ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ ﻭ ﺍﻟﺠﻴﻭﻴﺩ ﺘﺭﺠﻤﺔ ﻟﻤﻘﺎل ﺭﺍﺌﻊ ﻟﻠﺒﺭﻭﻓﻴﺴﻭﺭ ﺘﺸﺎﺭﻟﺯ ﻤﻴﺭﻱ ﻤﻥ ﺠﺎﻤﻌﺔ ﻜﺎﺒﺘﻭﻥ ﺒﺠﻨﻭﺏ ﺃﻓﺭﻴﻘﻴﺎ ﻨﺸﺭ ﻓﻲ ﻋﺩﺩ ﺃﻏﺴﻁﺱ ٢٠٠٨ﻡ ﻟﻤﺠﻠﺔ ﺍﻟﻤﺴﺎﺤﺔ ﺍﻟﻔﻨﻴﺔ ، Surveying Technicalﻭﺍﻟﻤﻘﺎل ﻴﺸﺭﺡ ﺍﻟﺠﻴﻭﻴﺩ ﻭﻋﻼﻗﺘﻪ ﺒﻘﻴﺎﺴﺎﺕ ﺃﻭ ﺍﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ. ________________________________________________________________ ﻴﻘﺩﻡ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ )ﻭﺒﺎﻗﻲ ﺍﻟﻨﻅﻡ ﺍﻟﻌﺎﻟﻤﻴﺔ ﺍﻷﺨﺭﻯ ﻟﺘﺤﺩﻴﺩ ﺍﻟﻤﻭﺍﻗﻊ ﺒﺎﻟﺭﺼـﺩ ﻋﻠـﻲ ﺍﻷﻗﻤـﺎﺭ ﺍﻟـﺼﻨﺎﻋﻴﺔ Global Navigation Satellite Systemsﺃﻭ ﺍﺨﺘﺼﺎﺭﺍ (GNSSﺍﻟﻤﻭﺍﻗﻊ ﺃﻭ ﺍﻹﺤـﺩﺍﺜﻴﺎﺕ ﺜﻼﺜﻴـﺔ ﺍﻷﺒﻌﺎﺩ :ﺨﻁ ﺍﻟﻁﻭل ﻭ ﺩﺍﺌﺭﺓ ﺍﻟﻌﺭﺽ ﻭ ﺍﻻﺭﺘﻔﺎﻉ .ﻟﻜﻥ ﺍﻟﻤﺭﺠﻊ ﺍﻟﺫﻱ ﺘﻨﺴﺏ ﻟﻪ ﺍﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﺠﻲ ﺒـﻲ ﺇﺱ ﻫﻭ ﺍﻟﺒﺴﻭﻴﺩ WGS84ﺃﻱ ﺃﻥ ﺍﻟﻘﻴﻡ ﺘﺩل ﻋﻠﻲ ﺍﺭﺘﻔﺎﻉ ﺍﻟﻨﻘﻁﺔ ﺃﻋﻠﻲ ﻤﻥ ﺴﻁﺢ ﻫﺫﺍ ﺍﻻﻟﺒﺴﻭﻴﺩ ]ﺍﻻﻟﺒﺴﻭﻴﺩ ﺃﻭ ﺍﻟﺸﻜل ﺍﻟﺒﻴﻀﺎﻭﻱ ﻫﻭ ﺸﻜل ﻨﻅﺭﻱ ﻤﻌﻠﻭﻡ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺭﻴﺎﻀﻴﺔ ﻴﻤﺜل ﺸـﻜل ﻭ ﺤﺠـﻡ ﻜﻭﻜـﺏ ﺍﻷﺭﺽ ، ﻭﻴﺴﺘﺨﺩﻡ ﻓﻲ ﺘﻨﻔﻴﺫ ﻤﻌﺎﺩﻻﺕ ﺤﺴﺎﺏ ﺍﻹﺤﺩﺍﺜﻴﺎﺕ ﻭ ﺇﺴﻘﺎﻁ ﺍﻟﺨﺭﺍﺌﻁ[ ،ﻭﻟﺫﻟﻙ ﺘﺴﻤﻲ ﺍﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﺒﺴﻭﻴﺩﻴﺔ ﺃﻭ ﺍﺭﺘﻔﺎﻋﺎﺕ ﺠﻴﻭﺩﻴﺴﻴﺔ .Ellipsoidal or Geodetic Heightsﻭﻤﻊ ﺃﻥ ﺍﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ ﺃﻗل ﺩﻗﺔ ﻤﻥ ﺍﻹﺤﺩﺍﺜﻴﺎﺕ ﺍﻷﻓﻘﻴﺔ )ﺨﻁ ﺍﻟﻁﻭل ﻭ ﺩﺍﺌﺭﺓ ﺍﻟﻌﺭﺽ( ﺒﻤﻌﺎﻤل ﻴﺘـﺭﺍﻭﺡ ﺒـﻴﻥ ١,٥ﻭ ، ٣ﺃﻻ ﺃﻥ ﻫـﺫﻩ ﺍﻻﺭﺘﻔﺎﻋﺎﺕ ﻤﻔﻴﺩﺓ ﺠﺩﺍ ﻓﻲ ﺘﻁﺒﻴﻘﺎﺕ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺴﺎﺤﻴﺔ. ﻟﻜﻥ ﻨﻭﻉ ﺍﻻﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﻤﻁﻠﻭﺏ ﻓﻲ ﻤﻌﻅﻡ ﺍﻟﺘﻁﺒﻴﻘﺎﺕ ﺍﻟﻬﻨﺩﺴﻴﺔ ﻫﻭ ﺍﻻﺭﺘﻔﺎﻉ ﺍﻟﻤﻘﺎﺱ ﻤﻥ ﻤﺘﻭﺴﻁ ﻤﻨـﺴﻭﺏ ﺴﻁﺢ ﺍﻟﺒﺤﺭ Mean Sea Levelﺃﻭ ﺍﺨﺘﺼﺎﺭﺍ ] MSLﻭﻫﻭ ﺍﻻﺭﺘﻔﺎﻉ ﺍﻟﺫﻱ ﻴﺄﺨﺫ ﺍﺴﻡ ﺍﻟﻤﻨـﺴﻭﺏ ﻓـﻲ ﻤﺼﻁﻠﺤﺎﺕ ﺍﻟﻤﺴﺎﺤﺔ [ ﻭﻴﻌﺭﻑ ﻫﺫﺍ ﺍﻟﻨﻭﻉ ﻤﻥ ﺍﻻﺭﺘﻔﺎﻋﺎﺕ ﺒﺎﺴﻡ ﺍﻻﺭﺘﻔﺎﻉ ﺍﻷﺭﺜـﻭﻤﺘﺭﻱ Orthometric .Heightﻭﻟﻨﻜﻭﻥ ﺃﻜﺜﺭ ﺘﺤﺩﻴﺩﺍ ﻭﺃﻜﺜﺭ ﺩﻗﺔ ﻓﺄﻥ ﺍﻟﻤﺭﺠﻊ ﻟﻬﺫﺍ ﺍﻟﻨﻭﻉ ﻤﻥ ﺍﻻﺭﺘﻔﺎﻋﺎﺕ ﻫﻭ ﺍﻟﺠﻴﻭﻴﺩ :Geoid ﺴﻁﺢ ﻤﺘﺴﺎﻭﻱ ﺍﻟﺠﻬﺩ ﻴﻘﺘﺭﺏ ﺒﻨﺴﺒﺔ ﻜﺒﻴﺭﺓ ﺠﺩﺍ ﻤﻥ ﻤﺘﻭﺴﻁ ﻤﻨﺴﻭﺏ ﺴﻁﺢ ﺍﻟﺒﺤﺭ ]ﻴﻌﺩ ﺍﻟﺠﻴﻭﻴﺩ ﻫﻭ ﺍﻟـﺸﻜل ﺍﻟﺤﻘﻴﻘﻲ ﻟﻸﺭﺽ ،ﻟﻜﻨﻪ ﻭﻟﻸﺴﻑ ﺍﻟﺸﺩﻴﺩ ﺴﻁﺢ ﻤﺘﻌﺭﺝ ﻏﻴﺭ ﻤﻨﺘﻅﻡ ﻟﻴﺱ ﻟﻪ ﻤﻌﺎﺩﻻﺕ ﺤﺴﺎﺒﻴﺔ ﻟﻭﺼـﻔﻪ ﻭ ﺒﺎﻟﺘﺎﻟﻲ ﻻ ﻴﻤﻜﻥ ﺍﺴﺘﺨﺩﺍﻤﻪ ﻓﻲ ﺤﺴﺎﺒﺎﺕ ﺍﻹﺤﺩﺍﺜﻴﺎﺕ ﻭ ﺇﺴﻘﺎﻁ ﺍﻟﺨﺭﺍﺌﻁ[. ﻨﺘﻴﺠﺔ ﻟﻌﺩﺓ ﻋﻭﺍﻤل – ﻤﻨﻬﺎ ﻋﺩﻡ ﺘﺠﺎﻨﺱ ﻜﺜﺎﻓﺔ ﻁﺒﻘﺎﺕ ﺍﻷﺭﺽ – ﻓﺄﻥ ﺍﻟﺠﻴﻭﻴـﺩ ﻴﺒﺘﻌـﺩ ﻋـﻥ ﺍﻻﻟﺒـﺴﻭﻴﺩ ﺒﻤﺴﺎﻓﺎﺕ ﺘﺼل ﺇﻟﻲ ١٢٠ﻤﺘﺭ ،ﻭﻫﺫﺍ ﺍﻟﻔﺭﻕ ﻴـﺴﻤﻲ ﺤﻴـﻭﺩ ﺍﻟﺠﻴﻭﻴـﺩ ﺃﻭ ﺍﺭﺘﻔـﺎﻉ ﺍﻟﺠﻴﻭﻴـﺩ Geoid Undulation or Geoid Heightﻜﻤﺎ ﻫﻭ ﻤﻭﻀﺢ ﻓﻲ ﺍﻟﺸﻜل .١ ﺸﻜل :١ﺍﺭﺘﻔﺎﻉ ﺍﻟﺠﻴﻭﻴﺩ ____________________ ﺘﺭﺠﻤﺔ ﺩ .ﺠﻤﻌﺔ ﻤﺤﻤﺩ ﺩﺍﻭﺩ ١ _______________________ ﺃﺴﺄﻟﻜﻡ ﺍﻟﺩﻋﺎﺀ ﺒﻅﺎﻫﺭ ﺍﻟﻐﻴﺏ ﻟﻜﻲ ﻴﺘﻡ ﺘﺤﻭﻴل ﺍﺭﺘﻔﺎﻉ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ )ﺍﺭﺘﻔﺎﻉ ﺠﻴﻭﺩﻴﺴﻲ( ﺇﻟﻲ ﺍﻻﺭﺘﻔﺎﻉ ﺍﻷﺭﺜﻭﻤﺘﺭﻱ )ﺍﻟﻤﻨـﺴﻭﺏ( ﻓﺄﻨﻨـﺎ ﻨﺤﺘﺎﺝ ﻟﻨﻤﻭﺫﺝ ﺩﻗﻴﻕ ﻤﻥ ﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ ]ﺃﻱ ﻤﻌﺭﻓﺔ ﻗﻴﻤﺔ ﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ ﻋﻥ ﻜل ﻨﻘﻁﺔ ﻤﻁﻠـﻭﺏ ﺘﺤﻭﻴـل ﺍﺭﺘﻔﺎﻋﻬﺎ ﺇﻟﻲ ﻤﻨﺴﻭﺏ[ .ﻭﻫﻨﺎ ﺘﺄﺘﻲ ﺍﻟﺼﻌﻭﺒﺔ .ﻴﻌﺩ ﺘﻁﻭﻴﺭ ﻨﻤﻭﺫﺝ ﺩﻗﻴﻕ ﻟﻠﺠﻴﻭﻴـﺩ ﻋﻤﻠﻴـﺔ ﺼـﻌﺒﺔ ﺠـﺩﺍ ﺴﻨﺘﻨﺎﻭﻟﻬﺎ ﻓﻲ ﺍﻷﺠﺯﺍﺀ ﺍﻟﺘﺎﻟﻴﺔ. ﻨﻤﺫﺠﺔ ﺍﻟﺠﻴﻭﻴﺩ Geoid Modelling ﻋﺎﻤﺔ ﺘﻭﺠﺩ ﻁﺭﻴﻘﺘﻴﻥ ﻟﺘﻁﻭﻴﺭ ﻨﻤﺎﺫﺝ ﺍﻟﺠﻴﻭﻴﺩ :ﺍﻟﻁﺭﻴﻘﺔ ﺍﻟﻬﻨﺩﺴﻴﺔ ،ﻁﺭﻴﻘﺔ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ. -١ﺍﻟﻁﺭﻴﻘﺔ ﺍﻟﻬﻨﺩﺴﻴﺔ: ﻴﻌﺩ ﻫﺫﺍ ﺍﻷﺴﻠﻭﺏ ﻫﻭ ﺍﻷﻤﺜل ﻟﻠﻤﺴﺎﺤﺔ ﺒﺎﻟﺠﻲ ﺒﻲ ﺇﺱ ﻭﺨﺎﺼﺔ ﻟﻠﻤﻨﺎﻁﻕ ﺍﻟﺼﻐﻴﺭﺓ )ﻤﻨﻁﻘﺔ ﺘﻐﻁﻲ ﻤـﺴﺎﺤﺔ ﻤﻥ ١٠ﺇﻟﻲ ٢٠ﻜﻴﻠﻭﻤﺘﺭ ﻤﺭﺒﻊ( .ﻴﺘﻡ ﺘﻨﻔﻴﺫ ﻗﻴﺎﺴﺎﺕ ﺠﻲ ﺒﻲ ﺇﺱ ﻋﻨﺩ ﻤﺠﻤﻭﻋـﺔ ﻤـﻥ ﺍﻟﻨﻘـﺎﻁ ﺍﻟﻤﻌﻠـﻭﻡ ﻤﻨﺴﻭﺒﻬﺎ ]ﻨﻘﺎﻁ ﺭﻭﺒﻴﺭﺍﺕ ﺃﻭ BMﺒﻠﻐﺔ ﺍﻟﻤﺴﺎﺤﺔ[ .