The shape radio_signals_wavefront_encountered_in_the_context_of_the_uhecr_radiodetection_experiments

Preprint typeset in JINST style - HYPER VERSION
The shape of the radio signals wavefront
encountered in the context of the ultra high energy
cosmic ray radio-detection experiments
Ahmed REBAIa∗, Tarek SALHIb
a SUBATECH IN2P3-CNRS/Université de Nantes/École des Mines de Nantes,
Nantes, France
b École des Mines de Nantes, Nantes, France,
Now at General Electric Oil&Gas, Algeria
E-mail: ahmed.rebai2@gmail.com
ABSTRACT: Ultra high energy cosmic rays are the most extreme energetic subatomic particles
in nature. Coming from the outer space, these particles initiate extensive air showers (EAS) in
the Earth’s atmosphere. The generated EAS produce elusive radio-transients in the MHz frequency
band measured by sensitive antenna arrays and radio telescopes. Theoretical developments indicate
that the EAS radio wavefront shape depends on the shower longitudinal development, it is waited
that the wavefront curved shape provides information to answer many fundamental questions about
UHECR nature and origins. In the ﬁrst part of this paper, we report on an investigation in the
wavefront shape, based on an already published sample of events collected between November
2006 and January 2010 at the CODALEMA II experiment located in the radioastronomy facility at
Nançay in France. We ﬁnd that measurements of individual air showers have been conclusive for
a non-planar shape which could be hyperbolical (further analysis are needed). By cons and in the
second part of this paper, a spherical shape of the wavefront for the anthropic radio-sources has been
proposed. Many studies have shown the strong dependence of the solution of the radio-transient
sources localization problem (the radio wavefront time of arrival on antennas TOA), such solutions
are purely numerical artifacts. Based on a detailed analysis of some published results of radio-
detection experiments around the world like : CODALEMA III in France, AERA in Argentina,
TREND in China and LUNASKA in Australia, we demonstrate the ill-posed character of this
problem in the sense of Hadamard. To support the mathematical studies, a comparison between the
experimental results and the simulations have been made.
KEYWORDS: UHECR; Radio emission; radio-detection; wavefront shape; antennas;
CODALEMA; non-convex analysis; optimization; ill-posed problem.
∗Ahmed REBAI, ahmed.rebai2@gmail.com

Contents
1. Introduction 2
2. Experimental evidences of the radio wavefront curvature 4
2.1 The CODALEMA facility 4
2.2 Case of radio transients initiated by air showers detected in a slave-trigger mode 5
2.3 Case of anthropogenic radio emission in a self-trigger mode 12
2.4 Discussion 15
3. A new hyperbolic model to ﬁt the EAS radio wavefront curvature 16
4. A spherical model to ﬁt the anthropic radio wavefront signals 23
4.1 Compilation of experimental results from self-triggered radiodetection experiments 23
4.2 Simulation studies of the localization of emission source with minimization algo-
rithms 27
5. Ill-posed formulation of the emission source localization for the spherical emission 34
5.1 Convexity property of the non-linear χ2 function 35
5.2 Critical points 35
5.3 The antennas array convex hull concept 36
6. Conclusion 37
7. Appendix 1 38
7.1 Symbolic calculus 38
7.2 Explicit calculus using the Taylor expansion 40
7.3 Study of the convexity property 41
8. Appendix 2 42
8.1 Degeneration line for a linear antenna array 42
9. Appendix 3 44
9.1 Convex hull for a linear antenna array 44
10. Appendix 4: Systematics due to atmospheric earth 46
– 1 –

