Time Reversal of Electromagnetic Fields and its Application to Lightning Location

Farhad Rachidi EMC Laboratory

Swiss Federal Institute of Technology (EPFL) Lausanne, Switzerland farhad.rachidi@epfl.ch

Marcos Rubinsein IICT

UAS Western Switzerland, HEIG-VD Yverdon-les-Bains, Switzerland marcos.rubinstein@heig-vd.ch

Abstract—In this invited lecture, we present a general survey of the electromagnetic time reversal and its application to locate lightning discharges.

Keywords—Lightning location systems (LLS), Electromagnetic time reversal (EMTR), Magnetic direction finding (MDF), Time of Arrival (ToA).

I. INTRODUCTION Time reversal has received increasing attention in the fields

of acoustics and electromagnetic applications in the past few decades following the work of Fink and co-workers (e.g., [1, 2]. Time reversal or T-symmetry describes the symmetry of physical laws under a time reversal transformation: t→−t .

Time reversal techniques have been applied in various fields of engineering. Recently, the electromagnetic time reversal (EMTR) technique was used as a means of locating lightning discharges ([3, 4].

In this lecture, we will present a general survey of the application of EMTR to the location of lightning discharges. The lecture will be organized as follows. In Section II, we will briefly describe the concept of time-reversal invariance with special attention to Maxwell’s equations. In Section III, we present an overview of the applications of the time-reversal technique in various fields of engineering. Special emphasis will be made to the application of this technique to the problem of fault detection in distribution networks. Section IV will give a general overview of present techniques used to detect and locate lightning discharges. The application of EMTR to locate lightning discharges will be discussed in Section V. Summary, conclusions and outlook will be given in Section VI.

II. TIME-REVERSAL INVARIANCE OF MAXWELL’S EQUATIONS

As discussed by Snieder [5], most laws of nature feature invariance under time reversal. In other words, the laws remain unchanged when running the clock backwards rather than forwards. Mathematically, time reversal implies making the substitution t→−t .

Let us consider Maxwell’s equations in vacuum:

∇⋅ ε (r )E(r ,t)( ) = ρ(r ,t) (1a)

∇⋅ µ(r )H (r ,t)( ) = 0 (1b)

∇×E(r ,t) = −µ(r ) ∂

H (r ,t)∂t

(1c)

∇×

H (r ,t) = ε(r ) ∂E(r ,t)∂t

+J (r ,t) (1d)

where E(r ,t) and

H (r ,t) are, respectively, the electric and

the magnetic field, ρ(r ,t) is the charge density, J (r ,t) is the

electric current density, and ε(r ) and µ(r ) are, respectively, the electric permittivity and the magnetic permeability.

Applying the time-reversal transformation to equations (1a-1d), we obtain

∇⋅ ε(r )

E(r ,−t)( ) = ρ(r ,−t) (2a)

∇⋅ µ(r ) −H (r ,−t)( )( ) = 0 (2b)

∇×E(r ,−t) = −µ(r ) ∂(−

H (r ,−t))∂(−t)

(2c)

∇× (−

H (r ,−t)) = ε(r ) ∂

E(r ,−t)∂(−t)

+ (−J (r ,−t)) (2d)

It can be readily seen that expressions (2a)-(2d) are identical to equations (1a)-(1d), except that the magnetic field and the current density have changed sign. In other words, to make the equations invariant under time reversal, the magnetic field and the electrical current density should change sign as well:

H (r ,t)→−

H (r ,−t) and

J (r ,t)→−

J (r ,−t) .

A discussion is in order on the necessity of the sign change for the magnetic field and the current density, which created some controversies in the literature (see e.g., [6-9]). Indeed, It was argued by Albert [8] that Maxwell’s equations are not time reversal invariant and the sign change of the magnetic field is just a mathematical trick to save the time reversal invariance (see discussion in [7]). This appears to be an incorrect statement because, as discussed by Snieder [5], there is a physical reason for which the electrical current density should change sign under time reversal: When the direction of time is

reversed, the velocity of the charges changes sign. As a result, the associated current density should change sign as well. Finally, a change of sign of the current density should result in a change of sign of the associated magnetic field. Hence, Maxwell’s equations are invariant under time reversal.

