Wave Motion 43 (2005) 123–131 Primary electromagnetic fi eld in the sea induced by a moving line of electric dipoles Edson E.S. Sampaio? Instituto de Geoci? encias da UFBA, Sala 213B, Campus Universit′ ario de Ondina, 40170-290 Salvador, Bahia, Brazil Received 19 January 2005; received in revised form 12 July 2005; accepted 8 August 2005 Available online 3 October 2005 Abstract The analysis of the primary electromagnetic fi elds caused by steady state or transient electric current fl owing along a cable of fi nite length moving with a constant velocity below the sea surface has several applications. It supports the analysis of submarine physical data and it is useful for protecting ships from the threat of sea mines. The usual approach to describe the primary fi eld starts from the solution of a magnetic vector potential in the frequency domain due to an electric dipole, which is obtained with themethodofGreen.SubsequenttransformationtothetimedomainemployingLaplacetransformaswellasintegrationalongthe cablelengthallowstodescribethetimevariationoftheprimaryelectromagneticinducedfi eldduetoalineofelectricdipoles.The result is applicable to both shallow and deep sea water environments. Because of the difference in velocity between source and receiver,acarefulapplicationoftheconvolutionintegralisnecessaryinordertoadaptthesourcepulsesolutiontoanytypeoftrans- mittingcurrentwaveform.Furthermoreitisshownthattheinfl uenceoftherelativevelocitybetweensourceandreceiverisnotnec- essarily negligible. Therefore it has to be considered in the interpretation of the secondary fi elds scattered by geological features. ? 2005 Elsevier B.V. All rights reserved. Keywords: Electromagnetic; Sea; Moving source; Line of dipoles 1. Introduction The analysis of electromagnetic fi elds caused by steady state or transient electric current fl owing along a cable of fi nite length placed in sea water has several applications. It supports the analysis of submarine physical data for environmental and shallow or deep crustal research, and it is also useful for protecting ships from the threat of sea mines. During the last two decades practical and theoretical efforts have led to the development of controlled source electromagnetic induction (EMI) techniques for surveying the sea substractum: [3,5,6,8,10]. Presently the surveys consist of towing a transmitting cable near the sea bottom while the receivers remain fi xed at the sea bottom. Besides the conductivity of the fl uid this procedure has a fundamental difference to ground and ? Tel.: +55 713 2038603; fax: +55 713 2038501. E-mail address:

[email protected] 0165-2125/$ – see front matter ? 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2005.08.001 124E.E.S. Sampaio / Wave Motion 43 (2005) 123–131 airborne surveys, where transmitters and receivers are on the same frame of reference. In this paper we investigate the role that the relative velocity between transmitter and receiver has on the space and time variation of the primary electromagnetic fi eld. A precise knowledge of the primary fi eld is important because the scattered fi eld usually represents a fraction of it. So, even minor differences in the primary fi eld caused by the differential velocity may render inadequate inverse modeling and interpretation of EMI data. We will not pursue dielectric and magnetic aspects here because of the following reasons. The fundamental fre- quency of the surveys are usually lower than 30Hz. Therefore the conduction current largely exceeds the displace- mentcurrentinthesea.Theseawatermagnetizationisvanishinglysmall.Sowecanputμ ≈ μ0= 4π × 10?7H/m. We will also neglect phase variation along the source cable, because its length of hundred of metres is much smaller