Accuracy of Normal Heights Determined Using Improved Geopotential Models

The accuracy of determining the Molodensky normal heights using geopotential models has been investigated. On the basis of geopotential model JGM-3/OSU91A the Molodensky normal heights can be computed with an accuracy of about ±35 cm at the GPS sites in the central part of Europe. On the basis of JGM-3/OSU91A improved the accuracy becomes higher, about ±14 cm.     

Determination of Geopotential Differences between Local Vertical Datums and Realization of a World Height System

The methodology developed for connecting Local Vertical Datums (LVD) was applied to the Australian Height Datum (AHD) and the North American Vertical Datum (NAVD88). The geopotential values at AHD and NAVD88 were computed and the corresponding vertical offset of 974 mm with rms 51 mm was obtained between the zero reference surfaces defined by AHD and NAVD88. The solution is based on the four primary geodetic parameters, the GPS/levelling sites and the geopotential model EGM96. The Global Height System (or the Major Vertical Datum) can be defined by a geoidal geopotential value used in the solution as the reference value, or by the geopotential value of the LVD, e.g. NAVD88.     

基于GRACE卫星重力数据确定地球重力场模型WHU-GM-05

基于卫星轨道运动的能量积分方程,可导出利用卫星跟踪卫星数据求解地球重力场的实用公式.本文在Jekeli给出的公式基础上导出了基于能量守恒方程利用两颗低-低卫星跟踪的扰动位差求解重力位系数的严密关系式.基于两颗GRACE卫星的观测数据,采用本文导出的严密能量积分方法求解得到120阶的GRACE地球重力场模型,命名为WHU-GM-05;将WHU-GM-05模型与国际上同类重力场模型EIGEN-GRACE系列和GGM02S分别在阶方差和大地水准面高等方面作了比较,并与美国和中国的部分地区GPS水准观测值进行了精度分析.结果表明基于本文推导的严密双星能量守恒方程得到的WHU-GM-05重力场模型精度与国际上同类重力场模型的精度相当.     

Investigation of topographic reductions and aliasing effects on gravity and the geoid over Greece based on various digital terrain models

The reduction of gravity-field related quantities (e.g., gravity anomalies, geoid heights) due to the topography plays a crucial role in both geodetic and geophysical applications, since in the former it is an intermediate step towards geoid prediction and in the latter it reveals lateral as well as radial density contrasts and infers the geology of the area under study. The computations are usually carried out by employing a DTM and/or a DBM, which describe the topography and bathymetry, respectively. Errors in these DTMs/DBMs will introduce errors in the computed topographic effects, while poor spatial resolution of the topography and bathymetry models will result in aliasing effects to both gravity anomalies and geoid heights, both influencing the accuracy of the estimated solutions. The scope of this work is twofold. First, a validation and accuracy assessment of the SRTM 3″ (90 m) DTM over Greece is performed through comparisons with existing global models as well as with the Greek 450 m national DTMs. Whenever a misrepresentation of the topography is identified in the SRTM data, it is “corrected” using the local 450 m DTM. This process resulted in an improved SRTM DTM called SRTMGr, which was then used to determine terrain effects to gravity field quantities. From the fine-resolution SRTMGr DTMs, coarser models of 15″, 30″, 1′, 2′ and 5′ have been determined in order to investigate aliasing effects on both gravity anomalies and geoid heights by computing terrain effects at variable spatial resolutions. From the results acquired in two test areas, it was concluded that SRTMGr provides similar results to the local DTM making the use of other older global DTMs obsolete. The study for terrain aliasing effects proved that when high-resolution and accuracy gravity and geoid models are needed, then the highest possible resolution DTM should be employed to compute the respective terrain effects. Based on the results acquired from two the test areas a corrected SRTMGr DTM has been compiled for the entire Greek territory towards the development of a new gravimetric geoid model. Results from that analysis are presented based on the well-known remove-compute-restore method, employing land and marine gravity data, EGM08 as a reference geopotential model and the SRTMGr DTM for the computation of the RTM effects.     

Comparison of global geopotential models from the champ and grace missions for regional geoid modelling in Turkey

