Spatial baseline determination is a key technology for interferometric synthetic aperture radar (InSAR) missions. Based on the intersatellite baseline measurement using dual-frequency GPS, errors induced by InSAR spatial baseline measurement are studied in detail. The classifications and characters of errors are analyzed, and models for errors are set up. The simulations of single factor and total error sources are selected to evaluate the impacts of errors on spatial baseline measurement. Single factor simulations are used to analyze the impact of the error of a single type, while total error sources simulations are used to analyze the impacts of error sources induced by GPS measurement, baseline transformation, and the entire spatial baseline measurement, respectively. Simulation results show that errors related to GPS measurement are the main error sources for the spatial baseline determination, and carrier phase noise of GPS observation and fixing error of GPS receiver antenna are main factors of errors related to GPS measurement. In addition, according to the error values listed in this paper, 1 mm level InSAR spatial baseline determination should be realized.

Close formation flying satellites equipped with synthetic aperture radar (SAR) antenna could provide advanced science opportunities, such as generating highly accurate digital elevation models (DEMs) from Interferometric SAR (InSAR) [

In order to realize the advanced space mission goal of InSAR mission, the high-precision determination of inter-satellite interferometric baseline [

The interferometric baseline is defined as the separation between two SAR antennas that receive echoes of the same ground area [

The spaceborne dual-frequency GPS measurement scheme [

In our research, impacts of the errors introduced by spatial baseline measurement are analyzed. This paper starts with a description of spatial baseline measurement using dual-frequency GPS. The baseline transformation from the relative position to spatial baseline is given. In a second step, errors are classified into two groups: errors related to GPS measurement and errors related to baseline transformation. The error characters are studied, and the impact of each error on spatial baseline determination is analyzed from theoretical aspect. Then the impacts of each error and total errors on spatial baseline determination are analyzed by single factor simulations and total error sources simulations. At last, conclusions are shown.

In preparation for latter description some coordinate systems are introduced at first, which are illustrated in Figure

Definitions of coordinate systems employed in this paper. CIRF, ITRF, satellite body coordinate system, and satellite orbit coordinate system are denoted as _{E}-_{CIRF}_{CIRF}_{CIRF}, _{E}-_{ITRF}_{ITRF}_{ITRF}, _{S}-_{Body}_{Body}_{Body}, and _{S}-_{Orbit}_{Orbit}_{Orbit}, respectively. _{E} is the geocenter, and_{S} is the mass center of satellite.

As the spatial baseline is determined by spaceborne dual-frequency GPS measurement scheme, the entire process of spatial baseline determination consists of relative positioning and baseline transformation. Figure

Geometric relation for spatial baseline determination.

Relative positioning is the determination of

From Figure

Flow chat of spatial baseline determination.

According to the generation of spatial baseline in Section

The relative positions of two satellites are determined by the reduced dynamic carrier phase differential GPS approach. In this approach, the absolute orbits of one reference satellite (Satellite 1) are fixed, which are determined by the zero-difference reduced dynamic batch least squares approach based on GPS measurements of single satellite. Only the relative positions are estimated by reduced dynamic batch least-squares approach based on differential GPS measurements. The integer double difference ambiguities for relative positioning are resolved by estimating wide-lane and narrow-lane combinations [

By differenced GPS observation, common errors can be eliminated or reduced. International GNSS Service (IGS) final GPS ephemeris product (orbit product and clock product) [

The quality of GPS carrier phase observation data used is of utmost importance for relative positioning. The noise of GPS carrier phase measurement belongs to random error, which cannot directly be eliminated by GPS differential observation. Take the BlackJack receiver and its commercial Integrated GPS and Occultation Receiver (IGOR) version, for example, which are widely used for geodetic grade space missions and exhibit a representative noise level of 1 mm for

The phase center location accuracy of the GPS receiver antenna will directly affect the veracity of GPS observation modeling. GPS receiver antenna phase center is the instantaneous location of the GPS receiver antenna where the GPS signal is actually received. It depends on intensity, frequency, azimuth, and elevation of GPS receiving signal.

