Pseudo-seismic wavelet transformation of transient electromagnetic response in engineering geology exploration

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L645, doi:.29/27gl36, 27 Pseudo-seismic wavelet transformation of transient electromagnetic response in engineering geology exploration G. Q. Xue, Y. J. Yan, 2 and X. Li 3 Received 25 June 27; revised 27 July 27; accepted August 27; published 3 August 27. [] This paper presents some new theoretical analysis and numerical simulations of that transient electromagnetic diffusion-field response is transformed into a pseudoseismic wavelet in engineering geology exploration. It can clearly reveal the electric interface under ground. To simplify the integral equation used in the transformation, the integral range is separated into seven windows, and each window is compiled into a group of integral coefficients. Then, the accuracy of the coefficients is tested, and the calculated coefficients are used to derive the pseudo-seismic wavelet by optimization method. Finally, several geoelectric models are designed, so that model responses are transformed into the pseudo-seismic wavelet. The transformed imaginary wave shows that some reflection and refraction phenomena appear when the wave meets the electric interface. This result supports the introduction of the seismic interpretation in data processing of transient electromagnetic method. Citation: Xue, G. Q., Y. J. Yan, and X. Li (27), Pseudo-seismic wavelet transformation of transient electromagnetic response in engineering geology exploration, Geophys. Res. Lett., 34, L645, doi:.29/27gl36.. Introduction [2] Recently, the developed and developing technologies in seismology are gradually applied to Transient Electromagnetic Method (TEM), such as pseudo-seismic interpretation and MT migration imaging technique etc. These methods consumedly extend the potential capacities of TEM, so that the clear information of ground target can be extracted from practical measured data. [3] Over the past decades, transient electromagnetic methods (TEM) have become widely used in the ground source exploration [Gunderson et al., 986], ground target mapping [Xue et al., 24], and borehole-based ore investigations. However, with the rapid development of engineering construction, the precision involved in TEM prospecting has not met the increasingly stringent demands of engineering geology. It is very necessary that some new methods must be developed to solve this problem. Geophysicists have been paying close attention to research on pseudo-seismic methods and electromagnetic migration imaging methods. Many important advances as the combined migration and inversion have been made in these fields in recent years. Levy et al. Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China. 2 Department of Engineering Mechanics, Northwestern Polytechnical University, Xi an, China. 3 School of Geology and Survey Engineering, Chang An University, Xi an, China. Copyright 27 by the American Geophysical Union /7/27GL36 [988] first investigated pseudo-seismic MT (Magnetotelluric). However, so far, research on TEM pseudo-seismic interpretation is still infrequent. [4] At present, the TEM pseudo-seismic interpretation has two approaches. One is to transform the data of the TEM diffusion field into MT data through the approximate formula t = (94 2)/f (t-ms, f-hz) [Maxwell, 996]. This was first used to correct the MT static shift according to TEM magnetic-field sounding data so that seismic-like data can be obtained via the MT pseudo-seismic method [Sternberg et al., 985]. [5] Another is the wave-field transformation method. Because there is some relationship between the diffusion and wave equations, the similarities between wave propagation and diffusion propagation should be. Lee et al. [987] formulated a complete imaging algorithm that is the EM equivalent of seismic migration. Based on the similarity of the Laplace transform for the diffusion and non-diffusion fields, Gershenson [993] presented a simple transformation from the diffusion field into wave field; and this was proposed as a means of interpreting TEM sounding data. The diffusion equation for a field H in time domain t can be transformed into a wave equation for a field U in a time-like variable [Lee et al., 989]. This involves transforming the TEM response into a pseudo-seismic wavelet and then interpreting the TEM sounding data according to plane-wave sounding theory.the pseudo-seismic wavelet is different with general wavelet transform. In fact, the former is only the echo wave similarly with earthquake. Based on the relationship between diffusion equation and wave equation, TEM sounding data can be converted into pseudo-seismic wavelet, and there will exist the certain relationship between converted imaginary wave and real seismic reflection wavelets. Thus, ones can equivalently regard effusion information as the seismic reflection signal which comes from underground interface. As a result, ones can interpret TEM sounding data according to well development seismic theory. [6] However, the transforming equation will give rise to difficulties in the numerical calculation, so efficient numerical solutions need still to be developed. Based on the above-mentioned wave-field transformation method, this paper shows how to realize the wave-field transformation and provides a more detailed solution to the pseudo-seismic wavelet inversion. In order to get good numerical solutions, the integral range is firstly separate into seven time windows, and integral coefficients are assigned to each time window using the same equation in all cases so that the precision in calculating the coefficients can be increased. Finally, an optimizing method is used to obtain a stable and reliable solution for the seismic-like wavelet. The example results have shown that the obtained pseudo-seismic curve L645 of5

