Performance of three types of Stokes's kernel in the combined solution for the geoid

When regional gravity data are used to compute a gravimetric geoid in conjunction with a geopotential model, it is sometimes implied that the terrestrial gravity data correct any erroneous wavelengths present in the geopotential model. This assertion is investigated. The propagation of errors from the low-frequency terrestrial gravity field into the geoid is derived for the spherical Stokes integral, the spheroidal Stokes integral and the Molodensky-modified spheroidal Stokes integral. It is shown that error-free terrestrial gravity data, if used in a spherical cap of limited extent, cannot completely correct the geopotential model. Using a standard norm, it is shown that the spheroidal and Molodensky-modified integration kernels offer a preferable approach. This is because they can filter out a large amount of the low-frequency errors expected to exist in terrestrial gravity anomalies and thus rely more on the low-frequency geopotential model, which currently offers the best source of this information. Received: 11 August 1997 / Accepted: 18 August 1998     

On determination of heights by using terrestrial clocks and GPS signals

 Considering a GPS satellite and two terrestrial stations, two types of equations are derived relating the heights of the two stations to the measured data (frequency ratio or clock rate differences) and the coordinates and velocity components of all three participating objects. The potential possibilities of using such relations for the determination of heights (in terms of geopotential numbers or orthometric heights) are discussed. Received: 6 December 2000 / Accepted: 9 July 2001     

Equilibria in the system Au–Ag–S–fluid: computed and experimental data

We present results of computations on the interaction of solid-phase electrum–argentite–pyrite (weight ratios 210⁻⁵/ 210⁻³/1 and 210⁻⁵/410⁻²/1) association with Cl-containing aqueous moderately acid solutions (0.5m NaCl, pH = 3.08) at 300 °C and 500 bars. These data are a physicochemical basis for predicting the geochemical behavior of Au and Ag during the hydrothermal-metasomatic transformation of Au-Ag-pyrite. We also propose a technique of study of this process based on the phase equilibria of the subsystem Au–Ag–S with the aqueous solution at different liquid/solid (l/s) ratios, with the use of new graphic diagrams. The relationship of the composition of the solid-phase association with l/s ratio in real boundary conditions (Au = 17 ppm, mAu/mAg = 10^–3.57–10^–2.28) is shown. The maximum l/s values for complete leaching of gold and silver (l/smax = 200–800) are estimated. It has been established that argentite is the first to dissolve when mAu/mAg(s) > mAu/mAg(sol), and electrum, when mAu/mAg(s) < mAu/mAg(sol).

The experimental results showed that at 300 °C, the conversion of electrum (N_Au = 300‰) nonequilibrated with pyrite into an Au-richer form (N_Au = 730‰) and argentite follows an intricate kinetic scheme. Using the Pilling-Bedwords kinetic equation for processing data yielded the process rate constant K = 2.8(±0.5)10⁻⁵ g²cm⁻⁴day⁻¹. With this equation, the time of the complete conversion of 200 μm thick flat gold grains is 604 days. These data evidence a significant role of kinetic factors in hydrothermal-metasomatic processes involving native gold, which requires combination of thermodynamic and kinetic approaches on the construction of geologo-genetic models for hydrothermal sulfide formation.     

A new high-resolution gravimetric geoid for South Spain and the Gibraltar Strait area: SOSGIS

A new gravimetric geoid is computed for South Spain and the Gibraltar Strait area. This geoid is located just in the junction between two tectonic plates (Euro-Asiatic and African plates) and in the junction of two gravimetric geoids: IGG2005 (the Iberian Gravimetric Geoid obtained in 2005) and MORGEO (the MORoccan GEOid). IGG2005 is the Iberian geoid and MORGEO is the Moroccan geoid, both geoids have been previously obtained. The new geoid is the gravimetric geoid solution that connects the two above-mentioned geoids, getting a more accurate and reliable picture of this area than the other previous geoids. The method used is the Stokes integral in convolution form, which shows to be an efficient method to reach the proposed objective. The terrain correction and the indirect effect have been taken into account. The new geoid is obtained as a regular grid (with a mesh size of 1.5′ × 1.5′) in the GRS80 reference system, covering the study area from 34° to 40° of latitude and from −8° to 0° of longitude. This gravimetric geoid and the previous geoids: IGG2005 and MORGEO; are compared to the geoid undulations derived at the validation points located on the study area (four GPS/levelling points measured on Morocco and five points of the European vertical reference network (EUVN) measured on Iberia). As it is expected, the new geoid is a more precise and reliable model, fitting the geoidal heights of these validation points with more accuracy than the other previous geoids. This new model will be useful for orthometric height determination by GPS in the mountains and remote areas, where levelling has many logistic problems. Also, it can be interesting for other geophysical purposes different to the height measurements, because it can provide a constraint for the density distribution, the thermal state of Lithosphere and the viscosity in the mantle. Such details can be inferred from a geoid model and the seismic velocity structure.     

