GPS / GLONASS frequency clock deviation modeling and forecasting method and device

Abstract

The invention discloses a GPS / GLONASS inter-frequency clock deviation modeling and forecasting method and device, and the method comprises the steps: obtaining positioning data received by a receiver from a satellite, and constructing a geometric-free ionosphere-free phase combination observation value according to the positioning data; an inter-epoch difference method is adopted to remove ambiguity constants absorbing phase hardware delay time invariants in the geometry-free ionosphere-free phase combinations, and a single observation station ionosphere-free combination IFCB is obtained; integrating observation data of multiple observation stations, and calculating a final IFCB deviation by adopting a multi-station weighted average mode; according to the IFCB data, carrying out harmonic analysis by adopting fast Fourier transform to obtain an accurate period of an IFCB sequence, and proposing an IFCB model based on a third-order polynomial and a seven-period harmonic function; and performing experiment forecasting according to the IFCB time sequence model, extracting and comparing periodic characteristics of IFCB deviation, and calculating an optimal forecasting period. According to the method, the observation data of the GPS / GLONASS system can be corrected in real time, so that the positioning result is accurately corrected to improve the positioning accuracy.

Description

Technical Field

[0001] This invention relates to the field of satellite positioning technology, and in particular to a method and apparatus for modeling and predicting clock offsets between GPS / GLONASS frequencies. Background Technology

[0002] Real-time three-frequency precise point positioning (PPP) or PPP-RTK (Precise Point Positioning - Real-Time Kinematic) requires real-time estimation of precise satellite orbits and clock bias products, as well as inter-frequency clock bias (IFCB) corrections. This significantly increases workload and impacts the efficiency of real-time data stream transmission. Establishing a high-precision IFCB prediction model and achieving high-precision modeling and prediction of IFCB bias can effectively improve the efficiency of real-time PPP calculations and reduce data preparation workload. Currently, the precise clock bias correction products for Global Navigation Satellite System (GNSS) satellites publicly released by the International GNSS Service (IGS) are calculated based on a specific frequency combination without an ionosphere. For example, the Global Positioning System (GPS) uses the L12 frequency combination without an ionosphere. Using this satellite clock bias product to correct the clock bias of a third frequency satellite will lead to inter-frequency clock bias problems. The existence of IFCB (Inverse Container Registry) prevents IGS satellite clock bias products from being directly applied to multi-frequency PPP (Poly-Popular Utility Modeling). IFCB processing is a key issue that needs to be addressed first in multi-frequency PPP. However, a high-precision IFCB bias model for the GLONASS system has not yet been established, and the IFCB modeling methods for GPS satellites have shortcomings and their modeling accuracy needs further improvement. Summary of the Invention

[0003] The purpose of this invention is to provide a method and apparatus for modeling and predicting clock offsets between GPS and GLONASS frequencies, in order to solve the problems of insufficient accuracy in GPS and GLONASS satellite ICB offset modeling and low positioning accuracy.

[0004] The technical solution adopted by this invention to solve its technical problem is:

[0005] In a first aspect, the present invention provides a method for modeling and predicting clock offsets between GPS / GLONASS frequencies, including:

[0006] The positioning data of the satellite is acquired by the receiver, and the positioning data includes the three-frequency carrier and pseudorange observations of the GPS / GLONASS system satellites;

[0007] Based on the positioning data, the phase hardware delay of the satellite end and the receiver end is constructed, and the phase hardware delay is divided into two parts: time-varying and time-invariant.

[0008] Construct a geometry-free and ionosphere-free GFIF combination based on known data, and determine the geometry-free and ionosphere-free GFIF combination and its variance; the geometry-free and ionosphere-free GFIF combination includes the ionosphere-free combination IFCB and an ambiguity constant that absorbs the phase hardware delay time invariant.

[0009] The ambiguity constants that absorbed the phase hardware delay time invariant in the geometrically un-ionospheric (GFIF) combination were removed by the interepoch difference method to obtain the ionospheric combination IFCB and variance of a single station.

[0010] Based on multi-station observation data, the satellite ionospheric IFCB is estimated by multi-station weighted average. The satellite ionospheric combined IFCB is obtained by summing, and the ionospheric combined IFCB is converted into non-difference non-combined IFCB corrections by correction relation.

[0011] Based on the IFCB bias data, harmonic analysis was performed using Fast Fourier Transform to obtain the precise period of the IFCB sequence. An IFCB bias model based on a third-order polynomial and a 7-period harmonic function was proposed. Experimental forecasts were conducted based on the IFCB time series model, and the periodic characteristics of the IFCB bias were extracted and compared to calculate the optimal forecast period.

[0012] Preferably, acquiring satellite positioning data via a receiver includes:

[0013] ;

[0014] in, and These represent the three-frequency carrier and pseudorange observations, respectively, with superscript indicating the superscript. and These represent the satellite PRN number and the GNSS system, respectively, with subscripts. and These represent the receiver and the frequency, respectively, where i = 1 and 2. Indicates the distance between the measuring station and the satellite. and For receiver clock bias and satellite clock bias, At the speed of light, and These are the zenith tropospheric wet delay and its projection function, respectively. Represents the ionospheric coefficient factor. For system frequency, The ionospheric parameters are for the first frequency. and These are wavelength and integer ambiguity, respectively. and These represent the phase hardware delays at the receiver and satellite ends, respectively. and These are the pseudorange hardware delays at the receiver and satellite ends, respectively.

[0015] For the GLONASS system, when i=1 or 2, When i=3, .

[0016] Preferably, the phase hardware delay between the satellite end and the receiver end is constructed based on the positioning data, and the phase hardware delay is divided into two parts: time-varying and time-invariant, including:

[0017] ;

[0018] in, and These represent the phase hardware delays at the satellite end and the receiver end, respectively. and These are the time-varying components of the phase hardware delay at the satellite end and the receiver end, respectively. and These are the time-invariant parts of the phase hardware delay at the satellite end and the receiver end, respectively.

