This paper presents S-inversion method to tunnel prediction to forecasting water-filled faults or fracture zones ahead of the front wall of a tunnel during tunnel excavation. S-inversion is an interpretation method of transient electromagnetic (TEM) data using the second derivative of the conductivity parameter based on the moving thin sheet approach. It is suggested for the technology of second derivative of vertical apparent conductivity, which is traditionally used in surface TEM data interpretation, the theoretical analyses and the method of numerical calculation in detail are also given. Finally, a real tunnel forecasting is studied for TEM surveys, and results show that the proposed method is effective and successful for exploring and predicting unfavorable geology during tunnel construction.
 It is a well-known fact that in the west-south of China, there are large numbers of mountains which lead to low economy development in the area, therefore, constructing railway is one of the most important tasks for government and designing a lot of tunnels is a good choice for railway construction. But the geological structure is very complex in the area. Especially, water-filled structure brings more difficulties to excavation. So, it is crucial to carry out tunnel prediction before excavation.
 At present, tunnel prediction based on geology is mainly attained through the combination of drill exploration and geophysical methods including Tunnel Seismic Prediction (TSP) [Dickmann and Sandeter, 1996] True Reflection Tomography (TRT) etc. However, there is no efficient way to detect water-filled geological features during tunnel based on geology prediction.
 TEM response is sensitive to low resistivity bodies and the received signal corresponding to the peak of the secondary field can provide useful information about the target. Because TEM can provide excellent resolution of conductive layers, the imaging method is capable of detecting thin low resistivity water-filled zones (such as fractures and faults).
 S-inversion is an interpretation method for transient electromagnetic (TEM) data using the second derivative of the conductivity parameter based on the moving thin sheet approach. The S in word “S-inversion” is a physical parameter, i.e. the conductance of belowground medium which will need to be calculated in this paper. The “inversion” means reversion transformation. Therefore, the S-inversion means the reversion transformation for the conductance of belowground medium. The S-inversion is a fast-imaging technique for TEM data interpretation based on the thin sheet model approach in earth-surface exploration. The foundations of the thin sheet model approach for conductivity-depth imaging depend on the theory of EM induction in thin conducting plate. This theory was first developed by Price  and Sheinmann . The theory for such models was extended by Berdichevsky and Zhdanov , Zhdanov , Tartaras and Zhdanov , and Singer and Green . In this technical method, the observed electromagnetic signal is used to invert the physical parameter of belowground medium, i.e. the conductance S(t) using the algorithm proposed in this paper. Then, the deposited distribution of ground conductivity medium can be interpreted using the analysis on the magnitude and distributing of the conductance S(t).
 In this paper, the conductive finite plate model [Singer and Green, 1998; Markku and Saurabh, 1998; Efthimios and Michael, 2000] is introduced to tunnel based geological prediction. A tunnel forecasting TEM method has been developed, where the response from a transmitter coil placed vertically on the front tunnel wall are measures by the established facility system. But there is the few report about the application in tunnel predication.
 Tunnel forecasting TEM has been carried out to predict water-filled structure in west of HuBei province in south-west of China. Figure 1 is the view of the tunnel location where the working area crosses interline of HuBei province and Chong Qing city. The main harmful geological bodies are crack fault and water-filled cave which may lead to water-jet and mud-jet.
2. Analyses of Equivalent Conductive Plate Solution
 In order to simplify the discussion, it is assumed that a circular loop located on the front wall of a tunnel is excited by a step current, with the power turned off at time t = 0. The current in the loop is defined by
 This current has an associated primary electromagnetic field to induce eddy currents when it is switched off. A secondary electromagnetic field associated with the eddy currents can be surveyed from the front wall of the tunnel. The measured field can be approximated by replacing a homogenous medium with an equivalent conductive plate.For a small current source Idl the vertical component of magnetic intensity can be written as
where I is the current intensity, l is the length of circle loop, R is the distance from survey point to small current source, a is the radius of circle loop. The moving speed of sheet is v = where σ is conductivity of thin sheet, z is the moving distance of conductive sheet, and,
The relationship of magnetic field Bz and the vertical component of magnetic intensity Hz is accord with Bz = μHz where the μ is magnetic permeability, Despite of that the Bz and Hz are multiple variable functions, such as the explored depth h, electric current intensity I and circle loop l and a. But, for the given circle loop, the explored medium and depth, the Bz and Hz only are the function of time t. Therefore, the Bz can be written as a single variable function Bz(t). The derivative of magnetic field Bz(t) can be expressed as
 The conductivity parameter σ is a sensitive parameter to water medium so it can be used to explore the water mishap of tunnel. On the other hand, the advanced imaging technique can be adopted using the conductivity parameter, at the same time, a hidden function can be established between the observed and educed parameters in mathematics so that the objective of ground source can be well carried out through the introduction of auxiliary function.
