Calibration of the Superconducting Gravimeter: Cantley Station, Canada
One of the most important operating procedures after the installation of a Superconducting Gravimeter (SG) is its frequent calibration. The calibration process can identify and evaluate any variability in the scale factor and in the hardware anti-aliasing filter response over time. The SG calibration is required in many studies, e.g., Earth elasticity and normal mode amplitudes. In this contribution, the SG installed in Cantley, Canada is calibrated to estimate its scale factor in time and frequency domains. In the time domain, we use the weighted linear regression method whereas in the frequency domain we use the least squares response method (LSRM). The long series of Superconducting and Absolute Gravimeters from 1997-2009 recorded with GWR TT70, JILA-2, and FG5-236, respectively at Cantley, are used to estimate the SG scale factor. Accurate and rigorous procedures are applied to define data disturbances, outliers and realistic data noise levels. Using only JILA-2 data, the SG scale factor is estimated in the time and frequency domains as -78.372 ± 0.0155 μGal/V and -78.403 ± 0.0750 μGal/V respectively; these two estimates are statistically incompatible. The relative accuracy in the time domain is 1.9 × 10-4, which is comparable with the absolute method of calibration of SG when using a moving platform. The FG5-236 short time series (May 2007 to February 2009) is used to re-estimate the scale factor in the time domain only with -78.407 ± 0.0615 μGal/V which is statistically compatible with the one from JILA-2 estimated in the frequency domain. We cannot identify any significant periodicity in the scale factor in this long time series. In addition, the hardware anti-aliasing filter response is tested by injecting known voltages (sine or square waves) into the control electronics of the system. Results from the frequency response analysis show that the anti-aliasing filter response is stable (gain and phase) and conforms to the Global Geodynamics Project (GGP) standards.
Modern Observations of the Chandler Wobble
Modern observations of polar motion with techniques such as Very Long Baseline Interferometry (VLBI) have error levels approaching three orders of magnitude below those of classical astronometric methods. In this paper we focus on the VLBI observations which are characteristically unequally spaced. We develop a very effective method of spectral analysis for unequally spaced time sequences. First, the least squares fit to the representation of the sequence by the Discrete Fourier Transform (DFT) is calculated, weighting the observations by the inverse square of the accompanying standard error. The coefficient matrix of the normal equations of this fit is nearly singular. It is subjected to a Singular Value Decomposition (SVD). In the usual application of SVD, singular values are eliminated in order to improve the stability of the numerical system but no criterion is given for how many singular values to eliminate. To overcome this shortcoming, we introduce the Parseval condition which relates the mean square in the time domain to that in the frequency domain. Singular values are eliminated until Parseval's theorem is satisfied. In the absence of excitation, the Chandler Wobble is closely a prograde motion along a circular arc. For a step excitation, the centre of the arc shifts, giving a secular motion but an equal and nearly opposite contribution to the Chandler Wobble occurs, giving only a second order discontinuity in the pole path. To detect excitation events, we fit circular arcs by least squares to the unequally spaced data, weighting by the inverse of the square of the accompanying standard errors. A break is determined if extrapolation along the circular arc leads to a forecast pole position for which the next measured position lies outside a circle of acceptance. We find that often for quite long periods of time, there seems to be relatively little continuous excitation, leading to the conclusion that much of the excitation comes from sudden events. In particular, we are encouraged that a break in the pole path was found eleven days before the December 26, 2004 Sumatra-Adaman Islands earthquake (M=9.1).
Qualitative Comparisons of Global Ocean Tide Models by Analysis of GRACE Intersatellite Ranging Data
Four global ocean tide models are compared in terms of their contribution to GRACE satellite-to-satellite tracking residuals. The residuals are computed relative to a comprehensive model of Earth's time-varying gravity, including allowance for mass motions in the atmosphere, ocean, terrestrial hydrology, and mantle, in addition to tides. For each analyzed tide model four years of GRACE range-rate data are processed. Range and range-acceleration residuals are tidally analyzed by geographic location. All four global tide models are shown to be error-prone in various ways, leaving tidally coherent residuals especially in polar regions, but also in some lower latitude regions. Considerable power in the solar semidiurnal S2 tide in low latitudes suggests errors in our adopted model of atmospheric tides, which is based on 3-hourly ECMWF operational analyses. Anomalies in the μ2 tidal constituent over some shallow seas suggest the presence of unmodeled nonlinear compound tides. Similarly, anomalies in the nonlinear M4 tide are seen if this constituent is omitted from the models. Errors in assumed seawater density may be contributing to some residuals.
