The geopotential from gravity measurements,levelling data and satellite results

The geodetic boundary value problem is formulated which uses as boundary values the differences between the geopotential of points at the surface of the continents and the potential of the geoid. These differences are computed by gravity measurements and levelling data. In addition, the shape of the geoid over the oceans is assumed to be known from satellite altimetry and the shape of the continents from satellite results together with three-dimensional triangulation. The boundary value problem thus formulated is equivalent to Dirichlet's exterior problem except for the unknown potential of the geoid. This constant is determined by an integral equation for the normal derivative of the gravitational potential which results from the first derivative of Green's fundamental formula. The general solution, which exists, of the integral equation gives besides the potential of the geoid the solution of the geodetic boundary value problem. In addition approximate solutions for a spherical surface of the earth are derived.     

Regularization of geopotential determination from satellite data by variance components

 Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming mission GRACE as well as gradiometer measurements for GOCE. In addition, the estimate of the geopotential has to be smoothed or regularized because of the inversion problem. It is proposed to solve these two tasks by Bayesian inference on variance components. The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large linear systems. The posterior density function for the variance components, weighting factors and regularization parameters are given in order to compute the confidence intervals for these quantities. Test computations with simulated gradiometer observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach. Received: 5 June 2001 / Accepted: 28 November 2001     

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.     

A new integral approach to geopotential coefficient determinations from terrestrial gravity or satellite altimetry data

The spherical harmonic coefficients of the Earth’s gravitational potential are conveniently determined by integration of gravity data or potential data (derived from satellite altimetry) over a sphere. The major problem of such a method is that the data, given on the non-spherical surface of the Earth, must be reduced to the sphere. A new integral formula over the surface of the Earth is derived. With this formula improved first order topographic corrections to the spherical formulas are obtained.     

A geopotential error analysis for a non planar satellite to satellite tracking mission

This study is concerned with gravity field recovery from low-low satellite to satellite range rate data. Compared against a coplanar mission an improvement is predicted in errors associated with certain parts of the geopotential by the separation of the orbital planes of the two satellites. Using Hill's equations an analytical scheme is developed to model the range rate residuals. It is flexible enough to model the residuals between pairs of satellites in the same orbital plane or whose planes are separated in right ascension. This scheme should allow the possibility of larger planar separations than previously since no small angle approximation is made. The effects of such an orientation on gravity field recovery can therefore be analysed by means of an extensive error analysis. The results of this analysis indicate that a significant improvement in the errors of the near sectorial coefficients are obtained when the satellite's orbital planes are separated. Received: 9 April 1996; Accepted: 26 September 1996     

A new technique to determine geoid and orthometric heights from satellite positioning and geopotential numbers

This paper takes advantage of space-technique-derived positions on the Earth’s surface and the known normal gravity field to determine the height anomaly from geopotential numbers. A new method is also presented to downward-continue the height anomaly to the geoid height. The orthometric height is determined as the difference between the geodetic (ellipsoidal) height derived by space-geodetic techniques and the geoid height. It is shown that, due to the very high correlation between the geodetic height and the computed geoid height, the error of the orthometric height determined by this method is usually much smaller than that provided by standard GPS/levelling. Also included is a practical formula to correct the Helmert orthometric height by adding two correction terms: a topographic roughness term and a correction term for lateral topographic mass–density variations.     

A study on the combination of satellite, airborne, and terrestrial gravity data

Satellite gravity missions, such as CHAMP, GRACE and GOCE, and airborne gravity campaigns in areas without ground gravity will enhance the present knowledge of the Earths gravity field. Combining the new gravity information with the existing marine and ground gravity anomalies is a major task for which the mathematical tools have to be developed. In one way or another they will be based on the spectral information available for gravity data and noise. The integration of the additional gravity information from satellite and airborne campaigns with existing data has not been studied in sufficient detail and a number of open questions remain. A strategy for the combination of satellite, airborne and ground measurements is presented. It is based on ideas independently introduced by Sjöberg and Wenzel in the early 1980s and has been modified by using a quasi-deterministic approach for the determination of the weighting functions. In addition, the original approach of Sjöberg and Wenzel is extended to more than two measurement types, combining the Meissl scheme with the least-squares spectral combination. Satellite (or geopotential) harmonics, ground gravity anomalies and airborne gravity disturbances are used as measurement types, but other combinations are possible. Different error characteristics and measurement-type combinations and their impact on the final solution are studied. Using simulated data, the results show a geoid accuracy in the centimeter range for a local test area.     