[0019] Preferably, the step of constructing a geometrically free and ionosphere-free GFIF combination based on known data, and determining the geometrically free and ionosphere-free GFIF combination and its variance, is as follows:

[0020] ;

[0021] ;

[0022] , , ;

[0023] in, This indicates a geometrically non-ionosphere (GFIF) combination. This represents the variance of the geometrically unbiased and ionosphere-free GFIF combination. The satellite clock bias is calculated from the L1 / L2 ionosphere-free combination. The satellite clock bias is calculated from the L1 / L3 ionosphere-free combination. Indicates the ionosphere-free composite IFCB. To incorporate the ambiguity constant that is invariant to phase hardware delay, , and These represent the integer ambiguity at three frequencies. , Indicates frequency, For the phase hardware delay of the third frequency receiver, , and All were set to 0.02 weeks.

[0024] Preferably, the ambiguity constants that absorb the phase hardware delay time invariant in the geometrically un-ionospheric (GFIF) combination are removed by the inter-epoch difference method to obtain the ionospheric combination IFCB and variance of a single station, as follows:

[0025] ;

[0026] in, This represents the ionosphere-free IFCB based on the interepoch difference method. and They represent Liyuanhe Geometrically and ionosphere-free combination of observations based on epoch frequencies L1, L2, and L3. for variance Indicates the number of epochs;

[0027] During the execution of the interepoch difference method, a detection threshold of three times the standard deviation is set.

[0028] Preferably, the step of estimating the satellite ionospheric non-IFCB using a multi-station weighted average method based on multi-station observation data, obtaining the satellite ionospheric non-combination IFCB through accumulation, and converting the ionospheric non-combination IFCB into a non-difference, non-combination IFCB correction value through a correction relation includes:

[0029] The satellite's ionospheric non-ionospheric IFCB was estimated using a multi-station weighted average method as follows:

[0030] ;

[0031] in, This indicates that the satellite has no ionosphere (IFCB). The weights related to the satellite's elevation angle are indicated. This is the satellite elevation angle. Indicates the number of stations;

[0032] The epoch is obtained through accumulation. Time Satellite The following are ionosphere-free IFCB combinations:

[0033] ;

[0034] in, Represents the epoch Time Satellite IFCB without ionosphere Indicates reference time period Time Satellite Ionosphere-free composite IFCB;

[0035] The corrections for converting the non-ionosphere combined IFCB to a non-differential, non-combined IFCB are as follows:

[0036] ;

[0037] in, This represents the non-difference, non-combination IFCB correction.

[0038] Preferably, the satellite observation arc is set to be abandoned when it is less than 10 minutes.

[0039] Set the satellite cutoff elevation angle to 10;

[0040] If the ionospheric combination IFCB of the same satellite at different stations has the same satellite elevation angle, then it is assigned the same weight.

[0041] Preferably, based on the IFCB deviation data, harmonic analysis is performed using Fast Fourier Transform to obtain the precise period of the IFCB sequence. An IFCB deviation model based on a third-order polynomial and a 7-period harmonic function is proposed as follows:

[0042] ;

[0043] In the formula, express Epoch IFCB bias, Represents a constant term. , and These are the coefficients of the first, second, and third order terms, respectively. For amplitude, For a period of time, Indicates phase, Indicates the order of the harmonic function;

[0044] Experimental forecasts were conducted based on the IFCB bias model, and the periodic characteristics of the IFCB bias were extracted and compared to calculate the optimal forecast period.

[0045] Secondly, the present invention provides a GPS / GLONASS frequency inter-frequency clock offset modeling and prediction device for implementing the above-mentioned GPS / GLONASS frequency inter-frequency clock offset modeling and prediction method, the device comprising:

[0046] The acquisition module is used to acquire satellite positioning data through a receiver. The positioning data includes three-frequency carrier and pseudorange observations of GPS / GLONASS system satellites.

[0047] The first determining module constructs a phase hardware delay between the satellite end and the receiver end based on the positioning data, and divides the phase hardware delay into two parts: time-varying and time-invariant.

[0048] The second determining module is used to construct a geometry-free and ionosphere-free GFIF combination based on known data, and to determine the geometry-free and ionosphere-free GFIF combination and its variance; the geometry-free and ionosphere-free GFIF combination includes an ionosphere-free combination IFCB and an ambiguity constant that absorbs the phase hardware delay time invariant.

[0049] The third determining module is used to remove the ambiguity constants that have absorbed the phase hardware delay time invariant in the geometrically non-ionospheric GFIF combination by using the interepoch difference method, so as to obtain the single-station non-ionospheric combination IFCB and variance.

[0050] The fourth determination module is used to estimate the satellite ionospheric IFCB by using a multi-station weighted average method based on observation data from multiple stations. It obtains the satellite ionospheric combined IFCB by accumulation and converts the ionospheric combined IFCB into a non-differential non-combined IFCB correction number through a correction relation.

[0051] The fifth determination module is used to perform harmonic analysis using fast Fourier transform based on IFCB deviation data, obtain the precise period of the IFCB sequence, propose an IFCB deviation model based on a third-order polynomial and a 7-period harmonic function, conduct experimental forecasting based on the IFCB time series model, extract and compare the periodic characteristics of the IFCB deviation, and calculate the optimal forecast period.

[0052] Thirdly, an electronic device is provided, comprising a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein the computer program, when executed by the processor, implements the steps of the method of the first aspect.

[0053] Fourthly, a computer-readable storage medium is provided on which a computer program is stored, which, when executed by a processor, implements the steps of the method of the first aspect.