The phase center locations can be described by the mechanical antenna reference point (ARP), a phase center offset (PCO) vector, and phase center variations (PCVs). The PCO vector describes the difference between the mean center of the wave front and the ARP. PCVs represent direction-dependent distortions of the wave front, which can be modeled as a consistent function that depends on azimuth and elevation of the observation from the position indicated by the PCO vector. The position of GPS receiver antenna phase center can be measured by ground calibration, such as using an anechoic chamber and using field calibration techniques [

Ground calibrated mean PCVs result of SEN67-1575-14+CRG antenna on ionosphere-free combination.

As there is a slim difference between the line of sight (LOS) vectors of two satellites during close formation flying, the common systematic errors of GPS receiver antenna phase center and near-field multipath can be eliminated by differential GPS observation. Therefore, the same type of GPS receiver antenna has to be selected for both formation satellites in order to reduce the impact of these errors.

Satellite attitude data are obtained from star camera observations and provided as quaternion. The error of satellite attitude measurement consists of a slowly varying bias and a random error. Its impact on GPS relative positioning appears on the correction for GPS observation data of single satellite, that is, the reference point of GPS observation data has to be corrected from GPS receiver antenna phase center to the mass center of satellite by satellite attitude data and GPS receiver antenna phase center data. Take Satellite 1 for instance. The correction in direction of LOS vector

Assuming that the Euler angles are

Assuming that the errors of Euler angle measurements are

Furthermore, the impact of Euler angle errors on

Assuming Euler angle errors of different axes are independent, we can get

As

Taking (

For differential GPS observation, the impact of attitude determination error on two satellites can be given as follows

According to the TanDEM-X missions, the attitude determination accuracy has a slowly varying bias of

Take the GPS receiver antenna ARP location of TanDEM-X mission for instance, that is,

The fixing error of GPS receiver antenna is caused by the inaccuracy of the fixed position of antenna onboard the satellite. This error is a random error for multiple repeated satellite missions. But for a single launch, it is considered to be a fixed bias vector in satellite body coordinate system during satellite flying.

The fixing errors of GPS receiver antenna in body coordinate system of two satellites are assumed as follows:

For a mutually observed GPS satellite

The impact of fixing error of GPS receiver antenna on differential GPS observation is

Due to the close separation of two satellites, we can assume

From (

As the magnitudes of

In addition, we can also draw a conclusion from the aforementioned analysis that the GPS receiver antenna bias caused by thermal distortions of satellites can be cancelled out by differential GPS observation.

From (_{1}_{1} and _{2}_{2}, which is caused by the inconsistency of two SAR antenna phase centers.

Take

Note that the transformation from CIRF to ITRF is in accordance with IERS 1996 conventions [

Hence, the impact of attitude determination errors on baseline transformation is given as follows:

Take the attitude determination accuracy of TanDEM-X mission for instance and select the magnitudes of

Unlike GPS receiver antenna, active phased array antenna is selected for SAR antenna. The phase center of the SAR antenna describes the variation of the phase curve within the coverage region against a defined origin, here the origin of the antenna coordinate system [

The inconsistency between receiver channels, which is introduced by manufacturing process, such as the instrument difference, machining art level, module assembling level and the work temperature difference, et al.

The inconsistency between the locations of apertures, which is mainly caused by the fixing flatness difference, relative dislocation difference, the deployment inconsistency of SAR antennas and the configuration distortions caused by different thermal circumstances, and others.

According to current ability of engineering, the phase inconsistency between T/R modules at X-band can be constrained to

Take the TanDEM-X mission, for example. Setting

The HELIX satellite formation is selected for the simulations and the orbit elements of two satellites are shown in Table

Orbit elements of formation satellites.

