TY - JOUR
AU - Lu,, Kailiang
AB - Abstract The high-performance transient electromagnetic method (TEM) excitation source is a new type of source that has been proposed for urban underground space exploration. This source is composed of two trapezoid plates. To ensure that the radiation field was focused in a certain direction, the two trapezoid plate-shaped antennas were arranged into a horn shape. This new source is characterised by high power, directional excitation and high resolution. The corresponding multi-component global apparent resistivity definition method is established for a high-performance transient electromagnetic excitation source. This method is studied using the inverse function theorem. Then, the monotonic relationship between components of the electromagnetic field and resistivity is analysed. For the fields that satisfy the monotonic relationship with half-space resistivity, the apparent resistivity can be calculated correctly to ignore the time period and location in the space. This means that this definition method can eliminate the limitation of early and late times, and the near and far zones. The apparent resistivity calculation results of the theoretical layered model reveal that global apparent resistivity curves show a regular change, which smoothly and comprehensively reflects the change of electrical information in the model. The experimental results of the 3D model show that the five-layer low-resistivity anomaly contained in the urban underground space designed in this paper exhibits an obvious response in the global apparent resistivity profile. It is concluded that a high-performance TEM excitation source possesses a high resolution, clearly reflecting all of the anomalies of a complex urban underground space model. high-performance TEM source, global apparent resistivity, multi-component, urban detection 1. Introduction Underground space resources are the third largest areas after space and marine resources (Qian 2000, 2016). They make an inevitable choice to transform the urban development mode and break through the bottleneck of urban development by fully using urban underground space resources (Cano-Hurtado & Canto-Perello 1999; Bobylev 2009, 2010; Tengborg & Sturk 2016; Andersen et al.2018). Fine detection in urban areas is a major issue that must be realised as the key to rational use of urban underground resources. Geophysical methods can be used as an effective means to explore underground targets (Metwaly et al.2014; Imposa et al.2015; Widodo et al.2016; Chandran & Anbazhagan 2017; Andersen et al.2018). However, many difficulties still remain in the detection of urban underground areas, as follows. (i) Artificial noise in the city is strong, resulting in a low signal-to-noise ratio of geophysical signals. (ii) The urban underground space requires a large depth of detection and high resolution. A transparency plan in urban areas requires an exploration depth of 200 m, and resolution requires the meter level, or even the sub-meter level; thus, this is a great challenge for traditional geophysical methods (Di et al.2019). (iii) The particularities of urban environment lead to strict limitations. For example, a drilling method could cause damage to underground buildings and foundations, such as pipelines and malls, which would have serious effects on residents’ daily lives, so use of geophysical methods is greatly restricted. To resolve these difficulties, a new method has been developed for urban underground space detection (Li et al.2018a, 2018b, 2019). Due to the fact that the electromagnetic (EM) method can perform non-destructive detection on an underground space with high efficiency (Xue et al.2013), it can be used in urban detection. Nevertheless, a magnetic source cannot transmit directional information. In addition, exploration is greatly affected in a strong interference environment. As a result, the fine detection of urban underground targets cannot be completed. With this issue in mind, a high-performance transient electromagnetic excitation source was designed for urban underground space detection (Li et al.2018a, 2018b, 2019). The new source was composed of a reflector antenna with two trapezoid plates. The radiant power is stronger due to the structure of the reflector antenna. The antenna was arranged into a horn shape making the distribution of radiation energy directional. However, traditional sources do not have this characteristic, so the high-performance TEM excitation sources have higher signal-to-noise ratios and accuracy than traditional sources (Li et al.2018a, 2018b). Subsequently, the team achieved the purpose of enriching the frequency components of excitation signals by transmitting complex waveforms, which means transmitting some different waveforms that contain different frequency components. For the results of different waveforms, a combined correlated superposition algorithm was used to get more abundant information regarding the underground target body to realise the multi-resolution detection of underground targets. Then, synthetic aperture technology was introduced and synthesised by means of multiple apertures