Geopositioning method using assistance data

Abstract

A method for determining assistance data to facilitate processing of radio-navigation signals from a set of radio-navigational satellites of a reference network, each radio-navigational satellite broadcasting at least a first radio-navigation signal on a first frequency and a second radio-navigation signal on a second frequency, the second frequency being distinct from the first frequency, is provided.

Claims

1. A method for determining assistance data to facilitate processing of radio-navigation signals from a set of radio-navigational satellites of a reference network, each radio-navigational satellite broadcasting at least a first radio-navigation signal on a first frequency and a second radio-navigation signal on a second frequency, the second frequency being distinct from the first frequency, the method comprising: recording, by the at least one signal receiver of the reference network, as radio-navigation signals any first radio-navigation signal and second radio-navigation signal broadcasted by each radio-navigational satellite; deriving, by the at least one signal receiver, code and phase measurements from the said radio-navigation signals recorded, the phase measurements having ambiguities, each of which is an a priori unknown integer number of cycles; resolving, by the at least one signal receiver, the ambiguities in the phase measurements on the first frequency in a same manner for all the satellites and reference receivers of the reference network to produce a resolution; deducing, by the at least one signal receiver, a set of iono-free transmitter phase clock values arising from the resolution and a set of iono-free receiver phase clock values arising from the resolution; calculating, by the at least one signal receiver, for each satellite-reference receiver pair, a carrier code shift combination value from the code and phase measurements for each respective satellite on the first frequency; calculating, by the at least one signal receiver, a plurality of measurement residues based on the carrier code shift combination value, a modelled geometric contribution, a contribution of the iono-free receiver phase clock value, an iono-free transmitter phase clock value, and a contribution of the carrier code shift combination value; separating, by the at least one signal receiver, in a same manner for all the satellites and receivers of the reference network, each of measurement residues into a transmitter part and a receiver part, and smoothing, by the at least one signal receiver, the transmitter parts of the plurality of measurement residues.

2. The method according to claim 1, wherein the iono-free transmitter phase clock values associated with the resolution of the ambiguities as well as the smoothed transmitter parts are provided to receivers outside the reference network as assistance data.

3. The method according to claim 1, wherein the iono-free transmitter phase clock values associated with the resolution of the ambiguities as well as the smoothed transmitter parts are provided to receivers outside the reference network as assistance data over a telecommunications channel.

4. The method according to claim 1, further comprising: calculating, by the at least one signal receiver, for each satellite of said set of satellites, a transmitter clock value associated with the carrier code shift combination as the sum of the iono-free transmitter phase clock value and the smoothed transmitter parts.

5. The method according to claim 4, wherein said transmitter clock values associated with the carrier code shift combination are provided to receivers outside the reference network as assistance data.

6. The method according to claim 4, wherein said transmitter clock values associated with the carrier code shift combination are provided to receivers outside the reference network as assistance data over a telecommunications channel.

7. The method according to claim 1, wherein the smoothing of the transmitter parts of the plurality of measurement residues is performed each time by fitting a sine function with a period of 12 hours to the transmitter parts of the plurality of measurement residues.

8. The method according to claim 1, wherein the first frequency is selected from a list including 1575.42 MHz, 1227.6 MHz, 1176.45 MHz, 1207.14 MHz, 1278.75 MHz, 1561.098 MHz, 1589.742 MHz, 1207.14 MHz, and 1268.52 MHz.

9. A computer program product comprising a non-transitory computer readable medium storing computer program code for determining assistance data to facilitate processing of radio-navigation signals from a set of radio-navigational satellites of a reference network, each radio-navigational satellite broadcasting at least a first radio-navigation signal on a first frequency and a second radio-navigation signal on a second frequency, the second frequency being distinct from the first frequency, the computer program code being executable by a processor of at least one signal receiver of the reference network to cause the processor to perform: recording as radio-navigation signals any first radio-navigation signal and second radio-navigation signal broadcasted by each radio-navigational satellite; deriving code and phase measurements from the said radio-navigation signals recorded, the phase measurements having ambiguities, each of which is an a priori unknown integer number of cycles; resolving the ambiguities in the phase measurements on the first frequency in a same manner for all the satellites and reference receivers of the reference network to produce a resolution; deducing a set of iono-free transmitter phase clock values arising from the resolution and a set of iono-free receiver phase clock values arising from the resolution; calculating for each satellite-reference receiver pair, a carrier code shift combination value from the code and phase measurements for each respective satellite on the first frequency; calculating plurality of measurement residues based on the carrier code shift combination value, a modelled geometric contribution, a contribution of the iono-free receiver phase clock value, an iono-free transmitter phase clock value, and a contribution of the carrier code shift combination value; separating in a same manner for all the satellites and receivers of the reference network, each of measurement residues into a transmitter part and a receiver part, and smoothing the transmitter part of the plurality of measurement residues.

