Geoid Height Research Articles

MARINE GEOID UNDULATION ASSESSMENT OVER SOUTH CHINA SEA USING GLOBAL GEOPOTENTIAL MODELS AND AIRBORNE GRAVITY DATA

Abstract. Global geopotential models (GGMs) are vital in computing global geoid undulations heights. Based on the ellipsoidal height by Global Navigation Satellite System (GNSS) observations, the accurate orthometric height can be calculated by adding precise and accurate geoid undulations model information. However, GGMs also provide data from the satellite gravity missions such as GRACE, GOCE and CHAMP. Thus, this will assist to enhance the global geoid undulations data. A statistical assessment has been made between geoid undulations derived from 4 GGMs and the airborne gravity data provided by Department of Survey and Mapping Malaysia (DSMM). The goal of this study is the selection of the best possible GGM that best matches statistically with the geoid undulations of airborne gravity data under the Marine Geodetic Infrastructures in Malaysian Waters (MAGIC) Project over marine areas in Sabah. The correlation coefficients and the RMS value for the geoid undulations of GGM and airborne gravity data were computed. The correlation coefficients between EGM 2008 and airborne gravity data is 1 while RMS value is 0.1499.In this study, the RMS value of EGM 2008 is the lowest among the others. Regarding to the statistical analysis, it clearly represents that EGM 2008 is the best fit for marine geoid undulations throughout South China Sea.

Jeoid Yüksekliklerinin Belirlenmesinde Yapay Sinir Ağları ve Kriging Enterpolasyon Yöntemlerinin Karşılaştırılması

Turkiye’de uydu tekniklerinin yayginlasmasi ile dusey datumlarin belirlenmesinde Kuresel Uydu Konumlama Sistemi (GNSS) 2001 yilindan itibaren kullanilmaya baslamistir. GNSS sisteminin verdigi yukseklikler elipsoidal yuksekliklerdir. Fakat Turkiye Ulusal Dusey Kontrol Agi-1999 (TUDKA-1999) ortometrik yukseklik sistemini esas almaktadir. GNSS olcmeleri ile elde edilen elipsoidal yuksekliklerden ortometrik yukseklige gecebilmek icin jeoid yuksekliklerine ihtiyac vardir. Bu calismanin amaci Yapay Sinir Aglari (YSA) ve Kriging Enterpolasyon yontemiyle jeoid yuksekliklerinin belirlenmesidir. Afyonkarahisar bolgesinde secilen noktalardan olusan test aginda yaygin olarak kullanilan enterpolasyon yontemlerinde biri olan Kriging yontemi ve Radyal Bazli Fonksiyon Sinir Agi (RBFSA) yontemleri karsilastirilmistir. RBFSA ve Kriging yontemleriyle hesaplanan jeoid yukseklikleri ile bilinen jeoid yukseklikleri arasindaki farklar karesel ortalama hata yonunden degerlendirilmis ve RBFSA yontemiyle daha iyi sonuclar elde edilmistir.

Evaluation of global and regional geoid models in South Korea by using terrestrial and GNSS data

In this study, 96 bench marks and unified control points across South Korea were selected as GPS/Leveling stations, GPS surveying was conducted in the Network-RTK (VRS) method and the results were utilized to calculate geometric geoidal heights. These geometric geoidal heights were utilized in the evaluation of the geoid models comprising global gravity field models, such as EGM2008 and EIGEN-6C4, and South Korean geoid models, such as KNGoeid13 and KNGeoid14. In addition, geoid model KNGeoid14, which was found to have the highest accuracy and precision through the evaluation, was utilized to calculate GPS derived orthometric heights, and the results were evaluated to review the possibility of orthometric height surveys using the GPS/Geoid method. As for the results of the study, in the evaluation of the global gravity field models, the standard deviations of EGM2008 and EIGEN-6C4 were shown to be almost the same as the values were 7.0 cm and 7.1 cm respectively, and geoid model KNGeoid14 was found to have been improved compared to KNGeoid13 as its RMS and standard deviation were shown to be smaller than those of KNGeoid13. In addition, differences in the orthometric height values of 96 points derived in the GPS/Geoid method were analysed using geoid model KNGeoid14, and, in the results, the average error was shown to be 3.9 cm and the standard deviations was shown to be 5.2 cm so that the possibility of utilization of the GPS/Geoid method in orthometric height surveys with the decimeter accuracy could be identified.

