the location accuracy. The second way the interferometer is used is with either a TDOA or FDOP measurement. For example, assume TDOA and AOA intercepts determine the emitter position. Then resolving the multiple intersections is accomplished by sequential dwells. As the LEO satellite moves the parabolic and hyperbolic LOP generated at each receiver dwell intersect near the emitter with an uncertainty due to their thickness. So in this case the AOA uncertainty is critical. 
         [0013]    Increasing the spacing between the antenna phase centers, i.e., increasing the baseline vector ({right arrow over (d)})  101  length, proportionally improves the LOP accuracy. However, increasing the baseline length beyond a half wavelength (λ/2) of the signal source generates phase measurement ambiguities  105 . Here n  105  is an integer reducing the true phase so the measured phase (φ)  104  lies in the region −π≦φ&lt;π. This means that interferometer phase is measured modulo 2π. 
         [0014]    Determining ambiguity integers to recover the true phase requires special processing, an example of which is described by Malloy in a 1983 IEEE ICASSP paper entitled “Analysis and Synthesis of General Planar Interferometer Arrays.” See also U.S. Pat. No. 6,421,008, “Method to Resolve Interferometric Ambiguities” Dybdal and Rousseau, and U.S. Pat. No. 5,572,220, “Technique to Detect Angle of Arrival with Low Ambiguity,” Khiem V. Cai. 
         [0015]    These results show eliminating the phase measurement integer ambiguities requires adding additional antennas between the outermost elements. Thus a linear interferometer installed to robustly and fully implement Aubain&#39;s method as taught in U.S. &#39;799 may actually resemble  FIG. 2  of applicant&#39;s drawings, not  FIG. 1 . In  FIG. 2 , antennas  203  and  204  are added to determine the integers n i    207  to resolve the phase measurement (φ 1 )  205 , and hence find the angle of arrival (θ)  206  from the true phase resolved phase vector ({right arrow over (φ)})  211 . This resolution is done by processing measurements by an ambiguity resolver  209  across the multiple baselines. An error in determining n i  on any baseline invalidates the subsequent relation between the resolved phase vector ({right arrow over (φ)}) and DOA unit vector ({right arrow over (u)})  212 , throwing off the subsequent estimate of AOA  206  by a large amount. Hence it is called a gross error. 
         [0016]    In doing the processing as indicated by reference numeral  209  of  FIG. 2 , Malloy obtains a uniform gross error rate across the frequency band of interest independent of the emitter relative bearing by only allowing antennas placed at relatively prime integer multiples of a fundamental spacing (d 0 ). He calls the discrete set of candidate points the array lattice.  FIG. 3  of subject applicant&#39;s drawings illustrates a simple linear three antenna array designed according to Malloy&#39;s approach. In this  FIG. 300  is the array lattice, and the antennas  301  have relative integer spacings 3 and 4 times the lattice spacing (d 0 )  302 . The lattice spacing is determined by the wavelength  304  at the highest frequency and the FOV constraint  303 . 
         [0017]    The fixed value of the gross error rate across frequency and AOA is a most desirable property. A further significant advantage of using Malloy&#39;s placement restrictions and his design method is they result in arrays maximizing the AOA or DOA accuracy while minimizing the gross error rate. Also, critical for satellite applications, his method does this using the minimum possible number of antennas. 
         [0018]    Malloy&#39;s approach thus produces the best sparse element interferometer, optimal in the sense that, for a given Gaussian phase error vector ({right arrow over (ε)}) as indicated by reference number  214  with covariance matrix R, the quadratic cost L which can be expressed as: 
         [0000]    
       
         
           
             
               
