On Practical Implementation of Electromagnetic Time Reversal to Locate Lightning H. Karami
F. Rachidi
Dept. Electrical Engineering Bu-Ali Sina University Hamedan, Iran
[email protected]
EMC Laboratory Swiss Federal Inst. Tech. (EPFL) Lausanne, Switzerland
[email protected]
Abstract— Electromagnetic Time Reversal (EMTR) has been recently proposed as a lightning location method. The method is based on the recording of the full electric or magnetic field waveform at multiple stations, time-reversing the recorded fields, and back-propagating them using numerical simulations into the location domain. The back-propagated fields will add up in phase at the lightning strike location. The implementation of an EMTRbased lightning location system requires that a certain number of practical difficulties be overcome, including the fact that most of the deployed lightning location networks do not record the complete electric or magnetic field waveforms. We propose a solution to this problem based on the use of matrix pencil method (MPM) to minimize the computer memory and transmission bandwidth requirements to store the measured electromagnetic field waveforms. The performance of the MPM method is evaluated using measured waveforms of electric and magnetic fields from distant natural lightning and it is shown that using only 46 poles and residues, it is possible to reproduce very accurately the measured waveforms. Keywords— Lightning location systems (LLS); Electromagnetic time reversal (EMTR); Matrix pencil method (MPM).
I.
stations, time-reversing the recorded fields, and using simulations to back-propagate them into the location domain. In this paper, we present the principle of the time reversal technique, we discuss the practical difficulties that need to be overcome to implement a real system and we propose solutions for them. II. ELECTROMAGNETIC TIME REVERSAL AND ITS APPLICATION TO LIGHTNING DETECTION AND LOCATION As discussed in [Rachidi and Rubinstein, 2013], it can be shown that Maxwell’s equations in vaccum are invariant under the time reversal transformation. This property has been used to develop a technique based on EMTR to locate lightning discharges [Lugrin et al., 2014; Mora et al., 2012]. The technique can be summarized as follows: -
Several sensors record the wideband electric or magnetic fields from a lightning return stroke.
-
The recorded waveforms are time reversed and transmitted back into the location domain by numerical simulation.
-
It can be shown that the back-propagated fields will add up in phase at the lightning strike location.
INTRODUCTION
Time reversal or T-symmetry describes the symmetry of physical laws under a time reversal transformation: t → −t . Time reversal has been an important element in the development of theoretical physics [Sachs, 1987]. However, it is in the past two decades or so that it has attracted considerable attention following the work of Fink and coworkers (e.g., [Fink, 1992]) in various fields of electrical engineering and acoustics (see, e.g., a non-exhaustive list in [Rachidi and Rubinstein, 2013]). Electromagnetic Time Reversal (EMTR) has been recently proposed as a lightning location method ([Lugrin et al., 2014; Mora et al., 2012]). The method is based on the recording of the full electric or magnetic field waveform at multiple
M. Rubinstein IICT Univ. Appl. Sci. W. Switzerland Yverdon, Switzerland
[email protected]
Lugrin et al. [Lugrin et al., 2014] mathematically demonstrated that the Difference in Time of Arrival (DToA) lightning location technique is a particular case of electromagnetic time reversal in the case of a perfectly conducting ground. III.
ON THE PRACTICAL IMPLEMENTATION OF AN EMTRBASED LIGHTNING LOCATION TECHNIQUE
Since EMTR takes advantage of the whole waveform of the measured fields (including amplitude and ToA), it can turn out
to be very promising in terms of achievable location accuracy and detection efficiency [Lugrin et al., 2014]. However, the implementation of an EMTR-based lightning location system requires that a certain number of practical difficulties be overcome: (1) The method requires a computer model to simulate the back propagation of the recorded fields in the location domain. Since the performance of the method depends on the accuracy of the simulation, the model should be accurate enough. At the same time, the simulations should be carried out in a relatively short time to be able to provide in a given time window an estimate of the location of the discharge. (2) The electromagnetic propagation involving a dissipative medium is not time reversal invariant unless an inverted-loss medium is considered for the reverse times. The requirement of back-propagation along an inverted-loss medium could represent a numerical issue, and necessitates the knowledge of the ground electrical parameters. (3) Many of the deployed lightning location networks, such as EUCLID [Diendorfer et al., 1998] and NLDN [Cummins and Murphy, 2009], do not record the complete electric or magnetic field waveforms. Instead, they extract and store only a number of signal parameters such as the triggering time, the rise-time, and the peak value. Note that this is not the case for other networks (e.g., GLD360 [Lojou et al., 2011]) or LINET [Schmidt et al., 2005] where the waveforms are generally stored for each flash.