Combination of GPS and Leveling Observations and Geoid Models Using Least-Squares Variance Component Estimation

AbstractIn the last few years, satellite-based positioning techniques, such as the Global Positioning System (GPS), proved their abilities in various geodetic fields. Height determination, as one of such applications, requires a more likely correct geoid model to provide reliable geoid heights for transformation of the ellipsoidal heights to orthometric heights. An important step is then to establish such a model by optimal combination of the available geoid models. This can be achieved via variance component estimation (VCE) methods, which provide appropriate weights to GPS and leveling observations as well as the geoid models. The authors demonstrate the efficacy of the least-squares VCE (LS-VCE) to this problem. The algorithm is applied to real data sets in Shahin-Shahr, Isfahan, Iran, to evaluate the EGM2008 and GGMplus Earth geopotential models and a regional geoid model (named IRGeoid10) in terms of agreement to the GPS/leveling observations and introduce the more likely correct model over the case-...

Thermochemical structure of the North China Craton from multi-observable probabilistic inversion: Extent and causes of cratonic lithosphere modification

We present a 3D thermochemical model of the North China Craton (NCC) from the surface down to 350km by jointly inverting surface wave phase velocity data, geoid height, surface heat flow and absolute elevation with a multi-observable probabilistic inversion method. Our model reveals a thin (~65–100km) and chemically fertile lithosphere (87<Mg#<90) beneath the Eastern NCC, consistent with independent results from mantle xenoliths, and supports the idea that the Eastern NCC experienced significant lithospheric destruction and refertilization during the Phanerozoic. In contrast, beneath the Trans-North China Orogen, Inner Mongolia Suture Zone and Yinshan belt, we observe a more heterogeneous (chemically and thermally) lithosphere, indicating that these areas have been partly involved in lithospheric modification and mechanical erosion at multiple scales. A cold and chemically refractory (Mg#>90) lithospheric mantle is imaged beneath the central TNCO and Ordos Block, reaching depths >260km. This lithospheric “keel” is surrounded to the east by a high-temperature sublithospheric anomaly that originates at depths >280km. The spatial distribution of this anomaly and its correlation with the location of recent volcanism in the region suggest that the anomaly represents a deep mantle upwelling being diverted by the cratonic keel and spreading onto regions of shallow lithosphere. Our results indicate that the present-day thermochemical structure beneath the NCC is the result of a complex interaction between a large-scale return flow associated with the subduction of the Pacific slab and the shallow lithospheric structure.

Using GIS-based digital raster analysis for improving harmonic models-derived geoidal heights

In this study, the GIS-based raster analysis principle was applied as a trial to enhance the local performance of geopotential models. Specifically, harmonic models-derived geoidal height raster layers, with respect to the WGS-84 reference ellipsoid, were investigated. As an illustrative case study, the Egyptian Territory was used as a test region. A set of geoidal height grids was synthesized from five geopotential harmonic models. The synthesis window was organized so as to avoid any subsequent edge effects during the analysis process. After importing such raster layers into a GIS-platform, an iterative series of linear high-frequency filters was applied. The progress of the spatial quality of the filtered layers, at independent GPS/Leveling points, was interactively monitored. Then, the optimal filtering radius and the relevant best attained spatial quality were picked out for each of the five raster data. The results revealed that the applied GIS-based sharpening algorithm works well, even for a satellite-only model. In general, large spatial quality enhancements were achieved. The maximal attained improvement was 21 and 29 %, in terms of standard deviation and range of differences, respectively, at the control points. So, it is recommended to try the GIS-based raster analysis scheme over other geographical windows. Also, linear and non-linear forms of high-frequency filters may be applied and tested, regarding the improvement of other types of synthesized gravitational signals.

