Abstract
The ill-posed problem is the key obstacle to obtain the accurate inversion results in the geophysical inversion field, and the Levenberg-Marquardt1, 2(hereinafter referred to as the L-M method) method has been widely used as it can effectively improve the ill-posed problems. However, the inversion results obtained by the L-M method are usually stable but incorrect, the reason is that the damping factor in the L-M method is difficult to solve, and it is usually approximated with a positive constant by experience or through some fitting methods. This paper uses the binary gravity model to demonstrate that the damping factor in the L-M method cannot be regarded as a positive constant only, it should have the following characteristics: (i) the damping factor is a vector, not just a constant; (ii) the values of the vector are composed of both positive and negative constants, not just positive constants; (iii) the corresponding value in the vector is close or equal to ∞ when the corresponding density block’s value is close or equal to zero. Even if the above characteristics have been found in the L-M method, it is difficult to reasonably estimate the damping factor as the damping factor oscillate severely due to the third characteristic, and the improved L-M method is proposed which effectively avoids the damping factor’s severe oscillation problem. The strategy of obtaining the reasonable damping factor is given finally.
Discussion on the damping factor and improvement of the Levenberg-Marquardt method