WO2006108227A1 - Improved phase ambiguity resolution using three gnss signals - Google Patents

Abstract

This invention presents a system for improved ambiguity resolution and positioning performed with three or more ranging signals from one or more GNSS constellations, such as modernized GPS and Galileo systems. For a given GNSS service with every three frequencies, Subsystem 1 of the system searches and selects at least two extra-widelane (wavelength of above 290 cm) and one narrow- lane (wavelength of about 11 cm) virtual signals that have the minimal ionospheric scale factors in magnitude against their wavelengths. As a result, these EWL and NL signals can tolerate much larger ionospheric errors than other signals, thus allowing for more reliable and rapid ambiguity resolution over a much longer base- to-rover distances. As estimation of ionospheric and tropospheric parameters are no longer needed, the computational effort for statistical search and validation is minimal. Subsystem 2 of the system is a set of 2 or more generic GNSS receiver capable of tracking GNSS satellites in view and outputs the code and carrier measurements on each of the three frequencies. Subsystem 3 of the system is a processing system that performs ambiguity resolution and position estimation using double-differenced phase and code measurements from at least two receivers. Subsystem 3 comprises four procedures: (1) resolving two select extra-widelane (EiWL) ambiguities using measurements from a single or multiple epochs at almost 100% certainty; (2) estimating the first-order ionospheric correction using the two ambiguity-fixed extra-widelane signals and applying the correction to the widelane (WL) signal; (3) resolving the select narrow-lane (NL) ambiguity over distances of up to a few hundreds of kilometers; and (4) using these ambiguity resolved virtual signals to perform real time kinematic positioning at the regional to global scale for decimeter accuracy, and the local to regional scale for centimeter accuracy. This invention provides the key technical algorithms for future generation local, regional and global GNSS services using three and more carrier signals.

Description

IMPROVED PHASE AMBIGUITY RESOLUTION USING

THREE GNSS SIGNALS

This invention presents a system for efficient Ambiguity Resolution (AR) and positioning performed with three or more ranging signals from Global Navigation Satellite Systems (GNSS) such as modernized GPS and Galileo satellite systems and Regional Navigation Satellite Systems (RNSS) such as Quasi-Zenith Satellite Systems (QZSS). The system consists of three subsystems, for GNSS service, GNSS receiver and user terminal respectively.

BACKGROUND OF THE INVENTION

1. Field of Invention The present invention concerns an ambiguity resolution (AR) method performed with three or more ranging signals from Global Navigation Satellite Systems (GNSS), examples being modernized GPS, Galileo and Quasi-Zenith Satellite Systems

The invention also concerns a positioning method using specifically combined GNSS signals whose phase ambiguities were correctly determined using the said AR method to support decimeter positioning services on regional to global scales and centimeter level positioning services locally to regionally.

The invention is described as a system consisting of three subsystems, using the given GNSS service frequency scheme, a receiver and user/application terminals.

2. Description of Prior Art

Global Navigation Satellite Systems (GNSS) include the Global Positioning Systems (GPS), Glonass and all future systems such as Galileo systems. Regional Navigation Satellite Systems (GNSS) include the planned Japanese Quasi- Zenith Satellite Systems (QZSS) and the possible Chinese Beidou 2 system.

Each GPS satellite transmits continuously using two radio frequencies in the L-band, referred to as L1 and L2, at respective frequencies of 1575.42 Mhz and 1227.60 Mhz. Each GPS signal has a carrier at the L1 and L2 frequency, pseudo-random noise (PRN) codes and satellite navigation data. Two different PRN codes are transmitted by each satellite C/A code (P1) and P2 code (PfY). The modernized GPS will introduce an additional frequency, L5 for 1197.45Mhz, then each GPS satellite will transmit 3 carrier phases and 3 codes.

The Galileo system is designed to provide four signals for commercial and civilian use centered on L1 (1575.42MHz), E6 (1278.750MHz), E5B (1207.140Mhz) and E5A (1176.450MHz). Each signal also transmits both carrier phase and PRN codes. The Galileo open service and safety-of-life service will be provided with L1 , E5A and E5B frequencies, while the Galileo commercial service can also access an additional frequency on E6.

The planned QZSS system will provide service with L1 , L2 and L5 frequencies as well on regional basis. It has been suggested the Chinese Beidou-2 system also transmits signals at three frequencies in L-band, although details of the frequency allocations remain unpublished.

Many techniques have been developed for reliable and rapid determination of the cycle ambiguity in carrier phase signals observed by two or more GPS receivers 400.

Ambiguity resolution techniques normally involve the use of unambiguous code observations and ambiguous phase observations on multiple satellites together to

• provide enough measurements for estimation of all the states (unknown parameters). The methods and apparatus being developed are classified into two types: Geometry-dependent and Geometry-free methods, which are examined respectively.

The geometry-dependent approach to estimate the carrier phase ambiguities is to construct a global estimator that includes four devices:

(1 ) Physical modeling that builds functional equations between all the observations and all states, including rover position, ambiguities and ionospheric biases etc;

(2) Stochastic modeling that gives the statistical knowledge or assumptions on the residual errors or noises, such as zero mean noise and covariance values

(3) Least-square estimator that gives approximate real values for all the parameters including the ambiguities and covariance matrix

(4) Least-squares integer search engine that perform statistical search over the potential ambiguity candidates to find the best set of integer candidates and validate the top set of ambiguity candidates

Fig 1 shows the general structure of an ambiguity resolution global estimator. Double-differenced code and phase data set for 2 or more frequencies from 2 or more GNSS receivers 400 will be supplied as ready data set 105 to a global filter 100. In the process 110, physical models defined by observation vector and matrix will be formed. The process 120 defines the a-priori and a-posteriori statistical information of the measurement noises. Estimator 130 supplies floating ambiguity estimates for all the carrier signals. The search engine 140 yields integer solutions for the ambiguity and integer-fixed position solutions for the current epochs, from which the residuals will be supplied to the process 120 for improved statistical information for use in future epochs. This process is optional for some software.

A prior-art solution for this class of methods may cover any parts of the global filter

100, regardless of using signals for dual or three or more carriers. Some software systems implement more efficient integer search algorithms in the engine 140, whilst others are superior in physical and stocastic modeling at 110 and 120. The most successful ambiguity software takes good care of all the parts of the system 100, this being especially true in the current GPS case, where only L1 and L2 carriers are available for ambiguity resolution.

Fig 1A shows a prior-art structure of the physical modeling process 110 in terms of measurements and parameters/biases of different types 111a, for the GPS dual case 112a and the GPS three frequency case 113a. In addition to user states and ambiguity parameters, the process either has to include the ionosphere biases as parameters to estimate or correct them with other means, in order to reduce the effects to a certain level, for instance, less than 0.5 cycle. The number of the ionospheric parameters is as high as the number of satellites in view minus one. The accuracy requirements for the ionospheric bias estimation/correction are difficult to meet. This is why the existing methods operate over short base-rover distances, typically 10-20km in the single-base real time kinematic (RTK) case. In the network- RTK case, the ionospheric corrections can be better estimated from multiple reference stations, therefore, the base-to-rover distance (equivalent) can be extended to 30 to 40 km.

The recent US patent application 2005010248 by Vollath Ulrich (2005) "Ambiguity estimation of GNSS signals for three and more carriers" provides a comprehensive review of prior-art for geometry-dependent methods. The work recognizes the need to deal with the large number of ionospheric biases/parameters in three frequency cases and presents computationally efficient techniques for obtaining good approximation of the carrier phase ambiguities for three and more frequency bands, greatly reducing efforts required for statistical search and validation process.

Other research on phase ambiguity resolution using three modernized GPS signals and four Galileo signals includes the contributions from Forssell et al (1997), Han and Rizos (1999), Hatch et al (2000), Isshiki, (2003, 2004), Werner et al (2003). These are mostly the mixture of geometry-free and geometry-dependent methods based on the idea of "bridging the wavelength gap", making use of the combined signals of different wavelengths to provide extra-wide, wide and medium lanes (Hatch, at el, 2000, Misra and Enge, 2004).