1. Introduction
Cosmic rays are defined as high-energy particles incident on the Earth from outer space. Although
their discovery dates back to a century ago, several fundamental questions remain open, especially
about their exact origin and their nature in the Greisen-Zatsepin-Kuzmin cutoff (GZK) region of
their energy spectrum. Then, the study of the ultra-high energy cosmic rays (UHECRs) is indeed
still a very open field, where it is essential to identify the composition on an event-by-event ba-
sis and to understand the hadronic interactions characteristics beyond the Large Hadron Collider
energy. The UHECRs chemical composition is determined from the measurement of the shower
maximum depth Xmax. This latter can be inferred with the fluoresence technique for each single
event, but generally the mass composition is statistically estimated by comparing the Xmax mean
value (< Xmax >) and its dispersion (RMS) σ(Xmax) to the extensive air showers (EAS) numerical
simulations. This statistical treatement is due to the fluctuations on the interactions models used
for the shower development phenomenology (especially for showers initiated by protons) and the
uncertainties provided by the Xmax experimental measurements. Till now, the determination of the
UHECRs composition has mainly used two experimental indirect methods of detection which are
the ground-based detectors array and the fluorescence light telescopes. The first technique has a
good duty cycle close to 100% but it has a strong dependence on hadronic models extrapolated
from the low energy regions available in terrestrial accelerators and it needs to deploy very large
areas (> 1000 km2) to collect a large number of air showers. The second technique, which images
the UV fluorescence of atmospheric nitrogen excited by the secondary charged particles, is model
independent and allows a large volume of detection but with a limited duty cycle around 10% (only
dark moonless nights), which drastically limits the number of collected EAS events.
Seeking to determine the nature of UHECRs, the Pierre Auger Observatory (PAO) located
in the southern hemisphere and the Telescope Array-High Resolution Fly’s Eye (TA-HiRes) ex-
periment which is the largest detector in the northern hemisphere, are collecting data since many
years. The newly reported experimental results have shown that in the case where TA-HiRes see a
lightening of the composition in function of the energy (above 1.6 EeV) compatible with a proton
primaries [1], PAO see a less fluctuating Xmax distribution than predictions from air shower simu-
lations for a pure proton composition above 1 EeV then a heavy composition compatible with iron
primaries [2]. It is interesting to note that the primary particle nature is still actually a hot scientific
topic since there is a disagreement between the two great UHECR observatories in the world. In
fact, an increase by a factor 3 in statistics is required in the northern hemisphere for a more accurate
measurements [3].
Faced with these difficulties in interpretation, the solution could be the use of a model in-
dependent technique which must collect a sufficient statistics within a reasonable time (a higher
duty-cycle technique). Several solutions are proposed and appear to be promising. May be men-
tioned the JEM-EUSO futur experiment which will detect the fluorescence and Cherenkov radia-
tion emitted into the atmosphere by the EAS from the space [4]. But more cheaper and proven,
the radiodetection of UHECRs has arised again, since the last decade, as a complementary detec-
tion method for these established techniques mentioned above. New promising approaches could
emerge from the exploitation of the radio-detection method which uses antennas to detect the ra-
dio signal initiated during the air shower development. Many experiments like CODALEMA [5]
– 2 –

