Statistical evaluation of primary relevellings

For the following problems

- estimating the statistical parameters of the precise levelling,

- adjusting the primary levelling networks and

- estimating vertical crustal movements

mathematical models are being sketched out. Results obtained in evaluating primary relevellings in the G.D.R. are reported.     

Prospects for gradiometric aiding of inertial survey systems

The state of current and proposed moving-base gravity gradiometer instruments is briefly reviewed. The review perspective is directed toward their deployment as a source of additional gravimetric data during inertial surveys. In such gradiometer-aided surveys, the additional gravity gradient information could be used to:

1. Improve surveyed gravity vector accuracy
2. Extend the interval between zero velocity update stops
3. Accomplish varying combinations of the above.

The paper examines potential survey improvements associated with gradiometers having noise levels observed in laboratory prototypes. The additional improvements possible with future gradiometers are also discussed. These results are interpreted in light of present and likely future inertial survey system technology.     

A note on adjustment of free networks

The present paper deals with the least-squares adjustment where the design matrix (A) is rank-deficient. The adjusted parameters \(\hat x\) as well as their variance-covariance matrix ( \(\sum _{\hat x} \) ) can be obtained as in the “standard” adjustment whereA has the full column rank, supplemented with constraints, \(C\hat x = w\) , whereC is the constraint matrix andw is sometimes called the “constant vector”. In this analysis only the inner adjustment constraints are considered, whereC has the full row rank equal to the rank deficiency ofA, andAC ^T =0. Perhaps the most important outcome points to the three kinds of results

1. A general least-squares solution where both \(\hat x\) and \(\sum _{\hat x} \) are indeterminate corresponds tow=arbitrary random vector.
2. The minimum trace (least-squares) solution where \(\hat x\) is indeterminate but \(\sum _{\hat x} \) is detemined (and trace \(\sum _{\hat x} \) corresponds tow=arbitrary constant vector.
3. The minimum norm (least-squares) solution where both \(\hat x\) and \(\sum _{\hat x} \) are determined (and norm \(\hat x\) , trace \(\sum _{\hat x} \) corresponds tow?0

     

GOCE gravitational gradiometry

GOCE is the first gravitational gradiometry satellite mission. Gravitational gradiometry is the measurement of the second derivatives of the gravitational potential. The nine derivatives form a 3 × 3 matrix, which in geodesy is referred to as Marussi tensor. From the basic properties of the gravitational field, it follows that the matrix is symmetric and trace free. The latter property corresponds to Laplace equation, which gives the theoretical foundation of its representation in terms of spherical harmonic or Fourier series. At the same time, it provides the most powerful quality check of the actual measured gradients. GOCE gradiometry is based on the principle of differential accelerometry. As the satellite carries out a rotational motion in space, the accelerometer differences contain angular effects that must be removed. The GOCE gradiometer provides the components V _xx , V _yy , V _zz and V _xz with high precision, while the components V _xy and V _yz are of low precision, all expressed in the gradiometer reference frame. The best performance is achieved inside the measurement band from 5 × 10^–3 to 0.1 Hz. At lower frequencies, the noise increases with 1/f and is superimposed by cyclic distortions, which are modulated from the orbit and attitude motion into the gradient measurements. Global maps with the individual components show typical patterns related to topographic and tectonic features. The maps are separated into those for ascending and those for descending tracks as the components are expressed in the instrument frame. All results are derived from the measurements of the period from November to December 2009. While the components V _xx and V _yy reach a noise level of about \({10\;\rm{\frac{mE}{\sqrt{Hz}}}}\), that of V _zz and V _xz is about \({20\; \rm{\frac{mE}{\sqrt{Hz}}}}\). The cause of the latter’s higher noise is not yet understood. This is also the reason why the deviation from the Laplace condition is at the \({20 \;\rm{\frac{mE}{\sqrt{Hz}}}}\) level instead of the originally planned \({11\;\rm{\frac{mE}{\sqrt{Hz}}}}\). Each additional measurement cycle will improve the accuracy and to a smaller extent also the resolution of the spherical harmonic coefficients derived from the measured gradients.     

La determination du rayon terrestre par J. PICARD en 1669–1671

The French astronomerJean PICARD (1620–1682) was certainly one of the leading scientists of his time. Friend of Huygens, of Hevelius, of Oldenburg, master of Römer, indefatigable traveller, he played a very important part in the development of positional astronomy and geodesy.

