Abstract:
A method is disclosed for estimating the location of a transmitter based on signals transmitted therefrom. The transmitted signals are received at a plurality of reception locations and are employed to compile data including a plurality of entries, where each entry includes a received signal level (RSL) of a received signal, a channel identification of the received signal, and position information corresponding to the respective reception location when the received signal was received. The entries are organized into groups according to channel identification, and, in each group, entries having the same position information are consolidated. Thereafter, for each entry, the distance DRT between the reception location and the transmitter is estimated, and the distance DRR between the reception location thereof and an adjacent reception location is also estimated. Then, for each group, the location of the transmitter corresponding to the channel identification of such group is estimated based on the distances DRT and DRR.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 60/191,269, filed Mar. 22, 2000 and entitled “Location Estimation of One or More Transmitters Based On Signals Transmitted Therefrom”, hereby incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to a system and method that estimates the location of a single or multiple transmitters. More particularly, the present invention relates to such an estimation method that uses measured data such as Received Signal Strength Indication (RSSI) data collected by a scanning and tracking receiver such as that which may be commercially available. 
     BACKGROUND OF THE INVENTION 
     The non-invasive estimation of transmitter location and configuration based on signals received from such transmitter(s) is considered to be highly useful in many circumstances. For example, an investment banker or the like may employ such information for preliminary asset determination among other things; a tower company or the like may employ such information for development of a business plan and guidance for site acquisition personnel among other things; a competitor wireless service provider or the like may employ such information to assess the overall system layout of other service providers in a particular market among other things; and a consultant or the like may employ such information to provide value added services to improve overall system performance to a client among other things. 
     One typical methodology that may be employed to non-invasively estimate the location of a particular transmitter is to measure signal strengths of signals received from such transmitter at several reception locations, to estimate distance based on such measured signal strengths, and then to triangulate based on such estimated distances. However, and importantly, many factors affect signal strength as measured at a particular reception location, including but not limited to transmission power, terrain, atmospheric conditions, interference, line of sight issues, etc. Thus, the distances estimated from such measured signal strengths have relatively large margins of error, and accordingly triangulation calculations based on such estimated distances have relatively large margins of error, too. Ultimately, the margin of error of such triangulation calculations is so significant as to render the result of the calculations highly suspect if not meaningless. 
     Accordingly, a need exists for a system and methodology for non-invasively estimating transmitter location and configuration information, whereby the margin of error of the result of the methodology is within a more acceptable range. 
     SUMMARY OF THE INVENTION 
     The present invention satisfies the aforementioned need by providing a method for estimating the location of a transmitter based on signals transmitted therefrom. The transmitted signals are received at a plurality of reception locations and are employed to compile data including a plurality of entries, where each entry includes a received signal level (RSL) of a received signal, a channel identification of the received signal, and position information corresponding to the respective reception location when the received signal was received. 
     In the method, the entries are organized into groups according to channel identification, and, in each group, entries having the same position information are consolidated. Thereafter, for each entry, the distance DRT between the reception location and the transmitter is estimated, and the distance DRR between the reception location thereof and an adjacent reception location is also estimated. Then, for each group, the location of the transmitter corresponding to the channel identification of such group is estimated based on the distances DRT and DRR. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing summary as well as the following detailed description of the present invention will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating the invention, there are shown in the drawings embodiments which are presently preferred. As should be understood, however, the invention is not limited to the precise arrangements and instrumentalities shown. In the drawings: 
     FIG. 1 is a data collection system employed in accordance with one embodiment of the present invention; 
     FIG. 2 is a diagram showing data points (P 1 , P 2 ) at which data was collected and possible transmitter locations (P 3 , P 3 ′) represented by the data; 
     FIG. 3 is a diagram employed to show how an iterative approach are used to solve for four isosceles triangles using distances derived from FIG. 2 in accordance with one embodiment of the present invention; 
     FIG. 4 is a diagram employed to show how transmitter locations are estimated from data points in accordance with one embodiment of the present invention; and 