10. The computer program product according to claim 9, wherein the iono-free transmitter phase clock values associated with the resolution of the ambiguities as well as the smoothed transmitter parts are provided to receivers outside the reference network as assistance data.

11. The computer program product according to claim 9, wherein the iono-free transmitter phase clock values associated with the resolution of the ambiguities as well as the smoothed transmitter parts are provided to receivers outside the reference network as assistance data over a telecommunications channel.

12. The computer program product according to claim 9, wherein the computer program code being executable by the processor of at least one signal receiver of the reference network to cause the processor to further perform: calculating, by the at least one signal receiver, for each satellite of said set of satellites, a transmitter clock value associated with the carrier code shift combination as the sum of the iono-free transmitter phase clock value and the smoothed transmitter parts.

13. The computer program product according to claim 12, wherein said transmitter clock values associated with the carrier code shift combination are provided to receivers outside the reference network as assistance data.

14. The computer program product according to claim 12, wherein said transmitter clock values associated with the carrier code shift combination are provided to receivers outside the reference network as assistance data over a telecommunications channel.

15. The computer program product according to claim 9, wherein the smoothing of the transmitter parts of the plurality of measurement residues is performed each time by fitting a sine function with a period of 12 hours to the transmitter parts of the plurality of measurement residues.

16. The computer program product according to claim 9, wherein the first frequency is selected from a list including 1575.42 MHz, 1227.6 MHz, 1176.45 MHz, 1207.14 MHz, 1278.75 MHz, 1561.098 MHz, 1589.742 MHz, 1207.14 MHz, and 1268.52 MHz.

17. A system comprising radio-navigational satellites and at least one signal receiver, the at least one signal receiver configured to determine assistance data to facilitate processing of radio-navigation signals from a set of radio-navigational satellites of a reference network, each radio-navigational satellite broadcasting at least a first radio-navigation signal on a first frequency and a second radio-navigation signal on a second frequency, the second frequency being distinct from the first frequency, the at least one signal receiver configured to: record as radio-navigation signals any first radio-navigation signal and second radio-navigation signal broadcasted by each radio-navigational satellite; derive code and phase measurements from the said radio-navigation signals recorded, the phase measurements having ambiguities, each of which is an a priori unknown integer number of cycles; resolve the ambiguities in the phase measurements on the first frequency in a same manner for all the satellites and reference receivers of the reference network to produce a resolution; deduce a set of iono-free transmitter phase clock values arising from the resolution and a set of iono-free receiver phase clock values arising from the resolution; calculate for each satellite-reference receiver pair, a carrier code shift combination value from the code and phase measurements for each respective satellite on the first frequency; calculate plurality of measurement residues based on the carrier code shift combination value, a modelled geometric contribution, a contribution of the iono-free receiver phase clock value, an iono-free transmitter phase clock value, and a contribution of the carrier code shift combination value; separate in a same manner for all the satellites and receivers of the reference network, each of measurement residues into a transmitter part and a receiver part, and smooth the transmitter part of the plurality of measurement residues.

18. The system according to claim 17, wherein the iono-free transmitter phase clock values associated with the resolution of the ambiguities as well as the smoothed transmitter parts are provided to receivers outside the reference network as assistance data.

19. The system according to claim 17, wherein the iono-free transmitter phase clock values associated with the resolution of the ambiguities as well as the smoothed transmitter parts are provided to receivers outside the reference network as assistance data over a telecommunications channel.

20. The system according to claim 17, wherein the at least one signal receiver is further configured to: calculate, for each satellite of said set of satellites, a transmitter clock value associated with the carrier code shift combination as the sum of the iono-free transmitter phase clock value and the smoothed transmitter parts.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other characteristics and features of the invention will become apparent from the detailed description of a preferred method of implementation given below, for illustration, with reference to the accompanying drawings. These show:

(2) FIG. 1: a schematic illustration of a network of reference receivers,

(3) FIG. 2: a flowchart of a method for determining the assistance data;

(4) FIG. 3: a flowchart of a geopositioning method using the assistance data that can be determined in the method of FIG. 2;

(5) FIG. 4: a graphical representation of the transmitter parts of two measurement residues each obtained by subtracting from a carrier code shift measurement the (modelled) geometric contribution, the contribution of the iono-free receiver and transmitter phase clock values and the contribution of the ambiguity of the phase measurement;

(6) FIG. 5: a graphical representation of the transmitter parts of two measurement residues of FIG. 4 after smoothing.