- He first, had the idea of comparing the length units to a reproductible physical quantity, namely the length of the one second pendulum at Paris, and measured that length.

- He conceived the first cross wire telescopes and adapted them on geodetic and astronomical instruments of his own, used throughout one century until 1780.

- He obtained the first really reliable value of the earth radius, in his famous measurement of the meridional arc PARIS-AMIENS, being the original cell of the French triangulations.

The following article is devoted to a recomputation and evaluation of the accuracy of that work, as compared with further operations, but independently concludes that this achievement gave the necessary impulse to the development of geodesy in France and probably abroad.     

Pre-zero-epoch covariance matrix in sequential analysis of deformations

A sequential adjustment procedure is proposed for the direct estimation of point—velocities in deformation analysis networks. At any intermediate stage of the adjustment the up-to-date covariance matrix of those velocities tells the evolving story of the network in terms of solvability and reliability. A pre-zero-epoch covariance matrix is utilized for a smooth and flexible treatment of two characteristic problems of deformation analysis:

- high turnover of points in the network

- processing variable and generally incomplete observational batches.

A small numerical example is presented at the end as an illustration.     

Remote Sensing Data Acquisition,Platforms and Sensor Requirements

Although data available from various earth observation systems have been routinely used in many resource applications, however there have been gaps, and data needs of applications at different levels of details have not been met. There is a growing demand for availability of data at higher repetivity, at higher spatial resolution, in more and narrower spectral bands etc. Some of the thrust areas of applications particularly in the Indian context are;

1. Management of natural resources to ensure sustainable increase in agricultural production,
2. Study the state of the environment, its monitoring and assessment of the impact of. various development actions on the environment,
3. Updating and generation of large scale topographical maps.
4. Exploration/exploitation of marine and mineral resources and
5. Operational meteorology and studying various land and oceanic processes to understand/predict global climate changes.

Each of these thrust area of application has many components, related to basic resource areas such as agriculture, forestry, water resources, minerals, marine resources etc. and the field of cartography. Observational requirements for major applications have been summarized as under. Monitoring vegetation health from space remains the most important observational parameter with applications, in agriculture, forestry, environment, hydrology etc. Vegetation extent, quantity and temporal changes are the three main requirements which are not fully realized with RS data available. Vegetation productivity, forest biomass, canopy moisture status, canopy biogeochemistry are some examples. Crop production forecasting is an important application area. Remotely sensed data has been used for identification of crops and their acreage estimation. Fragmented holdings, large spread in crop calendars and different management practices continue to pose a challenge lo remote sensing. Remotely sensed data at much higher spatial resolution than hitherto available as well as at greater repetivity are required to meet this need. Non-availability of cloud-free data in the kharif season is one of the serious problems in operational use of remote sensing for crop inventory. Synthetic aperture radar data al X & Ku bands is necessary to meet this demand. Nutrient stress/disease detection requires observations in narrow spectral bands. In case of forestry applications, multispectral data at high spatial resolution of the order of 5 to 10 metres is required to make working plans at forest compartment level. Observations from space for deriving tree height are required for volume estimation. Observations in the middle infrared region would greatly enhance capability of satellite remote sensing in forest fire detection. Temporal, spatial and spectral observational requirements in various applications on vegetation viewing are diverse, as they address processes at different spatial and time scales. Hence, it would be worthwhile to address this issue in three broad categories. a) Full coverage, moderate spatial resolution with high repetivity (drought, large scale deforestation, forest phenology....). b) Full coverage, moderate to high spatial resolution and high repetivity (crop forecasting, vegetation productivity). c) Selected viewing at high spatial resolution, moderate to high repetivity and with new dimensions to imaging (narrow spectral bands, different viewing angles). A host of agrometeorological parameters are needed to be measured from space for their effective use in development of yield models. Estimation of root-zone soil moisture is an important area requiring radar measurements from space. Surface meteorological observations from space at the desired spatial and temporal distributions has not developed because of heavy demands placed on the sensor as well as analytical operational models. Agrometeorology not only provides quantitative inputs to other applications such as crop forecasting, hydrological models but also could be used for farmer advisory services by local bodies. Mineral exploration requires information on geological structures, geomorphology and lithology. Surface manifestation over localized regions requires large scale mapping while the lithology can be deciphered from specific narrow bands in visible. NIR, MIR and TIR regions. Sensors identified for mapping/cartography in conjunction with imaging spectrometer would seem to cover requirements of this application. Narrow spectral bands in the short regions which provide diagnostics of relevant geological phenomenon are necessary for mineral exploration. Thermal inertia measurements help in better discrimination of different rock units. Measurements from synthetic aperture data which would provide information on geological structures and geomorphology are necessary for mineral exploration. The applications related to marine environment fall in three major areas: (i) Ocean colour and productivity, biological resources; (ii) Land-ocean interface, this includes coastal landforms, bathymetry, littoral transport processes, etc. and; (iii) Physical oceanography, sea surface temperature, winds, wave spectra, energy and mass exchange between atmosphere and ocean. Measurement of chlorophyll concentration accurately on daily basis, sea surface temperature with an accuracy of 0.5 °K. and information on current patterns arc required for developing better fishery forecast models. Improved spatial resolution data are desirable for studying sediment and other coastal processes. Cartography is another important application area. The major problems encountered in relation to topographic map updation are location and geometric accuracy and information content. Two most important requirements for such an application are high spatial resolution data of 1 to 2 metre and stereo capability to provide vertical resolution of 1 metre. This requirement places stringent demands on the sensor specifications, geometric processing, platform stability and automated digital cartography. The requirements for the future earth observation systems based on different application needs can be summarized as follows:

1. Moderate spatial resolution (l50-300m), high repetivity (2 Days), minimum set of spectral bands (VIS, NIR, MIR. TIR) full coverage.
2. Moderate to high spatial resolution (20-40m), high repetivity (4-6 Days), spectral bands (VIS, MR, MIR, TIR) full coverage.
3. High spatial resolution (5-10m) muitispectral data with provision for selecting specific narrow bands (VIS, N1R. MIR), viewing from different angles.
4. Synthetic aperture radar operating in at least two frequencies (C, X, Ku), two incidence angles/polarizations, moderate to high spatial resolution (20-40m), high repetivity (4-6 Days).
5. Very high spatial resolution (1-2m) data in panchromatic band to provide terrain details at cadastral level (1:10,000).
6. Stereo capability (1-2m height resolution) to help planning/execution of development plans.
7. Moderate resolution sensor operating in VIS, NIR, MIR on a geostationary platform for observations at different sun angles necessary for the development of canopy reflectance inversion models.
8. Diurnal (at least two i.e. pre-dawn and noon) temperature measurements of the earth surface.
9. Ocean colour monitor with daily coverage.
10. Multi-frequency microwave radiometer, scatterometer. altimeter, atmospheric sounder, etc.

     

Investigations on the accuracy concerning the derivation of absolute gravimetric height anomalies and deflections of the vertical based on the theory of Molodensky

The investigations refer to the compartment method by using mean terrestrial free air anomalies only. Three main error influences of remote areas (distance from the fixed point >9°) on height anomalies and deflections of the vertical are being regarded:

1. The prediction errors of mean terrestrial free air anomalies have the greatest influence and amount to about ±0″.2 in each component for deflections of the vertical and to ±3 m for height anomalies;
2. The error of the compartment method, which originates from converting the integral formulas of Stokes and Vening-Meinesz into summation formulas, can be neglected if the anomalies for points and gravity profiles are compiled to 5°×5° mean values.
3. The influences of the mean gravimetric correction terms of Arnold—estimated for important mountains of the Earth by means of an approximate formula—on height anomalies may amount to 1–2 m and on deflections of the vertical to 0″0.5–0″.1, and, therefore, they have to be taken into account for exact calculations.

The computations of errors are carried out using a global covariance function of point free air anomalies.     

Geology of the area around Katta,sindhudurg district,Maharashtra. Based on aerospace data

Geological studies of the area around Katta, in the southern part of the Ratnagiri District of Maharashtra, were carried out with the help of visual remote sensing techniques using LANDSAT imageries on 1:250,000 scale and aerial photographs on 1:60,000 scale. The major stratigraphic units represented in the area under study are the Archean Complex, Kaladgi Supergroup, Deccan Trap, Laterite and Alluvium. The Kaladgis unconformably overlie the Archean metasediments and also at places exhibit faulted contacts with the latter. The major part of the area is covered by a thick evergreen vegetation. The interpretation followed by field work and laboratory work revealed the following:

1. The different lithologic units could be delineated on the aerial photographs.
2. Different lineaments marked on the imagery were found to be due either to faults or fracture zones. Some of the older faults appear to have been rejuvenated after the formation of the laterites.
3. Some of the lithologic horizons can be identified on the Landsat imagery by virtue of their spatial signatures.