                 
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         [0000]    is minimized. D in equation (1) is the matrix of baseline vectors  216  shown in  FIG. 2 , where ({right arrow over (k)})  217  is the unit vector along the antenna phase centers and ({right arrow over (u)}) the DOA unit vector  212 . For planar arrays as shown in  FIG. 5  of this set of drawing figures and discussed hereinafter, (D) is the matrix  506  of baseline vectors ({right arrow over (d)} 1 )  508  forming the array. Planar arrays are used when DOA rather than AOA is required. Interferometers designed according to Malloy&#39;s method are thus ideal for satellite applications. 
         [0019]    None the less, utilizing a simple array as shown in  FIG. 3  of the set of drawing figures disclosed herein, or an equivalent with four antennas as taught by the satellite telecommunications system of Aubain (U.S. Pat. No. 6,417,799) presumably could be used, especially since in the Aubain approach the detection satellite could be dedicated to the emitter location problem and hence ab initio designed to support an optimal multi-antenna element linear array. But signal jamming is not just a problem for GEO telecommunication satellites. LEO communication satellites are also important. For example, U.S. Pat. No. 5,412,388, “Position Ambiguity Resolution,” Attwood, discloses an LEO communications satellites support radio telecommunications system operating with portable units. Portability requires low power battery operation, and small antennas. Such units can communicate with satellites in LEO orbits, but not GEO satellites 30× farther out. 
         [0020]    Aubain&#39;s multi-satellite technique does not adapt well to locating LEO jammers. For example, reversing the roles of the low earth orbit and geosynchronous satellites will not work because of the GEO detection difficulty Nor does using two LEO satellites work since simultaneous emitter detection is a significant problem in low earth orbit. It is a problem both because of weak signals, where one of the LEO satellites is far from the transmitter, and because the emitter may not be visible to all antennas in the detection satellite&#39;s array. 
         [0021]    Coordination is a further major problem. This is a difficulty for any multiple LEO satellite scheme, not just one based on Aubain&#39;s approach. Even when all the satellites detect the interfering signal, some central controller and processor must recognize that a jammer was detected, collect data of the same jammer signal from all the satellites, and then derive a geolocation estimate. To satisfy this need Aubain suggests downlinking and processing the data at a ground station. This is a satisfactory solution in Aubain&#39;s original scenario because of the presence of a geostationary telecommunications satellite and such a satellite can always have a ground station visible. But with only LEO satellites, it is much too restrictive to assume a ground station with the required processing facilities will be visible to any of the satellites let alone all when the jammer transmits. 
         [0022]    Because of these detection and coordination problems, the most robust operational approach to LEO geolocation of interfering emitters is to use a single satellite. Further, that satellite must be the one experiencing jamming, and must be able to autonomously derive the jammer location from onboard measurements. Attwood solved the emitter location problem with a single LEO satellite by making sequential time delay and frequency change measurements, but suggested synchronizing the frequency base and time base used by the satellite and transmitter to achieve the required signal stability. Jammers, however, are not so helpful. Because the jamming transmitter will not generally cooperate by providing signal stability, allowing the use of delta-Doppler, location must be derived solely from angle of arrival (AOA) or direction of arrival (DOA). AOA is adequate in Aubain&#39;s approach because it is used with time difference of arrival (TDOA) or frequency Doppler difference (FDOP). But when only angle measurements are available, DOA is far superior to AOA since it results in an actual emitter location estimate at each receiver dwell, rather than just the line-of-position on which the transmitter lies. Also, the transmitter may be on only briefly. Accordingly, using DOA greatly expedites the location process. 
         [0023]    Attempts have previously been made to measure DOA with a single antenna, and thus avoid the need for phase measurements across a two dimensional interferometer array. For example, U.S. Pat. No. 6,583,755, “Method and Apparatus for Locating a Terrestrial Transmitter from a Satellite,” Martinerie and Bassaler, discloses the concept of performing single platform LEO geolocation by measuring a plane the emitter&#39;s DOA vector lies in, called the propagation vector. However, it uses a single special antenna to make this measurement. The plane intersects the earth, resulting in a circle of position. The DOA plane is derived from measurements of the electromagnetic field&#39;s polarization, with the polarization being restricted to linear polarization. Unfortunately the restriction to one special type of electromagnetic wave polarization greatly limits the method&#39;s usefulness. Jamming signals are not restricted in their polarization. 
         [0024]    Accordingly, there is currently no robust way to perform single satellite geolocation other than by measuring DOA by implementing a multielement antenna array equivalent to a planar interferometer. And in particular, if a conventional planar array is chosen, an approach such as presented by Malloy appears to be the best available. As noted, such an array will generate the best DOA accuracy for the lowest gross error rate utilizing the minimum number of antennas. The antennas may be chosen to cover the frequency band of interest and respond to any transmitted polarization. Such an array, if large enough, could accurately locate emitters in a single receiver dwell by intersecting the DOA vector with the earth&#39;s surface. This method is called Az/El geolocation. However, the baselines for such an array are comparatively large and they typically are measured in meters rather than conventional centimeters. 
         [0025]    Hence implementing such an array generally requires specifically designing the satellite to support it, possibly using such specialized structures as that described in U.S. Pat. No. 6,016,999, “Spacecraft Platforms” by Simpson, McCaughey and Hall. Therefore, for the widest possible application, especially on existing satellite designs, smaller arrays must be considered. Such arrays, however, do not support Az/El geolocation, but do support locating emitters using some form of triangulation or bearings-only geolocation over several dwells as the satellite moves in its orbit. 
         [0026]    An example of such an array using Malloy&#39;s design approach is shown in  FIG. 4  of this specification. The points  400  of  FIG. 4  represent a lattice of antenna spacings for an optimal planar array analogous to the antenna spacings  300  in  FIG. 3  for the linear array. The points are now located by two fundamental lattice position vectors ({right arrow over (d)} 1 ) and ({right arrow over (d)} 2 ) shown by reference numeral  40 . At least nine (9) antennas  402  are typically required to provide a small gross error rate while providing DOA performance supporting bearings-only geolocation. The optimal array is designed by arranging the nine antennas on the lattice points in different configurations consistent with the relatively prime integer-multiple requirement and computing the quadratic cost (Equation 1) for each arrangement. Just as for the linear array, utilizing these lattice points and prime integer spacings guarantees an array having a gross error rate independent of frequency and signal angle of arrival. The configuration chosen would be one minimizing the cost while giving the required DOA accuracy with the lowest gross error rate. The configuration shown in  FIG. 4  is illustrative of how such a final design would look. 
         [0027]    The array  500  shown in  FIG. 5A  is a more pictorial representation showing its implementation with a two channel receiver. By intersecting the AOA cones in a DOA processor  501  across multiple antenna pairs, a unique DOA unit vector ({right arrow over (u)})  502  is found. The DOA unit vector  502  is also shown relative to the array  500 . 
         [0028]    Note that because of the phase measurement error vector ({right arrow over (ε)})  503  the DOA unit vector ({right arrow over (u)})  502  has an error cone  505  associated with it. This is usually taken to represent a 3σ error deviation, so the true DOA lies within this cone 98.9% of the time. Because of this error the range line  506  shown in  FIG. 5B  extending the DOA  52  to the earth&#39;s surface, thus providing a slant range to the emitter, does not usually intersect the earth at the emitter&#39;s true location. Rather a somewhat elliptical boundary  507  is formed by the cone  505 , shown in  FIG. 5A , and the emitter lies within this boundary all but about 1% of the time. 
         [0029]    When the emitter lies near the satellite&#39;s suborbital point, the boundary closely approximates an ellipse  600  shown in  FIG. 6A . But as emitters approach the satellite&#39;s horizon, the boundary becomes more distorted, larger and egg shaped as shown by reference numeral  601 . The increase in the area is due to the interaction of the earth&#39;s curvature  602  shown in  FIG. 6B  with the DOA error cone  603 . The earth&#39;s curvature  603  increases the range uncertainty to the emitter. Thus the slant range error  605  is significantly larger away from the suborbital point  604  compared with the error  606  for transmitters closer in. 
         [0030]    The error cone  603 , and hence slant range errors  605  and  606  are reduced by extending the outermost interferometer baseline spacings. So accurately locating any emitter within the satellite&#39;s field of view, particularly those near its horizon, requires increasing the outer antenna spacing in the array  500  shown in  FIG. 5A  to the maximum extent possible. But improving DOA accuracy introduces larger phase measurement ambiguities  507 , and hence requires more antennas to resolve them. The nine (9) antennas shown in  FIG. 5A  may thus actually represent a lower bound for the number required in a planar interferometer array supporting LEO bearings-only location for emitters far from the suborbital point. 
         [0031]    Hence even implementing an optimal interferometer with the minimal number of elements to solve the LEO location problem still generates what is derisively called by the satellite community an “antenna farm.” Accordingly, this solution is normally not practical or desirable to implement because of the space required on the satellite&#39;s surface. The space required is not the only restriction, however. Each antenna must have its own unobstructed field of view. This greatly limits what else can be installed. Therefore, implementing even a small array usually requires that the satellite still be specifically designed to support it. 