     FIG. 5 is a flow diagram detailing an algorithm performed in accordance with one embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Certain terminology may be used in the following description for convenience only and is not considered to be limiting. For example, the words “left”, “right”, “upper”, and “lower” designate directions in the drawings to which reference is made. Likewise, the words “inwardly” and “outwardly” are directions toward and away from, respectively, the geometric center of the referenced object. The terminology includes the words above specifically mentioned, derivatives thereof, and words of similar import. 
     In one embodiment of the present invention, a system and method are employed to estimate the location of a transmitter through an iterative application of received signal strength data (such as RSSI data), slow fading characteristics and geometric triangulation. In particular, and referring now to the drawings in detail, wherein like numerals are used to indicate like elements throughout, there is shown in FIG. 1 a data collection system  10  in accordance with one embodiment of the present invention. As seen, the system  10  includes a The essential elements of the equipment are a tunable receiver  12  with an appropriate antenna  12   a  for receiving RF transmissions (or the like) from one or more transmitters, a GPS (Global Positioning System) positioning device  14  with an appropriate antenna  14   a  for determining the global position of the system  10  at the time of each collection of data, and a recording device such as a computer  16  for recording the collected data from the receiver  12  and the corresponding positioning information from the GPS device  14 . Thus, the system  10  is employed to produce a data file containing a plurality of entries, where each entry includes a Received Signal Level (RSL) (perhaps expressed in dBm) as obtained from the receiver  12 , channel number as obtained from the receiver  12 , and corresponding GPS coordinates as obtained from the GPS device  14 , where all the data in the entries is collected from/in connection with an operating transmitter network with unknown transmitter locations. 
     Each of the aforementioned elements  12 ,  14 ,  16  of the system  10  may be any appropriate element without departing from the spirit and scope of the present invention, and may be appropriately configured in any particular arrangement, again without departing from the spirit and scope of the present invention. Moreover, the computer/recording device  16  may be running any appropriate software once again without departing from the spirit and scope of the present invention. The details and specifics of such elements, configuration, and software are known or should be apparent to the relevant public, and therefore need not be described further herein. 
     OVERVIEW 
     With the system  10  as shown in FIG. 1, the following method is employed in one embodiment of the present invention to estimate transmitter location. Briefly, in the method, the slow fading characteristics of the propagation environment are characterized by {P 1 , m} where P 1  represents the one mile intercept, expressed in dBm, and m is the decay slope, expressed in dB per decade. Under these conditions, the RSL from each entry of the aforementioned data file and the distance from the associated transmitter are directly related. Applying triangulation and this relationship, an estimate of the transmitter location may be obtained. 
     The slow fading characteristics of an RF signal in the land mobile environment may be expressed as L P ≈1/r n , where L P  represents the RF path loss, r represents the distance, and n is the slow fading decay constant for a specific morphology group. Expressed in the logarithmic domain, 1/r n  becomes −10 nlog(r). The decay slope, m, is equal to −10 n. The concept of the 1-mile intercept, P 1 , allows for convenient calculations in the logarithmic domain. An added benefit is the distance is great enough to ensure the main lobe of the antenna is considered thereby stabilizing the calculations. 
     The above concepts were applied with great success by W. C. Y. Lee in the equation:              RSL   =                P1   +     10        log        (       P   t       P   ref       )         +     m                   log        (   r   )         +                                α                 log        (       h   t       h   ref       )       +     β                   log        (       h   m       h   ref       )         +   APE   +   KED                                  
     where: 
     RSL=Received Signal Level (dBm) 
     P 1 =1 mile intercept (dBm) 
     P t =Effective Radiated Power (ERP) of the site 
     P ref =Reference ERP 
     m=slow fading decay slope (dB/decade) 
     r=distance 
     α=coefficient applied to transmitter height adjustment 
     h t =height above ground level of the base station transmit antenna 
     h ref =reference height of the base station transmit antenna 
     β=coefficient applied to receiver height adjustment 
     h m =height above ground level of the receiver (mobile) antenna 
     h ref =reference height of the receiver antenna 
     APE=antenna pattern effects (vertical and horizontal beamwidths) 
     KED=Knife Edge Diffraction 
     The primary contributors are RSL, P 1 , 10 log(P t /P ref ), mlog(r), and αlog(h t /h ref ). The remaining terms are either unknown or have little impact on the result. RSL is obtained from the collected data. P 1 , P t , m, α, and h t  are assumed. The distance r is to be calculated. The final equation to be used becomes:        r   =     10   ^     [       1   m          (     RSL   -   P1   -     10        log        (       P   t       P   ref       )         -     αlog        (       h   t       h   ref       )         )       ]                              
     Referring now to FIG. 2, it is seen that P 1  and P 2  represent the location of two measured data points given by: 
     
