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Figure (4.5): Distribution of Egyptian Gravity Data in A frican Gravity Project. [Source: Fairhead at el., 1997, subset from Fig. (1)].

Figure 22 (4.5): Distribution of Egyptian Gravity Data in A frican Gravity Project. [Source: Fairhead at el., 1997, subset from Fig. (1)].

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Figure (1.1): Research Methodology
Figure (1.1): Research Methodology
Figure (2.1): Attraction and Centrifugal Force.
Figure (2.1): Attraction and Centrifugal Force.
Figure (2.5): Terrain Correction (Hofmann-Wellenhof & Moritz 2005, p. 136)
Figure (2.5): Terrain Correction (Hofmann-Wellenhof & Moritz 2005, p. 136)
Figure (2.7): Pratt-Hayford Isostatic Compensation Model (Sanso &Sideris, 2013, p.349) A anrnminn tn BUnfmann_W allanhnaf and Manty (INN n1A1) tha
Figure (2.7): Pratt-Hayford Isostatic Compensation Model (Sanso &Sideris, 2013, p.349) A anrnminn tn BUnfmann_W allanhnaf and Manty (INN n1A1) tha
Figure (2.8): The geometry of the RTM reduction
Figure (2.8): The geometry of the RTM reduction
Figure (2.9): The Reference Surfaces and Height Systems.
Figure (2.9): The Reference Surfaces and Height Systems.
Table (2.2): Numerical Values of Some Parameters of WGS 84 Ellipsoid.
Table (2.2): Numerical Values of Some Parameters of WGS 84 Ellipsoid.
a) For gravity anomalies:
a) For gravity anomalies:
Figure (2.10): Signal (thick lines) and error (thin lines) amplitudes per degree in terms of gravity anomaly for the geopotential harmonic model EGM 2008 (Dotted), EGM96 (Dashed) and GOCO05s (Dash-dot).
Figure (2.10): Signal (thick lines) and error (thin lines) amplitudes per degree in terms of gravity anomaly for the geopotential harmonic model EGM 2008 (Dotted), EGM96 (Dashed) and GOCO05s (Dash-dot).
Figure (2.11) Cumulative RMS of Gravity Anomaly and Geoid Error.
Figure (2.11) Cumulative RMS of Gravity Anomaly and Geoid Error.
Figure (2.12): Essential parameters of covariance function
Figure (2.12): Essential parameters of covariance function
Figure (2.14): The First-order Levelling Network in Egypt (Saad, A., 1993).
Figure (2.14): The First-order Levelling Network in Egypt (Saad, A., 1993).
the n = m location. We have: Then, the diagonal elements corresponding to the diagonal passing through
the n = m location. We have: Then, the diagonal elements corresponding to the diagonal passing through
Table (4.1): Parameters of Earth Gravity Field Model GOCO05s.
Table (4.1): Parameters of Earth Gravity Field Model GOCO05s.
Table (4.2): Parameters of Earth Gravity Field Model EGM2008. The available gravity anomaly data that were necessary for the computation of the 5 arc-minute area mean values to develop the EGM2008 model, divide into three sub-divisions (see Pavlis et al., [2012, sections 3.5]), as shown in Fig. (4.1a):- a) Areas without any restrictions [are colored green in Fig. (4.1a)], most of this area is ocean areas, where the altimetry-derived gravity anomalies data.
Table (4.2): Parameters of Earth Gravity Field Model EGM2008. The available gravity anomaly data that were necessary for the computation of the 5 arc-minute area mean values to develop the EGM2008 model, divide into three sub-divisions (see Pavlis et al., [2012, sections 3.5]), as shown in Fig. (4.1a):- a) Areas without any restrictions [are colored green in Fig. (4.1a)], most of this area is ocean areas, where the altimetry-derived gravity anomalies data.
Figure (4.1): Geographic Display of the 5 Arc-minute Anomalies Used to Develop the EGM2008 model: (a) Data Availability. (b) Data Source Identification [source: Pavlis et al., 2012, Fig.(3)].
Figure (4.1): Geographic Display of the 5 Arc-minute Anomalies Used to Develop the EGM2008 model: (a) Data Availability. (b) Data Source Identification [source: Pavlis et al., 2012, Fig.(3)].
