Abstract:
A global positioning system comprising: (a) a GPS receiver configured to (1) receive a plurality of signals from a plurality of visible GPS satellites, and (2) produce a plurality of pseudorange measurements from the received signals, the pseudorange measurements being indicative of the GPS receiver&#39;s position and having an amount of ionospheric delay error contained therein; (b) a processor configured to estimate the amount of ionospheric delay in the pseudorange measurements by (1) estimating an amount of global ionospheric delay attributable collectively to the plurality of visible GPS satellites, and (2) estimating a plurality of amounts of local ionospheric delays, each local ionospheric delay being attributable to a different visible GPS satellite. Preferably the processor implements the ionospheric delay estimations using a modified Kalman filter. The ionospheric delay amounts may be determined as a function of a single GPS frequency, L 1  or L 2 , which may change over time as the frequency available to the GPS receiver changes. Preferably, the global ionospheric delay amount and the local ionospheric delay amounts are initially set equal to a predetermined value and those initial values are subsequently modified as a function of an obliquity factor and a scalar, the scalar being dependent upon the single GPS frequency.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention relates to global positioning system (GPS) technology. In particular, the present invention relates to an improved technique for estimating the ionospheric delay in GPS signals.  
         BACKGROUND OF THE INVENTION  
         [0002]    The use of GPS technology has become increasingly prevalent in the world over the past several years. Basic GPS technology is well-known in the art and need not be described in depth herein, but for the purpose of background, its basic principles will be set forth.  
           [0003]    GPS utilizes a constellation of satellites orbiting the Earth (28 total in operation with an additional 4 back-up satellites). The orbits of these satellites are arranged so that at any time, anywhere on Earth, 6 to 12 GPS satellites are visible by a GPS receiver. Each satellite broadcasts navigation signals of two frequencies, L 1  at 1575.42 MHz and L 2  at 1227.6 MHz. The signals are spread spectrum signals transmitted using code division multiple access (CDMA) techniques wherein each GPS satellite is assigned its own code. A GPS receiver operates to receive the signals transmitted by the GPS satellites. The coding present in the GPS satellite signals allows the GPS receiver to separate the different signals that correspond to each visible satellite. The coding also allows the GPS receiver to measure the signal propagation delay from the individual satellites and the GPS receiver. The signal delay along with a clock error shared by measurements to all satellites forms the GPS receiver&#39;s estimate of pseudorange . All of these processes are well-known in the art.  
           [0004]    The term “pseudorange” is used because this position estimate is obtained after making certain approximations regarding how the signal propagates from the satellite to the receiver. In actuality, there are several sources of error in this approximation, including ephemeris errors, clock errors, receiver noise, and most importantly for the present invention, atmospheric errors, all of which are well-known in the art. The atmospheric errors are due to the effects of both the troposphere and the ionosphere. The contribution of the troposphere to pseudorange error can be well accounted for using existing error modeling techniques. However, there is a need in the art for an improved method of correcting for ionospheric signal delay in pseudorange estimates.  
           [0005]    The ionosphere ranges between 100 kilometers to 1000 kilometers above the Earth in altitude. The ionosphere is a dispersive medium that introduces errors greater during the day than at night and greater for satellites near the horizon than for satellites near zenith. Uncompensated iono-errors can range anywhere from 1 meter to more than 60 meters. Typical iono-errors are 10 meters for satellites near zenith and 30 meters for satellites near the horizon. In most GPS applications, the iono-error represents the most dominant source of error in the GPS position estimation.  
           [0006]    A current approach used in commercial GPS receivers is to model the ionospheric delay in software using parameters transmitted every 12 minutes from the GPS satellites. This approach cuts the iono-error approximately in half. However, even at this lower error level, the ionosphere serves as a significant error source.  
           [0007]    Differential GPS receivers and associated software have the capability of accurately estimating ionospheric delays. However, such delay estimation requires the availability of base stations that transmit reference signals. Such base stations currently do not have global coverage and are costly to implement. As such, the extension of this type of technology to non-differential GPS technology suffers from shortcomings.  
