WO2017070732A1

WO2017070732A1 - A method of analysing a signal transmitted between a global satellite navigation satellite system and a receiver

Info

Publication number: WO2017070732A1
Application number: PCT/AU2016/050951
Authority: WO
Inventors: Peter J G TEUNISSEN; Dennis ODIJK; Robert ODOLINSKI; Amir KHODABANDEH; Baocheng Zhang
Original assignee: Spatial Information Systems Research Ltd
Priority date: 2015-10-27
Filing date: 2016-10-10
Publication date: 2017-05-04

Abstract

The present disclosure discloses a method of analysing a signal transmitted between a global navigation satellite system and a receiver to determine estimable parameter functions for use in an application. The method comprises the step of selecting the application and identifying parameters required by the application. The method further comprises the step of analysing the signal to identify observed quantities of the signal able to be associated with the parameters. The method also comprises determining any unknown parameters being associated with any one of the observed quantities and formulating these unknown parameters into parameter functions. The parameter functions are representative of all the unknown parameters without omitting any unknown parameters. In addition, the method comprises identifying the parameter functions to derive estimable parameter functions from which the unknown parameters can be determined.

Description

A method of analysing a signal tr nsmitted between a global satellite navigation system and a receiver

The resent invention relates to a method of analysing a signal transmitted between a global satellite navigation system and a receiver.

More particularly, the present invention relates to a method of analysing a signal transmitted between a global satellite navigation system (GNSS) and a receiver to identify and transform estimable parameter functions for use in GNSS applications . BACKGROUND

In precise point positioning (PPP) a user employs a single global navigation satellite system receiver that collects carrier-phase and code (pseudo-range) observations from a signal transmitted by a satellite, to determine the position of the receiver with centimetre to decimetre accuracy. For this purpose the user needs precise satellite orbit and clock products, which are obtained from an external provider, such as the International GNSS Service. The positioning accuracy that can be achieved with PPP strongly depends on the observation time over which the signal is accumulated by the receiver. Single- frequency GNSS PPP, which relies on the availability of ionospheric corrections (obtained from e.g. Global Ionospheric Maps), has a positioning accuracy which is typically at decimetre level after about 15 minutes. In the absence of such ionospheric corrections, but based on the ionosphere- free combination of dual- frequency observations, the accuracy of GNSS PPP can reach centimetre level, but this requires a much longer signal observation time span, e.g. at least one hour.

Although PPP is based on the very precise carrier-phase data, this high precision cannot be exploited as ambiguities in the carrier-phase data are not estimable as integers. In relative positioning techniques, where use is made of data from a reference station, such as Real-Time Kinematic (RTK) positioning, the carrier-phase ambiguities are estimable and can be resolved to integers, allowing for mm-cm level positioning accuracy of the receiver based on the high precision of phase observations. Special algorithms have been developed for the crucial integer ambiguity resolution process, of which the LAMBDA method is the de facto standard. However, integer ambiguity resolution is only feasible within short time spans for relatively short distances between the receiver and the reference station (i.e. less than 10 km), based on the assumption that the differential ionospheric delays can be ignored.

Several methods have been proposed for integer ambiguity resolution enabled PPP. These methods aim to obtain RTK-like positioning accuracy by resolving the phase ambiguities in the observations by the single receiver where the user incorporates additional satellite and receiver related corrections. In this sense PPP can be considered as a relative technique as well, as these orbit, clock and hardware bias corrections are determined by a global or regional network of reference stations.

However, these methods have restrictions in their applicability as most of them are restricted to dual- frequency observations and are based on ionosphere- free combinations, thereby a priori eliminating the

ionospheric delays. As with RTK, the forming of ionosphere-free combinations is clearly unfavourable for PPP because it is not possible to obtain integer ambiguity resolution within reasonably short

observation time spans. Additionally, methods based on the ionosphere- free combination are not suitable for single-frequency applications. These methods are therefore restrictive in the light of the development of new multi- frequency GNSS constellations, as well as due to PPP-RTK users requiring ionospheric corrections to obtain fast integer ambiguity resolution results.