The continuous efforts on establishment and modernization of the geodetic control in Turkey include a number of regional geoid models that have been determined since 1976. The recently released gravimetric Geoid of Turkey, TG03, is used in geodetic applications where GPS-heights need to be converted to the local vertical datum. To reach a regional geoid model with improved accuracy, the selection of the appropriate global geopotential model is of primary importance. This study assesses the performance of a number of recent satellite-only and combined global geopotential models (GGMs) derived from CHAMP and GRACE missions’ data in comparison to the older EGM96 model, which is the underlying reference model for TG03. In this respect, gravity anomalies and geoid heights from the global geopotential models were compared with terrestrial gravity data and low-pass filtered GPS/levelling data, respectively. Also, five new gravimetric geoid models, computed by the Fast Fourier Transform technique using terrestrial gravity data and the geopotential models, were validated at the GPS/levelling benchmarks. The findings were also compared with the validation results of the TG03 model. The tests showed that as it was expected any of the high-degree combined models (EIGEN-CG03C, EIGEN-GL04C, EGM96) can be employed for determining the gravity anomalies over Turkey. In the west of Turkey, EGM96 and EIGEN-CHAMP03S fit the GPS/levelling surface better. However, all the tested GGMs revealed equal performance when they were employed in gravimetric geoid modelling after de-trending the gravimetric geoid model with corrector surface fitting. The new geoid models have improved accuracy (after fit) compared to TG03.     

Mean Earth'S Equipotential Surface From Topex/Poseidon Altimetry

The geopotential value of W ₀ = (62 636 855.611 ± 0.008) m ² s ^–2 which specifies the equipotential surface fitting the mean ocean surface best, was obtained from four years (1993 - 1996) of TOPEX/POSEIDON altimeter data (AVISO, 1995). The altimeter calibration error limits the actual accuracy of W ₀ to about (0.2 - 0.5) m ² s ^–2 (2 - 5) cm. The same accuracy limits also apply to the corresponding semimajor axis of the mean Earth's level ellipsoid a = 6 378 136.72 m (mean tide system), a = 6 378 136.62 m (zero tide system), a = 6 378 136.59 m (tide-free). The variations in the yearly mean values of the geopotential did not exceed ±0.025 m ² s ^–2 (±2.5 mm).     

On The Determination Of The Earth's Model – The Mean Equipotential Surface

The sea surface cannot be used as reference for Major Vertical Datum definition because its deviations from the ideal equipotential surface are very large compared to rms in the observed quantities. The quasigeoid is not quite suitable as the surface representing the most accurate Earth's model without some additional conditions, because it depends on the reference field. The normal Earth's model represented by the rotational level ellipsoid can be defined by the geocentric gravitational constant, the difference in the principal Earth's inertia moments, by the angular velocity of the Earth's rotation and by the semimajor axis or by the potential (U ₀ ) on the surface of the level ellipsoid. After determining the geopotential at the gauge stations defining Vertical Datums, gravity anomalies and heights should be transformed into the unique vertical system (Major Vertical Datum). This makes it possible to apply Brovar's (1995) idea of determining the reference ellipsoid by minimizing the integral, introduced by Riemann as the Dirichlet principle, to reach a minimum rms anomalous gravity field. Since the semimajor axis depends on tidal effects, potential U ₀ should be adopted as the fourth primary fundamental geodetic constant. The equipotential surface, the actual geopotential of which is equal to U ₀ , can be adopted as reference for realizing the Major Vertical Datum.     

GRACE/GOCE扩展重力场模型确定我国1985高程基准重力位的精度分析

高精度高程基准重力位的确定往往依赖于高精度全球重力场模型,其对全球和区域高程基准的高精度统一非常关键,GRACE、GOCE卫星重力计划极大地提高了全球重力场模型中长波的精度.本文首先对GRACE/GOCE卫星重力场模型的内符合和外符合精度进行讨论分析,结果说明卫星重力模型的截断误差影响可达到分米级水平,在确定高程基准重力位时该影响不可忽略.利用EGM2008模型扩展GRACE/GOCE卫星重力场模型至2190阶,可有效减弱卫星重力模型的截断误差影响,但不同模型扩展时的最优拼接阶次不同,其中DIR-1、DIR-5模型对应的最优拼接阶次分别为180阶和220阶,以GPS水准数据检验,扩展模型在中国区域的精度均优于18cm.最后,基于最优拼接阶次获得的扩展重力场模型对我国1985高程基准重力位进行了估计,DIR-5和TIM-5模型对应数值分别为62636853.47m~2·s~(-2)和62636853.49m~2·s~(-2),精度均为1.51m~2·s~(-2);发现在中国区域模型大地水准面与GPS/水准数据的差值存在微弱的系统性倾斜,东西向倾斜约为9cm,南北向倾斜约为1.4cm,考虑倾斜改正后基于DIR-5和TIM-5模型估计我国1985高程基准重力位的精度提高了0.16m~2·s~(-2).     

Parameters of the Earth's tri-axial level ellipsoid

Summary Four parameters defining the Earth's tri-axial ellipsoid (E) have been derived on the basis of the condition that the gravity potential on E be constant and equal to the actual geopotential value (W₀) on the geoid. The geocentric gravitational constant, the angular velocity of the Earth's rotation, the actual 2nd degree geopotential Stokes parameters and W₀ are taken to be the primary geodetic constants defining E and its (normal) gravity field.     

