Singular value decomposition and cluster analysis as regression diagnostics tools for geodetic applications

It is well known that high-leverage observations significantly affect the estimation of parameters. In geodetic literature, mainly redundancy numbers are used for the detection of single high-leverage observations or of single redundant observations. In this paper a further objective method for the detection of groups of important and less important (and thus redundant) observations is developed. In addition, the parameters which are predominantly affected by these groups of observations are identified. This method thus complements other diagnostics tools, such as, e.g., multiple row diagnostics methods as described in statistical literature (see, e.g., Belsley et al. in Regression diagnostics: identifying influential data and sources of collinearity. Wiley, New York, 1980). The method proposed in this paper is based on geometric aspects of adjustment theory and uses the singular value decomposition of the design matrix of an adjustment problem together with cluster analysis methods for regression diagnostics. It can be applied to any geodetic adjustment problem and can be used for the detection of (groups of) observations that significantly affect the estimated parameters or that are of negligible impact. One of the advantages of the proposed method is the improvement of the reliability of observation plans and thus the reduction of the impact of individual observations (and outliers) on the estimated parameters. This is of particular importance for the very long baseline interferometry technique which serves as an application example of the regression diagnostics tool.     

The determination of potential difference by the joint application of measured and synthetical gravity data: a case study in Hungary

In an elementary approach every geometrical height difference between the staff points of a levelling line should have a corresponding average g value for the determination of potential difference in the Earth’s gravity field. In practice this condition requires as many gravity data as the number of staff points if linear variation of g is assumed between them. Because of the expensive fieldwork, the necessary data should be supplied from different sources. This study proposes an alternative solution, which is proved at a test bed located in the Mecsek Mountains, Southwest Hungary, where a detailed gravity survey, as dense as the staff point density (~1 point/34 m), is available along a 4.3-km-long levelling line. In the first part of the paper the effect of point density of gravity data on the accuracy of potential difference is investigated. The average g value is simply derived from two neighbouring g measurements along the levelling line, which are incrementally decimated in the consecutive turns of processing. The results show that the error of the potential difference between the endpoints of the line exceeds 0.1 mm in terms of length unit if the sampling distance is greater than 2 km. Thereafter, a suitable method for the densification of the decimated g measurements is provided. It is based on forward gravity modelling utilising a high-resolution digital terrain model, the normal gravity and the complete Bouguer anomalies. The test shows that the error is only in the order of 10⁻³mm even if the sampling distance of g measurements is 4 km. As a component of the error sources of levelling, the ambiguity of the levelled height difference which is the Euclidean distance between the inclined equipotential surfaces is also investigated. Although its effect accumulated along the test line is almost zero, it reaches 0.15 mm in a 1-km-long intermediate section of the line.     

Characterization of diurnal cycles in ZTD from a decade of global GPS observations

The diurnal cycle of the tropospheric zenith total delay (ZTD) is one of the most obvious signals for the various physical processes relating to climate change on a short time scale. However, the observation of such ZTD oscillations on a global scale with traditional techniques (e.g. radiosondes) is restricted due to limitations in spatial and temporal resolution. Nowadays, the International GNSS Service (IGS) provides an important data source for investigating the diurnal and semidiurnal cycles of ZTD and related climatic signals. In this paper, 10 years of ZTD data from 1997 to 2007 with a 2-hour temporal resolution are derived from global positioning system (GPS) observations taken at 151 globally distributed IGS reference stations. These time series are used to investigate diurnal and semidiurnal oscillations. Significant diurnal and semidiurnal oscillations of ZTD are found for all GPS stations used in this study. The diurnal cycles (24 hours period) have amplitudes between 0.2 and 10.9 mm with an uncertainty of about 0.5 mm and the semidiurnal cycles (12 h period) have amplitudes between 0.1 and 4.3 mm with an uncertainty of about 0.2 mm. The larger amplitudes of the diurnal and semidiurnal ZTD cycles are observed in the low-latitude equatorial areas. The peak times of the diurnal cycles spread over the whole day, while the peak value of the semidiurnal cycles occurs typically about local noon. These GPS-derived diurnal and semidiurnal ZTD signals are similar with the surface pressure tides derived from surface synoptic pressure observations, indicating that atmospheric tides are the main driver of the diurnal and semidiurnal ZTD variations.     