These methods compute the ambiguity of three independent signals in three major steps as follows.

Step 1 : the integer ambiguities for the extra-widelanes (EWL) as marked in Table 2 and Table 3 can be fixed by rounding off the difference in cycle unit between the virtual phase and selected virtual code measurements, where the first-order ionospheric delays were deleted. The total noise in the difference is 1/5 cycle or less.

The 100% success rate can be expected with the measurement from a single epoch or a few epochs. This has been the first step of almost all existing methods. But this method relies on the code measurements on both L2 and L5 signals, which may not be expected to be available in all circumstances or for all three frequency receivers.

Step 2: There are two methods developed for this step: a. Method 1 : After the EWL integer has been fixed, efforts are made to fix the integer for the widelane signals by rounding off the difference in cycle units between the widelane signal and the refined range with the integer-fixed extra- widelane phase measurement. Due to the presence of ionospheric errors in the difference, performance of this method depends on base-to-rover distance. b. Method 2. Applying the same process as Step 1 to the widelane signal instead. The total noise in the difference is about 1/3 to 1/2 cycle. Therefore, high success rate by rounding-off the measurement from a single epoch cannot be expected. Experimental results from GPS data have shown the success rate of fixing integers by rounding off is about 80% to 90 % with data from a single epoch and over 95% from multiple epochs. This estimation is distance- independent. Step 3: After the widelane integer has been fixed, the third step is to try to fix the integer for the medium-lane (ML) signal by rounding off the difference in cycle unit between the ML signal and the refined range with the integer-fixed wideiane phase measurement. The idea is the same as in Method 1 , Step 2. The problem is again the large effect of the ionospheric delay in the difference with respect to the wavelength, thus limiting the step to working over short distances only.

Vollath Ulrich (2005) outlined a prior-art solution intended for the proposed Galileo system with three carrier frequencies. The solution was founded in Laboratory

Experiments on Carrier Phase Positioning Techniques for GNSS-2 (TCAR-test),

Technical Note WP 2100: Use of Physical Space Information, ESA/ESTEC Contract

No12.406/77/NL/DS Rider 1 (1999). Ulrich pointed out that "a deficiency of the proposed approach to three carrier ambiguity resolution (TCAR) is that all errors are handled as noise, making it computationally less efficient than desired and producing ambiguity estimates which are worse than desired".

Feng (2005) and Feng and Rizos (2005) outlined a distance-independent, geometry- free method for performing ambiguity resolution with three GNSS signals, comprising three major processes:

Process 1 : Geometry-free estimation of the first and second integers: Fixing the integer ambiguities for the extra-widelane (EWL) and WL as marked in Tables 2 and 3 for GPS and Galileo service cases by rounding off the difference between the extra-widelane virtual phase and the virtual code double difference measurements. In these differences, the first-order ionospheric errors are eliminated. The success of the ambiguity fixing depends on the code noise level, which may contain effects of multipath. Process 2: Estimation of the first-order double-differenced ionospheric delay of any ray path from the two integer-fixed widelane signals. Depending on the phase noise level, including multipath effects, the uncertainty of the direct estimate from a single epoch measurement would range from a few centimeters to a few decimeters. The estimate can be improved by modeling over multiple epochs or recursively. This estimation is a distance-independent process.

Process 3: Geometry-free estimation of the third independent integer. The third virtual observable must be linearly independent of the two known combinations used in Process 1 , for instance, L1 , being the one in this category. Applying the known ionospheric correction to the third signal, the first-order ionospheric bias can be removed. As a result, this distance-sensitive bias term is replaced by the distance- independent noise term magnified by noise in the phase observable. The AR performance of the third signal integer mainly depends on the uncertainty of the ionosphere estimate/correction, which generally consists of bias and random noise components. The integer uncertainty in a cycle is 12 times the residual uncertainty in metres. This shows that, to achieve high AR reliability, it is necessary to either keep the standard deviation (STD) of the double-differenced ionospheric estimation as low as a few centimetres, or the residual ionospheric bias to the same order of magnitude. In the triple-frequency case this accurate estimate or correction may be obtained from within the receiver over a period of observation; hence the performance is distance-independent. However, in the dual-frequency case, where the ionospheric correction is obtained from one or multiple reference stations, the performance will depend on the spacing of the network stations or the base-to-rover distances (Rizos 2002b; Dai et al. 2003).

A significant advantage of the above method is the nature of distance-independence.

Both ionospheric and tropospheric delays themselves do not affect the ambiguity resolution results. This has never been achievable before in the current dual- frequency based GPS ambiguity resolution. In other words, the success of the first two and third signal ambiguities depends on code and phase noise levels. The problem is, however, that it is difficult to achieve desired success rates and reliability of fixing the second (widelane) integers and the third integers (medium-lane) if strong multipath effects are present in code and phase measurements.

SUMMARY OF THE INVENTION

This invention presents a system for efficient Ambiguity Resolution (AR) and positioning performed with three or more ranging signals from Global Navigation Satellite Systems (GNSS) such as modernized GPS and Galileo satellite systems and Regional Navigation Satellite Systems (RNSS) such as Quasi-Zenith Satellite Systems (QZSS). The systems consist of three subsystems for GNSS service, GNSS receiver and user terminal respectively. The high-level description of the system is shown in Fig 2.

It is the object of the present invention to provide a generic method to identify superior virtual combinations most effective for carrier phase ambiguity resolution using two, three or more carrier GNSS signals, from practically possible combinations. This method is described in Subsystem 1 of the invention, as shown in Fig 2.

The major advantages in accordance with this invention are the new design for the physical modeling process 110B as shown in Fig 1B. This is compared to the existing design of the physical modeling process 110A as shown in Fig 1A, where the emitting signals L1 , L2 and L5 ( or widelanes (WL)) are used directly. In the dual- frequency case 112B, the widelane and a specific narrow-lane are chosen instead of other possible combinations, so that the effects of the residual ionospheric biases are much less sensitive to the base-rover distances. In other words, the selected 2 signals can tolerate larger ionosphere errors, thus allowing for more reliable and rapid ambiguity resolution over longer distances. In the three frequency GPS case 113B, two EWL signals and 1 narrow-lane (NL) signal are chosen to form the physical model design, where the effects of ionospheric biases can be reduced to the level of 0.1 cycles or less, effectively through corrections by code measurements or differential or network-based ionospheric techniques. In summary, the ionosphere is no longer a dominating factor in the ambiguity resolution.

It is a further object of the present invention to outline efficient ambiguity resolution procedures using three chosen virtual combinations in the four GPS and Galileo service cases. The procedures include steps:

(1) to form double differenced phase measurements and linear equations for the 2 select EWL signals and 1 select NL signal;

(2) to determine the integer ambiguity of the EWL by rounding off the difference (in cycle units) between that EWL phase and the selected virtual code measurements;

(3) to resolve the integer ambiguity for the second of two select EWL signals using an existing least squares integer search procedure;

(4) to estimate the first-order ionospheric bias using the two ambiguity-resolved EWL signals or their linearly dependent derivatives; (5) to use the estimate of the first-order ionospheric delay to correct some ambiguity-resolved signals, such as the widelane signal, to allow for positioning processing over any distance.

An additional object of the present invention is to maximize the performance benefits of positioning solutions, achieved using the select GNSS signals whose phase ambiguities were correctly determined using the said steps.

Therefore, the invention can support at least the positioning applications as follows; (1 ) At the regional to global scale, real time kinematic positioning services at decimeter level with three carrier signals from observation of seconds to minutes (2) At the local to regional scale real time kinematic positioning services at centimeter level with three carrier signals from observations of seconds to minutes

The positioning performance will then depend on the phase noise level and tropospheric errors. The local, regional and global scales in this context are defined over the base-to-rover distance of tens, hundreds and thousands of kilometers.

This invention has significant implications for the future generation GNSS services using three and more frequency bands.