         [0032]    This is generally unacceptable since locating interfering emitters would be an auxiliary task, not the primary mission of most LEO satellites. So the array for locating an interfering transmitter must fit into an existing design, and not require a new one. The only way the array in  FIG. 5A , for example, repeated in  FIG. 7  by reference numeral  700 , can fit onto an existing design is by eliminating antennas. If only three antennas are retained, such as shown by reference numeral  701  in  FIG. 7B , the array has a much better chance of being acceptable. It will achieve the DOA accuracy required. But now, because of the phase measurement ambiguities  704 , spurious DOAs  705  are produced, one possible DOA for each integer pair (n, m)  706 . The ambiguous DOA vectors  702  each have an associated error cone  703 . 
         [0033]    This, of course, results in ambiguous emitter locations as shown in  FIG. 8B , where for clarity an array  800  identical to array  701  shown in  FIG. 7B  is reproduced. In  FIG. 8B , reference numeral  801  represents the range vector ({right arrow over (r)}) extended along the true DOA to an actual emitter location, while range vector  802  and the other range vectors indicate spurious locations. Note that even though the DOA associated with range line  802  is spurious, its error region  803  is formed by the associated error cone intersecting the earth&#39;s surface, just as for the true emitter. 
         [0034]    Since the reduced array size requires relative motion to locate the emitter, it is natural to ask if the ambiguous sites may be eliminated by the same satellite movement. A technique for exploiting platform motion to resolve phase measurement ambiguities does exist for the two element array shown in  FIG. 1 . In U.S. Pat. No. 3,935,574, “Signal Source Position-Determining Process,” Pentheroudakis discloses a method utilizing either rotational motion or translational motion of a two element ambiguous horizontal interferometer to pick the correct sequence of AOA cones over a series of receiver dwells and hence eventually resolve the linear array. 
         [0035]    At the first dwell Pentheroudakis establishes the set of possible ambiguity integers (n i ) and uses each integer from this set to produce a candidate resolved phase. The phase rate is then measured and the phase roll over is tracked or changed in each integer n i  as the array moves relative to the emitter. Under this phase unwrapping procedure, the correctly updated ambiguity integer sequence produces a stable location estimate that converges to the true emitter position. The other sequences eventually produce AOA that exhibit abrupt changes when the respective ambiguity integer gets incremented, causing the corresponding location estimates to diverge and hence eliminating those integer sequences as viable candidates. 
         [0036]    Pentheroudakis&#39; method requires almost continuous phase measurement updates to track the phase change and hence the integer roll over for each integer set or lobe. Although used to resolve a single two element interferometer, the method could be readily adapted to the three element planar array  800  shown in  FIG. 8A . Now the roll over in two sets of ambiguity integers (n i ,m i ) must be tracked. This greatly extends and complicates the software processing required, but does not alter the basic method of using phase tracking to update the ambiguity integers (n i ,m i ), and then updating the candidate emitter locations. 
         [0037]    A more significant drawback when trying to adapt the method to satellite applications is the extent of both the true and spurious error regions, e.g., regions  803  shown in  FIG. 8B , especially for candidate emitter locations  804  relatively far from the suborbital point of range line  802 . Also important is the fact the error regions for the ambiguous locations can overlap each other as shown by reference numeral  805 . 
         [0038]    The extent and overlap of the error regions  805  both have critical consequences. Reference will now be made to  FIG. 8C . First assume that region  815  is a true site. As the satellite moves in its orbit from position  806  to  807 , the spurious location  809  associated with DOA  808  may jump to new location  810  in a more erratic manner than estimates for the true site even before an incorrect integer roll over occurs. But this somewhat erratic movement is difficult or even impossible to detect with Pentheroudakis&#39; method because of the extent of the error region  816 . Further, the abrupt DOA jumps caused when phase tracking generates incorrect integer updates are a function of the measured phase change created by the true emitter DOA&#39;s movement relative to the interferometer baseline. This has a significant impact on ambiguity resolution performance. 
         [0039]    To understand this impact, assume now that the true emitter happens to lie far from the suborbital point, for example at point  811  rather than point  815 . Now a comparatively large satellite orbital translation and hence significant time is required to create enough true DOA change to trigger integer updates and thus cause eventual large abrupt jumps in the spurious DOA to occur at other distant sites such as site  809 . When such jumps occur, because of the overlap of error regions these jumps may not eliminate wrong locations. For example, if suborbital point  811  is the true emitter location and it occurs in an overlap region  812 , it cannot be easily statistically differentiated from spurious site  813 . Even a phase roll-over may not clearly differentiate the two. It should be noted that there can be a significant number of these overlap regions, particularly at higher frequencies. 
         [0040]    Another critical problem adapting Pentheroudakis&#39; method to satellite based emitter location is the requirement for almost continuous sampling of the phase measurement. For the planar array  800  shown in  FIG. 8A , the receivers for the two channels shown must switch via switch  814  between baselines to do the phase sampling, so there is a limit to the sample rate. But the most fundamental problem with continuous phase tracking is that the noncooperative emitter might not be transmitting or detected at each receiver dwell during the geolocation process. For example, a simple way to defeat the Pentheroudakis scheme is to blink the emitter with a duty cycle that still interferes with satellite communication, but does not allow unwrapping of the candidate phases. Also, as noted before, on a satellite all antennas in the array may not have the same unrestricted field of view. So even with the emitter continuously transmitting, satellite attitude may prevent phase measurements during a significant number of the receiver dwells. 
         [0041]    The present invention overcomes the inherent problems associated with applying the Aubain, Attwood, Martinerie, and Pentheroudakis approaches to locating an emitter interfering with a satellite in low earth orbit (LEO). In particular, both cooperative and non-cooperative emitters are located with equal facility. The present invention does not require an emitter to transmit continuously, or require the transmitter to have special stability or polarization characteristics. It does not require downlinking data to a ground station. It does not require simultaneous detection of the emitter by multiple satellites, although it can be incorporated in such methods to increase their versatility. 
         [0042]    The present invention is directed to an intrinsically low earth orbit technique that exploits the excellent DOA measurement capability of planar interferometer arrays. It can employ arrays designed using Malloy&#39;s optimal approach, but with the array not fully populated with antenna elements and hence producing ambiguous DOAs. If this is done special processing utilizing the antenna&#39;s relatively prime integer spacing in an array lattice can be incorporated, but this concept is not pursued in the present invention since it is not essential. Further, it is not an intrinsic requirement of the invention that the antenna element placement satisfy the Malloy, Dybdal and Rousseau or Cai restrictions. Their placement can be arbitrary, dictated by space available on the satellite. In fact the antenna spacing in the subject invention can vary from one measurement update to another and thus the interferometer array can be flexible or floating. 
         [0043]    In performing geolocation, the present invention uses a minimal number of antennas required to generate ambiguous emitter DOAs. If no satellite attitude change is allowed during a receiver dwell the minimal number is three. But if some restricted attitude change is feasible, a modification to the method described in the present inventor&#39;s U.S. Pat. No. 5,457,466 “Emitter Azimuth and Elevation Direction Finding Using Only Linear Interferometer Arrays” permits the use of only two antennas. Also only two antennas may be required, if not coboresited, when incorporating the method described in the inventor&#39;s U.S. Pat. No. 5,608,411 “Apparatus for Measuring a Spatial Angle to an Emitter Using Squinted Antennas.” 
         [0044]    The present invention does not generate a candidate ambiguity integer set at each receiver dwell by phase tracking. Thus maintaining a continuous common field of view for all the antennas is not a requirement. Gaps of many seconds can occur between phase updates. Instead of lobe tracking or phase unwrapping, the present invention uses the initial emitter locations obtained from the initial set of ambiguous DOA to predict ambiguity integers during the satellite&#39;s subsequent orbital motion. In other words, this invention does not use phase to predict location, but rather location to predict phase. 
       SUMMARY OF THE INVENTION 
       [0045]    Accordingly, it is an object of this invention to initially determine all possible emitter sites of an emitter from ambiguous phase measurements made in the first receiver location processing dwell, and then to use these estimated sites to resolve the phase measurements made in subsequent dwells. 
         [0046]    It is a further object of the invention to input the corresponding resolved phase to a location estimator for each site, and hence update the site&#39;s geolocation estimate based on the phase resolved by that site. 
         [0047]    The present invention thus effects both a feed forward process where the resolved phases drive the corresponding emitter locations, and a feed-back process where the updated emitter locations generate the cycle integers required to resolve the next set of measured phases. 
         [0048]    Another object of the invention is to assign a probability or likelihood to each individual site&#39;s location estimate based on the feed-forward and feed-back processing. 
         [0049]    Still another object of the invention is to use these probabilities for estimating an actual emitter location without necessarily determining which site correctly predicts the ambiguity resolution. 