in France, LOPES [6] in Germany, LOFAR [7] across Europe and AERA [8] in Argentina have
shown the feasibility and the potential of the method to estimate the EAS different parameters,
as the arrival direction, the shower core location at ground, the electric field lateral distribution
function (LDF) and the primary cosmic ray energy [9, 10, 11, 12, 13]. Neverthless, the temporal
radio wavefront characteristics remain still poorly determined although its knowledge is important
in estimating the arrival direction of the shower and the primary nature. Assuming a planar wave-
front, the arrival direction is given by the zenith angle θ and the azimuthal angle φ. But since the
emission is originated from a source distant only a few kilometers from the antenna array, the wave
front is curved. This concept is supported by previous simulations [14] and by recent experimental
results from CODALEMA [17], LOFAR [15] and LOPES [16]. In this paper, we confirm the re-
sults published by LOFAR on the hyperbolic shape of the radio wavefront. Furthermore, LOFAR
and LOPES experiments have demonstrated that the wavefront can be used to study the UHECRs
composition. Indeed, The importance of this information resides in its sensitivity to the nature of
the primary particle, especially because of the existence of a curvate radio wavefront that could
provide the location of the emission source main point, and possibly an estimation of Xmax, in an
event by event basis. The distribution of arrival times being defined by the radio signal maximum
amplitude is linked to a limited portion of the shower longitudinal development (and so notably at
the shower maximum region).
On the other hand, the migration from the present small scale radio-experiments arrays to large
scale experiments spread over surfaces of several tens of 1000km2 using self-triggered antennas, is
challenging. This technique is subjected to delicate limitations in regard to UHECRs recognition,
due to anthropic radio-sources induced by human activities (high voltage power lines, electric
transformers, cars, trains and planes) or by stormy weather conditions (lightning). The commonly
used technique relies on the minimization of an objective function which depends on the assumed
shape of the wavefront, using the arrival times and locations of the antennas.
The results and analysis presented in this paper, are based on data from CODALEMA II [10,
20] and CODALEMA III [22] experiments. Moreover, we will discuss some already published
results from the AERA [29] and the TREND [18] experiments. This paper is divided into three
parts: Firstly, we will demonstrate the existence of the radio wavefront curvature in two cases:
the case of radio emission of air showers and the case of anthropogenic radio sources (generally
static or moving with low velocity compared to c). In the second part, we will expose a hyperbolic
model to fit the curvature in the first case. We will show that this model allows to find the radius of
curvature and the air shower core at the ground. Consequently, the measurement can improve the
reconstruction of the electric field lateral distribution and then the primary energy estimation. In
the third part, we will study the second case sources encountered in self-trigger radio-experiments.
We will highlight that the minimization of a spherical model give an ill-posed problem and we will
show that it originates from strong dependencies of the minimization algorithms convergence with
initial conditions, from the existence of solutions degenerations (half lines) which can trap most of
the deterministic algorithms, and from the existence of bias in the reconstructed positions.
– 3 –

2. Experimental evidences of the radio wavefront curvature
2.1 The CODALEMA facility
Since 2002, CODALEMA hosted on the radio observatory site at Nançay in France with geo-
graphical coordinates (47.3◦N, 2.1◦E and 137 m above sea level), aims to study the potential of
the radiodetection technique in the 1016 eV energy range (detection threshold) to 1018 eV (upper
limit imposed by the area surface). Taking advantage of ultrafast electronic devices and a quiet ra-
dio environment from anthropic transmitters in the detection bandwidth, the CODALEMA facility
consists of three major experiments (see fig. 1).:
• (External triggering technique): The CODALEMA II experiment spreads over a surface of
about 1
4 km2 and it is made of 3 mains arrays of detectors [10]. The first array is built with 24
short active dipole antennas distributed on a cross geometry with dimensions 400 m by 600 m.
The dipole antenna is made by two radiator arms each 60 cm long at a height of 1.2 m. The
antenna design was optimized to reach an almost isotropic pattern. A low noise amplifier
(LNA) is used to amplify the electric signal. It is conceived to be sensitive to the radio
galactic background and is linear over a wide frequency band from 30 MHz up to 230 MHz.
The second array is a ground-based particle detector array formed by 17 plastic scintillators
placed on a square of 340 m side. It measures the primary particle energy and provides the
trigger signal to the other detector arrays. The third apparutus is the Nançay decametric
array formed by 144 conic logarithmic old antennas. The entire acquisition system (DAQ)
is triggered by the passage of secondary particles in coincidence through each of the five
central scintillators with a trigger detection threshold energy equal to 5.1015 eV. The radio
waves forms in each antenna is recorded in a 0-250 MHz frequency band during a 2.5 µs
time window with a 1GS/s sampling rate. The figure 2 shows an example of radio filtered
transients in the band [23 − 83] MHz for two selected events with different energies. Radio
events that are detected by dipole antenna array in coincidence with atmospheric shower
events are identified during offline analysis. After this analysis phase, a data set containing
the shower parameters reconstructed using the information provided by the particle detectors
(arrival times distribution, arrival directions, shower core on the ground and energy) and a
set of observables for each radio antennas (arrival times distribution, radio signal amplitudes
distribution) and the radio observables of the shower reconstructed by the use of radio data
alone (EAS direction, radio shower core on the ground, energy) are obtained event by event.
These observations are used to study the curvature of the radio wave front that could be one
of the discriminating parameters of the primary nature (Xmax estimation).
• (Self-triggering technique): The CODALEMA III experiment [22] is an autonomous antenna
array with a larger surface of about 0.5 km2 which can give more statistics in 1016 −1018 eV
energy range. The array is composed of 34 autonomous stations each one equipped with dual
polarization butterfly antenna. CODALEMA III was born from the idea of using radiodetec-
tion method in an extensive manner over large areas above 1000 km2 then, it is necessary
that radiodetection becomes autonomous and apply this throughout the entire technical as-
pects (triggering, energy consumption, signal processing and transmission). To achieve this,
– 4 –

