WO2021146775A1 - Systems and methods for processing gnss data streams for determination of hardware and atmosphere-delays - Google Patents

Abstract

Methods and systems for precise determination of hardware, ionosphere and troposphere delays in the Global Navigational Satellite System code and phase signals from a single reference receiver computing to network computing modes. The single receiver computing mode resolves SD and UD integers with properly combined signals. Substituting the SD and UD into the combined equations, one obtaining all independent uncalibrated hardware delay parameters and SD and slant ionosphere delay and troposphere delay terms epoch by epoch. The system creating state-space representation (SSR) corrections for fitted UHD delays and ionosphere and troposphere delays from a single reference receiver or a network of multiple receivers, these SSR corrections supporting both RTK positioning and PPP-RTK at the user terminals with single, dual and multiple frequency receivers. The SSR corrections being substituted into the original code and phase measurements of reference receivers as well, supporting the network for epoch-by-epoch estimations of satellite and receiver states under certain datum settings.

Description

Systems and methods for processing GNSS data streams for determination of hardware and atmosphere-delays

FIELD OF THE INVENTION

[0001] The present invention relates to generally processing of Global Navigation Satellite Systems (GNSS) data streams collected from every reference station and more specifically relates to determination of state-space representation corrections for satellite specific hardware delays, ionosphere- and troposphere-delays in GNSS code and phase signals. The invention more generally relates to a decomposed computing approach for kinematic estimations of satellite and receiver states.

BACKGROUND OF THE INVENTION

[0002] The Global Navigation Satellite Systems (GNSS) generally include the US Global Positioning System (GPS), Russia's GLONASS, Europe's Galileo and China's Beidou system. Each of these systems transmits code and phase signals at three or more frequencies. In the observation equations for a single receiver tracking S satellites with three code and phase signals, there are a total of 15S+10 parameters to be determined, including 4 satellite position and time states, 1 troposphere-delay, 1 ionosphere-delay, 6 satellite-specific hardware delays, 3 code offsets and 3 phase integer ambiguity terms for each satellite, 4 receiver position and time states and 6 receiver-specific hardware delays for the receiver. There are also 6S receiver noise and multipath terms. Each type of state parameters and delays has different characteristics and time variations. To determine these parameters in real time, there are basically two GNSS computing modes in processing GNSS observation data: network-based computing mode and single-receiver computing mode. Network-based computing makes use of data streams or data files from a large number of regionally or globally distributed receivers to precisely determine GNSS satellite orbits and receiver coordinates, satellite and receiver clock offsets, differential code biases alongside the global ionosphere-delay model, and uncalibrated hardware delays (UHD) in transmitters and receivers. To support single-receiver based processing, the redetermined biases of satellite orbits and clocks relative to broadcast ephemeris, UHDs for respective code and phase measurements, ionosphere-delay models for each line-of-sight and troposphere-delays for each station are grouped into State-Space Representation (SSR) corrections (Wubbena et al., 2005). The integer ambiguities in phase measurements are determined as intermediary parameters alongside the state and delay parameters. The network-based computing problems are complex and have drawn serious research and development efforts in both academia and industry communities worldwide since early 1990s. Single-receiver based computing can refer to a reference-receiver based computing and a user-receiver based computing. A reference receiver may simply output raw observations to users who may perform baseline processing, real time kinematic (RTK) or differential positioning. Alternatively, as shown in Feng et al., (2013), single-epoch data streams from each GNSS reference receiver station are processed to generate station-based solutions, or reference receiver-specific parameters. These may include precise receiver clock offset, zenith tropospheric delay (ZTD), Differential Code Biases (DCB), ambiguity parameters, vertical ionospheric delays. At the user end, a single-receiver computing may use the single- epoch data streams from a reference station or virtual reference station to determines the states of the receiver and integer ambiguities, such as RTK positioning mentioned above. If the single-receiver computing treats the satellites orbit, clock and hardware delays for a network or reference station as given values and determine the receiver states and integers, the processing is known as precise point positioning (PPP) with ambiguity resolutions (PPP- AR). If all SSR corrections from the reference network are provided to the user receiver as known values, the user positioning processing is known as PPP-RTK.

[0003] As part of the research and development outcomes, various network-based real time GNSS processing platforms and products have been developed for research and commercial PPP-AR services in the past decade. These may refer to the Centre National d'Etudes Spatiales (CNES)'s CLK93, (Laurichesse & Privat, 2015), Deutsches Zentrum fur Luft- und Raumfahrt (DLR)'s CLK22, Wuhan University phasebias (Geng et al.,2019) and Geoscience Australia's ACS services. These products are openly available for research and purposes and often used by hardware integrators and end users to generate PPP-AR solutions in emerging markets. In parallel with public offerings, there are commercial PPP-AR services with global coverage, such as Trimble's OmniSTAR, Fugro's Starfix and NanCom's StarFire (NavCom Technology, 2016) focusing on marine and agricultural applications. Trimble's RTX products provide PPP- AR and PPP-RTK services using dense regional networks (Chen et al., 2015; Trimble Navigation Limited, 2016). Veripos provides six additional PPP-AR and PPP-RTK products for marine applications (VERIPOS, 2017).

[00004] In general, a GNSS computing problem is to solve a subset set of parameters in the original code and phase signals, depending on users' applications, with the remaining parameters being constrained, given, or cancelled. The constrained or given parameters in a GNSS equation system play the role of datum, although the datum settings could be arbitrary, to certain extent, and can introduce errors or biases for the derived parameters. Cancellation of biases and errors in GNSS equations is a very effective way in GNSS computing and can be effectively realised by combination and differencing treatment among the original code and phase measurements. One example is the double-difference operation to remove satellite and receiver specific clock offsets and hardware delays common to the same code and phase signals in both reference and user receivers. As a result, RTK processing can determine the coordinates to the precision of centimetres to millimetres through kinematic or filtering processing. Theoretically if all the cancelled bias and delay terms are redetermined from the reference receiver or network epoch by epoch and applied on the same bias and delays terms in the user equations, the user PPP-RTK processing can achieve the same outcome as that RTK processing. In other words, with the same reference and user receiver data sets or data streams, RTK and PPP-RTK should give equivalent results. The fact is that the existing PPP- based processing takes longer time to resolve the ambiguity parameter than the RTK-based processing with the same reference stations. The hypothesis is that theoretically, there must be disputes in the estimation or treatment of the biases and delays in the network or single receiver-based processing.

[0005] Many recent research and development efforts and commercial systems turn to use UD and uncombined single-epoch code and phase measurements in their network-based computing problems (Gu et al., 2013; Lou et al., 2016; Zhao et al., 2018). One claimed benefit is to allow full exploitation of the information contained in each individual observation type and preserves the original measurement accuracy. However, the implications of UD and uncombined models must be clarified. First, mathematically, differenced observations can be completely equivalent to UD observations, while uncombined and linearly combined observations can be completely equivalent as well. All information and precision can be preserved unless a treatment is chosen not to. GNSS processing with uncombined and UD models must determine all or a subset of state and delay parameters altogether. The approach depends on precise modelling of every state and delay term and precisely given constraints for precise solutions. This could be possible for the satellite states based on well- understood satellite dynamics knowledges and static receiver states. However, for the ionosphere-delays subject to spatial and temporal variations, even complicated treatments with both deterministic and stochastic models do not necessarily lead to centimetre-level resolution needed for fast ambiguity resolution. In general, mismodelling of one type of delays may affect the estimation of all or other state or delay parameters in the whole network. The challenge is how to model different types of delays individually in time and geographic domains without affecting each other.