than the wavelengths under consideration. Furthermore, the time measurement accuracy of electromagnetic in- duction (EMI) surveys is of the order of 1?s. So we will not be concerned with the precise arrival times of the electromagnetic wave fronts. However we do not exclude this last point to be a matter of interest either for bistatic differential velocity ground probe radar (GPR) or for global positioning system (GPS) surveys, specially if we take into account dielectric and magnetic dispersion [2]. For a general and thourough discussion of the Doppler effect we refer to [4]. 2. Development of the solution We will not reproduce here the fundamental aspects of the electromagnetic theory. We refer the interested reader to [11,9]. For our purposes it will suffi ce to employ the magnetic vector potential. We shall express it as ? a(x,y,z,t), in the time domain. In this expression (x,y,z) represents the observation point and t, is time. Since we neglect displacement currents and magnetization, the magnetic fi eld, ? h(x,y,z,t), the electric fi eld, ? e(x,y,z,t), and the vector potential relate to each other according to: ? h = 1 μ0 ? × ? a,(1) ? e = 1 σ ? ×?h.(2) In Eq. (2) σ represents the electric conductivity. The magnetic vector potential obeys the inhomogeneous wave equation of pure electric conduction: ?2? a ? μ0σ ?? a ?t = ?μ0?js.(3) The source term in Eq. (3) is the electric current density in A/m2,?js. Let it be an electric dipole oriented along the x-direction and situated at a point (xc,y0,z0 ) of a homogeneous infi nite medium, such that xc= x0+ vt, as represented in Fig. 1. The electric current density is given by: ?js(x,y,z,t) = I(t)dx0δ(x ? xc)δ(y ? y0)δ(z ? z0)?i. (4) In Eq. (4), I(t) represents the electric current, dx0the elementary dipole length, and the δ(α ? α0) represents the Dirac delta functions with singularities in the points α = α0. Since the electric dipole has only an x component, Eq. (3) shows that the primary potential also has only an x component, which represents the particular solution of that equation. So ? a = ax?i. From now on we will drop the x, subscript of the scalar component ax. We obtain the solution of Eq. (3) in the time domain by application of Green’s method. Let v, x0, y0, and z0be constant. According to [1,4,7,9] we obtain in this case: E.E.S. Sampaio / Wave Motion 43 (2005) 123–131125 Fig. 1. An electric dipole Idx0?i at (xc,y0,z0 ), moving with a velocity v along the x-direction and a fi xed observation point at (x,y,z) in a homogeneous isotropic space. R is the distance between the dipole and the observation point and r is its horizontal projection: R = ? r2+ (z ? z0)2;r = ? (x ? x0? vt)2+ (y ? y0)2. a(x,y,z,t) = ? μ3 0σ dx0 8√π3 ? +∞ 0 I(t ? ξ)e?(μ0σ(R(t?ξ)) 2)/4ξ ? ξ3 dξ,(5) where R(t) = ? (x ? x0? vt)2+ (y ? y0)2+ (z ? z0)2. For a pulse waveform: I(t) = Cδ(t ? t0), Eq. (5) yields a(x,y,z,t + t0) = ? μ3 0σCdx0e?(μ 0σ(R(t0))2)/4t 8√π3t3 H(t).(6) Since t0 is completely arbitrary we can equate it to zero, and the solution is the same as for a fi xed source. The situation is quite different for the case of a general current waveform. We will analyze Eq. (5) for a causal current, I(t) =0, for t 0, fl owing along a cable of length L. In this case Eq. (5) yields: a(x,y,z,t) = μ0 8π ? t 0 e?(μ0σ((y?y0) 2+(z?z0)2))/4ξ ξ I(t ? ξ) × ? erf ??μ 0σ 4ξ (x + L ? v(t ? ξ)) ? ? erf ??μ 0σ 4ξ (x ? v(t ? ξ)) ?? dξ.(7) 3. Numerical analysis Let us compare the values of hyand ex between a moving and a fi xed line of electric dipoles. Applying Eqs. (1) and (2) in Eq. (7) we obtain the following expressions for ?L ≤ x0≤0: hy(R,t) = ?(z ? z0)μ0σ 16π ? t 0 e?(μ0σ((y?y0) 2+(z?z0)2))/4ξ ξ2 I(t ? ξ) × ? erf ??μ 0σ 4ξ (x + L ? v(t ? ξ)) ? ? erf ??