Electromagnetic Fields About a Moving Magnetic Dipole: Implications for Marine Controlled Source Electromagnetic Systems

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1 Index Table of contents Electromagnetic Fields About a Moving Magnetic Dipole: Implications for Marine Controlled Source Electromagnetic Systems E. E. S. Sampaio Centro De Pesquisa Em Geofísica E Geologia, Instituto De Geociências Da Universidade Federal Da Bahia, Rua Barão De Geremoabo S/N, , Salvador, Brazil ( Edson@Cpgg.Ufba.Br) Summary A thorough investigation of the role that the source velocity has on the space and time variation of the electromagnetic field scattered by the earth is necessary, because marine controlled source electromagnetic geophysical surveys employ moving sources. A first step towards this goal consists of the analysis of the difference between the scattered field due to a moving source and that due to a static source for this type of survey. The model which suffices to stress this cinematic aspect is a vertical magnetic dipole moving at a constant speed along a horizontal line in a homogeneous medium separated from two homogeneous half-spaces by horizontal boundaries both above and below the dipole. The results show that both the velocity and the relative displacement between the source and the medium cause a measurable variation relative to the static condition. Therefore, it is necessary to take them into account in geophysical interpretation and to adapt the concepts of time and frequency domain for electromagnetic systems with moving sources. Introduction Electromagnetic surveys may employ moving sources. In some of them, the receivers remain stopped while the transmitter is dragged such as in marine controlled source electromagnetic (MCSEM) (Cheesman et al., 1987; Edwards and Cheesman, 1987; Ellingsrud et al., 00). De Hoop (005) and Sampaio (006) analyzed the behavior of the primary field for the case of a moving source in a conductive medium. However, the geophysical literature lacks a thorough investigation of the role that the transmitter velocity has on the spatial and temporal variation of the electromagnetic field scattered by the earth and its consequence on EM data interpretation. This paper represents a first step toward this investigation. It introduces it by computing and analyzing the differences between the variation of the scattered field due to a moving source and the same variation caused by a stopped source. Because of the inherent difficulty of the problem and also because I intend to stress the cinematic aspect, I employ a simple model. The source is a vertical magnetic dipole moving at a constant speed along a horizontal line in a homogeneous medium. The geometry comprises a homogeneous and conductive medium separated from two homogeneous half-spaces by horizontal boundaries both above and below the dipole for the MCSEM case. The three media are non-magnetic. Even though the values of those differences may be two to three orders of magnitude smaller than the values for a system in which both the transmitter and the receivers are at rest, a geophysical survey would measure them. Also, the time variation of the cinematic and the static quantities are markedly distinct. So, it is necessary to redefine the concepts of frequency and time domain for systems with moving sources. Methodology and Results As Figure 1 shows, a vertical magnetic dipole moves horizontally with a velocity v 0 at the plane z = 0, inside a horizontal layer of thickness h 0 + h 1. The magnetic moment of the dipole is M = NAI 0. The layer h 0 < z < +h 1 represents a medium with a wave number u 1. Two half-spaces bound the layer: z < h 0, for a medium with a wave number u 0, and z > +h 1, for a medium with a wave number u.

2 Figure 1. Configuration of the model. I assume quasi-static conditions. Therefore, the square of the wave numbers are: κ 0 = µ 0 ε 0 ω ; κ 1 = iµ 0 σ 1 ω; and κ = iµ 0 σ ω. In these expressions: µ 0 is the free-space magnetic permeability; ε 0 is the free-space dielectric permittivity; and σ 1 and σ are, respectively, the conductivities of the sea water and of the sea substratum. By solving the correspondent wave equation for a Hertz potential of magnetic source due to this dipole we can determine the electromagnetic field components. I computed the difference between the values of the horizontal component of the scattered electric field of a MCSEM survey for two cases: v 0 = 10 m/s and v 0 = 0. I employed two functions of time for the current: a box function lasting 1s and a causal function, with a sin(πt) variation. For both functions, M = 8 π X 10 4 Am. Under the quasi-static assumption the expression of the electric field is the following: ρ = ((x x 0 + vt v 0 ξ) + (y y 0 ) ) 1/ and u j = (λ κ j ) 1/, j = 0,1,. I substituted in those equations: σ 1 = 3 S/m, h 0 = 80 m, h 1 = 0 m, y y 0 = 0, and z = 0 m.