       
         P 1 : Lat 1 , Lon 1 , RSL 1 , Chan 1   
       
     
     
       
         P 2 : Lat 2 , Lon 2 , RSL 2 , Chan 1   
       
     
     as derived from the aforementioned data file. Note here that Chan 1  was measured at P 1  and P 2 . P 3  represents the estimated location of the transmitter from which Chan 1  originated. P 3 ′ represents the mirror solution. D 12  is the distance between the two data points and may be calculated with a reasonable degree of accuracy. D 13  and D 23  are the respective distances from the data point to the transmitter location and may be estimated by applying an assumed set of propagation parameters, including the slow fading characteristics of the environment. However, since D 13  and D 23  are estimated, the direct use of D 12 , D 13 , and D 23  may not form a triangle that can be easily solved using the Law of Cosines. 
     In one embodiment of the present invention, then, and as shown in FIG. 3, an iterative approach of solving for four isosceles triangles using the three distances is implemented, thereby yielding 4 possible solutions (P 3   a,  P 3   b,  P 3   c,  P 3   d ). The coordinates of such isosceles triangles may be readily solved using the Law of Cosines. Performing the above procedure on N data points will yield 2N possible solutions. Such solutions are then divided into two groups, G 1  and G 2 , where G 1 : {P 3   a,  P 3   b } and G 2 : {P 3   c,  P 3   d}.  The arithmetic mean of each group is then calculated, and the average with the smallest variance is chosen to yield the estimated most likely location of the transmitter. 
     IMPLEMENTATION 
     In a transmitter network of a system such as a wireless communication systems, multiple channels are typically used multiple times (Frequency Reuse). In many cases, multiple directional antennas, each with its own channel, are assigned to an individual location (Sectorization). Also, the sampling rate of the scanning receiver is often greater than the rate of update of the GPS receiver, thereby producing multiple measurements at the same recorded location (Duplicate Coordinates). Therefore, the present invention must account for each of these real world conditions. 
     Duplicate Coordinates may be handled by either discarding subsequent entries or averaging them. Discarding subsequent entries is computationally more efficient than averaging but imposes significant risk of increased error due to limiting the information about the environment to a single data point. Averaging provides a mechanism to which more information about the environment is used, thereby decreasing the overall error imposed in the process. The computational cost of averaging is considerable small compared to the increased confidence associated with using all of the information. Therefore, averaging the RSL for duplicate coordinates is implemented in the present invention. 
     Frequency reuse is the retransmission of a given frequency, typically designated by a channel number, at multiple locations in a manner that introduces tolerable levels of interference into the communication path. In high capacity systems, reuse distances may be as small as two to three miles. An evaluation of the relative distances between successive coordinates of the same channel may be used to detect reuse. The limitation of this approach is that there is a risk of assigning a reuse characteristic to channels transmitted from a location with a discontinuous cover region. However, the solution for discontinuous coverage regions should converge to a relatively small area and may be treated as a sectored site. 
     Sectorization is a generic term that implies a location has multiple antennas transmitting unique frequencies. The solutions obtained for multiple channels from a single location should converge in a relatively small region. All solutions within a user-defined region will be attributed to a single location and assigned to the arithmetic average of all solutions within the region. This approach assumes the individual solutions are reasonably correct and will account for sites with discontinuous coverage areas. 
     ALGORITHM AND EXAMPLE 
     In one embodiment of the present invention, an algorithm such as that detailed in connection with the example set forth below is employed to perform transmitter location estimation for one or more transmitters. Of course, many variations may be made to the algorithm listed below without departing from the spirit and scope of the present invention. Note that each step of the algorithm is also shown in FIG.  5 . 
     Consider Table 1, as follows, which is derived from a sample data file containing the Latitude, Longitude, Channel, and RSL information therein: 
     
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Meas. 
                 Latitude 
                 Longitude 
                 Channel 
                 RSL 
                 Ref. Chan. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                  1 
                 45.51988 
                 −122.563 
                 857 
                 −53 
                 1 
               
               
                  2 
                 45.51988 
                 −122.563 
                 857 
                 −52 
                 1 
               
               
                  3 
                 45.51988 
                 −122.563 
                 857 
                 −52 
                 1 
               
               
                  4 
                 45.51988 
                 −122.563 
                 857 
                 −52 
                 1 
               
               
                  5 
                 45.51988 
                 −122.563 
                 101 
                 −48 
                 2 
               
               
                  6 
                 45.51988 
                 −122.563 
                 101 
                 −48 
                 2 
               
               
                  7 
                 45.51988 
                 −122.563 
                 101 
                 −48 
                 2 
               
               
                  8 
                 45.51988 
                 −122.563 
                 101 
                 −52 
                 2 
               
               
                  9 
                 45.51966 
                 −122.563 
                 101 
                 −44 
                 2 
               
               
                 10 
                 45.51943 
                 −122.563 
                 101 
                 −47 
                 2 
               
               
                 11 
                 45.51919 
                 −122.563 
                 101 
                 −50 
                 2 
               
               
                 12 
                 45.5191 
                 −122.563 
                 101 
                 −49 
                 2 
               
               
                 13 
                 45.51913 
                 −122.563 
                 857 
                 −48 
                 3 
               
               
                 14 
                 45.51994 
                 −122.564 
                 857 
                 −49 
                 3 
               
               
                 15 
                 45.52127 
                 −122.564 
                 857 
                 −52 
                 3 
               
               
                 16 
                 45.52284 
                 −122.564 
                 857 
                 −51 
                 3 
               
               
                 17 
                 45.5246 
                 −122.564 
                 857 
                 −44 
                 3 
               
               
                 18 
                 45.52569 
                 −122.564 
                 857 
                 −41 
                 3 
               
               
                 19 
                 45.52595 
                 −122.564 
                 857 
                 −39 
                 3 
               
               
                 20 
                 45.52595 
                 −122.564 
                 857 
                 −41 
                 3 
               
               
                 21 
                 45.52595 
                 −122.564 
                 857 
                 −41 
                 3 
               
               
                 22 
                 45.52595 
                 −122.564 
                 857 
                 −41 
                 3 
               
               
                 23 
                 45.52618 
                 −122.564 
                 857 
                 −36 
                 3 
               
               
                 24 
                 45.52689 
                 −122.564 
                 857 
                 −41 
                 3 
               
               
                 25 
                 45.52827 
                 −122.564 
                 857 
                 −40 
                 3 
               