Feasibility Study of Network-RTK(VRS) Surveying Inside and Outside of Korean CORS Network

This study aims to analyze the accuracy for feasibility study of Network-RTK(VRS) surveying inside and outside of Korean CORS network. The southwest coast of Korea where some part of mainland and islands are outside of CORS network is chosen as the test area. To evaluate the accuracy of VRS surveying at surveying points, several Unified Control Points (UCPs) inside and outside of Korean CORS network were selected as the points in the test area. The feasibility of VRS surveying was analyzed by investigating the errors related to the location of points inside and outside of CORS network and the difference of 3-dimensional coordinates observed on different date. As the results of this study, the orthometric height errors of points outside of CORS network based on KNGeoid14 were improved about 5.0 cm in RMSE in comparison with KNGeoid13. The height errors of VRS surveying were considered to be less relevant to the results from PDOP and number of satellites (GPS and GLONASS). However, the orthometric errors caused by the geoidal height of KNGeoid and the ellipsoidal height of VRS surveying at points located outside of CORS network need to be addressed carefully for control surveying. When a point surveyed twice on different date, the difference of the ellipsoidal height of the point outside of CORS network was larger than that of the point inside of CORS network.

Assessment of Gravity Field and Steady State Ocean Circulation Explorer (GOCE) geoid model using GPS levelling over Sabah and Sarawak

The GOCE satellite mission has significantly contributed to various applications such as solid earth physics, oceanography and geodesy. Some substantial applications of geodesy are to improve the gravity field knowledge and the precise geoid modelling towards realising global height unification. This paper aims to evaluate GOCE geoid model based on the recent GOCE Global Geopotential Model (GGM), as well as EGM2008, using GPS levelling data over East Malaysia, i.e. Sabah and Sarawak. The satellite GGMs selected in this study are the GOCE GGM models which include GOCE04S, TIM_R5 and SPW_R4, and the EGM2008 model. To assess these models, the geoid heights from these GGMs are compared to the local geometric geoid height. The GGM geoid heights was derived using EGMLAB1 software and the geometric geoid height was computed by available GPS levelling information obtained from the Department Survey and Mapping Malaysia. Generally, the GOCE models performed better than EGM2008 over East Malaysia and the best fit GOCE model for this region is the TIM_R5 model. The TIM_R5 GOCE model demonstrated the lowest R.M.S. of ± 16.5 cm over Sarawak, comparatively. For further improvement, this model should be combined with the local gravity data for optimum geoid modelling over East Malaysia.

Accurate Establishment of Error Models for the Satellite Gravity Gradiometry Recovery and Requirements Analysis for the Future GOCE Follow-On Mission

Firstly, the new single and combined error models applied to estimate the cumulative geoid height error are efficiently produced by the dominating error sources consisting of the gravity gradient of the satellite-equipped gradiometer and the orbital position of the space-borne GPS/GLONASS receiver using the power spectral principle. At degree 250, the cumulative geoid height error is 1.769 × 10−1 m based on the new combined error model, which preferably accords with a recovery accuracy of 1.760 ×10−1 m from the GOCE-only Earth gravity field model GO_CONS_GCF_2_TIM_R2 released in Germany. Therefore, the new combined error model of the cumulative geoid height is correct and reliable in this study. Secondly, the requirements analysis for the future GOCE Follow-On satellite system is carried out in respect of the preferred design of the matching measurement accuracy of key payloads comprising the gravity gradient and orbital position and the optimal selection of the orbital altitude of the satellite. We recommend the gravity gradient with an accuracy of 10−13−10−15 /s2, the orbital position with a precision of 1-0.1 cm and the orbital altitude of 200-250 km in the future GOCE Follow-On mission.

Water Surface Height Determination with a GPS Wave Glider: A Demonstration in Loch Ness, Scotland

AbstractA geodetic GPS receiver has been installed on a Wave Glider, an unmanned water surface vehicle. Using kinematic precise point positioning (PPP) GPS, which operates globally without directly requiring reference stations, surface heights are measured with ~0.05-m precision. The GPS Wave Glider was tested in Loch Ness, Scotland, by measuring the gradient of the loch’s surface height. The experiment took place under mild weather, with virtually no wind setup along the loch and a wave field made mostly of ripples and wavelets. Under these conditions, the loch’s surface height gradient should be approximately equal to the geoid slope. The PPP surface height gradient and that of the Earth Gravitational Model 2008 geoid heights do indeed agree on average along the loch (0.03 m km−1). Also detected are 1) ~0.05-m-sized height changes due to daily water pumping for hydroelectricity generation and 2) high-frequency (0.25–0.5 Hz) oscillations caused by surface waves. The PPP heights compare favorably (~0.02-m standard deviation) with relative carrier phase–based GPS processing. This suggests that GPS Wave Gliders have the potential to autonomously determine centimeter-precise water surface heights globally for lake modeling, and also for applications such as ocean modeling and geoid/mean dynamic topography determination, at least for benign surface states such as those encountered during the reported experiment.