It is to be understood that, if any prior art publication is referred to herein, such reference does not constitute an admission that the publication forms a part of the common general knowledge in the art in any country. In the specification hereinafter, reference is made to S-system theory as described in Teunissen, "Generalised inverses, adjustment, the datum problem and S-Trans formations ", Springer-Verlag, pp 11-55, 1985. SUMMARY

According to one aspect, there is provided a method of analysing a signa transmitted between a global navigation satellite system and a receiver to determine estimable parameter functions for use in an application, th method comprising the steps of:

selecting the application and identifying parameters required by the application;

analysing the signal to identify observed quantities of the signal able to be associated with the parameters;

determining any unknown parameters being associated with the observed quantities and formulating these unknown parameters into parameter functions, and

identifying the parameter functions to derive estimable parameter functions from which the unknown parameters can be determined.

The parameter functions are in one specific embodiment of the present invention representative of all unknown parameters associated with the observed quantities.

The signal may be transmitted by the global satellite navigation system and tracked by a single receiver. The signal may be a single frequency signal. The signal may be a multiple frequency signal.

The application may comprise determining location parameters for use in determining a location of the receiver.

The application may comprise determining atmospheric parameters for use in weather forecasting. The atmospheric parameters may comprise ionospheric and/or tropospheric delays.

The application may comprise determining coordinated universal time by estimating variations between the receiver' s time and the global navigation satellite system's time.

The application may comprise determining satellite bias parameters in a common clocks S-system. The satellite bias parameters in the common clocks S-system may comprise satellite orbits, satellite clocks, satellite phase biases, satellite code biases or satellite differential code biases .

The application may comprise determining receiver bias parameters in a common clocks S-system. The receiver bias parameters in the common clocks S-system may comprise receiver phase biases, receiver code biases or intersystem biases.

The unknown parameters may include ionospheric corrections. The ionospheric corrections may comprise an estimate of slant ionospheric delay . The step of resolving the parameter functions may comprise the steps of: representing the parameters in a design matrix;

identifying rank deficiencies within the design matrix and using the rank deficiencies to construct a null space basis matrix;

selecting a S-basis complementary to the null space basis matrix, wherein the S-basis is dependent on the application, and using the S- basis to construct a S-basis matrix;

calculating a S-trans formation matrix utilising the null space basis matrix and the S-basis matrix;

whereby the S-trans formation matrix enables the estimable parameter functions to be derived.

Construction of the design matrix may comprise ordering the parameters according to their parameter types. The ordering may be arranged by geometry parameters, receiver-dependent parameters, satellite-dependent parameters, ionospheric parameters, and ambiguity parameters. The rank deficiencies identifiable may comprise deficiencies between receiver and satellite clocks, deficiencies between receiver and satellite hardware biases, deficiencies between receiver clocks and receiver hardware biases, deficiencies between satellite clocks and satellite hardware biases, deficiencies between receiver phase biases and ambiguities or deficiencies between satellite phase biases and

ambiguities .

The step of identifying rank deficiencies may comprise the steps of identifying additional rank deficiencies and identifying additional columns of the null space basis matrix. The additional rank deficiencies may result from parametrization of slant ionospheric delays. The additional rank deficiencies may result from regional network

assumptions . BRIEF DESCRIPTION OF THE FIGURES

The present invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which:

Figure 1 shows a flow diagram of the steps applied in the method of analysing a signal;

Figure 2 shows a table providing a definition of some special scalars, vectors and matrices;

Figure 3 shows a flow scheme to determine additional rank

deficiencies in the absence of temporal (random-walk) constraints on one or more types of parameters;

Figure 4 shows a table representing columns to be added to a null space matrix V_net for a regional-sized network, in absence or presence of temporal constraints on geometry and satellite clocks, satellite hardware bias parameters, or ionospheric delays, or in case of a time-constant geometry;

Figure 5 shows a table providing a choice of S-basis constraints under a Common Clocks (Pivot) Receiver (CC-R) S-system;

Figure 6 shows a table providing additional constraints for the CC-R S-basis in case ionospheric slant delays are parametrized;

Figure 7 shows a table listing estimable parameter functions plus their interpretation in the CC-R S-system;

Figure 8 shows a table representing changes to estimable parameter functions in the CC-R S-system due to parametrization of ionospheric slant delays instead of vertical delays; and

Figure 9 shows a table representing changes of estimable parameter functions in the CC-R S-system with respect to the table in Figure 7 for a regional-sized network for which additional rank deficiencies occur that depend on the presence/absence of temporal constraints on

positions/ZTDs (geometry), satellite clocks, satellite hardware biases and vertical ionospheric delays. DETAILED DESCRIPTION