DETAILED DESCRIPTION

(7) A navigation signal receiver can perform (non-ambiguous) code measurements and phase measurements (ambiguous by an integer number of cycles) on the radio-navigation signals it receives from various visible satellites (i.e. satellites above the horizon). A multi-frequency receiver can perform these measurements on at least two distinct frequencies f.sub.1 and f.sub.2. Assuming a dual-frequency receiver, one therefore has, for each visible satellite and at each time step, two code measurements, denoted P.sub.1 and P.sub.2, and two phase measurements, denoted L.sub.1 and L.sub.2, on frequencies f.sub.1 and f.sub.2. At each time step (t.sub.k), one therefore obtains, in a receiver, a set of code measurements P.sub.1.sup.j(t.sub.k), P.sub.2.sup.j(t.sub.k), L.sub.1.sup.j(t.sub.k) and L.sub.2.sup.j(t.sub.k), where the superscript (j) indicates the satellite, whose signal is received by the receiver. As the satellites orbit the Earth, only some of them are visible at a given time from the location of the receiver. For ease of notation, the dependence of time and the satellite index will not always be explicitly stated in the following.

(8) We use the following notations:

(9) $\gamma =\frac{f_1^2}{f_2^2},{\lambda }_1=\frac{c}{f_1},{\lambda }_2=\frac{c}{f_2}$

(10) where c is the speed of light. For the L1 and L2 bands of the GPS system, we have, for example: f.sub.1=154 f.sub.0 and f.sub.2=120 f.sub.0, where f.sub.0=10.23 MHz. By convention, the code measurements P.sub.1, P.sub.2 are expressed in units of length, whereas the phase measurements L.sub.1, L.sub.2 are expressed in cycles.

(11) The code measurements and phase measurements satisfy the following equations (measurements on the left, model parameters on the right):
P.sub.1=D.sub.1+e+ΔH.sub.p,1
P.sub.2=D.sub.2+γe+ΔH.sub.p,2
λ.sub.1L.sub.1=D.sub.1+λ.sub.1W−e+ΔH.sub.1−λ.sub.1N.sub.1
λ.sub.2L.sub.2=D.sub.2+λ.sub.2W−γe+ΔH.sub.2−λ.sub.2N.sub.2   (E1)

(12) where D.sub.1 represents the propagation distance between the phase centres of the satellite and the receiver for the frequency f.sub.1, including tropospheric elongation, relativistic effects, etc. D.sub.2 is the analogous quantity for the frequency f.sub.2; e is the ionospheric extension term which varies with the square of the frequency and comes with an opposite sign between the code measurement and the phase measurement; N.sub.1 and N.sub.2 represent the (integer) phase ambiguities of the two carriers, W represents the contribution of the “wind-up” effect, in cycles, and ΔH.sub.p,1, ΔH.sub.p,2, ΔH.sub.1, ΔH.sub.2 are the differences between the receiver clock and the transmitter clock for the different measurements.

(13) Obviously, a single-frequency receiver can only make one code measurement and one phase measurement per satellite and per time step. Without loss of generality, we may assume that these measurements are P.sub.1 and L.sub.1 (instead of P.sub.2 and L.sub.2).

(14) One calls “iono-free” code combination and denotes P.sub.c the combination of the code measurements that eliminates the ionospheric contribution:

(15) $\begin{array}{cc}{P}_c=\frac{\gamma P_1-P_2}{\gamma -1}=\frac{\gamma D_1-D_2}{\gamma -1}+\frac{\mathrm{\gamma \Delta }{H}_{p,1}-\Delta H_{p,2}}{\gamma -1}& \left(\mathrm{E2}\right)\end{array}$

(16) An (ambiguous) “iono-free” phase combination can be defined analogously. The advantage of working with “iono-free” combinations resides mainly in the fact that the ionosphere is a relatively unknown source of error in the sense that the ionospheric contribution is difficult to model with adequate accuracy. However, the “iono-free” code and phase combinations are only available in a receiver that is at a minimum dual-frequency.