(1) where M is the number of system poles, and the complexvalued parameters si and Ri denote, respectively, the poles and residues of the system. The discretized waveform y(t) with a sampling period Ts can be written as (2)
The aim is to find the best estimates for M, si, and Ri. Using the samples given in (2), the following two matrices [Y1] and [Y2] can be built
(3)
and
Regarding point (1), the availability of efficient computer codes for electromagnetic propagation and the increasing computing capabilities offered by modern computers should make it possible to satisfy this requirement. Concerning point (2), as shown in [Lugrin et al., 2014], acceptable accuracies can be achieved even assuming a perfectly conducting ground in the back-propagation model. Finally, concerning point (3), the main reason for which not all the measured field waveforms are stored for each flash is simply due to computer memory requirements. The lack of full waveforms can be dealt with to some extent using extrapolation techniques, as shown by Lugrin et al. [Lugrin et al., 2014]. However, any extrapolation technique based on a limited number of waveforms parameters will not be able to recover the full waveform in a satisfactory manner, and it will thus affect the resulting accuracy of the technique.
(4)
where L is the pencil parameter [Sarkar and Pereira, 1995] and yk = y(kTs) for k = 0,1,…,N-1. [Y1] and [Y2] can be written as
[Y1 ] = [Z1 ][R][Z0 ][Z 2 ] [Y2 ] = [Z1][R][Z 2 ]
(5)
where
A solution to this problem is presented in Section IV. IV.
USE OF MATRIX PENCIL METHOD
In this paper, the matrix pencil method (MPM) [Sarkar and Pereira, 1995] is used to minimize the computer memory requirements to store the measured electromagnetic field waveforms. The theory of MPM is summarized in what follows. A. MPM Theory A system response in time-domain y(t) -such as a measured electromagnetic fields waveform- can be approximated by M exponential terms
(6)
where zi is defined in (2). The matrix pencil [Sarkar and Pereira, 1995] can be considered as
(
)
[Y1 ] − λ[Y2 ] = [Z1][R] [Z0 ] − λ [I ] [Z 2 ]
(7)
where [I] is the identity matrix. If M ≤ L ≤ N-M, then the rank of the matrix pencil ([Y1]-λ[Y2]) is M. However, if λ = zi, i =0, 2,…, M-1, the rank is reduced to M-1 since the ith row and column of [Z0]-λ[I] become zero. To obtain M, the singular value decomposition should be applied to the matrix pencil in a way that the singular values are arranged in a descending form. If σm is the mth element of the singular values, M is defined so that [Sarkar and Pereira, 1995]
σM = 10 − p σ max
(8)
B. MPM Application In order to evaluate the performance of MPM, we used experimental waveforms from distant natural lightning recorded during the Summer of 2006 [Mosaddeghi et al., 2009]. Fig. 1 shows a measured vertical electric field (solid/black line) together with its fitted curve (dash/red line) using MPM (p = 2 and M = 46). It can be seen from the figure that MPM can accurately approximate the measured data using 46 poles and residues. To quantify the difference between the two curves, we will use two quantities. The first one is the percent error in the energies of the two waveforms, which, for continuous waveforms, can be written as
∫ %error =
20 µ s
2 f meas (t )dt − ∫
0
∫
In (7), λ represents the generalized eigenvalues of the matrix pencil. Since [Z0] is a diagonal matrix, zi, which is equal to λ is found. Hence
[Y1 ][ri ] = [zi ][Y2 ][ri ]
2 f fitted (t )dt
0
20 µ s
2 f meas (t )dt
0
where σmax is the maximum singular value and p is the number of significant decimal digits specified for data accuracy. Singular values above M are set to zero and, hence, the number of singular values is equal to the selected value of M. Furthermore, p is also used to control the performance of higher order approximations.