Comparison of local geoid height surfaces, in the province of Trabzon

In this study, the estimation of a local geoid model in the area of Trabzon by using geometric methods is presented. The geometric method used in this study interpolates Global Positioning System-derived ellipsoidal and orthometric heights by using the weighted average interpolation, polynomial surface regression, and multiquadratic interpolation. The orthometric and ellipsoidal heights measured at 600 C3 order control stations within a reference benchmark network covering the entire area of Trabzon in Turkey were used in this study. From the existing orthometric and ellipsoidal heights at control stations, the geoid heights were determined. Control stations were divided into two sets, namely, “reference stations” and “testing stations,” to test the estimated geoid height surface. The three methods are compared from a practical point of view by using the descriptive statistics of the root mean square error, mean absolute error, maximum error, and minimum error of the geoid height residuals.

Gravity field anomalies from recent GOCE satellite-based geopotential models and terrestrial gravity data: a comparative study over Saudi Arabia

Global geopotential models (GGMs) provide important information about the Earth gravity field in terms of precise gravity anomaly, gravity disturbance, and geoid height on global and regional scales. The GGMs based on the Gravity Recovery and steady-state Ocean Circulation Explorer (GOCE) satellite gravity mission provided by the International Center for Global Earth Models (ICGEM) differ in their accuracy and spatial resolution. In this paper, we evaluate the free-air gravity anomalies and geoid heights determined from several recent GOCE-based GGMs using the corresponding gravity functions (i.e., gravity anomalies and geoid heights) obtained from the Earth Gravitational Model 2008 (EGM2008) over the Kingdom of Saudi Arabia (KSA). Moreover, free-air gravity anomalies determined from GOCE-based GGMs have also been compared with the corresponding ones obtained from ground-based “terrestrial” gravity measurements over the study area. The latter comparison, using the available 3500 terrestrial gravity observations, applies the spectral enhanced method (SEM). We use the SEM to compensate the missed high-frequency component of geoid heights in GOCE-based GGMs using EGM2008 and a high-resolution digital terrain model based on the Shuttle Radar Topography Mission (SRTM). The statistical comparisons of the computed results in terms of free-air gravity anomalies and geoid heights obtained from GOCE-based GGMs are in general agreement. The estimated standard deviations of differences are 5 mGal (20 cm), 6–7 mGal (22–24 cm), and 8 mGal (25.9 cm) at spherical harmonic (SH) degree/order (d/o) 200, 240, and 280, respectively. Free-air gravity anomalies computed from available terrestrial gravity data, using the SEM, have been compared with the corresponding ones estimated from GOCE-based GGMs, and the results indicate an improvement over EGM2008 in the medium wavelengths. Investigation of various releases of GOCE-based GGM, up to d/o 200, reveals that the fourth and fifth releases of GOCE-based GGMs developed with the use of time-wise solution strategies (TIM_R4 and TIM_R5) approximate the gravity field well over the KSA region, since they provide the least standard deviation differences at SH. Thus, the TIM_R5 and TIM_R4 are suggested as reference models for recovering the long wavelength up to SH d/o 200 and 240, respectively, when modeling the gravimetric quasi-geoid over the KSA area.

CHANGING THE NATIONAL HEIGHT SYSTEM AND GEOID MODEL IN LATVIA

According to the decision of IAG Reference Frame Sub-commission for Europe (EUREF) the EVRF2007 solution as the vertical reference has to be deployed in EU countries.The new height system LAS-2000,5 had been enacted as the European Vertical Reference System‘s EVRF2007 realization in Latvia and the new geoid model LV‘14 had been introduced by Latvian authority Latvian Geospatial Information Agency. However, the appreciation of the quality of quasi-geoid model LV‘14 is rather contradictious among the users in Latvia. The independent estimate and comparison of the two Latvian geoid models developed till now has been performed by the Institute of Geodesy and Geoinformatics. Previous geoid model LV98 which was developed for Baltic-1977 height system almost 20 years ago is outdated now. Preparatory actions described in order to fulfil the task of comparison the geoids in two different height systems. The equations and transformation parameters are presented in this article for the normal height conversion from Baltic-1977 height system to the Latvian realization named LAS-2000,5. The comparison is performed of both Latvian quasigeoid models – the new one LV‘14 and previous LV98. The quality of both models estimated by controlling the geoid heights at the properly densified GNSS/levelling network sites. The distribution of discrepancies in comparison with normal distribution N(x,μ,s) is depicted in corresponding figures. For LV‘14 quasi-geoid model the standard deviation of discrepancies is 3.2 cm, 75% of discrepancies x ≤ 3.2 cm. For LV98 quasigeoid model the standard deviation of discrepancies is 4.7 cm, 80% of discrepancies x ≤ 6 cm. Without doubt, the newly developed LV‘14 quasi-geoid model is of higher quality.