3 The curves of Figures and 3 display the differences of values of the scattered field between the two aforementioned cases for the box function and three values of σ : 1 S/m, 0.1 S/m, and 0.01 S/m. In Figure : v = 0 and x x 0 = 0 m when t = 0. The differences have about the same order of magnitude of the values of the field in the static condition. In Figure 3: v = v 0, which implies that x x 0 remains equal to 0m. In this case the differences are between the same order of magnitude and one order of magnitude lower than the values of the field in the static condition. Therefore, neglecting those differences will cause error of TDEM interpretation in either case. Figure. Variation of the TDEM field difference with time at a constant receiving point on the sea bottom. Solid line: σ = 1 S/m. Short dashed line: σ = 0.1 S/m. Long dashed line: σ = 0.01 S/m. Figure 3. Variation of the TDEM field difference with time at a receiving point in-tandem with the transmitter on the sea bottom. Solid line: σ = 1 S/m. Short dashed line: σ =0.1S/m. Long dashed line: σ = 0.01 S/m. The curves of Figures 4 and 5 display the same type of differences for the causal sinusoidal function. In Figure 4: v = 0, x x 0 = 100 m when t = 0 and σ equal to 1 S/m and 0.1 S/m. It shows that a

4 constant peak or r.m.s. value doesn t occur if the transmitter is moving. After s the differences are even larger than the values of the field for the static case. In Figure 5: v = v 0, which implies that x x 0 remains equal to 100 m, and σ equal to 1 S/m, 0.1 S/m, and 0.01S/m. The three curves show that, for each value of conductivity, a different period of time is necessary to reach a stationary regime. They also show that one may not disregard the magnitude of the differences relative to the static condition. Figure 4. Variation of the FDEM field difference with time at a constant receiving point on the sea bottom. Solid line: σ = 1 S/m. Short dashed line: σ = 0.1 S/m. Figure 5. Variation of the FDEM field difference with time at a receiving point in-tandem with the transmitter on the sea bottom. Solid line: σ =1S/m. Short dashed line: σ = 0.1 S/m. Long dashed line: σ = 0.01 S/m. Therefore, it is necessary to take into account the movement of the source and analyze the time variation of the waveform for several values of frequency, in order to perform adequate inverse modeling and interpretation of data in the FDEM realm. Conclusions I have determined the solution for the secondary electromagnetic field scattered by a layered medium under the action of a controlled source moving dipole. I performed a complete algebraic analysis of

5 the subject in terms of the space and time variation for a marine survey. I also computed and compared the values of the fields between the static and the cinematic situations. Both the velocity and the relative displacement between the source and the medium cause a measurable variation relative to the static condition even for this simple 1-D model of the earth. Therefore, it is necessary to take them into account in inverse modeling and in geophysical interpretation of EM data, especially under -D or 3-D conditions. Furthermore, I advocate from our analysis and results that it is necessary to adapt the concepts of time and frequency domain for electromagnetic systems with moving sources. Under the cinematic condition the terms modulus and phase or in-phase and in-quadrature lose their meaning. Hopefully, this study will stimulate further research on the subject. Acknowledgments This work has been supported with a grant and a fellowship from the Brazilian National Council of Research (CNPq). References Cheesman, S., Edwards, R. and Chave, A. [1987]. On the theory of seafloor conductivity mapping using transient electromagnetic systems. Geophysics 5, De Hoop, A. [005]. Fields and waves excited by pulsating sources in motion - The general 3D timedomain Doppler effect. Wave Motion 43, Edwards, R. and Cheesman, S. [1987] Two dimensional modeling of a towed transient magnetic dipole-dipole sea floor EM system. J. Geophys. 61, Ellingsrud, S., Eidesmo, T., Johansen, S., Sinha, M., MacGregor, L. and Constable, S. [00] Remote sensing of hydrocarbon layers by seabed logging (SBL): results from a cruise offshore Angola. The Leading Edge 1, Sampaio, E. [006] Contribution of the relative velocity between source and receivers to electromagnetic fields in the sea. Studia Geophysica et Geodaetica, 50,