of different sizes, so as to achieve the purpose of improving the horizontal and vertical resolutions, then realising multi-scale information extraction in urban underground space. However, the interpretation of high-performance transient electromagnetic excitation sources has not yet begun. In the processing of transient electromagnetic data, due to the extremely high computational complexity of 3D inversion, it is difficult to realise (Cox & Zhdanov 2008; Cox et al.2010; Zhdanov 2010). In the field, the observed EM fields cannot reflect the distribution of underground geoelectricity directly; however, the apparent resistivity can. Furthermore, due to the computation requirement of the apparent resistivity definition being much lower than 3D inversion, it will be easier to generalise. Therefore, this paper aims to study the multi-component apparent resistivity definition method ignoring early or late times, and far or near zones for high-performance transient electromagnetic excitation sources so as to obtain a more intuitive and reliable preliminary interpretation method (Zhou & Xue 2014; Xue et al.2015). The results not only provide more information for 3D interpretation technology and inversion, but also act as the basis for the analysis of virtual wave field velocity. The TEM response of a tilted electrical dipole on the ground was provided, then the TEM response of high-performance TEM source is obtained by the superposition of a tilted electric dipole. The relationship between the two components of the magnetic field and resistivity was monotonous. Therefore, the inverse apparent relationship can be directly used to define the global apparent resistivity. The three components of the electric field can be regarded as a single-peak function of the half-space resistivity. As the definition of global apparent resistivity is the inverse function, there is a one-to-one correspondence between the field value and resistivity. Consequently, for this unimodal function, when defining the apparent resistivity, the first goal is to determine the location of the peak point, then to divide the single-peak curve into two monotonic curves and to determine the solutions on the two sides of the peak point. Finally, the apparent resistivity of the series one-dimensional model and the complex three-dimensional model are calculated. 2. Theory and analysis on the field characteristics 2.1. TEM response of electric dipole under one-dimensional layered model As shown in figure 1, a layered model is established, and the parameters of each layer are determined by its resistivity and top surface coordinate. The right-handed rectangular coordinate system is used with the harmonic factor as |${e^{ - iwt}}$|⁠. Figure 1. Open in new tabDownload slide One-dimensionally layered model. Figure 1. Open in new tabDownload slide One-dimensionally layered model. The control equations for electromagnetic fields are as follows: $$\begin{equation} \hspace{-28pt}\nabla \times {{\bf E}} = i\omega {{\bf B}}, \end{equation}$$(1) $$\begin{equation} \nabla \times {{\bf B}} = \mu \sigma {{\bf E}} + \mu {{{\bf J}}_s}, \end{equation}$$(2) where E is the electric field, B the magnetic field and Js the imposed electric dipole source. Under the conditions of Coulomb, a magnetic vector potential is: $$\begin{equation} {{\bf B}} = \nabla \times {{\bf A}}, \end{equation}$$(3) $$\begin{equation} {{\bf E}} = i\omega {{\bf A}} + \frac{1}{{\mu \sigma }}\nabla (\nabla \cdot {{\bf A}}). \end{equation}$$(4) For the one-dimensional model displayed in figure 1, vector potential A is generated by the electric dipole at any position and can be obtained by the Hankel transform (Key 2009; Xue et al.2012): $$\begin{equation} {{\bf A}}({{\bf r}}) = \frac{1}{{2\pi }}\int\limits_{0}^{\infty }{{\skew3\hat{A}(\lambda ,z){{\rm J}_0}(\lambda r)}}\lambda d\lambda , \end{equation}$$(5) where |${J_0}$|is the zero-order of the first type of Bessel function, and |$r$|is the horizontal distance between the transmission and reception. The response of an arbitrary tilted electric dipole can be obtained by a combination of a vertical electric dipole and a horizontal electric dipole. The equidistant spacing of each triangular plate is then divided into five line sources, and the electromagnetic field response excited by the entire triangular plate is obtained by integrating the five line sources, as shown in figure 2. Figure 2. Open in new tabDownload slide Schematic diagram of the high-performance transient electromagnetic source from different points of view. (a) Front of source. (b) Side of source. Figure 2. Open in new tabDownload slide Schematic diagram of the high-performance transient electromagnetic source from different points of view. (a) Front of source. (b) Side of source. Equations (3) and (4) are substituted into equation (2), and Fourier transform was applied in the xy direction: $$\begin{equation} - \frac{{{d^2}{{\hat{\bf A}}}}}{{d{z^2}}} + {\gamma ^2}{{\hat{\bf A}}} = \mu {{{\bf J}}_s}, \end{equation}$$(6) where ${\gamma ^2}{\rm{ = }}{\lambda ^{\rm{2}}}{\rm{ - }}i\omega \mu \sigma \begin{array}{*{20}{c}} ,&{{\lambda ^{\rm{2}}} = k_x^2} \end{array} + k_y^2$ ⁠, kx and ky are wave numbers and the solution of equation (5) can be obtained by solving equation (6). For the horizontal electric dipoles, the vector potential is |${{\hat{\rm A} = }}(0,{\skew3\hat{A}_y},\skew3\hat{A}) = (0,{\skew3\hat{A}_y},\frac{\partial }{{{\partial _y}}}{\hat{\Lambda }_z}),$| and let: $$\begin{equation} {A_y}({{\bf r}}) = \frac{1}{{2\pi }}\int\limits_{0}^{\infty }{{{{\skew3\hat{A}}_y}(\lambda ,z){J_0}(\lambda r)\lambda d\lambda }}, \end{equation}$$(7) $$\begin{equation} {A_z}({{\bf r}}) = \frac{1}{{2\pi }}\frac{\partial }{{\partial y}}\int\limits_{0}^{\infty }{{{{\hat{\Lambda }}_z}(\lambda ,z){J_0}(\lambda r)\lambda d\lambda }}. \end{equation}$$(8) In the i-th layer, the vector potential takes the following form: $$\begin{equation} {\skew3\hat{A}_{y,i}} = {a_i}{e^{{\gamma _i}(z - {z_{i + 1}})}} + {b_i}{e^{ - {\gamma _i}(z - {z_i})}} + {\delta _{ij}}\frac{\mu }{{2{\gamma _j}}}{e^{ - {\gamma _i}\left| {z - {z_s}} \right|}}, \end{equation}$$(9) $$\begin{eqnarray} {\hat{\Lambda }_{z,i}} &=& {c_i}{e^{{\gamma _i}(z - {z_{i + 1}})}} + {d_i}{e^{ - {\gamma _i}(z - {z_i})}}\\ && - \frac{{{\gamma _i}}}{{{\lambda ^2}}}({a_i}{e^{{\gamma _i}(z - {z_{i + 1}})}} - {b_i}{e^{ - {\gamma _i}(z - {z_i})}}), \end{eqnarray}$$(10) where |${z_i}$| is the top surface depth of the i-th layer, |$\gamma _i^2\,{\rm{ = }}\,{\lambda ^2}{\rm{ - }}i\omega \mu {\sigma _i}$| and the dipole source is located in the j-th layer with the depth of |${{{z}}_{{s}}}$|⁠, which is then loaded into equation (9) through the Kronecker Function. It is worth mentioning that the upward decay coefficients a and c are defined at the bottom of layer, while downward decay coefficient is defined at the top. This has the advantage of avoiding the numerical overflow caused by an ultra-thick layer or ultra-large conductivity. For layers above the excitation source, there are recursive function|${\rm{s\ of\ \ }}R_i^ - = {b_i}/{a_i}$| and |$S_i^ - = {d_i}/{c_i}$|⁠; while at the layer below the excitation source, there are recursive functions of|${\rm{\ \ \ }}R_i^ + = {a_i}\ /{b_i}$| and |$S_i^ + = {c_{i/}}{d_i}$|⁠. R and S, respectively, represent the reflection coefficients of the transverse electric and magnetic fields. An expression of a recursive function can be obtained by imposing a tangential continuous condition to equations (9) and (10). $$\begin{equation} R_i^ \pm = \frac{{\left( {r_i^ \pm + R_{i \pm 1}^ \pm {e^{ - {\gamma _{i \pm 1}}{h_{i \pm 1}}}}} \right){e^{ - {\gamma _i}{h_i}}}}}{{1 + r_i^ \pm R_{i \pm 1}^ \pm {e^{ - {\gamma _{i \pm 1}}{h_{i \pm 1}}}}}}, \end{equation}$$(11) where $$\begin{equation} r_i^ \pm = \frac{{{\gamma _i} - {\gamma _{i \pm 1}}}}{{{\gamma _i} + {\gamma _{i \pm 1}}}}, \end{equation}$$(12) and $$\begin{equation} S_i^ \pm = \frac{{\left( {s_i^ \pm + S_{i \pm 1}^ \pm {e^{ - {\gamma _{i \pm 1}}{h_{i \pm 1}}}}} \right){e^{ - {\gamma _i}{h_i}}}}}{{1 + s_i^ \pm S_{i \pm 1}^ \pm {e^{ - {\gamma _{i \pm 1}}{h_{i \pm 1}}}}}}, \end{equation}$$(13) where $$\begin{equation} s_i^ \pm = \frac{{{\gamma _i}{\sigma _{i \pm 1}} - {\gamma _{i \pm 1}}{\sigma _i}}}{{{\gamma _i}{\sigma _{i \pm 1}} + {\gamma _{i \pm 1}}{\sigma _i}}}, \end{equation}$$(14) |${h_i} = {z_{i + 1}} - {z_i}$| was the i-th thickness, + denotes the layer below source and — denotes the layer above the source. On the outer boundary, |$R_1^ - = S_1^ - = 0$| and |$R_N^ + = S_N^ + = 0$|⁠. Then, the recursive equation of the reflection coefficient is obtained. $$\begin{equation} {a_j} = \left( {{e^{ - {\gamma _j}\left| {{z_{j + 1}} - {z_s}} \right|}} + R_j^ - {e^{ - {\gamma _j}\left| {{z_j} - {z_s}} \right|}}} \right)\frac{{R_j^ + {e^{{\gamma _j}{h_j}}}}}{{1 - R_j^ - R_j^ + }}\frac{\mu }{{2{\gamma _j}}}, \end{equation}$$(15) $$\begin{equation} {b_j} = \left( {R_j^ + {e^{ - {\gamma _j}\left| {{z_{j + 1}} - {z_s}} \right|}} + {e^{ - {\gamma _j}\left| {{z_j} - {z_s}} \right|}}} \right)\frac{{R_j^ - {e^{{\gamma _j}{h_j}}}}}{{1 - R_j^ - R_j^ + }}\frac{\mu }{{2{\gamma _j}}}, \end{equation}$$(16) $$\begin{equation} {c_j} = \left( { - {e^{ - {\gamma _j}\left| {{z_{j + 1}} - {z_s}} \right|}} + S_j^ - {e^{ - {\gamma _j}\left| {{z_j} - {z_s}} \right|}}} \right)\frac{{S_j^ + {e^{{\gamma _j}{h_j}}}}}{{1 - S_j^ - S_j^ + }}\frac{\mu }{{2{\lambda ^2}}}, \end{equation}$$(17) $$\begin{equation} {d_j} = \left( { - S_j^ + {e^{ - {\gamma _j}\left| {{z_{j + 1}} - {z_s}} \right|}} + {e^{ - {\gamma _j}\left| {{z_j} - {z_s}} \right|}}} \right)\frac{{S_j^ - {e^{{\gamma _j}{h_j}}}}}{{1 - S_j^ - S_j^ + }}\frac{\mu }{{2{\lambda ^2}}}. \end{equation}$$(18) For a vertical electric dipole, its vector potential is |${{\hat{\bf A}}} = (0,{\skew3\hat{A}_y},{\skew3\hat{A}_z})$| with a Hankel transformation: $$\begin{equation} {A_z}({{\bf r}}) = \frac{1}{{2\pi }}\int\limits_{0}^{\infty }{{{{\skew3\hat{A}}_z}(\lambda ,z){J_0}(\lambda r)\lambda d\lambda }}, \end{equation}$$(19) $$\begin{equation} {\skew3\hat{A}_{z,i}} = {c_i}{e^{{\gamma _i}(z - {z_{i + 1}})}} + {d_i}{e^{ - {\gamma _i}(z - {z_i})}} + {\delta _{ij}}\frac{\mu }{{2{\gamma _j}}}{e^{ - {\gamma _i}\left| {z - {z_s}} \right|}}. \end{equation}$$(20) This is also applicable to the recurrence formula in equation (13), after which: $$\begin{equation} {c_j} = \left( {{e^{ - {\gamma _j}\left| {{z_{j + 1}} - {z_s}} \right|}} + S_j^ - {e^{ - {\gamma _j}\left| {{z_j} - {z_s}} \right|}}} \right)\frac{{S_j^ + {e^{{\gamma _j}{h_j}}}}}{{1 - S_j^ - S_j^ + }}\frac{\mu }{{2{\gamma _j}}}, \end{equation}$$(21) $$\begin{equation} {d_j} = \left( {S_j^ + {e^{ - {\gamma _j}\left| {{z_{j + 1}} - {z_s}} \right|}} + {e^{ - {\gamma _j}\left| {{z_j} - {z_s}} \right|}}} \right)\frac{{S_j^ - {e^{{\gamma _j}{h_j}}}}}{{1 - S_j^ - S_j^ + }}\frac{\mu }{{2{\gamma _j}}}. \end{equation}$$(22) 2.2. Characteristic analysis of a high-performance TEM excitation source For the apparent resistivity definition, first we have to determine the functional relationship between the field and resistivity. The relative positions of the source and measuring point are shown in figure 3. Figure 3. Open in new tabDownload slide The location of the source and measuring points. (a) Sliced view and (b) top view of the relative positions of the source and measuring point. Figure 3. Open in new tabDownload slide The location of the source and measuring points. (a) Sliced view and (b) top view of the relative positions of the source and measuring point. The parameters in the calculation were as follows: the source length was 2 m, the opening width was 1 m, angle between the two triangular plates 90°, the current 1 A, the offset of the observation points M1 and M2 from the source were 10 and 50 m, and corresponding coordinates in the coordinate system were (10, 0, 0 m) and (50, 0, 0 m), respectively. Figure 4 shows the variation of the magnetic field response intensity with resistivity for the new source in different offsets in uniform half-space model. Figure 4 parts a–c are the correspondences between the Bx, By, Bz of M1 and resistivity, respectively. Figure 4 parts d–f are the correspondences between the Bx, By, Bz of M2 and resistivity. Figure 4 shows in the change range of resistivity (0.001–1000 Ω⋅m), under a small offset (10 m) or a large offset (50 m), By(t) and Bz(t) can be regarded as a monotonic function of the half-space resistivity at any time. The curve shape of Bx(t) with resistivity changes is shown to be complex. Figure 4. Open in new tabDownload slide Magnetic field response Bx(t), By(t) and Bz(t) at different offsets changing with the resistivity of the uniform half-space model at different times. (a) Offset of 10 m, Bx(t). (b) Offset of 10 m, By(t); (c) offset of 10 m, Bz(t); (d) offset of 50 m, Bx(t); (e) offset of 50 m, By(t) and (f) offset of 50 m, Bz(t). Figure 4. Open in new tabDownload slide Magnetic field response Bx(t), By(t) and Bz(t) at different offsets changing with the resistivity of the uniform half-space model at different times. (a) Offset of 10 m, Bx(t). (b) Offset of 10 m, By(t); (c) offset of 10 m, Bz(t); (d) offset of 50 m, Bx(t); (e) offset of 50 m, By(t) and (f) offset of 50 m, Bz(t). Figure 5 illustrates the variation of the electric field response intensity with resistivity for the new source in different offsets in uniform half-space model. Figure 5 parts a–c are the correspondences between the Ex, Ey, Ez of M1 and resistivity, respectively. Figure 5 parts d–f are the correspondences between the Ex, Ey, Ez of M2 and resistivity respectively. Figure 5 shows in change range of resistivity (0.001–1000 Ω⋅m), under a small offset (10 m) or large offset (50 m), Ey(t) and Ez(t) can be regarded as a monotonic function of the half-space resistivity at any time. As the global apparent resistivity definition is the inverse function, there is a one-to-one correspondence between the field and resistivity. Consequently, for this unimodal function, when defining the apparent resistivity, first we have to determine the location of the peak point, then divide the single-peak curve into two monotonic curves and last find the two solutions on the two sides of the peak point. For high-performance transient electromagnetic excitation sources, By(t), Bz(t) and Ez(t) are used for the global apparent resistivity definition. Figure 5. Open in new tabDownload slide Electric field response Ex(t), Ey(t) and Ey(t) at different offsets changing with the resistivity of the uniform half-space model at different times. (a) Offset of 10 m, Ex(t); (b) offset of 10 m, Ey(t); (c) offset of 10 m, Ez(t); (d) offset of 50 m, Ex(t); (e) offset of 50 m, Ey(t) and (f) offset of 50 m, Ez(t). Figure 5. Open in new tabDownload slide Electric field response Ex(t), Ey(t) and Ey(t) at different offsets changing with the resistivity of the uniform half-space model at different times. (a) Offset of 10 m, Ex(t); (b) offset of 10 m, Ey(t); (c) offset of 10 m, Ez(t); (d) offset of 50 m, Ex(t); (e) offset of 50 m, Ey(t) and (f) offset of 50 m, Ez(t). Figure 6. Open in new tabDownload slide The location of the source and measuring points. (a) Sliced view and (b) top view of the relative positions of the source and measuring point. Figure 6. Open in new tabDownload slide The location of the source and measuring points. (a) Sliced view and (b) top view of the relative positions of the source and measuring point. 