The present disclosure now is described more fully hereinafter with reference to the accompanying figures. The present disclosure provides a method to utilise undi fferenced and uncombined observation equations for the processing and analysis of multi- frequency carrier-phase and code data from global navigation satellite systems, tracked by a network of receivers or by a single receiver. By utilising all the parameters, the method is very flexible allowing for use thereof in a range of GNSS applications, including the generation of ionospheric corrections. An undi fferenced observation equation provides the advantage of being able to use a simple observational variance matrix, while having all the parameters remain available making further model strengthening possible. Parameters that are not considered of interest then can be relatively easily eliminated through reduction of the normal equations, instead of performing an a priori elimination of such parameters at the

observational level.

Due to working with undi fferenced observation equations, there is a need to account for rank deficiencies as not all unknown parameters can be estimated unbiasedly. This is achieved by applying S-system theory, originally developed for terrestrial geodetic networks, to solve for the rank-deficient system of observation equations and to allow for a proper interpretation of the estimable network and user parameters.

It will be appreciated that as the underlying network and user models are rank deficient, and even vary in their rank deficiencies depending on the chosen measurement set-up and/or modelling, different sets of estimable parameters, each with their own interpretation, exist, and each such set is defined by the chosen S-basis. For instance, after resolving the rank-deficient observation equations, one cannot speak any more of a satellite clock, or a satellite phase bias, or a receiver code bias. These parameters, although existing in their original physical form, simply cannot be estimated as such. What can be estimated are certain functions of the parameters; functions that then can be treated as representing a satellite clock, a satellite phase bias, or a receiver code bias. These parameters cannot be simply combined or equated, but with a careful application of S-system theory it is possible to give a clear description of the estimable parameters that are involved in the different network and user models. By means of the S-trans formation, the relation between the original "absolute" parameters and the estimable parameters then can be established. Due to the generality of S-system theory, any existing or future PPP-RTK model formulation can be cast in this framework, thereby directly providing the interpretation that should be given to the resolved parameters of the chosen formulation.

In the present disclosure S-system theory is used to provide a general multi- frequency, multi-epoch formulation, thereby creating a high level of flexibility. The derived models are therefore valid for any multi- frequency GNSS constellation incorporating the Code Division Multiple Access (CDMA) technology, such as GPS, BeiDou or Galileo and future GLONASS as well.

The method of the present disclosure allows different types of

information to be extracted from the GNSS signal for use in different types of applications. For example, the method to unbiasedly identify and transform parameter functions could be used in the following applications :

• Identification of position (location) information from GNSS

satellites through receiver technology. The method is applicable to all current GNSS-based positioning methods (i.e. DGNSS, PPP, RTK and PPP-RTK) .

• Estimation of atmospheric parameters for numerical weather

forecasting (ZTDs) and space weather forecasting (TEC) . · Determination of time transfer information between different

systems (i.e. determining coordinated universal time by estimating variations between the receiver' s time and the global navigation satellite system's time) .

• Estimation of satellite bias parameters (i.e. satellite orbits, clocks, phase biases, code biases or DCBs, ionospheric delays) based on data of either a network of receivers or even a single receiver . • Estimation of receiver bias parameters (i.e. receiver code and phase hardware bias parameters, as well as ISBs between different GNSSs), applicable to calibration purposes.

The method can be used in the above wide range of applications because the undifferenced, uncombined observation equations keep all the unknown parameters in the system (i.e. there is no prior elimination) . Depending on the desired application (which determines the desired type of GNSS information to be used from the GNSS signal) as well the required precision level of the information, a mathematical model is set up which relates the observed quantities (GNSS observables) to unknown parameters. Depending on the type of application this model is numerically solved to extract the estimated parameters (plus their precision description) . The estimated parameters (or a subset) may or may not form input for other GNSS-based applications. GNSS observables do not contain enough information to determine all unknown parameters unbiasedly. However, this problem is overcome in the present method by making use of S-system theory to estimate parameter functions. The type of parameter functions that are estimated depends on the type of application for the method and the types of GNSS observables that are required. For high-precision GNSS applications it is necessary to use carrier-phase measurement of the GNSS signals, whereas

pseudo-range measurements will usually suffice for applications that require less accuracy. By means of selecting the appropriate S-basis (i.e. the necessary and sufficient set of inestimable parameter functions) the estimable parameter functions are identified in the associated S-system. By means of an S-trans formation (which becomes known once the S-basis is selected), solutions in different S-systems can be directly transformed and compared.