           [0008]    Military GPS receivers (or dual-frequency GPS receivers) use measurements made simultaneously on both the L 1  and L 2  frequencies to correct iono-error (commercial GPS receivers can only operate on a single frequency). A typical error with this approach is 1-2 meters. At this level, the iono-error is commensurate with other GPS error sources. Including all error sources, a well-tuned military GPS receiver and associated software is typically capable of a total circular error probable (CEP) of approximately 4 meters.  
           [0009]    An example of a dual frequency approach to ionospheric delay estimation can be found in U.S. Pat. No. 5,876,411 issued Feb. 2, 1999 to Kumar, the entire disclosure of which is incorporated herein by reference.  
           [0010]    However, it must be noted that the dual frequency approach is problematic in GPS jamming scenarios. Particularly in military applications, a GPS jamming device may be used to jam the L 1  and L 2  signals. GPS anti-jam (AJ) systems are available, but the cost of an AJ system capable of counter jamming on both the L 1  and L 2  frequencies is almost twice the cost of a system that can counter jam only one frequency at a time. As such, because it is much more cost-effective to leave a single GPS frequency unjammed at a given time than to unjam both frequencies, a strong need in the art exists for a GPS system that can substantially reduce iono-errors using only a single GPS frequency.  
         SUMMARY OF THE INVENTION  
         [0011]    With this problem in mind, the inventors herein developed the present invention wherein the iono-error modeling software does not require simultaneous measurements in both the L 1  and L 2  frequencies. Because only a single GPS frequency is needed at any given time to produce highly accurate ionospheric delay estimates, the present invention is suitable for use in a wide variety of applications where jamming of GPS signals may occur, particularly GPS-guided munitions.  
           [0012]    One aspect of the invention is to model the ionospheric delay between the GPS receiver antenna and each GPS satellite such that the ionospheric delay consists of a “global” delay (common to all GPS satellites) and “local” delays (each applicable to an individual GPS satellite). The inventors herein have found that the accuracy of such an approach to ionospheric delay estimation is more accurate than the conventional dual frequency approach used in conventional military GPS technology and more accurate than single frequency approaches known to the inventors. The global iono delay is modeled as the delay that would be observed between the GPS receiver antenna and a GPS satellite for a satellite at a known position wherein the ionosphere is an ideal homogeneous layer at a predetermined height above the earth. The preferable known position of such a satellite is at zenith and the preferable predetermined height is 190 nautical miles.  
           [0013]    Accordingly, disclosed herein is a method for estimating ionospheric delays present in pseudorange measurements produced by a global positioning system (GPS) receiver configured to track a plurality of GPS satellites, the method comprising: (a) estimating an amount of global ionospheric delay attributable to all of the tracked GPS satellites; and (b) for each tracked GPS satellite, estimating an amount of local ionospheric delay attributable thereto.  
           [0014]    Also disclosed herein is a global positioning system comprising: (a) a GPS receiver configured to (1) receive a plurality of signals from a plurality of visible GPS satellites, and (2) produce a plurality of pseudorange measurements from the received signals, the pseudorange measurements being indicative of the GPS receiver&#39;s position and having an amount of ionospheric delay error contained therein; (b) a processor configured to estimate the amount of ionospheric delay in the pseudorange measurements by (1) estimating an amount of global ionospheric delay attributable collectively to the plurality of visible GPS satellites, and (2) estimating a plurality of amounts of local ionospheric delays, each local ionospheric delay being attributable to a different visible GPS satellite.  
           [0015]    Preferably the processor implements the ionospheric delay estimations using a modified Kalman filter. As noted above, the ionospheric delay amounts may be determined as a function of a single GPS frequency L 1  or L 2 . The particular frequency used may change over time as the frequency available to the GPS receiver changes. Further still, the present invention will also operate when both GPS frequencies are available.  
           [0016]    Preferably, the global ionospheric delay amount is initially set equal to a predetermined value and that initial value is subsequently modified as a function of an obliquity factor and a scalar, the scalar being dependent upon the single GPS frequency. The local ionospheric delay amounts are preferably initialized and updated in the same manner.  