These studies indicate that even in the area covered with thick vegetation, aerospace imagery in appropriate band and data scale can provide significant geological information.     

Combining consecutive short arcs into long arcs for precise and efficient GPS Orbit Determination

The final products of theCODE Analysis Center (Center for Orbit Determination in Europe) of theInternational GPS Service for Geodynamics (IGS) stem fromoverlapping 3-day-arcs. Until 31 December, 1994 these long arcs were computedfrom scratch, i.e. by processing three days of observations of about 40 stations (by mid 1995 about 60 stations were used) of the IGS Global Network in our parameter estimation program GPSEST. Becauseone-day-arcs have to be produced first (for the purpose of error detection etc.) the actual procedure was rather time-consuming. In the present article we develop the mathematical tools necessary to form long arcs based on the normal equation systems of consecutive short arcs (one-day-solutions in the case of CODE). The procedure in its simplest version is as follows:

• Each short arc is described bysix initial conditions and a number of dynamical orbit parameters (e.g. radiation pressure parameters). The resulting long arc in turn shall be based onn consecutive short arcs and described bysix initial conditions and again the same number of dynamical parameters as in the short arcs..
• By asking position and velocity to be continuous at the boundaries of the short arcs we obtain a long arc which is actually defined by one set of initial conditions andn sets of dynamical parameters (ifn short arcs are combined)..
• By asking the dynamical parameters to be identical in consecutive short arcs, the resulting long arc is characterized by exactly the same number of orbit parameters as each of the short arcs.
• This procedure isnot yet optimized becauseformally all n sets of orbit parameters have to be set up and solved for in the long arc solution (although they are not independent). In order to allow for an optimized solution we derive all necessary relations to eliminate the unnecessary parameters in the combination. Each long arc is characterized by the actual number of independent orbit parameters. The resulting procedure isvery efficient.

From the point of view of the result the new procedure iscompletely equivalent to an actual re-evaluation of all observations pertaining to the long arc. It is much more efficient and flexible, however because it allows us to construct 2-day-arcs, 3-day-arcs, etc. based on the previously stored daily normal equation systems without requiring much additional CPU time. The theory is developed in the first four sections. Technical aspects are dealt with in appendices A and B. The actual implementation into the Bernese GPS Software system and test results are given in section 5.     

Methodology and use of tensor invariants for satellite gravity gradiometry

Although its use is widespread in several other scientific disciplines, the theory of tensor invariants is only marginally adopted in gravity field modeling. We aim to close this gap by developing and applying the invariants approach for geopotential recovery. Gravitational tensor invariants are deduced from products of second-order derivatives of the gravitational potential. The benefit of the method presented arises from its independence of the gradiometer instrument’s orientation in space. Thus, we refrain from the classical methods for satellite gravity gradiometry analysis, i.e., in terms of individual gravity gradients, in favor of the alternative invariants approach. The invariants approach requires a tailored processing strategy. Firstly, the non-linear functionals with regard to the potential series expansion in spherical harmonics necessitates the linearization and iterative solution of the resulting least-squares problem. From the computational point of view, efficient linearization by means of perturbation theory has been adopted. It only requires the computation of reference gravity gradients. Secondly, the deduced pseudo-observations are composed of all the gravitational tensor elements, all of which require a comparable level of accuracy. Additionally, implementation of the invariants method for large data sets is a challenging task. We show the fundamentals of tensor invariants theory adapted to satellite gradiometry. With regard to the GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite gradiometry mission, we demonstrate that the iterative parameter estimation process converges within only two iterations. Additionally, for the GOCE configuration, we show the invariants approach to be insensitive to the synthesis of unobserved gravity gradients.     

On the use of heterogeneous noisy data in spectral gravity field modeling methods

Spectral methods have been a standard tool in physical geodesy applications over the past decade. Typically, they have been used for the efficient evaluation of convolution integrals, utilizing homogeneous, noise-free gridded data. This paper answers the following three questions:

1. Can data errors be propagated into the results?
2. Can heterogeneous data be used?
3. Is error propagation possible with heterogeneous data?