The method of the present invention receives an ^input' of the type of (multi-) GNSS application for which the estimable parameter functions need to be identified or transformed. Based on this input, the method applies a five step process as described below and illustrated in Figure 1.

1. Construction of the (rank-deficient) system of GNSS observation

equations for the application at hand. The system of GNSS observation equations are the fundaments of any GNSS observation model. Thus the application determines the design elements such as number of receivers, epochs, frequencies, types of parameters, dynamic model, type of ionospheric parameters - slant or model

coefficient .

Consider the equations that relate the GNSS observables to the unknown GNSS parameters that need to be determined:

These observation equations are linearized, because the receiver position is a nonlinear function of the receiver-satellite range. As GNSS observables are random variables, the observation equations are denoted as the mathematical expectation, denoted using E{.}, of the phase observables, denoted as A _rj ^S(i), and the code observables, denoted as Ap_rj^S i). Due to the linearization the observables appear as ^observed- minus-computed' in the observation equations.

Now an overview is given of the unknown parameters in the GNSS

observation equations. First, there are position coordinates (3 per receiver), as well as a zenith tropospheric delay (ZTD; 1 per receiver) . This zenith tropospheric delay is the result of the mapping of the slant tropospheric delays for one receiver to the local zenith of that receiver. For notational convenience, these are combined for one receiver in a 4-dimensional vector, denoted as x_r(i). Their

coefficients, i.e. the receiver-satellite line-of-sight vector for the position and the tropospheric mapping function for the ZTD, are stored in 4-dimensional vector g_r ^s (i) . For the applications of this invention, it is assumed that the position of the satellites are known (computed from either the satellite's navigation message, or externally provided by e.g. the International GNSS Service) and thus do not appear as unknown parameters. The second type of parameters are the receiver-dependent parameters: a receiver clock offset, denoted as dtr(i), a receiver phase bias (per frequency), denoted as 5_Irj(i), and a receiver code bias (per frequency), denoted as d_Irj(i). The third type of parameters are the satellite-dependent parameters. These are very similar to the receiver- dependent parameters: a satellite clock offset, denoted as dt^s (i) , a satellite phase bias (per frequency), denoted as 5_j ³ (i) , and a satellite code bias (per frequency), denoted as d_j ³ (i) . The fourth type of parameters are the ionospheric delays, denoted as _r ^s(i). Because of the dispersive character of the ionosphere, the ionospheric delays for all frequencies are mapped to the ionospheric delay of the first frequency. It is furthermore assumed that the ionospheric delays of different receivers to one satellite can be mapped to one vertical ionospheric delay parameter for that satellite, i.e. _r ^s(i) = _r ³(i) (i) , with the vertical ionospheric delay and the ionospheric mapping function. We will later describe the situation if this mapping is not done and slant ionospheric delays are estimated, i.e. a parameter per receiver-satellite combination. The frequency-ionospheric coefficient is denoted and defined as _j = (X_j/ λ₁)², with the signal's wavelength denoted as X_j . The fifth type of parameters only apply to the carrier-phase data: these are the ambiguities (per frequency), denoted as z_Irj ³. It is known that these ambiguities cannot take on arbitrary values, but only integer values .

All observables and parameters are expressed in meters, except for the phase unique parameters, i.e. receiver/satellite phase bias and ambiguities, which are expressed in cycles (one cycle corresponds to one wavelength) . Carrier-phase ambiguities are constant in time (in the absence of cycle slips; in the presence of cycle slips existing statistical methods of cycle slip repair are addressed to restore the time constancy of the ambiguities), whereas it is initially assumed there are dynamic models on all time-varying parameters (denoted with epoch index i), that link these parameters in time. Again, we will later describe the situation for applications that do not assume dynamic models on one or more types of parameters. It is assumed that the noise of the time-varying parameters can be described as a random walk stochastic process. This means that they are linked in time as follows : Here the epoch index i runs from the second to the kth epoch. The vector j denotes the state of one type of parameters, whereas ΔΧ;— X;— x denotes the state vector minus an a priori value denote as x . This subtraction is needed as the stochastic process should have zero mean. The vector W; denotes a (Gaussian) white noise random process having zero mean. As we assumed this stochastic process to be random walk, the above transition matrix equals an identity matrix: Φ;;_ι— I■

The indices r, s and j in above formulas denote the receiver index, satellite index and frequency index. They depend on the application at hand. For example, in the case of a single receiver tracking phase and pseudo-range observations from 8 satellites at two frequencies r=\ , s=\,..., 8 and j=l,2.