Description of Tables and Figures

Table 1 gives frequencies, wavelengths and assumed noise levels of three GPS civilian signals and four Galileo open and commercial service signals. Table 2 gives frequencies, wavelengths, assumed noise levels and success rates of three GPS virtual WL and EWL signals, of which two are independent. Table 3 gives frequencies, wavelengths, assumed noise levels and success rates of six Galileo virtual WL and EWL signals, of which 2 WL and 2 EWL signals are independent. Table 4 summarizes the error characteristics of the two select EWL and WL combinations for each of the four possible services offered by every combination of three GNSS signals. Of these combinations, two are independent and the first EWL integer can be estimated by a geometry-free method. It is seen that in each service there is one EWL signal whose noise to wavelength ratio, or total error in cycles based on assumed noises, is a minimum, usually less that 0.20 cycles. The signal is superior to all others for geometry-dependent ambiguity resolution With the global estimator 100.

Table 5 summarizes the error characteristics of the third NL signals combined from each three-signal combination for each GNSS service. Each of the optional signals is independent of the two EWL and /or WL signals in Table 4. Given the relative tropospheric error of 2 cm and phase noise level of 0.5 cm, the last column is the expected total noise in cycles. It is clear that one or two of the signals have the total noise level lower than 0.25 cycles, which appear superior to all others for ambiguity resolution. Table 6 gives smoothing time epochs for various positioning accuracy levels, showing the comparison between GPS pseudo-ranges and GPS/Galileo widelane phase measurements. It will be noted that the latter require approximately one order of magnitude smaller time duration.

Fig 1 shows the general structure of an ambiguity resolution global estimator with inputs of GNSS code and phase data sets for 2, 3 or more carriers and outputs of ambiguity-fixed phase measurements and positioning solutions. Fig 1A shows a prior-art structure of the physical modeling process in terms of measurements and parameters/biases of different types, given examples in both GPS dual case and the three frequency case.

Fig 1B shows a summary of the physical modeling process, showing the major difference with respect to the existing process as shown in Fig 1A, in accordance with embodiments of this invention. Fig 2 shows the major components and overall structure of an improved three carrier ambiguity resolution system in accordance with embodiments of the invention, consisting of 3 subsystems.

Fig 3 illustrates the key component of Subsystem 1 , a process that identifies the suitable combined/virtual signals for a given GNSS service with signals on 3 or more frequencies in accordance with an embodiment of this invention. Fig 3A is a further architecture in Subsystem 1 , showing the classifications of possible GNSS services in terms of frequencies in accordance with an embodiment of this invention.

Fig 4 is a structure of a GNSS receiver that tracks satellites and outputs the code and carrier measurements on 2, 3 and more frequencies for all the satellites in view, in accordance with embodiments of this invention. The Subsystem 2 is set of at least two GNSS receivers 400.

Fig 5 shows the structure of Subsystem 3, comprising a computing system that consists of procedures and algorithms to process the measurement outputs from 2 or more receivers for ambiguity resolution and position estimation in accordance with embodiments of this invention.

DETAILED DESCRIPTION OF EMBODIMENTS

The invention is a system for improved three carrier phase ambiguity resolution, comprising 3 subsystems. Subsystem 1 operates at a navigation satellite system level; Subsystem 2 operates at the receiver level; Subsystem 3 operates at the user/network terminal or application level. Fig 2 shows the major components and overall structure of an improved three carrier ambiguity resolution system in accordance with embodiments of the invention.

1. Subsystem 1 : A Processing Device for Identification of Useful GNSS Signals

1.1 Basic Equations for Multiple GNSS Signals

A one-way pseudo-range or phase is defined as the pseudo-range or phase measurement for one receiver-to-satellite pair, which may also be called a "line-of- sight" measurement or "zero difference" measurement. The one-way pseudo-range can be written as P₁ = P + c(dt - dT) + d_trop + d_orb + UiOn₁ + ε_Pi (1 )

Where i = subscript for a certain frequency of signal, for GPS, i=1 ,2,5 P = pseudo-range p = geometric distance between the receiver and satellite d_grb . = satellite orbit error

d_trop ⁼ tropospheric delay

dion = ionospheric delay c = velocity of light dt, dT receiver and satellite clock errors

ε_P = pseudo-range noise The one-way phase is described as

L₁

+ ε_Li (2)

Where/. = phase measurement (in range units) N¹ = non-integer phase ambiguity, which contains the integer ambiguity N and fractional phase expressed as N' = [N₁ +[φ_r(t₀,L_i) -φ_s(t_ς>,L_i)]} ,

where φ_r(t₀,L_c) and ^_s(r₀,Z,,) are receiver and satellite instrument

biases respectively, λ = the wavelength of the carrier signals ε_c = phase noise, including maltipath effect

d_orb , d_irop , dion , c , dt, dT are the same as for Equation (1 ).

In the three GPS signal case, for one-way carrier phase in length unit, the linear combination of three signals can be generally formulated as (Feng, 2004, Abidin, 2000) L(i,j,k) J ^■ A - ₅ . L₅

where i, j, k are any integer coefficients. This linearly combined signal has the virtual frequency: f(iJ,k) = i-A +J- fi +k- f₅, (4) and the combined wavelength,

and the cycle ambiguity

N(i,j,k) = i - N_{ + j - N₂ + k - N_{5 (6)}

Whilst the frequency-independent errors remain the same in this combination, the frequency-dependent errors will change their magnitude via this linear combination. The magnitude of the ionospheric effects on the linear combination can be expressed as

^{%m ~} i-f_l+j-A+*-f, fr^kiy)fϊ ^{■ (7)} where Ki is a constant depending on the slant total electron content (STEC); β_x is the ionospheric scale factor (ISF).

_2π , _ (i-f.) ^•<, +σ-/₂)' -< +-(*-/₅ ")- <ts

<i-f_x +j-Λ +k-f₅γ ₍₈₎

For pseudo-ranges, we have the similar combination

KM=¹^ ly- fi"+ J -'f?i +⁺k*-f/_s ^*-*' (9)

but, the integers i, j and k take the choices of 0 or 1 only. The magnitude of the ionospheric effects on the linear combination (7) can be expressed the same as (5). The noise level of the combined code is expressed as

_σ,_mjιk) (10)

In any dual-frequency case, all notations with the index (i,j,k) will automatically be replaced with the index (i,j), for instance, L(i,j), N(i,j) and P(i,j).

For any given GNSS service, Fig 3A shows the classifications of possible GNSS services in terms of service frequencies in accordance with an embodiment of this invention. A GNSS signal data set 310 gives the groups of possible services using 2 and 3 frequencies from the processing unit 310a. The data set 310b lists five dual- frequency and four three-frequency options:

(1 ) GPS service with L1 and L2 (current GPS service)

(2) GPS service with L1 and L5 (possible GPS service) (3) Galileo service with L1 and E6 (public regulated service)

(4) Galileo service with L1 and E5a (the same as GPS L1 and L5)

(5) Galileo service with L1 and E5B (Open)

(6) GPS service with L1 , L2 and L5 (GPS civilian)

(7) Galileo service with L1 , E5B, E5A (open and safe-of-life) (8) Galileo service with L1 , E6, E5A (commercial)

(9) Galileo service with L1 , E6, E5B (commercial)

For specific user receivers, there are probably 1 or 2 or 3 options only.

1.2 Identifying Optimal Combinations for Geometry-Dependent Ambiguity Resolution

Theoretically, there is an infinite number of linear combinations defined by Equation (3). To assure the integer nature of the combined signals, only combinations with integers i, j, and k are considered, of which three are linearly independent. The end goal of three carrier ambiguity resolution (TCAR) is to find three combinations, whose integer ambiguities can be easily and reliably determined, then leading to recovery of the integer ambiguity solutions of the three natural phase measurements, for instance, in the GPS case, L1, L2 and L5.

To find the potentially useful combinations, we characterize parameters for all the combined signals with wavelengths ranging from 0.10 to 20 m for ie [-50, 50], je [- 50,50] and ke [-50,50]. There are over 1 ,000,000 combinations in each of the GPS and Galileo service cases. For convenience of analysis, we classify the combinations into four categories against wavelengths in this context: a) Extra-widelane (EWL) for combined signals with wavelengths of 2.903m and over; b) Widelane (WL) for combined signals with wavelengths of 0.751m to 2.903m; c) Medium-lane (ML) for combined signals with wavelengths of 0.1903 m to 0.750 m d) Narrow-lane (NL). for combined signals with wavelengths of 0.10 m to

0.1903 m (L1 wavelength).