A Preliminary Analysis of Nonlinear Shallow Water Tides for the West Shelf Region of Canada by Topex/Poseidon and Jason-1 Data
In west shelf region of Canada, tidal height associated with nonlinear shallow water constituents (e.g., M4 with amplitudes exceeding 25 cm near Anchorage) may not be neglected and should be included in the tide model as corrections for altimeter data. In this paper, 13 years T/P and 3 years Jason-1 data (1992-2005) are analyzed to derive the nonlinear constituents in west shelf region of Canada using the harmonic analysis method both at along-track and crossover locations, taking the effect of annual variation in sea level into account. Taking advantage of T/P, shifting to sub-track from 2002 to 2005, more crossover points could be derived and the spatial resolution of data would increase. The size and effect of nontidal variability are evaluated at crossover locations. A method to reduce the effect of nontidal variability is discussed. The effect of nontidal variability is evaluated in order to assess which nonlinear constituent with minimum amplitude could be derived from T/P and Jason-1. The nonlinear constituents estimated in this paper will supplement and improve existing results. The improved results will be assimilated into a tide model in the future to improve the prediction accuracy, providing sufficient tide correction for satellite altimetry in the west shelf region of Canada.
Estimating Second Order Ionospheric Delays During GPS-LEO Radio Occultation Observations: An Alternative Approach
The demand for millimeter accuracy in Global Positioning System (GPS) applications (receiver and precise point positioning) has raised a considerable interest in studying second order ionospheric delays. Unlike first order, second order ionospheric delays are functions of both, the Total Electron Content (TEC), and the geomagnetic field along the integrated signal path between the transmitter (GPS) and the receiver. The standard method of computing second order ionospheric delays makes use of ionosphere (IRI-2007) and magnetic field models (IGRF-10). We propose a different approach in estimating second order ionospheric delays, based solely on dual frequency GPS raw ionospheric excess phase delays. Specifically, our goal is twofold: (1) compute TEC free from second order ionospheric errors and (2) calculate geomagnetic field values along the integrated signal path between a GPS transmitter and a receiver. We apply our method to near-real time observational data provided by the Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) Data Analysis and Archive Centre (CDAAC), from which second order ionospheric delays can be realized. Second order ionospheric delays estimated based on our model attains values between -9 mm and 16 mm, which is larger than the values obtained for ground-based GPS observations. Currently, TEC estimations are based on differencing dual frequency GPS ionospheric excess phase delays, without accounting for second order ionospheric effects. We propose an alternative approach according to which we model the observed carrier-phase observables (L1, L2) taking into account the geomagnetic field that is the source of the second order ionospheric effect. Subsequently, by combining linearly the GPS observables we can solve for the TEC that is free from the second order ionospheric affect. In a second step, we compute geomagnetic field values along the integrated signal path between a GPS transmitter and a receiver based on the Faraday rotation effect, taking advantage of the relation between the rotation angle of the polarization plane of the GPS signal and the Earth's magnetic field. We analyze 30 COSMIC ionospheric radio occultation (RO) events in 2006, 2007 and 2008 and show that our results are in a very good agreement with the IGRF-10 model within 0.2% and 9%. We also demonstrate that the weighted mean geomagnetic field along a GPS-LEO integrated signal path during an RO event is conservative, as it has been assumed by many researchers when estimating second order ionospheric delays, but never proven.
