Spatial Decoupling of Borehole Electromagnetic Measurement Signals Using Compressive Sensing

Borehole electromagnetic (EM) technology facilitates rapid and accurate underground geophysical prospecting. However, analytical solutions of the Helmholtz equation involving a layered cylindrical borehole model couple the spatial position information of sensors with the target information, which tends to distort the measurement signals obtained when multiple sensors are used to improve the signal-to-noise ratio (SNR). Moreover, applying Gauss–Legendre quadrature approximation to convert the spatially coupled sensor and target terms into a product between the target vector and an observation matrix containing the spatial position information of each sensor array element leads to an ill-posed problem, while applications of the compressive-sensing (CS) technique to address this ill-posed problem have failed to address equally serious problems associated with signal quality and the lack of signal processing to enhance the SNR. The present work addresses these issues by introducing a sparsity constraint based on the dimensionality reduction property of the CS technique to optimize the sparsity of the target vector containing the target information, which provides a solution to the ill-posed problem obtained when borehole EM measurement signals are spatially decoupled via Gauss–Legendre quadrature approximation, while simultaneously enhancing the SNR obtained from multiple sensors. The ability of the proposed CS optimization model to solve the ill-posed reconstruction problem is established based on the results of sparsity analysis for the target vector and correlation analysis for the observation matrix of a standard case as an example, and the effectiveness of the proposed approach is demonstrated based on the results of simulations and experiments.