b'David Annetts best of Exploration GeophysicsFeatureSattel, D. 1998. Conductivity information in three dimensions.interaction between conductors, their resistive limit responses Exploration Geophysics (this issue). are additive.Smith, R. S., A. P. Annan, J. Lemieux, and R. N. Pedersen. 1996. Application of a modified GEOTEM system to reconnaissanceSquare wave responseexploration for kimberlitcs in the Point lake area. NWT.The measured response of a square wave system can be obtained Canada: Geophysics 61: 8292. from an infinite sum of alternating step responses to giveSmith, R. S., R. N. Edwards, and G. Buselli. 1994. Automatic technique for presentation of coincidcnt-loop. impulse-response. transient, electromagnetic data. Geophysics 59:A(t,T)=A i (et/ i e(t+T/2)/ i +e(t+T)/ i e(t+3T/2)/ i .) (6)15421550. iStolz, E. S., and J. C. Macnae. 1998. Evaluating EM waveforms byThis infinite series can be analytically summed to givesingular-value decomposition of exponential basis functions. Geophysics 63: 6474. e-t/ iA(t,T)=A i 1+eT/2 i (7)Appendix A: Mathematical basis of arbitrary waveformidecomposition Sampled step and square wave responsesThis brief summary includes some sign corrections from theIn a practical TEM system, the signal is sampled in windows that article by Stolz and Macnae(1998): may extend from time to t kto t k+ 1. In this case the sampled response is given byRepresentation of time-domain step and frequencv-doniain response 1 t k+1A(T,t k ,t k+1 )= t k+1 t kt k A(t,T)dt (8)The step function response of an isolated conductor can be expressed as: where t kis defined with respect to the t = 0 transition. Integration A(t)=A i exp(t/ i ) (1) is easily performed to give, in the case of the square wave,i A(T,t k ,t k+1 )=i A i (et k / i et k+1 / i )where there is an upper limit on the range ofi , i.e., a maximum( t k+1 t k )( 1+eT/2 i ) (9)time constant. The inductive limit A 0 , which is the response at time t = 0, is given by Sampled periodic ramp responseWe can choose to represent arbitrary waveforms by a sequence A 0 =A i (2) of periodically repeated linear ramp responses. For each of i these responses it can be shown from simple algebra that the Physically at this limit, current flow is restricted to the surface ofsampled response in window t kto t k+ 1 from a linear primary the conductor such that the perpendicular component of thefield ramp, amplitude change Pover a time interval from t mmlocal magnetic field remains a constant. In the frequency domainto t m+ 1 (both of which are earlier in time than t k ) is given bythere is an exact decomposition of the response A() intoAPj i j i +2 i 2 A(T,t k ,t k+1 ,t m ,t m+1 )=( t k+1 i k )( t m+1 i m )( i e m i ) (10)t t 1+ -T/2A()=A i 1+ j i =A i 1+ j22 i (3) i t k / i t k +1/ i t m+1 / i t m / ii i (e e )(e e )where j= 1 and the A, are identical to those in equation (1). If the primary field change coincides exactly with the sampling time, the response can be separated into two components: In frequency domain, the concept of the resistive limit hasone from previous repetitions of the waveform given by the proved useful in interpretation; defined as the slope ofequation above with an appropriate substitution of (t m T/2) for t m ,(t m +1T/2) for t m +1 and P forP ; plus the single amplitude versusin the low-frequency limit: n msection where sampling and primary ramp are coincident in the 1A interval t n tot n +1 . For this last part is is possible to express the RL= lim j=A ii (4) response as an integral to obtainit n+1 [1-exp((tt 0 )/)]This, in the case of the step response formulation, can be seenC=Pt n (t n+1 t n )2 (11)to be exactly mRL=A ii =0A(t)dt (5) which has the solutioni C=P p[1+p(exp(1/p)1)] (12)mLamontagne (1975) has shown that for a ramp primarywhere p=/(t n+1 t n ).excitation, the resistive limit is simply the magnetic field of the steady-state current that will flow in the conductor afterFitting arbitrary waveform datatransients have decayed. With a coil receiver measuring the time derivative of the response, this steady-state current producesOnce the convolution of an exponential decay with an arbitrary zero response. Since at the limit 0 there is no magneticwaveform has been computed using the equations given above, OCTOBER 2020 PREVIEW 50'