Study on Recovering the Earth＇s Potential Field Based on GOCE Gradiometry

Given the second radial derivative Vrr（P） ｜δs of the Earth＇s gravitational potential V（P） on the surface δS corresponding to the satellite altitude, by using the fictitious compress recovery method, a fictitious regular harmonic field rrVrr（P）^＊ and a fictitious second radial gradient field V：（P） in the domain outside an inner sphere Ki can be determined, which coincides with the real field V（P） in the domain outside the Earth. Vrr^＊（P）could be further expressed as a uniformly convergent expansion series in the domain outside the inner sphere, because rrV（P）^＊ could be expressed as a uniformly convergent spherical harmonic expansion series due to its regularity and harmony in that domain. In another aspect, the fictitious field V^＊（P） defined in the domain outside the inner sphere, which coincides with the real field V（P） in the domain outside the Earth, could be also expressed as a spherical harmonic expansion series. Then, the harmonic coefficients contained in the series expressing V^＊（P） can be determined, and consequently the real field V（P） is recovered. Preliminary simulation calculations show that the second radial gradient field Vrr（P） could be recovered based only on the second radial derivative V（P）｜δs given on the satellite boundary. Concerning the final recovery of the potential field V（P） based only on the boundary value Vrr （P）｜δs, the simulation tests are still in process.

Given the second radial derivative Vr r ( P ) |S of the Earth’s gravitational potential V ( P ) on the surface S corresponding to the satellite altitude, by using the fictitious compress recovery method, a fictitious regular harmonic field rrVr r( P )* and a fictitious second radial gradient field Vr *r ( P) in the domain outside an inner sphere K i can be determined, which coincides with the real field Vr r( P ) in the domain outside the Earth. Vr *r ( P)could be further expressed as a uniformly convergent expansion series in the domain outside the inner sphere, because rrVr r( P )* could be expressed as a uniformly convergent spherical harmonic expansion series due to its regularity and harmony in that domain. In another aspect, the fictitious field V * ( P) defined in the domain outside the inner sphere, which coincides with the real field V ( P ) in the domain outside the Earth, could be also expressed as a spherical harmonic expansion series. Then, the harmonic coefficients contained in the series expressing V * ( P) can be determined, and consequently the real field V ( P ) is recovered. Preliminary simulation calculations show that the second radial gradient field Vr r( P ) could be recovered based only on the second radial derivative Vr r ( P )| S given on the satellite boundary. Concerning the final recovery of the potential field V ( P ) based only on the boundary value Vr r ( P )|S, the simulation tests are still in process.