[0054] The present invention has the following beneficial effects:

[0055] This invention acquires positioning data received by a receiver from satellites, constructs a phase hardware delay between the satellite and receiver based on the positioning data, and divides it into time-varying and time-invariant components. Based on the hardware delay data, it estimates the IFCB bias of GPS and GLONASS system satellites using a multi-station weighted average method. Based on the IFCB bias data, it performs harmonic analysis using Fast Fourier Transform (FFT) to obtain the precise period of the IFCB sequence, and proposes an IFCB bias model based on a third-order polynomial and a 7-period harmonic function. Based on the IFCB time series model, it performs experimental predictions, extracts and compares the periodic characteristics of the IFCB bias, and calculates the optimal prediction period. This invention proposes a GPS / GLONASS system satellite IFCB bias modeling and prediction method based on a third-order polynomial and a 7-period harmonic function, which can correct GPS / GLONASS system observation data in real time, thereby accurately correcting the positioning results to improve positioning accuracy. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of a GPS / GLONASS frequency clock offset modeling and prediction method provided in one embodiment of the present invention;

[0057] Figure 2 This is a schematic diagram of the modeling process of the IFCB deviation model provided in one embodiment of the present invention;

[0058] Figure 3 This is a schematic diagram of the IFCB deviation modeling result analysis process in one embodiment of the present invention;

[0059] Figure 4 This is a schematic diagram of the IFCB deviation prediction result analysis process in one embodiment of the present invention;

[0060] Figure 5 This is a schematic diagram of the GPS / GLONASS frequency clock deviation modeling and prediction device provided in one embodiment of the present invention;

[0061] Figure 6 This is a schematic diagram of the period of the first four largest amplitude components in the daily IFCB time series of the GPS system Block IIF satellite provided in this embodiment of the invention;

[0062] Figure 7 This is a schematic diagram of the period of the first four components with the largest amplitude in the daily IFCB time series of GLONASS system satellites provided in this embodiment of the invention;

[0063] Figure 8This is a schematic diagram of the IFCB modeling results of each group of GPS system G01 satellites with DOY=127 days in 2020 and the residuals between them and the estimated values, provided in an embodiment of the present invention.

[0064] Figure 9 This is an illustration of the IFCB modeling results of each group of schemes for the GLONASS R05 satellite with a DOY of 127 days in 2020, provided in an embodiment of the present invention, and the residuals between them and the estimated values.

[0065] Figure 10 This is a schematic diagram of the residuals (RMS) between the modeling results and estimated values ​​of various GPS system satellite schemes provided in this embodiment of the invention.

[0066] Figure 11 This is a schematic diagram of the residuals (RMS) between the modeling results and estimated values ​​of various GLONASS system satellite schemes provided in this embodiment of the invention.

[0067] Figure 12 This is a schematic diagram of the RMS residuals of the IFCB forecast values ​​of 12 GPS satellites provided in this embodiment of the invention;

[0068] Figure 13 This is a schematic diagram of the RMS residuals of the IFCB prediction values ​​of four GLONASS satellites provided in this embodiment of the invention. Detailed Implementation

[0069] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the embodiments and accompanying drawings. Here, the illustrative embodiments and descriptions of this invention are used to explain the invention, but are not intended to limit the invention.

[0070] It should also be noted that, in order to avoid obscuring the invention with unnecessary details, only the structures and / or processing steps closely related to the solution according to the invention are shown in the accompanying drawings, while other details that are not closely related to the invention are omitted.

[0071] It should be emphasized that the term "including / comprises" as used herein refers to the presence of a feature, element, step, or component, but does not exclude the presence or addition of one or more other features, elements, steps, or components.

[0072] It should also be noted that, unless otherwise specified, the term "connection" in this article can refer not only to a direct connection, but also to an indirect connection involving an intermediary.

[0073] In the following description, embodiments of the invention will be illustrated with reference to the accompanying drawings. In the drawings, the same reference numerals represent the same or similar parts, or the same or similar steps.

[0074] It should be emphasized here that the step markers mentioned below are not a limitation on the order of the steps, but should be understood as meaning that the steps can be executed in the order mentioned in the embodiments, or in a different order than in the embodiments, or several steps can be executed simultaneously.

[0075] To address the problems existing in the prior art, this invention provides a method for modeling and predicting clock offsets between GPS / GLONASS frequencies. The subject of this method is a satellite navigation receiver. The method is as follows... Figure 1 As shown, it includes:

[0076] S1: Acquire positioning data received by the receiver from the satellites. This positioning data includes three-frequency carrier and pseudorange observations from GPS / GLONASS system satellites.

[0077] (1)

[0078] In the formula, These represent the satellite PRN number, receiver, frequency (i=1, 2), and GNSS system, respectively. Indicates the distance between the measuring station and the satellite. and For receiver clock bias and satellite clock bias, At the speed of light, and These are the zenith tropospheric wet delay and its projection function, respectively. Represents the ionospheric coefficient factor. Here are the ionospheric parameters at the first frequency, where For oblique ionospheric delay of the station star, For system frequency; and These are wavelength and integer ambiguity, respectively. and These represent the phase hardware delays at the receiver and satellite ends, respectively. and These are the pseudorange hardware delays at the receiver and satellite ends, respectively.

[0079] Since the GLONASS system transmits FDMA signals at frequencies L1 and L2, and transmits CDMA observations at the new L3 frequency, for the GLONASS system, when i=1 or 2, equation (1) has the following meaning: When i=3, ,in These represent the pseudorange hardware delays at the receiver and satellite ends, respectively. These represent the phase hardware delay at the receiver and satellite ends, respectively.

[0080] S2: Construct the phase hardware delay for the satellite and receiver ends based on the acquired positioning data, and divide it into time-varying and time-invariant parts.

[0081] Specifically, the phase hardware delay between the satellite end and the receiver end is divided into time-varying... , And time unchanged , Part of it is as follows:

[0082] (2)

[0083] and These represent the phase hardware delays at the satellite end and the receiver end, respectively. For ease of representation, the GNSS system superscript "S" has been omitted from the above formula.

[0084] S3: Construct a geometric-free and ionospheric-free (GFIF) combination based on known data and determine the variance of the GFIF combination. The GFIF combination consists of two parts: an ionospheric-free combination ICB and an ambiguity constant that incorporates the phase hardware delay time invariant.

[0085] In this step, the combination of geometric-free and ionospheric-free (GFIF) layers and its variance are constructed as follows:

[0086] (3)

[0087] in, , and All are set to 0.02 cycles. The specific expressions for the other parameters in the formula are as follows:

[0088] (4)

[0089] In the formula, , , , , and These represent the integer ambiguity at three frequencies. This refers to the phase hardware delay of the receiver at the third frequency.