(17) In a single-frequency receiver, it is nevertheless possible to eliminate the ionospheric contribution by using measurements on the single frequency, thanks to the combination, denoted hereinafter as P.sub.e, that is called “carrier code shift combination” (also called GRAPHIC combination, which is the acronym of the term “GRoup and PHase Ionospheric Correction”) that uses the sum of the phase measurement and the code measurement. For the frequency f.sub.1, this combination can be written as the arithmetic mean of the code measurement and the phase measurement:

(18) $\begin{array}{cc}{P}_e=\frac{P_1-{\lambda }_1{L}_1}{2}& \left(\mathrm{E3}\right)\end{array}$

(19) One therefore has:

(20) $\begin{array}{cc}{P}_e=D_1+\frac{{\lambda }_{1}W}{2}+\frac{\Delta H_{p,1}+\Delta H_1}{2}-\frac{{\lambda }_1{N}_1}{2}& \left(\mathrm{E4}\right)\end{array}$

(21) The term ΔΘ=(ΔH.sub.p,1+ΔH.sub.1)/2 corresponds to the difference between a receiver clock associated with the carrier code shift combination (common to all channels of the receiver, denoted Θ.sub.rec) and a transmitter clock associated with the rec, carrier code shift combination (which depends on the satellite, denoted Θ.sub.eme). This transmitter clock Θ.sub.eme is a priori unknown for each satellite (for each channel of the receiver), which is why the combination of carrier code shift was previously of little practical interest in the case of non-differentiated measurements between satellites.

(22) It is the merit of the inventors to have recognised that the transmitter part of the term ΔΘ can be estimated in a consistent manner for all satellites. This estimation is carried out using a network of reference receivers (or reference stations). The transmitter part Θ.sub.emeΔΘ can be communicated to other receivers outside the network as assistance information. Thanks to this, a receiver that is external to the network can break down the term ΔΘ into its transmitter part Θ.sub.eme and its receiver part Θ.sub.rec, common to all channels, thereby reducing drastically the number of unknowns in the system of positioning equations. As the transmitter part Θ.sub.eme of ΔΘ is determined consistently for all satellites, the solution of the system of positioning equations highlights the integer ambiguities of the phase measurements N.sub.1 for the different channels (i.e. for the different satellites).

(23) We will now describe the method that determines the assistance data provided through a network of reference receivers, part of which is schematically shown in FIG. 1. A flowchart of this method is shown in FIG. 2. The reference receivers 10, 12, 14 each perform (non-ambiguous) code measurements and (ambiguous) phase measurements on the radio-navigation signals transmitted by the radio-navigation satellites 16, 18, 20. Each satellite transmits radio-navigation signals 22, 24 on at least the frequencies f.sub.1 and f.sub.2. (For reasons of clarity, FIG. 1 does not show all the possible satellite-receiver links.) At each time step (t.sub.k), one therefore obtains, for the entire network, a set of code measurements P.sub.1,1.sup.j(t.sub.k), P.sub.2,1.sup.j(t.sub.k), L.sub.1,1.sup.j(t.sub.k) and L.sub.2,1.sup.j(t.sub.k), where the superscript (j) indicates the satellite, whose signals were received and the subscript (i) indicates the receiver that provided the measurement (step S1 in FIG. 2). As the satellites orbit the Earth, only some of them are visible at a given time from the location of each reference receiver. The dependence of the time, the index of the receiver and the satellite index will not be listed explicitly, unless it is necessary for proper comprehension.

(24) It is necessary to define the second term on the right side of equation (E2) as the “iono-free” code clock difference, denoted by Δh.sub.p=h.sub.p,rec−h.sub.p,eme Similarly, we define an “iono-free” phase clock difference, denoted by Δh=h.sub.rec−h.sub.eme. The system of equations (E1) thus becomes:
P.sub.1=D.sub.1+e+Δh.sub.p+Δτ.sub.p
P.sub.2=D.sub.2+γe+Δh.sub.p+γΔτ.sub.p
λ.sub.1L.sub.1=D.sub.1+λ.sub.1W−e+Δh+Δτ−λ.sub.1N.sub.1
λ.sub.2L.sub.2=D.sub.2+λ.sub.2W−γe+Δh+γΔτ−λ.sub.2N.sub.2,   (E5)

(25) where Δτ=τ.sub.rec−τ.sub.eme is the differential of the clock bias between the receiver and the transmitter between the “iono-free” phase clock and the phase clock for the frequency f.sub.1-by construction, the amount corresponding to the frequency f.sub.2 is γΔτ; Δτ.sub.p=τ.sub.p,rec−τ.sub.p,eme is the differential of the clock biases between the receiver and the transmitter between the “iono-free” code clock and the code clock for the frequency f.sub.1-by construction, the amount corresponding to the frequency f.sub.2 is γΔτ.sub.p

(26) With this new notation, we can express the carrier code shift combination as follows:

(27) $\begin{array}{cc}{P}_e=D_1+\frac{{\lambda }_{1}W}{2}+\Delta h+\frac{\Delta h_p-\Delta h+{\mathrm{\Delta \tau }}_p+\mathrm{\Delta \tau }}{2}-\frac{{\lambda }_1{N}_1}{2}& \left(\mathrm{E6}\right)\end{array}$