Fig 3 illustrates the processing elements of the system 300 to identify the most useful virtual signals for the given service classified in the data set 310. The functions of the five elements in the system 300 are described as follows:

(1) Element 310 identifies the first EWL signal, which is L(0,1 ,-1), assuming that f(1 ,0,0)>f(0,1 ,0)>f(0,0,1) and [(f(θ^'i _>O)-f(O,O,1))]>[f(1 ,O,O)-f(O,1 _>O)]. For the four possible GPS and Galileo services, the characteristics of this EWL are provided in the first line of each block of Table 4. (2) Element 320 examines the options for the second most useful signals. Table

4 lists the characteristics of the mostly interesting EWL and/or WL combination options for all the four possible GPS/Galileo services offered by three GNSS carriers. In each service case, as long as any two combinations are given, any third one is linearly dependent and can be derived from the given two. For the best ambiguity resolution purposes in this invention, we select two specific combinations, in which the total effect of the ionospheric error/bias and phase noise reach a minimum with respect to the wavelength.

For each specific virtual signal, the virtual frequency and wavelength are given in Columns 4 and 5 of Table 4; the ionospheric scale factor (ISF) and the phase noise factor (PNF) in Columns 7 and 8. Columns 9, 10 and 11 give the total noise levels in cycle units computed from the ISF and PNF using three different assumptions for the ionospheric delay on the L1 carrier and the 1 σ phase noise level and residual tropospheric delay.

(3) Element 325 decides which is the optimal second EWL signal from Table 4. In each service there is one superior EWL having a minimum ionospheric scale factor, which reduces the effects of the first-order ionospheric delay to the factor of 0.0744 in the GPS case, and 0.3035, 0.2454 and 0.0227 in the three Galileo cases respectively. Ambiguity resolutions using these EWL signals should outperform the use of any other combinations over long distances. In additional to the select EWL signals, Element 325 also decides one more additional EWL signal, for consistence check purposes if applicable.

(4) Element 330 examines the options for the third most useful signals from a new category of combinations, of which each must be independent of the previous two or their derivatives. Table 5 lists the characteristics of the most interesting NL combinations whose ISF are much less than 0.10 in magnitude, and any one is linearly independent of the previous two EWL or WL signals in Table 4 within each service case. After examination of all the possible combinations at EWL, WL, ML and NL levels, it is recognized that there are many EWL, WL and ML signals in this category, but none of these is evidently more useful for ambiguity resolution purposes over longer distances.

For instance, in the GPS case, L(-3, 4,0) is a widelane of 1.628 m wavelength and independent of L(1 ,0,-1 ) and L(0, 1 ,-1) and their derivatives. With the ISF of 18.225, an ionospheric delay of 10 cm on L1 can lead to a 1.82 m error on L(-3,4,0). At the NL levels of the signals, there are thousands of combinations in each of the four possible GNSS services. Similarly, for each specific virtual signal, the virtual frequency and wavelength are given in Columns 5 and 6; the ionospheric scale factor (ISF) and the phase noise factor (PNF) in Columns 7 and 8. Columns 9 and 10 give the total noise levels in cycle units computed from the ISF and PNF using three different assumptions for ionospheric delay on the L1 carrier and the 1 σ phase noise level. Column 11 includes the effects of the relative tropospheric errors. From this table, we can obtain comments as follows: a. These NLs are nearly ionospheric-free, having the wavelength of about 11 centimeters. These interesting characteristics are useful for both ambiguity resolution and positioning, as compared to the ionospheric- free combination L(77,-66,0), whose wavelength is ^• too short for ambiguity resolution. b. Indeed in each group, there are a few useful NL signals, whose ISFs are often about 1% or less. As a result, an ionospheric delay of 1 m level leads to a ranging error of 1 cm or less in these NL measurements. With respect to the wavelength of 11 cm, the ionospheric effect is only less than 0.1 cycles. If the other effects such as relative tropospheric errors and phase noise are controlled within a few centimeters, these NLs can be used for more effective ambiguity resolution over longer baselines. Fortunately, the tropospheric delay is less distance-sensitive. c. Useful NL signals are also identified in a number of dual frequency service cases, such as GPS L1 and L2, GPS L1 and L5, Galileo L1 and E6, Galileo L1 and E5A and Galileo L1 and E5B. This implies that long-range ambiguity resolution may be achieved using dual-frequency receivers as well. For instance, in the GPS dual carrier case, the useful NLs are L(4,-3) and L(5,-4), where the respective ISFs are 0.0902 and 0.0708.

(5) Element 335 decides which NL signal is used as the third signal for ambiguity resolution. In principle, the one with minimal noise to wavelength ratio (total noise in cycle) is the best choice. In practice, the one with the next lowest noise to wavelength ratio may be selected for consistence check purposes if possible. a. In the GPS civilian services with L1 , L2 and L5 frequencies, two narrow-lane choices are L(4,0,-3) and L(3,6,-8) b. In the GPS service with L1 and L2 frequencies, the two narrow-lane choices are L(4,-3) and L(5,-4) c. In the Galileo service with L1 , E6 and E5A, the two narrow-lane choices are L(4,0,-3) and L(3,3,-5) d. In the Galileo service with L1, E6 and E5B, the two narrow-lane choices are L(5,-3,1 ) and L(4,1 ,-4) e. In the Galileo service with L1 , E5B and E5A, the two narrow-lane choices are L(4,0,-3) and L(4,-1 ,-2) f. In the Galileo service with L1 , E6, the narrow-lane choice is L(5,-4)

The results of the process 300 would be the data set 340, which gives 3 sets of coefficients for the 3 best EWL/NL signals for ambiguity resolutions.

2 Subsystem 2. A Set of Generic Multiple Frequency GNSS Receivers

Fig 4. schematically illustrates a generic multiple frequency scenario of GNSS receiving system, which is known as Subsystem 2 as shown in Fig 2. Receiver 400 receives GNSS signals from any number of satellites in view such as SV1 , SV2, SV K, shown respectively at 411 , 412 and 410. Each signal has 2, 3 or more than three frequencies, produces 2, 3, and more sets of code and carrier measurements. To have full capacity of ambiguity resolution, this Subsystem must have a set of at least two multiple frequency GNSS receiver, that can track signals from more than four GNSS satellites. Use of existing dual-frequency GPS receivers can also partially achieve improved ambiguity resolutions.

3. Subsystem 3: A Processing Device for Carrier Phase Ambiguity Resolution

Fig 5 shows the structure of Subsystem 3, which has a core processing device 500 comprised of procedures and algorithms to process the measurement outputs from 2 or more receivers for ambiguity resolution and position estimation in accordance with embodiments of this invention. Two data sets 430 and 340 will be supplied to the device 500. Data set 430 contains code and carrier phase measurements from 2 or more GNSS receivers 400. Data set 340 contains 3 sets of coefficient parameters for the select EWL and NL virtual signals as determined in Subsystem 1. The subsystem resolves three carrier ambiguities at two levels: a) At the regional-to-global level, estimate the first type of EWL and WL ambiguities using distance-independent geometry-free approaches; estimate the second type of EWL and WL ambiguities using geometry-dependent methods, over distances of up to a few thousands of kilometers; b) At the local level, estimate the third NL ambiguity using geometry-dependent methods, over baselines of up to a few hundred kilometers.

According to this categorization, current dual-frequency based ambiguity resolution is achieved at the sub-local level, ie, over baselines of up to tens of kilometers.

The device 500 completes ambiguity and positioning tasks with operation of six processing elements: 510, 515, 520, 525, 530 and 535.

Processing Element 510 prepares data sets of double-differenced code and phase measurements for the select EWL, WL and ML and NL signals for use in the generic process 100. Other elements are described in the following sections.