         [0050]    These and other objects are achieved by the subject invention which associates both a location estimation and location probability determination process with each possible emitter site and its corresponding error region, and uses both feedback and feed forward interaction between the location and phase ambiguity resolution processes to generate resolved phase from emitter location, update emitter location, and, subsequently, utilizing the probabilities, producing a single estimate of an emitter&#39;s location. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0051]    The exact nature of this invention, as well as the objects and advantages thereof, will become readily apparent from consideration of the following specification in conjunction with the accompanying drawings in which like reference numerals designate like parts throughout the figures thereof and wherein: 
           [0052]      FIG. 1  is illustrative of a related art two element interferometer where the antenna spacing may be more than half the signal wave length apart so the phase measurement is modulo 2 ambiguous, and further illustrates that the signal direction of arrival for an ideal linear interferometer even with no phase measurement error is indeterminate in the sense that it may lie anywhere on the angle-of-arrival cone. 
           [0053]      FIG. 2  shows how the phase measurement ambiguities in the two element interferometer of  FIG. 1  are eliminated by adding additional antennas at smaller baseline spacings. 
           [0054]      FIG. 3  depicts the restricted way additional antennas in  FIG. 2  must be added according to the known related art comprising an approach that produces an array having the lowest quadratic cost. 
           [0055]      FIG. 4  illustrates how generalizing the approach shown in  FIG. 3  to design an optimal planar interferometer array limits antenna placement. 
           [0056]      FIGS. 5A and 5B  show an implementation of the array shown in  FIG. 4  using a two channel receiver. 
           [0057]      FIGS. 6A and 6B  are illustrative of how the region of uncertainty grows due to the earth&#39;s curvature as transmitters move further and further from the satellite&#39;s suborbital position. 
           [0058]      FIGS. 7A and 7B  illustrate how simplifying the array of  FIG. 5A  by eliminating antennas from the array lattice shown in  FIG. 4  creates a set of ambiguous emitter directions of arrival. 
           [0059]      FIGS. 8A and 8B  show how the error cones associated with the ambiguous direction of arrival vectors for a three element array shown in  FIG. 7B  can create a pattern of large and overlapping error regions where they intersect the earth. 
           [0060]      FIG. 8C  illustrates how error regions of  FIG. 8B  can be so large and overlap in such a way that subsequent satellite motion does not provide a reliable means to distinguish spurious locations from the true one if phase tracking is used to generate the location estimates. 
           [0061]      FIG. 9  is a top level block diagram generally illustrative of the operation of the subject invention. 
           [0062]      FIG. 10  is a flow chart illustrative of the method of this invention, shown in  FIG. 9  and providing details of the processing carried out. 
           [0063]      FIGS. 11A-11D  are a set of bar graphs illustrative of the evolution of the location probabilities for an emitter located in relation to an initial satellite suborbital point, where these probabilities are used to show that the weighted average location estimate is generally superior to the maximum probability estimate before the array is resolved. 
           [0064]      FIG. 12  is a detailed block diagram showing the preferred embodiment of the subject invention. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0065]    Referring now to the top level method block diagram in  FIG. 9  and corresponding flow chart of  FIG. 10 , the following description is made first with reference to a three element array such as shown, for example, in  FIG. 8A  by reference numeral  800  which includes two interferometer baselines. A satellite at location  900  including a three element array  800  initially detects an emitter E whose true location is at location  901  shown in  FIG. 9 . 
         [0066]    Upon initial detection, a set of steps in a Measure and Resolve Interferometer Phase processes  902  and a set of steps in a Generate Location Estimate and Location Statistics processes  903  are carried out. Special operations involving steps  902  and  903  of  FIG. 9  unique to a first dwell step occur as shown in  FIG. 10  by reference numeral  1000 . All possible direction of arrival (DOA) vector ambiguity integer pairs (m, n) are found in step  1001  and each pair is associated with a unique Measure and Resolve Interferometer Phase process  902 . The measured phase of step  902  is resolved with an associated integer set (m, n) in step  1002  of  FIG. 10 . 
         [0067]    Next, all candidate DOA vectors  907  of an emitter E are computed per step  1003  from each resolved phase, along with estimates of their location error variance, and are used for a corresponding Generate Location Estimate process  903 . Each step of the Generate Location process  903  determines an emitter site per step  1004  for its associated DOA using Az/El ranging. This is the only time Az/El geolocation is used. 
         [0068]    Based on the DOA error variance input from process  902 , a location error variance for each site is computed in step  1005  of  FIG. 10 . This equivalently produces an error bound  904  for each estimated position  905  shown in  FIG. 9 . 
         [0069]    The error variances and location estimates generated initialize a corresponding set of recursive location estimators per step  1006  in  FIG. 10  that are used in subsequent dwell times  1007 . The likelihood weight computation of step  1006  is also initialized and carried out in the Determine Location Probability process  911  ( FIG. 9 ). A weight is assigned to each site per step  1008 , and represents the probability the site is the correct one. 
         [0070]    For the initial dwell time  1000  ( FIG. 10 ), the default is all sites equiprobable, so each site is given a likelihood inverse to the number of DOA vectors  907 . However, a priori knowledge of the emitter location can be incorporated in initializing the probabilities. Thus if certain regions  904  are deemed more probable, or are of heightened interest the weight initialization per step  914  can be adjusted. This adjustment will influence the emitter geolocation estimates  915  made from the first few subsequent phase measurements in successive dwell steps  1007  shown in  FIG. 10 . If accurate, it will accelerate convergence to the true emitter location. If wrong it will delay convergence, but not prevent it since after a transient the weight calculation forgets the initialization and depends only on current measurements and the processing of each individual estimator initialized in step  1006 . That the weights generated in Location Probability process  911  are a product of individual estimator processing step  1006  occurring in each associated member of the Location Estimation process  903  and not a driver of the processing or fed back to the processing is a fundamental aspect of the invention. 
         [0071]    After the satellite moves in its orbit to location  906 , a second receiver dwell time  1007  ( FIG. 10 ) occurs. The phase measurements are now again made in accordance with the process steps  902  and  903  to determine location probability  911 . A number of different coordinate systems may be used in making and processing the measurements to eventually generate the emitter location  915 . The use of no one particular system is essential to the invention, however, it has been found most convenient to use the well known earth centered inertial (ECI) coordinate system for storing the candidate emitter locations. Thus the emitter positions from the previous dwell are updated per step  1009  of  FIG. 10  to account for the earth&#39;s rotation between the dwells. 
         [0072]    After the satellite moves in its orbit  906 , a second receiver dwell occurs and the Process  902  is now repeated. Also in Process  902  new DOA vectors are generated for each site but in a way completely different than the method used in the first dwell time  1000 . Now DOA vectors are generated per step  1010  using each position vector from the site coordinates updated in step  1009  to the current satellite position  906  shown in  FIG. 9 . An example is a range vector  908  derived from previous estimated site  905 . This vector is normalized to produce a predicted DOA vector. A new ambiguity integer pair (m, n) is generated per step  1011  by forming the vector scalar product of the predicted DOA and each of the interferometer baselines. 
         [0073]    The measured phase pair is then resolved and the resolved phase is applied to an input to each corresponding location estimator process  903 . The site estimate is updated per step  1012  using a recursive filter and a new estimated site  909  ( FIG. 9 ) is generated. This estimated site differs from site  905  even though previous estimated site  905  was used to resolve the phase measurement ambiguity. Predicting the correct phase ambiguity integer does not require a close match between the predicted and resolved measured phase, and in fact even for the true site  901  the two typically are significantly offset. This aspect is an intrinsic element of the subject invention. The change in estimated position to emitter site  909  from site  905  reflects the difference. 
         [0074]    Thus the previous site estimates for an emitter E combined with the satellite position at the time of the phase measurements drive the ambiguity integer prediction. Hence the current phase measurement is resolved based on the previous location estimate, and this resolved phase drives the Probability Location Determination process  912  for each new emitter location update. This is a feed forward process from the old site estimate to the new one. The new site estimate is then returned per step  910  to the Measure and Resolve Phase process step  902  to predict the next ambiguity integer set. This is a feed back process from the current location estimate to the ambiguity resolution of the next phase input. The interaction of this feed-forward and feed-back process is a central aspect to the present invention. 
         [0075]    Note that because there is no phase tracking or need to follow integer roll-overs the measurement updates can be irregular and spaced far apart in time. In fact theoretically only two updates are required. But, to increase the accuracy of the probability determination  911 , more updates than this minimum are desirable. 