           [0017]    These and other advantages of the present invention will be in part apparent and in part pointed out in the following description, claims, and referenced figures. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0018]    [0018]FIG. 1 is a block diagram depicting an overview of a preferred system implementing the present invention;  
         [0019]    [0019]FIG. 2 depicts the state vector X;  
         [0020]    [0020]FIG. 3 depicts the state covariance matrix P;  
         [0021]    [0021]FIG. 4 depicts the state transition matrix (D;  
         [0022]    [0022]FIG. 5 depicts the process noise matrix Q;  
         [0023]    [0023]FIG. 6 depicts the measurement sensitivity matrix H;  
         [0024]    [0024]FIG. 7 illustrates preferred values for the obliquity factor as a function of satellite elevation;  
         [0025]    FIGS.  8 ( a ) and  8 ( b ) depict applications of the present invention in environments where GPS signal jamming is present;  
         [0026]    [0026]FIG. 9 illustrates an implementation of the present invention where the GPS receiver produces ionospheric delay-corrected pseudorange measurements; and  
         [0027]    [0027]FIG. 10 illustrates an implementation of the present invention wherein an inertial navigator is incorporated into the overall system. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0028]    [0028]FIG. 1 is a block diagram depicting the overall position estimating system. A GPS receiver  100  operates to receive GPS signals from orbiting GPS satellites (not shown) and provide pseudorange measurements  104  (found in Z, described below) to processor  102 . Processor  102  includes a Kalman filter  110  modified in accordance with the teachings of the present invention. The GPS receiver also preferably provides the processor  102  with satellite position data  106  that identifies the position of each satellite visible to the GPS receiver relative to the center of the earth. Further, the GPS receiver  100  preferably provides the processor  102  with receiver tracking status data  108 . The status data  108  includes a satellite ID (identifying which GPS satellite is transmitting) and an identification of the tracking frequency (L 1  or L 2 ).  
         [0029]    As noted above, the processor  102  implements a modified Kalman filter  110 . The processor may be any computing device capable of carrying out programmed instructions of the type set forth below. Preferred processors include the Intel Pentium processor, the Motorola Power PC, and the Texas Instruments TMS 320C6000 family. Typical Kalman filters are well-known in the art, and as such, their basics need not be elaborated upon herein.  
         [0030]    Pertinent to the present invention are the additional states that the inventors herein have added to the Kalman filter to model ionospheric delay. The Kalman filter processes the input data to generate estimates of the navigation states (the vector X described below) and estimates of the uncertainties in the navigation state estimates (the matrix P described below).  
         [0031]    Upon receipt of the initial data from the GPS receiver (including, in particular, pseudorange measurements and the satellite position data), the Kalman filter vectors and matrices are initialized. The elements of the state vector X are initialized as X 0  and the elements of the state covariance matrix P are initialized as P 0 . As is well-known in the art, the system model can be defined as follows:  
           X   k+1 =φ k   X   k   +W   k    
         
       Z 
       k 
       =H 
       k 
       X 
       k 
       +V 
       k  
     
         [0032]    wherein k indexes the time steps for the system, X represents the state vector, φ represents the state transition matrix, Z represents the measurement vector (essentially the pseudorange measurements), H represents the measurement sensitivity matrix, and W and V represent, respectively, the process noise vector and measurement noise vector. W k  is approximated by N(0,Q k ) and V k  is approximated by N(0,R k ). The notation N(μ,Σ) indicates the variable is a random variable having a normal (Gaussian) distribution with a mean μ and a covariance Σ. Q represents the process noise covariance matrix and R represents the measurement noise covariance matrix.  
