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To compute RTM omission error estimates, we construct the RTM as difference between the 30 arc second Shuttle Radar Topography Mission (SRTM) elevation model and the spherical harmonic expansion of the DTM2006.0 elevation data base (Pavlis et al. 2007), expanded to degree 2,160. Subtraction of DTM2006.0 spherical harmonic elevations from the SRTM elevations removes a large part of the “spectral information’ already implied by EGM2008 (expanded to degree 2,190). The SRTM—DTM2006.0 residual topography possesses spectral energy beyond the resolution of EGM2008 (in spectral band nz +1=2,191 to~21,600, which is equivalent to the SRTM resolution of 30 arc second).The computation scheme for applied SEM in evaluation of GGM with GPS/levelling stations, the geoid model for Egypt EGGMO08 and ground gravity anomalies points will discuss in the next sections.

Figure 33 To compute RTM omission error estimates, we construct the RTM as difference between the 30 arc second Shuttle Radar Topography Mission (SRTM) elevation model and the spherical harmonic expansion of the DTM2006.0 elevation data base (Pavlis et al. 2007), expanded to degree 2,160. Subtraction of DTM2006.0 spherical harmonic elevations from the SRTM elevations removes a large part of the “spectral information’ already implied by EGM2008 (expanded to degree 2,190). The SRTM—DTM2006.0 residual topography possesses spectral energy beyond the resolution of EGM2008 (in spectral band nz +1=2,191 to~21,600, which is equivalent to the SRTM resolution of 30 arc second).The computation scheme for applied SEM in evaluation of GGM with GPS/levelling stations, the geoid model for Egypt EGGMO08 and ground gravity anomalies points will discuss in the next sections.

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Civil Engineering- Public Work Department
Civil Engineering- Public Work Department
Table 2.1: Specifications of the CHAMP mission (Rummel etal., 2002). Figure 2.1 (a) SST-hl concept (Rummel et al., 2002). (b) CHAMP satellite system.
Table 2.1: Specifications of the CHAMP mission (Rummel etal., 2002). Figure 2.1 (a) SST-hl concept (Rummel et al., 2002). (b) CHAMP satellite system.
Table 2.2: Specifications of the GRACE mission (Rummel etal., 2002). GOCE (Gravity-field and steady-state Ocean Circulation Experiment) has been launched on March 17, 2009 by the ESA (European Space A gency), for the purpose of developing the
Table 2.2: Specifications of the GRACE mission (Rummel etal., 2002). GOCE (Gravity-field and steady-state Ocean Circulation Experiment) has been launched on March 17, 2009 by the ESA (European Space A gency), for the purpose of developing the
Figure 2.2: (a) SST-LL concept (Rummel et al., 2002). (b) GRACE satellite system.
Figure 2.2: (a) SST-LL concept (Rummel et al., 2002). (b) GRACE satellite system.
Figure 2.3: Picture of the Super STAR accelerometer.
Figure 2.3: Picture of the Super STAR accelerometer.
Figure 2.6: Complete gravity gradiometer (Rummel, 2005). Figure 2.5: The star camera sensor to provide the attitude.
Figure 2.6: Complete gravity gradiometer (Rummel, 2005). Figure 2.5: The star camera sensor to provide the attitude.
Figure 2.7: LAGRANGE GPS Receiver (Rummel, 2005).
Figure 2.7: LAGRANGE GPS Receiver (Rummel, 2005).
Figure 2.10: Magnetotorque (Rummel, 2005).
Figure 2.10: Magnetotorque (Rummel, 2005).
Figure 2.11: Top-down view on the GOCE satellite showing the accommodation of instruments. The acronyms in the figure stand for: star tracker (STR), coarse Earth-Sun sensor (CESS), magneto-torquers (MTR), magnetometers (MGM), The LAGRANGE GPS receiver (GPS), command and data management unit (CDMU), power conditioning and distribution unit (PCDU), and laser retn- reflector (LRR). The Xenon tank contains the fuel for the ion propulsion module while the Nitrogen tank contains the fuel for the cold-gas thrusters.
Figure 2.11: Top-down view on the GOCE satellite showing the accommodation of instruments. The acronyms in the figure stand for: star tracker (STR), coarse Earth-Sun sensor (CESS), magneto-torquers (MTR), magnetometers (MGM), The LAGRANGE GPS receiver (GPS), command and data management unit (CDMU), power conditioning and distribution unit (PCDU), and laser retn- reflector (LRR). The Xenon tank contains the fuel for the ion propulsion module while the Nitrogen tank contains the fuel for the cold-gas thrusters.