               
                 26 
                 45.5296 
                 −122.564 
                 857 
                 −45 
                 3 
               
               
                 27 
                 45.53179 
                 −122.564 
                 857 
                 −52 
                 3 
               
               
                 28 
                 45.53352 
                 −122.564 
                 857 
                 −59 
                 3 
               
               
                 29 
                 45.53509 
                 −122.563 
                 857 
                 −62 
                 3 
               
               
                 30 
                 45.53653 
                 −122.562 
                 857 
                 −70 
                 3 
               
               
                 31 
                 45.53837 
                 −122.562 
                 857 
                 −68 
                 3 
               
               
                 32 
                 45.53998 
                 −122.561 
                 857 
                 −64 
                 3 
               
               
                 33 
                 45.54144 
                 −122.561 
                 857 
                 −63 
                 3 
               
               
                 34 
                 45.54307 
                 −122.561 
                 857 
                 −62 
                 3 
               
               
                 35 
                 45.54483 
                 −122.56 
                 857 
                 −70 
                 3 
               
               
                 36 
                 45.54627 
                 −122.559 
                 857 
                 −73 
                 3 
               
               
                   
               
             
          
         
       
     
     1. Assign Reference Channels (RefChan) to raw data: 
     As seen in Table 1, Measurements 1 through 4 are all from channel  857  and are assigned Reference Channel  1 ; Measurements 5 through 12 are all from channel  101  and are assigned Reference Channel  2 ; and Measurements 13 through 36 are all from channel  857  and are assigned Reference Channel 3. 
     2. Consolidate the Raw Data to remove identical coordinates for the same Reference Channel: 
     For each group of data entries having a common assigned Reference Channel, the average RSL is calculated as:            x   _     =       1   N            ∑     i   =   1     N          x   i           ,                          
     where {overscore (x)} is the average RSL, N is the number of measurements with identical Reference Channel and GPS Coordinates, x i  is the individual measurement. The standard deviation, sigma (σ), is given by:        σ   =       1     N   -   1              ∑     i   =   1     N            (       x   i   2     -     x     -   2         )     .                                
     If there is only one sample, the standard deviation is theoretically infinity but is determined to be 0 for practical purposes. The standard deviation as calculated here is not further employed in the algorithm of the present invention, but does aid in determining the reliability or ‘spread’ of the RSL data. The consolidated raw data with the average RSL and standard deviation appears in Table 2, below. 
     3. Estimate the distance from the transmitter for each consolidated data entry by applying a set of assumed reference conditions: 
     The following reference conditions are employed in connection with the present example, and are typical of an 850 MHz system operating in a suburban environment. Of course, other reference conditions may be employed without departing from the spirit and scope of the present invention. 
     P t =100 Watts 
     P ref =100 Watts 
     P 1 =−59 dBm 
     m=38.4 dB/decade 
     h t =130′ 
     h ref =150′ 
     α=15 
     Applying the above conditions to the consolidated data file using the equation set forth above for r yields the results given in Table 2, as follows: 
     
       
         
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                   
                   
                   
                   
                 Ave. 
                   
                   
               
               
                   
                   
                   
                   
                 RSL 
                   
                   
               
               
                 RefChan 
                 # Meas. 
                 Latitude 
                 Longitude 
                 (dBm) 
                 sigma 
                 r (miles) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 1 
                 4 
                 45.5199 
                 −122.563 
                 −52.25 
                   0.5 
                 0.63087 
               
               
                 2 
                 4 
                 45.5199 
                 −122.563 
                 −49 
                 2 
                 0.51917 
               
               
                 2 
                 1 
                 45.5197 
                 −122.563 
                 −44 
                 0 
                 0.38468 
               
               
                 2 
                 1 
                 45.5194 
                 −122.563 
                 −47 
                 0 
                 0.46049 
               
               
                 2 
                 1 
                 45.5192 
                 −122.563 
                 −50 
                 0 
                 0.55125 
               
               
                 2 
                 1 
                 45.5191 
                 −122.563 
                 −49 
                 0 
                 0.51917 
               
               
                 3 
                 1 
                 45.5191 
                 −122.563 
                 −48 
                 0 
                 0.48895 
               
               
                 3 
                 1 
                 45.5199 
                 −122.564 
                 −49 
                 0 
                 0.51917 
               
               
                 3 
                 1 
                 45.5213 
                 −122.564 
                 −52 
                 0 
                 0.62149 
               
               
                 3 
                 1 
                 45.5228 
                 −122.564 
                 −51 
                 0 
                 0.58532 
               
               
                 3 
                 1 
                 45.5246 
                 −122.564 
                 −44 
                 0 
                 0.38468 
               
               
                 3 
                 1 
                 45.5257 
                 −122.564 
                 −41 
                 0 
                 0.32135 
               
               
                 3 
                 4 
                 45.526 
                 −122.564 
                 −40.5 
                 1 
                 0.31185 
               
               
                 3 
                 1 
                 45.5262 
                 −122.564 
                 −36 
                 0 
                 0.2381 
               
               
                 3 
                 1 
                 45.5269 
                 −122.564 
                 −41 
                 0 
                 0.32135 
               
