Analytical Solution of the Full-Space Effect of Transient Electromagnetic Response from Anomalous Bodies in Tunnels

The transient electromagnetic method (TEM) is one of the most common and effective techniques for detecting anomalies in tunnel engineering. Traditional TEM interpretations often assumed a half-space model, which can be problematic in enclosed environments such as tunnels. This limitation often results in poor inversion accuracy, especially when full-space effects are pronounced. Utilizing thin-layer theory, this study derived analytical expressions for TEM responses under both full-space and half-space conditions and defined a ratio coefficient relating full-space and half-space responses in the presence of anomalous bodies. Furthermore, a sensitivity analysis of this ratio coefficient was conducted under varying conditions of surrounding rock and anomalous bodies, exploring the laws governing the full-space effect for anomalous bodies in tunnels. The research results indicate that when changes in surrounding rock or anomalous body conditions enhance the TEM response—such as when the anomalous body is closer to the receiver coil, thicker, or exhibits a larger conductivity contrast with the surrounding rock—the ratio coefficient between full-space and half-space responses decreases, potentially approaching 1.0. Conversely, when these changes weaken the anomalous body response, the ratio coefficient increases, approaching approximately 2.5, as observed under homogeneous medium conditions. Moreover, the off-time exerts a significant influence on the ratio coefficient. The earlier off-times yield larger coefficients, whereas the coefficient gradually approaches 1.0 with increasing time. Finally, this study elucidates the dynamic characteristics of the full-space effect and establishes a functional relationship describing the primary factors influencing the ratio coefficient. The findings provide a theoretical foundation for improving TEM detection of anomalous bodies in tunnels.