Table (4.3): Statistics of the 5 Arc-minute Anomaly Data Selected by the Merging Procedure Used to Develop the EGM2008 Model, Unit is mGal [source: Pavlis et al., 2012, Table (2)]. Over areas where only lower resolution gravity data were available
Table (4.3): Statistics of the 5 Arc-minute Anomaly Data Selected by the Merging Procedure Used to Develop the EGM2008 Model, Unit is mGal [source: Pavlis et al., 2012, Table (2)]. Over areas where only lower resolution gravity data were available
Figure (4.2): Data Availability for Egypt Used to Develop The EGM2008 Model [source: subset from Fig. (4.1)].
Figure (4.2): Data Availability for Egypt Used to Develop The EGM2008 Model [source: subset from Fig. (4.1)].
Table (4.4): Parameters of Earth Gravity Field Model EGM96.
Table (4.4): Parameters of Earth Gravity Field Model EGM96.
Accordingly, from Fig. (4.3), and referring to Table (4.5), most of Egypt is located in green area file (A), blue area file (B) and a small part in gray area file (D) as shown in Fig. (4.4). This data obtained from African Gravity Project (AGP) (Fairhead & Watts, 1989; Merry, 2003; Merry et al., 2005).
Accordingly, from Fig. (4.3), and referring to Table (4.5), most of Egypt is located in green area file (A), blue area file (B) and a small part in gray area file (D) as shown in Fig. (4.4). This data obtained from African Gravity Project (AGP) (Fairhead & Watts, 1989; Merry, 2003; Merry et al., 2005).
Figure (4.3): Geographic Locations and Identification of Merged 30 Arc- minute Gravity Anomalies Used to Develop The EGM96. (Source: Lemoine et al., 1998, p.8-21).
Figure (4.3): Geographic Locations and Identification of Merged 30 Arc- minute Gravity Anomalies Used to Develop The EGM96. (Source: Lemoine et al., 1998, p.8-21).
shows the statistics of this heights without marine areas. In this study, the DTM2006.0 model, complete to degree and order 190, has been used to create a 5’x5’ mean heights for Egypt Fig. (4.6) by sing harmonic synth program (Holmes & Pavlis, 2006), in the area ounded by latitudes 212 to 332 and longitudes 24° and 382, and Table (4.6) haAtare tha ectatchre nf thic haimhte waithait marina araac
shows the statistics of this heights without marine areas. In this study, the DTM2006.0 model, complete to degree and order 190, has been used to create a 5’x5’ mean heights for Egypt Fig. (4.6) by sing harmonic synth program (Holmes & Pavlis, 2006), in the area ounded by latitudes 212 to 332 and longitudes 24° and 382, and Table (4.6) haAtare tha ectatchre nf thic haimhte waithait marina araac
Figure (4.8): Distribution of NRIAG Gravity Data.
Figure (4.8): Distribution of NRIAG Gravity Data.
Hence, each gravity stations of the ENGSN97 precisely have a three-dimensional geodetic position (latitude, longitude, and geodetic height) from GPS data, as well as vertical position made by the orthometric height from precise levelling. The standard deviation of the final solution of the ENGSN97 network is range from 0.002 to 0.048 mGal (Dawod, 1998). Figure (4.9) Egyptian National Gravity Standardization Network 1997. (Dawod, 1998)
Hence, each gravity stations of the ENGSN97 precisely have a three-dimensional geodetic position (latitude, longitude, and geodetic height) from GPS data, as well as vertical position made by the orthometric height from precise levelling. The standard deviation of the final solution of the ENGSN97 network is range from 0.002 to 0.048 mGal (Dawod, 1998). Figure (4.9) Egyptian National Gravity Standardization Network 1997. (Dawod, 1998)
Figure (4.12): Distribution of Available Measurements of Gravity in the Database of BGI. (Balmino G.at el., 1999).
Figure (4.12): Distribution of Available Measurements of Gravity in the Database of BGI. (Balmino G.at el., 1999).
Figure (4.13): Distribution of BGI Gravity Data.
Figure (4.13): Distribution of BGI Gravity Data.
Figure (4.14): Geographic Locations of Available Gravity Data for Egypt.