[0006] This invention aims to address the above problems through both single-receiver computing and network-based computing procedures. The single-receiver computing approach regenerate the SSR corrections for satellite-specific uncalibrated hardware delays (UHD) in GNSS code and phase signals, as well as ionosphere- and troposphere-delays. These SSR components are derived with respect to a set of given precise GNSS orbits, clocks and widelane and narrowlane UHD delays. With the SD integer solutions from PPP-AR, the SD ionosphere-delays are directly determined with the integer-fixed phase combination and the SD troposphere-delays with the geometry-based ionosphere-free phase combination. The remaining UHD delays are derived from geometry-free and ionosphere-free combinations epoch by epoch. The single-receiver kinematic computing approach is then extended to the network-based computing mode. The SD and Unidifferenced (UD) integers and UHD delays from a network of multiple stations are processed to derive DD integers, DD code offsets, DD ionosphere-delays and DD troposphere-delays for all the essential baselines. Appending the UD quantities for each receiver-reference satellite, SD from one receiver and DD quantities from baselines, are mapped to all line-of-sight (LOS) directions. Thus, the single-epoch network-adjusted SSR corrections are obtained to retain the consistence of corrections for network-based RTK services. Fitted or filtered SSR corrections then support network-based PPP-AR and PPP-RTK services.

SUMMARY OF THE INVENTION

SUBSTITUTE SHEET (Rule 26) RO/AU [0007] In one aspect, the invention comprises the system of using combined GNSS measurements from a single receiver to determine the hardware delays and atmosphere delays in code and phase signals at dual or three frequencies. The hardware refers to satellite signal transmitters and antenna. The atmosphere-delays comprise ionosphere and troposphere-delays. The resulting delays can form three types of state space representation (SSR) corrections from dual- or triple frequency data streams from a single receiver and supporting both PPP-RTK and RTK positioning services. The method comprises the steps of; a. reformatting the original code and phase observables into three groups of linear combinations, including geometry-free ionosphere-free (GFIF) combinations, geometry-free ionosphere-present (GFIP) combinations, and geometry-based ionosphere-free (GBIF) combinations, the combinations comprising only linearly independent combinations for the processing in two cases: o For a triple frequency receiver and in each Line of Sight (LOS), the six independent combinations include 4 GFIF observables, 1 GFIP observable, and 1 GBIF observable. They are mathematically equivalent to the original 3 code and 3 phase signals. Of the six combinations, there are three-code-dominated combinations from GFIF group and three phase-only combinations from GFIF, GFIP and GBIF groups, thus preserving the phase precision of the states and delays to be determined. o For a dual-frequency receiver in each LOS, the combinations include 2 GFIF, 1 GFIP and 1 GBIF combinations, there two code-dominated combinations and two phase-only combinations. b. Making use of three GFIF models to determine Undifferenced (UD) integers for the receiver-to-satellite directions and SD integers between satellites. The initial combined UHD solutions from the previously determined or externally provided sources are applied to reduce the biases in the float UD and SD integer ambiguities. For each receiver, using a least-square estimation, the UD float ambiguities can be adjusted with SD integers as constraints and result in UD integers in LOS directions. c. Giving broadcast or precise GNSS orbits and clock products and station coordinates and two initial phase hardware delays for each LOS from the receiver, determining

SUBSTITUTE SHEET (Rule 26) RO/AU slant and SD ionosphere and troposphere-delays with phase-only measurements, and computing UD uncalibrated hardware delays (UHD) in code and phase signals epoch by epoch in all LOS receiver-satellite directions.

[0008] In one embodiment, the obtained single-epoch time series for UHDs, SD ionosphere- delays, SD troposphere-delays from a single receiver form a set of single-epoch SSR corrections, which can be accessed by a user receiver to correct the same bias terms in the user measurements. In another embodiment, the obtained single-epoch UHD corrections are fitted as functions of time, which can then be accessed by a user receiver to calibrate the same bias terms in the user measurements. a. wherein a set of single-epoch SSR corrections are applied to calibrate the SD code and phase measurements of an user receiver, users obtain DD linear equations for original phase measurements at single, dual or triple frequency, so the users can perform single- reference RTK positioning with single-frequency, dual-frequency or triple frequency measurements. b. wherein the fitted UHD corrections and single-epoch ionosphere and troposphere corrections are applied to correct the same delay terms in the SD code and phase measurements, the users obtain SD linear equations for original code and phase measurements for single, dual or triple frequency signals, so the users can perform single-reference PPP-RTK positioning with single-frequency, dual-frequency or triple frequency measurements c. In both cases, the inter-receiver code offsets can be estimated with long-term or historical data before the receiver is entered the RTK or PPP-RTK services with respect to the reference receiver, thus could be removed from the code signals.

[0009] In another aspect, the invention comprises an additional system of using combined GNSS measurements from a plurality of receivers to obtain DD integers and adjust the UHD in LOS code and phase signals at dual or three frequencies. As a result, the resulting UHD corrections over a network of receivers can improve precision with respect to the LOS UHD samples. The adjusted and fitted satellite specific UHDs and slant ionosphere and troposphere-delays can together support PPP-RTK and standard RTK positioning services within a network coverage. The additional system comprising, a. Resolving the DD integers for the necessary baselines using GFIF or GBIF models b. Mapping the baseline DD integers and the first station SD integers to obtain consistent LOS integers by appending the UD integers of each station to the reference satellite. c. Performing DD operation onto code UHD measurements and obtaining DD code UHD offset samples. Taking average or fitting over historical data period and gives precise estimation of the DD code offsets. d. Removing the obtained LOS integers, slant ionosphere-delay and slant troposphere- delays from the selected UD combined measurements, forming linear equations for all satellite and receiver specific UHDs and performing least-square adjustment to derive the satellite and receiver specific UHD time sequences of the whole network. e. Converting combined UHDs to original code and phase UHDs and performing fitting process to each combined or UHD time series for a function of time.

[0010] In one embodiment wherein the obtained single-epoch and adjusted time series for satellite specific UHDs, slant ionosphere-delays, slant troposphere-delays from the network of receivers are grouped as the single-epoch and network adjusted SSR corrections. For user receivers within the network coverage, wherein there are two options: a. users detect consistence among slant ionosphere and troposphere delays and perform interpolations using the consistent single-epoch corrections from the surrounding stations and satellite specific UHDs to calibrate the same bias terms in the measurements. Appropriate interpolation models, such as linear combination models and low order surface models (Dai, 2001) should be applied at user end, along with algorithms for detection of outliers in case of inconsistent integers between the delays from different stations. b. users form observation equations with different sets of inter-receiver code offsets and integer parameters with respect to each reference station. The ionosphere and troposphere delays from the surrounding stations and fitted satellite-specific UHDs are used to calibrate the same bias terms in the measurements and proceed with multiple-reference RTK or PPP-RTK positioning with single-frequency, dual-frequency or triple frequency measurements. [0011] In another embodiment wherein the obtained single-epoch and adjusted time series for satellite and receiver specific UHDs, slant ionosphere-delays, slant troposphere-delays from the network of receivers are substituted into the original code and phase equations, obtaining the clean observations free of delays. The clean measurements have the uniform function models and different noise characteristics, supporting the network-based kinematic computing for the satellite and receiver states under appropriate datum setting. a. Applying double-differencing operation on the functional models and deleting the satellite and receiver clocks offsets, the states of satellites' or receivers' antennas can be determined epoch by epoch, along with the zenith troposphere delays. b. Given precise satellite and receiver states, the satellite and receiver clocks biases can be determined epoch by epoch, along with the zenith troposphere delay.

Modelling of these kinematic state solutions can then be performed individually in time domain.