Determination of deflections of the vertical using a combination of spherical harmonics and gravimetric data for the area of Greece

The determination of gravimetric deflections of the vertical for the area of Greece is attempted by combining a spherical hamonics model and gravity nomalies using the method of least squares collocation. The components of deflections of the vertical are estimated on a grid with spacing 15′ in latitude and 20′ in longitude covering only the continental area of Greece, where a sufficient number of point gravity anomalies is available. In order to test the accuracy of the determination, gravimetric deflections of the vertical are computed at stations where astrogeodetic data are available. The results show that in a large region of rugged topography and irregular potential field, the prediction is possible with a standard deviation of 18% ... 28% of the root mean square variation of the observations, without taking into account the topography. Furthermore, the estimation of some systematic differences between observed and computed deflections of the vertical is attempted.     

The downward continuation of aerial gravimetric data without density hypothesis

Techniques for downward continuing gravity field data from aircraft altitude are well established if the earth's surface is either approximated by a plane or sphere, or is replaced by one of these surfaces using terrain corrections based on a density model. An alternative procedure is presented here where the continuation is analytical to a nontrivial surface, such as the actual topographic surface. Terrain elevations must be given, but density information is specifically not used.     

Marine gravity and geoid determination by optimal combination of satellite altimetry and shipborne gravimetry data

. Satellite altimetry derived geoid heights and marine gravity anomalies can be combined to determine a detailed gravity field over the oceans using the least-squares collocation method and spectral combination techniques. Least-squares collocation, least-squares adjustment in the frequency domain and input-output system theory are employed to determine the gravity field (both geoid and anomalies) and its errors. This paper intercompares these three techniques using simulated data. Simulation studies show that best results are obtained by the input-output system theory among the three prediction methods. The least-squares collocation method gives results which are very close to but a little bit worse than those obtained using input-output system theory. This slightly poorer performance of the least-squares collocation method can be explained by the fact that it uses isotropic structured covariance (thus approximate signal PSD information) while the system theory method uses detailed signal PSD information. The method of least-squares adjustment in the frequency domain gives the poorest results among these three methods because it uses less information than the other two methods (it ignores the signal PSDs). The computations also show that the least-squares collocation and input-output system theory methods are not as sensitive to noise levels as the least-squares adjustment in the frequency domain method is. Received 19 January 1996; Accepted 17 July 1996     

On the recovery of geopotential coefficients using satellite-to-satellite range-rate data on a sphere

If satellite range-rate information is given continuously on an extra terrestrial sphere (S), the spherical harmonic coefficients of the gravitational potential of the Earth can be determined by direct integration of the data. An exact solution is given in the case when circular satellite orbits crosses S in all directions. A simpler acquisition of the data is achieved when the observations are restricted to polar satellite orbits. However, in this case the solutions become more complex due to attenuation. A solution to order sin Δ, where Δ is the separation angle of the two satellites, is given for the zonal harmonics. For tesseral hamonics a zero-order solution is derived. Corrections are given for observations out of S. Finally, the along track configuration of the satellite pair is compared with a radially designed satellite system (one above the other). The former is found most favourable for the recovery of geopotential coefficients.     

