Adaptive filtering of GOCE-derived gravity gradients of the disturbing potential in the context of the space-wise approach

Filtering and signal processing techniques have been widely used in the processing of satellite gravity observations to reduce measurement noise and correlation errors. The parameters and types of filters used depend on the statistical and spectral properties of the signal under investigation. Filtering is usually applied in a non-real-time environment. The present work focuses on the implementation of an adaptive filtering technique to process satellite gravity gradiometry data for gravity field modeling. Adaptive filtering algorithms are commonly used in communication systems, noise and echo cancellation, and biomedical applications. Two independent studies have been performed to introduce adaptive signal processing techniques and test the performance of the least mean-squared (LMS) adaptive algorithm for filtering satellite measurements obtained by the gravity field and steady-state ocean circulation explorer (GOCE) mission. In the first study, a Monte Carlo simulation is performed in order to gain insights about the implementation of the LMS algorithm on data with spectral behavior close to that of real GOCE data. In the second study, the LMS algorithm is implemented on real GOCE data. Experiments are also performed to determine suitable filtering parameters. Only the four accurate components of the full GOCE gravity gradient tensor of the disturbing potential are used. The characteristics of the filtered gravity gradients are examined in the time and spectral domain. The obtained filtered GOCE gravity gradients show an agreement of 63–84 mEötvös (depending on the gravity gradient component), in terms of RMS error, when compared to the gravity gradients derived from the EGM2008 geopotential model. Spectral-domain analysis of the filtered gradients shows that the adaptive filters slightly suppress frequencies in the bandwidth of approximately 10–30 mHz. The limitations of the adaptive LMS algorithm are also discussed. The tested filtering algorithm can be connected to and employed in the first computational steps of the space-wise approach, where a time-wise Wiener filter is applied at the first stage of GOCE gravity gradient filtering. The results of this work can be extended to using other adaptive filtering algorithms, such as the recursive least-squares and recursive least-squares lattice filters.     

Group delay variations of GPS transmitting and receiving antennas

GPS code pseudorange measurements exhibit group delay variations at the transmitting and the receiving antenna. We calibrated C1 and P2 delay variations with respect to dual-frequency carrier phase observations and obtained nadir-dependent corrections for 32 satellites of the GPS constellation in early 2015 as well as elevation-dependent corrections for 13 receiving antenna models. The combined delay variations reach up to 1.0 m (3.3 ns) in the ionosphere-free linear combination for specific pairs of satellite and receiving antennas. Applying these corrections to the code measurements improves code/carrier single-frequency precise point positioning, ambiguity fixing based on the Melbourne–Wübbena linear combination, and determination of ionospheric total electron content. It also affects fractional cycle biases and differential code biases.     

A spatial error model with continuous random effects and an application to growth convergence

We propose a spatial error model with continuous random effects based on Matérn covariance functions and apply this model for the analysis of income convergence processes (\(\beta \)-convergence). The use of a model with continuous random effects permits a clearer visualization and interpretation of the spatial dependency patterns, avoids the problems of defining neighborhoods in spatial econometrics models, and allows projecting the spatial effects for every possible location in the continuous space, circumventing the existing aggregations in discrete lattice representations. We apply this model approach to analyze the economic growth of Brazilian municipalities between 1991 and 2010 using unconditional and conditional formulations and a spatiotemporal model of convergence. The results indicate that the estimated spatial random effects are consistent with the existence of income convergence clubs for Brazilian municipalities in this period.     

GLONASS phase bias estimation and its PPP ambiguity resolution using homogeneous receivers

Integer ambiguity resolution (IAR) appreciably improves the position accuracy and shortens the convergence time of precise point positioning (PPP). However, while many studies are limited to GPS, there is a need to investigate the performance of GLONASS PPP ambiguity resolution. Unfortunately, because of the frequency-division multiple-access strategy of GLONASS, GLONASS PPP IAR faces two obstacles. First, simultaneously observed satellites operate at different wavelengths. Second and most importantly, distinct inter-frequency bias (IFB) exists between different satellites. For the former, we adopt an undifferenced method for uncalibrated phase delay (UPD) estimation and proposed an undifferenced PPP IAR strategy. We select a set of homogeneous receivers with identical receiver IFB to perform UPD estimation and PPP IAR. The code and carrier phase IFBs can be absorbed by satellite wide-lane and narrow-lane UPDs, respectively, which is in turn consistent with PPP IAR using the same type of receivers. In order to verify the method, we used 50 stations to generate satellite UPDs and another 12 stations selected as users to perform PPP IAR. We found that the GLONASS satellite UPDs are stable in time and space and can be estimated with high accuracy and reliability. After applying UPD correction, 91 % of wide-lane ambiguities and 99 % of narrow-lane ambiguities are within (?0.15, +0.15) cycles of the nearest integer. After ambiguity resolution, the 2-hour static PPP accuracy improves from (0.66, 1.42, 1.55) cm to (0.38, 0.39, 1.39) cm for the north, east, and up components, respectively.     

Error analysis of the 3D similarity coordinate transformation

The 3D similarity coordinate transformation with the Gauss–Helmert error model is investigated. The first-order error analysis of an analytical least-squares solution to this problem is developed in detail. While additive errors are assumed in the translation and scale estimates, a 3 × 1 multiplicative error vector is defined to effectively parameterize the rotation matrix estimation error. The propagation of the errors in the coordinate measurements to the errors in the estimated transformation parameters is derived step-by-step, and the formulae for calculating the variance–covariance matrix of the estimated parameters are presented.     

