In this paper, we consider a joint-inversion problem using different types of geophysical data: gravity and magnetism. We first formulate two kinds of inverse problems in the famework of the first kind Fredholm integral equations, and then build up a sparse inversion model combining the two inverse problems as well as the cross-gradient term. The cyclic gradient method for quadratic function minimization is extended for solving the corresponding optimization problem. We update the stepsizes in a cyclic way, by combining the approximated Cauchy steps and the fixed steplengths. Theoretical analysis shows that the algorithm converges R-linearly. Experimental tests show that the proposed joint inversion sparse model as well as the proposed cyclic gradient method improve the numerical performances effectively, compared to the state of the art.

在本文中，我们考虑使用不同类型的地球物理数据（重力和磁场）的联合反演问题。我们首先在第一类Fredholm积分方程的框架中制定了两种反问题，然后结合这两个反问题以及交叉梯度项建立了一个稀疏反演模型。扩展了用于二次函数最小化的循环梯度方法，以解决相应的优化问题。通过组合近似的柯西步长和固定步长，我们以循环方式更新步长。理论分析表明，该算法线性收敛。实验测试表明，与现有技术相比，所提出的联合反演稀疏模型以及所提出的循环梯度方法有效地改善了数值性能。