         [0033]    As new GPS data is provided to the processor, the state variables and matrices are updated as follows:  
         X=φX; (Propagate state to measurement time)  
           P=φPφ   T   +Q ; (Propagate covariance to measurement time)  
           K=PH   T   [HPH   T   +R]   −1 ; (Compute Kalman gain)  
           X=X+K[Z−HX] ; (Update the state estimate)  
         P  P=[I−KH]P ; (Update the covariance estimate)  
         [0034]    wherein K represents the Kalman gain vector, I represents the identity matrix, and superscript T indicates matrix transpose. The matrix R is a standard m by m measurement noise covariance matrix.  
         [0035]    These Kalman filter equations are well understood in the art (see, for example, Gelb et al.,  Applied Optimal Estimation , MIT Press, 1974, the entire disclosure of which is incorporated herein by reference). As can be seen, these equation variables are generally vectors and matrices, meaning that the implied operations are matrix-vector operations. Formulations for implementing Kalman filter equations other than those described herein exist, and may also be used in the practice of the present invention. Accordingly, the implementation described herein is illustrative of only one possible Kalman filter equation formulation.  
         [0036]    A. State Vector X:  
         [0037]    [0037]FIG. 2 illustrates the matrix X, whose elements represent the Kalman filter states. The first n states (x 1  through x n ) are the baseline states. These states are the states existing in a standard, unmodified Kalman filter. These states, which account for, at minimum, position, but preferably also include velocity, attitude, and sensor biases. The number n of baseline states, the particular coordinate system used, and whether complete values or deviations from a nominal value are used will vary by implementation according to the application involved. These baseline states are readily understood in the art and need not be further elaborated upon herein. The value of n will depend upon the number of baseline states in the original Kalman filter for the system. A preferred n value is 17.  
         [0038]    The next state (x n+1 ) is the global iono state. The global iono state X n+1  (or X GLOBAL ) represents the ionospheric delay at zenith at the L 1  frequency for an ideal, homogenous ionosphere modeled as a thin shell at a height of 190 nautical miles. In the present invention, this state is deemed common to all GPS satellites, and should correct for the largest part of the ionospheric delay. It should be noted that, although undesirable, the global ionospheric delay can also be modeled to represent the ionospheric delay at zenith at the L 2  frequency for the idealized frequency.  
         [0039]    The remaining m states of X (states X n+2  through X n+m+1 ) represent the local iono states. Each local iono state corresponds to a particular GPS satellite. Preferably, each state is assigned to a satellite in a consistent manner such that once a local iono state is assigned to a particular satellite, that assignment remains the same as long as the GPS receiver tracks that particular satellite. The local iono states represent deviations (typically less than 2 meters of uncertainty) from the idealized global iono state for each tracked satellite. The value for m may be determined by a practitioner of the invention and will depend upon how many satellites a designer of the system wishes for the GPS receiver and Kalman filter to track. Typically, each GPS satellite is identified by a number from 1 to 32. Up to approximately 28 satellites are in orbit at any one time, and of these, from 6 to 12 are visible to a GPS receiver at any one location on the earth at any one time. Thus, the preferable range of m values is 6 to 12, and a preferred m value is 12.  
         [0040]    (i) Initialization of X:  
         [0041]    The baseline states (x l  through x n ) are preferably initialized as known in the art.  
         [0042]    Preferably, for applications using GPS-aided navigation, the global iono state is initialized once, at the start of the GPS-aided navigation. The initial global iono state can be selected or computed by a practitioner of the invention. A preferred selected value is 5 meters. If the practitioner prefers to compute the initial global iono state (among the factors that affect the iono delays are the 11 year solar cycle and the daily day-night cycle), the preferable inputs used to initialize the global iono state are as follows:  
         [0043]    Y=Year (preferably whole year, e.g., 2003);  
         [0044]    D=Day, fraction of a year (a value between 0 and 1,  
         [0045]    e.g., June 30≈0.5)  
         [0046]    T=UTC time of day (hours expressed as values 0-24);  
         [0047]    φ=Latitude (expressed in degrees, −90 to 90);  
         [0048]    λ=Longitude (expressed in degrees, −180 to 180);  
         [0049]    The latitude and longitude values may be derived from the pseudoranges and satellite position data provided by the GPS receiver using standard techniques.  