ﻓﺈﺫﺍ ﺭﻤﺯﻨﺎ ﻟﻼﺭﺘﻔﺎﻉ ﺍﻟﺠﻴﻭﺩﻴﺴﻲ ﺒﺎﻟﺭﻤﺯ hﻭﻟﻠﻤﻨﺴﻭﺏ ﺒﺎﻟﺭﻤﺯ Hﻭﻟﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ ﺒﺎﻟﺭﻤﺯ Nﻓﺄﻨﻨﺎ ﻴﻤﻜﻨﺎ ﻜﺘﺎﺒﺔ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺎﻟﻴﺔ )ﺃﻨﻅﺭ ﺍﻟﺸﻜل :(١ N=h–H )(1 ﻓﻲ ﺃﺒﺴﻁ ﺍﻟﺼﻭﺭ ﻓﻴﻤﻜﻥ ﺒﺎﺴﺘﺨﺩﺍﻡ ﻨﻘﻁﺔ ﻭﺍﺤﺩﺓ ﻓﻘﻁ ﻤﻌﺭﻓﺔ ﺍﻟﻔﺭﻕ ﺒﻴﻥ ﺴﻁﺤﻲ ﺍﻻﻟﺒﺴﻭﻴﺩ ﻭ ﺍﻟﺠﻴﻭﻴﺩ ،ﺇﻻ ﺃﻥ ﺭﺼﺩ ﺠﻲ ﺒﻲ ﺇﺱ ﻋﻨﺩ ٣ﺭﻭﺒﻴﺭﺍﺕ ﻴﻌﺩ ﻭﻀﻌﺎ ﺃﻓﻀل ﺒﺎﻟﺘﺄﻜﻴﺩ .ﻭﺠﻭﺩ ٣ﻨﻘﺎﻁ ﻤﻌﻠﻭﻡ ﻟﻬﻡ ﻜﻼ ﻤـﻥ h ﻭ Hﺴﻴﻤﻜﻨﻨﺎ ﻤﻥ ﺤﺴﺎﺏ ٣ﻤﻌﺎﻤﻼﺕ ) ﺍﻟﻤﻴل tiltﻓﻲ ﺍﺘﺠﺎﻩ ﺍﻟﺸﻤﺎل ،ﺍﻟﻤﻴل ﻓﻲ ﺍﺘﺠﺎﻩ ﺍﻟﺸﺭﻕ ،ﺍﻟﻔﺭﻕ ( ﻟﻭﺼﻑ ﺍﻟﻔﺭﻭﻕ ﺒﻴﻥ ﻜﻼ ﺍﻟﺴﻁﺤﻴﻥ .ﺃﻱ ﺃﻥ ﺍﻟﺠﻴﻭﻴﺩ ﻴﺘﻡ ﺘﻤﺜﻴﻠﻪ ﻤﻥ ﺨﻼل ﺴﻁﺢ ﺃﻭ ﻤﺴﺘﻭﻱ ﻤﺎﺌل tilted . planeﻭﺒﻌﺩ ﺫﻟﻙ ﻴﻤﻜﻥ ﺍﺴﺘﺨﺩﺍﻡ ﻫﺫﺍ ﺍﻟﻨﻤﻭﺫﺝ ﺃﻭ ﻫﺫﺍ ﺍﻟﻤﺴﺘﻭﻱ ﻟﻜﻲ ﻨﺤﻭل ﺍﺭﺘﻔﺎﻉ ﺍﻟﺠﻲ ﺒـﻲ ﺇﺱ ﻷﻱ ﻨﻘﻁﺔ ﺠﺩﻴﺩﺓ ﻤﺭﺼﻭﺩﺓ ﺇﻟﻲ ﻤﻨﺴﻭﺒﻬﺎ .ﻭﺒﺎﻟﻁﺒﻊ ﻴﻤﻜﻥ ﺍﺴﺘﺨﺩﺍﻡ ﺃﻜﺜﺭ ﻤﻥ ٣ﻨﻘﺎﻁ )ﻤﻌﻠـﻭﻡ ﻋﻨـﺩﻫﺎ hﻭ (H ﻭﺫﻟﻙ ﻟﻠﺤﺼﻭل ﻋﻠﻲ ﻤﺼﺩﺍﻗﻴﺔ ﺃﻜﺜﺭ more reliabilityﻟﻨﺘﺎﺌﺞ ﺍﻟﻤﺴﺘﻭﻱ ﺍﻟﻤﺎﺌل ]ﺍﺴﺘﺨﺩﻡ ٣ﻨﻘﺎﻁ ﻤﻌﻠﻭﻤﺔ ﻓﻘﻁ ﻴﻌﻁﻲ ٣ﻤﻌﺎﺩﻻﺕ ﻤﻁﻠﻭﺏ ﺤﻠﻬﻡ ﻓﻲ ٣ﻗﻴﻡ ﻤﺠﻬﻭﻟﺔ ﺃﻱ – ﺭﻴﺎﻀﻴﺎ ﻭ ﺇﺤﺼﺎﺌﻴﺎ -ﻻ ﻴﻭﺠﺩ ﺃﻱ ﺘﺤﻘﻴﻕ checkﻟﻠﻨﺘﺎﺌﺞ .ﺒﻴﻨﻤﺎ ﺍﺴﺘﺨﺩﺍﻡ ﺃﻜﺜﺭ ﻤﻥ ٣ﻨﻘﺎﻁ ﺴﻴﻌﻁﻲ ﻋﺩﺩ ﻤﻌﺎﺩﻻﺕ ﺃﻜﺒﺭ ﻤﻥ ﻋﺩﺩ ﺍﻟﻤﺠﺎﻫﻴـل ﻤﻤـﺎ ﺴﻴﻨﺘﺞ ﻋﻨﻪ ﻭﺠﻭﺩ ﺘﺤﻘﻴﻕ ﻭﻤﺅﺸﺭﺍﺕ ﺇﺤﺼﺎﺌﻴﺔ ﻟﺠﻭﺩﺓ ﺍﻟﻨﺘﺎﺌﺞ ﺍﻟﻤﺤﺴﻭﺒﺔ[ .ﺃﻴﻀﺎ ﻴﻤﻜﻥ ﺍﺴـﺘﺨﺩﺍﻡ ﻨﻤـﺎﺫﺝ ﺭﻴﺎﻀﻴﺔ ﺃﻜﺜﺭ ﺩﻗﺔ )ﻤﻥ ﻨﻤﻭﺫﺝ ﺍﻟﺴﻁﺢ ﺍﻟﻤﺎﺌل( ﺒﻔﺭﺽ ﻭﺠﻭﺩ ﻋﺩﺩ ﺃﻜﺒﺭ ﻤﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﻤﻌﻠﻭﻤﺔ )ﻤﻌﻠﻭﻡ ﻟﻬﺎ h ﻭ (Hﻟﻜﻨﻬﺎ ﺘﺤﺘﺎﺝ ﺨﺒﺭﺓ ﺠﻴﻭﺩﻴﺴﻴﺔ ﺃﻜﺒﺭ ﻟﺩﻱ ﺍﻟﻤﺴﺘﺨﺩﻡ. ﺸﻜل :٢ﻓﻜﺭﺓ ﺍﻟﻤﺴﺘﻭﻱ ﺍﻟﻤﺎﺌل ﻟﺘﻤﺜﻴل ﺍﻟﺠﻴﻭﻴﺩ ____________________ ﺘﺭﺠﻤﺔ ﺩ .ﺠﻤﻌﺔ ﻤﺤﻤﺩ ﺩﺍﻭﺩ ٢ _______________________ ﺃﺴﺄﻟﻜﻡ ﺍﻟﺩﻋﺎﺀ ﺒﻅﺎﻫﺭ ﺍﻟﻐﻴﺏ ﺃﻫﻡ ﻤﻌﻭﻗﺎﺕ ﺍﻟﻁﺭﻴﻘﺔ ﺍﻟﻬﻨﺩﺴﻴﺔ: ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺭﻴﺎﻀﻲ ﺍﻟﻤﺴﺘﻨﺒﻁ ﻴﺼﻠﺢ ﻓﻘﻁ ﻟﻠﻤﻨﻁﻘﺔ ﺍﻟﻤﺤﺼﻭﺭﺓ ﺒﺎﻟﻨﻘـﺎﻁ ﺍﻟﻤﻌﻠﻭﻤـﺔ )ﻤﺤﺎﻭﻟـﺔ ﺍﺴـﺘﻨﺒﺎﻁ extrapolationﻗﻴﻤﺔ Nﺨﺎﺭﺝ ﺍﻟﻤﻨﻁﻘﺔ ﻟﻥ ﺘﻜﻭﻥ ﺠﻴﺩﺓ ﻋﻠﻲ ﺍﻹﻁﻼﻕ(. ﻨﻤﻭﺫﺝ ﺍﻟﻤﺴﺘﻭﻱ ﺍﻟﻤﺎﺌل – ﻨﻤﻭﺫﺝ ﺒﺴﻴﻁ ﺭﻴﺎﻀﻴﺎ – ﻭﻴﺼﻠﺢ ﻓﻘﻁ ﻟﻤﻨﺎﻁﻕ ﺼﻐﻴﺭﺓ )ﺸﻜل ﻭﺘﻐﻴﺭ ﺍﻟﺠﻴﻭﻴﺩﺃﻜﺜﺭ ﺘﻌﻘﻴﺩﺍ ﻤﻥ ﻤﺤﺎﻭﻟﺔ ﻭﺼﻔﻪ ﺒﺴﻁﺢ ﻤﺎﺌل!(. ﻋﻤﻠﻴﺎ ﻗﺩ ﻴﻜﻭﻥ ﻤﻥ ﺍﻟﺼﻌﺏ ﺇﻴﺠﺎﺩ ﻨﻘﺎﻁ ﻤﻌﻠﻭﻤﺔ ﺍﻟﻤﻨﺴﻭﺏ )ﺭﻭﺒﻴﺭﺍﺕ ﺃﻭ (BMﻓﻲ ﺍﻟﻤﻨﻁﻘﺔ ﺍﻟﻤﻁﻠـﻭﺏﺍﻟﻌﻤل ﻓﻴﻬﺎ. -٢ﻁﺭﻴﻘﺔ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ: ﻴﻘﺩﻡ ﻫﺫﺍ ﺍﻷﺴﻠﻭﺏ ﺸﺒﻜﺔ ﻤﻨﺘﻅﻤﺔ ﻤﻥ ﻗﻴﺎﺴﺎﺕ ﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ ﻓﻲ ﻤﻨـﺎﻁﻕ ﻜﺒﻴـﺭﺓ ﺃﻭ ﺸﺎﺴـﻌﺔ ﺒﺎﻟﻤﻘﺎﺭﻨـﺔ ﺒﻤﺠﻤﻭﻋﺔ ﻨﻘﺎﻁ ﻤﺘﻔﺭﻗﺔ ﻓﻲ ﻤﻨﻁﻘﺔ ﺼﻐﻴﺭﺓ ﻜﻤﺎ ﻓﻲ ﺍﻟﻁﺭﻴﻘﺔ ﺍﻟﻬﻨﺩﺴﻴﺔ .ﻟﻜﻥ ﻫﺫﻩ ﺍﻟﻁﺭﻴﻘﺔ – ﻓﻲ ﺍﻟﻤﻘﺎﺒل – ﻟﻴﺴﺕ ﺴﻬﻠﺔ ﺭﻴﺎﻀﻴﺎ ﺤﻴﺙ ﺃﻨﻬﺎ ﺘﺘﻁﻠﺏ ﻋﻤل ﺘﻜﺎﻤـل integrationﻟﻘـﻴﻡ ﺸـﺫﻭﺫ ﺍﻟﺠﺎﺫﺒﻴـﺔ ﺍﻷﺭﻀـﻴﺔ Gravity Anomaliesﻟﻴﻤﻜﻥ ﺤﺴﺎﺏ ﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ .ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻷﺴﺎﺴﻴﺔ ﻓﻲ ﻫﺫﻩ ﺍﻟﻁﺭﻴﻘﺔ ﻤﻌﺭﻭﻓﺔ ﺒﺎﺴﻡ ﻤﻌﺎﺩﻟﺔ ﺴﺘﻭﻜﺱ Stokesﻨﺴﺒﺔ ﻟﻠﻌﺎﻟﻡ ﺍﻟﺠﻴﻭﺩﻴﺴﻲ ﺍﻟﺫﻱ ﺃﺒﺘﻜﺭﻫﺎ: N = ( R / 4π ) ∫∫ Δg S(ψ) dσ )(2 ﻭﺩﻭﻥ ﺍﻟﺩﺨﻭل ﻓﻲ ﺍﻟﺘﻔﺎﺼﻴل ﺍﻟﻔﻨﻴﺔ ﻟﻬﺫﻩ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻓﺄﻥ ﺍﻷﺭﺼﺎﺩ ﺃﻭ ﺍﻟﻘﻴﺎﺴﺎﺕ ﺍﻟﻤﻁﻠﻭﺒﺔ ﻫﻲ ﻤﺎ ﺘﻌﺭﻑ ﺒﺎﺴﻡ ﺸﺫﻭﺫ ﺍﻟﺠﺎﺫﺒﻴﺔ Δgﻭﻫﻲ ﺘﻤﺜل ﺍﻟﻔﺭﻕ ﺒﻴﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺎﺴﺔ ﻟﻠﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ]ﻨﺴﺘﺨﺩﻡ ﺃﺠﻬﺯﺓ ﺨﺎﺼﺔ ﻟﻘﻴﺎﺱ ﻗﻴﻤﺔ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﺘﺴﻤﻲ [ Gravimeterﻭ ﻗﻴﻤﺔ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻟﻨﻅﺭﻴـﺔ ]ﻴﻤﻜـﻥ ﺤـﺴﺎﺒﻬﺎ ﺭﻴﺎﻀـﻴﺎ ﺒﻤﻌﺎﺩﻻﺕ ﺘﻌﺘﻤﺩ ﻓﻘﻁ ﻋﻠﻲ ﻨﻭﻉ ﺍﻻﻟﺒﺴﻭﻴﺩ ﺍﻟﻤﺴﺘﺨﺩﻡ ﻟﺘﻤﺜﻴل ﺸﻜل ﺍﻷﺭﺽ[ .ﻭﻜﻤﺎ ﻨﺭﻱ ﻓﻲ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻓـﺄﻥ ﺍﻟﺘﻜﺎﻤل ∫∫ ﻴﺘﻡ ﻋﻠﻲ ﻜل ﺴﻁﺢ ﺍﻷﺭﺽ ،ﺃﻱ ﺃﻨﻪ ﻟﺤﺴﺎﺏ ﻗﻴﻤﺔ ﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ Nﻋﻨﺩ ﻨﻘﻁﺔ ﻭﺍﺤﺩﺓ ﻓﻴﻠﺯﻤﻨﺎ ﻋﺸﺭﺍﺕ ﺍﻵﻻﻑ ﻤﻥ ﻗﻴﺎﺴﺎﺕ ﺸﺫﻭﺫ ﺍﻟﺠﺎﺫﺒﻴﺔ .