BRIEF DESCRIPTIONS OF THE DRAWINGS

[0012] In the drawings, like elements are assigned like reference numerals. The drawings are not necessarily to scale, with the emphasis instead placed upon the principles of the present invention. Additionally, each of the embodiments depicted are but one of several possible arrangements utilising the fundamental concepts of the present invention. Several aspects of the present invention are illustrated by way of example, and not by way of limitation, in detail in the figures, wherein;

[0013] FIG.l is a representative illustration of a Global Navigation Satellite System (GNSS) with 4 satellites in view. Each satellite transmits code and phase signals at dual or multiple frequencies.

[0014] FIG.2 is a schematic flowchart showing the overall structure of the sequential GNSS computing system of implementing the present invention.

[0015] FIG.3 is a schematic flowchart showing the major steps of forming combined UD and SD observation equations that implementing the present invention. [0016] FIG. 4 is a schematic flowchart showing the major steps of the computing UD integers and SD integers of implementing the present invention.

[0017] FIG. 5 is a schematic flowchart showing the major steps of the computing slant ionosphere and troposphere-delays of implementing the present invention.

[0018] FIG. 6 is a flowchart of a method for determination of UHD in multiple frequency code and phase signals and fitting them as function of time of implementing the present invention.

[0019] FIG. 7 is a flowchart of a method for determination of LOS integers for multiple stations, then recomputing slant ionosphere and troposphere-delays in implementing the present invention.

[0020] FIG. 8 is a flowchart of a network-based computing method for redetermination of satellite and receiver specific UHD in multiple frequency code and phase signals and fitting them as function of time of implementing the present invention.

[0021] FIG. 9 is a flowchart of a system for estimation of user states and integers with single-epoch and fitted SSR correction messages from a single reference receiver and user- observation data streams that implementing the present invention.

[0022] FIG. 10 is a flowchart of a system for estimation of user states and integers with SSR correction messages from a network of reference receivers and user-observation data streams that implementing the present invention.

BRIEF DESCRIPTIONS OF THE TABLES

[0023] In the tables, like equations are assigned like reference numerals. Each of the Tables lists some of possible linear combinations in each type or group utilising the fundamental concepts of the present invention. They are given by way of example, and not by way of limitation, in detail in the tables, after wherein.

[0024] Table 1 lists the original code and phase observation equations in three frequencies represented byfi, f2and f3, and the description of each notation or each term in the equations. [0025] Table 2 gives the definitions of all combined code and phase signals, observation equations, and descriptions of the notations to be used in implementation of this invention.

DETAILED DESCRIPTION OF VARIOUS EMBODIMENTS

[0026] The invention relates to a system and method for processing GNSS data files and streams from a single receiver or multiple receivers. When describing the present invention, all terms not defined herein have their common art-recognised meanings. To the extent that the following description is of a specific embodiment or a particular use of the invention, it is intended to be illustrative only, and not limiting of the claimed invention. The following description is intended to cover all alternatives, modifications and equivalents that are included in the spirit and scope of the invention, as defined in the appended claims.

[0027] FIG. 1 illustrates a representative view of Global Navigation Satellite System (GNSS) 10. In one embodiment, the GNSS 10 comprises the Global Positioning System (GPS). The GNSS 10 includes a plurality of satellites 11 orbiting the earth with each satellite 11 transmitting signals 12. The GNSS 10 may broadcast the signals 12 on multiple frequencies. For example, if the GNSS 10 is a GPS system, each satellite 11 can broadcast the signals 12 using two or three frequency (i.e. the fi, f2 and f3 frequencies used by satellites in the GPS constellation). A GNSS receiver 15 is provided that is operative to receive the signals 12 transmitted by the plurality of satellites 11. A receiver tracks dual or triple frequency GNSS signal 12 continuously from one more GNSS constellations. The constellations can include any combinations of GPS, Galilleo, Beidou, Glonass, and QZSS.

[0028] FIG. 2 is a schematic illustration of the overall embodiment of the GNSS computing system of the present invention that processes the data files or data streams 110 for a single GNSS receiver 15 as shown in FIG. 1. The single receiver computing embodiment 100 can include the preparation method 200 for physical observation equations for selected combined signals as shown in FIG. 3, Integer ambiguity resolution subsystem 300 as shown in FIG 4, the computing subsystem 400 for determination of slant ionosphere-delays and troposphere-delay as shown in FIG 5, the computing subsystem 500 for determination of single-epoch satellite-specific UHDs and their fitting or filtering processing as shown in FIG. 6, the computing subsystem 600 for network-based treatment for DD integers and consistent LOS integers as shown in FIG.7; the computing subsystem 700 for determination of network- adjusted single-epoch UHD observables and fitted/filtered UHD results, the user-end RTK and PPP-RTK processing module 800 with SSR corrections from a single reference receiver and the user-end RTK and PPP-RTK processing module 900 with SSR corrections from a network of multiple reference receivers. The methods and computing subsystems are described in detail in the rest of the description.

[0029] FIG.3 is a schematic flowchart of Method 200 comprising a process to establish six linearly independent combinations over three code and phase observations in each line of sight (LOS) direction and SD equations with respect to a reference satellite.

[0030] Table 2 lists the key combinations to be used in this invention, including geometry- free and ionosphere-free (GFIF) combinations 21, 22, 23, 24, geometry-based ionosphere- free (GBIF) combination 25, and geometry-free and ionosphere-preserved (GFIP) combination 27. Such a set of the six types of combinations is completely equivalent to the set of six original code and phase observables for each receiver-satellite path.

For the selected combinations, the observation equation system for all types of delays and noise terms is expressed as follows:

1b) where the subscript “r" represents a single receiver until otherwise specified. is the slant troposphere-delays to all the satellites,

computed with the same empirical model. For instance, the model adopted by the Space- Based Augmentation Systems (SB.AS) standards [RTCA-MOPS, 2006] is as follows:

2) where T_z,_dry and T_z,wet are calculated from the receiver's height and estimates of five meteorological parameters: pressure [P(mbar)], temperature [T(K)], water vapour pressure [e (mbar)], temperature "lapse" rate [β(K/m)] and water vapour "lapse rate" [λ (dimensionless)]. The obliquity factor is the Black and Eisner mapping function (Black and Eisner, 1984):

3)

The term dT^s represents a residual troposphere-delay with respect to the computed value from the above empirical model.

[0031] Step 210 computes the matrix A as follows:

4)

The matrix A is invertible. For a triple frequency GPS signals, fi=1557.42 MHz, f₃=1227.6 MHz, f₂= 1176.45Hz, the matrix A is given as follows

Therefore, once the six UHD delays in the combined signals are determined, the UHDs in the original code and phase signals can be uniquely obtained, while the noise terms in the original signals can be fully propagated to the combined signals.

[0032] Comparing to use of the original signals, the combined models in (Equation 1) allow various types of parameters to be estimated sequentially or separately, while preserving all parameters and precision. In the dual frequency case, the second and fourth combinations in Equation 1 do not exist or can be removed.

[0033] Step 220 applies single-difference (SD) between satellites "s" and the reference satellite "1" and linearization, six SD linear observation equations for the position states, SD integer ambiguities, SD hardware delays, SD ionosphere and troposphere-delays will be obtained.

[0034] The computed SD linear questions for each satellite are expressed as follows:

where the symbol

stands "single difference (SD)" between satellites. is the SD

computed range with precise satellite orbits and clocks and tropospheric model corrections between sth and 1^st satellites at the rth receiver,

6)

Other vectors are expressed as follows:

where is a 1-by-3-dimensional vector for three partial derivatives of the SD range with respect to 3D receiver coordinates; dX is the 3-dimensional column vector for the derivations of the receiver coordinates with respect to the initial states.

the SD troposphere- delay and is the SD ionosphere-delay stand for the SD integers.