Regional precipitation and temperature scenarios for climate change

Abstract

To investigate the consequences of climate change on the water budget in small catchments, it is necessary to know the change of local precipitation and temperature. General Circulation Models (GCM) cannot provide regional climate parameters yet, because of their coarse resolution and imprecise modelling of precipitation. Therefore downscaling of precipitation and temperature has to be carried out from the GCM grids to a small scale of a few square kilometres. Daily rainfall and temperature are modelled as processes conditioned on atmospheric circulation. Rainfall is linked to the circulation patterns (CPs) using conditional probabilities and conditional rainfall amount distribution. Both temperature and precipitation are downscaled to several locations simultaneously taking into account the CP dependent spatial correlation. Temperature is modelled using a simple autoregressive approach, conditioned on atmospheric circulation and local areal precipitation. The model uses the classification scheme of the German Weather Service and a fuzzy rule-based classification. It was applied in the Aller catchment for validation using observed rainfall and temperature, and observed classified geopotential pressure heights. GCM scenarios of the ECHAM model were used to make climate change predictions (using classified GCM geopotential heights); simulated values agree fairly well with historical data. Results for different GCM scenarios are shown.     

Accuracy Estimates of Normal Heights Computed at GPS Sites

The dependence of accuracy of determining normal heights at GPS sites on the density of distribution of the sites within the GPS/levelling frame has been investigated. If the density amounts to one GPS/levelling site per 400 km ² , the accuracy of about ±11.5 cm in the central part of Europe. If the density is higher, e.g. one GPS/levelling site per 200 km ² , an accuracy of about ±5 cm can be achieved.     

Computation of Ray Amplitudes in Inhomogeneous Media with Curved

The TOPEX/POSEIDON (T/P) satellite altimeter data from January 1, 1993 to January 3, 2001 (cycles 11–305) was used for investigating the long-term variations of the geoidal geopotential W ₀ and the geopotential scale factor R ₀ = GM÷W ₀ (GM is the adopted geocentric gravitational constant). The mean values over the whole period covered are W ₀ = (62 636 856.161 ± 0.002) m²s^-2, R ₀ = (6 363 672.5448 ± 0.0002) m. The actual accuracy is limited by the altimeter calibration error (2–3 cm) and it is conservatively estimated to be about ± 0.5 m²s^-2 (± 5 cm). The differences between the yearly mean sea surface (MSS) levels came out as follows: 1993–1994: –(1.2 ± 0.7) mm, 1994–1995: (0.5 ± 0.7) mm, 1995–1996: (0.5 ± 0.7) mm, 1996–1997: (0.1 ± 0.7) mm, 1997–1998: –(0.5 ± 0.7) mm, 1998–1999: (0.0 ± 0.7) mm and 1999–2000: (0.6 ± 0.7) mm. The corresponding rate of change in the MSS level (or R ₀) during the whole period of 1993–2000 is (0.02 ± 0.07) mm÷y. The value W ₀ was found to be quite stable, it depends only on the adopted GM, and the volume enclosed by surface W = W ₀. W ₀ can also uniquely define the reference (geoidal) surface that is required for a number of applications, including World Height System and General Relativity in precise time keeping and time definitions, that is why W ₀ is considered to be suitable for adoption as a primary astrogeodetic parameter. Furthermore, W ₀ provides a scale parameter for the Earth that is independent of the tidal reference system. After adopting a value for W ₀, the semi-major axis a of the Earth's general ellipsoid can easily be derived. However, an a priori condition should be posed first. Two conditions have been examined, namely an ellipsoid with the corresponding geopotential which fits best W ₀ in the least squares sense and an ellipsoid which has the global geopotential average equal to W ₀. It is demonstrated that both a-values are practically equal to the value obtained by the Pizzetti's theory of the level ellipsoid: a = (6 378 136.7 ± 0.05) m.     

Geopotential model testing sites in the central part of Europe

Summary Sixteen geopotential model testing sites in the central part of Europe, coinciding with the first-order leveling network points, have been established. The geopotential values for these sites were determined with an accuracy not limiting the testing procedure. Tests have been carried out for models GEM-Tl, GEM-T3, JGM-l, JGM-2, JGM-3 and OSU91A.     

The gravity field and GGOS

The gravity field of the earth is a natural element of the Global Geodetic Observing System (GGOS). Gravity field quantities are like spatial geodetic observations of potential very high accuracy, with measurements, currently at part-per-billion (ppb) accuracy, but gravity field quantities are also unique as they can be globally represented by harmonic functions (long-wavelength geopotential model primarily from satellite gravity field missions), or based on point sampling (airborne and in situ absolute and superconducting gravimetry). From a GGOS global perspective, one of the main challenges is to ensure the consistency of the global and regional geopotential and geoid models, and the temporal changes of the gravity field at large spatial scales. The International Gravity Field Service, an umbrella “level-2” IAG service (incorporating the International Gravity Bureau, International Geoid Service, International Center for Earth Tides, International Center for Global Earth models, and other future new services for, e.g., digital terrain models), would be a natural key element contributing to GGOS. Major parts of the work of the services would, however, remain complementary to the GGOS contributions, which focus on the long-wavelength components of the geopotential and its temporal variations, the consistent procedures for regional data processing in a unified vertical datum and Terrestrial Reference Frame, and the ensuring validations of long-wavelength gravity field data products.     