3.1 Estimation of the first EWL phase ambiguity at the global level

Processing Element 520 is an estimator of the ambiguity of the first and second EWL signals comprising three steps. Step 1 of Processing Element 520 performs determination of the first EWL ambiguity using the geometry-free method as outlined in Feng (2004) and other authors (Hatch, 2000) etc. For double-differenced phase observations, the first EWL is constructed as follows:

where the double differenced range is expressed as:

Δ Vp = Δ Vp + AVd_lrop + AVd_orb ^■ (12)

The corresponding linear combinations for pseudo-ranges as well

^{A Vr}(O,l,l) - ^{l ύ})

It is clearly seen that the difference [ΔV P(0, 1 ,1)- ΔV L(0, 1 ,-1 ) ]will cancel out the first-order ionospheric delay, which is frequency-dependent. Therefore, we can compute the ambiguity as follows:

^"ΔVP(0,l,l) - ΔVZ(0,l,-l)^"

ΔVJV(0,l,-l) = (14)

/1(0,1,-1) J roundoff Successful fixing of the EWL integer by Equation (14) depends on the effects of the code noise. The standard deviation (STD) of the estimate in cycle units, for instance, is expressed as

. (1 5)

Given the levels of code and carrier phase noise, the STDs of the double- differenced integer estimates for one GPS and three Galileo EWL signals can be

given. For a given ^σ _ΔV# , the chance of correct integers by rounding can be given.

Provided that the error of the estimate is white Gaussian noise, ie, *~N(0, ^σ _ΔV^ ), the probability of the correct integer can be given as follows:

P(AVN = correct value) = P[round (x) = 0] (16) Tables 2 and 3 summarise GPS and Galileo virtual widelane and EWL signals, noises in double-differenced (DD) codes and success rate for fixing integers by rounding. Columns 7 and 8 in Table 3 give the success rates for fixing three WL and

EWL integers with two STDs: ^σ _ΔV^ for measurements from a single epoch and— ^σ _AVft for improved measurements (for instance, by averaging over several

epochs and any other methods). As seen, for the assumed levels of code noise for GPS and Galileo signals, the 100% success rate for fixing the extra-widelanes with the wavelengths of 2.93 m and over can be achieved easily using the estimate (14).

The above process is a geometry-free and distance-independent estimator, giving the correct integers in the simplest manner. This method is desirable and reliable for these specific EWLs, as the distance-related ionospheric errors do not have any effects. The problem is that this method requires the code measurements on the second and third emitting signals, which may not be available at all times or in all circumstances. Therefore, this step is optional.

3.2 Estimation of the Second EWL Integer Ambiguity

Step 2 of Processing Element 520 is a physical modeling process for the second select EWL signals. According to Table 4, the preferred second EWL signals in the four GPS/Galileo services studied are

• L(1 ,-6, 5), with the wavelength of 3.2561 m and ISF of -0.0744

• L(1 ,-3,2), with the wavelength of 3.2561 m and ISF of -0.3035 • L(1 ,-10,9), with the wavelength of 3.256 m and ISF of -0.0227.

• L(1 ,-4,3), with the wavelength of 3.663 m and ISF of 0.2454.

The linear equation for least squares estimation of any of the above EWL integer from one epoch data, for instance, ΔV N(1 , -6, 5) for L(1 ,-6,5), can be expressed as

ΔVP_I/2 =ΔV;5+£_ΔVPi/2

ΔVI(l,-6,5) = ΔVp+0.0744^-A(l,-6,5)ΔVN(l,-6,5)+^_ΔVi(I^_{5) (1 7)} where ΔVP_1/2 is the double-differenced, ionospheric-free code measurements. If we

let the ionospheric delay ¹- be corrected using the code measurements P1 and

P2, ie,

AVK, fl _r(ΔVP₂ -AVP₁) ' (18) f² f² - f²

The estimation error due to the code noises in P1 and P2 remains, which is expressed as

which is about 1.3 m if cr_ΔW, =0.60 m. Therefore, if the magnitude of is over a

few meters, correction with Equation (18) cold be an advantage. As a consequence, the second term of the right-hand side would be random in nature, subject to the code noise level. The ambiguity resolution with Equation (17) is then distance- independent. Linearising the above Equation (17), we have the following linear equation system

Δ

where δX is the 3-by-1 state correction term to be estimated.

Over shorter base-to-rover distances, the effects of the first-order differential ionospheric error on ΔVP₍₁ , ₀₎ results in lower code noise level than that of the

ionospheric-free code measurement ΔVP_1/2. Therefore, ΔVP_(Ui0) may be used

instead of ΔVP, 1/2

^■ ΔVP_(Ui0) - ΔVp(X_s , X₀) ^{■ '}A 0 T δX

+ 'ΔVP, 1(1.1 ) (21 ) AVL(\,-6,5) -AVp(X_s,X₀) A -A(l -6,5)lΔW(l -6,5) ^JΔVl(l,w₆,5) When the first EWL ambiguity ΔV N(0,1 , -1) cannot be determined in Step 1 of Processing Element 520, the following equations are used to allow ΔV N(0, 1 ,-1) and ΔV N(1 , -6, 5) to be estimated together or separately, using the geometry-dependent process. The physical observation models are:

ΔVP_1/2 =ΔVp+s_ΔWu2

Linearising the above Equation (22), we have the following linear equation system

ΔVf^> _/2 -ΔVp(X,,X₀) A 0 ΔVL(1 -6,5) -AVp(J^X₀) A -A(l,-6,5) (23) ΔVL(0,l,-l)-ΔV/?pς,X₀) A

Where δX\s the 3-by-1 state correction term to be estimated. The ionospheric effect on the L(0, 1 ,-1 ) is much larger in its value in length unit, but still small in cycle respected to the wavelength (5.82m in the GPS case), as shown in Table 4.

Step 3 of the Processing Element 520 performs estimation of the ambiguities ΔV N(0, 1 ,-1) and ΔV N(1 , -6, 5) using an existing global estimator as schematically shown in Fig 1. Section 3.5 outlines the popularly used least square ambiguity decorrelation adjustments (LAMDA) procedures as one option. Any other existing improved method can be employed.

For the three Galileo services, the EWLs L(1-3,2), L(1 ,-10,9) and L(1 ,-4,3) are used instead of L(1 ,-6,5).

3.3 Positioning with Ambiguity-Fixed WL observations After the 2 EWL signal ambiguities are correctly resolved, Processing Element 530 will perform the positioning estimation using ambiguity fixed observations. This process comprises three steps.

Step 1 of Processing Element 530 estimates the 1st-order ionospheric-bias using two ambiguity-fixed observations. First of all, with the correct EWL integers for

ΔV N(0, 1 ,-1 ) and ΔV N(1 ,-6,5), etc, we can derive some useful WL integer solutions, for instance, in the GPS case, we have

ΔV N(1 ,-1 ,0)= ΔV N(1 , -6, 5)+5 ΔV N(0,1 ,-1 ) , Λ(1 ,-1 ,0)=0.8619m ΔV N(1 ,0 ,-1 )= ΔV N(1 , -6, 5)+6 ΔV N(0,1 ,-1 ) , /1(1 ,0,-1 )=0.7514m As a result, the first-order ionospheric bias can be given as follows (Feng, 2004)

ΔV_tfM=^= y . r(AVZ(l,0-l)+A(l,0-l)ΔVN(l,0-l))-(ΔVZ(l,-l,^Q)+Λ(l-l,0)ΔVN(l-l,0)]

/, fiifi-fs) (24)

This equation allows the first-order ionospheric delay to be estimated directly from the three carrier signals in receiver, playing an important role in three carrier ambiguity resolution and positioning. The effects of the second-order ionospheric errors could be partially reduced as well over shorter base-to-rover distances. The problem, however, is the large uncertainty of the estimate (24) due to carrier phase multipath and receiver noise in the widelane combinations ΔV L(1 ,-1 ,0) and ΔV L(1 ,0,-1 ). This uncertainty can reach the level of a few to several decimeters, for σ_ΔVil ranging from 2 mm to 5 mm.