Figure 1. Set up of CODALEMA facility showing the layout of the 17 particle detectors array (blue circles),
the CODALEMA II array which contains short dipole antennas (yellow circles for East-West polarization
and orange circles for North-South polarization) and the CODALEMA III array formed by a 34 self-triggered
antennas with 2 polarisations EW and NS (white squares).
the CODALEMA III autonomous stations differ from the CODALEMA II simple anten-
nas. In fact, compared to the previous dipole antenna used in CODALEMA II, the new
butterﬂy antenna has been conceived to be more sensitive (more height above average ter-
rain than dipole antenna) at low frequencies for detecting air showers at large distance from
its core on the ground. Furthermore, the CODALEMA III triggering strategy is different
from CODALEMA II. This shift is part of the ongoing experimental efforts towards the full
characterization of UHECR properties only throug radio transients. Indeed, instead of the
external trigger, CODALEMA III is using a simple threshold voltage level in a 45−55 MHz
band. Unfortunately, we’ll see that under this transition, new problems has emerged. These
problems are related to the detection, recognition, localization and the suppression of the
noisy background sources induced by human activities (such as high power lines, electric
transformers, cars, trains and planes). In this regard, one of our major problems is trying to
solve the localization problem which belongs to a class of more general problems usually
termed as inverse problem.
• The EXTASIS project is a dedicated experiment working below 20 MHz and aims to observe
the radio signal emitted by the extinction of air showers at the ground level [23]. The pre-
dicted signal is due to several mechanisms like the charged particles transition radiation and
the sudden death mechanism. This signal could be particularly promising for estimating the
nature of the primary.
2.2 Case of radio transients initiated by air showers detected in a slave-trigger mode
The concept of slave-trigger mode in radiodetection experiments is quite oldish since the ﬁrst ob-
– 5 –

Figure 2. Filtred radio waveforms detected by CODALEMA II, in the [23 − 83] MHz frequency band and
associated with two extensive air showers with different energies. In the top and middle figures, example of
two selected events with primary particle energy respectively of Ep = 8∗1016eV and Ep = 1.6∗1018eV. In
this case, the antenna has detected the pulse with a good signal-to-noise ratio. At the bottom, the signal is
merged into the background noise which is mainly due to the galactic radio emission.
servation of radio transients from EAS made by Jelley and his collaborators in the 60s of the last
century [26], has used the same technique with a Geiger counters array. It allows to not directly
fire on the air shower but instead to product a trigger signal from another particle detector array.
The slave mode has several advantages such as the purity and the quality of the detected transients
which are correlated with real air showers. Indeed, the majority of noisy signals are avoided, like
the atmospheric electricity discharges (due to local weather conditions or ionospheric reflections)
and Man-made RF sources (like automobile ignition noise, police communications and radio sta-
tions). The figure 2 shows examples of radio pulses detected by CODALEMA II after an offline
– 6 –