Local geoid determination from airborne vector gravimetry

Methods are illustrated to compute the local geoid using the vertical and horizontal components of the gravity disturbance vector derived from an airborne GPS/inertial navigation system. The data were collected by the University of Calgary in a test area of the Canadian Rocky Mountains and consist of multiple parallel tracks and two crossing tracks of accelerometer and gyro measurements, as well as precise GPS positions. Both the boundary-value problem approach (Hotines integral) and the profiling approach (line integral) were applied to compute the disturbing potential at flight altitude. Cross-over adjustments with minimal control were investigated and utilized to remove error biases and trends in the estimated gravity disturbance components. Final estimation of the geoid from the vertical gravity disturbance included downward continuation of the disturbing potential with correction for intervening terrain masses. A comparison of geoid estimates to the Canadian Geoid 2000 (CGG2000) yielded an average standard deviation per track of 14 cm if they were derived from the vertical gravity disturbance (minimally controlled with a cross-over adjustment), and 10 cm if derived from the horizontal components (minimally controlled in part with a simulated cross-over adjustment). Downward continuation improved the estimates slightly by decreasing the average standard deviation by about 0.5 cm. The application of a wave correlation filter to both types of geoid estimates yielded significant improvement by decreasing the average standard deviation per track to 7.6 cm.     

The analytical continuation bias in geoid determination using potential coefficients and terrestrial gravity data

One important application of an Earth Gravity Model (EGM) is to determine the geoid. Since an EGM is represented by an external-type series of spherical harmonics, a biased geoid model is obtained when the EGM is applied inside the masses in continental regions. In order to convert the downward-continued height anomaly to the corresponding geoid undulation, a correction has to be applied for the analytical continuation bias of the geoid height. This technique is here called the geoid bias method. A correction for the geoid bias can also be utilised when an EGM is combined with terrestrial gravity data, using the combined approach to topographic corrections. The geoid bias can be computed either by a strict integral formula, or by means of one or more terms in a binomial expansion. The accuracy of the lowest binomial terms is studied numerically. It is concluded that the first term (of power H²) can be used with high accuracy up to degree 360 everywhere on Earth. If very high mountains are disregarded, then the use of the H² term can be extended up to maximum degrees as high as 1800. It is also shown that the geoid bias method is practically equal to the technique applied by Rapp, which utilises the quasigeoid-to-geoid separation. Another objective is to carefully consider how the combined approach to topographic corrections should be interpreted. This includes investigations of how the above-mentioned H² term should be computed, as well as how it can be improved by a correction for the residual geoid bias. It is concluded that the computation of the combined topographic effect is efficient in the case that the residual geoid bias can be neglected, since the computation of the latter is very time consuming. It is nevertheless important to be able to compute the residual bias for individual stations. For reasonable maximum degrees, this can be used to check the quality of the H² approximation in different situations.Acknowledgement The author would like to thank Prof. L.E. Sjöberg for several ideas and for reading two draft versions of the paper. His support and constructive remarks have improved its quality considerably. The valuable suggestions from three unknown reviewers are also appreciated.     

A new strategy for the atmospheric gravity effect in gravimetric geoid determination

Prior to Stokes integration, the gravitational effect of atmospheric masses must be removed from the gravity anomaly g. One theory for the atmospheric gravity effect on the geoid is the well-known International Association of Geodesy approach in connection with Stokes integral formula. Another strategy is the use of a spherical harmonic representation of the topography, i.e. the use of a global topography computed from a set of spherical harmonics. The latter strategy is improved to account for local information. A new formula is derived by combining the local contribution of the atmospheric effect computed from a detailed digital terrain model and the global contribution computed from a spherical harmonic model of the topography. The new formula is tested over Iran and the results are compared with corresponding results from the old formula which only uses the global information. The results show significant differences. The differences between the two formulas reach 17 cm in a test area in Iran.     