ﺃﻴﻀﺎ ﻓﺄﻥ ﻗﻴﺎﺴﺎﺕ ﺸﺫﻭﺫ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺘﺤﺘﺎﺝ ﺘﺼﺤﻴﺤﺎ ﺇﻀـﺎﻓﻴﺎ ﻴﻌﺘﻤﺩ ﻋﻠﻲ ﻤﻌﺭﻓﺔ ﺘﻀﺎﺭﻴﺱ ﺍﻷﺭﺽ ﻤﻤﺎ ﻴﺘﻭﺠﺏ ﻤﻌﻪ ﺃﻨﻨﺎ ﻨﺤﺘﺎﺝ ﻨﻤﻭﺫﺝ ﺍﺭﺘﻔﺎﻋـﺎﺕ ﺭﻗﻤـﻲ Digital Elevation Modelﺃﻭ ] DEMﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﻫﺫﻩ ﺍﻟﻁﺭﻴﻘﺔ ﻓﻲ ﻤﺼﺭ ﺒﻌﺩ ﺍﻜﺘﻤﺎل ﺭﺼﺩ ﺍﻟﺸﺒﻜﺔ ﺍﻟﻘﻭﻤﻴـﺔ ﺍﻟﻤﺼﺭﻴﺔ ﻟﻠﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻓﻲ ﻋﺎﻡ ١٩٩٨ﻡ[ . ﻴﻭﺠﺩ ﺒﺩﻴل ﻟﺘﻤﺜﻴل ﺸﻜل ﺍﻟﺠﻴﻭﻴﺩ ﻭﻫﻭ ﺍﺴﺘﺨﺩﺍﻡ " ﻨﻤﻭﺫﺝ ﺘﻤﺜﻴل ﻜﺭﻭﻱ ﻤﺘﻨﺎﺴﻕ ﻟﻤﺠـﺎل ﺠﻬـﺩ ﺍﻷﺭﺽ " . Spherical harmonic expansion of the Earth's geopotential fieldﻏﺎﻟﺒﺎ ﻴﺘﻡ ﺘﻁﻭﻴﺭ ﻤﺜل ﻫﺫﺍ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﻌﺎﻟﻤﻲ ﻭ ﺘﺤﺩﻴﺩ ﻗﻴﻡ ﻤﻌﺎﻤﻼﺘﻪ ] ﻴﺒﻠﻎ ﻋﺩﺩﻫﺎ ﺃﻻﻑ ﺍﻟﻤﻌﺎﻤﻼﺕ ﻭﻟﻴﺱ ﻓﻘﻁ ٣ﻤﻌﺎﻤﻼﺕ ﻤﺜـل ﻁﺭﻴﻘﺔ ﺍﻟﻤﺴﺘﻭﻱ ﺍﻟﻤﺎﺌل[ ﻤﻥ ﺨﻼل ﺘﺤﻠﻴل ﻤﺩﺍﺭﺍﺕ ﻨﻭﻋﻴﺔ ﺨﺎﺼﺔ ﻤـﻥ ﺍﻷﻗﻤـﺎﺭ ﺍﻟـﺼﻨﺎﻋﻴﺔ ﻤﻨﺨﻔـﻀﺔ ﺍﻻﺭﺘﻔﺎﻉ. ﺤﺎﻟﻴﺎ ﻨﻁﺒﻕ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﻌﺎﻟﻤﻴﺔ ﻟﺘﻤﺜﻴل ﻜﺭﻭﻱ ﻤﺘﻨﺎﺴﻕ ﻟﻤﺠﺎل ﺠﻬﺩ ﺍﻷﺭﺽ ﺒﺄﺤﺩ ﺃﺴﻠﻭﺒﻴﻥ: ____________________ ﺘﺭﺠﻤﺔ ﺩ .ﺠﻤﻌﺔ ﻤﺤﻤﺩ ﺩﺍﻭﺩ ٣ _______________________ ﺃﺴﺄﻟﻜﻡ ﺍﻟﺩﻋﺎﺀ ﺒﻅﺎﻫﺭ ﺍﻟﻐﻴﺏ ﺃ -ﺍﺴﺘﺨﺩﺍﻡ ﺍﻟﻨﻤﺎﺫﺝ ﻟﺤﺴﺎﺏ ﺍﻟﺘﻐﻴﺭﺍﺕ ﺃﻭ ﺍﻟﻘﻴﻡ ﺍﻟﺭﺌﻴﺴﻴﺔ ﻟﺸﻜل ﺍﻟﺠﻴﻭﻴﺩ ﺜﻡ ﺇﻀﺎﻓﺔ ﺍﻟﺘﻐﻴﺭﺍﺕ ﺍﻟﺩﻗﻴﻘﺔ ﺍﻟﺘـﻲ ﻴﺘﻡ ﺤﺴﺎﺒﻬﺎ ﺒﺎﺴﺘﺨﺩﺍﻡ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺭﻗﻡ ) (٢ﻟﻘﻴﺎﺴﺎﺕ ﺘﻤﺕ ﻓﻲ ﺍﻟﻤﻨﻁﻘﺔ ﺍﻟﻤﻁﻠﻭﺒﺔ ﻓﻘﻁ ﻭ ﻟﻴﺱ ﻟﻸﺭﺽ ﻜﻠﻬـﺎ ، ﻭﺒﺫﻟﻙ ﻨﺤﺼل ﻋﻠﻲ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ. ﺏ -ﺍﺴﺘﺨﺩﺍﻡ ﺒﻴﺎﻨﺎﺕ ﺍﻷﻗﻤﺎﺭ ﺍﻟﺼﻨﺎﻋﻴﺔ ﻭ ﻗﻴﺎﺴﺎﺕ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻤﻌﺎ )ﻓﻲ ﻨﻔﺱ ﺍﻟﺒﺭﻨـﺎﻤﺞ( ﻟﺤـﺴﺎﺏ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ ،ﻟﻜﻥ ﻫﺫﺍ ﺍﻷﺴﻠﻭﺏ ﻴﺘﻁﻠﺏ ﺃﺠﻬﺯﺓ ﻜﻤﺒﻴﻭﺘﺭ ﺒﻤﻭﺍﺼﻔﺎﺕ ﺘﻘﻨﻴﺔ ﻋﺎﻟﻴﺔ ﺠﺩﺍ ﻻ ﺘﺘﻭﺍﻓﺭ ﺇﻻ ﻟﺩﻱ ﺍﻟﻤﺅﺴﺴﺎﺕ ﺍﻟﻌﺎﻟﻤﻴﺔ ﺍﻟﻜﺒﺭﻯ ﻭﻤﻊ ﺃﻥ ﻁﺭﻴﻘﺔ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﺘﺘﻴﺢ ﺍﻟﺤﺼﻭل ﻋﻠﻲ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ ﺘﻔﺼﻴﻠﻲ ﻟﻤﻨﻁﻘﺔ ﺸﺎﺴﻌﺔ ﺇﻻ ﺃﻥ ﻟﻬـﺎ ﺒﻌﺽ ﺍﻟﻤﻌﻭﻗﺎﺕ ﻤﻨﻬﺎ: ﺃﻨﻬﺎ ﻁﺭﻴﻘﺔ ﻤﻌﻘﺩﺓ ﺭﻴﺎﻀﻴﺎ ﻭﺤﺴﺎﺒﻴﺎ ﺃﻴﻀﺎ. ﺩﻗﺔ ﺍﻟﻨﺘﺎﺌﺞ ﺘﻌﺘﻤﺩ ﻋﻠﻲ ﺩﻗﺔ ﻗﻴﺎﺴﺎﺕ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ،ﻜﻤﺎ ﺃﻥ ﻫﻨﺎﻙ ﻤﻨﺎﻁﻕ ﻜﺜﻴﺭﺓ ﻤﻥ ﺍﻟﻌﺎﻟﻡ ﻻ ﺘﻭﺠﺩﺒﻬﺎ ﻗﻴﺎﺴﺎﺕ ﺠﺎﺫﺒﻴﺔ ﺃﺭﻀﻴﺔ ﺘﻔﺼﻴﻠﻴﺔ ﻤﻤﺎ ﻴﻨﺘﺞ ﻋﻨﻪ ﻓﺭﺍﻏﺎﺕ ﻓﻲ ﻗﺎﻋﺩﺓ ﺒﻴﺎﻨﺎﺕ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻭ ﺘـﺅﺩﻱ ﻟﻔﺭﺍﻏﺎﺕ ﺃﻭ ﻋﺩﻡ ﺩﻗﺔ ﺍﻟﺠﻴﻭﻴﺩ ﺍﻟﻤﺤﺴﻭﺏ ﻋﻨﺩ ﻫﺫﻩ ﺍﻟﻤﻨﺎﻁﻕ ]ﻟﻸﺴﻑ ﺍﻟﺸﺩﻴﺩ ﻓﺄﻥ ﻜل ﺍﻟﺩﻭل ﺍﻟﻌﺭﺒﻴﺔ ﺘﻌﺘﺒـﺭ ﻗﻴﺎﺴﺎﺕ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻟﺩﻴﻬﺎ ﻭﻜﺄﻨﻬﺎ ﺃﺴﺭﺍﺭ ﻋﺴﻜﺭﻴﺔ ﻭ ﻻ ﺘﺴﻤﺢ ﺒﻨﺸﺭﻫﺎ ﺃﻭ ﺍﻟﻤﺴﺎﻫﻤﺔ ﺒﻬﺎ ﻓﻲ ﺘﻁﻭﻴﺭ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﻌﺎﻟﻤﻴﺔ ﻤﻤﺎ ﻴﺠﻌل ﻤﻌﻅﻡ ﻫﺫﻩ ﺍﻟﻨﻤﺎﺫﺝ ﻏﻴﺭ ﺩﻗﻴﻘﺔ ﻟﻼﺴﺘﺨﺩﺍﻡ ﻓﻲ ﺤﺴﺎﺏ ﺍﻟﺠﻴﻭﻴﺩ ﻓﻲ ﻫﺫﻩ ﺍﻟﺩﻭل[. ﻴﺘﻌﺭﺽ ﺍﻟﺠﻴﻭﻴﺩ ﺍﻟﻤﺤﺴﻭﺏ ﻤﻥ ﻫﺫﻩ ﺍﻟﻁﺭﻴﻘﺔ ﻟﻌﺩﺩ ﻤﻥ ﺍﻷﺨﻁﺎﺀ ﻨﺘﻴﺠـﺔ ﻷﺨﻁـﺎﺀ ﻤـﺩﺍﺭﺍﺕ ﺍﻷﻗﻤـﺎﺭﺍﻟﺼﻨﺎﻋﻴﺔ ﻭﺃﻴﻀﺎ ﻨﺘﻴﺠﺔ ﺍﻟﻔﺭﺍﻏﺎﺕ ﻓﻲ ﻗﺎﻋﺩﺓ ﺒﻴﺎﻨﺎﺕ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ. ﺍﻷﺴﻠﻭﺏ ﺍﻟﺘﻜﺎﻤﻠﻲ Combination Approach ﻤﻥ ﺍﻟﻤﻤﻜﻥ ﺍﻟﺩﻤﺞ ﺒﻴﻥ ﺍﻷﺴﻠﻭﺏ ﺍﻟﻬﻨﺩﺴﻲ ﻭ ﺃﺴﻠﻭﺏ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻟﺘﻁﻭﻴﺭ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ .ﺃﻭﻻ ﻴـﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﻁﺭﻴﻘﺔ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻹﻨﺘﺎﺝ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ ﺃﻭﻟﻲ ،ﺜﻡ ﻴﺘﻡ ﻤﻌﺎﻴﺭﺘﻪ ﻋﻨﺩ ﻨﻘﺎﻁ ﻤﺴﺎﺤﻴﺔ ﻤﻌﻠـﻭﻡ ﻋﻨﺩﻫﺎ ﻗﻴﻡ ﺤﻴﻭﺩ ﺍﻟﺠﻴﻭﻴﺩ )ﺃﻱ ﻤﻌﻠﻭﻡ ﻟﻬﺎ ﺍﻻﺭﺘﻔﺎﻉ ﺍﻟﺠﻴﻭﺩﻴﺴﻲ hﻭ ﺍﻟﻤﻨـﺴﻭﺏ .(Hﻭﻫـﺫﻩ ﺍﻟﻤﻌـﺎﻴﺭﺓ ﺃﻭ ﺍﻟﺘﺼﺤﻴﺢ ﻤﻥ ﺍﻟﻤﻤﻜﻥ ﺃﻥ ﻴﺘﻡ ﻤﻥ ﺨﻼل ﻨﻤﻭﺫﺝ ﺭﻴﺎﻀﻲ ﺒﺴﻴﻁ ﻤﺜل ﺍﻟﻤﺴﺘﻭﻱ ﺍﻟﻤﺎﺌل ،ﺃﻭ ﻨﻤﺎﺫﺝ ﺭﻴﺎﻀـﻴﺔ ﺃﻜﺜﺭ ﺘﻌﻘﻴﺩﺍ ﻋﻨﺩ ﺤﺴﺎﺏ ﺍﻟﺠﻴﻭﻴﺩ ﻟﻤﻨﺎﻁﻕ ﻭﺍﺴﻌﺔ ﺃﻭ ﻜﺒﻴﺭﺓ .