The_Code_for_High_Order_Analytical_Continuation - Jian Ma (19747060) (2024)
The analytical continuation based on Taylor series expansion plays a vital role in the construction of the Earth gravitational model, the determination of regional geoid and so on. The high-order continuation receives far less attention than the one-order analytical continuation nowadays. This study delves into the high-order analytical continuation algorithm, with a particular focus on the computation of its high-order radial derivatives. General closed formulae for the radial derivatives of the harmonic function and gravity disturbance are derived with use of a direct derivation method. Specific formulae for the first through sixth-order radial derivatives based on the direct derivation method are then presented according to the general formulae. Additionally, formulae for the first through sixth-order radial derivatives are also derived by a recursive method. Numerical test in a mountain area reveals theoretical defects exist in the high-order radical derivatives based on the direct derivation method, whereas the recursive method proves to be effective and practical. The optimal order of the analytical continuation algorithm is contingent upon the noise level of gravity data. For the downward continuation of poor-quality gravity data, the increase of the order of analytical continuation algorithm is unfavorable for the improvement of the continuation accuracy. When 3000-meter-high gravity data including noise of 1mGal are continued downward, the 2-order analytical continuation achieves the highest accuracy, surpassing the 1- and 3-order analytical continuation by 20.25% and 0.88%, respectively. However, for 3000-meter-high gravity data including noise of 2mGal, the 1-order analytical continuation outperforms the 2- and 3-order continuations by 2.13% and 9.45%, respectively.