Precise gravity-field modeling in the area of the Japanese Antarctic station Syowa and evaluation of recent EGMs

By combining a Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) Earth Gravity Model (EGM) and in situ gravity data obtained from the Japanese Antarctic Research Expedition (JARE) surveys, we estimated the regional gravity field in the area of Syowa Station, a Japanese research station located in Lützow-Holm Bay, East Antarctica. In situ data sets that were used consisted of land gravity data collected since 1967, shipborne data collected since 1985 and airborne gravity data collected in 2006. The GOCE direct (DIR) solution release 5 (R5) model was used as the long-wavelength reference of the gravity field. Using these data sets, we calculated gravity anomalies and geoid heights at 1-by-1′ grid by means of least-squares collocation.The resulting geoid height at Syowa Station was compared with a local height based on GPS, spirit leveling and tide gauge data. The result suggests that the sea surface height at Syowa Station is −1.57 m, which is consistent with a dynamic ocean topography model. During this investigation, we also evaluated GOCE EGMs and other recent EGMs by comparing them with the airborne gravity data. The results indicate that the GOCE DIR R5 produced the smallest RMS (Root Mean Square) differences and that the newer models performed nearly as well. These comparisons demonstrate the importance of using reliable in situ data when evaluating satellite-only EGMs.

On the Accuracy Assessment of the Latest Releases of GOCE Satellite-Based Geopotential Models with EGM2008 and Terrestrial GPS/Levelling and Gravity Data over Egypt

The Global Geopotential Models (GGMs) of GOCE (Gravity Recovery and steady- state Ocean Circulation Explorer) differ globally as well as regionally in their accuracy and resolution based on the maximum degree and order (d/o) of the fully normalized spherical harmonic (SH) coefficients, which express each GGM. The main idea of this study is to compare the free-air gravity anomalies and quasi geoid heights determined from several recent GOCE-based GGMs with the corresponding ones from the Earth Gravitational Model 2008 (EGM2008) over Egypt on the one hand and with ground-based measurements on the other hand. The results regarding to the comparison of GOCE-based GGMs with terrestrial gravity and GPS/levelling data provide better improvement with respect to EGM2008. The 4th release GOCE-based GGM developed with the use of space-wise solution strategy (SPW_R4) approximates the gravity field well over the Egyptian region. The SPW_R4 model is accordingly suggested as a reference model for recovering the long wavelength (up to SH d/o 200) components of quasi geoid heights when modelling the gravimetric quasi-geoid over the Egypt. Finally, three types of transformation models: Four-, Five- and Seven-parameter transformations have been applied to reduce the data biases and to provide a better fitting of quasi geoid heights obtained from the studied GOCE-based GGMs to those from GPS/levelling data. These models reveal that the standard deviation of vertical datum over Egypt is at the level of about 32 cm.

관측조건에 따른 GPS 타원체고 결정의 정확도 분석

GNSS/Leveling technology makes it possible to get geoidal height geometrically using GNSS and Leveling technology. GNSS/Geoid technology refers to a technology for obtaining orthometric height by subtracting geoidal height achieved by Geoid technology from ellipsoidal height achieved by GNSS technology. The purpose of this study is to verify the accuracy of the ellipsoidal height determination in order to verify the accuracy of the orthometric height determination by the GNSS/Geoid technology. For the study, a test bed was selected in Kyungnam province and GNSS Static surveying was accomplished in the test bed and then the GNSS data was processed in accordance with various analysis conditions. So, it was verified the accuracy of the ellipsoidal heights determination in accordance with the surveying conditions under the GNSS Static surveying. According to the research results, to ensure the 3cm goal accuracy of the ellipsoidal height determination, it should be surveyed by four fixed points on the survey area periphery and more than two hours of the GNSS occupation time, And also, it was found that should be limited to a baseline distance of 20km under the GNSS Static surveying.