ﻭﺒﻌﺩ ﺘﻁﻭﻴﺭ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ ﻟﺩﻭﻟﺔ – ﻤـﺜﻼ – ﻴﻤﻜﻥ ﺍﺴﺘﺨﺩﺍﻤﻪ ﻓﻲ ﺘﺤﻭﻴل ﺍﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ ﺇﻟﻲ ﻤﻨﺎﺴﻴﺏ. ﻨﻤﺎﺫﺝ ﺍﻟﺠﻴﻭﻴﺩ ﺍﻟﻌﺎﻟﻤﻴﺔ ﺍﻟﺤﺎﻟﻴﺔ: ﺤﺘﻲ ﻭﻗﺕ ﻗﺭﻴﺏ ﻜﺎﻥ ﺃﻓﻀل ﻨﻤﺎﺫﺝ ﺍﻟﺠﻴﻭﻴﺩ ﺍﻟﻌﺎﻟﻤﻴﺔ ﻫﻭ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﻤﻌﺭﻭﻑ ﺍﺨﺘﺼﺎﺭﺍ ﺒﺎﺴـﻡ EGM96 ﺍﻟﺫﻱ ﻁﻭﺭﺘﻪ ﻫﻴﺌﺔ ﺍﻟﻤﺴﺎﺤﺔ ﺍﻟﻌﺴﻜﺭﻴﺔ ﺍﻷﻤﺭﻴﻜﻴﺔ ﻭﻗﺴﻡ ﺍﻟﻤﺴﺎﺤﺔ ﺍﻟﺠﻴﻭﺩﻴﺴﻴﺔ ﻓﻲ ﺠﺎﻤﻌﺔ ﻭﻻﻴـﺔ ﺃﻭﻫـﺎﻴﻭ ﺍﻷﻤﺭﻴﻜﻴﺔ ]ﻟﻲ ﻜل ﺍﻟﺸﺭﻑ ﺃﻥ ﺃﻜﻭﻥ ﺃﺤﺩ ﺨﺭﻴﺠﻲ ﻫﺫﺍ ﺍﻟﻘﺴﻡ[ ﻓﻲ ﻋﺎﻡ ١٩٩٦ﻡ .ﻭﻜﺎﻨﺕ ﻜـل ﺃﺠﻬـﺯﺓ ﻭ ﺃﻴﻀﺎ ﺒﺭﺍﻤﺞ ﺤﺴﺎﺒﺎﺕ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ softwareﺘﺤﺘﻭﻱ ﺩﺍﺨﻠﻬﺎ ﻋﻠﻲ ﻫﺫﺍ ﺍﻟﻨﻤﻭﺫﺝ ﻟﻜﻲ ﺘﺴﺘﻁﻴﻊ ﺘﺤﻭﻴـل ﺍﺭﺘﻔﺎﻉ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ ﺇﻟﻲ ﻤﻨﺴﻭﺏ ﻭﺒﺩﻗﺔ ﻤﺘﺭ ﺃﻭ ﺃﻜﺜﺭ ﻗﻠﻴﻼ ]ﺘﻡ ﺘﻘﺩﻴﺭ ﺩﻗﺔ ﻨﻤﻭﺫﺝ EGM96ﻓﻲ ﻤﺼﺭ – ﻜﻤﺜﺎل -ﺒﺤﻭﺍﻟﻲ ١ﻤﺘﺭ ﻓﻘﻁ ﻓﻲ ﺍﻟﻤﺘﻭﺴﻁ[. ﻨﻤﻭﺫﺝ ﺍﻟﺠﻴﻭﻴﺩ EGM96ﻭﺃﻤﺜﺎﻟﻪ ﻴﺘﻤﻴﺯﻭﺍ ﺒﺩﺭﺠﺔ ﺘﻤﺜﻴل degreeﺘﺴﺎﻭﻱ ، ٣٦٠ﺃﻱ ﺃﻨﻪ ﻴﻤﻜﻨﻪ ﺘﺤﺩﻴـﺩ ﻗﻴﻤﺔ ﺍﻟﺠﻴﻭﻴﺩ ﻜل ١ﺩﺭﺠﺔ ﻤﻥ ﺨﻁﻭﻁ ﺍﻟﻁﻭل ﻭﺩﻭﺍﺌﺭ ﺍﻟﻌﺭﺽ .ﻭﺒﻤﻌﻨﻲ ﺁﺨﺭ ﻓﺄﻥ ﺍﻟﺘﻐﻴﺭ ﻓﻲ ﻗﻴﻤﺔ ﺤﻴـﻭﺩ ____________________ ﺘﺭﺠﻤﺔ ﺩ .ﺠﻤﻌﺔ ﻤﺤﻤﺩ ﺩﺍﻭﺩ ٤ _______________________ ﺃﺴﺄﻟﻜﻡ ﺍﻟﺩﻋﺎﺀ ﺒﻅﺎﻫﺭ ﺍﻟﻐﻴﺏ ﺍﻟﺠﻴﻭﻴﺩ ﻓﻴﻤﺎ ﻫﻭ ﺃﻗل ﻤﻥ ﺩﺭﺠﺔ )ﺃﻱ ﺤﻭﺍﻟﻲ ١٠٠ﻜﻴﻠﻭﻤﺘﺭ( ﻟﻥ ﻴﻤﻜﻥ ﺘﺤﺩﻴﺩﻩ ﺒﺩﻗـﺔ .ﻭﻨﺘﻴﺠـﺔ ﺃﺨﻁـﺎﺀ ﻤﺩﺍﺭﺍﺕ ﺍﻷﻗﻤﺎﺭ ﺍﻟﺼﻨﺎﻋﻴﺔ ﻭ ﺍﻟﻔﺭﺍﻏﺎﺕ ﻓﻲ ﻗﺎﻋﺩﺓ ﺒﻴﺎﻨﺎﺕ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ ﻓﺄﻥ ﺩﻗﺔ ﻨﻤـﻭﺫﺝ EGM96 ﻓﻲ ﺍﻟﻤﺘﻭﺴﻁ ﻭﻋﻠﻲ ﺍﻟﻤﺴﺘﻭﻱ ﺍﻟﻌﺎﻟﻤﻲ ﺘﺒﻠﻎ ٤٠ﺴﻨﺘﻴﻤﺘﺭ ﺘﻘﺭﻴﺒﺎ )ﻤﻊ ﻭﺠﻭﺩ ﻓﺭﻭﻕ ﻜﺒﻴﺭﺓ ﺃﻭ ﻤﺴﺘﻭﻴﺎﺕ ﺩﻗﺔ ﺃﺴﻭﺃ ﻟﻸﺠﺯﺍﺀ ﻜﺒﻴﺭﺓ ﻤﻥ ﻗﺎﺭﺓ ﺃﻓﺭﻴﻘﻴﺎ ﻭ ﺍﻟﺸﺭﻕ ﺍﻷﻭﺴﻁ ﺃﻴﻀﺎ(. ﺤﺩﻴﺜﺎ – ﻓﻲ ﻤﻨﺘﺼﻑ ﻋﺎﻡ ٢٠٠٨ﻡ – ﺃﻁﻠﻘﺕ ﻭﻜﺎﻟﺔ ﺍﻻﺴﺘﺨﺒﺎﺭﺍﺕ ﺍﻷﺭﻀﻴﺔ ﺍﻟﻭﻁﻨﻴﺔ ﺍﻷﻤﺭﻴﻜﻴـﺔ ]ﺍﻻﺴـﻡ ﺍﻟﺠﺩﻴﺩ ﻟﻬﻴﺌﺔ ﺍﻟﻤﺴﺎﺤﺔ ﺍﻟﻌﺴﻜﺭﻴﺔ ﺍﻷﻤﺭﻴﻜﻴﺔ ﺒﻌﺩ ﺩﻤﺠﻬﺎ ﻤﻊ ﺠﻬﺎﺕ ﻋﻠﻤﻴﺔ ﺃﻤﺭﻴﻜﻴﺔ ﺃﺨﺭﻱ[ ﻨﻤـﻭﺫﺝ ﺠﺩﻴـﺩ ﻟﻠﺠﻴﻭﻴﺩ ﺃﺴﻤﺘﻪ .EGM2008ﺃﻋﺘﻤﺩ ﺍﻟﺠﻴﻭﻴﺩ ﺍﻟﺠﺩﻴﺩ ﻓﻲ ﺘﻁﻭﻴﺭﻩ ﻋﻠﻲ ﻗﻴﺎﺴﺎﺕ ﺠﺎﺫﺒﻴﺔ ﺃﺭﻀـﻴﺔ ﻋﺎﻟﻤﻴـﺔ ﺘﻤﺕ ﻤﻥ ﺨﻼل ﻨﻭﻉ ﺠﺩﻴﺩ ﻤﻥ ﺍﻷﻗﻤﺎﺭ ﺍﻟﺼﻨﺎﻋﻴﺔ GRACEﺒﺎﻹﻀﺎﻓﺔ ﻟﻘﻴﺎﺴﺎﺕ ﺠﺎﺫﺒﻴﺔ ﺃﺭﻀﻴﺔ ﻷﺠـﺯﺍﺀ ﻜﺒﻴﺭﺓ ﻤﻥ ﺍﻟﻌﺎﻟﻡ .ﻭﺘﺒﻠﻎ ﺩﺭﺠﺔ ﺘﻤﺜﻴل ﻨﻤﻭﺫﺝ ) ٢١٩٠ EGM2008ﺒﺎﻟﻤﻘﺎﺭﻨﺔ ﺒﺩﺭﺠﺔ ﺘﻤﺜﻴل = ٣٦٠ﻓﻘﻁ ﻟﻠﻨﻤﻭﺫﺝ ﺍﻟﻘﺩﻴﻡ (EGM96ﺤﻴﺙ ﻴﺒﻠﻎ ﻋﺩﺩ ﻤﻌﺎﻤﻼﺕ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺠﺩﻴﺩ ٤,٧ﻤﻠﻴﻭﻥ ﻗﻴﻤﺔ ،ﻤﻤﺎ ﻴﺩل ﺒـﺼﻔﺔ ﻋﺎﻤﺔ ﻋﻠﻲ ﺃﻥ EGM2008ﺃﻜﺜﺭ ﺩﻗﺔ ﻭﺃﻜﺜﺭ ﺘﻔﺼﻴﻼ ﻤﻥ .EGM96ﻟﻜﻥ ﻻ ﻴﻤﻜﻨﻨﺎ ﺘﺤﺩﻴﺩ ﺩﻗﺔ ﺍﻟﺠﻴﻭﻴـﺩ EGM2008ﺇﻻ ﺒﻌﺩ ﺍﺨﺘﺒﺎﺭﻩ ﺃﻭ ﻤﻌﺎﻴﺭﺘﻪ ﻟﻜل ﺩﻭﻟﺔ ﻋﻠﻲ ﺤﺩﻱ ]ﻗﻤﺕ ﻤﻊ ﺯﻤﻴﻠﺘﻴﻥ ﻟﻲ ﺒﻌﻤل ﺒﺤﺙ ﻋـﻥ ﻫﺫﺍ ﺍﻟﺠﻴﻭﻴﺩ ﻭﻭﺠﺩﺕ ﺃﻥ ﺩﻗﺘﻪ ﻓﻲ ﻤﺼﺭ ﺘﻜﺎﺩ ﺘﺴﺎﻭﻱ ٠,٢٣ﻤﺘﺭ ﻓﻘﻁ ،ﻭﺴﻴﺘﻡ ﻨﺸﺭ ﺍﻟﺒﺤﺙ ﻓـﻲ ﺍﻟﻌـﺩﺩ ﺍﻟﻘﺎﺩﻡ ﻤﻥ ﻤﺠﻠﺔ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺴﺎﺤﻴﺔ ﺍﻟﺘﻲ ﻴﺼﺩﺭﻫﺎ ﺍﻻﺘﺤﺎﺩ ﺍﻷﻤﺭﻴﻜﻲ ﻟﻠﻤﻬﻨﺩﺴﻴﻥ ﺍﻟﻤﺩﻨﻴﻴﻥ[. ﺍﻟﺨﻼﺼﺔ: ﻴﻤﻜﻥ ﻟﻤﻬﻨﺩﺴﻲ ﺍﻟﻤﺴﺎﺤﺔ ﺃﻥ ﻴﺠﻨﻭﺍ ﻤﻤﻴﺯﺍﺕ ﺍﻟﺒﻌﺩ ﺍﻟﺜﺎﻟﺙ ﻟﺘﻘﻨﻴﺔ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ )ﺍﻻﺭﺘﻔﺎﻉ( ﻋﻨﺩ ﺤـﺼﻭﻟﻬﻡ ﻋﻠﻲ ﻨﻤﻭﺫﺝ ﺩﻗﻴﻕ ﻟﻠﺠﻴﻭﻴﺩ .ﻓﻲ ﺍﻟﻤﻨﺎﻁﻕ ﺍﻟﺼﻐﻴﺭﺓ ﻭﻋﻨﺩ ﻭﺠﻭﺩ ﻋﺩﺩ ﻤﻌﻘﻭل ﻤﻥ ﻨﻘﺎﻁ ﺍﻟﺜﻭﺍﺒﺕ ﺍﻷﺭﻀـﻴﺔ ﺍﻟﻤﺴﺎﺤﻴﺔ )ﻤﻌﻠﻭﻡ ﻟﻬﺎ ﺍﺭﺘﻔﺎﻉ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ ﻭ ﺍﻟﻤﻨﺴﻭﺏ ﺃﻴـﻀﺎ( ﻴﻤﻜـﻥ ﺍﺴـﺘﺨﺩﺍﻡ ﺍﻟﻁﺭﻴﻘـﺔ ﺍﻟﻬﻨﺩﺴـﻴﺔ ﻭﺍﻟﺤﺼﻭل ﻋﻠﻲ ﻨﺘﺎﺌﺞ ﺠﻴﺩﺓ ﻟﺘﺤﻭﻴل ﺍﺭﺘﻔﺎﻋﺎﺕ ﺍﻟﺠﻲ ﺒﻲ ﺇﺱ ﺇﻟﻲ ﻤﻨﺎﺴﻴﺏ .ﻭﻓﻲ ﺤﺎﻟﺔ ﻋﺩﻡ ﻭﺠـﻭﺩ – ﺃﻭ ﻗﻠﺔ ﻋﺩﺩ – ﻨﻘﺎﻁ ﺍﻟﺜﻭﺍﺒﺕ ﺍﻷﺭﻀﻴﺔ ﺍﻟﻤﺴﺎﺤﻴﺔ ﻓﻴﻜﻭﻥ ﻀﺭﻭﺭﻴﺎ ﺘﻁﻭﻴﺭ ﻨﻤﻭﺫﺝ ﺠﻴﻭﻴﺩ ﻤﺤﺴﻭﺏ ﻤﻥ ﺃﺭﺼﺎﺩ ﺍﻟﺠﺎﺫﺒﻴﺔ ﺍﻷﺭﻀﻴﺔ .ﻭﻤﻥ ﺃﻥ ﺠﻬﻭﺩﺍ ﻜﺒﻴﺭﺓ ﻗﺩ ﺒﺫﻟﺕ ﻟﺯﻴﺎﺩﺓ ﺩﻗﺔ ﻨﻤﺎﺫﺝ ﺍﻟﺠﻴﻭﻴﺩ ﺍﻟﻌﺎﻟﻤﻴﺔ ﺇﻻ ﺍﻨـﻪ ﻤﺎﺯﺍﻟـﺕ ﻫﻨﺎﻙ ﺤﺎﺠﺔ ﻹﻀﺎﻓﺔ ﺘﺤﺴﻴﻨﺎﺕ ﺠﺩﻴﺩﺓ ﻋﻠﻴﻬﺎ. ________________________________________________________________ ﻤﻠﺤﻭﻅﺔ :١ﺍﻹﻀﺎﻓﺎﺕ ﺒﻴﻥ ﺍﻟﻘﻭﺴﻴﻥ ] [ ﻟﻠﻤﺘﺭﺠﻡ ﻭ ﻟﻴﺱ ﻟﻠﻤﺅﻟﻑ ﺍﻷﺼﻠﻲ ﻟﻠﻤﻘﺎل. ﻤﻠﺤﻭﻅﺔ :٢ﺘﻡ ﺘﺠﺎﻫل ﺍﻟﻔﻘﺭﺍﺕ ﺍﻟﺘﻲ ﺘﺘﺤﺩﺙ ﻋﻥ ﺘﻁﺒﻴﻘﺎﺕ ﺍﻟﺠﻴﻭﻴﺩ ﻓﻲ ﺩﻭﻟﺔ ﺠﻨﻭﺏ ﺃﻓﺭﻴﻘﻴﺎ )ﻤﻭﻁﻥ ﻜﺎﺘﺏ ﺍﻟﻤﻘﺎل( ﻟﻌﺩﻡ ﺃﻫﻤﻴﺘﻬﺎ ﻟﻠﻘﺎﺭﺉ ﺍﻟﻌﺭﺒﻲ. ﻤﻠﺤﻭﻅﺔ :٣ﻟﻠﻘﺭﺍﺀﺓ ﺃﻜﺜﺭ ﻋﻥ ﺘﻁﻭﻴﺭ ﺍﻟﺠﻴﻭﻴﺩ ﻓﻲ ﺠﻤﻬﻭﺭﻴﺔ ﻤﺼﺭ ﺍﻟﻌﺭﺒﻴﺔ ﺃﻨﻅﺭ: Geoid of Egypt (English): http://gomaa.dawod.googlepages.com/geoidofegypt Geoid of Egypt (Arabic): http://gomaa.dawod.googlepages.com/egypt.geoid.arabic ________________________________________________________________ ____________________ ﺘﺭﺠﻤﺔ ﺩ .