Figure (4.14): Geographic Locations of Available Gravity Data for Egypt.
Figure (4.16): Components of Deflections of The Vertical Data in Egypt.
Figure (4.16): Components of Deflections of The Vertical Data in Egypt.
In each of the above three GPS project, the WGS84 reference frame was utilized to be the basic datum of the produced coordinates which were gained directly from GPS.
In each of the above three GPS project, the WGS84 reference frame was utilized to be the basic datum of the produced coordinates which were gained directly from GPS.
‘7&é PyGravsoft Launcher - Gravity field programs Figure (4.19) Main Programs Package of GRAVSOFT Software.
‘7&é PyGravsoft Launcher - Gravity field programs Figure (4.19) Main Programs Package of GRAVSOFT Software.
Table (4.10): Statistics of the selection Egyptian Point Gravity Anomalies.
Table (4.10): Statistics of the selection Egyptian Point Gravity Anomalies.
Table (4.11): Statistics of The Egyptian 5'x5' Mean Free-air Gravity Anomalies. Figure (4.21): The Egyptian 5'x5' Mean Free-air Gravity Anomalies Interpolated by Least Squares Collocation.
Table (4.11): Statistics of The Egyptian 5'x5' Mean Free-air Gravity Anomalies. Figure (4.21): The Egyptian 5'x5' Mean Free-air Gravity Anomalies Interpolated by Least Squares Collocation.
Figure (4.22): Histogram of Residual between The Gravity Anomalies Derived from Original Points and 5 Arc-minute Gravity Anomalies 4.8.4 Preparation of Deflections of the vertical Data
Figure (4.22): Histogram of Residual between The Gravity Anomalies Derived from Original Points and 5 Arc-minute Gravity Anomalies 4.8.4 Preparation of Deflections of the vertical Data
Figure (4.23): Components of Deflections of The Vertical Used for The Computation.
Figure (4.23): Components of Deflections of The Vertical Used for The Computation.
Differences between tide-free and mean tide system for the ellipsoidal heights er a Figure (4.24): Differences between tide- Free and mean tide system for the ellipsoidal heights of HARN stations
Differences between tide-free and mean tide system for the ellipsoidal heights er a Figure (4.24): Differences between tide- Free and mean tide system for the ellipsoidal heights of HARN stations
Figure (5.1): Iteration Scheme for Tailored geopotential Model.
Figure (5.1): Iteration Scheme for Tailored geopotential Model.
Figure(5.2): RMS of The Differences of The Egyptian 5 Arc-minute Mean Gravity Anomalies Using EG1GOC5s Tailored Model with respect to The Iteration Number.
Figure(5.2): RMS of The Differences of The Egyptian 5 Arc-minute Mean Gravity Anomalies Using EG1GOC5s Tailored Model with respect to The Iteration Number.
Figure (5.3): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Oniginal Model GOCO05s (b) Histogram of These Differences in 10 mGal bins. (Iteration Number =0)
Figure (5.3): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Oniginal Model GOCO05s (b) Histogram of These Differences in 10 mGal bins. (Iteration Number =0)
Figure (5.4): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EG1GOC5s Tailored Model (b) Histogram of These Differences in 10 mGal bins. (Iteration Number =5)
Figure (5.4): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EG1GOC5s Tailored Model (b) Histogram of These Differences in 10 mGal bins. (Iteration Number =5)
Figure (5.5): The Egyptian 5'x5' Mean Gravity Anomalies Derived by Using EG1GOC5s Model Completed to Degree and Order 280.
Figure (5.5): The Egyptian 5'x5' Mean Gravity Anomalies Derived by Using EG1GOC5s Model Completed to Degree and Order 280.
Figure(5.6): RMS of The Differences of The Egyptian 5 Arc-minute Mean Gravity Anomalies Using EGMO8F Tailored Model with respect to The Iteration Number.
Figure(5.6): RMS of The Differences of The Egyptian 5 Arc-minute Mean Gravity Anomalies Using EGMO8F Tailored Model with respect to The Iteration Number.