[0035] For the lumped code and phase UHD terms b_L and B_L as defined in (3), SD operation between satellites eliminates the receiver-specific components. Therefore, the SD code and phase UHDs are specified by

for L=1,2,3. Three receiver states are receiver-specific and common to all SD measurement vectors. All other SD parameters are satellite-specific and frequency-specific. It is noted that the inter-receiver code-offset term V

is also receiver-specific and cannot be separated from the satellite- specific code hardware delay in single receiver computing, thus being written in two

components. However, for the receivers "r" and "1", the code-offset can be determined or calibrated at the user-end data processing.

[0036] Step 230 establishes the covariance matrix for the combined measurement error term in Equation 5. To bring together all the SD elements in one observational equation system for a single station, a single-difference matrix D is set for all satellites in view, where the first satellite is set to the reference satellite.

where 1₆ is the 6-dimensional identity matrix; the symbo

is the Kronecker product. For a receiver tracking S satellites, there are a total of 6S original or 6S combined measurements, respectively. The combined noise vector δ and original noise vector e are related as follows:

9)

[0037] Therefore, the covariance matrix of the SD combined models (Equation 4) can be obtained by variance propagation from the six original code and phase measurements.

10)

[0038] The equivalence between SD combined equation 4 and their original SD models in terms of receiver states and integer ambiguity parameters is then completely preserved due to use of the observation vector (Equation 5) and covariance matrix (Equation 10) altogether.

[0039] FIG. 4 describes the system 300 making use of precise orbits and clocks, and initial phase UHD corrections for integer ambiguity resolution.

[0040] In another embodiment, Step 310 may implement the geometry-free approach with (Equation 1) to fix the UD and SD integers. With GFIF models 21 and 22 and 24, the UD extra- wide-lane, wide-lane and narrow-lane floating ambiguities can be obtained the average over their samples of size M over a data window,

10a)

where or the combined terms are initial UHD values derived from the

recent window through Equation 61 or external products for PPP-AR processing such as Centre CNES's CLK93 (Laurichesse & Privat, 2015) and Wuhan University's phasebias products (Geng et al.,2019). Removing the effects of these initial combined UHD values can reduce biases in the floating ambiguity solutions so that the UD integers can more reliably and stably fixed. are the UD integers by rounding the float UD ambiguity solutions . For

the rounding integer is denoted by N₃ .The data window for averaging may be set, so that the effective independent sample size M could be over 100. As a result, the uncertainty of the average values for the narrow lane integer in Equation 10c can be reduced to be lower than 0.2 cycles. [0041] Single differencing (SD) between satellites can delete the receiver-specific hardware delays and clocks-bias. Step 320 determines the SD integers by applying SD operation on Equations 10a, 10b and 10c. That is, SD integers are similarly obtained from SD code and phase measurements. The SD extra-wide-lane, wide-lane and narrow-lane floating ambiguities can be obtained by the average over their samples of size M over a data window,

where are initial UHD values which may be obtained from the most recent SSR corrections or a well-designed PPP-AR processing platform.

[0042] Step 330 performs the least square adjustment to obtain the consistence between each set of UD integers, through the following equation system for each frequency with the wavelength l

12)

where N° is a S-dimensional float ambiguity vector, and its initial UD integer vector by rounding is N°; VN is the known (S-l)-dimensional SD integer vector and Si is their (S-1)-by-S matrix. The UD ambiguity correction term is represented by dN, its least square with constraint conditions is solved as follows:

13)

[0042] Therefore the LOS integer vector can be obtained as follows,

14) [0043] The results 340 include 3(S-1) SD integers and S consistent UD integers for all directions.

[0044] FIG. 5 illustrates the method 400 comprising necessary steps of computing slant ionosphere and troposphere-delays.

[0045] After the SD integers are correctly determined, Step 410 computes the slant ionosphere-delay for the reference satellite path with GFIP 25

15a) and (S-1) SD ionosphere-delays

15b) where s=2,3,..., S, the terms represent the initial UHD

values, which may vary with time, or remain a constant. However, they play the roles of the datum in the determination of the slant and SD ionosphere-delays in Equations 15a and 15b.

[0046] Step 420 computes the slant troposphere-delay for the reference satellite path with the narrow-lane GBIF model 25 in Table 2,

16a) and the SD residual troposphere-delays

(Equation 16b) where the term as shown in Equation 2 contains the effects of satellite orbit and

clocks errors, and receiver position coordinate errors as well the troposphere model errors. The terms play the roles of the datum in the slant

and SD troposphere-delays.

[0047] Meanwhile, these terms serve as boundary conditions for UHD determination,

17a)

17b)

17d) where s=2,3,...,S. These terms serve as boundary conditions for UHD determination and remain unchanged for both reference-station-based and user-based computing. One straightforward option is to set

18a)

18b)

(Equation 18d)

[0047] The SD ionospheric delay and tropospheric delay terms have the precision of serval millimetres. [0048] Step 430 compute the slant ionosphere-delays by appending the slant delay for reference satellite path to all SD ionosphere-delays gives the slant ionosphere-delays in all LOS directions, as follows:

19)

[0049] Step 440 computes the slant troposphere-delay for the receiver-reference satellite direction and appends the UD delays to all SD delays to give the slant troposphere-delays in all LOS directions, with the following equation:

20)

[0050] FIG. 6 outlines the procedures of Method 500 for determining the UHDs in code and phase measurements from the outputs of Method 300 and Method 400: UD and SD integers and slant ionosphere and slant troposphere-delays.

[0051] Step 510 moves all the knowns or determined UD quantities to the left-hand-side of Equation 1 for the reference satellite path and the SD quantities to the left-hand side of Equation 5, the measurement equations for code and phase UHD parameters for the slant path is obtained as follows:

21a)

For each SD with respect to a reference satellite "1", the measurement equations for code and phase UHD parameters are:

21b)

[0052] Step 520 inversely determines the UHD values in the original SD code and phase signals through the inverse of the matrix A,

22)

[0053] Due to the structure of A matrix, three phase UHD terms can be determined with the three phase-only combinations epoch-by-epoch. But remain unchanged and The precision of single-epoch phase solutions from (Equation 23) is in the same order as the

precision of phase measurements.

[0054] Step 530 performs fitting or filtering processes gives smoothed estimates for SD code and phase UHD V as functions of time. Fitting does

not necessarily result in much higher precision but allows the UHD values to be interpolated for real time applications.

[0055] Fitting or filtering processes over a moving time window will give smoothed estimates as functions of time,

which results in much higher precision and allows the code UHD values to be predicted or interpolated for real time applications.

[0056] Step 540 collects all SD single-epoch UHDs and fitted UHDs for each reference receiver to form single-epoch SSR corrections and fitted SSR

corrections

[0057] Step 550 collects all LOS single-epoch UHDs and all slant ionosphere and slant troposphere-delays and prepare for the system 600. For every epoch, the combined UHDs in the first to sixth equation can be obtained. For convenience, for each receiver "r", we specify the S-by-1 vector l_r for slant ionosphere-delays, the vector dT_r for slant troposphere-delays and the S-by-6 matrix Z_r for the LOS UHDs. For all receivers, we also specify the RS-by-6 matrix Z for LOS UHDs and the RS-by-1 vectors I and dT for all slant ionosphere and troposphere- delays in the network, respectively:

23)

[0058] The precision of all single-epoch delay samples for LOS ionosphere, troposphere and phase UHDs are all in the order of several millimetres to 1 centimetre, while the single-epoch code UHD samples contain the effects of code noise and multipath errors, thus having the uncertainty in the order of a few to several decimetres. The precision of the fitted or filtered code UHD over a data window of tens of minutes will be in the order of a few to several centimetres. The SD formations of the above delays from a single-receiver can be used as part of SSR corrections for RTK and PPP-RTK services.