3. Definition method of global apparent resistivity Due to the complex implicit function relation between the electromagnetic field excited by high-performance TEM source and resistivity, it is impossible to directly and explicitly define the apparent resistivity by electric field or magnetic field intensity (Zhang et al.2015; Wu et al.2017). However, By(t) and Bz(t) can be regarded as a monotonic function of the half-space resistivity at any time, which creates conditions for the definition of apparent resistivity based on the concept of inverse function. When defining the apparent resistivity, various parameters such as the position coordinates and time are taken into account, and finally the definition of global apparent resistivity is realised regardless of time or offset. Due to the fact that the curve shape of Bx changing with resistivity is relatively complex, in this paper Bx is not used to define apparent resistivity. For Ex(t), Ey(t) and Ez(t), this can be regarded as a unimodal function of half-space resistivity. As the concept of inverse function is adopted in the definition of the global apparent resistivity, there must be a one-to-one correspondence between the field and resistivity. Therefore, for this unimodal function, when defining apparent resistivity, first we have to determine the location of the peak point, then divide the single-peak curve into two monotonic curves and last find the two solutions on the two sides of the peak point. Finally, to select the optimal solution. In this paper, By(t), Bz(t) and Ez(t) are used to define the global apparent resistivity. 3.1. Definition of apparent resistivity with y and z components of a magnetic field Since the observation field By(t) and Bz(t) is a monotonic function of resistivity, according to the inverse function theorem, there must be a resistivity value that uniquely corresponds to a By(t) or Bz(t) value. Therefore, the expression of By(t) and Bz(t) can be expanded into order form by the Taylor expansion method. $$\begin{eqnarray} \begin{array}{@{}l@{}} {B_p}(\rho ,C,t) = {B_p}(\rho _\tau ^{(0)},C,t) + {B_p}^\prime (\rho _\tau ^{(0)},C,t)(\rho - \rho _\tau ^{(0)})\\ \quad\quad\quad\quad\quad+\, \frac{{{B_p}^{\prime \prime }(\rho _\tau ^{(0)},C,t)}}{{2!}}{(\rho - \rho _\tau ^{(0)})^2} + \cdots .\\\quad\quad\quad\quad\quad +\, \frac{{{B_p}^{(n)}(\rho _\tau ^{(0)},C,t)}}{{n!}}{(\rho - \rho _\tau ^{(0)})^n} + {R_n}(\rho ) \end{array} \end{eqnarray}$$(23) Meanwhile, the high-order terms are neglected, while only the linear main part is retained. $$\begin{eqnarray} {B_p}(\rho ,C,t) &\approx& {B_p}(\rho _\tau ^{(0)},C,t) + {B_p}^\prime (\rho _\tau ^{(0)},C,t)\\&&\times\,(\rho - \rho _\tau ^{(0)})(p = x,z). \end{eqnarray}$$(24) Through transformation, the following can be obtained: $$\begin{equation} \rho = \frac{{{B_p}(\rho ,C,t) - {B_p}(\rho _\tau ^{(0)},C,t)}}{{{B_p}^\prime (\rho _\tau ^{(0)},C,t)}} + \rho _\tau ^{(0)}(p = x,z). \end{equation}$$(25) The iterative form can then be written as shown in equation (26): $$\begin{equation} \left\{ \begin{array}{@{}l@{}} \rho _\tau ^{(i + 1)} \approx \Delta \rho _\tau ^{(i)} + \rho _\tau ^{(i)}(i = 0,1,2,...)\\ \Delta \rho _\tau ^{(i)} = \frac{{{B_p}(\rho ,C,t) - {B_p}(\rho _\tau ^{(i - 1)},C,t)}}{{{B_p}^\prime (\rho _\tau ^{(i - 1)},C,t)}}(p = x,z) \end{array} \right.. \end{equation}$$(26) Finally, by setting iteration termination conditions, $$\begin{equation} \left| {\frac{{{B_p}(\rho ,C,t) - {B_p}(\rho _\tau ^{(i)},C,t)}}{{{B_p}(\rho ,C,t)}}} \right| < \varepsilon (p = x,z). \end{equation}$$(27) The global apparent resistivity of the model is obtained. 3.2. Definition of apparent resistivity with Ez B(t) was substituted by Ez(t) in equation (27). Section 2.2 shows that Ez(t) is a unimodal function of resistivity. Due to the fact that global apparent resistivity definition adopts the idea of inverse function, there must be a one-to-one correspondence between the field value and the resistivity. As a result, for this unimodal function, when defining the apparent resistivity, first, it is necessary to determine the location of the peak point, then to divide the unimodal curve into two monotone curves, then to obtain the two solutions on two sides of the peak point and finally to select the optimal solution. Further research shows that to obtain a smooth apparent resistivity curve with gradual change complete from Ez(t) data, the optimisation problem must be further decomposed into the following three situations: when the difference between the two solutions is small (e.g. 10%), the average of the two solutions is taken as the solution of the time channel; when the difference between the two solutions is large (e.g. 300%), the solution closer to the solution of the adjacent time channel is selected as the solution of the time channel; while all other cases are considered as problems without proper solutions. For such problems, to obtain a complete apparent resistivity curve, the minimum curvature interpolation method is used to compensate for the missing data in this paper, which can ensure that the curve changes according to the minimum curvature, and that the curve remains smooth. $$\begin{eqnarray} &&\rho _\tau ^{\left( k \right)}\left( i \right) = - \frac{1}{6}\left\{ {\rho _\tau ^{\left( {k - 1} \right)}\left( {i + 2} \right) + \rho _\tau ^{\left( k \right)}\left( {i - 2} \right)}\right.