[0090] The ionosphere-free combination IFCB is defined as the difference between the satellite clock bias calculated from the L1 / L3 ionosphere-free combination and the satellite clock bias calculated from the L1 / L2 combination. This is an ambiguity constant that incorporates the phase hardware delay time invariant.

[0091] Therefore, the ionosphere-free IFCB can be expressed as:

[0092] (5).

[0093] S4: Based on the interepoch difference method, the ambiguity constants that have absorbed the phase hardware delay time invariant are removed to obtain the ionospheric combination IFCB and its variance for a single station. When executing the estimation program, a three-times standard difference detection threshold is set for quality control.

[0094] Specifically, the epoch-difference (ED) method is used to remove the epochs in equation (5). As a constant, the ionosphere-free composite IFCB and its variance based on the inter-epoch difference method can be expressed as follows:

[0095] (6)

[0096] In the formula, Indicates the number of epochs. express The observations consist of a geometrically and ionospherically unaffected combination of frequencies at epochs L1, L2, and L3. For quality control, a three-times-standard-to-detection threshold is set during the estimation process; if the calculated result exceeds this threshold, the observation for that epoch is excluded from the calculation.

[0097] S5: Based on the observation data from different stations, the satellite ionospheric free IFCB is estimated by multi-station weighted averaging. The ionospheric free combined IFCB is obtained by summing. The ionospheric free (IF) combined IFCB is converted into non-differential non-combined IFCB corrections by correction relations, thus obtaining the complete GNSS IFCB estimation method and its correction model in the three-frequency non-differential non-combined model.

[0098] Specifically, the satellite's ionospheric non-ionospheric IFCB is estimated using a multi-station weighted average method as follows:

[0099] (7)

[0100] in, The weights related to the satellite's elevation angle are indicated. This is the satellite elevation angle. This indicates the number of monitoring stations.

[0101] In addition, for quality control, observations are abandoned when the satellite observation arc is less than 10 minutes, and the satellite cutoff elevation angle is set to 10 degrees. It is important to note that if the ICB (Integrated Focal Flow Coefficient) of the same satellite from different stations has the same corresponding satellite elevation angle, it is assigned the same weight.

[0102] Then, the epoch can be obtained by accumulation. Time Satellite The following are ionosphere-free IFCB combinations:

[0103] (8)

[0104] in, Indicates reference time period The ionosphere-free IFCB at time 0. In this invention, time 0 ( Setting the IFCB of ) to 0 will introduce a common bias to all epochs. However, this bias will be absorbed into the ambiguity parameter and therefore will not affect subsequent applications.

[0105] Here are some points to note when estimating the ionosphere-free combination IFCB using the inter-epoch difference method:

[0106] a. As can be seen from the above formula, the accuracy of IFCB estimation largely depends on the number of observations. Therefore, it is necessary to ensure that at least dozens or even hundreds of stations participate in IFCB calculation.

[0107] b. Existing research indicates that the time-varying component of the receiver's IFCB is sufficiently small; therefore, only the satellite's IFCB needs to be considered.

[0108] c. Since IFCB estimation is limited to the phase end and is independent of the receiver, the GLONASS system maintains consistency with CDMA systems such as GPS in terms of IFCB estimation algorithms and models.

[0109] In data processing, the atx files released by IGS are generally used to correct the phase center offsets and variations (PCO / PCV) of satellite (receiver) antennas at L1 and L2 frequencies. However, IGS has not yet released a usable PCO / PCV correction product for the third frequency. Therefore, the correction value for the second frequency is generally used to correct the third frequency, which may result in residual PCO / PCV error in the estimated IFCB product. However, since the user end also uses the same PCO / PCV correction method when correcting IFCB deviations using the aforementioned IFCB product, it will not affect the multi-frequency GNSS data processing results. In addition, since the IFCB obtained here is the combined IFCB, in order to eliminate the non-differential, non-combined IFCB deviation, it is necessary to convert the combined IFCB to the non-differential, non-combined IFCB correction value. The conversion relationship can be obtained by the following formula:

[0110] (9)

[0111] This represents the non-difference, non-combination IFCB correction.

[0112] Thus far, the embodiments of the present invention have provided a complete method for estimating GNSS IFCB and its correction model in a three-frequency non-differential non-combined model.

[0113] Furthermore, based on the IFCB deviation data, this invention employs Fast Fourier Transform (FFT) for harmonic analysis to obtain the precise period of the IFCB sequence, and proposes an IFCB deviation model based on a third-order polynomial and a 7-period harmonic function, as detailed below:

[0114] The IFCB deviation exhibits distinct sinusoidal variation characteristics, making it possible to model with high accuracy. Previous studies have used a model combining a linear function and a multi-harmonic function to describe the IFCB variation, which can be expressed as follows:

[0115] (10)

[0116] Previous studies achieved a modeling accuracy of approximately 4 mm using a 6-period IFCB model. However, this study found that these models primarily rely on harmonic functions to describe the IFCB variation period of GPS, but in reality, certain non-harmonic variation characteristics still exist in the IFCB sequence, especially in the GLONASS system. Modeling these non-harmonic characteristics using only a linear function would obviously result in significant accuracy loss. Therefore, this invention employs Fast Fourier Transform (FFT) for harmonic analysis to obtain the precise period of the long-period IFCB sequence (GPS: T1=12, T2=8, T3=6, T4=4.8, T5=4, T6=3, T7=2.4; GLONASS: T1=12, T2=8, T3=6, T4=4.8, T5=4, T6=3.249, T7=3). Based on this, a method using a 3rd-order polynomial and a 7th-order harmonic function to describe the IFCB variation is proposed as follows:

[0117] (11)

[0118] In the formula, express IFCB of the epoch, Represents a constant term. , and These are the coefficients of the first, second, and third order terms, respectively. For amplitude, For a period of time, Indicates phase, This represents the order of the harmonic function. To clarify the polynomial order and period number, it is necessary to analyze and judge in conjunction with experimental results. Then, experimental forecasts are performed based on the IFCB time series model. The periodic characteristics of the IFCB deviation are extracted and compared, and the optimal forecast period is calculated: the forecast accuracy of the GPS system's IFCB decreases as the forecast time interval increases, and the forecast accuracy is highest when the forecast time interval is 1 day, approximately 1.6 cm; the GLONASS system reaches its highest forecast accuracy, approximately 5 cm, when the IFCB forecast time interval is consistent with its nadir orbit revisit period (8 days). The extraction of the periodic term and the analysis of the forecast period are described below.