Parameters of satellites | Satellite 1 | Satellite 2 |
---|---|---|

Semimajor axis | 6886478 m | 6886478 m |

Inclination | 97.4438° | 97.4438° |

Eccentricity | 0.00117 | 0.001073 |

Argument of perigee | 90° | 90° |

Right ascension of ascending node | 0° | 0.01171° |

True anomaly | 269.99677° | 270.00622° |

The entire simulation consists of GPS measurement simulation and baseline transformation simulation. The flow chart of GPS observation data simulation is shown in Figure

Error accuracies and modeling descriptions in simulations.

Error type | Error accuracy | Modeling description |
---|---|---|

GPS code measurement noise | 0.5 m (1 | Gaussian white noise model with mean value of 0 m and standard deviation of 0.5 m |

GPS carrier phase measurement noise | 0.002 m (1 | AR(2) model with mean value of 0 m and standard deviation of 0.002 m |

Ground calibration error of GPS receiver antenna phase center | — | ARP data, PCO data, and PCVs data (mean value data and RMS data) of SEN67-1575-14+CRG |

Attitude measurement error | Fixed bias of 0.005° in the yaw, pitch, and roll components plus a 0.003° (1 | Gaussian white noise model with mean value of 0.005° and standard deviation of 0.003° in the yaw, pitch, and roll components |

Fixing error of GPS receiver antenna | 0.5 mm (3 | A fixed vector with direction randomly drawn in unit ball and magnitude of 0.5 mm in each satellite body coordinate system |

Consistency error of SAR antenna phase center | 0.25 mm (3 | A fixed vector with direction randomly drawn in unit ball and magnitude of 0.25 mm in body coordinate system of Satellite 1 |

Flow chart of GPS observation data simulation.

Baseline transformation simulation is the process that the spatial baseline in ITRF is obtained by mass center data of formation satellites in ITRF, attitude simulation data, and SAR antenna phase center simulation locations in satellite body coordinate system. The real SAR antenna phase center simulation location in satellite body coordinate system is (1.2278 m, 1.5876 m, 0.0223 m). The error accuracies and models in the simulations are shown in Table

Each error related to GPS measurement is analyzed by single factor simulation, which is intended to obtain its impact on relative positioning based on dual-frequency GPS. The impact of each error is drawn by the comparison residuals between the relative position solutions determined by GPS observation data and relative positions obtained by standard orbits of formation satellites. The relative position solutions are implemented in the separate software tools as part of the NUDT Orbit Determination Software 1.0. The GPS observation data processing consists of GPS observation data preprocessing [

The noises of GPS carrier phase (

From the following formula

One instance of carrier phase noise simulation is shown in Figure

One instance of carrier phase measurement noise simulation.

50 groups of 24 h GPS observation data (interval of 30 s) for two formation satellites are simulated by only adding the noises of GPS carrier phase (

Simulation results of GPS carrier phase measurement noise for relative positioning.

Ground calibration error of GPS receiver antenna phase center is mainly caused by PCVs. The PCVs values are described by the mean value and RMS value corresponding to the direction of received signal. The PCV value corresponding to the direction of received signal is simulated by Gaussian white noise with mean value and RMS value obtained from ground calibration result of GPS receiver antenna system SEN67-1575-14+CRG.

The GPS observation data are simulated only considered ground calibration error of GPS receiver antenna phase center. By the precise orbit determination for single satellite, the mean RMS values of comparison residuals of orbit solutions in ITRF are 4.018 mm of

The Euler angle errors are simulated by Gaussian white noise with 0.005° of mean value and 0.003° of standard deviation, and 50 groups of 24 h GPS observation data for two formation satellites are simulated by only adding the attitude measurement errors. The mean RMS values of comparison residuals of relative position solutions in ITRF (Figure

Simulation results of satellite attitude measurement error for relative positioning.

The fixing error of GPS receiver antenna belongs to systematic error and it is a fixed bias vector in satellite body coordinate system. At first, four representatively “extreme” circumstances of fixing errors of GPS receiver antennas onboard two formation satellites are simulated. The so-called “extreme” circumstance is that the directions of two fixed bias vectors are opposite. Four representatively “extreme” circumstances of fixing errors of GPS receiver antennas here are directions along

Relative positioning results for four representatively “extreme” circumstances of fixing errors of GPS receiver antenna.