(28) Note that the value of this combination (left side of equation (E6)) is calculated directly with the code and phase measurements (step S2 in FIG. 2), whereas the values of the modelled parameters (right side of equation (E6)) are unknown at the outset. It is recognised that:

(29) $\begin{array}{cc}\mathrm{\Delta \Theta }=\Delta h+\frac{\Delta h_p-\Delta h+{\mathrm{\Delta \tau }}_p+\mathrm{\Delta \tau }}{2}& \left(\mathrm{E7}\right)\end{array}$

(30) To calculate the values of Θ.sub.eme for the different satellites, one determines, in a first step, the values h.sub.eme in a consistent manner for the network of receivers. This method (step S3 in FIG. 2) is described in the patent application FR 2 914 430 and in the article “Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and Its Application to PPP” by D. Laurichesse and F. Mercier in “Proceedings of ION GNSS 2007”, Forth Worth, Tex., September 25-28, pp. 839-848.

(31) The (non-ambiguous) phase differences between the emission of the signals by the satellite and the reception by the receiver can be written L.sub.1+N.sub.1 and L.sub.2+N.sub.2, where N.sub.1 and N.sub.2 represent the (integer) ambiguities. We set N.sub.w=N.sub.2−N.sub.1, N.sub.w being called the (integer) widelane ambiguity.

(32) One calculates the ionospheric code delay by:

(33) $\begin{array}{cc}\mathrm{eP}=\frac{P_1-P_2}{1-\gamma }.& \left(\mathrm{E8}\right)\end{array}$

(34) We set:

(35) $\begin{array}{cc}{\tilde{N}}_1=\frac{P_1-2\mathrm{eP}}{{\lambda }_1}-L_1\mathrm{et}{\tilde{N}}_2=\frac{P_2-2\gamma \mathrm{eP}}{{\lambda }_2}-L_2.& \left(\mathrm{E9}\right)\end{array}$

(36) These quantities depend only on the measurements.

(37) However, the code measurement noises are such that, at the scale of one pass (time of visibility of a satellite), the estimated Ñ.sub.1 and Ñ.sub.2 suffer from a noise of the order of tens of cycles. We use the code measurements to determine the widelane ambiguity.

(38) We introduce the gross value Ñ.sub.w (which is an estimated value) of the widelane ambiguity by:
Ñ.sub.w=Ñ.sub.2−Ñ.sub.1   (E10)

(39) Substituting (E5) in (E9) we find for N.sub.w an expression of the form:
Ñ.sub.w=N.sub.w+d+μ.sub.rec−μ.sub.eme.   (E11)

(40) where μ.sub.rec is a linear combination of τ.sub.rec, τ.sub.p,rec, h.sub.rec−−h.sub.p,rec, μ.sub.eme is a linear combination of τ.sub.eme, τ.sub.p,eme, h.sub.eme−h.sub.p,eme and d is proportional to the difference between D.sub.1 and D.sub.2. The value of d being generally less than 0.1 widelane cycles, this quantity can be neglected in what follows. By calculating the average over one pass, we have:
<Ñ.sub.w>=N.sub.w+<μ.sub.rec>−<μ.sub.eme>  (E12)

(41) On the scale of one pass, Ñ.sub.w presents a sufficiently low noise (less than the fraction of cycle) to make a correct estimate of N.sub.w and thus of μ.sub.rec and μ.sub.eme (the widelane bias μ.sub.rec and μ.sub.eme remain constant over long periods).

(42) In the absence of additional hypotheses, this mixed integer-real problem is singular: it is possible to shift N.sub.w by one integer if we change the difference μ.sub.rec−μ.sub.eme at the same time. Moreover, μ.sub.rec and μ.sub.eme are defined only to within one real constant.

(43) The calculation process is started by choosing a first network station, preferably one where it is known that the μ.sub.rec are stable over time. For this station, the value of μ.sub.rec is fixed arbitrarily, for example by setting μ.sub.rec=0. One then goes through the passes of the satellites that are visible from this station. For each pass, one has) Ñ.sub.w=N.sub.w−μ.sub.eme, by definition from the first station (with μ.sub.rec=0). One therefore breaks down (Ñ.sub.w into an arbitrary integral quantity (e.g. the nearest integer number), denoted by N.sub.w, and a quantity that is not necessarily an integer and that corresponds to the difference N.sub.w−Ñ.sub.w, denoted by μ.sub.eme. This provides the μ.sub.eme of the satellites that are visible from the first station.