Figure 2.12: Scientific applications of the global gravity field. (Rummel, 2005)
Figure 2.12: Scientific applications of the global gravity field. (Rummel, 2005)
Figure 2.14: Water Cycle.
Figure 2.14: Water Cycle.
Figure 2.16: Altimetry principle.
Figure 2.16: Altimetry principle.
An improvement of the geoid accuracy will lead to a better understanding of the physics behind orbit perturbations. An improved separation of perturbations from gravity field on the one hand, and on the other hand from other perturbing forces, such as atmospheric drag, solar radiation pressure, etc., will be possible. Thus, a reduction of orbital errors is feasible, leading, for example, to reduce radial orbit errors for altimetric satellites.
An improvement of the geoid accuracy will lead to a better understanding of the physics behind orbit perturbations. An improved separation of perturbations from gravity field on the one hand, and on the other hand from other perturbing forces, such as atmospheric drag, solar radiation pressure, etc., will be possible. Thus, a reduction of orbital errors is feasible, leading, for example, to reduce radial orbit errors for altimetric satellites.
Figure 2.18: Overview of GOCE applications (ESA, 2010).
Figure 2.18: Overview of GOCE applications (ESA, 2010).
Figure 3.1: Illustration of the geoid, reference ellipsoid, and the vectors of the gravity and normal gravity. If Wp = Wo = Ug , the equations (3.9) can be rewritten as (Bruns’s formula):
Figure 3.1: Illustration of the geoid, reference ellipsoid, and the vectors of the gravity and normal gravity. If Wp = Wo = Ug , the equations (3.9) can be rewritten as (Bruns’s formula):
Figure 3.2: The geometry of the RTM reduction.
Figure 3.2: The geometry of the RTM reduction.
Figure 3.4: The Pratt-Hayford model of isostatic compensation.
Figure 3.4: The Pratt-Hayford model of isostatic compensation.
Figure 3.5: Local and regional compensation. 3.2.5 Rudzki’s Inversion
Figure 3.5: Local and regional compensation. 3.2.5 Rudzki’s Inversion
Figure 3.6: Geometry of Rudzki reduction in spherical approximation.
Figure 3.6: Geometry of Rudzki reduction in spherical approximation.
Figure 3.9: Representation of spherical coordinates.
Figure 3.9: Representation of spherical coordinates.
Figure 3.11: Practical computation of the geoid. The remove-compute-restore technique depends on the combination of the different gravity field wavelength to determine the geoid. The window technique (A bd-Elmotaal and K ithtreiber, 2003) has been suggested to get rid of the double consideration of the topographic-isostatic masses within the data window in the framework of the remove-restore technique. The modified Strokes’ integral kernel has been suggested to possibly combine the local data signals with the global geopotential models. 3.5.1 The Window Technique
Figure 3.11: Practical computation of the geoid. The remove-compute-restore technique depends on the combination of the different gravity field wavelength to determine the geoid. The window technique (A bd-Elmotaal and K ithtreiber, 2003) has been suggested to get rid of the double consideration of the topographic-isostatic masses within the data window in the framework of the remove-restore technique. The modified Strokes’ integral kernel has been suggested to possibly combine the local data signals with the global geopotential models. 3.5.1 The Window Technique
The window technique can be overcome this problem by adapt the used global geopotential model. The adapted model can be obtained by subtracting the effect of the topographic-isostatic masses of the data window, in terms of potential coefficients, from the reference field coefficients. Thus, removing the long wavelength part by using this adapted reference field does not lead to a double consideration of a part of the topographic- isostatic masses (no double hatched area in Fig. 3.13) (Abd-Elmotaal and Kiihtreiber, 2003).
The window technique can be overcome this problem by adapt the used global geopotential model. The adapted model can be obtained by subtracting the effect of the topographic-isostatic masses of the data window, in terms of potential coefficients, from the reference field coefficients. Thus, removing the long wavelength part by using this adapted reference field does not lead to a double consideration of a part of the topographic- isostatic masses (no double hatched area in Fig. 3.13) (Abd-Elmotaal and Kiihtreiber, 2003).
Figure 3.13: The window remove-restore technique.
Figure 3.13: The window remove-restore technique.
Figure 4.2: Distribution of the free-air gravity anomalies stations in Egypt.
Figure 4.2: Distribution of the free-air gravity anomalies stations in Egypt.