Testing global geopotential models through comparison of a local quasi-geoid model with GPS/leveling data

The satellite missions CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) provide accurate data that are routinely inverted into spherical harmonic coefficients of the geopotential forming a global geopotential model (GGM). Mean square errors of these coefficients, in some cases even entire covariance matrices, are included in the GGM. Due to estimation procedures with a large redundancy and insufficiently propagated observation errors, they often do not represent the actual accuracy of the harmonic coefficients, thus also gravity field parameters synthesized from the respective GGM. Since in most cases standard methods validating the GGMs reached their limits, new procedures and independent data are being currently sought. This article discusses an alternative validation procedure based on comparison of the GGMs with independent data represented by a set of GPS/leveling stations. Due to a different spectral content of the height anomalies synthesized from the GGMs and of those derived by combination of GPS-based ellipsoidal and leveled normal heights, the GGM-based low frequency height anomaly is enhanced for a high frequency component computed from local ground gravity and elevation data. The methodology is applied on a set of selected points of the European Vertical Reference Network and Czech trigonometric stations. In accordance with similar tests based on entirely independent data of cross-over altimetry, obtained results seem to indicate low-frequency deficiencies in the current GGMs, namely in those estimated from data of single-satellite missions.     

Zero-frequency tidal distrotion due to the orbital elements of tide-forming bodies

Summary It has been proved that orbital elements of perturbing bodies should be taken into account when 10^–10 accuracy is required for zero-frequency tidal distortion in the second zonal Stokes parameter of the geopotential. A solution at the 10^–15 level of magnitude has been presented. The zero-frequency tidal distortion in the fourth zonal Stokes parameter has been derived as 1·3×10^–10 if the secular Love number is unity. It should be reflected in the geopotential models respecting the 10^–10 level of magnitude.     

Regional geoid modeling in the area of subglacial Lake Vostok,Antarctica

We present a geoid model for the area of Lake Vostok, Antarctica, from a combination of local airborne gravity, ice-surface and ice-thickness data and a lake bathymetry model. The topography data are used for residual terrain modeling (RTM) in a remove–restore approach together with GOCE satellite data. The quasigeoid is predicted by least-squares collocation (LSC) and subsequently converted to geoid heights. Special aspects of that method in presence of an ice sheet are discussed.It is well known that a body freely floating in water is in a state of hydrostatic equilibrium (HE). This usually applies, e.g., to ice shelves or sea ice. However, it has been shown that this is valid also for the ice sheet covering the subglacial Lake Vostok. Thus, we demonstrate the use of such a refined regional geoid model for glaciological and geophysical applications by means of the HE surface of that lake. The mean quadratic residual geoid signal (0.56 m) w.r.t. the GOCE background model exceeds the residual variations of the estimated apparent lake level (ALL) (0.26 m) within the central part of the lake. An approach considering the actual geopotential at the ALL has been derived and subsequently applied. In this context, downward continuation of the potential field within the ice sheet as well as the latitudinal tilt of off-geoid equipotential surfaces are discussed. In view of the accuracy of the ice-thickness measurements that dominate the total error budget of the estimated ALL these effects are negligible. Thus, the HE surface of subglacial lakes may safely be described by a constant height bias in small-scale regional applications. However, field continuation is significant with respect to the formal uncertainty of the quasigeoid, which is at the level of 5 cm given that accurate airborne gravity data (±2 mGal) are available.     

On the analysis of temporal geoid height variations obtained from GRACE-based GGMs over the area of Poland

Temporal mass variations in the Earth system, which can be detected from the Gravity Recovery and Climate Experiment (GRACE) mission data, cause temporal variations of geoid heights. The main objective of this contribution is to analyze temporal variations of geoid heights over the area of Poland using global geopotential models (GGMs) developed on the basis of GRACE mission data. Time series of geoid height variations were calculated for the chosen subareas of the aforementioned area using those GGMs. Thereafter, these variations were analyzed using two different methods. On the basis of the analysis results, models of temporal geoid height variations were developed and discussed. The possibility of prediction of geoid height variations using GRACE mission data over the area of Poland was also investigated. The main findings reveal that the geoid height over the area of Poland vary within 1.1 cm which should be considered when defining the geoid model of 1 cm accuracy for this area.     

Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.