ﺠﻤﻌﺔ ﻤﺤﻤﺩ ﺩﺍﻭﺩ ٥ _______________________ ﺃﺴﺄﻟﻜﻡ ﺍﻟﺩﻋﺎﺀ ﺒﻅﺎﻫﺭ ﺍﻟﻐﻴﺏ Surveying technical GPS and the geoid by Prof. Charles Merry, University of Cape Town GPS provides full 3D positions. However, the heights use the WGS84 ellipsoid as a reference surface, whereas the user almost invariably requires heights above sea level (the geoid). T his article discusses the relationships between GPS heighting and the geoid and the status of geoid modelling, with particular reference to southern Africa. GPS (and other GNSS systems) provides fully three-dimensional positions – latitude, longitude and height. The reference surface used is the WGS84 ellipsoid, and heights are heights above this surface – ellipsoidal heights. Due to a combination of unmodelled residual refraction and weak geometry (satellites below the horizon are not visible) GPS-determined heights are generally less accurate than horizontal positions (by a factor ranging from 1,5 to as much as 3). Nevertheless, there are many applications of GPS where heights are useful, if not essential, especially in engineering surveys. However, the type of height that is needed is a height above mean sea level (MSL) – commonly called orthometric height. To be more specific, the reference surface employed for heighting is the geoid, a level surface that corresponds very closely with MSL. Due to inhomogeneities in the density of the Earth's crust and upper mantle the geoid departs from the ellipsoid by up to 120 m. This separation is known as geoidal height (or geoidal undulation) – N (see Fig. 1). In order to convert GPS-derived ellipsoidal heights to orthometric heights we need an accurate model of geoidal heights. This is where the difficulty arises – it is extremely difficult to determine such a model, or at least to determine an accurate model. In the next section we will discuss some of these methods and their limitations. Geoid modelling In essence, there are only two methods – geometric and gravimetric: Geometric This approach is the one most favoured for GPS surveys of small extent (area covered less than 10 - 20 km2). 18 Fig. 1: Geoidal height. GPS measurements are taken at benchmarks with known orthometric height H. The difference between the GPS-derived ellipsoidal height h and the orthometric height H provides the geoidal height N at that point (see Fig. 1): N=h-H In the simplest case a single control point would provide a constant height shift. However, the most common case involves three benchmarks at which GPS measurement are made. This allows two tilts (North-South and East-West) and a shift to be determined. In effect, the geoid is modelled by a tilted plane (Fig. 2). These tilts and shift can then be applied to the heights determined from GPS at any new points in the area. The "calibration" of RTK GPS prior to the start of a survey incorporates this model, as well as a model for horizontal shifts and a rotation in azimuth. It is possible to take GPS measurements at more than three benchmarks. In this case the commercial software will use these data to get a more reliable model of the