It is finally worth noting that the electromagnetic propagation involving a dissipative medium is not time reversal invariant unless an inverted-loss medium is considered for the reverse times. This point will be further discussed in Section V.

III. ENGINEERING APPLICATIONS OF EMTR As mentioned in the Introduction, time reversal has

received much attention in the field of acoustics (e.g., [2, 10-12]) and, more recently, in electromagnetics, power and communications. A few interesting applications are as follows:

- Focusing of electromagnetic waves [13] and applications in biomedical engineering (e.g., [14]);

- Target imaging in a cluttered environment (e.g., [15];

- Landmine detection (e.g., [16, 17];

- Communications and radar (e.g, [18]);

- Lightning location ([3, 4]).

Recently, electromagnetic time reversal was applied to the problem of fault location in power networks [19]. Fault location is an important function in power systems upon which depend the security and the quality of energy supply. In what follows, the basic principles of this method will be described. Interested readers are referred to [19, 20] for further details of the method and its implementation. Consider the case of a lossless, two-conductor line (or a single wire above a perfectly conducting ground). The Telegrapher’s equations describing the wave propagation along the line are given by

∂v(x,t)∂x

+ L ' ∂i(x,t)∂t

= 0 (3)

∂i(x,t)∂x

+C ' ∂v(x,t)∂t

= 0 (4)

in which v(x,t) and i(x,t) are voltage and current waves along the line, and where L ' and C ' are the per-unit-length inductance and capacitance of the line, respectively.

Applying the time reversal transformation to the Telegrapher’s equations and changing the sign of the current

i(x,t)→−i(x,−t) (see discussion in Section II), we get

∂v(x,−t)∂x

+ L ' ∂(−i(x,−t))∂(−t)

= 0 (5)

∂(−i(x,−t))

∂x+C ' ∂v(x,−t)

∂(−t)= 0 (6)

Expressions (5)-(6) are identical to equations (3)-(4).

Hence, as discussed in [19], the Telegrapher’s equations are invariant under a time-reversal transformation for lossless (non-dissipative) lines. Based on this observation, Razzaghi et al. [19] proposed a technique to locate faults in a network using EMTR. The method requires the measurement of the transients associated with the fault at one location along the system. The

observed transients are stored, time-reversed and transmitted back into the system, using numerical simulations for different guessed fault locations. Finally, the fault location is determined by identifying the point characterized by the highest energy concentration associated with the back-injected time-reversed fault current [20]. The authors present also an experimental validation of the proposed technique using a reduced-scale network on which faults were hardware-emulated and fault-originated transients were recorded. Then, time-reversed transient waveforms were generated using an arbitrary waveform generator and injected into the network. The fault current was measured at different guessed locations along the line and it was shown that the energy of the fault current maximizes at the real fault location. Other advantages of the method include high accuracy and robustness (against the a priori assumed fault impedance), applicability to inhomogeneous networks (comprising both overhead lines and underground cables), to active networks and to series compensated transmission lines.

IV. LIGHTNING DETECTION AND LOCATION Lightning location systems (LLS) have undergone major

expansion during the past decades and they are being extensively used throughout the globe to detect lightning discharges and to provide information including the location, the lightning type, its intensity, and the movement of thunderstorms (e.g. [21]). In this part of the lecture, we will provide a brief overview of the most widely used techniques to locate lightning discharges, concentrating on the location of cloud-to-ground lightning. This will provide a basis for comparison with the presentation in Section V of the use of electromagnetic time reversal in lightning location.

A. Direction Finding Direction finding uses sensors that can estimate the

azimuth to the lightning strike point. The first investigations on the location of lightning by direction finding were carried out on the decade of the fifties by Horner [22-24].