Step 2 of Processing Element 530 is to correct the ionospheric bias in the Widelane signals and refine WL measurements with phase smoothing procedures. Substituting the known-integers and the ionospheric estimate (24) to the L(1 ,0,-1), L(1 ,-1 ,0) or L(0, 1 ,-1 ), L(1 ,-6,5), the user state can be estimated through one of these WL or EWL measurements. We choose to use the WL signals L(1 ,-1 ,0) or L(1 ,0,-1 ) for positioning, the observation equations are:

ΔVZ{l,-l,0)+/?_(IH,0)ΔV4,, +A(l,-l,0)ΔVN(l,-l,0)=ΔVp+ΔV^_r6+ΔV^+_y5_(I,0,.₁₎ —^ +ΔV_%_,_,0)

Λ (25) In general, for all combinations in the category, the observation equation is expressed as:

Am,J,®+fa_jήwL+KU,kym_m =Wp+AVd_oΛ+Wd_lnp+Q v_l,m +ΔVε ¹W, jM)

(26)

where a and b are integers. Expanding the last two terms of the

right-hand side of Equation (26) and then grouping like terms, the ranging standard deviation for GPS signals is expressed as

' ΔV m.j.k) = 110σ ΔVIl (27)

And for Galileo L1 , E6 and E5A signals

' ΔV UUIJt) = 61. Zσ Δ. Vil (28)

After this correction processing, all the measurements in this category hold in the approximately same uncertainty. Therefore, any one is equivalent for positioning at this level. While the effects of the orbital and tropospheric errors are systematic in nature and may be reduced to certainty by other modelling or estimation techniques, the effects of phase noise ε_AVL , with standard deviations expressed by (27) or (28)

are distance-independent and random in nature. It can be reduced via the carrier- phase smoothing technique using ionosphere-free phase measurements, such as

AVZ(77,-6Q0)+^(77,-6Q0)(77A^V; -6(AW₂)=A^+^₇^_Q0)

(29)

The phase noise standard deviation of this phase measurement is about 3 σ_\_. The

difference between time epochs can be used in smoothing:

SWL(Il -60,0) = δtNp + *_ΛVWβ) (30)

The variance of the smoothed phase measurements Δ VL(I, -1,0) is expressed as follows

⁰⁷AVL(I,-!,)

<y Δ_ΛTW,τ, (1,-1,0) + σ ΔVL(77,-60,0) (31 ) n where n is the number of measurement epochs. The technique was first proposed by Hatch (1982) for smoothing pseudo-range measurements and can be readily generalized to provide real time recursive estimation of the ambiguity-fixed phase measurements, or over a sliding/moving observation window. Table 6 outlines the minimum number of samples or time epochs required to achieve the different accuracy levels of double-differenced (DD) measurements given in the first column. It is seen that, to achieve the same DD ranging accuracy, the smoothing period required for the widelane phase measurements is typically over 10 times smaller than required for smoothing the code measurements. As a result, decimeter positioning accuracy can be achieved within 1 to 3 minutes with the ambiguity- resolved widelane phase measurements, instead of 10 to 30 minutes of convergence time to the same levels as compared to use of DD ionosphere-free code measurements. This technological advance is remarkable, promising much wider regional and global scale applications which have not been offered by existing GPS technologies.

Step 3 of Processing Element 530 carries out estimation of the user states using the ionosphere corrected and/or smoothed WL measurements using least-square estimation. After the first two EWL ambiguities ΛvN(o,i,-i) and ΔVN(i,-6,5) or their

derivatives are obtained, positioning can be done without the resolution of the third ambiguity, if only decimeter accuracy is concerned. This positioning is performed over any distances.

3.4 Resolving the Third Integer Ambiguity

There are two choices of processes for resolutions of the third ambiguity with or without resolutions of 2 EWL ambiguities. Processing Elements 515 and 535 deal with both cases respectively. In the Processing Element 515, the estimation is based on the models using one NL measurement in Table 5, for instance, L(4,0,-3). This narrow-lane must be used together with other virtual measurements to allow the over-determined ambiguity estimation with measurements from a single epoch.

The first step of this Element 515 is to form the physical models with one WL and one NL signals. The observation models for ambiguity resolution are directly written as

ΔW(l,O,l)= ΔVJσ + +s_ΔW(1Λ1)

ΔVL( 1 ,0,- 1 >= AVp - A(I₅O -l)ΔVN(l,0 -1) + ε_AVLW) . ΔVZ(4,0 -3) = ΔVp - 2(4,0 -3)ΔVN(4,0 -3) + *_ΔVIWW) (32) Their linear equation system is

(33)

Other choices of NL signals for different GPS and Galileo services can be found in Table 5. Other choice of virtual code measurements may be P(1 ,0,1 ) or P(1 ,1 ,0). Different choices for code measurements will made little difference for the performance of the ambiguity resolution.

Step 2 of Processing Element 315 is estimation of the ambiguities ΔV N(1 ,0,-1) and ΔV N(4, 0, -3) using an existing a global estimator as schematically shown in Fig 1. Section 3.5 outlines the least square ambiguity decorrelation adjustments (LAMDA) procedures as one of the option to complete this estimation.

The success in estimation of the integer ambiguity of L(4,0,-3) depends upon the quality of the WL and NL measurements. The WL measurements L(1 ,-1 ,0) or L(1 ,0,- 1 ), if the amplification of phase noise is estimated to be at the several to 10 centimeters, could provide precise positional constraints on the NL integers. For the NL measurement L(4,0,-3), the direct phase noise would be typically at the level of 1 to 2 centimeters while the ionospheric term can be controlled to 1 cm or less, either via bounding the ionospheric error on L1 to 1 m, or bounding the base-to-rover baseline. The tropospheric error seems the most dominating factor. Therefore, taking care to bound the effects of tropospheric errors to centimeters (less than half a cycle), or the Zenith Tropospheric Delay (ZTD) error to 1 to 2 cm, the successful resolution of the NL ambiguities can be anticipated at a high level. According to Zheng (2004) and Zheng at el (2004), this ZTD error could allow a distance of 100 to 200 kilometers.

Processing Element 515 covers the case of Dual Carrier Ambiguity Resolution (DCAR ) using the WL and NL signals.

Existing GPS-based dual carrier ambiguity resolution (DCAR) methods are mostly based on P1, L1 , P2 and L2 measurements (Misra, and Enge, 2004). The widelane

L(1 ,-1) measurements may be determined first and the positioning results are used as constraints for reduced integer search space. In this study, the Widelane L(1,-1) and Narrow-laneNarrow-lane L(4,-3) or L(5,-4) are used together to resolve the ambiguities for GPS L1 and L2 signals. The measurements include

ΔVK1

ΔVP(l,l) = ΔV^ + +1.2833 ^"*^{" S} AVP(IA)

ΔVKl

ΔVL(1,-1) = ΔVp + 1.2833 - A(I - 1)Δ VN(I,- l) + f_ΔVL(1,-₁,o, f.² (34)

ΔVZ(4 -3) = ΔVp - 0.0902 ^L - A(4,-3)ΔVN(4,-3) + £_ΔVL(4,.₃₎

The linear equations are as follows:

AVP(I₅I) - AVP(X₅₃ X₀) 0

ΔVL(l,-l) - ΔVp(X_s,X₀) A -X '₁(i.-i) (35) ΔVZ(4 -3) - AVp(X₅₅X₀) A -U "₁(,4,-3) 3/L

In the dual frequency case, the ionospheric delay is unknown. However, over shorter distances, the magnitude of the ionospheric error on L1 of about 1 cycle (19 cm) cause the error on L(1 ,-1) to be about 26 cm, and on L(4,-3) about 1.8 cm respectively. With respect to their wavelengths of 86.2 cm and 11 cm, the effects are 1/3 or 1/5 cycles respectively. This compares with the effects of the same level ionospheric delay on L1 and L2 of about 1 cycle. Therefore, theoretically, Equation (26) can be used to resolve ambiguities more efficiently, if the effect of multipath in carrier phase is normal.