3 dimension/mm | ||||
---|---|---|---|---|

0.512 | 0.499 | 0.701 | 1.002 | |

0.677 | 0.694 | 0.133 | 0.979 | |

0.072 | 0.083 | 0.081 | 0.136 | |

Diagonal | 0.502 | 0.503 | 0.429 | 0.830 |

From Table

In practice, the occurrence of “extreme” circumstances is extremely low and they are just analyzed as the ultimate circumstances. For multiple repeated satellite missions, the fixing error of GPS receiver antenna is a random error. So this error can be simulated as a fixed vector with direction randomly drawn from unit ball and magnitude of 0.5 mm in each satellite body coordinate system. 50 groups of 24 h GPS observation data for two formation satellites are simulated by only adding the simulations of fixing error of GPS receiver antenna. The mean RMS values of comparison residuals of relative position solutions in ITRF (Figure

Simulation results of fixing error of GPS receiver antenna for relative positioning.

From aforementioned simulations of each error related to GPS measurement, it is shown that the impacts of GPS carrier phase measurement noise and fixing error of GPS receiver antenna on GPS relative positioning are much bigger than other errors related to GPS measurement and these two errors are the main factors of errors related to GPS measurement.

In this section, the impact of each error on baseline transformation is obtained by single factor simulation. Each impact is given by the comparison between the spatial baseline solutions obtained with and without errors.

The satellite attitude simulation data used here are the same as Section

Simulation results of satellite attitude measurement error for baseline transformation.

It is shown by the analysis in Section

Simulation results of consistency error of SAR antenna phase center for baseline transformation.

In this section, all the errors are added to the flow of spatial baseline determination simulations according to the error models listed in Table

Simulation results of total error sources for spatial baseline determination.

In addition, the impact of total errors related to GPS measurement on GPS relative positioning in ITRF (Figure

Simulation results of total errors related to GPS measurement for relative positioning.

Simulation results of total errors related to baseline transformation for baseline transformation.

It is shown by the simulations of total error sources that errors related to GPS measurement are the main error sources for the spatial baseline determination and 1 mm level InSAR spatial baseline determination can be realized according to current simulation conditions.

In this paper, the errors introduced by spatial baseline measurement for InSAR mission are deeply studied. The impacts of errors on spatial baseline determination are analyzed by single factor simulations and total error sources simulations. The main conclusions are drawn as follows.

The spatial baseline measurement errors can be classified into two groups: errors related to GPS measurement and errors related to baseline transformation. By simulations, the three-dimensional impacts of these errors on spatial baseline determination in ITRF are 0.755 mm and 0.334 mm, respectively. It is shown that the errors related to GPS measurement are the main influence on spatial baseline determination.

By the results of single factor simulations, the three dimensional impacts of GPS carrier phase measurement noise and the fixing error of GPS receiver antenna on GPS relative positioning in ITRF are 0.560 mm and 0.495 mm, respectively. These two errors are the main factors of errors related to GPS measurement.

It is shown by total error sources simulations that the impact of all the errors on spatial baseline determination in ITRF is 0.500 mm of

The mean antenna phase center description for the Sensor Systems SEN67157514 antenna has been contributed by the German Space Operations Center (GSOC), Deutsches Zentrum für Luft- und Raumfahrt (DLR), Wessling, to enable the simulation of antenna phase center data of dual-frequency GPS receiver. Precise GPS ephemerides for use within this study have been obtained from the Center for Orbit Determination in Europe at the Astronomical Institute of the University of Bern (AIUB). The authors extend special thanks to the support of the above institutions. This paper is supported by the National Natural Science Foundation of China (Grant no. 61002033 and no. 60902089) and Open Research Fund of State Key Laboratory of Astronautic Dynamics of China (Grant no. 2011ADL-DW0103).