digital filtering in the [23 − 83] MHz frequency band. We note that the net improvement of the
signal-to-noise ratio with the primary particle energy is a well-known behaviour due to the coher-
ence of the radio emission which implies that the electric field scales linearly as the shower energy
increases [12].
On the other hand, according to signal theory, these coherent radio pulses can be characterized
by three main physical parameters which are: the time delay, the signal maximum amplitude and
the phase. Thus in this article, we will use the time delay distribution to reconstruct the spatial
position of the emission sources. At this point, the most important issue to be considered is the
relevance of the curved radio wavefront assumption. In many published papers, the reconstruction
procedure is based on the assumption of a far-field sources. Under such hypothesis, the radio
shower front is assumed to be a plane perpendicular to the shower axis so only the primary particle
direction can be determined directly by triangulation using the time of flight between different
antennas. Then if we take the first tagged antenna, in each event, as a reference for arrival time
and we study the theoretical time delay ∆ttheo as a function of the experimental time delay ∆texp,
one must observe a straight alignment between the two quantities. In the next paragraphs, we will
describe the method of calculation of these temporal variables:
• ∆ttheo: We assume that the radio wavefront is a plan perpendicular to shower axis which
starts propagation in phase with the primary cosmic ray and moves parallel to this axis with
the speed of light in the vacuum c. The plan equation can be written as:
u.x+v.y+w.z+γ = 0
with (u,v,w) = (cos(φ).sin(θ),sin(φ).sin(θ),cos(θ)) are the coordinates of the unit vector
n normal to the plane. Now we take the first tagged antenna (fta) as reference to calculate the
constant γ. the equation becomes:
u.x+v.y+w.z−(u.xfta +v.yfta +w.zfta) = 0
The distance between this plane and the other tagged antennas located at positions (xi,yi,zi)
with i = 1,...,N is given by:
di =
|u.xi +v.yi +w.zi −(u.xfta +v.yfta +w.zfta)|
√
u2 +v2 +w2
The plan arrival instant on each antenna is:
tpred
i = tfta +
di
c
which allows to reproduce the plane wave propagation from the first tagged antenna until
other antennas. Finally theoretical delay is written:
∆ttheo
i = tpred
i −tfta =
di
c
• ∆texp : It is an experimental quantity based on the detected transients which are subjected to
a digital fitler in [23−83] MHz frequency band (see the section 2.1). This filtering procedure
– 7 –

produces signals that oscillate with periods varying between 10 ns and 40 ns. Thus, we
consider empirically that this procedure gives an average statistical error on time of about
10 ns. In fact, if a secondary extremum (maximum or minimum ) is marked instead of the true
extremum, a 10 ns error can be occured. Note that, in CODALEMA experiment we have no
major systematic effects: the antennas positions and the cables delays are measured precisely
and the resulted systematic shifts are corrected during the off-line analysis, according to the
antenna, they are varying between a few ns and 20 ns. Finally, The ﬁltred pulses maximum
in each antenna enables the determination of the experimental arrival time “the real time”
noted tmax
i (see ﬁgure 2). Experimental delay is then written:
∆texp
i = tmax
i −tmax
fta
– 8 –

0 2 4 6
x 10
−7
0
1
2
3
4
5
6
7
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
Run 936 Event 1434
Multiplicity = 20 Antennas
θ = 35.60 ◦
φ = 326.25 ◦
Ep = 1017.75
eV
20
i |∆ttheo
i − ∆texp
i | = 286.4ns
0 1 2 3 4 5
x 10
−7
0
1
2
3
4
5
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
Run 835 Event 411
θ = 35.34◦
φ = 312.8◦
Ep = 1016.81
eV
10
i |∆ttheo
i − ∆texp
i | = 203.5ns
0 2 4 6 8
x 10
−7
0
1
2
3
4
5
6
7
8
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
Run 921 Event 1282
θ = 35.78◦
φ = 30.08◦
Ep = 1016.90
eV
19
i |∆ttheo
i − ∆texp
i | = 233.4ns
0 1 2 3 4 5 6
x 10
−7
0
1
2
3
4
5
6
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
Run 902 Event 1379
θ = 29.44◦
φ = 325.85◦
Ep = 1017.76
eV
18
i |∆ttheo
i − ∆texp
i | = 230.3ns
0 0.5 1 1.5 2 2.5 3
x 10
−7
0
0.5
1
1.5
2
2.5
3
x 10
−7
∆ t
i
theo
(s)
∆t
i
exp
(s)
Run 999 Event 1172
θ = 12.78◦
φ = 164.06◦
Ep = 1017.34
eV
17
i |∆ttheo
i − ∆texp
i | = 411ns
0 0.5 1 1.5 2
x 10
−7
0
0.5
1
1.5
2
x 10
−7
∆ t
i
theo
(s)
∆t
i
exp
(s)
θ = 10.13◦
φ = 97.98◦
Ep = 1018.21
eV
16
i |∆ttheo
i − ∆texp
i | = 2.3 ∗ 10−7
s
Run 998 Event 635
Figure 3. The black line presents the plane wave best line ﬁt, we see that despite the error bars of 10 ns on
both axes. Many points deviate systematically from the line which shows that the wavefront is not a plan.
The data in the top ﬁgure are from the CODALEMA II experiment [20]
– 9 –

