Benedikt Soja

Application of Kalman filtering in VLBI data analysis

In this paper, we demonstrate the advantage of applying a Kalman filter for the parameter estimation in very-long-baseline interferometry (VLBI) data analysis. We present the implementation of a Kalman filter in the VLBI software VieVS@GFZ. The performance is then investigated by looking at the accuracy obtained for various parameters, like baseline lengths, Earth Orientation Parameters, radio source coordinates, and tropospheric delays. The results are compared to those obtained when the classical least squares method (LSM) is applied for the parameter estimation, where clocks and zenith wet delays are estimated with 30-min intervals and gradients with 120-min intervals. We show that the accuracy generally is better for the Kalman filter solution, for example, the baseline length repeatabilities are on average about 10 % better compared to the LSM solution. We also discuss the possibilities to use the Kalman filter to estimate sub-diurnal station position variations and show that the variations caused by solid Earth tides can be retrieved with an accuracy of about 2 cm.

Tropospheric delay determination by Kalman filtering VLBI data

The troposphere is one of the most important error sources for space geodetic techniques relying on radio signals. Since it is not possible to model the wet part of the tropospheric delay with sufficient accuracy, it needs to be estimated from the observational data. In the analysis of very long baseline interferometry (VLBI) data, the parameter estimation is routinely performed using a least squares adjustment. In this paper, we investigate the application of a Kalman filter for parameter estimation, specifically focusing on the tropospheric delays. The main advantages of a Kalman filter are its real-time capability and stochastic approach. We focused on the latter and derived stochastic models for VLBI zenith wet delays, taking into account temporal and location-based differences. Compared to a static noise model, the quality of station coordinates, also estimated in the Kalman filter, increased as a result. In terms of baseline length and station coordinate repeatabilities, this improvement amounted to 2.3 %. Additionally, we compared the Kalman filter and least squares results for VLBI with zenith wet delays derived from GPS (Global Positioning System), water vapor radiometers, and ray tracing in numerical weather models. The agreement of the Kalman filter VLBI solution with respect to water vapor radiometer data was larger than that of the least squares solution by 6–15 %. Our investigations are based on selected VLBI data (CONT campaigns) that are closest to how future VLBI infrastructure is designed to operate. With the aim for continuous and near real-time parameter estimation and the promising results which we have achieved in this study, we expect Kalman filtering to grow in importance in VLBI analysis.

Probing the solar corona with very long baseline interferometry

Understanding and monitoring the solar corona and solar wind is important for many applications like telecommunications or geomagnetic studies. Coronal electron density models have been derived by various techniques over the last 45 years, principally by analysing the effect of the corona on spacecraft tracking. Here we show that recent observational data from very long baseline interferometry (VLBI), a radio technique crucial for astrophysics and geodesy, could be used to develop electron density models of the Sun’s corona. The VLBI results agree well with previous models from spacecraft measurements. They also show that the simple spherical electron density model is violated by regional density variations and that on average the electron density in active regions is about three times that of low-density regions. Unlike spacecraft tracking, a VLBI campaign would be possible on a regular basis and would provide highly resolved spatial–temporal samplings over a complete solar cycle.

Improving the modeling of the atmospheric delay in the data analysis of the Intensive VLBI sessions and the impact on the UT1 estimates

The very long baseline interferometry (VLBI) Intensive sessions are typically 1-h and single-baseline VLBI sessions, specifically designed to yield low-latency estimates of UT1-UTC. In this work, we investigate what accuracy is obtained from these sessions and how it can be improved. In particular, we study the modeling of the troposphere in the data analysis. The impact of including external information on the zenith wet delays (ZWD) and tropospheric gradients from GPS or numerical weather prediction models is studied. Additionally, we test estimating tropospheric gradients in the data analysis, which is normally not done. To evaluate the results, we compared the UT1-UTC values from the Intensives to those from simultaneous 24-h VLBI session. Furthermore, we calculated length of day (LOD) estimates using the UT1-UTC values from consecutive Intensives and compared these to the LOD estimated by GPS. We find that there is not much benefit in using external ZWD; however, including external information on the gradients improves the agreement with the reference data. If gradients are estimated in the data analysis, and appropriate constraints are applied, the WRMS difference w.r.t. UT1-UTC from 24-h sessions is reduced by 5% and the WRMS difference w.r.t. the LOD from GPS by up to 12%. The best agreement between Intensives and the reference time series is obtained when using both external gradients from GPS and additionally estimating gradients in the data analysis.