[0059] FIG. 7 is a schematic illustration of the embodiment of the GNSS computing system 600 of the present invention that processes the data files or data streams 110 for a plurality of GNSS receivers 15 shown in FIG 1 and the processing outcomes from the single reference receiver embodiment 100. The system 600 is a network-based computing embodiment comprising double-differenced treatments to maintain the consistence between all LOS integers over a network of multiple stations, thus the consistence between LOS ionosphere and LOS troposphere-delays.

[0060] Step 610 restructures LOS integers for all the LOS directions for a network of R stations and S satellites, by introducing the following notations:

24) where P, F are the RS-dimensional vectors for code and phase measurements, N standards for the RS-dimensional integer vector for all LOSs at each frequency; b for the RS-dimensional code UHD vector and B for phase UHDs for all LOS at each frequency, from the column of the matrix Z in (23). [0061] Defining the UD, SD and DD operator matrices s as follows

25)

26)

27) where D₀ is a R-by-RS matrix; D₁ is a (S-l)-by-RS matrix and D₂ is a (R-l)(S-l)-by-RS matrix.

[0062] Applying D₀ and D₁ operators to the UD vector N , the following results are obtained:

28) [0063] Applying the D2 operation to the UD vector N , the following results are obtained:

29) where DoN gives the R-dimensional UD integer vector, D₁N is the (S-l)-dimensional SD integer vector for the first station D₂N is the (R-l)(S-l)-dimensional DD integer vector. If it the DD integer vector D₂N are obtained more reliably, it can adjust the LOS integer vector.

[0064] Step 620 maps UD, SD, and DD integers to all LOS directions for LOS integer consistence. Inverting the matrix D and portioning the inverse matrix into three components according to the column sizes of

30) the RS-dimensional LOS integers can be computed with the following equation:

31)

[0065] Step 630 adjusts the LOS UHDs with adjusted LOS integers from consistence. The adjusted B is given by

32) where the over-line of N and B indicates the network adjusted LOS integers and phase delays.

[0066] FIG. 8 illustrates the computing system 700 that comprises steps to compute the satellite and receiver specific hardware delays in combined or original signals. [0067] Step 710 of FIG. 8 obtains the code UHD offsets in DD code signals. Substituting the LOS integers in the code dominated combinations, performing DD operations, the code offsets are obtained as follows:

33)

34)

35)

[0068] These code offsets remain constants or slowly varying, thus being fitted as function of time over a long period of observations or updated daily and represented by

[0069] Step 720 forms the observation vectors for six types of combined UHD state vectors by removing the code-offsets from the UHD equations,

36)

37)

38)

39)

40)

41) [0070] The observation equations and statistic models for satellite and receiver specific UHDs.

42) where

43)

[0071] The term in Equation 51 are unrelated, but their error terms are generally

cross correlated. For Equation 51, it is assumed that error vectors are contemporaneously correlated, the covariance matrix between two USD observable vectors are

44) where I _RS is the RS-dimensional unit matrix.

[0072] Step 730 performs least square estimation to obtain time series for each satellite and receiver specific UHD states.

[0073] Because the noise vectors in Equation 42 could be cross correlated, theoretically run the weighted least squares estimation should be applied to solve all the UHD vectors in Equation 42 together as above. However, due to the same matrix H for the USD parameters in all the models, the weighted least square solutions turn out to be theoretically identical to the least squares estimate without considering the correlation between models (Davidson, R., MacKinnon J. G., 1993). That is, we have

45) and the residual vector for the Jth equation

46)

[0074] The residuals contain the multipath errors and computed geometric range errors. In an alternative embodiment, the historical r, data of a moving window of hours may be used to train the error patterns and generate corrections for these errors for these residual terms in the next few hours. As a result, the systematic residual errors in Equation 51 can be removed before the least-square estimation with Equation 45.

[0075] There are six residual vectors for all UHD solutions. The estimators of the covariance components are given by

47)

[0076] The covariance matrix for the estimate Z_{UHD T} for each GF equation is

48)

[0077] The network-adjustment solution (Equation 54) gives the time series for each element of Z _{UHD J}(t) . The covariance matrix Equation 48 shows the performance factor of the network adjustment results from a single epoch. The Network Dilution of Precision (NDOP) defined as the square root of the diagonal elements of the matrix (H^TH)^_1 can describe this performance factor. For a satellite-specific UHD parameter, NDOP depends on both the number of stations and the number of satellites commonly observed. This means that network-adjustment process can also improve the accuracy or precision of single-epoch UHD samples at every epoch. It must be noted that in the receiver specific UHDs from Ys(t), [UHD₂,UHD₃,...UHD_R]₅ , are defined with respect to their receiver clock offsets.

[078] Step 740 converts the UHD estimates in six combined signals into the UHD estimates in the original code and phase signals

49) where is the (S+ R-1)-by-1 UHD vector, include S satellite-specific UHDs and (R-1)

receiver specific UHDs.

[0079] Step 750 collects the single-epoch satellite specific UHDs and fitted UHDs for each reference receiver to form single-epoch SSR corrections for 6S satellite specific UHDs out of for J = P₁,P₂,P₃,Φ₁,Φ₂,Φ₃ and a RS-dimensional slant ionosphere-delay I(t) and

RS-dimensional troposphere vector JT(f) .

[0080] Step 760 estimates of UHDs parameters in the time domain with the network-adjusted UHD samples at each epoch. Regardless of post-processing or real-time processing, the UHDs of six original signals may be modelled over a continuous observation period to achieve filtered results. How to model the UHDs in time domain is a less studied issue. In this treatment, in order to model the possible time variations of UHDs, each element of the state vector X_UHDJ is more generally augmented by a P_J -degree polynomial function of time with

(p_J+1) parameters, J = P₁,P₂,P₃,Φ₁,Φ₂,Φ₃ . We introduce the following matrix and vector notations:

51)

52)

[0081] Considering the independent samples between epochs and the cross-correlation of between the error time series after the network adjustment for each USD component, we have the linear equations:

53) and the covariance matrices

54) where m_d=R+S-l is the dimension of ; n represents the samples if the whole data period ti to t_n.

[0082] The weighted least square solutions of the problem (Equation 53) are given by

55)

56)

[0083] Defining the residual vector from the ordinary least square solution,

the estimators of the covariance components over the whole data period are given by

)

At all the time instants ti to t_n, the estimated

its covariance matrix can be expressed by

This processing is run for. J = P₁,P₂,P₃,Φ₁,Φ₂,Φ₃ .

[0084] Step 780 coverts the fitted to the fitted can be also obtained through the

matrix A. As a result, the total hardware delays in the combined GFIF measurements can be obtained as follows:

[0085] Their predicted values can be used as the initial values in the determination of LOS and integers with Equations 10a, 10b and 10c.

[0086] FIG. 9 illustrates the user receiver embodiment 800 comprising the methods 810 and 820 for estimation of user states and integers with SSR correction messages 540, and with the user data streams from a single, dual or triple frequency receiver 805. The subsystems 810 and 820 comprises steps of user receiver precise state estimation by applying the SSR corrections from a single reference station to calibrate the same error terms in the GNSS code and/or phase observables at a single, dual and triple frequency with a user receiver.