0 2 4 6
x 10
−7
0
1
2
3
4
5
6
7
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
Simulation based on Event 1434 Run 936
R = 10 km
R = 5 km
R = 3 km
0 0.5 1 1.5 2
x 10
−7
0
0.5
1
1.5
2
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
R = 10 km
R = 5 km
R = 3 km
0 1 2 3 4 5 6
x 10
−7
0
1
2
3
4
5
6
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
R = 10 km
R = 5 km
R = 3 km
0 2 4 6 8
x 10
−7
0
1
2
3
4
5
6
7
8
x 10
−7
∆ t
i
theo
(s)
∆t
i
exp
(s) Simulation based on Event 1282 Run 921
R = 10 km
R = 5 km
R = 3 km
0 1 2 3
x 10
−7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
−7
∆ t
i
theo
(s)
∆ti
exp
(s)
R = 10 km
R = 5 km
R = 3 km
0 1 2 3 4 5 6
x 10
−7
0
1
2
3
4
5
6
x 10
−7
∆ t
i
theo
(s)
∆t
i
exp
(s)
R = 10 km
R = 5 km
R = 3 km
Figure 4. Simulations based on real events geometry. The emission center is distant of 3, 5 and 10 km from
the ground
– 10 –

Interpretation of the time delay: In the following we will discuss some interpretations about
the shape of the radio shower. Since the late 1950s [24], the temporal structure of showers and its
front curvature have been experimentaly investigated and since then many studies have concerned
the particles (include hadrons, electrons and muons) front, the fluorescence and Cherenkov pho-
tons front. Besides, it is well known that the propagation of a wavefront from a localized source
produces a curved front at large distances (see chapter 9 of [25]). As described next, this fact will
be tested on radio emission. So if the EAS radio wavefront had a plane shape then, when we study
∆texp in function ∆ttheo we should observe an alignment with the plane wave best line fit. But,
the figure 3 shows that data points deviate from this line despite the 10 ns experimental timing
uncertainty.
XXX Dans le figure XXX, nous avons presente 6 evenements issues des donnees CODALEMA.
Ces evenements ont ete choisis a la base de 3 criteres faible et grande multiplicite (figure en haut et a
gauche et ...), l’energie de la particule primaire qui varie entre le seuil de detection de l’experience
et la valeur maximale de l’evenement le plus energetique 1018 (figures au milieu a gauche et a
droite) et l’inclinaison de la gerbe (faible θ et grand θ).
As mentioned above 2.2, this deviation from planarity is not a systematic bias on time mea-
surements but the proof of a curvature in the wavefront shape then the maximum signal generation
region in the shower was located at a distance Rc from the ground with respect to the arrival di-
rection. To verify this effect, simulations of wave propagation from this emission center have been
performed with the triple goal of reproducing event per event the geometric configuration, using
of a spherical wave shape for simplicity reasons and approaching the real detection conditions in
terms of time resolution by random number generator. Figure 4 illustrates an example of simulation
that reproduce the same event parameters (for data see fig. 3) and with an emission center distant of
3, 5 and 10 km from the ground. According to the data recorded in codalema II, we can conclude
two important facts: the simulations reproduced the data in the context that the wavefront shape is
different from a plane and when emission center is moving away from the ground more points in
the figure approaches from the best fit line is a clear tendency to the normal plane wave model (far
field region).
Time delay was studied according to different shower parameters, as the arrival direction,
multiplicity, the energy and the antenna position. By definition, for the first impacted antenna
both temporal variables are equal to zero ns, ∆texp = ∆ttheo = 0 ns. Note also that the time delays
distributions have a general form than can be fitted to a Γ−probability distribution function of the
form: f(∆t) = Γ(∆t) = a(∆t)be−c.∆t where, a, b, and c are adjustable parameters.
The distributions have long, thinly populated tails, These authors found that the local delay
distributions of the different particle groups are substantially different, yet the general shape which
is given by the dominating electrons at this radial interval can be fitted to a. This figure shows that
the deviation from the plane wave antenna by antenna exists.
On remarque que le retard temporel est accentue par le nombre d’antenne cad plus le nbre
augmente plus il y a du retard temporel. Du coup, il est interessent d’avoir des evenements a
grande multiplicite. Ce retard cumule peut atteindre 411 ns apres pour les valeurs les plus grandes
on risque d’avoir un biais systematique due a la non-compte des retards des cables. XXX Je peux
parler un peu de la chaine de traitement du signal dans codalema XXX. On remarque aussi que le
retard depend de plusieurs facteurs la direction d’arrivee, la multiplicite, l’energie, de la position
– 11 –