Reference Frame-Induced Errors in VLBI Earth Orientation Determinations

This paper presents how Very Long Baseline Interferometry (VLBI) realizes the Earth Orientation Parameters (EOP) and which accuracy can be theoretically reached depending on the involved reference frames. The definition of EOP is based on the transformation between the Geocentric Celestial Reference System (GCRS) and the International Terrestrial Reference System (ITRS). The ITRS part is in common for all the space geodetic techniques. The method applied here utilizes the uncertainty of the orientation of the actual set of radio sources of a VLBI observing session related to GCRS and the uncertainty of the orientation of the set of terrestrial network stations related to ITRS for the assessment of the uncertainty of the EOP. The uncertainty of the initial orientation of the GCRS is about 35 � as and the uncertainty of the orientation stability is about 0.7 � as=year. In addition small systematics are present due to the aberration caused by the rotation of our galaxy at the level of 5 � as=year. The uncertainty of the initial orientation of the ITRS of about 800 � as is about 20 times larger, while the uncertainty of the orientation rate of about 80 � as=year is about 100 times larger compared to the corresponding celestial values. The initial orientation and the orientation stability of ITRS could be much more precisely defined by constraining it to the GCRS via the EOP.

Solar Corona Electron Densities from VLBI and GIM Data

The electron density of the solar corona can be determined by multi-frequency radio measurements, e.g. to spacecraft during superior solar conjunctions. Recently, also Very Long Baseline Interferometry (VLBI) has been successfully used to estimate coronal electron densities. The greatest challenge was to separate the dispersive effects of the solar corona and the Earth’s ionosphere. Here, we developed and applied another approach including global ionospheric maps (GIM) to eliminate the effect of the ionosphere. By using such an external data set, an independent validation of the previous results is possible. The models of the electron density derived by these two approaches agree well: the electron density at the Sun’s surface is calculated as (1. 24 ± 0. 42) × 1012 m−3 (VLBI only) and (1. 31 ± 0. 51) × 1012 m−3 (VLBI + GIM). The results are compared to external information about indicators of solar activity (e.g. Sunspot numbers), coronagraph images as well as to models of the electron density determined by measurements to spacecraft.

Optimized Parameterization of VLBI Auxiliary Parameters in Least-Squares Adjustment: Preliminary Results

In a general parameter estimation model, a priori information is used to linearize the system of equations being solved so that just offsets to the a priori values need to be estimated. A priori information used in Very Long Baseline Interferometry (VLBI) data analysis is additionally needed for modeling and constraining the auxiliary parameters, i.e. zenith wet delays, clocks, and troposphere gradients, in order to stabilize the parameter estimation. In our study we investigate the modelling of the auxiliary parameters.

Water Vapor Radiometer Data in Very Long Baseline Interferometry Data Analysis

We investigate the possibilities to use data from water vapor radiometers (WVR) to calibrate the wet tropospheric delays in geodetic Very Long Baseline Interferometry (VLBI) observations. We test three methods: (1) direct calibration using WVR measurements aquired in the directions of the VLBI observations, (2) estimating zenith wet delays and gradients from the WVR data and use these to correct the VLBI data, and (3) including the WVR measurements as additional observations in the VLBI data analysis. Furthermore, in all cases we model the WVR calibration errors in the data analysis. We test the three methods using data from the continuous VLBI campaigns CONT02–CONT14. We find clear improvements when applying methods 1 and 3 for CONT05 campaign, however, the results are degraded for the other campaigns.

Rapid UT1 Estimation by Combining VLBI Intensives with GNSS

We present a Kalman filter for combining dUT1 from the VLBI Intensive sessions with GNSS results for rapid estimation of dUT1. In order to be able to also combine polar motion, pre-reduced normal equations for the Intensive sessions are used in the Kalman filter. We validate our results by comparing with dUT1 estimates from standard global 24-h VLBI sessions. It is found that the Kalman filter is able to use the polar motion measured by GNSS to properly correct the errors in dUT1 caused by inaccurate a priori polar motion. Furthermore, we investigate how the coordinates of the Tsukuba VLBI station can be handled in the analysis after the Tōhoku (Japan) Earthquake in 2011.