(44) For the set of satellites for which one now knows the internal delays μ.sub.eme, one estimates the delays μ.sub.rec of the other stations. This time, in the equation Ñ.sub.w=N.sub.w+μ.sub.rec−μ.sub.eme, the value of μ.sub.eme is known. Ñ.sub.w+μ.sub.eme is then broken down into an arbitrary integer N.sub.w (of the new station) and the corresponding station delay μ.sub.rec. These steps are repeated for all satellites and all stations in the reference network. One thus obtains consistent values of μ.sub.rec across the entire reference network. The values μ.sub.eme can be considered constant over at least one day.

(45) After the widelane ambiguity has been determined, the ambiguity N.sub.1 remains unknown.

(46) Because the widelane ambiguity is known, solving the ambiguity of phase N.sub.1 or N.sub.2 (narrowlane ambiguity) is significantly easier, especially with regard to the precision of the required modelling.

(47) The code measurements P.sub.1 and P.sub.2 depend on several factors including the geometric distance between the points of transmission and reception, the ionospheric effects, the tropospheric effects, and the transmitter and receiver clocks. To identify the remaining ambiguities, one needs to have a sufficiently accurate model of these quantities, which will require a comprehensive resolution for the network of receivers to be treated, because of the clocks.

(48) We set:

(49) $\begin{array}{cc}{\hat{Q}}_c=\frac{{\mathrm{\gamma \lambda }}_1\left(L_1+{\hat{N}}_1\right)-{\lambda }_2\left(L_2+{\hat{N}}_1+N_w\right)}{\gamma -1}& \left(\mathrm{E13}\right)\end{array}$

(50) where {circumflex over (N)}.sub.1 is an integer estimation of N.sub.1, for example the nearest integer to Ñ.sub.1. {circumflex over (N)}.sub.1 can be removed from the true value of N.sub.1 by ten or so cycles because of code measurement noise.

(51) {circumflex over (Q)}.sub.c represents therefore an estimate of the non-ambiguous iono-free phase combination Q.sub.c (which is not directly measurable):

(52) 0$\begin{array}{cc}{Q}_c=\frac{{\mathrm{\gamma \lambda }}_1\left(L_1+N_1\right)-{\lambda }_2\left(L_2+N_1+N_w\right)}{\gamma -1}& \left(\mathrm{E14}\right)\end{array}$

(53) By setting δN.sub.1=N.sub.1−{circumflex over (N)}.sub.1, one obtains the system:
P.sub.c=D+Δh.sub.p
Q.sub.c=D.sub.w+Δh+λ.sub.cδN.sub.1,   (E15)

(54) where λ.sub.c=(γλ.sub.1−λ.sub.2)/(γ−1), D=(γD.sub.1−D.sub.2)/(γ−1) (geometric distance between the phase centres, ionospheric contribution offset), and D.sub.w=λ.sub.cD

(55) Instead of directly calculating N.sub.1, one first determines δN.sub.1. Doing this requires an accurate modelling of D.sub.w, which in particular uses the following elements: combination of the dual-frequency phase centres: this is the iono-free combination of the receiver and transmitter antenna phase centres (L1 and L2); the precise orbits of the satellites; law of satellite attitude (law of nominal yaw attitude); relativistic effects due to the eccentricity of the satellites; accurate modelling of the receiver position (with model of Earth tides); modelling of the tropospheric extension (a vertical lengthening per station with the lowering function depending on the site as defined in STANAG); modelling of Wind-up (geometric rotation of phase).

(56) The parameters estimated by the filter are: at each time step, the clocks h.sub.eme and h.sub.rec of the satellites and stations, for each pass, a constant phase ambiguity δN.sub.1 (without the constraint that this is an integer) a vertical tropospheric extension for each station, with a slow variation over time (typically a constant segment every 4 hours); precise satellite orbits (if precise orbits are not provided as input data).

(57) The filter can be in as least squares formulation or in Kalman formulation, which is more compatible with real-time processing. The input values used by the filter are the iono-free code and iono-free phase values, with their respective noise, which are the order of 1 m for the code and 1 cm for the phase.

(58) After this step, one obtains estimates of the identified residues δN.sub.1 calculated by ({circumflex over (Q)}.sub.c−D.sub.w−Δh)/λ.sub.c. Examples of residues δN.sub.1 are shown in FIG. 1. (The δN.sub.1 are not integers because no assumption about integers was made during the filtering.)

(59) This filtering step is used primarily to calculate correctly the term D, (geometric modelling). The clocks identified at this stage are subsequently used as initial values, thereby allowing the small clock variations to be worked on subsequently, but this is not essential.