de l’antenne, de la chronologie de la gerbe lorsqu’elle frappe le reseau mais une etude plus poussee
reste a faire pour quantifier et tirer des conclusions plus precises de cette deviation.
La figure 6 montre que par rapport au lot d’evenements etudie et pour toutes les antennes,
on remarque tout le temps la presence d’une deviation par a l’onde plane. On peut conjecturer
l’existence d’un biais systematique mais la bonne resolution angulaire du reseau d’antenne prouve
que l’absence de ce biais temporelle.
XXX Remarques importantes: a propos de l’ajustement avec cftool, un bon ajustement est
lorsque on obtient SSE proche de 0, un R-square proche de 1, un adjusted R-square proche de 1 et
un RMSE proche de 0. On remarque que lorsqu’on etudie les histogrammes de la valeurs moyenne
du biais on trouve que le maximum pointe a 10 ns. Une autre remarque on a besoin d ameliorer la
methode de tagging du maximum pour diminuer l erreur temporelle vers moins de 10 ns. XXX
XXX La seule possibilite est que le front d’onde est different de l’onde plane. Cette derniere
figure montre que la deviation par rapport a l’onde plane existe antenne par antenne XXX
Finalement, nous avons etudie la deviation par rapport a plusieurs facteurs: energie, multi-
plicite, geometrie de l’evenement, antenne par antenne et nous avons superpose plusieurs effets a
la fois. mais on ne sait pas encore la cause de l’oscillation vu dans la figure 6 mais on pense quand
meme que c’est du a la disposition speciale des antennes sur une croix.
To quantify signal processing efficiency, we generated the expected signal, fed it through the
same preamplifier and filter configurations used for data acquisition, and superposed it on records
otherwise free of transients. These signals were taken to have the form f(t) = θ(t)At2(e−Bt −e−Dt)
with the coefficient C chosen so that f(t) has no DC component, and D corresponding to a long
duration of the negative amplitude component. For all pulses we chose D = B/20, so that C =
1/8000 cancels the DC component.
Let’s consider that the shape of the pulse is not affected by preamplification and filtration for
0 50 100 150 200 250 300 350 400
10
0
10
1
10
2
N
i |∆ttheo
i − ∆texp
i | (ns)
Numberofevents
Multiplicity ≥ 18 [73 events]
0 10 20 30 40 50 60
0
20
40
60
80
100
120
< τ >= N
i |∆ttheo
i − ∆texp
i |/N (ns)
Numberofevents
Selection Cuts
N
i |∆ttheo
i − ∆texp
i | < 400 ns
Multiplicity ≥ 5
Fit
Γ − function = a ∗ tb
∗ e(−c∗t)
Coefficients (with 95% confidence bounds) :
a = 0.6009 (0.352, 0.8497)
b = 3.582 (3.306, 3.858)
c = 0.3224 (0.2999, 0.3448)
Goodness of fit:
R-square: 0.9853
Figure 5. Stacked histograms of the cumulative sum of the difference between theoretical delay ∆ttheo
i and
experimental delay ∆texp
i in function of the events multiplicity. We see that the deviation from the planar
model increases with the high multiplicity.
2.3 Case of anthropogenic radio emission in a self-trigger mode
In the context of the transition to an autonomous radio trigger technique based solely on the imple-
– 12 –