Table 4.1. The statistics of the gravity anomalies at ground gravity stations. Table 4.2.shows the statistics of gravity anomalies difference between the GGMs and ground gravity stations before applying the SEM. The result shows that the first generation of GOCE time-wise approach is better than the other time wise generations with standard deviation 16.98 mgal. Similarly, the first generation of GOCE combined model (GOCO) is better than the second and third generations. The same previous results the first generation of GOCE direct approach models DIR 1 is better than other models with standard deviation 16.15 mgal. For GOCE space-wise models, the second generation is better than the first generation, the standard deviation of SP2 is 16.62 mgal compared to 18.43 mgal for SP1. Figure 4.5 shows the gravity anomalies difference between the DIR1 and ground gravity stations before applying the SEM.
Table 4.1. The statistics of the gravity anomalies at ground gravity stations. Table 4.2.shows the statistics of gravity anomalies difference between the GGMs and ground gravity stations before applying the SEM. The result shows that the first generation of GOCE time-wise approach is better than the other time wise generations with standard deviation 16.98 mgal. Similarly, the first generation of GOCE combined model (GOCO) is better than the second and third generations. The same previous results the first generation of GOCE direct approach models DIR 1 is better than other models with standard deviation 16.15 mgal. For GOCE space-wise models, the second generation is better than the first generation, the standard deviation of SP2 is 16.62 mgal compared to 18.43 mgal for SP1. Figure 4.5 shows the gravity anomalies difference between the DIR1 and ground gravity stations before applying the SEM.
Table 4.3. The statistics of gravity anomalies difference between the GG Ms and ground gravity stations after applying SEM. Table 4.4 shows the statistics of gravity anomalies difference between the GGMs and ground gravity stations after remove the topographic —isostatic masses. The result shows that the first generation of GOCE time-wise approach is better than the other time wise generations with standard deviation 22.48 mgal. The first generation of GOCE combined model (GOCO) is better than the other generations. The same of previous results the first generation of GOCE direct approach models DIR 1 is better than the other models with standard deviation 21.58 mgal. For GOCE space- wise models, the second generation is better than the other generations.
Table 4.3. The statistics of gravity anomalies difference between the GG Ms and ground gravity stations after applying SEM. Table 4.4 shows the statistics of gravity anomalies difference between the GGMs and ground gravity stations after remove the topographic —isostatic masses. The result shows that the first generation of GOCE time-wise approach is better than the other time wise generations with standard deviation 22.48 mgal. The first generation of GOCE combined model (GOCO) is better than the other generations. The same of previous results the first generation of GOCE direct approach models DIR 1 is better than the other models with standard deviation 21.58 mgal. For GOCE space- wise models, the second generation is better than the other generations.
Figure 4.5: The gravity anomalies difference between the DIR1 and ground gravity stations before applying the SEM.
Figure 4.5: The gravity anomalies difference between the DIR1 and ground gravity stations before applying the SEM.
Figure 4.7. Distribution of the 17 GPS/leveling benchmarks in Egypt. The statistics of the comparisons of the GGMs after applying SEM are given in Tables 4,7, 4.8 and 4.9. According to Tables 4.7, 4.8 and 4.9, GPS/leveling comparisons suggest ¢ geoid agreement of 0.48 m (DIR 1) to 0.39 m (ITG-GO) for the highest expansions. Figures 4.8, 4.9, 4.10, 4.11 and 4.12 shows the GPS/leveling differences of the highest expansions of the GGMs for Egypt.
Figure 4.7. Distribution of the 17 GPS/leveling benchmarks in Egypt. The statistics of the comparisons of the GGMs after applying SEM are given in Tables 4,7, 4.8 and 4.9. According to Tables 4.7, 4.8 and 4.9, GPS/leveling comparisons suggest ¢ geoid agreement of 0.48 m (DIR 1) to 0.39 m (ITG-GO) for the highest expansions. Figures 4.8, 4.9, 4.10, 4.11 and 4.12 shows the GPS/leveling differences of the highest expansions of the GGMs for Egypt.
Table4.5: The evaluation of GOCE models with GPS/leveling over the world (RMS).
Table4.5: The evaluation of GOCE models with GPS/leveling over the world (RMS).
Table 4.7; GPS/leveling differences, in m, of the highest expansions of GOCE direct approach and GOCE combined models (GOCO).
Table 4.7; GPS/leveling differences, in m, of the highest expansions of GOCE direct approach and GOCE combined models (GOCO).
The GOCE short arc approach (ITG-GO) have standard deviation 0.39 m, and GOCE combined models (DGM and EIGEN-6S) standard deviation are 0.40 m and 0.42 m respectively. From the previous results the best model in this assessment is ITG-GO model with standard deviation 0.39m. The main differences occurs in the north of the western desert.