\\ &&\quad- 4\left.\left[ {\rho _\tau ^{\left( {k - 1} \right)}\left( {i + 1} \right) + \rho _\tau ^{\left( k \right)}\left( {i - 1} \right)} \right] \right\}\left( {i = 1, \cdot \cdot \cdot ,M} \right), \end{eqnarray}$$ $$\begin{equation} \left\{ {\begin{array}{@{}*{1}{c}@{}} {{\rho _\tau }\left( {i - 1} \right) = 2{\rho _\tau }\left( i \right) - {\rho _\tau }\left( {i + 1} \right)\left( {i = 1} \right)}\\ {{\rho _\tau }\left( {i + 1} \right) = 2{\rho _\tau }\left( i \right) - {\rho _\tau }\left( {i - 1} \right)\left( {i = M} \right)}\\ {{\rho _\tau }\left( {i - 2} \right) = {\rho _\tau }\left( {i + 2} \right) - 2\left[ {{\rho _\tau }\left( i \right) - {\rho _\tau }\left( {i - 1} \right)} \right]\left( {i = 1} \right)}\\ {{\rho _\tau }\left( {i + 2} \right) = {\rho _\tau }\left( {i - 2} \right) + 2\left[ {{\rho _\tau }\left( i \right) - {\rho _\tau }\left( {i - 1} \right)} \right]\left( {i = M} \right)} \end{array},} \right. \end{equation}$$(28) where k represents the number of iterations and M represents total number of participating interpolations. In general, two points are taken from the left and right ends of the area to be interpolated to participate in the calculation. 4. Analysis of numerical simulation results To verify the correctness of global apparent resistivity, a series of one-dimensional models are selected for numerical calculation. The one-dimensional models mainly include two-layered models and three-layered models. Finally, a complex three-dimensional urban underground space model is designed. The results further illustrate the fact the new source possesses a high resolution. 4.1. Analysis of layered model results A two-layered model is designed to change the resistivity of the second layer, to verify global apparent resistivity definition method. The model parameters were as follows: the source length was 2 m, opening width was 1 m, angle between the two triangular plates was 90°, the current was 1 A, the offset of observation point M2 from the source was 50 m and the corresponding coordinates in the coordinate system were (50, 0, 0 m). The relative position of the source and measuring point are shown in figure 6. The resistivity of the first layer ρ1 = 100 Ω⋅m, thickness h1 = 20 m and changing resistivity ρ2 was 2, 5, 10, 30, 80, 200, 500 and 800 Ω⋅m, respectively. The multi-component global apparent resistivity curves are defined by By(t), Bz(t) and Ez(t). Figure 7 shows that with the change of the second layer resistivity, the apparent resistivity curves reveal regular changes that gradually tend to the resistivity of the first and second layers of the model, respectively, in the early and late times. And they completely, gradually and smoothly reflect the changes of the electrical information for the model. Figure 7. Open in new tabDownload slide Global apparent resistivity curve as defined by By(t), Bz(t) and Ez(t) when the offset is 50 m. (a) Apparent resistivity curve obtained by By(t). (b) Apparent resistivity curve obtained by Bz(t). (c) Apparent resistivity curve obtained by Ez(t). Figure 7. Open in new tabDownload slide Global apparent resistivity curve as defined by By(t), Bz(t) and Ez(t) when the offset is 50 m. (a) Apparent resistivity curve obtained by By(t). (b) Apparent resistivity curve obtained by Bz(t). (c) Apparent resistivity curve obtained by Ez(t). A three-layered model is designed to verify the definition method of the multi-component global apparent resistivity. The parameters of the model were as follows: the source length was 2 m, opening width was 1 m, angle between the two triangular plates was 90°, the current 1 A, the offset of observation point M2 from the source 50 m and the corresponding coordinates in the coordinate system were (50, 0, 0 m). The resistivity of the second layer of (H, K, A, Q) had changed. For the H-type model, the resistivity of the first layer and third layers was 100 Ω⋅m, the resistivities of the second layer were 5, 10, 30 and 50 Ω⋅m, respectively, and the thicknesses of the first and second layers were 20 and 10 m, respectively. Figure 8 parts a–c show the global apparent resistivity curves of the H model, as respectively defined by By(t), Bz(t) and Ez(t). For the K-type model, the resistivities of the first and third layers were 10 Ω⋅m, those of the second layer were 30, 60, 100 and 200 Ω⋅m, respectively, and the thicknesses of the first and second layers were 10 and 20 m, respectively. Figure 8 parts d–f show the global apparent resistivity curves of the K-type models as respectively defined by By(t), Bz(t) and Ez(t). For the A-type model, the resistivities of the first and third layers were 30 and 100 Ω⋅m, respectively, those of the second layer were 50, 60, 70 and 80 Ω⋅m and the thicknesses of the first and second layer were 10 and 20 m. Figure 8 parts g–i show the global apparent resistivity curves of the A-type model as defined by By(t), Bz(t) and Ez(t), respectively. For the Q-type model, the resistivities of the first and third