Errors of target localization with the traditional magnetic gradient tensor mainly comes from three aspects, namely, the large error of the magnetic gradient tensor for shallow targets, the low signal-to-noise ratio (SNR) of the response for deep targets, and the overlapping responses of multi-targets. In this study, a towed transient electromagnetic sensor with a $3\times3$ receiving coils array is constructed. On the basis of the sensor array, an improved magnetic gradient tensor is proposed to accurately locate targets. For shallow targets, the magnetic gradient tensor is constructed using the responses of four adjacent receiving coils to reduce the error of the magnetic gradient tensor. For deep targets, all the responses of nine receiving coils are used to improve the SNR. Both the early and late time responses are used to roughly estimate the positions of multi-targets to improve the localization accuracy. Experimental results show that for underground targets within 2 m, the depth errors of the targets do not exceed 10 cm, and the horizontal errors of the targets are mostly within 10 cm, even if the responses of two adjacent targets overlap each other, indicating that the proposed method can effectively improve the localization accuracy of underground targets.

Differential evolution (DE) algorithm, which is a global convergence algorithm, is often used to estimate the position of underground targets detected with a portable transient electromagnetic (TEM) system. The DE algorithm is extremely time-consuming due to thousands of iterations. A new algorithm for fast localization of an underground target by magnetic gradient tensor and Gaussian-Newton algorithm with a portable TEM system is proposed. First, the gradient tensor of an underground target is constructed with the differential responses received by the portable sensor. Gradient tensor, commonly used in magnetic detection, is applied for the first time in TEM detection to estimate the target position for each measurement. Then, all the estimated positions are averaged to reduce the localization error. Taking the averaged position as the initial value, the Gaussian-Newton algorithm can complete iterations within dozens of times, which can effectively improve the speed and accuracy of the algorithm. Finally, the performance of the new method has been verified in the test-stand and field experiments. Results show that the errors of averaged positions by gradient tensor are no more than 8 cm in the horizontal direction. The errors of the estimated positions, inclination, and rotation angles by the Gaussian-Newton algorithm are no more than 4 cm, 6°, and 5°, respectively. The statistical running time of the proposed method takes approximately tens of milliseconds, accounting for about 7% of the DE algorithm. The proposed method can achieve fast and accurate localization and characterization of targets and has an important significance to the digging and recognition of underground targets.

The magnetic gradient tensor algorithm, which can quickly locate underground targets without iteration, is constrained by positioning errors when detecting unexploded ordnance with a transient electromagnetic system due to the large sensor baseline distance. An innovative method based on a cross-shaped portable transient electromagnetic system is proposed to improve the positioning accuracy of magnetic gradient tensor localization. First, the magnetic gradient tensor algorithm is introduced to estimate target positions and positioning errors. Second, an error correction model is constructed, with its parameters fitted using the least squares algorithm. Finally, the estimated target position obtained from the magnetic gradient tensor algorithm is corrected based on the error correction model. Experimental results show that the proposed method can notably improve the positioning accuracy of underground targets, achieving a horizontal position error of no more than 10 cm and a depth position error of no more than 5 cm. The inversion time takes approximately 25 ms, providing a novel method for real-time detection and rapid positioning of unexploded ordnance near the surface.