The GOCE short arc approach (ITG-GO) have standard deviation 0.39 m, and GOCE combined models (DGM and EIGEN-6S) standard deviation are 0.40 m and 0.42 m respectively. From the previous results the best model in this assessment is ITG-GO model with standard deviation 0.39m. The main differences occurs in the north of the western desert.
(b) Geoid undulation differences GPS/leveling stations and the DIR 2.
(b) Geoid undulation differences GPS/leveling stations and the DIR 2.
(b) Geoid undulation differences GPS/leveling stations and the GOCO 2.
(b) Geoid undulation differences GPS/leveling stations and the GOCO 2.
(c) Geoid undulation differences GPS/leveling stations and the ITG-GO.
(c) Geoid undulation differences GPS/leveling stations and the ITG-GO.
Figure 4.13: EGGM08 geoid model for Egypt (A bd-Elmotaal, 2008).
Figure 4.13: EGGM08 geoid model for Egypt (A bd-Elmotaal, 2008).
Figure 4.14; Geoid undulations difference between EGGMO08 and DIR 1.
Figure 4.14; Geoid undulations difference between EGGMO08 and DIR 1.
Figure 5.1: DIR 1 gravity anomalies in Egypt. Contour interval: 20 mgal. USA government, and the German and Italian Space Agencies. On February 11, 2000, the Shuttle
Figure 5.1: DIR 1 gravity anomalies in Egypt. Contour interval: 20 mgal. USA government, and the German and Italian Space Agencies. On February 11, 2000, the Shuttle
Figure 5.3: Distribution of the free-air gravity anomalies stations used in geoid determination.
Figure 5.3: Distribution of the free-air gravity anomalies stations used in geoid determination.
Figure 5.4; 30 arc second SRTM + for Egypt.
Figure 5.4; 30 arc second SRTM + for Egypt.
Table5.1: Statistics of the gravity anomalies in Egypt (991 gravity stations)
Table5.1: Statistics of the gravity anomalies in Egypt (991 gravity stations)
Figure 5.6: The topographic- isostatic corrections for Egypt. Contour interval: 10 mgal. The effect of topographic —isostatic masses according to Airy-Heiskanen model has been computed by tcfour- program (Forsberg et al. 2008) for all masses within a radius of 167 km around the computational point. A constant value for the density of the topography of Po =2.67gcm~3. A value of .4 gcm~? used for the density contrast between the crust and mantle and a normal crustal thickness Ty of 30 km has been used. Figure 5.6 shows the topographic- isostatic corrections for window ( 20° N to 34° N, 24° E to 38 °E).
Figure 5.6: The topographic- isostatic corrections for Egypt. Contour interval: 10 mgal. The effect of topographic —isostatic masses according to Airy-Heiskanen model has been computed by tcfour- program (Forsberg et al. 2008) for all masses within a radius of 167 km around the computational point. A constant value for the density of the topography of Po =2.67gcm~3. A value of .4 gcm~? used for the density contrast between the crust and mantle and a normal crustal thickness Ty of 30 km has been used. Figure 5.6 shows the topographic- isostatic corrections for window ( 20° N to 34° N, 24° E to 38 °E).
Table5.2: Statistics of the empirical tests for the cap size yy, at the GPS stations.
Table5.2: Statistics of the empirical tests for the cap size yy, at the GPS stations.
Figure 5.8: The indirect effect of the topographic isostatic reduction. isostatic reduction. The contribution of the reference geopotential model (DIR 1) Ney was computed according to E.q 3.66. (See. Figure 5.2)
Figure 5.8: The indirect effect of the topographic isostatic reduction. isostatic reduction. The contribution of the reference geopotential model (DIR 1) Ney was computed according to E.q 3.66. (See. Figure 5.2)
Table5.3: Statistics of the remaining differences at the 16 GPS stations. Figure 5.10: The geoid difference (Ngps — Ngeoia ) before fit. Contour interval: 0.1 m.
Table5.3: Statistics of the remaining differences at the 16 GPS stations. Figure 5.10: The geoid difference (Ngps — Ngeoia ) before fit. Contour interval: 0.1 m.
Figure 5.12: Geoid difference for Egypt between EGGM08 model and computed geoid undulations.
Figure 5.12: Geoid difference for Egypt between EGGM08 model and computed geoid undulations.
Figure 5.14: Geoid difference for Egypt between EGM96 model and DIR1.
Figure 5.14: Geoid difference for Egypt between EGM96 model and DIR1.
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