               
                 3 
                 1 
                 45.5283 
                 −122.564 
                 −40 
                 0 
                 0.30264 
               
               
                 3 
                 1 
                 45.5296 
                 −122.564 
                 −45 
                 0 
                 0.40845 
               
               
                 3 
                 1 
                 45.5318 
                 −122.564 
                 −52 
                 0 
                 0.62149 
               
               
                 3 
                 1 
                 45.5335 
                 −122.564 
                 −59 
                 0 
                 0.94563 
               
               
                 3 
                 1 
                 45.5351 
                 −122.563 
                 −62 
                 0 
                 1.13201 
               
               
                 3 
                 1 
                 45.5365 
                 −122.562 
                 −70 
                 0 
                 1.82887 
               
               
                 3 
                 1 
                 45.5384 
                 −122.562 
                 −68 
                 0 
                 1.62218 
               
               
                 3 
                 1 
                 45.54 
                 −122.561 
                 −64 
                 0 
                 1.27624 
               
               
                 3 
                 1 
                 45.5414 
                 −122.561 
                 −63 
                 0 
                 1.20196 
               
               
                 3 
                 1 
                 45.5431 
                 −122.561 
                 −62 
                 0 
                 1.13201 
               
               
                 3 
                 1 
                 45.5448 
                 −122.56 
                 −70 
                 0 
                 1.82887 
               
               
                 3 
                 1 
                 45.5463 
                 −122.559 
                 −73 
                 0 
                 2.18931 
               
               
                   
               
             
          
         
       
     
     4. Estimate distances between measurements for same Reference Channel: 
     The triangulation procedure requires at least two consolidated measurements with the same reference channel. Since Reference Channel  1  has only one consolidated measurement, no estimate can be made. However, estimates may be made for Reference Channels  2  and  3 . 
     First, the distance between measurements of the same reference channel must be determined. The constants of interest are: 
     Polar circumference of the earth (C Pol )=2*π*3949.9028 miles=817.9712377 miles 
     Equatorial circumference of the earth (C Equ )=2*π*3963.1906 miles=901.4609475 miles 
     Latitudinal Distance—consider two lines of latitude (La 1 , La 2 ). Since the lines of latitude are equally spaced, the latitudinal distance may be easily calculated by:          D   Lat     =         C   Pol     *          La1   -   La2            360                            
     Longitudinal Distance—consider two lines of longitude (Lo 1 , Lo 2 ). Since the distance between lines of longitude are dependant on the corresponding latitudes, longitudinal distance is calculated by:          D   Lon     =           C   Pol     *          Lo1   -   Lo2            360            cos   deg          (       La1   +   La2     2     )                                
     Final Distance—the final distance is given by the hypotenuse of the right triangle formed by D Lat  and D Lon  and is calculated using the Pythagorean Theorem as follows. 
     
       
           D =( D   Lat   2   +D   Lon   2 ) ½   
       
     
     Applying the above equations to the data in Table 2 yields the data in Table 3, below. Note that since each D Lat  and D Lon  calculation requires a pair of latitude—longitude position coordinates, the calculation for each data entry (row) employs the coordinates of that row and of the next row below. In the case of the last data entry (row) for a particular reference channel, no calculation is performed since no next row exists for such last row. 
     
       
         
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 Ref. 
                 # 
                   
                   
                   
                 D Lat   
                 D Lon   
                   
               
               
                 Ch. 
                 Ms. 
                 Lat. 
                 Long. 
                 r (miles) 
                 (miles) 
                 (miles) 
                 D (miles) 
               
               
                   
               
             
             
               
                 1 
                 4 
                 45.5199 
                 −122.563 
                 0.63087 
                 N/A 
                 N/A 
                 N/A 
               
               
                 2 
                 4 
                 45.5199 
                 −122.563 
                 0.51917 
                 0.013788 
                 0 
                 0.01379 
               
               
                 2 
                 1 
                 45.5197 
                 −122.563 
                 0.38468 
                 0.020682 
                 0 
                 0.02068 
               
               
                 2 
                 1 
                 45.5194 
                 −122.563 
                 0.46049 
                 0.013788 
                 0 
                 0.01379 
               
               
                 2 
                 1 
                 45.5192 
                 −122.563 
                 0.55125 
                 0.006618 
                 0.00974 
                 0.01178 
               
               
                 2 
                 1 
                 45.5191 
                 −122.563 
                 0.51917 
                 N/A 
                 N/A 
                 N/A 
               
               
                 3 
                 1 
                 45.5191 
                 −122.563 
                 0.48895 
                 0.056323 
                 0.01478 
                 0.05823 
               
               
                 3 
                 1 
                 45.5199 
                 −122.564 
                 0.51917 
                 0.091482 
                 0.0033 
                 0.09154 
               
               
                 3 
                 1 
                 45.5213 
                 −122.564 
                 0.62149 
                 0.108096 
                 0.00111 
                 0.1081 
               
               
                 3 
                 1 
                 45.5228 
                 −122.564 
                 0.58532 
                 0.121263 
                 0.00111 
                 0.12127 
               