μ 0σ 4ξ (x ? v(t ? ξ)) ?? dξ,(8) 126E.E.S. Sampaio / Wave Motion 43 (2005) 123–131 ex(R,t) = μ0 16π ? t 0 (2ξ ? μ0σ((y ? y0)2+ (z ? zo)2))e?(μ0σ((y?y0) 2+(z?z0)2))/4ξ ξ3 I(t ? ξ) × ? erf ??μ 0σ 4ξ (x + L ? v(t ? ξ)) ? ? erf ??μ 0σ 4ξ (x ? v(t ? ξ)) ?? dξ.(9) InEqs.(8)and(9)wewillkeepaconstantvalueforthefollowingparameters:σ = 3S/m;μ0= 4π × 10?7H/m, y ? y0= 20m, z ? z0= 20m, and the length of the source cable L = 300m. We will compute both hyand exfor three values of v: 0, 5, and 10m/s. We will employ two conditions for the longitudinal distance between the receiver and the closer extremity of the line of dipoles: Fig. 2. Transient variation of hyin mA/m and of exin mV/m as a function of time in second, after turning off a square wave source current of 500A fl owing along a 300m cable: on time of 2s for (a) and (c); and on time of 0.5s for (b) and (d). The receiver is fi xed at (20m, 20m, 20m). The cable ends are at (vt, 0, 0) and (vt ? 300m, 0, 0). Solid line for v = 0; short dashed line for v = 5m/s; and long dashed line for v = 10m/s. E.E.S. Sampaio / Wave Motion 43 (2005) 123–131127 ? A fi xed receiver, such that the longitudinal distance between the receiver and the moving transmitter will vary in time. In this fi rst condition we will substitute x =20m, in Eqs. (8) and (9). ?A constant source-receiver longitudinal distance. In this second condition we will substitute x ? vt = 20m in Eqs. (8) and (9). Wewillanalyzethemagneticfi eldhy inmA/mandtheelectricfi eldexinmV/mfortwocausalcurrentfunctions. Fig. 3. Transient variation of the difference in hyin mA/m and ex in mV/m between a moving and a fi xed source as a function of time in second, after turning off a square wave source current of 500A fl owing along a 300m cable: on time of 2s for (a) and (c); and on time of 0.5s for (b) and (d). The receiver is at (20m + vt, 20m, 20m). The cable ends are at (vt, 0, 0) and (vt ? 300m, 0, 0). Solid line for v = 5m/s; and short dashed line for v = 10m/s. 128E.E.S. Sampaio / Wave Motion 43 (2005) 123–131 4. Time domain electromagnetic (TDEM) The fi rst causal current function relates to TDEM data and consists of a square wave current, I1(t) = 500A, for 0 t t0, and I1(t) = 0 otherwise. We selected two values for t0: 2 and 0.5s, to visualize the contribution of the length of the on time on the variation of the fi eld values. TDEM geophysical data consist of readings usually taken during the off times, and stacked during several on and off times. Sinceofftimesareusuallyequalorlargerthantheontimes,wemayassociatet0= 2stoafundamentalfrequency f ≤ 0.25Hz and t0= 0.5s to a fundamental frequency f ≤ 1Hz. Values of t0larger than 2s increase the problem we are analyzing and, at the same time, diminish the contribution of the conductivity. Values of t0smaller than 0.5s increase the attenuation and consequently decrease the depth of exploration. Fig. 4. Variation of hyin mA/m and of exin mV/m as a function of time in second for a causal sine wave source current with an amplitude of 500A fl owing along a 300m cable: period of 4s for (a) and (c); and period of 1s for (b) and (d). The receiver is fi xed at (20m, 20m, 20m). The cable ends are at (vt, 0, 0) and (vt ? 