Fitting of NWM Ray-traced Slant Factors to Closed-form Tropospheric Mapping Functions
Ray-tracing in numerical weather models (NWM) is a promising solution for describing the elevation angle- and azimuth-dependence of tropospheric delay, especially at very low elevation angles, in an attempt to de- correlate vertical position and zenith tropospheric delay during GPS estimation. On the other hand, mapping functions expressed in closed form remain imperative, demanded by the need for (i) fast processing and (ii) convenient distribution to end-users, who employ a variety of third-party GPS processing packages. We investigate the fitting of ray-tracing results to closed-form expressions. We neglect the variation of the tropospheric delay with latitude, longitude, and height, offering a mapping function valid for a specific station site (similarly as done for VMF1-Site [Boehm et al., 1996]). We focus on the variation of the delay with time, elevation angle, and azimuth. For the time-dependence, we choose to work with slant factors instead of slant delays, because the former are more stable in time than the latter; that is a consequence of the normalization by zenith delays which removes the bulk of the variation with time. For the elevation angle-dependence we compare the continued form fraction of Yan and Ping  with that of Marini  (normalized to yield unity at zenith, as given by Herring ). The latter is more commonly used, but the former is expected to provide a better fit at elevation angles below five degrees. Since the ray-tracing results do not necessarily assume azimuthal symmetry, we have to account for the azimuth-dependence. For that we compare the single-direction model of Davis et al.  with the inclusion of secondary directions [Seko et al., 2004] and arbitrary spherical harmonics [Böhm and Schuh, 2001]). We also assess whether physically-oblivious models (i.e., not derived from analytical idealized atmospheric models), such spline or polynomials, as suggested by Rocken et al. , are adequate.
Analysis and Implementation of NOAA NWP-Based Tropospheric Correction Model
Tropospheric delay is one of the dominant Global Positioning System (GPS) errors, which degrades the positioning accuracy. Recent developments in tropospheric modeling rely on implementation of more accurate Numerical Weather Prediction (NWP) models. In North America one of the NWP-based tropospheric correction models is the NOAA model, which has been developed by the US National Oceanic and Atmospheric Administration (NOAA). Because of its potential to improve the GPS positioning accuracy, the NOAA tropospheric correction model became the focus of many researchers. In this research, we examined the performance of the NOAA model and studied its effect on the GPS positioning accuracy. We generated a three-year-long tropospheric zenith total delay (ZTD) data series for the NOAA, Hopfield, and the IGS final tropospheric correction product, respectively. These three sets of data series were generated at different geographical locations represented by ten IGS reference station spanning Canada and the United States. We analyzed the NOAA ZTD data series and compared it with those of the Hopfield model. The IGS final tropospheric product was used as a reference. The analysis showed that the NOAA model is largely season dependent and its performance was superior to the Hopfield model. We further investigated the effect of implementing the NOAA model on the positioning accuracy, which again behaved better than the Hopfield model.
Development of a Frequency Dependent INS/GPS System Response Model for Bridging GPS Outages
The integration of Inertial Navigation System (INS) and Global Positioning System (GPS) architectures can be achieved through the use of many time domain filters such as, an extended Kalman filter, an unscented Kalman filter, divided difference filter, and particle filter. The main objective of all the above filters is to achieve precise fusion of the data from GPS and INS to provide INS only navigation solution during GPS outages. The prediction mode performance of all state of the art time domain filters is poor with significant drift in the INS only solution. In this paper, a new frequency domain dynamic response method with variable frequency bandwidth is proposed to model the INS/GPS system. The Least Squares Spectral Analysis (LSSA), Parzen window based smoothing, and the Inverse Least Squares Fourier Transform (ILSFT) are employed to develop the INS/GPS system frequency response (transfer function). The input to this dynamic system is the INS only solution and the output is the INS/GPS integration solution. The discrete inverse ILSFT of the transfer function is applied to estimate the impulse response of the INS/GPS system. The focus of this paper is the improvement in velocity solution, which leads to almost the same level of improvement in the position solution in an INS/GPS system. To examine the performance of the proposed approach, a kinematic dataset (Dual frequency GPS data from a Trimble BD950 receiver and inertial data from DQI100 IMU) is collected in Hamilton Harbour, Ontario, onboard a hydrographic surveying vessel owned by the Canadian Hydrographic Service. The loosely coupled INS/GPS with unscented Kalman filter is developed to obtain an INS/GPS integrated navigation solution and an INS only solution. Then, the INS/GPS and INS only navigation solutions are used to develop the impulse response of the INS/GPS system. It is shown that the developed impulse response can be used to detect and recover the long term motion dynamics of DQI100 IMU during 300s GPS outages with about 65% dynamic recovery of the north velocity and 45% dynamic recovery of east velocity solution when compared with the INS only solution. We will present and discuss many examples from a variety of GPS outages that exemplify the effectiveness of our method.