               
                 3 
                 1 
                 45.5246 
                 −122.564 
                 0.38468 
                 0.075419 
                 0 
                 0.07542 
               
               
                 3 
                 1 
                 45.5257 
                 −122.564 
                 0.32135 
                 0.017924 
                 0.00107 
                 0.01796 
               
               
                 3 
                 4 
                 45.526 
                 −122.564 
                 0.31185 
                 0.016063 
                 0 
                 0.01606 
               
               
                 3 
                 1 
                 45.5262 
                 −122.564 
                 0.2381 
                 0.048878 
                 0.00407 
                 0.04905 
               
               
                 3 
                 1 
                 45.5269 
                 −122.564 
                 0.32135 
                 0.094653 
                 0.00184 
                 0.09467 
               
               
                 3 
                 1 
                 45.5283 
                 −122.564 
                 0.30264 
                 0.091826 
                 0.00145 
                 0.09184 
               
               
                 3 
                 1 
                 45.5296 
                 −122.564 
                 0.40845 
                 0.150907 
                 0.00296 
                 0.15094 
               
               
                 3 
                 1 
                 45.5318 
                 −122.564 
                 0.62149 
                 0.119678 
                 0.00518 
                 0.11979 
               
               
                 3 
                 1 
                 45.5335 
                 −122.564 
                 0.94563 
                 0.108096 
                 0.03033 
                 0.11227 
               
               
                 3 
                 1 
                 45.5351 
                 −122.563 
                 1.13201 
                 0.099134 
                 0.02364 
                 0.10191 
               
               
                 3 
                 1 
                 45.5365 
                 −122.562 
                 1.82887 
                 0127054 
                 0.02587 
                 0.12966 
               
               
                 3 
                 1 
                 45.5384 
                 −122.562 
                 1.62218 
                 0.110923 
                 0.02219 
                 0.11312 
               
               
                 3 
                 1 
                 45.54 
                 −122.561 
                 1.27624 
                 0.10072 
                 0.01846 
                 0.1024 
               
               
                 3 
                 1 
                 45.5414 
                 −122.561 
                 1.20196 
                 0.112301 
                 0.02514 
                 0.11508 
               
               
                 3 
                 1 
                 45.5431 
                 −122.561 
                 1.13201 
                 0.121263 
                 0.03401 
                 0.12594 
               
               
                 3 
                 1 
                 45.5448 
                 −122.56  
                 1.82887 
                 0.099134 
                 0.02737 
                 0.10284 
               
               
                 3 
                 1 
                 45.5463 
                 −122.559 
                 2.18931 
                 N/A 
                 N/A 
                 N/A 
               
               
                   
               
             
          
         
       
     
     5. Estimate Transmitter Position for each Reference Channel: 
     The position estimation is accomplished using D/2 and r from Table 3, above, to form a series of right triangles. Consider the diagram shown in FIG. 4, where P 1 , and P 2  are measured data points given by (La 1 , Lo 1 ) and (La 2 , Lo 2 ) respectively (La is latitude, Lo is longitude). P 3  is the midpoint between the P 1  and P 2  along the line D. E 1  and E 2  are estimated transmitter locations using the distance calculated from the RSL of P 1 . E 1  and E 2  lie along a line that is perpendicular to and bisects D at P 3  and are given by (ELa 1 , ELo 1 ) and (ELo 2 , ELo 2 ) respectively. The distance from P 3  to either E 1  or E 2  is given by          r   perp     =         r   2     -       (     D   2     )     2                                
     The slope of r perp  is the negative reciprocal of the slope m of line D. Applying the above properties, the coordinates of E 1  and E 2  may be calculated as follows:        ELa1   =     La3   +       360   *     r   perp         m   *     C   Pol                   ELo1   =     Lo3   -       360   *   m   *     (     ELa1   -   ELa3     )         C   Equ                 ELa2   =     La3   -       360   *     r   perp         m   *     C   Pol                   ELo2   =     Lo3   +       360   *   m   *     (     ELa2   -   ELa3     )         C   Equ                                
     Applying the above equations to the data from Table 3 yields Table 4A and 4B (where the first three columns thereof are identical), as follows: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                 TABLE 4A 
               
               
                   
               
               
                 Ref Chan 
                 r (miles) 
                 D (miles) 
                 La3 
                 Lo3 
                 r perp   
                 m 
               
               
                   
               
             
             
               
                 1 
                 0.63087 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
               
               
                 2 
                 0.51917 
                 0.01379 
                 45.5198 
                 −122.563 
                 0.51898 
                 2000000 
               
               
                 2 
                 0.38468 
                 0.02068 
                 45.51955 
                 −122.563 
                 0.38412 
                 3000000 
               
               
                 2 
                 0.46049 
                 0.01379 
                 45.5193 
                 −122.563 
                 0.46029 
                 2000000 
               
               
                 2 
                 0.55125 
                 0.01178 
                 45.519152 
                 −122.56310 
                 0.55112 
                 0.47761 
               
               
                 2 
                 0.51917 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
               