Geoid height computation in strip-area project by using least-squares collocation

Geoid height computation in strip-area project by using least-squares collocation

A research of consistencies and progresses of geoid models in Turkey

A number of geoids with various data and names have been determined up to present day in Turkey. In this paper, all Turkish geoids have been reviewed. For this purpose, ten local geoid models have been examined. These are 1976 Turkish Geoid (TG-76), Turkish Geoid-1991 (TG-91), Turkish Doppler Geoid-1992 (TDG-92), Turkish PseudoWGS84Doppler Geoid-1992 (TPDG-92), Turkish Astrogeodetic Geoid-1994 (TAG-94), Turkish GPS/Leveling Geoid, Updated Turkish Geoid-1999 (TG-99A), Turkish Geoid-2003 (TG-03), Turkish Geoid-2009 (TG-09), and Turkish Hybrid Geoid-2009 (THG-09). Within the numerical applications carried out, 30 test benchmarks of the Turkish National Fundamental GPS Network have been selected and geoid heights from each geoid model over these points have been calculated. Later, all of the models have been compared with each other in terms of the criteria based on their developments and geoid height differences have been calculated using local models at the test points. Through these comparisons, the causes of the inconsistencies between the various geoid models are explored while their overall accuracy is determined as well.

Extraction of geoid heights from shipborne GNSS measurements along the Weser River in northern Germany

Abstract In-land geoid models rely on several measuring techniques. The quality of those models is directly related to the spatial resolution of the measurement data. Occasionally, a local geoid model does not cover the coastal area at all and a local marine geoid simply does not exist. ShipborneGNSS measurementsmay provide away of overcoming this problem in coastal areas. However, several corrections to the raw measurements must be applied in order to account for systematic effects induced by ship dynamics and other static and dynamic impacts from tides, atmospheric pressure or wind stress. This paper presents the theoretical background for the method and the results of a case study in the estuary of the Weser River in Germany. A series of GNSS measurements were carried out aboard a ship and the approximate geoid height along the river was derived. For accuracy assessments of this method, the resultswere compared to the German Combined QuasiGeoid 2011 (GCG2011). The results are very promising and indicate the ability to extract geoid heights from shipborne GNSS measurements.

On the integral inversion of satellite-to-satellite velocity differences for local gravity field recovery: a theoretical study

The gravity field can be recovered locally from the satellite-to-satellite velocity differences (VDs) between twin-satellites moving in the same orbit. To do so, three different integral formulae are derived in this paper to recover geoid height, radial component of gravity anomaly and gravity disturbance at sea level. Their kernel functions contain the product of two Legendre polynomials with different arguments. Such kernels are relatively complicated and it may be impossible to find their closed-forms. However, we could find the one related to recovering the geoid height from the VD data. The use of spectral forms of the kernels is possible and one does not have to generate them to very high degrees. The kernel functions are well-behaving meaning that they reduce the contribution of far-zone data and for example a cap margin of $$7^{\circ }$$ is enough for recovering gravity anomalies. This means that the inversion area should be larger by $$7^{\circ }$$ from all directions than the desired area to reduce the effect of spatial truncation error of the integral formula. Numerical studies using simulated data over Fennoscandia showed that when the distance between the twin-satellites is small, higher frequencies of the anomalies can be recovered from the VD data. In the ideal case of having short distance between the satellites flying at 250 km level, recovering radial component of gravity anomaly with an accuracy of 7 mGal is possible over Fennoscandia, if the VD data is contaminated only with the spatial truncation error, which is an ideal assumption. However, the problem is that the power of VD signal is very low when the satellites are close and it is very difficult to recognise the signal amongst the noise of the VD data. We also show that for a successful determination of gravity anomalies at sea level from an altitude of 250 km mean VDs with better accuracy than 0.01 mm/s are required. When coloured noise at this level is used for the VDs at 250 km with separation of 300 km, the accuracy of recovery will be about 11 mGal over Fennoscandia. In the case of using the real velocities of the satellites, the main problems are downward/upward continuation of the VDs on the mean orbital sphere and taking the azimuthal integration of them.