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Time Reversal of Electromagnetic Fields and its
Application to Lightning Location
Farhad Rachidi
Marcos Rubinsein
EMC Laboratory
Swiss Federal Institute of Technology (EPFL)
Lausanne, Switzerland
[email protected]
IICT
UAS Western Switzerland, HEIG-VD
Yverdon-les-Bains, Switzerland
[email protected]
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)
(
)
(
)
 
 
 ∂H ( r,t)
∇ × E( r,t) = − µ ( r )
(1c)
  ∂t
 
 ∂ E( r ,t)  
∇ × H ( r ,t) = ε ( r )
+ J ( r ,t)
(1d)
∂t
 
 
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 (1a1d), we obtain
  

∇ ⋅ ε ( r ) E( r ,−t) = ρ ( r ,−t)
(2a)
(
)
 

∇ ⋅ µ ( r ) − H ( r,−t) = 0
(
(
))
(2b)
 
 
 ∂(− H ( r,−t))
∇ × E( r,−t) = − µ ( r )
(2c)
∂(−t)
 
 
 
 ∂ E( r ,−t)
∇ × (− H ( r ,−t)) = ε ( r )
+ (− J ( r ,−t))
(2d)
∂(−t)
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, 1012]) 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)
∂i(x,t)
+ L'
=0
(3)
∂x
∂t
∂i(x,t)
∂v(x,t)
+C'
=0
(4)
∂x
∂t
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)
∂(−i(x,−t))
+ L'
=0
∂x
∂(−t)
(5)
∂(−i(x,−t))
∂v(x,−t)
+C'
=0
∂x
∂(−t)
(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 faultoriginated 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 timereversed 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 .
 
component, the electric field En (r, t) , measured by the sensor

at location rn is equal to:
 
⎛
r −r ⎞
ϕ ⎜t − n s ⎟
 
c ⎠
⎝
E(rn , t) =
 
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 timereversed by replacing t by T - t, where T is the duration of the
signal. The result is given in Equation (8).
 
r −r ⎞
⎛
ϕ ⎜T − t + n s ⎟
 
c ⎠
⎝
En (r,T − t) =
 
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 timereversed 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 timereversed 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
 
 
⎛
rn − rs
r − rn ⎞
ϕ
T
−
t
+
−
⎜⎝
N
 
c
c ⎟⎠
ETR1 (r, t) = ∑
(9)
 
rn − rs
n=1


At the strike point, r = rs and (9) equation reduces to
N
 

1
ETR1 (rs , t) = ϕ (T − t ) ∑  
r
n=1 n − rs
(10)
The contributions of the different sensors add in phase at
the source point, making the back-propagated field maximum
at that point.


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
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 backpropagated 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
r − rn
−
=K
c
c
(11)
Writing (11) for two different sensors i and j, and
rearranging terms, we can write
 
 
ri − rs
r − ri
+
=T −K
c
c
 
 
rj − rs
r − rj
+
=T −K
c
c
(12)
(13)
Subtracting (13) from (12) and moving the terms

dependent on r to the left hand side, we get
 
 
 
 
r − rj
rj − rs
r − ri
r −r
−
=
− i s
c
c
c
c
(14)
Model 1: Perfect Ground Back-Propagation Model
In this model, one disregards the losses in the backpropagation. 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 timereversed field is given by
E!"#$ r, ω = E∗! (ω)e!!!
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:
!!!
!
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) ∙
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 backpropagation 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.
!
!! !!!
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
!
! !!
!!! ! !
!
(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
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.
!!!!
!
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)
!∗! ( !!!! ,!)
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.
Substituting (15) into (20), we obtain at the stroke location
VI.
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!,!"#$%&%' (!))
(22)
!"#! (E! (!))
where max! (∙) is the maximum over time of the function and
H(!)
E!,!"#$%&%' (𝑡) is the field E! (𝑡) after filtering by
.
S! (!!"# ,!)
!!!!
!
∙
!(!)
S! !!!! ,!
∗
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 timereversal 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.
The characteristic equation of this third model reads
E!"#$ r, ω = E∗! ω e!!!
SUMMARY, CONCLUSIONS AND FUTURE OUTLOOK
!
!!
(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
ACKNOWLEDGMENT
The authors are grateful to Prof. Mario Paolone, Gaspard
Lugrin, Nicolas Mora and Reza Razzaghi for their valuable
suggestions and comments.
!!
frequency and, therefore, it does not affect the ability of the
back-propagating model to detect the source.
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