The answer to the above questions is yes and is illustrated for the case of two input data sets and one output. Firstly, a solution is obtained in the frequency domain using the theory of a two-input, single-output system. The assumption here is that both the input signals and their errors are stochastic variables with known PSDs. The solution depends on the ratios of the error PSD and the signal PSD, i.e., the noise-to-signal ratios of the two inputs. It is shown that, when the two inputs are partially correlated, this solution is equivalent to stepwise collocation. Secondly, a solution is derived in the frequency domain by a least-squares adjustment of the spectra of the input data. The assumption is that only the input errors are stochastic variables with known power spectral density functions (PSDs). It is shown that the solution depends on the ratio of the noise PSDs. In both cases, there exists the non-trivial problem of estimating the input noise PSDs, given that we only have available the error variances of the data. An effective but non-rigorous way of overcoming this problem in practice is to approximate the noise PSDs by simple stationary models.     

Some problems related to satellite gradiometry

Observation equations for a gradiometer in inertial flight are described, and the effects of orbit and attitude errors are modelled, showing that, when high precision direction observations are missing, the use of rotation invariant equations is advantageous. The invariant form of the equations is analysed by demonstrating the necessity of using as observables the eigenvalues of the gravitational tensor instead of classic invariants. The singularity of the problem in spherical approximation is shown, and consequently the necessity of taking into accountJ ₂ in the reference field used in the linearization.     

The gravity field model IGGT_R1 based on the second invariant of the GOCE gravitational gradient tensor

Based on tensor theory, three invariants of the gravitational gradient tensor (IGGT) are independent of the gradiometer reference frame (GRF). Compared to traditional methods for calculation of gravity field models based on the gravity field and steady-state ocean circulation explorer (GOCE) data, which are affected by errors in the attitude indicator, using IGGT and least squares method avoids the problem of inaccurate rotation matrices. The IGGT approach as studied in this paper is a quadratic function of the gravity field model’s spherical harmonic coefficients. The linearized observation equations for the least squares method are obtained using a Taylor expansion, and the weighting equation is derived using the law of error propagation. We also investigate the linearization errors using existing gravity field models and find that this error can be ignored since the used a-priori model EIGEN-5C is sufficiently accurate. One problem when using this approach is that it needs all six independent gravitational gradients (GGs), but the components \(V_{xy}\) and \(V_{yz}\) of GOCE are worse due to the non-sensitive axes of the GOCE gradiometer. Therefore, we use synthetic GGs for both inaccurate gravitational gradient components derived from the a-priori gravity field model EIGEN-5C. Another problem is that the GOCE GGs are measured in a band-limited manner. Therefore, a forward and backward finite impulse response band-pass filter is applied to the data, which can also eliminate filter caused phase change. The spherical cap regularization approach (SCRA) and the Kaula rule are then applied to solve the polar gap problem caused by GOCE’s inclination of \(96.7^{\circ }\). With the techniques described above, a degree/order 240 gravity field model called IGGT_R1 is computed. Since the synthetic components of \(V_{xy}\) and \(V_{yz}\) are not band-pass filtered, the signals outside the measurement bandwidth are replaced by the a-priori model EIGEN-5C. Therefore, this model is practically a combined gravity field model which contains GOCE GGs signals and long wavelength signals from the a-priori model EIGEN-5C. Finally, IGGT_R1’s accuracy is evaluated by comparison with other gravity field models in terms of difference degree amplitudes, the geostrophic velocity in the Agulhas current area, gravity anomaly differences as well as by comparison to GNSS/leveling data.     

Self-consistent treatment of tidal variations in the geocenter for precise orbit determination

We show that the current levels of accuracy being achieved for the precise orbit determination (POD) of low-Earth orbiters demonstrate the need for the self-consistent treatment of tidal variations in the geocenter. Our study uses as an example the POD of the OSTM/Jason-2 satellite altimeter mission based upon Global Positioning System (GPS) tracking data. Current GPS-based POD solutions are demonstrating root-mean-square (RMS) radial orbit accuracy and precision of \({<}1\)  cm and 1 mm, respectively. Meanwhile, we show that the RMS of three-dimensional tidal geocenter variations is \({<}6\)  mm, but can be as large as 15 mm, with the largest component along the Earth’s spin axis. Our results demonstrate that GPS-based POD of Earth orbiters is best performed using GPS satellite orbit positions that are defined in a reference frame whose origin is at the center of mass of the entire Earth system, including the ocean tides. Errors in the GPS-based POD solutions for OSTM/Jason-2 of \({<}4\)  mm (3D RMS) and \({<}2\)  mm (radial RMS) are introduced when tidal geocenter variations are not treated consistently. Nevertheless, inconsistent treatment is measurable in the OSTM/Jason-2 POD solutions and manifests through degraded post-fit tracking data residuals, orbit precision, and relative orbit accuracy. For the latter metric, sea surface height crossover variance is higher by \(6~\hbox {mm}^{2}\) when tidal geocenter variations are treated inconsistently.     