2. Construction of the (rank-deficient) design matrix.

The design matrix of a mathematical model relates the undi fferenced, uncombined (multi) GNSS observables to the unknown parameters. This matrix is rank deficient, which means that it contains columns that are linear dependent since it is not possible to uniquely solve for all parameters. Thus these rank deficiencies need to be overcome in order to identify the model's estimable parameter functions. To construct the design matrix on the basis of the observation equations and dynamic state-space equations of step 1, a certain order of observables and parameters is fixed. Assuming the following ordering of observables: first all phase observables, then all code observables and after that the random-walk temporal constraints. Within phase/code observables and temporal constraints the order is: epochs - frequencies - receivers - satellites (as the temporal constraints depend on the parameter type they are not all frequency, receiver and satellite dependent at the same time) . The unknown parameters follow the order of the temporal constraints. The order of parameter types is: geometry parameters (containing receiver positions and ZTDs) . - receiver-dependent parameters - satellite-dependent parameters - ionospheric parameters - ambiguity parameters. To be able to identify the rank deficiencies, we combine all GNSS observables and random-walk (temporal) constraints into one vector, denoted as y_net and write the model in the following (batch) form: E(y_net) = A_net x · The rank-deficient design matrix in generic form then reads as follows:

Here ( . ) ^T denotes a transposed matrix and ( . ) ^_1 an inverse matrix. This 5 design matrix, generally denoted as A_net, can be seen as being built up of different sub-matrices. For an efficient and compact notation, these sub-matrices are denoted using a Kronecker product. This Kronecker product is defined as follows. Let A be an m x n matrix and B be a p x q matrix. The mp x nq matrix defined by (A) _i;jB is called the Kronecker 10 product and it is denoted as A ® B = (A)_i;jB. Properties of the Kronecker product are (assuming all matrices have the appropriate dimensions):

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IInn tthhee ccoommppaacctt ddeessiiggnn mmaattrriixx ssoommee ssccaallaarrss,, vveeccttoorrss aanndd mmaattrriicceess aarree uusseedd 1155 tthhaatt aarree ddeeffiinneedd iinn tthhee ttaabbllee sshhoowwnn iinn FFiigguurree 22.. IInn aaddddiittiioonn,, tthheerree aarree mmaattrriicceess FF_ggeeoo aanndd FF_iioonn tthhaatt aarree aa ffuunnccttiioonn ooff tthhee rreecceeiivveerr--ssaatteelllliittee ggeeoommeettrryy aanndd iioonnoosspphheerriicc mmaappppiinngg,, rreessppeeccttiivveellyy.. FFiirrsstt,, tthhee ggeeoommeettrryy mmaattrriixx iiss ddeeffiinneedd aass

^λλbbllkkddiiaagg'' ddeennootteess aa block-diagonal matrix and:

20 with receiver-dependent matrices G_r(i) containing the geometry vectors for receiver r and m satellites. In a similar way, the ionosphere matrix is denoted by

Fi i.on = blkdiag [F_ion(l), ... , F_ion(k)], with:

with matrices M_r(i) containing the ionospheric mapping functions for receiver r and m satellites.

The design matrix parts corresponding to the different type of parameters are the 'columns' within above matrix separated by the vertical lines, thus A_geo for the position and ZTD for all receivers; A_rec all pure receiver dependent parameters, i.e. receiver clocks, receiver phase biases and receiver code biases; A_sat for all pure satellite dependent parameters, i.e. satellite clocks, satellite phase biases and satellite code biases; A_ion for all ionospheric parameters and A_amb for the ambiguities .