Processing Element 535 resolve the NL ambiguities after the process 530, where two EWL ambiguities (their dependent WL ambiguities) have been fixed and WL have been corrected and refined through Step 2. Therefore, the observation models for ambiguity resolution are directly written as:

ΔVL(l,0,-l)= ΔVp + _{£ΔVL(1Λ.1)} ΔVI(4,0 -3) = ΔVp - λ(4 ,0 -3)ΔVN(4,0 -3) + ^_Vi(40._{3) (36)}

Their linear equation system is

AVL(IA-I)-AVp(X₅₅X₀)^" ΔVL(4,0-3)-ΔVp(X_i5X₀) A

Other choices of NL signals for different GPS and Galileo services can be found in Table 5. Higher precision of the WL measurements may lead to more reliable ambiguity resolution for the NL signals over even further long distances.

The final step of Processing Element 535 performs the positioning process after the third ambiguity is resolved. At this end, the ambiguities for the three original signals (eg L1 , L2, and L5 ) can be recovered uniquely. Therefore, the first-order ionospheric delay can be estimated precisely and correction can apply to any combinations. To achieve the best possible positioning accuracy and other applications, those with minimum phase noise factor should be used for positioning. Good examples are the ionospheric free measurements L(77,-60,0) with a phase noise factor of 3 and L(4,0,-3) with a phase noise factor of 2.6, while worse examples are the ionospheric free signal L(0,24,-23) with a phase noise factor 16.

If the effect of phase noise is 3σ or less, the effect of the tropospheric error will be the dominant factor. In other words, if the effect can be restricted to the centimeter level, centimeter positioning performance is achievable. An efficient method is to restrict the base-to-rover baselines to 100 km or so, over which the relative tropospheric biases in the double differenced ranges could be bounded to a few centimeters.

3.5 Least Square Ambiguity Search Approach: LAMBDA Method

Choosing the superior virtual EWL and NL signals is the key to efficient and reliable ambiguity resolution using three frequency signals as outlined in Fig 2. To complete the geometry-dependent based ambiguity resolution process, we turn to introduce the least-square based ambiguity resolution formulae, following the well-known Least-squares ambiguity decorrelation adjustment (LAMBDA) method by Teunissen, P. J. G. (1995). Any improved version of LAMBDA method will be an additional advantage.

The linear observation models for ambiguity estimation, such as (21), (33) and (37) can in general be rewritten as follows: L = AδX + BN+ ε (38a)

Cov(ε) = σ²W^~λ . (38b)

The float solutions for the position and ambiguities and their covariance matrix can be written as follows δX ^~A^TWA A⁷WB o~]~ r ^~A AT^TnWrLr

(39)

N _B^TWA B^TWB _B^TWL_

Ambiguity search of the LAMBDA method is to find the integer vector N, which minimizes the cost function.

where

The central point of the LAMBDA method is to introduce a hypothetical transformation Z in the restricted class of transformation which diagonalises the matrix WN. Let

M = ZN and M = ZN (43) where the full rank Z and inverse matrix Z^'1 both have integer entries. The cost function in transfer space is c{M) = (M - M)(Z-⁷W_f1Z^'1 (M - M) (44)

(Z^~TW^Z^~l ) is supposed to be diagonal, thus we can find the solution for M by

rounding off each element of M to _. give the integer solution M. Inverse transformation takes M to N, giving the integer solution of the original problem.

3.6. Recovering the Integer Ambiguities for Three Emitting Carriers

Processing Element 525 is responsible for recovering the integer ambiguities of three emitting signals. It is the final process for complete ambiguity resolutions. In dual-frequency cases, given two known integers, for instance, ΔVN(l,-l) and

ΔVN(4,-3) , we can uniquely determine the integer ambiguities of the emitting signals

L1 and L2, the relation is expressed as follows rΔW(l,0) -3 lT AV-V(L-I)

(45) |_AV_V(O,1)_ - 4 l_||_ΔVN(4,-3)_ Similarly, In three frequency cases, given three known integers, for instance, ΔVN(0,l,-l) , ΔV_V(l,-6,5) and ΔVN(4,0,-3) we can uniquely determine the integer

ambiguities of the emitting signals L1, L2 and L5. The relation is expressed as follows AV_V(L0,0)^'" - 18 -3 1 ^'AVN(OX-I)

ΔvN(o,i_so) - 23 - 4 1 ΔVN(l,-6,5) (46). ΔW(O,O,I) -24 -4 1 ΔVN(4,0,-3) 4. Some Comments on Overall Performance of the System

The present invention for three carrier ambiguity resolution use two select EWL signal and one NL signal instead of other possible combined signals. These EWL and NL signals can tolerate much larger ionospheric errors than others, thus allowing for more reliable and rapid ambiguity resolution over longer distances. In the least-square estimator, no ionospheric and tropospheric parameters are needed, the computational effort for statistical search and validation is minimal.

The subsystem resolves three carrier ambiguities at two levels

(1) At the regional-to-global level, estimate the first type of EWL and WL ambiguities using distance-independent geometry-free approaches; estimate the second type of EWL and WL ambiguities using geometry-dependent methods, over the distances of up to a few thousands of kilometers; (2) At the local to regional level, estimate the third NL ambiguity using geometry- dependent methods, over baselines of up to a few hundred kilometers.

This is significant improvement as . compared to the current dual-frequency based ambiguity resolution achieved at the local level, ie, over baselines of up to tens of kilometers.

The above improved ambiguity resolution techniques support kinematic positioning services at the regional to global scales for decimeter accuracy and at the local to regional scales for centimeter accuracy, achieved in real time or post-processing.

The present invention will have significant impact on the future GNSS services at all global, regional and levels. REFERENCES

Abidin, H (2000), Triple Frequency GPS Measurements and Their Combinations, The Directions of GPS: The 2000 GPS Lecture Series, School of Geomatic Engineering, The University of New South Wales, 5-6 October 2000.

Feng, Y (2004), A Complete Geometry-Free Approach to Three Carrier Ambiguity Resolutions, GNSS 2004 Sydney, 6-8 December 2004, submitted to Journal of Geodesy

Feng, Y (2005), Long-Range Kinematic Positioning Made Easier Using Three Frequency GNSS Signals, ION National Technical Meeting, San Diego, January, 2005.

Forssell, B.; Martin-Neira, M.; Harris, R. A. (1997):Carrier Phase Ambiguity Resolution in GNSS-2. Proceedings of ION GPS-97, Kansas City, September 16-19, pp. 1727-1736.

Han S. and C Rizos (1999), The Impact of Two Additional Civilian GPS Frequencies on Ambiguity Resolutions Strategies, Proceedings of ION Annual Technical Meeting.

Hatch , R., J, Jung, P. Enge and B Pervan (2000), Civilian GPS: The Benefits of Three Frequencies, GPS Solution, VoI 3, No 4, pp1-9.

Isshiki, H., (2003a), An application of widelane to long baseline GPS measurements (3), ION GPS/GNSS 2003, The Institute of Navigation

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Isshiki, H., (2004b), Widelane Assisted Long Baseline High Precision Kinematic Positioning by GNSS, The 2004 International Symposium on GNSS/GPS Werner, W, J. Winkel (2003), TCAR and MCAR Options, With Galileo and GPS, Proceedings of ION GPS/GNSS 2003, Portland, OR, USA, 9 to 11 September 2003, pp790-800

Zheng Y (2004) Interpolating Residual Zenith Tropospheric Delays for Improved Wide Area Differential GPS Positioning, Proceedings of ION GNSS 2004, Long Beach, LA, September 21-24

Mirsa, P, P Enge (2004), Global Positioning Systems, Signals, Measurements and Performance, Ganga-Jamuna Press. P227-254.

Dai L, Wang J, Rizos C, Han S (2003) Predicting Atmospheric Biases for Real-Time Ambiguity Resolution in GPS/GLONASS Reference Station Networks, Journal of Geodesy, 76, 617-628.

Feng, Y and C Rizos (2005), "Three Carrier Approaches for Future Global, Regional and Local GNSS Positioning Services: concepts and performance perspectives", p2277-2787, Proceedings of ION GNSS 2005, September 2005.