Homogeneous reprocessing of GPS,GLONASS and SLR observations

The International GNSS Service (IGS) provides operational products for the GPS and GLONASS constellation. Homogeneously processed time series of parameters from the IGS are only available for GPS. Reprocessed GLONASS series are provided only by individual Analysis Centers (i. e. CODE and ESA), making it difficult to fully include the GLONASS system into a rigorous GNSS analysis. In view of the increasing number of active GLONASS satellites and a steadily growing number of GPS+GLONASS-tracking stations available over the past few years, Technische Universität Dresden, Technische Universität München, Universität Bern and Eidgenössische Technische Hochschule Zürich performed a combined reprocessing of GPS and GLONASS observations. Also, SLR observations to GPS and GLONASS are included in this reprocessing effort. Here, we show only SLR results from a GNSS orbit validation. In total, 18 years of data (1994–2011) have been processed from altogether 340 GNSS and 70 SLR stations. The use of GLONASS observations in addition to GPS has no impact on the estimated linear terrestrial reference frame parameters. However, daily station positions show an RMS reduction of 0.3 mm on average for the height component when additional GLONASS observations can be used for the time series determination. Analyzing satellite orbit overlaps, the rigorous combination of GPS and GLONASS neither improves nor degrades the GPS orbit precision. For GLONASS, however, the quality of the microwave-derived GLONASS orbits improves due to the combination. These findings are confirmed using independent SLR observations for a GNSS orbit validation. In comparison to previous studies, mean SLR biases for satellites GPS-35 and GPS-36 could be reduced in magnitude from \(-35\) and \(-38\)  mm to \(-12\) and \(-13\)  mm, respectively. Our results show that remaining SLR biases depend on the satellite type and the use of coated or uncoated retro-reflectors. For Earth rotation parameters, the increasing number of GLONASS satellites and tracking stations over the past few years leads to differences between GPS-only and GPS+GLONASS combined solutions which are most pronounced in the pole rate estimates with maximum 0.2 mas/day in magnitude. At the same time, the difference between GLONASS-only and combined solutions decreases. Derived GNSS orbits are used to estimate combined GPS+GLONASS satellite clocks, with first results presented in this paper. Phase observation residuals from a precise point positioning are at the level of 2 mm and particularly reveal poorly modeled yaw maneuver periods.     

Reducing the draconitic errors in GNSS geodetic products

Systematic errors at harmonics of the GPS draconitic year have been found in diverse GPS-derived geodetic products like the geocenter $Z$ -component, station coordinates, $Y$ -pole rate and orbits (i.e. orbit overlaps). The GPS draconitic year is the repeat period of the GPS constellation w.r.t. the Sun which is about 351 days. Different error sources have been proposed which could generate these spurious signals at the draconitic harmonics. In this study, we focus on one of these error sources, namely the radiation pressure orbit modeling deficiencies. For this purpose, three GPS+GLONASS solutions of 8 years (2004–2011) were computed which differ only in the solar radiation pressure (SRP) and satellite attitude models. The models employed in the solutions are: (1) the CODE (5-parameter) radiation pressure model widely used within the International GNSS Service community, (2) the adjustable box-wing model for SRP impacting GPS (and GLONASS) satellites, and (3) the adjustable box-wing model upgraded to use non-nominal yaw attitude, specially for satellites in eclipse seasons. When comparing the first solution with the third one we achieved the following in the GNSS geodetic products. Orbits: the draconitic errors in the orbit overlaps are reduced for the GPS satellites in all the harmonics on average 46, 38 and 57 % for the radial, along-track and cross-track components, while for GLONASS satellites they are mainly reduced in the cross-track component by 39 %. Geocenter $Z$ -component: all the odd draconitic harmonics found when the CODE model is used show a very important reduction (almost disappearing with a 92 % average reduction) with the new radiation pressure models. Earth orientation parameters: the draconitic errors are reduced for the $X$ -pole rate and especially for the $Y$ -pole rate by 24 and 50 % respectively. Station coordinates: all the draconitic harmonics (except the 2nd harmonic in the North component) are reduced in the North, East and Height components, with average reductions of 41, 39 and 35 % respectively. This shows, that part of the draconitic errors currently found in GNSS geodetic products are definitely induced by the CODE radiation pressure orbit modeling deficiencies.     