2 L645 XUE ET AL.: PSEUDO-SEISMIC WAVELET TRANSFORMATION L645 can distinguish the electric interface more clearly than traditional data-processing curve. 2. Theoretical and Numerical Analyses [7] From the Maxwell equation, the partial differential equation of the magnetic field can be expressed as rrh m ðr; H mðr; tþ ¼ ; ðþ [4] Equation (7) is the formula of the wave-field transformation from a transient electromagnetic field component to a pseudo-seismic wavelet. [5] If the integral coefficients are known, we can obtain the wave field value U(t j ) from the surveyed transient electromagnetic field value H m (t) by performing an inverse transformation of equation (7). Consequently, the integral coefficient h j is more important. [6] Let u(t) =, equation (6) have the following form where H m is the magnetic field intensity, m is magnetic permeable, s is conductivity, and r is the distance from the source to the field point. [8] The initial and boundary conditions can be written as H m ðr; Þ ¼ ; H m j G ¼ H m ðr ; tþ; t > ; ð2þ where G is the boundary of volume V at r =r b. We introduce U(r, t) that satisfies the following wave equations rrur; ð 2 Ur; ð t Þ ¼ ; ð3þ Z According to the special integral equation (8) becomes Z xe ax2 dx ¼ 2a ; Z te t2 =4t dt: te t2 =4t dt ¼ p ffiffiffiffi pt ð8þ ð9þ ðþ Ur; ð Ur; ð tþj t¼ ¼ ; ð4þ Uj r ¼ Ur ð b ; tþ; t > ; ð5þ where t is the variable of pseudo-time in the wave domain. [9] The unique relationship between the time-domain diffusion field H m (r, t) and the imaginary t domain wave field U(r, t) is as following Z te t2 4t UðtÞdt: This transformation equation between the diffusion field H m (t) and the pseudo-wave field U(t) involves only time t and parameter t which is independent of r. [] According to equation (6), the value of the pseudowave field U(t) can be calculated using the surveyed value of the diffusion field H m (t). However, the equation is illposed and the unknown U(t) is hidden in the integral equation. Consequently, a suitable numerical scheme must be used to solve the problem. [] For a homogeneous whole space, the equation can be simplified so that a simple relationship between the diffusion and wave fields can be determined. For example, a standard finite-difference method can be used to obtain the solution of the pseudo-wave field. [2] In this section, to obtain the solution of the half space, we will seek a group of integral coefficients to linearize the integral functions. [3] We can rewrite equation (6) as follows: H m ðt i Þ ¼ Xn j¼ where a(t i, t j )= pffiffiffiffi 2 i ð6þ U t j ati ; t j hj : ð7þ t 2 j 4t t j e i, h j is the integral coefficient. and the right-hand term of equation (7) can be written as X n j¼ at i ; t j hj : ¼ p ffiffiffiffiffi : ðþ Integral coefficients can be obtained from equation () using optimization method. We rewrite equation () as a matrix equation: T ¼ a h þ a 2 h 2 þþa N h N T ¼ a 2 h þ a 22 h 2 þþa 2N h N T M ¼ a M h þ a M2 h 2 þþa MN h N ð2þ where T i = p ffiffiffiffi, a i,j = a(t i, t j ). Solving equation (2), the integral coefficients h j can be obtained using optimization method. [7] During the forward transformation of wave-field, in order to ensure transient electromagnetic field calculation precision, the two-step optimized method is adopted. This method can deal well with ill-posed question produced by large amount of integral coefficients and depress the illposed problem of the first calculator function. In wave-field inverse transformation, the normalized method are adopted, optimizing normalized parameters are selected by the deviation theory and the Newton iterative form to make the transformed wave field stable and reliable. [8] Because the temporal scope of transient electromagnetic field data is wide from several tens ms to several tens ms, the number of matrix coefficients will become excessive large. In order to overcome this trouble, the entire time range ( ms) is separated into seven time ranges so as to calculate the coefficients by solving equation (2) for each window. For every time window, the n in equation () is selected as 2, thus 2 integral coefficients (h h 2 ) can be obtained by solving equation (2). All results of the acquired integral coefficients are listed in Table. 2of5