[0119] The following example obtains station observation data, uses the above method to predict satellite frequency clock offset, and analyzes the number of periods and polynomial order in the model. See [link to example]. Figure 2 The specific implementation process is as follows:

[0120] A1. Obtain observation data.

[0121] In this embodiment, the IFCB time series of the GPS system is estimated using GPS tri-frequency observations recorded by 120 MGEX stations over a two-year period from January 1, 2017 to December 30, 2018. The observation intervals are 30 seconds, and the cutoff elevation angle is 10°. The IFCB sequences of all 12 GPS Block IIF satellites (G01, G03, G06, G08, G09, G10, G24, G25, G26, G27, G30, and G32) are estimated during these two years. In addition, 135 globally distributed MGEX stations with DOY=1 to 170 days in 2020 are selected for GLONASS system IFCB estimation. These stations can all observe GLONASS L3 frequency observations.

[0122] A2. Obtain the precise period of the IFCB sequence.

[0123] In this embodiment, the proposed harmonic function model based on a third-order polynomial and a 7-period is used to model the ICBs of GPS and GLONASS system satellites, respectively. First, harmonic analysis is performed using Fast Fourier Transform (FFT) to obtain the precise period of the ICB sequence. This work is carried out using the MATLAB open-source software package.

[0124] A3. Analysis of IFCB deviation modeling results; see the detailed implementation process below. Figure 3 ,include:

[0125] A31. Extracting the period from the IFCB sequence

[0126] First, the periodic signal in the IFCB sequence is extracted using the FFT function. The Fast Fourier Transform can describe the periodic changes of the IFCB time series through frequency and amplitude, giving the spectral characteristics of the IFCB sequence in the frequency domain, and its periodic characteristics can be modeled and described accordingly. Figure 6 The period (T1, T2, T3, T4) corresponding to the first four largest amplitude components in the daily ICB time series of GPS Block IIF satellites from 2017 to 2018 is given. As shown in the figure, the ICB of GPS satellites exhibits seven periods: 12, 8, 6, 4.8, 4, 3, and 2.4 hours. Among these seven periods, the 12, 8, 6, and 4 periods are the most significant, but for high-precision modeling, the other three periods should not be ignored.

[0127] Similar to GPS satellites, the periods of the first four components with the largest amplitudes in the IFCB time series of the four GLONASS satellites are... Figure 7The figure shows that GLONASS satellites also exhibit seven epochal variations: 12, 8, 6, 4.8, 4, 3.429, and 3 hours. Comparing this to the GPS system, the 3.429-hour epoch is unique to GLONASS, and while some of these epochs share similarities with GPS, relating to internal satellite temperature variations, others require further explanation. Therefore, considering these seven epochal signals, a 24-hour data arc modeling method is used instead of a 72-hour arc. Furthermore, because the first epoch constraint is set to 0 during IFCB estimation, noticeable abrupt changes occur between days.

[0128] A32. Set up five sets of comparative experiments considering different numbers of periods and different polynomial orders.

[0129] Specifically, five sets of comparative experiments were set up for GPS and GLONASS systems, and the specific experimental schemes are shown in Table 1. As can be seen from Table 1, the difference between the schemes lies in the number of periods considered and the order of the polynomials. Scheme 1 is set to consider 2 periods and a first-order polynomial, while Scheme 5 uses a combination of seven periods and a third-order polynomial. Then, the harmonic coefficients and polynomial coefficients of the corresponding equation (11) for the five schemes are estimated by the least squares fitting method.

[0130] Table 1. GPS / GLONASS Comparison Experiment Scheme Setup

[0131]

[0132] A33. Calculate the IFCB estimates for GPS system G01 satellite in 2017 (DOY=127 days) and GLONASS system R05 satellite in 2020 (DOY=127 days), as well as the modeling values ​​for each scheme.

[0133] Specifically, taking the GPS system G01 satellite with DOY=127 days in 2017 as an example, we calculate the IFCB estimate of the satellite on that day and the modeling values ​​for each scheme, such as... Figure 8 As shown in the figure. The upper subplot represents the IFCB modeling results for each scheme of the GPS system G01 satellite, where the black line represents the estimated value of IFCB for that day. Figure 8It can be seen that the scheme using 7 periods plus a 3rd-order polynomial has the best expected fit, especially at the peaks and troughs. The subplot below shows the residual sequence between the modeled values ​​and the IFCB estimates for each scheme. The scheme using 7 periods plus a 3rd-order polynomial has the smallest residual, with a root mean square error (RMS) of only 1.4 mm. Furthermore, the IFCB modeling results and residuals for each scheme of the GLONASS R05 satellite with DOY=127 days in 2020 are also presented, as shown below. Figure 9 As shown in the figure, similar to the GPS system modeling results, the scheme using 7 periods plus a 3rd order polynomial has the highest degree of agreement, with the smallest residual RMS of only 2.0 mm.

[0134] A34. Calculate the residual sequence between the modeling results of the GPS system Block IIF satellite and the GLONASS system satellite using the above five schemes and the estimated values ​​for the day, and calculate the root mean square error of the residuals for each day.