The minimum number of of sensors required to get a fix with direction finding is two, as long as the strike point is not in line with the two sensor’s baseline and the lower part of the lightning channel can be assumed as vertical. To increase accuracy, a network of more than two sensors is generally used in commercial systems.

One advantage of direction finding for the location of return strokes is the fact that the systems can tolerate clock synchronization of the order of a millisecond or so, as opposed to the time of arrival technique that will be described in the next subsection.

The most commonly used direction finding sensors nowadays use the knowledge that, at ground level, the radiation component of the magnetic field from the return stroke phase of cloud-to-ground lightning is perpendicular to the direction of propagation of the return stroke wavefront. These sensors use two perpendicular loops to measure two components of the wideband horizontal magnetic field. When these types of sensors are used, the systems are said to use the magnetic direction finding technique (MDF).

It should be noted here that it is also possible to estimate the direction to a radiation source by using antenna arrays that detect the phase difference between the narrowband radiation wavefronts received at the different antennas of the array. This method is commonly known as Interferometry in the context of lightning location.

B. Difference in Time of Arrival The technique known as Difference in Time of Arrival, or

simply Time of Arrival uses a number of time-synchronized sensors that detect the arrival time of the return stroke to each one of them. The time difference of arrival in 2D defines two hyperbolic branches for each pair of sensors and, in order to eliminate ambiguities that appear in some regions of the location space, a minimum of four sensors needs to be used (see, for instance, [25])

V. APPLICATION OF EMTR TO LOCATE LIGHTNING

A. General Methodology The use of time reversal to locate lightning has been

described in [3, 4, 26]. To apply the technique, several sensors are used to record the wideband electric or magnetic fields from a return stroke. A simulation program is then used to time-reverse the recorded waveforms and to retransmit the thus processed waveforms back into the location network domain. Under ideal conditions, the back-propagated time-reversed fields will add up in phase at the lightning strike location. Thus, the amplitude of the resulting field will be maximized at the source location and this criterion can be used to determine the location of the lightning strike point.

B. Implementation Algorithm Consider Fig. 1, where a lightning strike is represented at

point rs and where N sensors are represented at locations

r1 to rN .

Fig. 1. N sensors at positions r1 to

rN represented by dots and the strike point

represented by an asterisk at rs . (Adapted from [4])

Assuming that the propagation distance is high enough for

the field to be essentially determined by the radiation

component, the electric field En (r, t) , measured by the sensor

at location rn is equal to:

E(rn, t) =

ϕ t −

rn −rs

c⎛⎝⎜

⎞⎠⎟

rn −rs

(7)

where

ϕ is a function that depends on the nature of the

source. For a lightning return stroke, the form and amplitude of ϕ will generally depend on the return stroke speed and on

the behavior of the lightning return stroke current as it travels up the channel. To use EMTR for lightning location, the field given in (7) is recorded at the nth sensor. It is then time-reversed by replacing t by T - t, where T is the duration of the signal. The result is given in Equation (8).

En (r ,T − t) =

ϕ T − t +rn −rs

c⎛⎝⎜

⎞⎠⎟

rn −rs

(8)

Note that, although we have written the functional

dependence of the field on distance, all we are assuming is that the field waveform was measured, stored and time-reversed at each sensor. These operations do not require knowledge of

ϕ or any other details of the source. If the

sensors are now used as emitters that reradiate the time-reversed version of the electric field they have received, the total back-propagated field, which is simply the sum of the contributions of each sensor, is given by

ETR1(

r, t) =

ϕ T − t +

rn −rs

c−r − rnc

⎛⎝⎜

⎞⎠⎟

rn −rsn=1

N

∑

(9)

At the strike point,

r = rs and (9) equation reduces to

ETR1(

rs, t) =ϕ T − t( ) 1

rn −rsn=1

N

∑ (10)

The contributions of the different sensors add in phase at

the source point, making the back-propagated field maximum at that point.