layers were 100 and 10 Ω⋅m, respectively, those of the second layer were 20, 30, 50 and 80 Ω⋅m, respectively, and the thicknesses of the first and second layers were 20 and 10 m, respectively. Figure 8 parts j–l show the global apparent resistivity curves of the Q-type models as respectively defined by By(t), Bz(t) and Ez(t). It can be seen that the global apparent resistivity curves of the four types models show regular changes. These gradually tend to the resistivities of the first and last layer, respectively, in the early and late times, and completely, gradually and smoothly reflect the changes of the electrical information for the model. Figure 8. Open in new tabDownload slide Global apparent resistivity curve defined by By(t), Bz(t) and Ez(t) when the offset is 50 m. (a) Obtained by By(t) for the H model. (b) Obtained by Bz(t) for the H model. (c) Obtained by Ez(t) for the H model. (d) Obtained by By(t) for the K model. (e) Obtained by Bz(t) for the K model. (f) Obtained by Ez(t) for the K model. (g) Obtained by By(t) for the A model. (h) Obtained by Bz(t) for the A model. (i) Obtained by Ez(t) for the A model. (j) Obtained by By(t) for the Q model. (k) Obtained by Bz(t) for the Q model. (l) Obtained by Ez(t) for the Q model. Figure 8. Open in new tabDownload slide Global apparent resistivity curve defined by By(t), Bz(t) and Ez(t) when the offset is 50 m. (a) Obtained by By(t) for the H model. (b) Obtained by Bz(t) for the H model. (c) Obtained by Ez(t) for the H model. (d) Obtained by By(t) for the K model. (e) Obtained by Bz(t) for the K model. (f) Obtained by Ez(t) for the K model. (g) Obtained by By(t) for the A model. (h) Obtained by Bz(t) for the A model. (i) Obtained by Ez(t) for the A model. (j) Obtained by By(t) for the Q model. (k) Obtained by Bz(t) for the Q model. (l) Obtained by Ez(t) for the Q model. Figure 9. Open in new tabDownload slide The location of the source and measuring points. (a) Sliced view and (b) top view of the relative positions of the source and measuring point. Figure 9. Open in new tabDownload slide The location of the source and measuring points. (a) Sliced view and (b) top view of the relative positions of the source and measuring point. To explain the influence of offset in the global apparent resistivity definition, the following model was designed: source length was 2 m, opening width was 1 m, the angle between the two triangular plates was 90°, the current was 1 A and the offsets were 10, 50 and 100 m. The corresponding coordinates in the coordinate system were (10,0,0), (50,0,0) and (100,0,0). The relative position of the source and measuring point are shown in figure 9. The resistivities ρ1 = 100 Ω⋅m, ρ2 = 10 Ω⋅m and ρ3 = 100 Ω⋅m and thicknesses were h1 = 20 m and h2 = 10 m for the three-layered model. It can be seen from figure 10 that the global apparent resistivity definition method is applicable to both the large and small offsets. The curves can completely, gradually and smoothly reflect the changes of the underground electrical information, approaching the resistivity of the top and bottom layers in the early and late times. This proves that the apparent resistivity of high-performance transient electromagnetic excitation source can be defined, so as to ignore the offset. Through comparison, it can be seen that the changes of the offset have little influence on the apparent resistivity curves of each component. Figure 10. Open in new tabDownload slide Effect of the offset variation on the multi-component global apparent resistivity curve of the high-performance source (a) Effect of the offset on the By(t) apparent resistivity curve. (b) Effect of the offset on the Bz(t) apparent resistivity curve. (c) Effect of the offset on the Ez(t) apparent resistivity curve. Figure 10. Open in new tabDownload slide Effect of the offset variation on the multi-component global apparent resistivity curve of the high-performance source (a) Effect of the offset on the By(t) apparent resistivity curve. (b) Effect of the offset on the Bz(t) apparent resistivity curve. (c) Effect of the offset on the Ez(t) apparent resistivity curve. 4.2. Analysis of the 3D complex urban underground space model results To verify the validity of the global apparent resistivity definition of the new source, a complex urban underground space model was designed as shown in figure 11. The background resistivity was 150 Ω⋅m, source was located on the ground, the offset was 10 m and the current 10 A. The vector finite element method was used for three-dimensional forward modeling. Figure 11. Open in new tabDownload slide Schematic of urban model. (a) 3D schematic diagram. (b) Sliced view. Figure 11. Open in new tabDownload slide Schematic of urban model. (a) 3D schematic diagram. (b) Sliced view. In figure 11, five common urban underground models are created, i.e. pipeline, mall, passages, warehouse and active fault, the respective buried depths of which are 5, 10, 30, 50 and 70 m. Table 1 lists the resistivity parameters of the models. Table 1. Resistivity of urban model. Structure . Resistivity (Ω⋅m) . Structural description . Pipeline (pipe wall) 0.1 Tube wall thickness