300m, 0, 0). Solid line for v = 0; short dashed line for v = 5m/s; and long dashed line for v = 10m/s. E.E.S. Sampaio / Wave Motion 43 (2005) 123–131129 The curves of Figs. 2 and 3 are typical of TDEM data. They show a 1s transient behaviour of either hyor ex, after turning off the square wave current. Fig. 2(a) and (c) describes the condition of a fi xed receiver and t0= 2s. For Fig. 2(b) and (d) the receiver is also fi xed and t0 =0.5s. We observe a large difference in the fi eld values and that the gradient of the curves of both hy and exbecomes steeper as the values of v and t0increase. Fig. 3(a) and (c) describes the condition of a constant source-receiver longitudinal distance and t0= 2s. For Fig. 3(b) and (d) the source-receiver longitudinal distance is also constant and t0= 0.5s. We observe that by compensating the distance, the effect of varying t0is smaller than any practical accuracy. However the contribution Fig. 5. Variation of the difference of hyin mA/m and of ex in mV/m between a moving and a fi xed source as a function of time in second for a causal sine wave source current with an amplitude of 500A fl owing along a 300m cable: period of 4s for (a) and (c); and period of 1s for (b) and (d). The receiver is at (20m + vt, 20m, 20m). The cable ends are at (vt, 0, 0) and (vt ? 300m, 0, 0). Solid line for v = 5m/s; and short dashed line for v = 10m/s. 130E.E.S. Sampaio / Wave Motion 43 (2005) 123–131 of the velocity yields a difference larger than 100?A/m for transient times less than 10ms for the magnetic fi eld, and larger than 2?V/m for transient times less than 1ms for the electric fi eld. Though smalls, these values are of the same order of magnitude of the contribution from deep scatterers. 5. Frequency domain electromagnetic (FDEM) The second causal current function relates to FDEM data and consists of a sine wave I2(t) = 500 sin(2πft)A, for 0 ≤ t ≤ 10s and I2(t) = 0, for t ≤ 0. To conform the TDEM and FDEM analysis we selected two values for the frequency: 0.25 and 1Hz. FDEM geophysical data consist of readings taken over several periods integrated for the r.m.s. value, to obtain the amplitude at a certain frequency. Phase differences between the fi eld and the source current are also measured, but we will not compute or discuss them here. The curves of Figs. 4 and 5 are typical of FDEM data. They show a 10s cyclical behaviour of either hyor ex. In Fig. 4(a) and (c) the receiver point is fi xed and f = 0.25Hz. In Fig. 4(b) and (d) the receiver point is fi xed and f = 1Hz.Weobservealargedifferenceofthefi eldvaluesasafunctionofbothvandfforthistypeofsourcecurrent too. Furthermore, the amplitude of each one of the curves related to a mobile source varies with time. Therefore, the r.m.s value of the curves due to a mobile source is a function of the number and the range of cycles employed in its determination. InFig.5(a)and(c)aconstantlongitudinaldistancebetweentransmitterandreceiverisassumedandf = 0.25Hz. In Fig. 5(b) and (d) a constant longitudinal distance between transmitter and receiver is assumed and f = 1Hz. We observe that by compensating the distance, the differences are practically the same for the two values of frequency. However the amplitude varies with v by more than 10?A/m and 4?V/m, respectively, for the magnetic and the electric fi eld. 6. Conclusion Our results show that the relative velocity between the source and the receiver affects the primary fi eld. Conse- quently it will also affect the fi eld scattered by geological structures. Since the voltages measured by induction coils allow to measure the magnetic fi eld with a precision of 1?A/m, the velocity of the source affects in a measurable manner the primary fi eld value whether we consider the observation point at a constant or a variable distance from thesource.Asimilarbehaviouroccurswiththeelectricfi eld.ThereforeitisnecessarytoanalyzeTDEMandFDEM controlledsourcegeophysicaldatacarefully.Thisfacthastobetakenintoaccountintheseparationofthesecondary fi eld and in the identifi cation of its actual space–time position. Otherwise one may do gross interpretation errors. Acknowledgments This work has been supported with a grant and a fellowship from CNPq. References [1] M. Abramowitz, I. Stegun, Laplace transfoms, in: Handbook of Mathematical Fun