C. Relation between EMTR and ToA Lugrin et al. [4] showed that the Difference in Time of

Arrival lightning location technique already discussed in Section IV.B is a particular case of electromagnetic time reversal in the case of a perfectly conducting ground. In their proof, Lugrin et al. [4] use the condition that all of the back-propagated wavefronts must arrive at the source location in phase. To do this, they set the phase terms in Equation (9) equal to an unknown constant K for every sensor:

T −rn −rs

c−r − rnc

= K (11)

Writing (11) for two different sensors i and j, and

rearranging terms, we can write

ri −rs

c+r − ric

= T −K

(12)

rj −rs

c+r − rjc

= T −K (13)

Subtracting (13) from (12) and moving the terms

dependent on r to the left hand side, we get

r − ric

−r − rjc

=rj −rs

c−ri −rs

c (14)

The left-hand side in (14) represents the difference in ToA from a general point

r to sensors i and j. The right-hand side, on the other hand, is the difference in ToA from the strike point

rs to the same sensors i and j. The equation defines a hyperbola and it is the basic equation used to locate lightning using the Difference in Time of Arrival technique.

D. Treatment of losses Since, as mentioned in Section I, propagation over a

finitely conducting ground is not time-reversal invariant, three back propagation models have been proposed by Lugrin et al. [4] to deal with the ground losses and dispersion while using EMTR for lightning location. These will be presented in the reminder of this subsection.

To account for the propagation along a lossy ground, the vertical electric field En (ω ) produced by a lightning return stroke at location

rs and measured by a sensor at position rn

can be represented in the frequency domain as follows:

Ε!(ω) = S!( 𝑟! − 𝑟! ,𝜔)!

!!!!!φ(ω)e!!!

!!!!!! (15)

where S𝑓( 𝑟𝑛 − 𝑟𝑠 ,𝜔) represents a filter function that depends on the propagation distance and on the parameters of the soil [27]. The field in (15) is the field received after forward propagation form the strike point to each sensor and it is the basis for all three methods to be presented, which will assume three different strategies for the modeling of the back-propagation of the field. Note that, in the frequency domain, the time reversal operation, denoted by the use of an asterisk as a superscript, is equivalent to a complex conjugate operation.

Model 1: Perfect Ground Back-Propagation Model

In this model, one disregards the losses in the back-propagation. The error incurred with this approximation will depend on the actual conductivity of the ground and on the propagation distance from the lightning strike point and the sensors. The contribution of the nth sensor to the time-reversed field is given by

E!"#$ r,ω = E!∗ (ω)e!!!

!!!!! (16)

where E!∗ 𝜔 is the time-reversed version of the field captured by the sensor at location 𝑟!. Note that, in this model, only the time delay !!!!

! is taken into account in the back-

propagation. Inserting (15) into (16), we obtain at the stroke location:

 

  E!"#$ r!,ω = S!∗( r! − r! ,ω) ∙ !

!!!!!φ∗(ω) (17)

Unlike the case of a lossless ground, one can see from (17)

that the propagation filter influences both the amplitude and the phase of the field computed at the strike location. The field will, in general, not be in phase with the fields computed for other sensors.

Simulations show that, for a conductivity of 0.002 S/m, this first method offers the worst performance compared to the one presented in the next two sub-sections, with errors if the order of a few hundred meters. Model 2: Lossy Ground Back-Propagation Model

In this model, the propagation loss is taken into account in the calculation of the back-propagated fields as follows:

E!"#$ r,ω = E!∗ (ω)e!!!

!!!!! ∙ S!( r − r! ,ω)   (18)  

Inserting (15) into (18), we obtain, at the stroke location

E!"#$ r!,ω = S!

!( r! − r! ,ω) ∙ !

!!!!!φ∗(ω) (19)

Since the propagation filter is taken into account twice, one of them mathematically conjugated with respect to the other, the losses appear as a purely real amplitude factor and do not affect the phase of the wavefronts arriving at the strike point (although they do change the amplitude). Here, the amplitude filter S!

! modifies the shape of the time-domain signal.

From simulations with a ground conductivity of 0.002 S/m, the accuracy exhibit by this second method is of the order of a few hundred meters but somewhat better than that observed for Method 1.