[0135] Specifically, to further study the modeling effect of IFCB, the residual sequence of the modeling results of 12 GPS system Block IIF satellites using the above five schemes and the estimated values ​​for the day was first calculated. Then, the root mean square error (RMS) of the residuals for each day was calculated, and the RMS sequences of the residuals for each scheme were plotted as follows. Figure 10 As shown in the figure, the average RMS residuals for each scheme over 2 years are also presented. The figure reveals significant differences in modeling accuracy among the different modeling schemes, all roughly at the millimeter level. Compared to the modeling results considering two periods, the average RMS residual of the model considering six periods decreased by approximately 1.3–2.5 mm. Further considering a seventh small period of 2.4 hours, the average RMS residual decreased slightly by approximately 0.2 mm. Based on this, the modeling accuracy of the IFCB for each satellite was further significantly improved by adopting a multi-order polynomial model. Compared to Scheme 3, Scheme 4, which considers a combination of a second-order polynomial and a 7-period model, further reduced the average RMS modeling error by approximately 1.3–2.4 mm. Scheme 5, which considers a third-order polynomial model, achieved the best modeling accuracy, improving by 0.2–0.9 mm compared to Scheme 4.

[0136] Figure 11The figure presents the RMS residuals between the modeling results and estimated values ​​for each group of GLONASS satellite models, along with the mean RMS of the modeling errors for each group. As shown in the figure, the modeling accuracy of the GLONASS satellite IFCB is mostly within a few centimeters, slightly worse than the GPS system. This may be partly due to the higher accuracy of the GPS IFCB estimate and the more pronounced periodicity of IFCB changes. For the GLONASS system, similar to the GPS modeling results, the scheme using a third-order polynomial with 7 periodic terms achieved the best modeling accuracy. The mean RMS of the modeling errors for R05, R12, R21, and R26 satellites were 0.5, 2.16, 0.61, and 0.93 cm, respectively, showing a significant improvement over the other schemes. The scheme using 2 periodic terms produced the worst modeling results.

[0137] A35. Calculate the mean of the modeling residuals for all satellites in each scheme of the GPS and GLONASS systems.

[0138] Specifically, the mean values ​​of the modeling residuals for all satellites in each scheme of the GPS and GLONASS systems are shown in Table 2. For the GPS system, Scheme 5 achieved the highest modeling accuracy, with a mean RMS modeling residual of only 2.46 mm, effectively reducing the RMS by 63.82% compared to Scheme 1's 6.80 mm, significantly better than the 13.5 mm and 4 mm modeling accuracies given in existing literature. Compared to Scheme 1, which considers two-period signals, Scheme 2, which considers four more-period signals, reduced the modeling error from 6.8 mm to 4.79 mm. After considering an additional seventh small-period signal for 2.4 h, the modeling accuracy improved slightly to 4.61 mm, but the improvement was limited. Compared to Scheme 3, which uses a first-order polynomial, using a second-order polynomial modeling method can improve the modeling accuracy to 2.87 mm; when further using a third-order polynomial modeling method, the modeling accuracy can reach 2.46 mm. The overall modeling accuracy of the GLONASS system is worse than that of the GPS system. Scheme 5 also achieved the highest modeling accuracy of 1.05 cm, which is an effective improvement of 70.17% compared to Scheme 1's 3.52 cm. Compared to Scheme 2, which considers a 6-cycle signal, Scheme 3, which additionally considers a seventh small cycle, shows a smaller improvement in modeling accuracy. However, when multi-order polynomial modeling is used, the modeling accuracy is significantly improved.

[0139] The results above demonstrate that the IFCB model employing a third-order polynomial and a 7-period signal achieves the highest modeling accuracy. Compared to the 2-period IFCB model, the 6-period IFCB model significantly improves modeling accuracy, but the performance improvement from considering a seventh small period is relatively limited. Compared to the IFCB model using a first-order polynomial, the second-order polynomial model significantly improves modeling accuracy, but the performance improvement is less significant when using a third-order polynomial model. Overall, the proposed IFCB modeling method using multi-order polynomials and a 7-period signal achieves high-precision modeling results, representing a significant improvement over traditional methods.

[0140] Table 2. Statistics on modeling accuracy of each experimental scheme in GPS / GLONASS groups

[0141]

[0142] A4. Analysis of IFCB deviation forecast results; see detailed implementation steps. Figure 4 ,

[0143] As can be seen from the above analysis, the IFCB model consisting of a third-order polynomial and a seventh-order harmonic function can achieve a GPS system IFCB modeling accuracy of about 2 mm and a GLONAS system IFCB modeling accuracy of about 1 cm. Considering its own order of magnitude of 2 dm and 5 dm, this modeling accuracy can meet the application requirements of various occasions.

[0144] A41. Calculate the RMS residuals of the IFCB prediction values ​​for the 12 satellites of the GPS system.

[0145] Specifically, the RMS residuals of the IFCB prediction values ​​for 12 GPS satellites are given first. Figure 12 As shown in the figure, the mean RMS residuals for each satellite at different prediction time intervals are also presented. Furthermore, Table 3 shows the mean RMS residuals for all satellites at different prediction time intervals. As the figure shows, when the prediction time interval is 1 day, the IFCB forecast residual RMS is generally better than 2 cm. During the shadow period, the residual RMS exhibits significant jumps. As the prediction time interval increases, the IFCB forecast residuals also increase. The statistical data in Table 3 shows that as the prediction interval increases from 1 day to 9 days, the mean IFCB forecast residual RMS decreases from 1.69 cm to 2.49 cm.

[0146] A42. Calculate the RMS residuals of the IFCB predictions for the four GLONASS satellites.

[0147] Specifically, the RMS residuals of the IFCB prediction values ​​for the four GLONASS satellites are given (e.g., Figure 13 The table shows the RMS mean residuals of the GLONASS satellites and all their satellites. As can be seen from the figure, unlike the GPS system, the IFCB prediction accuracy is 9.1 cm when the prediction interval is 1 day. As the prediction interval is extended, the IFCB prediction accuracy continuously decreases, reaching a minimum of 18.88 cm when the prediction interval is 4 days. Subsequently, further extending the prediction interval improves the IFCB prediction accuracy, reaching a maximum of 5.14 cm when the prediction interval is 8 days. Table 3 shows that when the prediction interval exceeds 8 days, the prediction accuracy decreases again to 10.76 cm. Therefore, for GLONASS satellites, the prediction accuracy is highest when the interval is 8 days.