         [0050]    Using these inputs, the following values are computed:  
         μ( Y, D )=0. 1624{24+5 cos [2π( Y+D− 1957.8)/11]}{1+0.12 cos[2π D ]−0.2 cos[4π D]}   
         [0051]    [0051]         f        (   φ   )       =     1   +       2        (       90      °     -        φ          )         90      °                   TOD        (     T   ,   λ     )       =     T   +     λ   15                               
         [0052]    wherein TOD(T,λ) represents the local time of day, and (1) if TOD&lt;0, then TOD=TOD+24; and (2) if TOD&gt;24, then TOD=TOD−24;  
           g        (   TOD   )       =     cos        [       2        π        (     TOD   -   14     )         30     ]         ;       for                 7     &lt;   TOD   &lt;   21                           
 
         [0053]    g(TOD)=0.1045; for other TOD values  
         [0054]    Using these computed values, the initial value for the global iono state X GLOBAL (0) (or x n+1 (0)) can be computed as follows:  
           X   GLOBAL (0)=μ( Y,D ) f (φ) g ( TOD ( T ,λ))  
         [0055]    The term μ is the estimated mean global ionospheric delay at zenith for the L 1  frequency, which roughly matches historical ionospheric measurements for 11-year, annual and semi-annual cycles. The f (.) function scales the global mean delay by latitude such that equatorial latitudes have larger mean ionospheric delays than more extreme latitudes. The g(.) function scales the global mean ionospheric delay at latitude by time of day using a cosine function such that daytime delays are larger than night-time delays.  
         [0056]    The m local iono states (x n+2  through X n+m+1 ) are also preferably initialized at the beginning of GPS-aided navigation and, for a particular local state, whenever there is a change in the status of the tracked satellite (as various satellites become visible and invisible to the GPS receiver). Preferably, the local iono states are initialized to zero.  
         [0057]    To properly index each local iono state with the appropriate satellite, the following rules are preferably implemented in software:  
         [0058]    (1) each local iono state index (n+2 through n+m+1 for m total, referred to hereinafter as a slot) shall cross-reference a particular satellite number to a local iono state;  
         [0059]    (2) a slot cross-reference shall remain the same for as long as the satellite associated therewith is in the list of m slots;  
         [0060]    (3) each satellite currently tracked by the GPS receiver shall have a corresponding slot, subject to the proviso of rule ( 5 );  
         [0061]    (4) even if a satellite is no longer tracked by the GPS receiver, the slot assignments are maintained until a more newly-tracked satellite forces the limit of m slots to be exceeded;  
         [0062]    (5) if all m slots currently have a satellite assigned thereto, and a new satellite tracked by the GPS receiver causes the limit to m slots to be exceeded, then the slot corresponding to the “oldest” satellite (such an “oldest” satellite being determined by the time of the last measurement update therefrom) shall be re-assigned to the new satellite;  
         [0063]    (6) upon re-assignment of a slot, the corresponding local iono state and local covariance state shall be re-initialized.  
         [0064]    B. State Covariance Matrix P:  
         [0065]    [0065]FIG. 3 illustrates the preferred state covariance matrix P. The preferable size of P is n+m+1 by n+m+1. P comprises a baseline matrix P BASELINE  of size nxn, augmented by additional diagonal elements corresponding to the global covariance estimate and m local covariance estimates. P BASELINE  is readily understood in the art and need not be elaborated upon herein as it is left unchanged in the practice of the present invention. All off-diagonal elements for the augmented part of P (the part of P not belonging to P BASELINE ) are set to zero. The augmented covariance element corresponding to the global iono covariance is P n+1,n+1 . Its value is σ G   2 ,with σ G   2  preferably being initialized to (5 meters)  2 , or P n+1,n+1 (0)=25. The diagonal states corresponding to the local iono covariances are pi,i wherein i ranges from n+2 to n+m+1. Each local iono covariance in P corresponds to a local iono state in X such that pi,i corresponds to x i . Also, each local iono covariance σ L   2  is preferably initialized with the same value, preferably σ L   2 (0)=(2 meters) 2 =4.  