20 µ s
×100 (12)
The percent energy error gives an indication of the similarity between the overall amplitudes of the waveforms. However, it does not measure how close the shapes of the waveforms are. For that reason, we will use the angle between the waveforms based on the inner product of functions to measure the similarity between the shapes of the two waveforms. The cosine of the angle α between two continuous functions is given by (see, for instance, [Williams, 2009])
(9)
∫
cos(α ) =
20 µ s
0
∫
where [ri] are the generalized eigenvectors corresponding to zi, or in the equivalent form
20 µ s
0
f
f meas (t ) f fitted (t )dt
2 meas
(t )dt
∫
20 µ s
0
f
2 fitted
(13)
(t )dt
(10)
The closer the angle is to zero degrees, the better the waveform match.
where the superscript “ † ” shows the Moore-Penrose pseudoinverse operation. Thus, zi=λ can be obtained using the
A measured magnetic field and its approximation using MPM (p = 2, M= 46) are depicted in Fig. 2, where it is obvious that, as in the case of the electric field, the measured and fitted curves are in excellent agreement.
{[Y ] [Y ] − [z ][I ]}[r ] = 0 †
2
1
i
i
†
eigenvalues of [Y2 ] [Y1 ] . Hence, the complex-valued si is directly calculated from (2). Having obtained zi and M, the complex-valued Ri can be easily calculated by solving a least square problem as
200
Measured Fitted (p=2, M=46)
150
(11)
Ez (V/m)
100 50 0 -50 -100 0
The matrix equation (11) is solved using QR decomposition. In MPM, the parameter L is set between N/3 to N/2. More details on MPM can be found in [Sarkar and Pereira, 1995].
2
4
6
8
10 12 time (µs)
14
16
18
20
Fig. 1. Vertical electric field from distant lightning. Solid/black line: measured waveform, dashed/red line: fitted with MPM.
0.1
Measured Fitted (p=2, M=46)
H (A/m)
0.05
0
-0.05
The performance of MPM method was evaluated using measured waveforms of electric and magnetic fields from distant natural lightning and it was shown that using only 46 poles and residues, it is possible to reproduce very accurately the measured waveforms with percent errors in energies lower than 0.12 % and inner-product based angles between the waveforms of 2o or less.
-0.1
0
2
4
6
8
10 12 time (µs)
14
16
18
20
Fig. 2. Magnetic field from distant lightning. Solid/black line: measured waveform, dashed/red line: fitted with MPM.
The percent error in energies for corresponding to each of the waveforms in Fig. 1 and Fig. 2 are 0.12% and 0.10%, respectively. The angle between the measured and the fitted waveforms is 2o for the electric field in Fig. 1 and 1.9o for the magnetic field in Fig. 2, indicating and excellent fit. The memory requirements to store the lightning remote electromagnetic field can be easily reduced using MPM. In both examples, only 46 complex-valued poles and residues are needed to store to recover the waveforms with very high accuracy. V.
CONCLUSIONS
Electromagnetic Time Reversal (EMTR) has been recently proposed as a lightning location method. The method is based on the recording of the full electric or magnetic field waveform at multiple stations, time-reversing the recorded field, and back-propagating them using numerical simulations into the location domain. The back-propagated fields will add up in phase at the lightning strike location. Since EMTR takes advantage of the whole waveform of the measured fields, it can turn out to be very promising in terms of achievable location accuracy and detection efficiency. However, the implementation of an EMTR-based lightning location system requires that a certain number of practical difficulties need to be overcome, including the fact that most of the deployed lightning location network do not record the complete electric or magnetic field waveforms, because of prohibitive computer memory requirement. In this paper, we proposed a solution to this problem which consists of the use of matrix pencil method (MPM) to minimize the computer memory requirements to store the measured electromagnetic field waveforms.
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