(60) With the value of D.sub.w obtained by filtering, one now looks for the integer values of δN.sub.1 at the level of the reference network. Once again, one uses the equation
{circumflex over (Q)}.sub.c−D.sub.w=λ.sub.cδN.sub.1+h.sub.rec−h.sub.eme,   (E16)

(61) where D.sub.w now takes the value found by the filtering. Note that the equation has a global unobservability. Indeed, one can shift the values δN.sub.1 for a given transmitter and the corresponding values h.sub.eme and/or h.sub.rec while keeping the equation valid:
{circumflex over (Q)}.sub.c−D.sub.w=λ.sub.c(δN.sub.1+α)+(h.sub.rec−λ.sub.cα)−h.sub.eme   (E17)

(62) At this stage, one iteratively calculates the values h.sub.eme starting with a first station (a first reference receiver), whose clock is taken as the reference clock, and successively adding stations in order to complete the entire network.

(63) For the first station, one chooses δN.sub.1=0 and h.sub.rec =0. This choice is arbitrary and results in a set of h.sub.eme for the satellites that are visible from the first station, such that equation (E16) holds.

(64) The addition of a station is carried out as follows. With the set of h.sub.eme known before the addition of the station we calculate the residues δN.sub.1+h.sub.rec/λ.sub.c which must be expressed as an integer value per pass (the δN.sub.1), and a real value for each time step (corresponding to the clock h.sub.rec of the added station). FIG. 2 shows the residues δN.sub.1+h.sub.rec/λ.sub.c for a newly added station. Note that the residues are spaced by integer values and their offset from the nearest integer value is the same. We can therefore suppose that the offset between the residue and the nearest integer value corresponds to h.sub.rec/λ.sub.c and the integer value itself to δN.sub.1.

(65) Note that for a new station, the satellite clocks, and thus the residues δN.sub.1+h.sub.rec/λ.sub.c are only known, a priori, for a part of the passes. But as δN.sub.1 is constant per pass (cycle breaks are included in {circumflex over (Q)}.sub.c−D.sub.w), it can be extended to the entire pass. The times at which a given satellite is visible from a station correspond only partially to the times at which the satellite is visible from a neighbouring station. The greater the distance between stations, the shorter the length of common observation time. This implies that one always adds a station neighbouring with at least one of the previous stations.

(66) Note that together with the set of integer δN.sub.1, one also obtains a set of consistent satellite clocks h.sub.eme and receivers h.sub.rec, having the first station's clock as its reference clock.

(67) Using the values {μ.sub.eme.sup.j} (valid for at least one day) and {h.sub.eme.sup.j} (needing to be updated each epoch), a dual-frequency receiver outside the network can determine the ambiguities N.sub.1 for the satellites in view in an efficient manner.

(68) On the other hand, to find the quantity Θ.sub.eme, one must also know the transmitter part of the second term on the right side of the equation (E7), that is (h.sub.p,eme−h.sub.eme+τ.sub.p,eme+τ.sub.eme)/2, which will be denoted C.sub.eme hereafter. We furthermore define:
C.sub.rec=(h.sub.p,rec−h.sub.rec+τ.sub.p,rec+τ.sub.rec)/2 and ΔC=C.sub.rec−C.sub.eme.

(69) In the network of reference receivers, the values of μ.sub.rec, N.sub.w, N.sub.1, h.sub.rec and h.sub.eme are known thanks to the preceding calculations. The quantity ΔC is thus observable:

(70) $\begin{array}{cc}\Delta C=\frac{P_1+{\lambda }_1{L}_1}{2}-D_1-\frac{{\lambda }_{1}W}{2}-\Delta h+\frac{{\lambda }_1{N}_1}{2}=C_{\mathrm{rec}}-C_{\mathrm{eme}}& \left(\mathrm{E18}\right)\end{array}$

(71) Note that ΔC corresponds to the measurement residue obtained by subtracting from the carrier code shift combination (i.e. from the term (P.sub.1+λ.sub.1L.sub.1)/2), the (modelled) geometric contribution which includes the distance and the windup effect (i.e. the term D.sub.1+λ.sub.1W/2), the contribution of the receiver and transmitter iono-free phase clock values (i.e. the term Δh) and the contribution of the ambiguity of the phase measurement (i.e. the term −λ.sub.1N.sub.1/2). One can calculate the receiver and transmitter parts C.sub.rec and C.sub.eme of the measurement residue for example by the method of least squares at each time step (step S4 in FIG. 2). Taking into account all receivers in the network, we have a system of equations of the form:
{ΔC.sub.i.sup.j=C.sub.rec,i−C.sub.eme.sup.j}.sub.i,j,   (E19)

(72) which can be written {right arrow over (ΔC)}=Γ{right arrow over (x)}, where Γ is the matrix of partial derivatives and {right arrow over (x)} is the vector of the unknowns: {right arrow over (x)}=(C.sub.rec,1, . . . ,C.sub.rec,I,C.sub.eme.sup.1, . . . ,C.sub.eme.sup.J).sup.T with i=number of receivers in the network and J=number of satellites.