         [0076]    Also of particular note is the fact that Process  911  determines the emitter site probabilities by comparing results from the feed-forward and feedback processing. As part of the location estimation process in  903 , error variances are produced for the prior site update range vector  908 . These error variances are input  913  and used to bound the difference between the measured resolved phase  916  and the phase predicted by  908 . The signal to noise ratio (SNR) associated with the actual phase measurement is used in the bound and a probability then assigned to the site based on the statistical closeness of the predicted and measured phases. 
         [0077]    As noted there is no feedback from the probability estimation process  911  to the location process  903  or ambiguity resolution process  902 . Nor is there any cross processing in  1008  ( FIG. 10 ) determining the probability involving the other sites. After the probabilities are updated per step  1012 , Process  914  generates a unique emitter position estimate. There are two ways to do this: (a) weight each estimate with its probability and form a weighted average per step  1013 ; or, (b) pick the single emitter site having the highest probability per step  1014 . If enough receiver dwells are made, it is not critical which method is used since both will eventually converge to the same answer. In fact, convergence is defined to occur when it is determined both location estimates are the same to within statistical error at step  1015 . 
         [0078]    However, if the overlaps persist dwell to dwell, this set tends to be clustered around the true emitter position, so the weighted average provides an accurate estimate. Thus it is the weighted average that is reported after each update in step  1016 . Hence a significant improvement is provided in that an accurate emitter estimate can be provided without determining the correct DOA. i.e., the emitter E is located without completely resolving the array. 
         [0079]    Reference to the graphs shown in  FIGS. 11A-11D  illustrates this important behavior and shows the probability evolution for a 6 GHz emitter, for example, located 926 km from the satellite suborbital point. In this simulation phase measurements were made on a three element equilateral array with 45.7 cm baselines and 5 second updates. The phase error on each measurement had a 1σ value of 15 electrical degrees (edeg). Initially  38  of the array ambiguity integer pairs (m, n) produced DOAs intersecting the earth&#39;s surface. After three further updates, spaced five seconds apart, as shown in  FIGS. 11B ,  11 C and  11 D, the sites numbered  23 ,  25  and  31  in  FIG. 11D  all had significant probabilities. Note that the numbering of the sites does not necessarily indicate geographical closeness, but in this case all three sites were clustered comparatively close together. The weighted average over all the sites using the probabilities shown at the 15 second update produced an error of 8.6 km. 
         [0080]    This example was with a comparatively small array. Extending the baseline lengths or reducing the phase error would reduce the location error proportionally, as will now be shown in the following description of the preferred embodiment of the invention. 
         [0081]    Referring now to  FIG. 12 , shown thereat is a detailed block of the preferred embodiment of the subject invention. Antennas  1200   1 ,  1200   2  and  1200   3  are chosen to be responsive to a wide range of emitter polarizations across all frequencies of interest. Generally circularly polarized antennas are preferred because they are also responsive to linear and elliptical wave polarizations. However, such antennas are either left (L) or right (R) circularly polarized (CP). Typically one type of CP predominates in the emitters of interest, so this restriction is not a problem. But if it is, then baselines formed from both left (LCP) and right (RCP) antennas can be used. Since the dual polarization antennas are essentially colocated this will not usually create an installation difficulty. Switching between the antenna pairs, however, increases the time required to cycle through the band and update phase measurements. But the present invention supports an extended and irregular time between dwells, so this is not a problem. 
         [0082]    If dual polarization LCP and RCP antennas are required there is a way to generate the ambiguous DOA using only two antennas that are not co-boresited. This method, as previously noted, is described in U.S. Pat. No. 5,608,411, “Apparatus for Measuring a Spatial Angle to an Emitter Using Squinted Antennas” issued to the present inventor on Mar. 4, 1997. Implementing such an arrangement does not require special satellite attitude changes, and so is consistent with the operation of the subject invention, which is to use only translational motion to geolocate. Such a specialized implementation will not be pursued here. Instead three antennas  1200   1 ,  1200   2  and  1200   3  as shown in  FIG. 12  form two interferometer baselines  1206   1  and  1206   2 . If the three antennas  1200   1 ,  1202   2  and  1203   3  are placed at the vertices of an equilateral triangle as shown in  FIG. 12 , the array symmetry provides robust performance. 
         [0083]    A tolerance of extended and irregular phase measurement sample times is provided by the subject invention. Accordingly, a two channel receiver  1201  is used to make phase measurements in a single phase detector  1199 . This saves weight, power and cost. Switch  1231  determines the baseline  1206   1  or  1206   2  across which phase is measured. To obtain emitter DOA, phase must be measured across both baselines in a single receiver dwell. Calibrating out the phase mistrack between the channels CH 1  and CH 2  is essential. The phase mistrack between the two channels depends on emitter frequency, phase amplitude and ambient temperature. To reduce this mistrack error a calibration signal is injected via a CAL circuit  1202 , and the result of this calibration essentially provides a residual error having a fixed component no more than 5 edeg and varying part no more than 1.5 edeg. 
         [0084]    Phase measurements outputted from the phase detector  1230  on data line  1203  are time tagged via timing signals from a system clock  1209  on clock signal line  1204  and stored in a memory  1205 . Interferometer baseline vectors of the baselines  1206   1  and  1206   2  at the phase measurement times are also stored in a memory  1207 , where they are transformed from sensor to ECI coordinates using onboard navigation measurements from the NAV system  1208 . Time tags are provided by the precision system clock  1209  by way of clock signal line  1204 . This clock provides the equivalent of Universal Time or so called UT1 time since inaccuracies in the clock create effectively larger phase measurement errors. 
         [0085]    Using the stored phase measurements and baseline vectors from memories  1205  and  1207 , a set of all possible DOAs is generated in process step  1003  ( FIG. 10 ) by ambiguous DOA signal generator  1210 . First the field of view (FOV) limits are established by finding the maximum emitter-to-satellite angle at the satellite. Then the ambiguity integer pairs (m, n) are found. This can be done by simply modulating down d/λ, i.e. the ratio of baseline length to signal wave length. Doing this for both baselines and then forming the direct product of the two integer sets gives all possible integer pairs (m, n). The measured phase vector is resolved with each integer pair, and the corresponding DOA found. If the DOA is within the field of view limits it is retained, otherwise discarded. Special test are needed for DOA not intersecting the earth near the horizon to assure valid DOA are not eliminated because of measurement noise. 
         [0086]    The adjustment for emitters at the horizon takes place after correcting the resolved phase with antenna calibration data from a calibration table circuit  1211 . Calibration is used to reduce the fixed bias part of the antenna mistrack error. The antenna errors are DOA dependent, and so a different correction is provided for each hypothesized emitter position. The initial DOA estimate for an uncalibrated phase on signal line  1212  from the ambiguous DOA generator  1210  is used to find the DOA dependent error correction signal in signal line  1213  in the database. This is added to the resolved phase and the DOA recomputed in the ambiguous DOA generator  1210 . This iterative method is accurate enough to reduce the fixed error to about 0.5 edeg. Multipath from scatter off the satellite and refractive effects can be accounted for in this manner, as well as radome errors. But because the initial DOA input to the calibration table circuit  1211  table is not corrected the variable part of the error is still fairly large, about 2 edeg. However, since the process is dwell-to-dwell random, the impact can be substantially reduced in the recursive estimation that occurs in a signal filter  1216  which provides an update location to each site. 
         [0087]    The set of ambiguous DOAs appearing on an output lead  1214  from DOA generator  1210  are passed to the ambiguous location estimator  1215 . In this process the unit DOA vectors are extended from the satellite, for example, the satellite shown at location  806  in  FIG. 8C , using satellite position data from an ephemeris signal block  1217 , a table listing current and future positions of certain celestial objects, including the satellite, relative to the earth&#39;s surface. This data may have to be converted from the ephemeris coordinates, typically perifocal, to ECI. If so, this is also done in the processor  1215  The surface is typically modeled according to the well known WGS84 ellipsoid; however, when desirable, any suitable model can be used. In particular significant terrain elevations can be incorporated. Further test are also done here on emitters near the horizon by checking the angle between the normal to the earth&#39;s surface at the candidate emitter and line of sight to the satellite. If the angle is greater than 90°, but pulling the location toward the edge of the error bound closest to the satellite reduces the angle, the potential emitter site is retained. 
         [0088]    The candidate sites initialize a bank of recursive estimators or filters in the processor  1216  and are stored in a memory  1218 . These stored sites are used when outputted on data line  1219  at the next phase measurement to predict the corresponding set of ambiguity location integers in signal block  1215 , whereupon subsequent processing again occurs in the ambiguous DOA generator  1210 . But it is essentially different from the processing for the first phase measurement vector described above, and this difference is a critical aspect of the invention. The stored site for candidate location i, {right arrow over (R)} e     i   , is used with the satellite position at the time of the updated phase measurement, {right arrow over (r)} s  as shown by reference numerals  607  and  608  in  FIG. 6B , obtained from the ephemeris, to compute the predicted DOA unit vector {right arrow over (u)} i  according to the expression: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       ⇀ 
                     