3. Identification of the rank deficiencies and null space.

Having constructed the design matrix, the next step is the identification of its rank deficiencies to enable the construction of a basis matrix of the null space. The analytically constructed null space forms the foundation of the identification and transformation of the parameter functions. In general, the following five types of rank deficiencies are identifiable, for the GNSS model that has random-walk constraints on all parameters (except ambiguities that are time constant):

I. Between receiver and satellite dependent parameters, i.e.:

X . Between receiver clocks and satellite clocks

ii. Between receiver hardware biases and satellite hardware biases (for each frequency; phase and code)

II . Between receiver clocks and receiver hardware (phase and code) biases III. Between satellite clocks and satellite hardware (phase and code) biases

IV. Between receiver phase biases and ambiguities

V. Between satellite phase biases and ambiguities Additional rank deficiencies however show up in case the model

assumptions are changed in the following ways, as they cause a change in the design matrix:

• Parametrization of slant ionospheric delays instead of vertical

ionospheric delays In that case the ionospheric delays are not mapped to vertical delays per satellite, but the original receiver-satellite specific slant ionospheric delays are maintained as parameters. In terms of the design matrix this means that the part A_ion is replaced by the following matrix:

Due to this reparametrization the following additional rank deficiencies occur :

I. Between receiver hardware (phase and code) biases and slant

ionospheric delays.

II. Between satellite hardware (phase and code) biases and slant ionospheric delays.

• Regional-sized networks

For regional-sized networks all receivers experience approximately parallel line-of-sight vectors. In the limiting case, assuming identical line-of-sight vectors for all receivers, this implies that Gi(i) = ■■■= G_n(i) = G(i) for the geometry matrices, as well as Ά_λ(±) = ...=

M_n(i) = M(i) for the ionospheric mapping function matrices. This then has as consequence that the geometry and ionosphere submatrices in the design matrix are replaced by:

This regional-network assumption causes additional rank deficiencies, but only in absence of temporal constraints on the geometry parameters and/or satellite-dependent parameters:

I. Between positions/ZTDs and satellite clocks/hardware biases (occurs in presence/absence of geometry temporal constraints but in absence of satellite clocks/hardware bias temporal constraints)

II. Between satellite hardware biases and vertical ionospheric delays (occurs in absence of satellite hardware bias constraints but in presence/absence of vertical ionospheric temporal

constraints )

• Absence of random-walk constraints on one or more parameter types In general, irrespective of the regional-network assumption, the rank deficiencies will alter in the absence of the temporal constraints on one or more parameter types. The additional rank deficiencies are however always a combination of the earlier identified types. Figure 3 presents a scheme from which the additional rank deficiencies can be identified for each situation, depending on the combination of parameter types with or without random-walk constraints. The starting point of this scheme is the size of the rank deficiency in the network model in presence of all random-walk constraints. The additional multi-epoch rank deficiencies are denoted in the scheme using an asterisk. Not all combinations of parameter types without temporal constraints lead to additional rank deficiencies. The total amount of the additional rank deficiencies follows from adding up the amounts corresponding to the number (plus letter; look up at the bottom of the scheme) for the random walk constraints that are left out in the model. Note I: -geo = no geometry constraints; -rcl = no receiver clock constraints; -scl = no satellite clock constraints; -hw = no receiver and satellite hardware bias constraints; -ion = no ionospheric delay constraints. Note II: a rank deficiency within a square only occurs in case of a "regional- sized" network, whereas a rank deficiency within a circle only occurs in case of a slant (instead of vertical) ionospheric parametrization . Note III: in absence of ionospheric random-walk constraints, the additional rank deficiencies of types 3b* and 0c* cannot occur at the same time, as they are similar types of rank deficiencies. Thus, in case of a slant ionospheric parametrization type 3b* occurs (and not 0c* even for a regional-sized network) , whereas in case of vertical ionospheric parameters type Oc* occurs (and not 3b*), only if the network is regional .

After identification of the rank deficiencies the null space is constructed. For the general design matrix (including all random-walk constraints) a basis matrix of this null space can be identified as:

Here the five types of rank deficiencies each correspond to a ' column' (separated by vertical lines) in the above matrix. For the above matrix it holds that A_net V_net = 0.

In the situation that the network model parameterizes slant instead of vertical ionospheric delays, the rank deficiency of the model changes by additional rank deficiencies, resulting in the following 'columns' being added to above null space matrix:

In the situation of the ^xregional-sized' network assumption, the columns to be added to the null space matrix depend on the presence or absence of temporal constraints on the geometry parameters, as well as satellite clocks, satellite hardware biases and vertical ionospheric parameters. The table shown in Figure 4 presents an overview of the different situations that can occur.

4. Choice of S-basis and construction of the S-transformation matrix.

To overcome the identified rank deficiencies a choice of S-basis has to be made. This S-basis choice is driven by the application at hand, as certain choices may be more advantageous than others. Having selected the S-basis and constructed the S-basis matrix, with the use of the null space matrix the S-trans formation matrix can be constructed.