Preprocessing of gravity gradients at the GOCE high-level processing facility

One of the products derived from the gravity field and steady-state ocean circulation explorer (GOCE) observations are the gravity gradients. These gravity gradients are provided in the gradiometer reference frame (GRF) and are calibrated in-flight using satellite shaking and star sensor data. To use these gravity gradients for application in Earth scienes and gravity field analysis, additional preprocessing needs to be done, including corrections for temporal gravity field signals to isolate the static gravity field part, screening for outliers, calibration by comparison with existing external gravity field information and error assessment. The temporal gravity gradient corrections consist of tidal and nontidal corrections. These are all generally below the gravity gradient error level, which is predicted to show a 1/f behaviour for low frequencies. In the outlier detection, the 1/f error is compensated for by subtracting a local median from the data, while the data error is assessed using the median absolute deviation. The local median acts as a high-pass filter and it is robust as is the median absolute deviation. Three different methods have been implemented for the calibration of the gravity gradients. All three methods use a high-pass filter to compensate for the 1/f gravity gradient error. The baseline method uses state-of-the-art global gravity field models and the most accurate results are obtained if star sensor misalignments are estimated along with the calibration parameters. A second calibration method uses GOCE GPS data to estimate a low-degree gravity field model as well as gravity gradient scale factors. Both methods allow to estimate gravity gradient scale factors down to the 10⁻³ level. The third calibration method uses high accurate terrestrial gravity data in selected regions to validate the gravity gradient scale factors, focussing on the measurement band. Gravity gradient scale factors may be estimated down to the 10⁻² level with this method.     

Phase center modeling for LEO GPS receiver antennas and its impact on precise orbit determination

Most satellites in a low-Earth orbit (LEO) with demanding requirements on precise orbit determination (POD) are equipped with on-board receivers to collect the observations from Global Navigation Satellite systems (GNSS), such as the Global Positioning System (GPS). Limiting factors for LEO POD are nowadays mainly encountered with the modeling of the carrier phase observations, where a precise knowledge of the phase center location of the GNSS antennas is a prerequisite for high-precision orbit analyses. Since 5 November 2006 (GPS week 1400), absolute instead of relative values for the phase center location of GNSS receiver and transmitter antennas are adopted in the processing standards of the International GNSS Service (IGS). The absolute phase center modeling is based on robot calibrations for a number of terrestrial receiver antennas, whereas compatible antenna models were subsequently derived for the remaining terrestrial receiver antennas by conversion (from relative corrections), and for the GNSS transmitter antennas by estimation. However, consistent receiver antenna models for space missions such as GRACE and TerraSAR-X, which are equipped with non-geodetic receiver antennas, are only available since a short time from robot calibrations. We use GPS data of the aforementioned LEOs of the year 2007 together with the absolute antenna modeling to assess the presently achieved accuracy from state-of-the-art reduced-dynamic LEO POD strategies for absolute and relative navigation. Near-field multipath and cross-talk with active GPS occultation antennas turn out to be important and significant sources for systematic carrier phase measurement errors that are encountered in the actual spacecraft environments. We assess different methodologies for the in-flight determination of empirical phase pattern corrections for LEO receiver antennas and discuss their impact on POD. By means of independent K-band measurements, we show that zero-difference GRACE orbits can be significantly improved from about 10 to 6 mm K-band standard deviation when taking empirical phase corrections into account, and assess the impact of the corrections on precise baseline estimates and further applications such as gravity field recovery from kinematic LEO positions.     