 Let S(t) = σ denote the equivalent apparent conductive of medium. Equation (5) becomes
Seemingly, the second derivative of S(t) can be calculated from the third derivative of Bz(t). In fact, it is very difficult to calculate directly using the because there is the F() related to time t in the denominator.In this paper, an improved method of calculating the second derivative is developed, and the is not needed. The proposed method is following as:
 In order to calculate F(), a auxiliary function () is introduced. Let
 5) Substitute F() and into equation (10) to determine vertical apparent conductivity S(t)
 6) Calculate the parameter of from S(t)
3. Numerical Simulation of Parameter and Water-Filled Fault
 In order to show the advantage of using the second derivate of conductivity, we encode program for calculation based on above-mentioned suggestion. Figure 2a shows of H-type geo-electric section, the parameter is ρ1 = 200Ω·m, ρ2 = 50Ω·m, ρ3 = 200Ω·m, h1 = h2 = 200m where Curve 1 is the S curve, curve 2 is the curve and curve 3 is the curve, respectively. As shown in Figure 2a, it is clear that after derivation processing to apparent vertical conductivity curve, the original curve turning point is corresponding to maximum point of the second derivate curve which shows the electric interface more directly.
 In order to examine the validity of the S-inversion scheme, a H-type geo-electric model for numerical simulation of water-filled fault is designed as shown in Figures 2b and 2c, which parameters are ρ1 = ρ.3 = 100Ω·m, ρ.2 = 10Ω·m and h.1 = 100m, h2 = 3m. A water-filled fault will be regarded as a thin conductive sheet, corresponding to the h2 = 3m in the model. By using one-dimension forward calculation program, we get the data of voltage decay curve in Figure 2b, the second derivative of apparent conductivity has been calculated according to the method analyzed in section 2. Imaging section has been drawn by using the parameter of second derivative of apparent conductivity in Figure 2c. The conductive plate can not be directly distinguished in Figure 2b. However, the second derivative image section gives clear information of interface where the red peaks show geo-electric interface.
 The proposed TEM has been used to forecast the harmful geological body in the tunnel. Figure 3 presents the schematic view of configuration. Figure 3a describes a sketch of TEM configuration located the surface of tunnel where a square transmitter loop (3m × 3m with 10 turns) is placed vertically on the front tunnel wall. A special antenna with 210m2 area is placed in the center of the transmitter loop. Figure 3b is the sketch of the special antenna. The circle plate in up-part is a shield sheet which be used to get rid of the signals come from the back of tunnel.
 This facility is used to take a surveying at numerous positions across the front wall of the tunnel. Data were recorded in the time delay range of 0.008ms–0.96ms with sampling frequency 225Hz. The output current was 10A across a voltage of 24V. Figure 4 shows the result of prediction survey. Figure 4a presents apparent resistivity section and Figure 4b shows secondary derive of apparent vertical conductive imaging section. From Figures 4a and Figure 4b, it is estimated that there is a low resistance geo-electric boundary at 28m ahead of the tunnel, which indicates a water filled fault of fracture zone is filled with mud or silt. Table 1 lists the location of the fault, and also the error and precision of TEM forecasting by comparing with ex-cave result. The error is only 2m and the precision is 97.2%, which demonstrate that TEM S-inversion method can be used to tunnel prediction.
Table 1. Result Comparing TEM Forecasting With Ex-Cave
DK109 + 014
DK109 + 012
 The obtained results have been checked by ex-cave recorder as shown in Figure 4c, where dash line denotes TEM result and red zone denotes ex-cave result. There is only small error between both results.
 TEM has been used extensively for surface exploration in China over the past two decades. However, there are few reports of TEM being applied in tunnel forecasting. In this paper, it is developed that the special configuration put a transmitter loop vertically in front of tunnel wall to receive TEM response with a special antenna. The equivalent conducting sheet solution for the TEM water-structure forecasting is proposed and the method of calculating from collected data is also given.
 The completed model simulation and experimental results show that S-inversion method can more successfully determine the distance to thin conductive sheets than traditional apparent conductivity curves. The proposed new method can give out more clear information of low resistance interface. Obviously, this technique can more successfully detect water or mud-filled, faults or fracture zones ahead of the front wall of a tunnel during construction.
 This work was supported by China post-doctoral fund (2005038388), the Knowledge Innovation Project of Chinese Academy of Science (KZCX2-VW-113), K. C. Wong Education Foundation, Hong Kong (20050919081006) and National Natural Science Fund (50539080).