tilted plane (i.e. the geoid is still assumed to take this shape over the region of interest). It is possible to use the extra data to generate a more complex model for the geoid, using for example polynomials, splines or Kriging, but this will require some expertise on the part of the user. The chief limitations of the geometric case are: • The model is only valid over the area encompassed by the known benchmarks (extrapolation beyond this area is highly inadvisable). • The simple tilted plane model can only be safely used over a small area (the geoid is much more complex in shape). • It is not always possible (or convenient) to find sufficient benchmarks with known orthometric heights in the area of interest. Gravimetric This approach provides a uniform grid of geoidal heights over a large area, in contrast to the scattered point values PositionIT - July/August 2008 SURVEYING Fig. 2: Tilted plane model of the geoid. over a limited area provided by the geometric method. However, it requires a complex numerical integration of gravity anomalies to determine a geoidal height. The key formula is that due to Stokes: Without going into the details of the other terms, the core "observation" is the gravity anomaly ∆g, which represents the difference between observed gravity and a theoretical gravity value. The integration takes place over the entire surface of the Earth, so that to get a single geoidal height tens of thousands of gravity observations are needed. The gravity anomaly also requires a correction for the effect of terrain variations, so that a detailed digital elevation model is also needed. An alternative representation of the geoid is to use a spherical harmonic expansion of the Earth's geopotential (a spherical harmonic model is basically a two-dimensional Fourier series). Typically the coefficients of this model are determined from the analysis of the perturbations in the orbits of low-orbiting satellites. Nowadays it is common to use these two representations together in one of two ways: • Use the satellite-based spherical harmonic expansion to determine the long wavelength (low frequency) component of the geoid, and a modified version of Stokes' formula to determine the high frequency component, integrating over a spherical "cap", not the entire Earth. • Use both satellite data and terrestrial gravity anomalies in a single solution for a high-order spherical harmonic expansion of the Earth's gravity field. This requires significant computer resources to compute PositionIT - July/August 2008 the coefficients, and is beyond the capability of all but the largest agencies. Although the gravimetric approach provides good spatial coverage over large areas and can provide the detailed structure of the geoid, it suffers from some drawbacks: • It is computationally intensive and mathematically complex. • The results are only as good as the underlying gravity measurements. If there are errors in the gravity data, there will be errors in the geoid. More seriously, there are many parts of the world where gravity data are sparse or non-existent, leading to gaps or smoothing in the geoid model. • The gravimetric geoid is susceptible to biases and tilts, due to errors in satellite orbit modelling