Global morphology of ionospheric F-layer scintillations using FS3/COSMIC GPS radio occultation data

We report on the FormoSat-3/Constellation Observing System for Meteorology, Ionosphere and Climate (FS3/COSMIC) limb-viewing observations of GPS L-band scintillations since mid-2006 and propose to study global F-layer irregularity morphology. The FS3/COSMIC has generally performed more than 1000 ionospheric radio occultation (RO) observations per day. We reprocess 1-Hz amplitude data and obtain complete limb-viewing profiles of the undersampling (sampling frequency lower than Fresnel frequency) S4 scintillation index from about 80% of the RO observations. There are a few percent of FS3/COSMIC RO observations having greater than 0.09 undersampling S4max values on average. However, seven identified areas, Central Pacific Area (?20° to 20° dip latitude, 160°E–130°W), South American Area (?20° to 20° dip latitude, 100°W–30°W), African Area (?20° to 20° dip latitude, 30°W–50°E), European Area (30°–55°N, 0°–55°E), Japan Sea Area (35°–55°N, 120°–150°E), Arctic Area (>65° dip latitude), and Antarctic Area (<?65° dip latitude), have been designated to have a much higher percentage of strong limb-viewing L-band scintillations. During the years in most of the last sunspot cycle from mid-2006 to the end 2014, the scintillation climatology, namely, its variations with each identified area, season, local time, magnetic activity, and solar activity, have been documented.     

GLONASS pseudorange inter-channel biases considerations when jointly estimating GPS and GLONASS clock offset

GLONASS clock offset estimation is affected by the inter-channel biases (ICBs) caused by frequency division multiple access technique. The effect of ICBs on joint GPS/GLONASS clock offset estimation is analyzed. An efficient approach for joint estimation of GPS/GLONASS satellite clock offset is applied to the generation of 30-s clock offset products. During the estimation, the following three ICB handling strategies were tested: calculating ICBs for each GLONASS signal channel, calculating ICBs for each GLONASS satellite and neglecting ICBs. The behavior of ICBs under different strategies was statistically stable. Subsequently, the clock offset products using different ICB strategies were evaluated. The evaluation shows that consideration of the ICB is important when estimating the clock offset. Furthermore, estimating one ICB for each GLONASS satellite is better than estimating one for each GLONASS signal channel because, with the former strategy, the clock offset products behave more smoothly and have higher accuracy compared with products from the International GNSS Service Analysis Center. In addition, precise point positioning, using clock offsets based on one ICB for each GLONASS satellite, has the highest positioning accuracy.     

Global ionosphere map constructed by using total electron content from ground-based GNSS receiver and FORMOSAT-3/COSMIC GPS occultation experiment

Effects of rapidly changing ionospheric weather are critical in high accuracy positioning, navigation, and communication applications. A system used to construct the global total electron content (TEC) distribution for monitoring the ionospheric weather in near-real time is needed in the modern society. Here we build the TEC map named Taiwan Ionosphere Group for Education and Research (TIGER) Global Ionospheric Map (GIM) from observations of ground-based GNSS receivers and space-based FORMOSAT-3/COSMIC (F3/C) GPS radio occultation observations using the spherical harmonic expansion and Kalman filter update formula. The TIGER GIM (TGIM) will be published in near-real time of 4-h delay with a spatial resolution of 2.5° in latitude and 5° in longitude and a high temporal resolution of every 5 min. The F3/C TEC results in an improvement on the GIM of about 15.5%, especially over the ocean areas. The TGIM highly correlates with the GIMs published by other international organizations. Therefore, the routinely published TGIM in near-real time is not only for communication, positioning, and navigation applications but also for monitoring and scientific study of ionospheric weathers, such as magnetic storms and seismo-ionospheric anomalies.     

Assessment of PPP integer ambiguity resolution using GPS,GLONASS and BeiDou (IGSO,MEO) constellations

Although integer ambiguity resolution (IAR) can improve positioning accuracy considerably and shorten the convergence time of precise point positioning (PPP), it requires an initialization time of over 30 min. With the full operation of GLONASS globally and BDS in the Asia–Pacific region, it is necessary to assess the PPP–IAR performance by simultaneous fixing of GPS, GLONASS, and BDS ambiguities. This study proposed a GPS + GLONASS + BDS combined PPP–IAR strategy and processed PPP–IAR kinematically and statically using one week of data collected at 20 static stations. The undifferenced wide- and narrow-lane fractional cycle biases for GPS, GLONASS, and BDS were estimated using a regional network, and undifferenced PPP ambiguity resolution was performed to assess the contribution of multi-GNSSs. Generally, over 99% of a posteriori residuals of wide-lane ambiguities were within ±0.25 cycles for both GPS and BDS, while the value was 91.5% for GLONASS. Over 96% of narrow-lane residuals were within ±0.15 cycles for GPS, GLONASS, and BDS. For kinematic PPP with a 10-min observation time, only 16.2% of all cases could be fixed with GPS alone. However, adding GLONASS improved the percentage considerably to 75.9%, and it reached 90.0% when using GPS + GLONASS + BDS. Not all epochs could be fixed with a correct set of ambiguities; therefore, we defined the ratio of the number of epochs with correctly fixed ambiguities to the number of all fixed epochs as the correct fixing rate (CFR). Because partial ambiguity fixing was used, when more than five ambiguities were fixed correctly, we considered the epoch correctly fixed. For the small ratio criteria of 2.0, the CFR improved considerably from 51.7% for GPS alone, to 98.3% when using GPS + GLONASS + BDS combined solutions.