Figure (5.7): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Original Model EGM2008 (till degree 360) (b) Histogram of These Differences in 10 mGal bins.(Iteration Number =0)
Figure (5.7): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Original Model EGM2008 (till degree 360) (b) Histogram of These Differences in 10 mGal bins.(Iteration Number =0)
Figure (5.8): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EGM08F Tailored Model (b) Histogram of These Differences in 10 mGal bins.(Iteration Number =5)
Figure (5.8): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EGM08F Tailored Model (b) Histogram of These Differences in 10 mGal bins.(Iteration Number =5)
Figure (5.9): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Original Model EGM2008 (Max. degree. 2190) (b) Histogram of These Differences in 15 mGal bins.
Figure (5.9): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Original Model EGM2008 (Max. degree. 2190) (b) Histogram of These Differences in 15 mGal bins.
Figure (5.10): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EGTM0818 Tailored Model (Max. degree. 2190) (b) Histogram of These Differences ir 15 mGal bins.
Figure (5.10): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EGTM0818 Tailored Model (Max. degree. 2190) (b) Histogram of These Differences ir 15 mGal bins.
Figure (5.11): The Egyptian 5’x5’ Mean Gravity Anomalies Derived By Using EGTM0818 Model Complete to Degree and Order 2190.
Figure (5.11): The Egyptian 5’x5’ Mean Gravity Anomalies Derived By Using EGTM0818 Model Complete to Degree and Order 2190.
Table (5.10) shows the differences between the geoid heights derived from 17 GPS/levelling stations of HARN project and those derived from the fitted geoid models after removing trend surface.
Table (5.10) shows the differences between the geoid heights derived from 17 GPS/levelling stations of HARN project and those derived from the fitted geoid models after removing trend surface.
Table (5.10): Differences between The Geoid Models and The 17 GPS/Levelling of HARN Stations after Removing Trend Surface. In order to validate an external check of the quality of the fitted geoid models, the 9 GPS/levelling stations were used Table (5.11), which was not used for the geoid fitting in the previous section. In addition, the EGM96 geopotential model (cf. Fig. 4.5), complete to degree and order 360, has also been used to compute geoid model for Egypt for comparison
Table (5.10): Differences between The Geoid Models and The 17 GPS/Levelling of HARN Stations after Removing Trend Surface. In order to validate an external check of the quality of the fitted geoid models, the 9 GPS/levelling stations were used Table (5.11), which was not used for the geoid fitting in the previous section. In addition, the EGM96 geopotential model (cf. Fig. 4.5), complete to degree and order 360, has also been used to compute geoid model for Egypt for comparison
The result of check points shows that the geoid models derived from
The result of check points shows that the geoid models derived from
Finally, Fig. (5.16) and Fig. (5.18) show the geoid model for Egypt derived from harmonic coefficients of both EG1GOC5s and EGTM0818 tailored geopotential models, respectively, while Fig. (5.17) and Fig. (5.19) show their 3D fitted geoid model for both geoid models Negcicocss and Necrmosis- Table (5.12): Statistics of The 5'x5' Fitted Geoid Models for Egypt.
Finally, Fig. (5.16) and Fig. (5.18) show the geoid model for Egypt derived from harmonic coefficients of both EG1GOC5s and EGTM0818 tailored geopotential models, respectively, while Fig. (5.17) and Fig. (5.19) show their 3D fitted geoid model for both geoid models Negcicocss and Necrmosis- Table (5.12): Statistics of The 5'x5' Fitted Geoid Models for Egypt.
Figure (5.16): Geoid Model for Egypt Derived from Tailored Geopotential Model EG1GOC5s (till degree and order 280) after Removing Trend Surface Using The 4-parameter Model and The Weighted-Mean. Contour interval 1.0 m
Figure (5.16): Geoid Model for Egypt Derived from Tailored Geopotential Model EG1GOC5s (till degree and order 280) after Removing Trend Surface Using The 4-parameter Model and The Weighted-Mean. Contour interval 1.0 m
jure (5.17): 3D - Fitted Geoid Model for Egypt Derived from Tailored Geopotential Model EG1GOC5s.
jure (5.17): 3D - Fitted Geoid Model for Egypt Derived from Tailored Geopotential Model EG1GOC5s.