3 L645 XUE ET AL.: PSEUDO-SEISMIC WAVELET TRANSFORMATION L645 Table. Integral Conversion Coefficients h j 2 in Different Time Range Time Number of Time Gate Range [9] In order to validate the accuracy of the numerical calculating scheme, the acquired integral coefficients of every time window are inversed into the wave-field value. For the wave-field value u(t) =, we let the electromagnetic field value be H m (t i )= p ffiffiffiffi then, equation () can be rewritten as p ffiffiffiffiffi ¼ Xn j¼ U t j ati ; t j hj : ð3þ Equation (3) is solved using the calculated integral coefficients, so the wave-field value u(t) can be induced. Finally, the numerical results of wave-field value are compared with its theoretical value. Results of the inverse wave-field value and the theoretical wave-field value corresponding to each time window are shown in Figure. In Figure, the dotted lines represent the inverse wave-field value, while solid lines represent the theoretical wave-field value (u(t) = ). The inversion results and level of obtained accuracy demonstrate that it is appropriate to separate the time range into seven time windows and use the coefficients in calculating the wave-field values. It is also shown that relative errors for the 7 time windows are 5.7%, 6.7%, 5.4%, 6.%, 5.2%, 5.7%, and 4.5%, respectively. In general, this error level is acceptable in practice engineering issues. the results of wave-field transformation of the G-type model at different depths h. [24] During the forward calculation, the transmitter square loop is set to 2 m, the transmitter current equals to A, and the recorder time ranges from 72 ms to8msin Figure 2. However, information related to the geo-electric interface cannot be directly observed. On the other hand, Figure 2b shows that with the increase of h, the arrival time of the pseudo-wavelet wave becomes later, and the amplitude decreases, but the width of the pseudo wave-let increases. [25] In order to improve the identifying ability for the geo-electric interfaces using the proposed numerical analysis, the four more complex models with three-layered geoelectric interfaces are designed. The four models are labeled as H-type, K-type, A-type, and Q-type. For the H-type, r = 25W m and h =6m, r 2 =5W m and h 2 =6m, r 3 = 5W m. For the K-type, r =5W m and h =6m, r 2 =5W m and h 2 =6m, r 3 =5W m. For the A-type, 3. Model Examples [2] In this section, the 6 geo-electric models are given to obtain the pseudo-seismic wavelet using the proposed numerical approach. First, the two models with the twolayers, which are G-type model and D-type model, are designed using a center loop configuration that can be used to survey field data in a similar way to seismic exploration by self-exciting and a self-receiving configuration. Before doing the wave-field transformation, the response of the half-space from the resistivity of the first layer is subtracted using forward-calculated data. [2] For G-type model, the resistivity of the first layer is r =5Wm, and that of the second layer is r 2 = 5W m. For D-type model, the resistivity of the first layer is r = W m, and that of the second layer is r 2 =W m. [22] Based on equation (7), results of the wave-field transformation of the G-type model with the two-layer geo-electric section at different depth h, where the h equals to 4 m, 6 m, 8 m, m, 2 m, 4 m, 6 m, 8 m and 2 m in turn are showed in Figure 2. [23] Figure 2a is the forward calculations of the decay curve of the G-type model at different depths. Figure 2b is 3of5 Figure. Results of the wave-field inversion.

4 L645 XUE ET AL.: PSEUDO-SEISMIC WAVELET TRANSFORMATION L645 r =5W m and h =6m, r 2 =5W m and h 2 =6m, r 3 = 5W m. For the Q-type, r =25W m and h =6m, r 2 = 5W m and h 2 =6m, r 3 =W m. [26] The typical forward-calculated decay curves and their apparent resistivity of the H-type models are shown in Figures 3a and 3b. They cannot still reveal the geoelectric interface. However, the pseudo-wave field curves transformed from the center loop TEM responses of the model is plotted, as shown in Figure 3c, it can clearly depict geo-electric interfaces. Results also show that the magnitude of the first interface is greater than that of the second. Even for a body buried at a depth of 2 m, TEM is still able to distinguish the electric interface. 4. Conclusion [27] The proposed method of TEM pseudo-seismic interpretation represents an advance in the available technology in which we use a special equation to transform TEM field data into pseudo-seismic wavelet data. As the equation that links the time-domain diffusion field H m (r, t) and the pseudo wave field is generally ill-posed, we took the steps of determining a group of integral coefficients and linear integral equation, separating the integral range into seven time windows, and adopting an optimizing method to obtain Figure 3. Three-layer H-type geo-electric section: (a) decay curve, (b) apparent resistivity curve, and (c) result of wavefield transformation. stable wave field values. Tests of the accuracy of the model calculations indicate that the method is sound. [28] As the wavelength of the electric field is longer than that of the real seismic wave, the length of the transformed wave-field wavelet is also large. [29] The calculated result reveals that the transformed imaginary wave shows features of propagation during travel through a conductive medium: there are reflection and refraction phenomena when the wave meets the electric interface. This result supports the introduction of the seismic interpretation method into TEM data-processing. [3] Acknowledgments. This work was supported by the Knowledge Innovation Project of the Chinese Academy of Sciences (KZCX2-YW- 3). The authors gratefully acknowledge the support of the K. C. Wong Education Foundation, Hong Kong (259986). This project was also supported by the China Postdoctoral Science Foundation ( ) and National Natural Science Fund (55398). Figure 2. Results for the transformation of the two-layer G-type model: (a) decay curves derived and (b) results of wave-field transformation. References Gershenson, M. (993), Simple interpretation of time domain electromagnet sounding using similarities between wave and diffusion propagation [J], Geophysics, 62, of5

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