[0148] The above research shows that the prediction accuracy of the GPS system's ICB decreases as the prediction time interval increases. The prediction accuracy is highest when the prediction time interval is 1 day, which is about 1.6 cm. The highest prediction accuracy is achieved when the GLONASS system's ICB prediction time interval is consistent with its nadir orbit revisit period (8 days), which is about 5 cm.

[0149] Table 3. Prediction accuracy of GPS / GLONASS system for different prediction time intervals.

[0150]

[0151] A5. Optimal combination of ICB deviation model coefficients for GPS / GLONASS systems

[0152] Based on the above experimental results, this invention proposes a high-precision modeling and prediction model for GPS / GLONASS system satellite IFCB based on a third-order polynomial and a 7-period harmonic function. Results show that the proposed IFCB model achieves a modeling accuracy of approximately 2.5 mm for GPS satellites, significantly better than the 13.5 mm and 4 mm modeling results given in existing literature, while achieving a modeling accuracy of approximately 1 cm for GLONASS system satellites. Regarding IFCB prediction, the prediction accuracy of the GPS system decreases with increasing prediction time intervals, reaching 1.6 cm when the prediction time interval is 1 day. The highest prediction accuracy, approximately 5 cm, is achieved when the GLONASS system's IFCB prediction time interval matches its nadir orbit revisit period (8 days). Using this model can effectively improve positioning accuracy and reliability.

[0153] Based on the above-mentioned inventive concept, the present invention also provides a GPS / GLONASS frequency inter-frequency clock offset modeling and prediction device, see [link to relevant documentation]. Figure 5 The device includes:

[0154] The acquisition module is used to acquire satellite positioning data through a receiver. The positioning data includes three-frequency carrier and pseudorange observations of GPS / GLONASS system satellites.

[0155] The first determining module constructs a phase hardware delay between the satellite end and the receiver end based on the positioning data, and divides the phase hardware delay into two parts: time-varying and time-invariant.

[0156] The second determining module is used to construct a geometry-free and ionosphere-free GFIF combination based on known data, and to determine the geometry-free and ionosphere-free GFIF combination and its variance; the geometry-free and ionosphere-free GFIF combination includes an ionosphere-free combination IFCB and an ambiguity constant that absorbs the phase hardware delay time invariant.

[0157] The third determining module is used to remove the ambiguity constants that have absorbed the phase hardware delay time invariant in the geometrically non-ionospheric GFIF combination by using the interepoch difference method, so as to obtain the single-station non-ionospheric combination IFCB and variance.

[0158] The fourth determination module is used to estimate the satellite ionospheric IFCB by using a multi-station weighted average method based on observation data from multiple stations. It obtains the satellite ionospheric combined IFCB by accumulation and converts the ionospheric combined IFCB into a non-differential non-combined IFCB correction number through a correction relation.

[0159] The fifth determination module is used to perform harmonic analysis using fast Fourier transform based on IFCB deviation data, obtain the precise period of the IFCB sequence, propose an IFCB deviation model based on a third-order polynomial and a 7-period harmonic function, conduct experimental forecasting based on the IFCB time series model, extract and compare the periodic characteristics of the IFCB deviation, and calculate the optimal forecast period.

[0160] It is worth noting that this device embodiment corresponds to the above method embodiment. The implementation methods of the above method embodiments are all applicable to this device embodiment and can achieve the same or similar technical effects, so they will not be described in detail here.

[0161] The present invention also provides a computer-readable storage medium storing one or more programs, the one or more programs including instructions that, when executed by a computing device, cause the computing device to perform the above-described GPS / GLONASS frequency inter-frequency clock offset modeling and prediction method.

[0162] The present invention also provides a computing device comprising one or more processors, a memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, and the one or more programs include instructions for performing the above-described GPS / GLONASS frequency inter-frequency clock offset modeling and prediction method.

[0163] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0164] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0165] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0166] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0167] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A method for modeling and predicting clock offset between GPS / GLONASS frequencies, characterized in that, include: The positioning data of the satellite is acquired by the receiver, and the positioning data includes the three-frequency carrier and pseudorange observations of the GPS / GLONASS system satellites; Based on the positioning data, the phase hardware delay of the satellite end and the receiver end is constructed, and the phase hardware delay is divided into two parts: time-varying and time-invariant. Construct a geometry-free and ionosphere-free GFIF combination based on known data, and determine the geometry-free and ionosphere-free GFIF combination and its variance; the geometry-free and ionosphere-free GFIF combination includes the ionosphere-free combination IFCB and an ambiguity constant that absorbs the phase hardware delay time invariant. The ambiguity constants that absorbed the phase hardware delay time invariant in the geometrically un-ionospheric (GFIF) combination were removed by the interepoch difference method to obtain the ionospheric combination IFCB and variance of a single station. Based on multi-station observation data, the satellite ionospheric IFCB is estimated by multi-station weighted average. The satellite ionospheric combined IFCB is obtained by summing, and the ionospheric combined IFCB is converted into non-difference non-combined IFCB corrections by correction relation. Based on the IFCB bias data, harmonic analysis was performed using Fast Fourier Transform to obtain the precise period of the IFCB sequence. An IFCB bias model based on a third-order polynomial and a 7-period harmonic function was proposed. Experimental forecasts were conducted based on the IFCB time series model, and the periodic characteristics of the IFCB bias were extracted and compared to calculate the optimal forecast period.

2. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 1, characterized in that, The acquisition of satellite positioning data via receiver includes: ； in, and These represent the three-frequency carrier and pseudorange observations, respectively, with superscript indicating the superscript. and These represent the satellite PRN number and the GNSS system, respectively, with subscripts. and These represent the receiver and the frequency, respectively, where i = 1 and 2. Indicates the distance between the measuring station and the satellite. and For receiver clock bias and satellite clock bias, At the speed of light, and These are the zenith tropospheric wet delay and its projection function, respectively. Represents the ionospheric coefficient factor. For system frequency, The ionospheric parameters are for the first frequency. and These are wavelength and integer ambiguity, respectively. and These represent the phase hardware delays at the receiver and satellite ends, respectively. and These are the pseudorange hardware delays at the receiver and satellite ends, respectively. For the GLONASS system, when i=1 or 2, When i=3, .

3. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 2, characterized in that, Based on the positioning data, the phase hardware delay at the satellite end and receiver end is constructed, and the phase hardware delay is divided into two parts: time-varying and time-invariant, including: ； in, and These represent the phase hardware delays at the satellite end and the receiver end, respectively. and These are the time-varying components of the phase hardware delay at the satellite end and the receiver end, respectively. and These are the time-invariant parts of the phase hardware delay at the satellite end and the receiver end, respectively.

4. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 3, characterized in that, The following steps describe the construction of a geometry-free and ionosphere-free GFIF combination based on known data, and the determination of the geometry-free and ionosphere-free GFIF combination and its variance: ； ； ， ， ； in, This indicates a geometrically non-ionosphere (GFIF) combination. This represents the variance of the geometrically unbiased and ionosphere-free GFIF combination. The satellite clock bias is calculated from the L1 / L2 ionosphere-free combination. The satellite clock bias is calculated from the L1 / L3 ionosphere-free combination. Indicates the ionosphere-free composite IFCB. To incorporate the ambiguity constant that is invariant to phase hardware delay, , and These represent the integer ambiguity at three frequencies. , Indicates frequency, For the phase hardware delay of the third frequency receiver, , and All were set to 0.02 weeks.

5. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 4, characterized in that, The epoch-difference method is used to remove the ambiguity constants that absorb the phase hardware delay time invariant in the geometrically unspatial GFIF combination, resulting in the IFCB and variance of the single-station ionospheric combination, as follows: ； in, This represents the ionosphere-free IFCB based on the interepoch difference method. and They represent Li Yuanhe Geometrically and ionosphere-free combination of observations based on epoch frequencies L1, L2, and L3. for variance Indicates the number of epochs; During the execution of the interepoch difference method, a detection threshold of three times the standard deviation is set.

6. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 5, characterized in that, The process involves estimating the satellite's ionospheric non-ionospheric IFCB using a multi-station weighted average based on multi-station observation data, obtaining the satellite's combined ionospheric IFCB through summation, and converting the combined ionospheric IFCB into a non-differential, non-combined IFCB correction value using a correction relation. This includes: The satellite's ionospheric non-ionospheric IFCB was estimated using a multi-station weighted average method as follows: ； in, This indicates that the satellite has no ionosphere (IFCB). The weights related to the satellite's elevation angle are indicated. This is the satellite elevation angle. Indicates the number of stations; The epoch is obtained through accumulation. Time Satellite The following are ionosphere-free IFCB combinations: ； in, Represents the epoch Time Satellite IFCB without ionosphere Indicates reference time period Time Satellite Ionosphere-free composite IFCB; The corrections for converting the non-ionosphere combined IFCB to a non-differential, non-combined IFCB are as follows: ； in, This represents the non-difference, non-combination IFCB correction.

7. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 6, characterized in that, Set the observation period to be abandoned if the satellite observation arc is less than 10 minutes. Set the satellite cutoff elevation angle to 10; If the ionospheric combination IFCB of the same satellite at different stations has the same satellite elevation angle, then it is assigned the same weight.

8. The method for modeling and predicting inter-frequency clock offsets between GPS and GLONASS frequencies according to claim 7, characterized in that, Based on the IFCB deviation data, harmonic analysis is performed using Fast Fourier Transform to obtain the precise period of the IFCB sequence. An IFCB deviation model based on a third-order polynomial and a 7-period harmonic function is proposed as follows: ； In the formula, express Epoch IFCB bias, Represents a constant term. , and These are the coefficients of the first, second, and third order terms, respectively. For amplitude, For a period of time, Indicates phase, Indicates the order of the harmonic function; Experimental forecasts were conducted based on the IFCB bias model, and the periodic characteristics of the IFCB bias were extracted and compared to calculate the optimal forecast period.

9. A GPS / GLONASS frequency clock offset modeling and prediction device, characterized in that, The apparatus for implementing the GPS / GLONASS frequency inter-frequency clock offset modeling and prediction method according to any one of claims 1 to 8 includes: The acquisition module is used to acquire satellite positioning data through a receiver, the positioning data including the three-frequency carrier and pseudorange observations of GPS / GLONASS system satellites; The first determining module constructs the phase hardware delay between the satellite end and the receiver end based on the positioning data, and divides the phase hardware delay into two parts: time-varying and time-invariant. The second determining module is used to construct a geometry-free and ionosphere-free GFIF combination based on known data, and to determine the geometry-free and ionosphere-free GFIF combination and its variance; the geometry-free and ionosphere-free GFIF combination includes an ionosphere-free combination IFCB and an ambiguity constant that absorbs the phase hardware delay time invariant. The third determining module is used to remove the ambiguity constants that have absorbed the phase hardware delay time invariant in the geometrically non-ionospheric GFIF combination by using the interepoch difference method, so as to obtain the single-station non-ionospheric combination IFCB and variance. The fourth determination module is used to estimate the satellite ionospheric IFCB by using a multi-station weighted average method based on observation data from multiple stations. It obtains the satellite ionospheric combined IFCB by accumulation and converts the ionospheric combined IFCB into a non-differential non-combined IFCB correction number through a correction relation. The fifth determination module is used to perform harmonic analysis using fast Fourier transform based on IFCB deviation data, obtain the precise period of the IFCB sequence, propose an IFCB deviation model based on a third-order polynomial and a 7-period harmonic function, conduct experimental forecasting based on the IFCB time series model, extract and compare the periodic characteristics of the IFCB deviation, and calculate the optimal forecast period.

10. A computer-readable storage medium for storing one or more programs, characterized in that, The one or more programs include instructions that, when executed by a computing device, cause the computing device to perform any of the methods according to claims 1 to 8.

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