[0087] Method 810 is a single-reference RTK method with the single-epoch SSR corrections. Referencing to Equation (21b), the SSR calibration forms the observed -minus-computed terms of SD equations for the reference station as follows.

(Equation 62)

(Equation 63)

Step 812 applies the single-epoch SSR corrections to calibrate the same terms in code and phase signals of a user receiver. Noting the code UHD has two terms and the phase UHD takes the satellite specific only as shown in Equation 7, the linear observation equations of use receivers are expressed as follows:

for L=1,2,3; s=2,3,...,S (Equation

64)

where the and are the inter-receiver code offsets that still exist in the code SD

UHDs

[0088] The equation 64 and 65 are the exact forms of DD code and phase equations for RTK processing where the ionosphere and troposphere delays are corrected. But the equation (64) explicitly exhibits the effects of inter-receiver code offset in the DD code

measurements. Calibration of the code-offset term can be done from time to time with respect to the reference receiver after the integers in (65) are correctly fixed. This shows how the single-epoch SD SSR corrections can support RTK positioning which is based on DD code and phase measurements. But using the single-epoch SSR corrections instead of single-epoch measurements from the reference stations can reduce the data size because the single-epoch SSR components vary slowly and the lower data rates can be adopted when being delivered to users.

[0089] Step 814 performs the determination and/or calibration processes for the code-bias offset with respect to the reference station, that is, Determination is

performed from the difference between (equation 64) and (equation 65) after the integer term are correctly resolved with the historical data over longer period.

Calibration is to apply a known code-bias to remove its effect when the same receiver and same reference stations are used in RTK services.

[0090] Step 816 performs the standard RTK processing with single-frequency, or widelane, dual-frequency measurements.

[0091] Method 820 is a PPP-RTK processing approach with the fitted SSR corrections. Step 822 applies the fitted or predicted SSR to calibrate the same terms in code and phase signals of a user receiver, to enable user-end PPP-RTK processing.

[0092] When fitted or filtered UHD and single-epoch SD ionosphere and troposphere-delays from a reference receiver are applied to calibrate the same terms in code and phase signals of a user receiver, the linear observation equations are expressed as follows:

[009S] The equations 65 and 66 are the forms of SD code and phase equations for PPP-RTK processing. The precision of DD and SD phase measurements Equations (65) and (67) are almost the same after SSR corrections being applied, but the code measurements Equation 66 will have higher precision than that of Equation 64, due to calibrating with fitted

[0094] Step 824 performs the determination and/or calibration processes for the code-bias offset with respect to the reference station, Determination simply takes

average of the difference between Equations 66 and 65 with the historical data over which the integers are correctly fixed as well. Calibration is to apply a known code-bias to remove its effect.

[0095] Step 826 performs the RTK and PPP-RTK processing with single-frequency code and phase measurements, dual-frequency or triple-frequency code and phase measurements. For long user- reference distance where the effects of the distance-dependent errors on the rover receiver grow beyond several centimetres, there are dual-frequency method and triple-frequency methods for achieving reliable ambiguity fixing and decimetre positioning results with widelane phase combinations. In the step 826 the dual-frequency method makes use of more precise combined code measurements P₁₃ and widelane phase measurements. The observation equations for each set of SD code and phase signals are as follows:

68)

for s=2,3,..., S (Equation

69)

[0096] The widelane integer ambiguity term should be reliably fixed almost instantlydue

to the long wavelength (0.862 m with GPS signals) and the low noise of code measurement P₁₃. The user position estimation with Equation 67 can achieve to the accuracy of ten to a few tens of centimetres immediately, for normal the dilution of precision (DOP) and growth of distance dependent errors within 10 centimetres. [0099] FIG. 10 illustrates the user receiver embodiment 900 comprising the methods 920 and 930 for estimation of user states and integers by with SSR correction messages, and with the user data streams from single, dual, and triple frequency receivers. The system 900 comprises subsystems and steps of user receiver precise state estimation by applying the SSR corrections from a network of reference stations to GNSS code and/or phase observables at a single, dual, and triple frequency with a user receiver.

[0100] Step 910 detects the consistence of the slant ionosphere delays between the surrounding stations and the consistence of the slant troposphere-delays between the surrounding reference stations, respectively. If they are consistent, steps 910 builds the interpolation models for the single-epoch slant ionosphere-delays and the troposphere delays, respectively, then proceed with Step 922 for the network-based RTK or PPP-RTK user positioning with these interpolated SSR corrections and UHDs. If they are not consistent due to incorrect integers, proceed with the step 932 for the multiple reference stations RTK positioning mode.

[0101] Method 920 is a network-based user RTK procedure. Step 922 applies the adjusted UHD and interpolated ionosphere and troposphere SSR to calibrate the same terms in code and phase signals of a user receiver, to support network-based user RTK positioning processing. The SD linear observation equations are formed as follows:

for L=1,2,3; s=2,3,...,S (Equation 71) where represent the interpolated values that are expressed as functions

of time and user location, where "x" indicates the horizontal locations of a receiver "u within the interpolation coordinate frame.

[0102] The equations 70) and 71 are equivalent to the network-based RTK positioning models, but using the fitted code and phase UHD instead. In addition, using the interpolated ionosphere and troposphere corrections from multiple stations instead of single-epoch measurements from a single receiver and can improve the precision of the ionosphere and troposphere-delays or increase the inter-receiver distances.

[0103] Step 924 performs the determination and/or calibration processes for the inter- receiver code-bias offset with respect to one of the reference station "r". Determination of the inter-receiver code-bias offsets can simply take the average of the difference between Equations 71 and 70 over the historical data and updated regularly. Calibration is to apply the determined or updated code-bias to remove its effect.

[0104] Step 926 completes the user-end RTK or PPP-RTK processing with single-frequency, or widelane, dual-frequency measurements. If the inter-stations distance are very long and, users can use the widelane phase measurements for RTK or PPP-RTK for stable decimetre accuracy positioning.

[0103] Step 924 performs the determination and/or calibration processes for the code-bias offset with respect to the reference station. Determination simply takes average of the difference between Equations 71 and 70 over the historical data and updated regularly. Calibration is to apply the determined or updated code-bias to remove its effect.

[0104] Step 926 perform the standard RTK processing with single-frequency, or widelane, dual-frequency measurements. Users obtain the network based RTK positioning solutions.

[0105] Method 930 is a multiple-reference RTK or PPP-RTK procedure. Step 932 applies the

same terms in code and phase signals of a user receiver.

[0106] There are Ri sets of linear observation equations expressed as follows:

for L=1,2,3;s=2,3,...,S, and r=1,2,..., R1 (Equation 72)

for L=1,2,3;s=2,3,...,S, and r=1,2,..., R1 (Equation

73) [0107] Equations 72 and 73 represent the single-reference RTK or PPP-RTK equations with respect to more than 1 reference stations. The user states dX are common, but the inter- receiver code-offsets and double-differenced integers are different for different reference receivers.

[0108] Step 934 performs the determination and/or calibration processes for the code-bias offset with respect to R1 reference receivers one-by-one. Determination of the inter-receiver code-bias offsets can simply take the average of the difference between Equations 72 and 73 over the historical data and updated regularly. Calibration is to apply the determined or updated code-bias to remove its effect.

[0109] Step 936 performs the PPP-AR processing with single-frequency, or widelane, dual- frequency measurements at a user-end. This is to obtain the PPP-AR and PPP-RTK solutions.