               
                 3 
                 0.48895 
                 0.05823 
                 45.5195345 
                 −122.56362 
                 0.48547 
                 −2.67869 
               
               
                 3 
                 0.51917 
                 0.09154 
                 45.5206065 
                 −122.56374 
                 0.51103 
                 19.51471 
               
               
                 3 
                 0.62149 
                 0.10810 
                 45.522054 
                 −122.56370 
                 0.61201 
                 68.17391 
               
               
                 3 
                 0.58532 
                 0.12127 
                 45.5237175 
                 −122.56370 
                 0.57262 
                 −76.47826 
               
               
                 3 
                 0.38468 
                 0.07542 
                 45.525144 
                 −122.56371 
                 0.37721 
                 −10940000 
               
               
                 3 
                 0.32135 
                 0.01796 
                 45.525821 
                 −122.56372 
                 0.32084 
                 −11.81818 
               
               
                 3 
                 0.31185 
                 0.01606 
                 45.5260675 
                 −122.56373 
                 0.31144 
                 −2330000 
               
               
                 3 
                 0.23810 
                 0.04905 
                 45.5265385 
                 −122.56377 
                 0.23300 
                 −8.44048 
               
               
                 3 
                 0.32135 
                 0.09467 
                 45.5275795 
                 −122.5638 
                 0.30708 
                 36.13158 
               
               
                 3 
                 0.30264 
                 0.09184 
                 45.528932 
                 −122.56376 
                 0.28837 
                 44.40000 
               
               
                 3 
                 0.40845 
                 0.15094 
                 45.5306925 
                 −122.56372 
                 0.37954 
                 35.88525 
               
               
                 3 
                 0.62149 
                 0.11979 
                 45.532655 
                 −122.56363 
                 0.60983 
                 16.22430 
               
               
                 3 
                 0.94563 
                 0.11227 
                 45.534307 
                 −122.56327 
                 0.93895 
                 2.50479 
               
               
                 3 
                 1.13201 
                 0.10191 
                 45.53581 
                 −122.56271 
                 1.12741 
                 2.94672 
               
               
                 3 
                 1.82887 
                 0.12966 
                 45.5374505 
                 −122.56220 
                 1.82426 
                 3.45131 
               
               
                 3 
                 1.62218 
                 0.11312 
                 45.5391765 
                 −122.56170 
                 1.61823 
                 3.51310 
               
               
                 3 
                 1.27624 
                 0.10240 
                 45.5407115 
                 −122.56128 
                 1.27212 
                 3.83465 
               
               
                 3 
                 1.20196 
                 0.11508 
                 45.5422565 
                 −122.56083 
                 1.19644 
                 3.13873 
               
               
                 3 
                 1.13201 
                 0.12594 
                 45.5439505 
                 −122.56022 
                 1.12498 
                 2.50570 
               
               
                 3 
                 1.82887 
                 0.10284 
                 45.545549 
                 −122.55959 
                 1.82597 
                 2.54513 
               
               
                 3 
                 2.18931 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
               
               
               
             
           
               
                 TABLE 4B 
               
               
                   
               
               
                 Ref 
                 r 
                 D 
                   
                   
                   
                   
               
               
                 Chan 
                 (miles) 
                 (miles) 
                 Ela1 
                 ELo1 
                 ELa2 
                 Elo2 
               
               
                   
               
             
             
               
                 1 
                 0.63087 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
               
               
                 2 
                 0.51917 
                 0.01379 
                 45.5198 
                 −122.5631088 
                 45.5198 
                 −122.5628912 
               
               
                 2 
                 0.38468 
                 0.02068 
                 45.51955 
                 −122.5630806 
                 45.51955 
                 −122.5629194 
               
               
                 2 
                 0.46049 
                 0.01379 
                 45.5193 
                 −122.5630965 
                 45.5193 
                 −122.5629035 
               
               
                 2 
                 0.55125 
                 0.01178 
                 45.53589026 
                 −122.5632161 
                 45.50241374 
                 −122.5629849 
               
               
                 2 
                 0.51917 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
               
               
                 3 
                 0.48895 
                 0.05823 
                 45.51690558 
                 −122.5637303 
                 45.52216342 
                 −122.5635267 
               
               
                 3 
                 0.51917 
                 0.09154 
                 45.52098636 
                 −122.5638542 
                 45.52022664 
                 −122.5636398 
               
               
                 3 
                 0.62149 
                 0.10810 
                 45.52218422 
                 −122.5638298 
                 45.52192378 
                 −122.5635732 
               
               
                 3 
                 0.58532 
                 0.12127 
                 45.52360889 
                 −122.5638216 
                 45.52382611 
                 −122.5635814 
               
               
                 3 
                 0.38468 
                 0.07542 
                 45.525144 
                 −122.5637921 
                 45.525144 
                 −122.5636339 
               
               
                 3 
                 0.32135 
                 0.01796 
                 45.5254272 
                 −122.5637913 
                 45.5262148 
                 −122.5636567 
               
               
                 3 
                 0.31185 
                 0.01606 
                 45.5260675 
                 −122.5638003 
                 45.5260675 
                 −122.5636697 
               