The method allows for user-freedom in choosing the S-basis for the application at hand. For example, for a GNSS network that has the goal to generate PPP-RTK products, a decision for a certain S-basis can be to stick as close as possible to the PPP products of the International GNSS Service (provision of ionosphere-free satellite clocks, Differential Code Biases) . The choice can furthermore be driven by the requirement to obtain estimable receiver and satellite clocks that are common for the phase and code observables at different frequencies. The above

considerations would then result in a Common Clocks S-system. Another example is that the estimable ambiguity parameters are only a combination of the ambiguity terms themselves, such that they are directly estimable as integer parameters.

For example, the table shown in Figure 5 presents the S-basis constraints corresponding to a Common Clocks (Pivot) Receiver (CC-R) S-system. This choice of S-basis results in estimable clock parameters that are common for the phase and code observables. The term "Pivot Receiver" refers to the choice to constrain parameters corresponding to one of the receivers (i.e. the pivot; denoted using subscript 1 in the table) in the network.

This choice of CC-R S-basis constraints can then be cast in the following S-basis matrix:

With this choice all ingredients are available to calculate the S- trans formation matrix. For this example this results in the following matrix :

In presence of additional rank deficiencies due to parameterization of slant instead of vertical ionospheric delays, additional parameters need to be constrained in the S-basis. In case of the CC-R S-basis these additional S-basis constraints could be those as given in the table shown in Figure 6.

In case of the ^xregional-sized' network assumption the additional rank deficiency is overcome by constraining the coordinates+ZTDs of the pivot receiver and/or the geometry- free satellite code biases (similar to the S-basis constraints in Figure 6) . 5. Identification of the estimable parameter functions . Having constructed the S-trans formation in the previous step, the estimable parameter functions automatically follow from the rows of the S-trans formation matrix. For the example of the CC-R S-system these estimable parameter functions are presented in the table shown in Figure 7 for the model in which random-walk constraints are included for all types of parameters (except the ambiguities) .

In the cases of the additional rank deficiencies some of the estimable parameter functions change (i.e. their estimability or the 'conditions' as in Figure 7, as well as their interpretation) . For example, when the ionospheric parameters are modelled as slant instead of vertical delays, this affects the estimable parameter functions for receiver hardware biases, satellite hardware biases as well as the ionospheric delays. The table shown in Figure 8 presents these changed parameter functions.

In case of the 'regional-sized' network assumption, the altered estimable parameter functions in the CC-R S-system are presented in the table shown in Figure 9 as a consequence of the additional rank deficiencies that occur .

Thus, the output of the method consists on the one hand of the estimable parameter functions and on the other hand the S-trans formation matrix that is needed to transform these parameter functions to other S-systems.

The above method provides the following advantages over known precise point positioning systems:

It is flexible, as the method is generic in the following ways. o It is valid for:

any number of satellites types (for single or multiple

GNSS constellations)

any number of receivers (i.e. network/single

baseline/single receiver)

any distance between the receivers

(local/ regional /global)

o In addition, the method allows for dynamic state-space models on some, all or none of the parameters. o The method also allows for applications for which some parameters are absent as the measurements have been corrected for them.

Thus the method is applicable to any GNSS-based application, whereby it is not restricted to single GNSS applications, as it has the capability to deal with multi-GNSS combinations as well.

• It allows for an evaluation of the estimable parameters plus their precision description in the chosen S-system. Furthermore, the interpretation of the estimable parameters unambiguously follows from the identified functions of parameters.

• It is efficient, as only the appropriate S-trans formation (which follows from the S-basis choice) is needed to transform the GNSS parameters (plus their precision description) to any other S- system. The S-trans formation provides the correct method of comparing solutions obtained in different S-systems.

The foregoing is illustrative of the present invention and is not to be construed as limiting thereof. Although a few exemplary embodiments of this invention have been described, those skilled in the art will readily appreciate that many modifications are possible in the exemplary embodiments without materially departing from the novel teachings and advantages of this invention. Accordingly, all such modifications are intended to be included within the scope of this invention as defined in the claims. For example, although this work has focussed the S-system theory on positioning, due to its generality the same method can be applied to non-positioning applications as well, such as GNSS-based ionospheric studies, time transfer or receiver bias calibration.