         [0066]    C. State Transition Matrix φ):  
         [0067]    [0067]FIG. 4 depicts the preferred state transition matrix Φ. The preferable size of Φ is n+m+1 by n+m+1. Φ comprises a baseline matrix Φ BASELINE  of size nxn, augmented by additional diagonal elements corresponding to the global transition and m local transitions. Φ BASELINE  is readily understood in the art and need not be elaborated upon herein. All off-diagonal elements for the augmented part of Φ (the part of Φ not belonging to Φ BASELINE ) are set to zero. Each diagonal term of the augmented part of the matrix is set equal to p, wherein p=e −T/τ . T is the filter propagation period (the time since the last propagation. The variable τ is the time constant of the ionospheric error. The values for T and τmay be selected by the user, and the preferable values are 1 second for T and 1800 seconds (30 minutes) for τ. Further, the time constant for the global element may be different than the time constant for the local elements, but it is preferred that the same value be used.  
         [0068]    D. Process Noise Matrix Q:  
         [0069]    [0069]FIG. 5 depicts the preferred process noise matrix Q. The preferable size of Q is n+m+1 by n+m+1. Q comprises a baseline matrix Q BASELINE  of size nxn, augmented by additional diagonal elements corresponding to the global transition and m local transitions. Q BASELINE  is readily understood in the art and need not be elaborated upon herein. All off-diagonal elements in the augmented part of Q are set to zero. The global diagonal element (q n+1,n+1 ) is preferably set equal to σ G   2 (1−ρ 2 ) and the local diagonal elements (q n+2,n+2  through q n+m+1,n+m+1 ) are preferably set equal to σ L   2 (1−ρ 2 ).  
         [0070]    E. Measurement Sensitivity Matrix H:  
         [0071]    [0071]FIG. 6 depicts the preferred measurement sensitivity matrix H. H is preferably constructed by rows, with one row for each pseudorange measurement from a satellite. Each row of H will comprise n+m+1 elements. Further, each row of H corresponding to the pseudorange measurement from a particular satellite is dependent upon (1) the elevation of the satellite, as determined with standard techniques using satellite position data from the GPS receiver and estimated GPS position data available from the GPS receiver and/or the baseline filter states of X, (2) whether the measurement was formulated while tracking the L 1  or L 2  frequency (the particular tracking frequency may change over time), and (3) and the indexing scheme that is used to associate each satellite with one of the local iono states.  
         [0072]    The preferable size of H is m by n+m+1. Each row of H comprises a vector H BASELINE  (columns 1 through n for each row of H) , augmented by additional elements corresponding to global and local iono elements. H BASELINE  is readily understood in the art and need not be elaborated upon herein. The construction of each pseudorange row of H is as follows:  
           Row  ( i )=[H BASELINE αβ αβH   i   LOCAL ] 
         [0073]    wherein H BASELINE  has a length of n, and H i   LOCAL  is a 1xm vector with all elements being zero except for the element that corresponds to the satellite associated with the row. Thus, for row 1 (which is associated with satellite 1) , H LOCAL  will be represented by [1 0 0 . . . 0], for row 2, H LOCAL  will be [0 1 0 . . . 0], and so on for each row.  
         [0074]    The variable α represents the iono “obliquity” factor. The obliquity factor generally represents the ratio between the iono delay at a given satellite elevation and the iono delay for a satellite at zenith. It can be calculated as:  
       α   =       [     1   -     0.8981          cos   2          (   El   )           ]       -     1   2                               
 
         [0075]    wherein the variable “El” is the elevation of the particular satellite the corresponds with the row (defined so that the horizon is zero elevation and zenith is 90 degrees). FIG. 7 illustrates the preferred obliquity factor values as a function of satellite elevation. Preferably, the satellite elevation is calculated in software using normal geometric relationships (based on the satellite position data and position estimates (from either the Kalman filter states or the GPS receiver pseudorange measurements) that are readily known in the art.  