Figure (5.18): Geoid Model for Egypt Derived from Tailored Geopotential Model EGTM0818 (till degree and order 2190) after Removing Trend Surface Using The 4-parameter Model and The Weighted-Mean. Contour interval 1.0 m
Figure (5.18): Geoid Model for Egypt Derived from Tailored Geopotential Model EGTM0818 (till degree and order 2190) after Removing Trend Surface Using The 4-parameter Model and The Weighted-Mean. Contour interval 1.0 m
Figure (5.19): 3D - Fitted Geoid Model for Egypt Derived from Tailored Geopotential Model EGTM0818.
Figure (5.19): 3D - Fitted Geoid Model for Egypt Derived from Tailored Geopotential Model EGTM0818.
After the remove-step, the residual height anomalies, as well as residual gravity anomalies, are estimated at the surface of the Earth from the residual gravity Table (5.13) or a combining between a residual gravity
After the remove-step, the residual height anomalies, as well as residual gravity anomalies, are estimated at the surface of the Earth from the residual gravity Table (5.13) or a combining between a residual gravity
Now all tools available for using 3D LSC for geoid solutions; the residual quantities, the covariance function, and the predictions surface area. The rest is to establish the normal equations Eq. (3.52) and compute predictions and error estimates, Eq. (3.53) and (3.56) without parameters (X) i.e. the output of 3D LSC called absolute solution. This may be done using GRAVSOFT program GEOCOL.
Now all tools available for using 3D LSC for geoid solutions; the residual quantities, the covariance function, and the predictions surface area. The rest is to establish the normal equations Eq. (3.52) and compute predictions and error estimates, Eq. (3.53) and (3.56) without parameters (X) i.e. the output of 3D LSC called absolute solution. This may be done using GRAVSOFT program GEOCOL.
Figure (5.22): Difference in Geoid Heights between The Gravimetric Geoid Norav-a and The GPS/Levelling Geoid before Removing Trend Surface. Contour interval = 30 cm.
Figure (5.22): Difference in Geoid Heights between The Gravimetric Geoid Norav-a and The GPS/Levelling Geoid before Removing Trend Surface. Contour interval = 30 cm.
Figure (5.23): Difference in Geoid Heights between The Gravimetric Geoid Nerav-s and The GPS/Levelling Geoid before Removing Trend Surface. Contour interval =30 cm.
Figure (5.23): Difference in Geoid Heights between The Gravimetric Geoid Nerav-s and The GPS/Levelling Geoid before Removing Trend Surface. Contour interval =30 cm.
Table (5.20): Statistics of Gravimetric Geoid Ncray-g for Egypt. Table (5.19): Statistics of Gravimetric Geoid Ncrav-a for Egypt.
Table (5.20): Statistics of Gravimetric Geoid Ncray-g for Egypt. Table (5.19): Statistics of Gravimetric Geoid Ncrav-a for Egypt.
Figure (5.24): Gravimetric Geoid Norav-a for Egypt. Contour interval 1.0 m.
Figure (5.24): Gravimetric Geoid Norav-a for Egypt. Contour interval 1.0 m.
Figure (5.25): Gravimetric Geoid Norav-8 for Egypt. Contour interval 1.0 m.
Figure (5.25): Gravimetric Geoid Norav-8 for Egypt. Contour interval 1.0 m.
Figure (5.28): Difference Gravimetric minus Combined Geoid Solution Computed by Using Tailored Geopotential Model EG1GOC5s in Remove-Restore Technique. Contour interval 5 cm.
Figure (5.28): Difference Gravimetric minus Combined Geoid Solution Computed by Using Tailored Geopotential Model EG1GOC5s in Remove-Restore Technique. Contour interval 5 cm.
Figure (5.29): Difference Gravimetric minus Combined Geoid Solution Computed by Using Tailored Geopotential Model EGTM0818 i Remove-Restore Technique. Contour interval 5 cm.
Figure (5.29): Difference Gravimetric minus Combined Geoid Solution Computed by Using Tailored Geopotential Model EGTM0818 i Remove-Restore Technique. Contour interval 5 cm.
From Table (5.21), the accuracy of both combined geoids is about
From Table (5.21), the accuracy of both combined geoids is about
Related topics:
Physical GeodesyEarth's gravity fieldGeoid DeterminationGravimetric GeoidDeflection of the vertical (DOV)Global Geopotential Models
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