Model 3: Inverted Losses Back-Propagation Model

The third model to be considered uses an equalization filter S! to simulate propagation over a fictitious ‘inverted-loss’ ground as follows:

E!"#$ r,ω = E!∗ (ω)e!!!

!!!!! ∙ !

!!∗( !!!! ,!)

(20)

Substituting (15) into (20), we obtain at the stroke location

E!"#$ r!,ω = !!!!!!

φ∗(ω) (21)

As can be seen from (21), the effect of losses is absent in (16) as the back-propagation model has equalized the forward propagation. This requires knowledge of the electrical and geometrical characteristics of the ground and, furthermore, an accurate model to account for the propagation effects.

A further drawback of this model is the fact that the equalization factor !

!! tends to infinity as ω goes to infinity.

This can be dealt with by introducing a low-pass filter H(𝜔) so that the expression H(!)

!!( !!!! ,!) decreases as ω increases to

infinity. To maintain the accuracy of the method, H(𝜔) must be a zero-phase filter so that the phase is not modified.

Still another potential drawback comes from the fact that the amplitude of the back-propagated signal increases with propagation distance. Under these conditions, the maximum amplitude of the total back-propagated field is no longer usable as a means to detect the stroke location.

It is possible to work around this difficulty by dividing by a suitable factor 𝐴! to maintain the amplitude constant, independent of the propagation distance.

The following value can be chosen for 𝐴! (in the time domain):

A! =!"#!(E!,!"#$%&%'(!))

!"#!(E!(!)) (22)

where max!(∙) is the maximum over time of the function and E!,!"#$%&%'(𝑡) is the field E!(𝑡) after filtering by H(!)

S!(!!"#,!).

The characteristic equation of this third model reads

E!"#$ r,ω = E!∗ ω e!!!

!!!!! ∙ !(!)

S! !!!! ,!

∗!!!

(23)

with H(𝜔) and 𝐴! chosen as described above.

Inserting (7) into (23), it can readily be seen that the effect of losses will disappear for frequencies inside the band of H(𝜔). Note that the additional term !

!! does not depend on the

frequency and, therefore, it does not affect the ability of the back-propagating model to detect the source.

Note finally that, if the losses are neglected, (S!(𝜔) ≡1,  H(𝜔) ≡ 1), all three models become identical.

Simulations for a ground conductivity of 0.002 S/m show that the accuracy of the Inverted Losses Back-Propagation Model is better than that of the other two, with errors of the order of e few tens of meters and, under ideal conditions, with errors limited only by the size of the back-propagation calculation mesh.

VI. SUMMARY, CONCLUSIONS AND FUTURE OUTLOOK A discussion was presented on the basis of time reversal

and the concept of time-reversal invariance in physics with emphasis on the invariance of Maxwell’s equations.

An overview was given of the application of the time-reversal technique in various fields of engineering and, in particular, its application to the problem of fault detection in distribution networks. A general overview of the most commonly used lightning location techniques was presented. The application of EMTR to locate lightning discharges was also discussed, showing first that the Time of Arrival technique is a subset of time reversal and then giving three strategies to deal with the non-invariance of wave propagation over a lossy ground.

Time reversal can potentially increase the accuracy of lightning location systems as it uses the complete waveform of the field to calculate the location of the strike point.

The implementation of time reversal for lightning location requires sensors capable of recording either the full electric or magnetic field waveforms or at least of a subset of waveform characteristics that can yield accurate localization results.

The following subjects need to be investigated to be able to design an optimum EMTR lightning location system:

- The necessary accuracy with which the electrical and topological characteristics of the propagation paths in the location domain.

- The required accuracy of the propagation models to be used in the back propagation.

- Efficient ways to record the waveforms of fields at sensors.

Work is in progress to deal with the above-mentioned subjects.

ACKNOWLEDGMENT The authors are grateful to Prof. Mario Paolone, Gaspard

Lugrin, Nicolas Mora and Reza Razzaghi for their valuable suggestions and comments.

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