Patterns of carbon sequestration in forests of western ghats and study of applicability of remote sensing in generating carbon credits through afforestation/reforestation

Satellite remote sensing is a proven tool for mapping landuse patterns and estimating vegetation biomass/carbon. Present study aims at estimating the potential of forests of Radhanagari WLS (Western Ghats, India) to sequester the atmospheric carbon-di-oxide, using ground based observations coupled with satellite remote sensing. The study area was stratified for dominant forest types based on the structure and composition of vegetation and elevation variations. Permanent sample plots were laid down in these homogeneous vegetation strata (HVS) to make different observations during time 1 and time 2. Carbon sequestration by plantations was also studied and compared with natural forests. Species and area-specific biomass equations were used for estimating carbon pool and sequestration. Among natural forests ‘mixed moist deciduous’ forests exhibited highest sequestration rate (8%), whereas, plantation as obvious had a comparatively higher sequestration rate than natural forests (20.27%). Total carbon sequestration by forests of the Radhanagari WLS between 2004 and 2006 is 78742.09 tons. Eligible land for reforestation activity under clean development mechanism (CDM) of Kyoto Protocol was identified using satellite remote sensing using 1989 and 2005 datasets and it was observed that the potential land that can be used for reforestation activity is 10080 ha.     

Efficient propagation of error covariance matrices of gravitational models: application to GRACE and GOCE

We have applied efficient methods for computing variances and covariances of functions of a global gravity field model expanded in spherical harmonics, using the full variance–covariance matrix of the coefficients. Examples are given with recent models derived from GRACE (up to degree and order 150), and with simulated GOCE derived solutions (up to degree and order 200).     

On the topographic bias in geoid determination by the external gravity field

The topographic bias is defined as the error/bias committed by continuing the external gravity field inside the topographic masses by a harmonic function. We study the topographic bias given by a digital terrain model defined by a spherical template, and we show that the topographic bias is given only by the potential of an inner-zone cap, and it equals the bias of the Bouguer shell, independent of the size of the cap. Then we study the effect on the real Earth by decomposing its topography into a template, and we show also in this case that the topographic bias is that of the Bouguer shell, independent of the shape of the terrain. Finally, we show that the topographic potential of the terrain at the geoid can be determined to any precision by a Taylor expansion outside the Earth’s surface. The last statement is demonstrated by a Taylor expansion to fourth order.     

Long term shoreline oscillation and changes of Cauvery delta coastline inferred from satellite imageries

Coastal zone is highly volatile ecosystem which is always in adjustments. Loss of shore line will cause severe impact on human life and as well as their properties. Remote sensing is a reliable technique to study the historical shoreline changes. Therefore in this paper long term shoreline oscillations of Cauvery delta shorelines at Poompuhar, Tharangambadi and Nagapattinam were studied using satellite imageries and the same was physically observed at the above three locations with the help of reference pillars and compared mutually. It was observed that the shoreline at Poompuhar is under accretion at the rate of 1.79m/ year and other shoreline stretches at Tharangambadi and Nagapattinam were under erosion at 0.4888m/ year and 0.4985m/ year respectively. It was also observed that the remote sensing study qualitatively matches with the physical observation for all the three coastal stretches of the study area.     

Using gravity and topography-implied anomalies to assess data requirements for precise geoid computation

Many regions around the world require improved gravimetric data bases to support very accurate geoid modeling for the modernization of height systems using GPS. We present a simple yet effective method to assess gravity data requirements, particularly the necessary resolution, for a desired precision in geoid computation. The approach is based on simulating high-resolution gravimetry using a topography-correlated model that is adjusted to be consistent with an existing network of gravity data. Analysis of these adjusted, simulated data through Stokes’s integral indicates where existing gravity data must be supplemented by new surveys in order to achieve an acceptable level of omission error in the geoid undulation. The simulated model can equally be used to analyze commission error, as well as model error and data inconsistencies to a limited extent. The proposed method is applied to South Korea and shows clearly where existing gravity data are too scarce for precise geoid computation.