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         [0000]    The phase φ i  is then predicted, using the interferometer baseline vector {right arrow over (d)} stored in memory  1207  at the time of the current phase measurement, by: 
         [0000]    
       
         
           
             
               
                 
                   
                     φ 
                     
                       pred 
                       i 
                     
                   
                   = 
                   
                     
                       
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         [0000]    The predicted phase φ predi  is then modulo&#39;d down to get the ambiguity integer. This is done for each baseline pair at each site {right arrow over (R)} e     i   . The resulting integer pairs generate a resolved phase φ i  for each site i. 
         [0089]    These resolved phases, not DOA estimates, outputted on line  1214  now drive the emitter update carried out in signal processors  1215  and  1216 . This method of generating resolved phase is used at each subsequent measurement update. The feed-forward and feedback process involved is clear from signals appearing on output signal lines  1214  and  2119  of  FIG. 12 . 
         [0090]    Although estimates are ultimately in ECI, the estimators in processor  1216  themselves are cycled in topocentric-horizon south-east-up or SEZ coordinates local to each hypothesized emitter position. This is done to avoid the singularity problems that arise if earth-centered inertial is solely used, and to more easily incorporate the oblate spheroidal earth flattening constraint. 
         [0091]    The SEZ coordinates are taken to have their origin at the initial location for each site, i.e. 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    and subsequent updates refine this estimate by ΔR e , where 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    Thus the elements of the filter state vector are the emitter south-offset Δs and east-offset Δe. The location state update model equation takes the simple form: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    The subscript k−1|k−1 denotes the estimate at phase sample k−1 after filter update with the phase measurement made at that time, while k|k−1 refers to extrapolation to the sample time at the k th  update from the sample time at the k−1 update. This is standard notation taught, for example, by Gelb in  Applied Estimation Theory , M.I.T Press, Cambridge 1974, but this notation may possibly be misleading. Because the generally variable sample time T is not incorporated in the notation it can be misconstrued as implying T is fixed and samples occur at regularly spaced times kT, with k=1, 2, . . . That is not the case in the subject invention, and the integer k simply refers to the update number in the sequence of measurements. 
         [0092]    Equation 3 is the phase measurement equation associated with location state model Equation 6. It is inputted on signal lead  1214  to processor  1215  where it is associated with a correct filter in Update Location Each Site processor  1216 . Because the measurement equation is a nonlinear function of the state elements it must be linearized to implement the filters, as explained in Gelb, supra. The linearization is not just a technical detail, but important in the computation of the Bayesian site probabilities in processor  1220 . So it is expressed here as: 
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         [0000]    where 
         [0000]    
       
         
           
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                             k 
                             ) 
                           
                         
                         - 
                         
                           
                             
                               R 
                               ⇀ 
                             
                             e 
                           
                            
                           
                             ( 
                             
                               k 
                               | 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                     T 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             r 
                             ⇀ 
                           
                           s 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                       - 
                       
                         ( 
                         
                           k 
                           | 
                           
                             k 
                             - 
                             1 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 ) 
               
               
                 1 
                 2 
               
             
           
         
       
     
         [0000]    is the estimated slant range,
 
with {right arrow over (r)} s (k) the observer&#39;s location at sample k and,
 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             c 
                             s 
                           
                         
                       
                       
                         
                           
                             c 
                             e 
                           
                         
                       