is 5 cm Shopping mall (outer wall) 0.01 Outer wall thickness is 60 cm Underground passage (outer wall) 0.001 Outer wall thickness is 60 cm Warehouse (outer wall) 0.01 Outer wall thickness is 60 cm Faults 10 Structure . Resistivity (Ω⋅m) . Structural description . Pipeline (pipe wall) 0.1 Tube wall thickness is 5 cm Shopping mall (outer wall) 0.01 Outer wall thickness is 60 cm Underground passage (outer wall) 0.001 Outer wall thickness is 60 cm Warehouse (outer wall) 0.01 Outer wall thickness is 60 cm Faults 10 Open in new tab Table 1. Resistivity of urban model. Structure . Resistivity (Ω⋅m) . Structural description . Pipeline (pipe wall) 0.1 Tube wall thickness is 5 cm Shopping mall (outer wall) 0.01 Outer wall thickness is 60 cm Underground passage (outer wall) 0.001 Outer wall thickness is 60 cm Warehouse (outer wall) 0.01 Outer wall thickness is 60 cm Faults 10 Structure . Resistivity (Ω⋅m) . Structural description . Pipeline (pipe wall) 0.1 Tube wall thickness is 5 cm Shopping mall (outer wall) 0.01 Outer wall thickness is 60 cm Underground passage (outer wall) 0.001 Outer wall thickness is 60 cm Warehouse (outer wall) 0.01 Outer wall thickness is 60 cm Faults 10 Open in new tab Figure 12 shows a cross-sectional view of the global apparent resistivity as defined by Ez. When x is equal to 5 m, the early channel shows obvious low resistance anomaly, which is the abnormal response of the pipeline. Starting from the fifth time channel, there is a mall anomaly that continues until reaching the 20th time channel. Then the anomaly of the underground passage occurs at x = ±18 m and lasts until the 40th time channel. The anomaly of the underground warehouse occurs at the 30th time channel. From the 45th time channel, the anomaly of the underground active faults occurs at x = ±8 m. This result shows that the five-layer low-resistivity anomalies included in the urban underground space designed in this paper exhibit obvious responses in the global apparent resistivity. This proves the validity of the global apparent resistivity definition of high-performance transient electromagnetic sources, and also provides a qualitative and intuitive interpretation method for the new source. It is also shown that the new source possesses high resolution, which can clearly reflect all of the anomalies of the complex urban underground space model. This also provides a new method for fine detection in urban, thereby leading to a likelihood of ‘transparency’ of urban underground space. Figure 12. Open in new tabDownload slide Global apparent resistivity profile. Figure 12. Open in new tabDownload slide Global apparent resistivity profile. 5. Conclusions For a high-performance TEM excitation source, a global apparent resistivity definition method is studied in this paper. Through the introduction of a theoretical formula, as well as the calculation method and numerical simulation and analysis of the theoretical model, the following conclusions are preliminarily obtained: Based on the concept of inverse function, the global apparent resistivity definition method for the new source is studied by using a magnetic field and electric field. This realises the global apparent resistivity calculation regardless of time or offset, which can completely, smoothly and gradually reflect the change of the underground electrical information. The influence of offset variation on the global apparent resistivity is also analysed. Next, a complete set of definition methods for the global apparent resistivity of high-performance transient electromagnetic excitation sources is established, which provides an intuitive, basic and qualitative interpretation method for new sources. The results of the three-dimensional model experiments reveal that the five-layer low-resistivity anomalies included in the urban underground space designed in this paper exhibit obvious responses in the global apparent resistivity cross-section, thus proving the validity of global apparent resistivity definition for high-performance TEM source. This result illustrates the new source has a higher resolution, while clearly reflecting all of the anomalies of the complex urban underground space models. This result also provides a new method for the fine detection in urban, thereby leading to a likelihood of ‘transparency’ of urban underground space. The global apparent resistivity definition method of high-performance transient electromagnetic excitation source provides more abundant information for three-dimensional interpretation technology and inversion, and acts as the basis for virtual wave field velocity analysis. 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Published by Oxford University Press on behalf of the Sinopec Geophysical Research Institute. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
TI - Global apparent resistivity definition for a high-performance TEM excitation source
JF - Journal of Geophysics and Engineering
DO - 10.1093/jge/gxaa024
DA - 2020-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/global-apparent-resistivity-definition-for-a-high-performance-tem-rdtnapICtZ
SP - 718
EP - 729
VL - 17
IS - 4
DP - DeepDyve
ER -