Estimation of satellite position,clock and phase bias corrections

Precise point positioning with integer ambiguity resolution requires precise knowledge of satellite position, clock and phase bias corrections. In this paper, a method for the estimation of these parameters with a global network of reference stations is presented. The method processes uncombined and undifferenced measurements of an arbitrary number of frequencies such that the obtained satellite position, clock and bias corrections can be used for any type of differenced and/or combined measurements. We perform a clustering of reference stations. The clustering enables a common satellite visibility within each cluster and an efficient fixing of the double difference ambiguities within each cluster. Additionally, the double difference ambiguities between the reference stations of different clusters are fixed. We use an integer decorrelation for ambiguity fixing in dense global networks. The performance of the proposed method is analysed with both simulated Galileo measurements on E1 and E5a and real GPS measurements of the IGS network. We defined 16 clusters and obtained satellite position, clock and phase bias corrections with a precision of better than 2 cm.     

An a priori solar radiation pressure model for the QZSS Michibiki satellite

It has been noted that the satellite laser ranging (SLR) residuals of the Quasi-Zenith Satellite System (QZSS) Michibiki satellite orbits show very marked dependence on the elevation angle of the Sun above the orbital plane (i.e., the \(\beta \) angle). It is well recognized that the systematic error is caused by mismodeling of the solar radiation pressure (SRP). Although the error can be reduced by the updated ECOM SRP model, the orbit error is still very large when the satellite switches to orbit-normal (ON) orientation. In this study, an a priori SRP model was established for the QZSS Michibiki satellite to enhance the ECOM model. This model is expressed in ECOM’s D, Y, and B axes (DYB) using seven parameters for the yaw-steering (YS) mode, and additional three parameters are used to compensate the remaining modeling deficiencies, particularly the perturbations in the Y axis, based on a redefined DYB for the ON mode. With the proposed a priori model, QZSS Michibiki’s precise orbits over 21 months were determined. SLR validation indicated that the systematic \(\beta \)-angle-dependent error was reduced when the satellite was in the YS mode, and better than an 8-cm root mean square (RMS) was achieved. More importantly, the orbit quality was also improved significantly when the satellite was in the ON mode. Relative to ECOM and adjustable box-wing model, the proposed SRP model showed the best performance in the ON mode, and the RMS of the SLR residuals was better than 15 cm, which was a two times improvement over the ECOM without a priori model used, but was still two times worse than the YS mode.     

Improved method for estimating the inter-frequency satellite clock bias of triple-frequency GPS

Considering the contribution of the hardware biases to the estimated clock errors, an improved method for estimating the satellite inter-frequency clock bias (IFCB) is presented, i.e., the difference in the satellite clock error as computed from ionospheric-free pseudorange and carrier phase observations using L1/L2 and P1/P2 versus L1/L5 and P1/P5. The IFCB is composed of a constant and a variable part. The constant part is the inter-frequency hardware bias (IFHB). It contains the satellite and receiver hardware delays and can be expressed as a function of the DCBs [DCB (P1 ? P2) and DCB (P1 ? P5)]. When a reference satellite is selected, the satellite IFHB can be computed but is biased by a reference satellite IFHB. This bias will not affect the utilization of IFCB in positioning since it can be absorbed by the receiver clock error. Triple-frequency observations of 30 IGS stations between June 1, 2013, and May 31, 2014, were processed to show the variations of the IFHB. The IFHB values show a long-term variation with time. When a linear and a fourth-order harmonic function are used to model the estimated IFCB, which contains contributions of the hardware delays and clock errors, the results show that 89 % of the IFCB can be corrected given the current five triple-frequency GPS satellites with the averaged fitting RMS of 1.35 cm. Five days of data are processed to test the estimated satellite clock errors using the strategy presented. The residuals of P1/P5 and L1/L5 have a STD of <0.27 m and 0.97 cm, respectively. In addition, most predicted satellite IFCBs reach an accuracy of centimeter level and its mean accuracy of 5 days is better than 7 cm.