                       
                         
                           
                             c 
                             z 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         D 
                         T 
                       
                        
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 1 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       f 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       D 
                       · 
                       
                         
                           R 
                           e 
                         
                          
                         
                           ( 
                           
                             k 
                             | 
                             
                               k 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where f is the ellipsoid flattening constant. 
         [0093]    In equation (8), D the rotation from SEZ to earth fixed geocentric can be expressed as, 
         [0000]    
       
         
           
             
               
                 
                   D 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             lat 
                              
                             
                                 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             lon 
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             lon 
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             lat 
                              
                             
                                 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             lon 
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             late 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             lon 
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             lon 
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             lat 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             lon 
                           
                         
                       
                       
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             lat 
                           
                         
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             lat 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with lat and lon being the latitude and longitude corresponding to R e (k|k−1). The vectors {right arrow over (i)} 1  and {right arrow over (i)} 2  are the unit vectors along the interferometer baselines at sample k stored in memory  1207 . Each is analogous to {right arrow over (k)}, shown in  FIG. 2  by reference number  217 . The computation of these unit vectors is done in ambiguous DOA generators  1210  and input to the estimators via signal line  1214 . 
         [0094]    The phase measurement, 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       ϕ 
                       1 
                     
                   
                 
                 
                   
                     
                       ϕ 
                       2 
                     
                   
                 
               
               ] 
             
             k 
           
         
       
     
         [0000]    input to the filter is normalized by 2π/λ. Hence the notation is modified from the φ used to denote the unnormalized phase. The subscript in the phase errors: 
         [0000]    
       
         
           
             [ 
             
               
                 
                   
                     ɛ 
                     
                       d 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         aoa 
                         1 
                       
                     
                   
                 
               
               
                 
                   
                     ɛ 
                     
                       d 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         aoa 
                         2 
                       
                     
                   
                 
               
             
             ] 
           
         
       
     
         [0000]    also reflects this normalization. 
         [0095]    After cycling the filters for each site, the updated SEZ locations are input to processor  1215  where they are converted to ECI, and stored in memory  1218 . These are not tested for the horizon constraint. An emitter that is beyond the horizon will produce a small probability in processor  1220  and be essentially neglected when performing the weighted average in Weighted Location Estimate processor  1221 . 
         [0096]    The probability determination in  1220  uses results from the SEZ iterative filters in  1216 , particularly the predicted phase and statistics of the phase noise. It also uses the measured phase from memory  1205 , passed through process  1216 . For convenience assume the phase error to have the same channel-to-channel variance σ φ   2 , determined from the signal SNR. With this simplifying assumption the phase errors from phase detector  1230  have a variance: 
         [0000]    
       
         
           
             
               
                 
                   
                     Λ 
                     k 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             .5 
                           
                         
                         
                           
                             .5 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
                         
                           σ 
                           φ 
                           2 
                         
                          
                         
                           ( 
                           
                             λ 
                             
                               2 
                                
                               π 
                             
                           
                           ) 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the quantity, 
         [0000]      Σ={right arrow over (φ)} meas −{right arrow over (φ)} pred     i     (11) 
         [0000]    has the theoretical variance of: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       
                         〈 
                         
                           Σ 
                           2 
                         
                         〉 
                       
                       i 
                     
                   
                   = 
                   
                     
                       
                         H 
                         k 
                         T 
                       
                        
                       E 
                        
                       
                         〈 
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         s 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         e 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 | 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 e 
                               
                               ] 
                             
                           
                           
                             
                               k 
                               - 
                               1 
                             
                             | 
                             
                               k 
                               - 
                               1 
                             
                           
                         
                         〉 
                       
                        
                       
                         H 
                         k 
                       
                     
                     + 
                     
                       Λ 
                       k 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    based on the estimator performance for the corresponding site in processor  1216 . This variance estimate is passed on signal lead  1222  to Compute Bayesian Probabilities processor  1220 . The predicted phase {right arrow over (φ)} pred  is applied on input data line  1223  from the DOA processor  1210 , and measured phase {right arrow over (φ)} meas  is applied on input data line  1224  from stored phase memory  1205 . 
         [0097]    Processor  1220  then generates for each site the Bayesian probability that the site is correct by comparing the actual measured offset (Equation 11) to the theoretical variance of the offset obtained by Equation 12, i.e., it generates the ratio: 
         [0000]    
       
         
           
             
               
                 
                   
                     ϒ 
                     i 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               ϕ 
                               → 
                             
                             meas 
                           
                           - 
                           
                             
                               ϕ 
                               → 
                             
                             
                               pred 
                               i 
                             
                           
                         
                         ) 
                       
                       T 
                     
                      
                     E 
                      
                     
                       
                         〈 
                         
                           Σ 
                           2 
                         
                         〉 
                       
                       i 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             ϕ 
                             → 
                           
                           meas 
                         
                         - 
                         
                           
                             ϕ 
                             → 
                           
                           
                             pred 
                             i 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and applying Bayes Rule, computes the probability for the site recursively by the expression: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       prob 
                        
                       
                         ( 
                         
                           site 
                           i 
                         
                         ) 
                       
                     
                     k 
                   
                   = 
                   
                     E 
                      
                     
                       
                         〈 
                         
                           Σ 
                           2 
                         
                         〉 
                       
                       i 
                       
                         - 
                         1 
                       
                     
                      
                     
                        
                       
                         - 
                         
                           ϒ 
                           2 
                         
                       
                     
                      
                     
                       
                         prob 
                          
                         
                           ( 
                           
                             site 
                             i 
                           
                           ) 
                         
                       
                       