and to gaps in terrestrial gravity data sets. Combination approach It is possible to combine the gravimetric and geometric approaches. The gravimetric geoid is computed first, and then "calibrated" by using GPS-determined geoidal heights at discrete points over the entire region. This calibration would, in the simplest case, consist of a bias and two tilts. However, much more complex correction surfaces are possible, and in the case of continental geoid models desirable. This approach also compensates for any tilts or biases that may exist in the precise levelling networks and enables the user to transform his GPS-derived heights directly to the national levelling datum. Current geoid models Until very recently the standard model for the geoid has been the Earth Geopotential Model 1996 technical (EGM96), developed jointly by the US Department of Defense's National Geo-Spatial Intelligence Agency (NGA), NASA and the Ohio State University [1]. A simplified gridded version of EGM96 is embedded in most handheld GPS receivers to enable the user to get "heights above MSL" from his GPS (at an accuracy level of a few metres). EGM96 is also incorporated in most GPS processing packages used for surveying. It uses a spherical harmonic expansion to degree 360 (137 000 coefficients), based upon a combination of satellite tracking data and terrestrial gravity anomalies. The expansion to degree 360 means that the effective "wavelength" of this geoid model is one degree - i.e. any variation in the geoid shape smaller in extent than one degree (about 100 km) will not be modelled. In addition, because of gaps and errors in the data, the inherent accuracy of this model is only around 40 cm (worse over large parts of Africa). This is not as bad as it sounds, as precise GPS surveying uses differential GPS and correspondingly it is differences in geoidal height that are important. EGM96 can represent these with an accuracy of around 10 cm plus 2 parts per million (ppm) of the distance between points. At the University of Cape Town (UCT) we have used EGM96 as a basis for more detailed models of the geoid over Africa. Our approach is the first of the combination methods mentioned in the previous section (EGM96 provides the low frequency contribution, terrestrial gravity anomalies in Stokes' formula provide the high frequency contribution). More recently we have replaced the EGM96 spherical harmonic expansion with that deduced from the GRACE satellite mission [2]. GRACE consists of a pair of satellites in a low orbit ― small variations in the range between them are a measure of changes in the Earth's gravity field [3]. GRACE has improved our knowledge of the low frequency coefficients of this field by an order of magnitude over EGM96. The UCT model (AGP2007) provides a detailed 5' grid of the geoid over Africa. However, its accuracy and detail are limited in areas where no gravity data exist. Earlier this year NGA introduced its new geopotential model – EGM2008 [4]. This model, based upon a combination of GRACE data and 19 SURVEYING technical terrestrial gravity anomalies, is a spherical harmonic representation of the geopotential to degree 2190 (over 4,7-million coefficients). A contour representation of