               
                 3 
                 0.23810 
                 0.04905 
                 45.52613808 
                 −122.5638259 
                 45.52693892 
                 −122.5637281 
               
               
                 3 
                 0.32135 
                 0.09467 
                 45.52770278 
                 −122.5638644 
                 45.52745622 
                 −122.5637356 
               
               
                 3 
                 0.30264 
                 0.09184 
                 45.52902621 
                 −122.5638265 
                 45.52883779 
                 −122.5637055 
               
               
                 3 
                 0.40845 
                 0.15094 
                 45.53084592 
                 −122.5638001 
                 45.53053908 
                 −122.5636409 
               
               
                 3 
                 0.62149 
                 0.11979 
                 45.53320023 
                 −122.5637644 
                 45.53210977 
                 −122.5635086 
               
               
                 3 
                 0.94563 
                 0.11227 
                 45.53974458 
                 −122.5634669 
                 45.52886942 
                 −122.5630731 
               
               
                 3 
                 1.13201 
                 0.10191 
                 45.54135981 
                 −122.5629494 
                 45.53026019 
                 −122.5624766 
               
               
                 3 
                 1.82887 
                 0.12966 
                 45.54511775 
                 −122.5625846 
                 45.52978325 
                 −122.5618194 
               
               
                 3 
                 1.62218 
                 0.11312 
                 45.54585818 
                 −122.5620454 
                 45.53249482 
                 −122.5613666 
               
               
                 3 
                 1.27624 
                 0.10240 
                 45.54552366 
                 −122.5615533 
                 45.53589934 
                 −122.5610197 
               
               
                 3 
                 1.20196 
                 0.11508 
                 45.54778583 
                 −122.5610874 
                 45.53672717 
                 −122.5605856 
               
               
                 3 
                 1.13201 
                 0.12594 
                 45.55046305 
                 −122.5604619 
                 45.53743795 
                 −122.5599901 
               
               
                 3 
                 1.82887 
                 0.10284 
                 45.55595586 
                 −122.5599754 
                 45.53514214 
                 −122.5592096 
               
               
                 3 
                 2.18931 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
               
               
                   
               
             
          
         
       
     
     6. Determine final location for each Reference Channel: 
     The final location for each reference channel is determined by averaging the results of E 1  and E 2  for each reference channel and assessing the spread of results for each E 1  and E 2 . Such assessment is accomplished by treating the standard deviation of the latitude and longitude calculations as independent and calculating the composite standard deviation σ as a root mean square, as follows: 
     
       
         σ={square root over (σ La   2 +σ Lo   2 )} 
       
     
     The composite standard deviations a for the solutions of reference channels  2  and  3  are: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                 TABLE 5 
               
               
                   
               
               
                 Ref. 
                 ELa1 
                 ELo1 
                 sigma 
                 Ela2 
                 ELo2 
                 sigma 
               
               
                 Ch. 
                 Ave. 
                 Ave. 
                 1 
                 Ave. 
                 Ave. 
                 2 
               
               
                   
               
             
             
               
                 2 
                 45.523635 
                 −122.56313 
                 0.0081729 
                 45.515266 
                 −122.56292 
                 0.0085707 
               
               
                 3 
                 45.533952 
                 −122.56299 
                 0.0115133 
                 45.528903 
                 −122.56266 
                 0.0020679 
               
               
                   
               
             
          
         
       
     
     The above results are then examined to determine from the sigmas  1  and  2  for each reference channel which of E 1  and E 2  is the more likely true location of the transmitter corresponding to such reference channel. That is, a smaller sigma indicates a smaller spread of the associated data and therefore a higher reliability in the results of such data. In particular, the results indicate that the location of the reference channel  2  transmitter is most likely E 1 , which has a sigma  1  less than the sigma  2  of E 2 , and E 1  has the coordinates shown. The most likely location estimate for the reference channel  3  transmitter is E 2  with a sigma  2  much less than the sigma  1  of E 1 . 
     It is to be noted that the method of the present invention may be performed manually, although with considerable effort and time required to do so. Of course, such method may also be performed in an automated and faster manner by way of a computer or processor  18  or the like, as is shown in FIG.  1 . Such computer or processor  18  may be any appropriate computer or processor without departing from the spirit and scope of the present invention. In fact, the computer  18  may be the recording device/computer  16 , which is to say that the method of the present invention may be performed on the computer  16 , thus obviating the need for an additional computer to be computer  18 . Moreover, the software employed to perform the method on the computer  18  may be any appropriate software, again without departing from the spirit and scope of the present invention. The details of such software are known or should be apparent to the relevant public, and therefore need not be described herein in further detail. 
     In the foregoing description, it can be seen that the present invention comprises a new and useful method for estimating the position of one or more transmitter locations based on signals transmitted from such transmitter(s). It should be appreciated that changes could be made to the embodiments described above without departing from the inventive concepts thereof. It should be understood, therefore, that this invention is not limited to the particular embodiments disclosed, but it is intended to cover modifications within the spirit and scope of the present invention as defined by the appended claims.