                         k 
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The probabilities computed at update k are stored in memory  1218  via data line  1225  for use in the next recursion. The set of all updated probabilities is passed to Find Weighted Location Estimate calculation means  1221  and Find Maximum Likelihood Location Estimate calculation means  1226 . 
         [0098]    The ratio Υ i  (Equation 13) measures the accuracy of the phase prediction φ predi  (Equation 3). The theoretical variance values generated in accordance with Equation 12 assume the ambiguity resolution is perfect. If a significantly wrong site provides the DOA vector projected onto the interferometer baseline to predict the phase and hence ambiguity integer, Υ i  will be large, and the recursion computed per Equation 14 will reduce the probability at each update. 
         [0099]    But predicting the correct ambiguity integer does not require precise emitter location. Therefore, at an emitter frequency where the satellite baseline lengths generate many ambiguous sites, several sites near the correct location will predict the correct ambiguity pair over successive updates. So a second important property of the ratio Υ i  is that it acts like a vernier, measuring the fidelity of the site location given the correctly resolved phase. 
         [0100]    Because the probabilities will vary over the correctly resolved sites according to their update-to-update performance, this vernier property ensures that doing a weighted average in processor  1221  of  FIG. 12  gives an accurate emitter estimate before the array is resolved. 
         [0101]    Thus the dual behavior of the probability calculation, in essence weighing sites according to both an error in ambiguity resolution, or, if that is correct, in site initialization, is essential to the subject invention. In particular it allows the use of calibrated but unresolved baselines to generate relatively high location accuracy even at high frequencies because the many spurious sites near the true site of the emitter will not significantly degrade the final weighted location estimate provided by processor  1221 . 
         [0102]    Processor  1226  determines the maximum probability and outputs an estimate computed in processor  1216  for the corresponding site. This is compared in processor  1228  with the weighted estimate calculated in estimator  1221 , and if the two are statistically close, then a single site is applied on signal line  1229  for use in all ensuing updates, for example, in processors  1215  and  1216 . The estimation collapses from a bank of filters to the single filter for that site. This greatly aids throughput processing in a dense emitter environment. It is thus important to take advantage of the statistical equivalence of the two estimates when it occurs, but this feature is not intrinsically required by the subject invention. The comparison is performed in processor  1228  because this process also transforms the estimates from ECI to geodetic coordinates for the signal output appearing on signal lead  1227  of  FIG. 12 . 
         [0103]    As noted above, a significant benefit of the subject invention is that it provides accurate geolocation before the comparison in processor  1228  determines the array is essentially resolved. This is vital because noncooperative emitters may not remain on, or may not be in the array&#39;s field of view (FOV) long enough to resolve the array. Therefore it is essential that an embodiment of the subject invention include this constraint. The following example illustrates the preferred approach in the system design process. The following example further illustrates how the subject invention can achieve the performance of a fully resolved planar interferometer array with only three antennas. A significant benefit is obtained when the design is carried out iteratively. 
         [0104]    The first step in this iterative process is to determine the most difficult transmitters to locate for a particular application. These transmitters are the set formed by those at the lowest frequency, shortest transmit times, and farthest from the satellite suborbital point. This set is referred to as the design determining set, or DDS. The number of transmitters that must be detected and geolocated, called the emitter density, establishes the average emitter revisit rate or expected interval between phase measurement sample times for the same emitter by the receiver  1201  shown in  FIG. 12 . For typical densities, the sample times can be on the order of 1 to 5 seconds. The emitters with the shortest transmit times then determine the smallest number of iterations available to cycle the estimator filter block  1216 . For example, if the lower bound for on-times is 10 seconds, then a sample rate of 0.5 Hz means  5  measurement samples must be available, or equivalently  5  location filter iterations must occur. Using this information in combination with data on the lowest frequency and farthest transmitters allows a hypothetical design of a conventional interferometer array to solve the geolocation problem. 
         [0105]    The best method for designing this hypothetical array is to utilize Malloy&#39;s optimal approach. Such an array will not be implemented; however, the predicted location performance against the DDS emitters using a single estimator and a fully resolved array provides the basis in the subject invention for placing antennas  1200   1 - 1200   3  in a three element ambiguous array along with the requirement of determining the receiver calibration and array calibration with calibration elements  1202  and  1211  as shown in  FIG. 12 . 
         [0106]    This is achieved as follows. The three antenna elements  1200   1 ,  1200   2  and  1200   3  of the ambiguous array are first taken as a subset of an optimal array. Predicted geolocation performance against the design driving set is thus generated. In doing this the implementation is constrained to use the same number of iterations against each emitter used by the optimal array. From the slant range accuracies thus produced, and the corresponding results for the optimal resolved array, a ratio is computed for each DDS emitter. This ratio is the ambiguous array slant range error divided by the resolved array slant range error. 
         [0107]    Using the largest of these ratios, the baselines for the three element array baselines are increased. This scaling now takes into account viable antenna placement on the satellite. Ideally the baselines will be scaled according to the largest ratio. But typically other installations on the satellite&#39;s surface do not allow this. So the calibration procedures carried with the calibration elements  1202  and  1211  must be enhanced to proportionally compensate. The result of this step is a system producing location estimates from an unresolved array that performs the same as a conventional installation using a fully resolved interferometer. 
         [0108]    Table I and Table II show results using the design method described above for the simple case of a single 6 GHz emitter and where the only trade-off is performance at various ranges. Table I summarizes the results of the first step of generating DDS ratios. 
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 Ambiguous Interferometer Baseline Length: 45.7 cm 
               
               
                 Ambiguous Array Phase error RMS Standard Deviation: 15.3° 
               
               
                 Resolved Interferometer Baseline Length: 45.7 cm 
               
               
                 Resolved Array Phase error RMS Standard Deviation: 15.3° 
               
               
                 Frequency 6 GHz 
               
             
          
           
               
                 Range from Sub 
                 Resolved Array 
                 Ambiguous Array 
                 DDS 
               
               
                 Orbital Point (nmi) 
                 Performance (km) 
                 Performance (km) 
                 Scale Factors 
               
               
                   
               
             
          
           
               
                 400 
                 1.6 
                 4.3 
                 2.7 
               
               
                 800 
                 3.4 
                 9.5 
                 2.8 
               
               
                 1200 
                 3.8 
                 13.1 
                 3.4 
               
               
                   
               
             
          
         
       
     
         [0109]    Note the scale factors are not the same for all ranges. Suppose the transmitter at 1200 km is the most important. Then the location performance of the ambiguous array lags that of the hypothetical optimal array by a factor of 3.4. The difference in performance is due to the ambiguous array not having enough receiver dwells, and hence location iterations  1216  to converge to a single emitter in processor  1228 . So the difference is due to “smearing” in the weighted average estimate found in computation block  1221  compared to the optimal array&#39;s single site estimate. However, performance can be adjusted by a combination of proportionally increasing the baseline lengths and decreasing the phase error. 
         [0110]    In this example, assume adequate space exist on the satellite surface to double the baseline lengths. This will not reduce the slant range error to the value desired so the phase error must also be reduced. The 15.3 edeg error assumes only very rudimentary antenna calibration in processor  1211 . If this calibration is enhanced by generating a larger cal table during the installation process, a phase error standard deviation of 10.1° can be achieved. This gives the result shown in Table II. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE II 
               
             
             
               
                   
               
               
                 Ambiguous Interferometer Baseline Length: 92 cm 
               
               
                 Ambiguous Array Phase error RMS Standard Deviation: 10.1° 
               
               
                 Resolved Interferometer Baseline Length: 45.7 cm 
               
               
                 Resolved Array Phase error RMS Standard Deviation: 15.3° 
               
               
                 Frequency 6 GHz 
               
             
          
           
               
                 Range from Sub 
                 Resolved Array 
                 Ambiguous Array 
               
               
                 Orbital Point (nmi) 
                 Performance (km) 
                 Performance (km) 
               
               
                   
               
             
          
           
               
                 400 
                 1.6 
                 0.67 
               
               
                 800 
                 3.4 
                 3.4 
               
               
                 1200 
                 3.8 
                 4.1 
               
               
                   
               
             
          
         
       
     
         [0111]    Although a simple example is shown, it should be clear from this how flexibility in antenna placement combined with well established calibration and installation techniques in the subject invention allow the geolocation of emitters to whatever practical accuracy is desired. 
         [0112]    The foregoing detailed description merely illustrates the principles of the invention. It will thus be appreciated that those skilled in the art will be able to devise various arrangements which, although not explicitly described or shown herein, embody the principles of the invention and are thus within its spirit and scope.