In the claims which follow and in the preceding description of the invention, except where the context requires otherwise due to express language or necessary implication, the word "comprise" or variations such as "comprises" or "comprising" is used in an inclusive sense, i.e. to specify the presence of the stated features but not to preclude the presence or addition of further features in various embodiments of the invention . List of abbreviations:

GNSS = Global Navigation Satellite System

GPS = Global Positioning System

DGNSS = Differential GNSS

PPP = Precise Point Positioning

RTK = Real-Time Kinematic

PPP-RTK = Precise Point Positioning Real-Time Kinematic ZTD = Zenith Tropospheric Delay

TEC = Total Electron Content

DCB = Differential Code Bias

ISB = Inter-System Bias

Claims

A method of analysing a signal transmitted between a global navigation satellite system and a receiver to determine estimable parameter functions for use in an application, the method comprising the steps of:

determining any unknown parameters being associated with the observed quantities and formulating these unknown parameters into parameter functions; and

identifying the parameter functions to derive estimable parameter functions from which the unknown parameters can be determined .

The method of claim 1 wherein the parameter functions are

representative of all unknown parameters associated with the observed quantities.

The method as claimed in any one of the preceding claims, wherein the signal is transmitted by the global satellite navigation system and tracked by a single receiver

The method as claimed in any one of the preceding claims, wherein the signal is a single frequency signal.

The method as claimed in any one of claims 1 to 3, wherein the signal is a multiple frequency signal.

The method as claimed in any one of the preceding claims, wherein the application comprises determining location parameters for use in determining a location of the receiver.

7. The method as claimed in any one of the preceding claims, wherein the application comprises determining atmospheric parameters for use in weather forecasting.

The method as claimed in claim 6, wherein the atmospheric parameters comprise ionospheric and/or tropospheric delays.

9. The method as claimed in any one of the preceding claims, wherein the application comprises determining coordinated universal time by estimating variations between the receiver' s time and the global navigation satellite system's time.

10. The method as claimed in any one of the preceding claims, wherein the application comprises determining satellite bias parameters in a common clocks S-system.

11. The method as claimed in claim 10, wherein the satellite bias

parameters in the common clocks S-system comprise satellite orbits, satellite clocks, satellite phase biases, satellite code biases or satellite differential code biases.

12. The method as claimed in any one of the preceding claims, wherein the application comprises determining receiver bias parameters in a common clocks S-system.

13. The method as claimed in claim 12, wherein the receiver bias

parameters in the common clocks S-system comprise receiver phase biases, receiver code biases or intersystem biases.

14. The method as claimed in any one of the preceding claims, wherein the unknown parameters include ionospheric corrections.

15. The method as claimed in claim 13, wherein the ionospheric

corrections comprise an estimate of slant ionospheric delay.

16. The method as claimed in any one of the preceding claims, wherein the step of resolving the parameter functions comprises the steps of:

representing the parameters in a design matrix;

identifying rank deficiencies within the design matrix and using the rank deficiencies to construct a null space basis matrix; selecting a S-basis of S-trans formation complementary to the null space basis matrix, wherein the S-basis is dependent on the application, and using the S-basis to construct a S-basis matrix; calculating a S-trans formation matrix utilising the null space basis matrix and the S-basis matrix;

17. The method as claimed in claim 16, wherein construction of the

design matrix comprises ordering the parameters according to their parameter types .

The method as claimed in claim 17, wherein the ordering is arranged by geometry parameters, receiver-dependent parameters, satellite- dependent parameters, ionospheric parameters, and ambiguity parameters .

The method as claimed in claim 17 or 18, wherein the rank

deficiencies identifiable comprise deficiencies between receiver and satellite clocks, deficiencies between receiver and satellite hardware biases, deficiencies between receiver clocks and receiver hardware biases, deficiencies between satellite clocks and satellite hardware biases, deficiencies between receiver phase biases and ambiguities or deficiencies between satellite phase biases and ambiguities .

The method as claimed in any one of claims 17 to 18, wherein the step of identifying rank deficiencies comprises the steps of identifying additional rank deficiencies and identifying additional columns of the null space basis matrix.

21. The method as claimed in claim 20, wherein the additional rank

deficiencies result from parametrization of slant ionospheric delays .

22. The method as claimed in claim 20 or 21, wherein the additional rank deficiencies result from regional network assumptions.