The solving of third geodetic boundary value problem need to gravity anomalies continued from surface of the Earth down to their mean values on geoid. Downward continuation (DC) is the most challenging part of precise geoid determination. The inverse of Poisson’s integral are frequently used by researchers for DC. In this paper, the planar approximation of Poisson’s integral is used which provides the same accuracy respect to other higher approximations such as, spherical or ellipsoidal.
The DC problem is inherently ill-posed being highly sensitive to high frequencies part of gravity signal. The DC process is ill-posed in its continuous mode. For the numerical evaluation, the process as a linear system (*Ax=b*) will be well-posed if the Hadamard conditions: existence of solution, uniqueness and stabilities are fulfilled. The existence and uniqueness of solution is guaranteed physically, but the process may be unstable, i.e., the solution does not grasp continuously on the data (*b*). The continuous problems must be discretized in order to prepare for a numerical evaluation. The discretization form of an ill-posed problem may turn to a well-posed depending on the discretization step. In DC process, the spacing of gravity anomalies is a major factor for conception of instability.
The discretization of Poisson’s integral equation can be done using two different mean (grid) and point (scatter) schemes. Usually, the gravity data are observed at scattered point such as at leveling benchmarks. Then, the mean gravity anomalies are predicted/averaged on regular mesh. DC of gridded gravity anomalies are much easier to implement and more stable due to the attenuating of the high frequency by averaging. In addition, the stability of linear equation systems is increased by removing the very close observations. However, the useful local part of gravity signal are lost by averaging and mean anomalies are unavoidably affected by perdition error particularly in regions of poor data coverage. The mean gravity anomalies on geoid can be directly computed from DC of observed gravity anomalies. This process leads to ill-condition linear system in most cases. Hence the some appropriate regularization methods need to obtained the desired accuracy. The DC of scattered data has some advantages such as, there is not prediction error in them or they contain all frequencies of gravity filed.
In this study, the accuracy and stability of DC of scattered and gridded anomaly are investigated. The discrete Picard condition is utilized to study the ill-poseness and instability of the DC linear equations system. Numerical examination is done in two mountainous test areas in Iran with a poor gravity data coverage and in the USA with dense gravity observations. Numerical results in both test areas show that the DC of scattered anomalies is an ill-posed due to closeness of point anomalies in some areas such as along levelling lines. Whereas the DC of 5'×5' gridded anomaly is a well-posed and stable problem. The DC of EGM08 synthetic gravity anomalies indicates that despite the presence of prediction error in gridded anomalies and the removing some useful high frequencies, their results are more accurate than scattered anomalies. |