         [0076]    The scale factor β adjusts the obliquity factor to account for frequency. If the satellite is being tracked on the L 1  frequency, the scalar β preferably equals 1. If the satellite is being tracked on the L 2  frequency, the scalar β preferably equals 1.6469.  
         [0077]    It should be noted, that in setting up the matrices, vectors, and variables for the modified Kalman filter, the values for τ, σ G , σ L , initial global iono state X GLOBAL  (0), and initial global iono covariance P GLOBAL  (0) can be important tuning parameters that optimize performance for a specific application. While the specific values set forth herein are preferred, practitioners of the present invention may use different values of their choosing to tailor the filter&#39;s performance to their particular end uses.  
         [0078]    The present invention is particularly suitable for use in environments where a jammer is used to disrupt either or both of the L 1  and L 2  GPS signals. Because only a single GPS frequency is needed by the present invention at any given time, the present invention maintains the ability to accurately estimate ionospheric delay (and thereby reduce the CEP of the determined position) using a low cost single frequency (L 1  or L 2 ) anti-jamming (AJ) system. A single frequency AJ system costs much less than a dual frequency one. FIGS.  8 ( a ) and  8 ( b ) illustrate the suitability of the present invention in situations where (a) a jammer 200 is used to jam both the L 1  and L 2  GPS signals and a single frequency AJ system  202  is used to free one of the two GPS signals (in the case of FIG. 8( a ), the L 2  signal), and (b) where a jammer  200  is used to jam one of the two GPS signals (in the case of FIG. 8 ( b ), the L 1  signal).  
         [0079]    Also, it should be noted that the preferred GPS receiver  100  of the present invention does not perform its own ionospheric delay corrections. Preferably, the pseudorange measurements produced by the GPS receiver includes an uncorrected amount of ionospheric delay, and the processor  102  estimates this delay amount. However, the GPS receiver  100  may be of the type that produces iono-corrected pseudorange measurements. With such GPS receivers, the receiver will also communicate its ionospheric delay correction amounts. Because of the high accuracy level of the present invention, a practitioner may wish to rely on the delay estimates generated by the processor  102  rather than whatever correction is provided internally by the GPS receiver. In such situations, with reference to FIG. 9, the processor  102  can also receive data  112  from the GPS receiver  100  that is indicative of the receiver&#39;s internal ionospheric delay correction amounts. Preferably, the processor  102  then operates to remove these correction amounts from the pseudorange measurements  102 , thereby creating uncorrected pseudorange measurements  114 . Then, the ionospheric delay for these uncorrected pseudorange measurements  114  may be generated by the processor in accordance with the teachings of the present invention.  
         [0080]    Further still, a GPS receiver may implement on-board internal ionospheric delay estimation in accordance with the teachings of the present invention. In these situations, the processor  102  would be internal to the GPS receiver  100 . Such a use of the present invention may be particularly suitable for commercial GPS receivers where only a single GPS frequency is available.  
         [0081]    Moreover, it should be noted that the system of the present invention may also incorporate other sensors, such as an inertial navigator  300 , as shown in FIG. 10. In applications such as GPS-guided munitions where the GPS vehicle will be in motion, an inertial navigator  300  provides improved estimation of vehicle attitude (roll, pitch, yaw) and helps maintain position and velocity estimates for highly dynamic vehicles and during GPS outages. These interactions with the navigator  300  via feedback loops such as those shown in FIG. 10 are understood in the art (see, for example, Farrell, J. A. and Barth, M.,  The Global Positioning System  &amp;  Inertial Navigation , McGraw-Hill 1999, the entire disclosure of which is incorporated herein by reference), and would benefit from the improved ionospheric delay estimates provided by the present invention.  
         [0082]    While the present invention has been described above in relation to its preferred embodiment, various modifications may be made thereto that still fall within the invention&#39;s scope, as would be recognized by those of ordinary skill in the art upon review of the teachings herein. As such, the full scope of the present invention is to be defined solely by the appended claims and their legal equivalents.