Fast Non-Convex Low-Rank Matrix Decomposition For Separation Of Potential Field Data Using Minimal Memory

A fast non-convex low-rank matrix decomposition method for potential field data separation is presented. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block Hankel matrix randomized singular value decomposition algorithm is integrated into the Altproj algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The integration of this improved estimation for the partial singular value decomposition avoids the construction of the trajectory matrix in the robust principal component analysis optimization problem. Hence, gravity and magnetic data matrices of large size can be computed and potential field data separation is achieved with better computational efficiency. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The performance of the algorithm, with and without the efficient estimation of the low rank matrix, is contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. These results demonstrate that the presented algorithm is not only computationally more efficient but it is also more accurate. Moreover, it is possible to solve far larger problems. As an example, for the adopted computational environment, matrices of sizes larger than 205x205 generate "out of memory" exceptions without the improvement, whereas a matrix of size 2001x2001 can now be calculated in 1062:29s. Finally, the presented algorithm is applied to separate real gravity and magnetic data in the Tongling area, Anhui province, China. Areas which may exhibit mineralizations are inferred based on the separated anomalies.