State-of-the-art of elevation extraction from satellite SAR data

Relative or absolute elevation extraction from satellite radar data has been an active research topic for more than 20 years. Various investigations have been made on different methods depending on the predominant “fashion” and data availability, leading each time to new developments to improve the capability and the applicability of each method. The paper presents an update of the state-of-the-art of elevation extraction from satellite SAR data. The performance and limitations of four different methods (clinometry, stereoscopy, interferometry and polarimetry) are reviewed, as well as their applicability to different satellite SAR sensors. Their advantages and disadvantages and how they are addressed during the data processing are also analysed. Finally, concluding remarks look at the complementarity aspects of each method to make the best use of the existing and future radar data for elevation extraction.     

Mapping and monitoring of Rann (tidal) ingress in Banni Plains,Kachchh, Gujarat using multi-temporal satellite data

The Rann ingress into the ‘Banni Plains’ of Kachchh district, Gujarat which is famous for the sprawling grasslands supporting significant cattle population is taking place at an alarming rate. This has resulted in the desertion of many villages due to intrusion of Rann waters. In the present investigation multi-date satellite data has been analysed to map and monitor the Rann ingress during the period 1975 to 1989. Detailed analysis has brought out that the inundation is quite rapid affecting about 244 sq. km area during 1960–1989 period. Mapping of the creeks has also been done to understand the possible mechanism of ‘Rann ingress’.     

Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data

The well-known statistical tool of variance component estimation (VCE) is implemented in the combined least-squares (LS) adjustment of heterogeneous height data (ellipsoidal, orthometric and geoid), for the purpose of calibrating geoid error models. This general treatment of the stochastic model offers the flexibility of estimating more than one variance and/or covariance component to improve the covariance information. Specifically, the iterative minimum norm quadratic unbiased estimation (I-MINQUE) and the iterative almost unbiased estimation (I-AUE) schemes are implemented in case studies with observed height data from Switzerland and parts of Canada. The effect of correlation among measurements of the same height type and the role of the systematic effects and datum inconsistencies in the combined adjustment of ellipsoidal, geoid and orthometric heights on the estimated variance components are investigated in detail. Results give valuable insight into the usefulness of the VCE approach for calibrating geoid error models and the challenges encountered when implementing such a scheme in practice. In all cases, the estimated variance component corresponding to the geoid height data was less than or equal to 1, indicating an overall downscaling of the initial covariance (CV) matrix was necessary. It was also shown that overly optimistic CV matrices are obtained when diagonal-only cofactor matrices are implemented in the stochastic model for the observations. Finally, the divergence of the VCE solution and/or the computation of negative variance components provide insight into the selected parametric model effectiveness.     

Estimating regional deformation from a combination of space and terrestrial geodetic data

We discuss an approach for efficiently combining different types of geodetic data to estimate time-dependent motions of stations in a region of active deformation. The primary observations are analyzed separately to produce loosely constrained estimates of station positions and coordinate system parameters which are then combined with appropriate constraints to estimate velocities and coseismic displacements. We define noninteger degrees of freedom to handle the case of finite constraints and stochastic perturbation of parameters and develop statistical tests for determining compatibility between different data sets. With these developments, we show an example of combining space and terrestrial geodetic data to obtain the deformation field in southern California. Received: 23 January 1997 / Accepted: 30 September 1997     

GPS接收机(OEM)二进制文件向RINEX文件的转换

介绍GPS接收机数据的RINEX标准交换格式,讨论二进制数据格式文件向文本文件转换的方法及需要注意的问题,通过分析HemisphereJ、avad、AC12、NavCom 4种不同OEM板二进制数据格式,指出GPS接收机的二进制数据文件向RINEX文件转换的一般方法,并编程实现所有程序,验证方法的正确性。     

卫星测高数据处理的理论与方法

本文针对测高卫星轨道及地面轨迹的独有特点,较系统地研究了卫星测高数据处理的理论与方法,内容包括测高卫星径向轨道误差的时域特征分析与空域特征分析、共线轨迹与交叉点分布等的谱特征、整体求解法的研究以及ＴＯＰＥＸ测高数据的初步计算与分析。（１）系统论述了卫...