[0110] Method 1000 is to calibrate the original code and phase measurements of all the reference receivers, obtaining the clean observations of the whole network for follow-on kinematic estimation of states of satellites and receivers. These states comprise 3- dimensional satellite and receiver coordinates, clock biases and troposphere delays. After substituting the derived SSR corrections into the original observations of the reference receivers, the functional models should be free of these delays and expressed as follows:

75) where L=1,2,3, s=1,2,..., S, r=1,2, ...R; is the residual integer term and can be

redetermined in the remaining network-based kinematic computing if necessary. Given the position states of satellites and receivers, Equations (74) and (75) can be used to kinematically redetermine clock biases along with a residual ZTD for each station under necessary datum settings or constraints. On the other hand, DD treatments can enable kinematic estimation of satellites or receiver antenna states and station ZTDs. Modelling of these states can then be performed individually in the time domain.

SUBSTITUTE SHEET (Rule 26) RO/AU Table 1.

Definition of original code and phase signals

Table 2. Definition of combined code and phase GNSS signals

REFERENCES

1. Blewitt G (1989) Carrier phase ambiguity resolution for the global positioning system applied to geodetic baselines up to 2000 km. J Geophys Res 94(B8):10187-10203.

2. Black, H.D and A. Eisner (1984). "Correcting satellite Doppler data for tropospheric effects". Journal of Geophysical Research, Vol. 89, No. D2, pp. 2616-2626

3. Chen, J., Zhang, Y., Yang, S. & Wang, J. 2015, A new approach for satellite based GNSS augmentation system: From sub-meter to better than 0.2 meter era, in Proceedings of the ION 2015 Pacific PNT Meeting, Honolulu, Hawaii, April 2015, pp. 180-184.

4. Collins P, Bisnath S, Lahaye F, Heroux P (2010) Undifferenced GPS ambiguity resolution using the decoupled clock model and ambiguity datum fixing. J Inst Navigat 57(2):123-135.

5. Davidson, R., MacKinnon J. G. (1993). Estimation and inference in econometrics. Oxford University Press.

6. Feng, Y, Gu S, Shi C, Rizos C (2013). A reference station-based GNSS computing mode to support unified precise point positioning and real-time kinematic services J Geod (2013), 87, pp 945-960

7. Feng, Y., Wang, Y., Gong, X., and Gu, S. (2017) Wide-area Kinematic Positioning with BeiDou Triple-frequency Signals. J Inst Navig, 64: 139- 148. doi: 10.1002/navi.174

8. Ge M, Gendt G, Rothacher M, Shi C, Liu J (2008) Resolution of GPS carrier-phase ambiguities in precise point positioning (PPP) with daily observations. J Geod 82(7):389-399.

9. Geng J, Meng X, Dodson A H, Teferle F N (2010) Integer ambiguity resolution in precise point positioning: method comparison. J Geod 84(9):569-581.

10. Geng, J. Teferle F N , Meng X, Dodson AH (2011) Towards PPP-RTK: Ambiguity resolution in real-time precise point positioning, Advances in Space Research, 47 (10):1664-1673.

11. Geng, J., Chen, X., Pan, Y. et al. A modified phase dock/bias model to improve PPP ambiguity resolution at Wuhan University. J Geod 93, 2053-2067 (2019). https://doi.org/10.1007/s00190-019-01301-6

12. Geng, J., Chen, X., Pan, Y. et al. PRIDE PPP-AR: an open-source software for GPS PPP ambiguity resolution. GPS Solut 23, 91 (2019). https://doi.org/10.1007/s10291-019· 0888-1. Gu S, Shi C, Lou Y, Feng Y, Ge M (2013) Generalized-positioning for mixed-frequency of mixed-GNSS and its preliminary applications. In J Sun, Jiadong, Jiao, Wenhai, Wu, Haitao, & Shi, Chuang (Eds.) China Satellite Navigation Conference (CSNC) 2013 Proceedings, Lecture Notes in Electrical Engineering 244, Springer Berlin Heidelberg, Chapter 35 : 399-428. Gu S, Lou Y, Shi C, Liu J (2015) BeiDou phase bias estimation and its application in precise point positioning with triple-frequency observable. J Geod DOI 10.1007/s00190-015-0827-z. Laurichesse D, Mercier F, Berthias JP, Broca P, Cerri L (2009) Integer ambiguity resolution on undifferenced GPS phase measurements and its application to PPP and satellite precise orbit determination. Navigation 56(2):135-149. Laurichesse, D. & Privat A. 2015, 'An open-source PPP client implementation for the CNES PPP-WIZARD demonstrator', in Proceedings of the 28th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2015), Tampa, Florida, September 2015, pp. 2780-2789. Liu T, Yuan Y, Zhang B, Wang N, Tan B, Chen Y. (2016). Multi-GNSS precise point positioning (MGPPP) using raw observations. Journal of Geodesy. Lou Y, Gong X, Gu S, Zheng F and Feng Y (2016), Assessment of code-bias variation of BDS triple-frequency signals and their impacts on ambiguity resolution over long baselines. GPS Solution DOI 10.1007 / sl0291 - 016 - 0514-4. Lou Y, Zheng F, Gu S, Wang C, Guo H, Feng Y. (2015). Multi-GNSS precise point positioning with raw single-frequency and dual-frequency measurement models.

GPS Solutions, 20, 849-862. doi: 10.1007/sl0291-015-0495-8. Montenbruck O., Hauschild, A. (2013) Code Biases in Multi-GNSS Point Positioning, Proceedings of the 2013 International Technical Meeting of The Institute of Navigation, San Diego, California, January 2013: 616-628. Montenbruck, O., Hugentobler, U., Dach, R., Steigenberger, P., Hauschild, A. (2012). Apparent clock variations of the Block IIF-1 (SVN62) GPS satellite. GPS Solutions, 16(3), 303-313. DOI: 10.1007/sl0291-011-0232-x NAVCOM TECHNOLOGY 2016, NavCom Cadastral & Boundary StarFire™, viewed 7 January 2020,

Stonemann E, Becker, M., Springer, T (2011) A new approach for GNSS analysis in a multi-GNSS and multi-signal environment. Journal of Geodetic Science. 1(3), 204- 214, doi:10.2478/vl0156-010-0023-2. Shi J, Gao Y (201S) A comparison of three PPP integer ambiguity resolution methods. GPS Solutions 18(4): 519-528. Shi J, Yuan X, Cai Y, Wang G (2016) GPS real-time precise point positioning for aerial triangulation. GPS Solutions 1-10. doi: 10.1007/sl0291-016-0532-2. Teunissen P J G, Khodabandeh A (2014). Review and principles of PPP-RTK methods, Journal of Geodesy, 2014, 89(3): 217-240. Tegedor, Javier, Liu, Xianglin, de Jong, Kees, Goode, Matthew, Øvstedal, Ola, Vigen, Erik, "Estimation of Galileo Uncalibrated Hardware Delays for Ambiguity-Fixed Precise Point Positioning", NAVIGATION, Journal of The Institute of Navigation, VoL 63, No. 2, Summer 2016: 173-179. TRIMBLE NAVIGATION LIMITED 2016, Subscription services, viewed 7 January 2020, http://www.omnistar.com/SubscriptionServices.aspx. TRIMBLE INC. 2019, Trimble RTX frequently asked questions, pamphlet, PN 022517- 231D (11/19), viewed 7 January 2020, https://positioningservices.trimble.com/wpcontent/uploads/2019/02/trimble-RTX-

FAQ-2020-Brochure.pdf VERIPOS. 2017, Services, viewed 7 January 2020, https://veripos.com/services. Wanninger L and Beer S (2015) BeiDou Satellite-induced code pseudorange variations: diagnosis and therapy. GPS Solution, doi 10.1007/sl0291-014-0423-3. Wubbena G (1985) Software developments for geodetic positioning with GPS using TI-4100 code and carrier measurements. In: Proceedings of first international symposium on precise positioning with the global positioning system, Rockville, USA: 403-412. Zhao Q, Wang Y, Gu S, Zheng F, Shi C, Ge M, Schuh H. (2018). Refining ionospheric delay modeling for undifferenced and uncombined GNSS data processing. Journal of Geodesy, doi: 10.1007/s00190-018-1180-9.