(73) Note once again that the C.sub.eme depend only on the satellites whereas the C.sub.rec depend only on the receivers. Since the system (E19) includes one equation less than unknowns, we need to add a constraint equation, which fixes for example the sum or average of the C.sub.eme to 0:
Σ.sub.jC.sub.eme.sup.j=0   (E20)

(74) The values C.sub.eme that we find are very noisy because of the code noise on the measurements. Two examples of plots 26, 28 of C.sub.eme are shown in FIG. 4. Preference is given to smoothing the values C.sub.eme by a model (step S5 in FIG. 2). It was found that it is preferable to use a sine function with a period of 12 hours to adjust it to each C.sub.eme. Thus one finds the smoothed values C′.sub.eme.sup.j(t). FIG. 5 shows the plots of FIG. 4 after smoothing with the sine functions.

(75) As assistance data, one finally has (step S6 in FIG. 2) iono-free transmitter clock values h.sub.eme (a value by time step and not by satellite) and the values C′.sub.eme (identified by the parameters of the sine functions) that can be transmitted to a receiver outside the network and used by it to fix the ambiguities N.sub.1.

(76) A method for geopositioning in a receiver outside the reference network, which provides the values h.sub.eme and C′.sub.eme is shown schematically in FIG. 3. The receiver performs code and phase measurements for satellites visible from its geographical location (step S7 in FIG. 3). In addition, it receives the values h.sub.eme and C′.sub.eme (step S8 in FIG. 3). With the code and phase measurements on the frequency f.sub.1, the receiver can derive the carrier code shift observable (step S9 in FIG. 3). Furthermore, by using the assistance data, the receiver can then calculate the quantities:

(77) $\begin{array}{cc}\frac{P_1^j+{\lambda }_1{L}_1^j}{2}+h_{\mathrm{eme}}^j+C_{\mathrm{eme}}^{\prime j}=D_1^j+\frac{{\lambda }_1{W}^j}{2}-\frac{{\lambda }_1{N}_1^j}{2}+h_{\mathrm{rec}}+C_{\mathrm{rec}}& \left(\mathrm{E21}\right)\end{array}$

(78) In system (E21), the satellite indices were used to illustrate more clearly what terms are common to all the satellites (all the channels of the receiver). Note that the term h.sub.rec+C.sub.rec is common to all the receiver channels and is equivalent to a global clock, to be estimated at each time step (i.e. for each measurement). The phase ambiguity N.sub.1, which depends on the satellite, only takes a single value per pass (time of satellite visibility) if we assume that the observation is not interrupted during this time and that the phase jumps are detected and included in the phase measurement L.sub.1. It is worth noting that it is sufficient for the receiver to know the values h.sub.eme+C′.sub.eme; the values h.sub.eme and C′.sub.eme do not therefore need to be transmitted separately. Note also that h.sub.rec+C.sub.rec corresponds to Θ.sub.rec as defined above; in the same way: h.sub.eme+C′.sub.eme=Θ.sub.eme.

(79) If the receiver position and the corresponding tropospheric extension are known (i.e. if D.sub.1 is known), the quantities (P.sub.1λ.sub.1L.sub.1)/2 −λ.sub.1W/2−D.sub.1h.sub.eme+C.sub.eme aggregate around values separated by intervals equal to integer multiples of λ.sub.1/2.

(80) If one wishes to determine the position of the receiver (D.sub.1 in this case being unknown at the start), we can, for example, solve a least squares problem over a certain period (e.g. a few hours) in which the following parameters are to be determined (step S10 in FIG. 3): the position of the receiver (included in the modelling of D.sub.1); the vertical tropospheric extension (included in the modelling of D.sub.1); the clock receiver associated with the carrier code shift combination (h.sub.rec+C.sub.rec); and the phase ambiguities (one value per pass).

(81) The ambiguities can then be determined by a “bootstrap” mechanism: an ambiguity is fixed to an arbitrary integer, after which the other ambiguities aggregate around integer values and can be determined iteratively.

(82) Using the information h.sub.eme and C′.sub.eme, the carrier code shift measurement becomes an unambiguous observable, free from the ionospheric contribution (“iono-free”), and having a noise equal to half the code noise. The PPP (“Precise Point Positioning”) has an accuracy of 20 to 50 cm in purely stochastic positioning. Its accuracy increases rapidly if one performs a static positioning of the receiver by storing several minutes' worth of measurements. For example, one obtains a positioning accuracy of about 10 cm for a quarter of an hour of measurements and about 2 cm for half an hour of data (with a receiver that is not too affected by multipath). The boot time is around an hour.