this model for southern Africa is given in Fig. 3. EGM2008 is more detailed and more accurate than EGM96. It is difficult to say with any certainty how accurate it is, but some initial testing indicates that the accuracy could be better than 10 cm over North America and Europe. In South Africa, some limited testing (using GPS measurement at precise levelling benchmarks) shows an accuracy of around 15 cm. It could be much worse in other parts of Africa. Fig. 4 shows the difference between the EGM2008 and EGM96 geoid models in southern Africa. For most of the region the differences are small and reflect the improvement achieved by using a more detailed representation. However, in central Mozambique the differences reach 3 m and cover a large region. Correspondence with the developers of EGM2008 (S. Kenyon, personal communication, 2007) indicates that (for this model) terrestrial gravity anomalies were predicted in central Mozambique using a digital elevation model. No measured gravity anomalies were used. Whether this approach is correct remains to be seen – the shape of the geoid in central Mozambique as shown by EGM2008 may not be real. It would be very interesting to see what would result from taking GPS measurements at levelling benchmarks (if there are any) in that region. Two of the goals of the Chief Directorate: Surveys and Mapping (CDSM) in modernising the national height network are to re-adjust the precise levelling network and to provide an accurate calibrated geoid model for all users. The production of a set of calibration points (GPS at levelling benchmarks) for this purpose is almost complete (R. Wonnacott, personal communication, 2008). Fig. 4: EGM2008 minus EGM96 geoid models for southern Africa. Conclusions [1] Surveyors will only be able to take full advantage of the third dimension of GPS if they have an accurate and reliable model of the geoid. In small regions with an adequate supply of height control points it is possible to use the geometric approach and get good results. Where height control 20 Fig. 3: EGM2008 geoid model for southern Africa. is inadequate or non-existing then a good gravimetric model is essential. Although substantial improvements in modelling the geoid using gravity data have been made, there is still room for further improvement. [2] CL Merry: "An updated geoid model for Africa", presented at Symposium G2, XXIV General Assembly of the IUGG, Perugia, Italy, July 2007. [3] B Tapley and C Reigber: "The GRACE mission: status and future plans", EOS, Trans. AGU, 82(47), 2001. References [4] NK Pavlis, SA Holmes, SC Kenyon, and JK Factor: "An Earth Gravitational Model to Degree 2160: EGM2008", presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 2008. FG Lemoine, SC Kenyon, JK Factor, RG Trimmer, NK Pavlis, DS Chinn, CM Cox, SM Klosko, SB Luthcke, MH Torrence, YM Wang, RG Williamson, EC Pavlis, RH Rapp, TR Olsen: "The development of the joint NASA GSFC and NIMA geopotential model EGM96", NASA Technical Publication NASA/TP-1998-206861, July 1998, Goddard Space Flight Center, Greenbelt, MD, USA. Contact Prof. Charles L Merry, School of Architecture, Planning & Geometrics, UCT, Tel 021 650-3576, charles.merry@uct.ac.za PositionIT - July/August 2008