Claims

1. A method of processing data files or streams from a single GNSS receiver for generation of space-state representation correction components for the receiver tracking a plurality of satellites transmitting code and phase signals at two or more frequencies, the method including the steps of: a. reformatting the code and phase signals into three groups of linear combinations, including geometry-free and ionosphere-free (GFIF) combinations, geometry-free and ionosphere-present (GFIP) combinations, and geometry-based and ionosphere- free (GBIF) combinations; b. using the known broadcast or precise GNSS orbits, clock and station coordinates, initial hardware delay products sequentially reprocessing the data streams from the receiver, determining undifferenced (UD) and single-differenced (SD) integer ambiguities, and slant and SD ionosphere and troposphere delays with phase-only GFIP and GBIF measurements, respectively and recomputing uncalibrated hardware delays (UHD) in code and phase signals epoch by epoch, which are known as raw or single-epoch state-space representation (SSR) corrections; c. applying the raw and fitted SSR corrections for code and phase UHDs, slant or SD ionosphere and troposphere delays to calibrate the same delay terms in the user code and phase measurements to enable Real Time Kinematic (RTK) and PPP-RTK positioning services, based on user's single-frequency, dual-frequency or triple- frequency data; d. substituting the fitted SSR corrections for hardware and atmosphere- delays adjusted from a network of reference receivers into the original code and phase observation equations of each reference receiver, to correct the same delay terms, to enable recovering the integer-fixed original phase measurements and the code-offset corrected original code measurements, and obtain uniformed function models for all original code and phase measurements which include mainly the effects of multipath errors and random observation noises.

2. The method of Claim 1 wherein the single-receiver based computing makes use of the GFIF and GBIF observables to resolve UD and SD integers further comprising: a. using three GFIF models to determine the UD float and integer ambiguities for each receiver-satellite direction, but the models may be corrected with initial UD code and phase biases to reduce the biases of the floating ambiguities; b. performing a least-squares adjustment with known SD integers as constraints to obtain the consistent Line-of-Sight (LOS) integers for each reference receiver; c. using the integer-fixed GFIP phase measurements and initial UHD products to compute SD ionosphere-delay for a reference receiver; d. computing a slant ionospheric delay of the receiver to the reference satellite path with respect to zero UD UHD value or an external ionospheric model value as the datum of the slant ionosphere-delay for each receiver; e. mapping the UD and SD ionosphere delays into all LOS directions; f. using the integer-fixed GBIF phase measurements to compute SD troposphere-delays for a reference receiver; g. setting the slant residual troposphere-delay in the reference satellite direction with respect to zero UD HUD value or a different troposphere-model as the datum of slant troposphere-delay for each receiver and mapping the UD and SD troposphere delays into all LOS directions; h. removing the determined SD integers from all selected SD GFIF, GFIP and GBIF observables, removing the SD ionosphere-delays from the GFIP observables, and removing the SD troposphere delays from the GBIF measurements, then forming linearly independent equations for combined UHD equations for all SD observables; i. converting the combined UHD observables to UHDs in the original code and phase signals; j. performing a fitting or filtering process for each single epoch UHD time series for forming function of time for prediction and interpolation; k. formatting the SD delays and SD UHD function of time into SSR correction formats;

L. mapping all SD integers and delays to LOS directions by appending the UD quantities, then obtaining slant delays and LOS UHDs for SSR corrections; and m. Implementing the above steps in a built-in processor of dual or multiple frequency receiver processor or an external computing device.

3. The method of Claim 1 wherein the network-based computing makes use of single-epoch combined UHD samples from each single receiver to derive adjusted double-differenced (DD) integers for the necessary baselines, which are then mapped to all LOS directions, by appending UD and SD integers from the single-receiver based processing the method comprising the steps of: a. performing DD operations over the obtained LOS UHD and LOS integers to derive DD integers for all necessary baselines to enable consistence between all LOS integers; b. mapping the DD integers and the first station SD integers to adjusted LOS integers by appending the LOS integers for all receiver-reference satellite directions; c. applying the adjusted LOS integers to lumped sums of LOS integers and UHDs to adjust the LOS UHDs; d. performing DD operation onto code LOS UHD time series and obtaining DD code UHD offsets for the network with long-term or historical data; e. removing LOS integers, slant ionosphere delay and slant troposphere delays from combined measurements of all LOSs, forming linear equations for satellite-specific and receiver-specific UHDs and performing least-square estimation to determine satellite- and receiver specific UHDs; f. computing the residuals of the least square solutions of the above linear equations and obtaining the covariance matrices of satellite-specific and receiver specific UHDs; and g. converting the obtained combined UHDs to original code and phase UHDs and performing fitting process to each original and combined UHD time series for a function of time.

4. A method of applying SSR corrections from a single reference receiver or a network of multiple receivers to enable both user-end RTK and PPP-RTK processing and improve their performance, the method comprising the steps of: a. directly applying the single-epoch or fitted SSR corrections from each reference receiver to calibrate user receiver's original or combined code and phase measurements, so the user receiver can form delay-free observational equations and perform single-reference RTK or PPP-RTK positioning with various types of measurements, such as code only, single-frequency, dual-frequency, or triple frequency measurements; b. applying the network-adjusted single-epoch or fitted SSR corrections from multiple reference receivers to user receiver original or combined measurements, the user receiver can form corrected observational equations and perform multi-reference RTK or PPP-RTK positioning processing with various types of measurements, such as code-only, single-frequency, dual-frequency or triple frequency measurements; and c. calibrating inter-receiver code offsets with respect to each reference receiver from the difference between corrected code and phase linear observational equations.

5. The method of Claim 4 comprising the further benefits of: a. the single-epoch SSR corrections have the same precision as the raw code and phase OSR measurements from the reference stations but can support both RTK and PPP- RTK processing with various types of measurements. The fitted UHD corrections have higher precision than that of the raw code measurements in OSR messages from the same reference stations, thus improving RTK and PPP-RTK performance; b. using the determined single-epoch SSR corrections, the user-end RTK and PPP-RTK positioning can give the completely consistent results; c. calibrating inter-receiver code offsets in original or combined code measurements with respect to a reference receiver can support differential GNSS positioning for decimetre accuracy over long distances and can contribute to shortening the convergency time of PPP-RTK solutions and integer ambiguity resolution; d. using the determined single-epoch or fitted SSR corrections instead of the raw code and phase measurements (OSR) from the reference stations, the size of SSR messages is one-sixth to one-fourth of the size of the OSR messages over a period of operation, thus the required bandwidth for data transmission for RTK and PPP-RTK services is significantly reduced.

6. The method of Claim 1 wherein the network-based computing approach makes use of integer-fixed and delay-free clean measurements to significantly simplify the estimation of satellite and receive states and clocks, the method comprising the treatments of: a. Using the double-differenced functional models to remove the satellite and receiver clock biases, determining the states of satellite antennas epoch by epoch with given receiver coordinates, or determining the states of receiver antennas epoch by epoch with given satellite states. Station-based ZTD states should be estimated along with state estimation in both cases; and b. with given satellite and receiver states, the satellite and receiver clock biases can be determined along with ZTD parameter epoch by epoch; and c. modelling or filtering individually in time domain without affecting other states and delays.

SUBSTITUTE SHEET (Rule 26) RO/AU