The Northern and Southern Scandes — structural differences revealed by an analysis of gravity anomalies, the geoid and regional isostasy

We investigate the Scandes mountain range by analysing the gravity field, the geoid heights and the degree of isostatic compensation of the lithosphere. Topographically, the Scandes mountain range can be divided in the Northern and Southern Scandes. Comparisons between the present topographic expression and the gravity field and the geoid show that the axis of highest elevation in the Northern Scandes is shifted eastwards compared to the minimum of the Bouguer anomaly, while the two coincide perfectly in the Southern Scandes. Geoid heights reduced by the effect of topographic masses show a large-scale minimum in the Northern Scandes, but no anomaly in the Southern Scandes.Regional, flexural isostatic calculations yield a flexural rigidity of D = 10²³ Nm for the lithosphere of the Southern Scandes and the isostatic gravity and geoid residuals point to additional isostatic support by low-density rocks below the Moho. On the other side, for the lithosphere in the Northern Scandes no significant flexural rigidity can be resolved. Here, the Bouguer anomaly is best modelled with a small flexural rigidity, indicating nearly Airy isostatic behaviour. Local subsurface loading and horizontal tectonic forces overprint the isostatic compensations and increase the tectonic complexity of the Northern Scandes. These distinctive features of the Scandes cannot be explained by currently existing models of the present and Neogene uplift and the isostatic mechanism of the Scandes.     

Paleolithospheric structure revealed by continental geoid anomalies

Lithospheric geoid anomalies record changes in elevation and potential energy experienced by continental lithosphere. Estimates of local isostatic equilibration and potential energy, in tandem with lithosphere-related geoid anomalies, can be used to estimate paleolithospheric thickness, providing a clearer understanding of how and why continental topography is developed. We employ several simplifying assumptions about the crustal and mantle lithosphere density and structure (and readily acknowledge that our results are therefore first-order approximations) to predict the pre-orogenic structure of the lithosphere. At the outset we emphasize that while this approach does not provide an exhaustive evaluation of the deformation mechanism, it does serve to quantify the relative role played by the variations in the crustal and upper mantle components of the lithosphere. In this way we are able to use independent measurement of lithospheric geoid anomalies, current (post-orogenic) elevation and lithospheric structure, and paleoelevation information to estimate topographic development and structural support over time. Application of this technique to the southwestern United States indicates that the uplift of the Colorado Plateau is the result of processes in both the crust and mantle lithosphere and that the lithosphere of the pre-orogenic Southern Basin and Range was thinned relative to the Northern Basin and Range and Colorado Plateau. Although we use the southwestern U.S. as an example, this method can help constrain uplift mechanisms for any region for which the structure and geoid anomaly of the modern lithosphere is well understood.     

大地水准面高对InSAR大范围地壳形变监测的影响分析

由于InSAR数据处理所用的WGS84参考椭球系统与通用的DEM高程系统(EGM96大地水准参考面)不一致,在InSAR形变监测分析中会引入大地水准面高导致的误差.本文利用覆盖青藏高原北部阿尔金断裂带西段的27景Envisat ASAR宽幅模式数据和44景条带模式数据,研究了大地水准面高与InSAR大范围形变测量不确定性的关系:(1)模拟分析表明对于100 m的垂直基线,8.8 m的DEM测量误差,若研究区域存在20 m的大地水准面高的变化,对宽幅或条带模式InSAR形变测量造成的影响将由3 mm增至10 mm左右;(2)实例验证表明对于不同的研究区域,大地水准面高与该地区地形变化存在较大相关性,对于同一研究区域,垂直基线的大小决定了大地水准面高对InSAR不确定性的影响程度;(3)对于大地水准面高有较大梯度变化的研究区域,组合短基线方法与去除轨道平面的方法难以消除大地水准面高的影响.使用基于WGS84高程系统的DEM,可以为InSAR形变测量分析提供统一的高程基准,有效避免大地水准面高误差的影响.