Teunissen, P. J. G. (1995) , The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. Journal of Geodesy, Vol. 70, No. 1-2, pp. 65-82.

Vollath Ulrich (2005) "Ambiguity estimation of GNSS signals for three and more carriers" provides a comprehensive review of prior-art for geometry-dependent methods, The recent US patent application 2005010248, p1-40.

Claims

The claims defining the invention

1. A system for efficient ambiguity resolution (AR) and positioning performed with three or more ranging signals from one or more global/regional navigation satellite system (GNSS) constellations, comprising three subsystems as shown in Fig. 2, in particular:

(1 ) Subsystem 1 , a processing device that searches/identifies the most useful linear combinations of the signals emitted at two, three or more than three frequencies from GNSS satellites and classifies these combinations into two categories: Extra- Widelane/Widelane(EWL,WL) category and Narrow-lane (NL) category. These linearly combined signals are also known as virtual signals.

(2) The device of Subsystem 1 , further selecting three combinations in which the magnitude of the ionospheric scale factors ((which is defined with respect to the delay on L1 carrier (which is defined with respect to the delay on L1 carrier) are relatively small against their wavelengths. Of the three independent combinations, two from the extra-widelane category and one from the narrow-lane category.

(3) Subsystem 2, a set of 2 or more generic GNSS receivers that tracks any number of GNSS satellites in view and outputs the code and carrier measurements on two, three, or more than three frequencies;

(4) Subsystem 3, a data processing system that is a generic global estimator performing ambiguity resolution and positioning using the selected extrd- widelane/widelane and narrow-lane signals, using the data sets from at least one base receiver and one rover receiver;

(5) The device of Subsystem 3, further estimating the integer ambiguity for the first extra-widelane signal using either an existing or future geometry-free procedure and/or geometry-dependent procedure over any distance, subject to a minimum of four satellites commonly tracked from both base and rover locations

(6) The device of Subsystem 3, further resolving the integer ambiguity for the second select extra-widelane signal using an existing or future least-squares ambiguity estimation/search procedure over any distance, subject to a minimum of four satellites commonly tracked from both base and rover locations.

(7) The device of Subsystem 3, further resolving the third integer ambiguity of any one select narrow-lane signal in the NL category, using an existing or future least- squares ambiguity estimation/search procedure.

(8) The device of Subsystem 3, further recovering uniquely the integer ambiguities for three emitting signals from the three known ambiguity integers, such as the two extra-widelane and one narrow-lane signals.

2. The Subsystem 1 of Claim 1 , wherein the select extra-widelane signals identified for different GNSS services are summarized in Table 4. Other examples for new GNSS services will emerge as future GNSS or RNSS systems are brought into service. The notation L(l,j,k) is defined by Equation (3), where the indices l,j,k are integers. Two most preferred EWL signals include, but not limited to, the following choices:

(1) In the GPS service with L1 , L2 and L5 frequencies, the first extra-widelane L(0, 1,-1) has a wavelength of 5.82 m; the second extra-widelane is L(1 ,-6,5), which has a wavelength of 3.265m, and the ionospheric scale factor (which is defined with respect to the delay on L1 carrier, all the same below) of -0.0744.

(2) In the Galileo service with L1 , E6 and E5A frequencies, the first extra-widelane L(0, 1 ,-1) has a wavelength of 2.903m; the second extra-widelane is L(1 ,-3,2), which has a wavelength of 3.256m and the ionospheric scale factor of -0.3035.

(3) In the Galileo service with L1 , E6 and E5B frequencies, the first extra-widelane L(0, 1 ,-1) has a wavelength of 4.1865 m; the second extra-widelane is L(1 ,-4,3), which has a wavelength of 3.6633m, and the ionospheric scale factor of -0.2454.

(4) In the Galileo service with L1, E5B and E5A frequencies, the first extra-widelane L(0, 1 ,-1) has a wavelength of 9.7684m; the second extra-widelane is L(1 ,-10,9), which has a wavelength of 3.256m, and the ionospheric scale factor is -0.0227.

3. The Subsystem 1 of Claim 1 , wherein one or more narrow-lanes (about 11 cm) are identified for each GNSS service given in Table 5, in particular,

(1) In the GPS service with L1 , L2 and L5 frequencies, the two preferred narrow-lanes choices are L(4,0,-3) or L(3,6,-8); (2) In the GPS service with L1 and L2 frequencies, the two preferred narrow-lanes choices are L(4,-3) or L(5,-4) ;

(3) In the Galileo service with L1 , E6 and E5A frequencies, the two preferred narrow- lane choices are L(4,0,-3) or L(3,3,-5);

(4) In the Galileo service with L1 , E6 and E5B frequencies, the two preferred narrow- lane choices are L(5,-3,1 ) or L(4,1 ,-4);

(5) In the Galileo service mode with L1 , E5B and E5A frequencies, the two preferred narrow-lane choices are: L(4,0,-3) or L(4,-1 ,-2);

(6) In the Galileo service with L1 , E6 frequencies, the preferred narrow-lane choice is L(5,-4).

4. The subsystem 3 of Claim 1 , further comprising steps of

(1) Applying correction for the ionospheric-bias to the selected extra-widelane signal methods including the group delay computed from any two code measurements for the same satellite, turning the systematic effects of the ionospheric term to random noise nature, which is then distance-independent;

(2) Using any existing or future least-square estimation/integer search algorithms to resolve the integer ambiguity of the second extra-widelane identified in Claim 2, Alternatively, the models of ionospheric-free code measurements is used together to allow the ambiguity estimation using measurements of a single epoch.

5. The Subsystem 3 of Claim 1 , further comprising steps of

(1) Estimating the first-order ionospheric bias using the two ambiguity-resolved extra- widelane signals or their derivative WL signals.

(2) Using the above estimate of the first-order ionospheric delay to correct some ambiguity-resolved signals, such as a widelane signal, to allow for performance of positioning processing over any distances without effects of the ionospheric-biases

6. The subsystem 3 of Claim 1 , further comprising steps of estimation of the select narrow-lane ambiguity using an existing or future least-square based integer search procedure. This is achieved using the linear equations for one widelane signal and one select narrow-lane signal with or without knowledge of the ambiguity of that widelane and its first-order ionospheric delay. (1 ) If the widelane ambiguity is fixed and the ionospheric bias is known, the ambiguity for the select narrow-lane signals can be resolved independently ;

(2) If the widelane ambiguity is unknown, and/or the ionospheric bias is not given, the ambiguities for both widelane and narrow-lane signals can be resolved jointly;

(3) The ambiguity resolution performance for the select NL signal¹ depends on effects of residual ionospheric errors, phase noise and relative tropospheric delay. High success rates are expected achievable over base-to-rover distances of much longer than the current tens of kilometers.

7. The method of Claim 6, including the procedures of resolution of two phase ambiguities from one widelane and one narrow-lane signals, where the narrow-lane signals include, but not limited to the narrow-lane signals L(5,-4,0), 1.(4,-3,0) and L(4,0,- 3) in the dual-frequency GNSS services with L1 and L2, L1 and L5/E5A, L1 and E6, respectively.

8. The method of Claim 5, further comprising steps of making use of measurements from multiple epochs to improve the estimate of the first-order ionospheric terms and positioning accuracy recursively. An existing carrier phase smoothing procedure can apply.

9. The method of Subsystem 3, wherein the global estimator does not need to estimate any ionospheric and troposphere parameters, the computation effort required for integer search and validation is minimal.

10 The method of Claim 5, wherein positioning estimation and global/regional scale GNSS analysis can be performed after the two select EWL ambiguities are determined, supporting kinematic positioning services in the regional to global scales for decimeter accuracy in either real time or post-processing mode.

11. The method of Claim 6, wherein positioning estimation and regional/local GNSS scale data analysis can be performed using the third ambiguity-resolved narrow-lane or ionospheric-free combinations, supporting kinematic positioning services in the local to regional scales for centimeter accuracy using the measurements of a single or multiple epochs.