0 1 2 3 4 5 6 7 8 9 10
x 10
−7
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
mean(∆texp
i − ∆ttheo
i ) (s)
Antennas
NS1
NS2
NS3
NS4
NS5
NS6
NS7
E01
E02
E03
E04
E05
E06
E07
D98d
D32d
NS2b
NS3b
NS4b
E04b
D98e
NE1
NE2
N01
N02b
SE1
Figure 6. The green squares are the mean values of the differences between theoretical delay and the
experimental delay for each antenna where the concerned antenna is not the first antenna touched by the
radio shower. Errors bars are calculated using the standard deviation of the time gaps. The dashed red line
shows the case where the gap is zero, ie there is no curvature in the wavefront.
mentation of an amplitude threshold on the detected transient, our goal is to differentiate signals
from noisy sources from the radio transients emitted by air showers. First we discuss the relevance
of the assumption of a spherical wavefront. Then, we will examine a compilation of experimental
results in the case of background sources localization within three independent experiments using
an autonomous radio trigger.
XXXXXXXXXXXXXXXXXX Most of papers in the state of the art are based on the as-
sumption of far-field sources. In this case, the front of the received signals is assumed to be planar:
only the arrival directions are reconstructed. However, in radiodetection the antennas-sources dis-
tances are not larger enough compared to the signal wavelength. In this consideration, the far field
assumption is no longer valide. The consideration of spherical waves (near field hypothesis) is
necessary.
In the scientific literature, the most papers are based on the assumption of a far-field sources.
La plupart des travaux dans la litterature se basent sur l’hypothese des sources en champ loin-
tain. Dans ce cas, le front des signaux recus est suppose plan: les positions des sources sont, peu
consequent, caracterisees simplement par leur direction d’arrivee. Cependant, dans la radiodetec-
tion les distances sources-antennes ne sont pas suffisamment grandes par rapport a la longueur
d’onde du signal. Dans cette consideration, l’hypothese de champ lointain n’est plus valable.
La consideration des ondes spheriques (hypothese de champ proche) est necessaire.
– 13 –

XXXXXXXXXXXXXXX The space surrounding the emission region is usually subdivided
into three regions: La definition du champ proche est d’abord specifiee a partir du principe du ray-
onnement electromagnetique d’une source. Les termes champ proche et champ lointain d’une
source de rayonnement electromagnetique sont les regions autour de la source ou les champs
electromagnetiques sont d’intensites plus et moins importante. XXXXimportant voir le BANAL-
ISXXXX Le Balanis rapporte trois zones autour d’une source. Les frontieres entre ces zones sont
definies selon la longueur d’onde Lambda, la dimension caracteristique de l’observateur D et la
distance entre l’observateur et l’element rayonnant r. Zone de champ lointain: Dans cette zone ou
r > 2D2/λ, les champs sont entierement radiatifs. Zones de champ proche reactif et radiatif. D’un
point de vue du traitement d’antennes, la principale difference entre les deux regions reside dans le
front d’onde a la reception. Si les recepteurs sont en champ lointain, le diametre du front est grand.
Dans ce cas, le front d’onde peut etre considere comme etant plan. Lorsque les recepteurs sont en
champ proche, le diametre du front d’onde est petit. L’hypothese de l’onde plane n’est donc plus
valable dans ce cas. Le front d’onde en champ proche est de forme spherique. Il est necessaire de
prendre en compte la sphericite du front d’onde due a la propagation en champ proche. Comparing
the far field with the plane wave, we can see that they are basically the same except that the far-
field amplitude is inversely proportional to the distance whilst the amplitude of the plane wave is
constant. Thus, the far field can be considered a local plane wave. It should be pointed out that the
far-field condition is actually not that straightforward. The condition of r >> λ
2π was introduced
for electrically small antennas and is just a function of the frequency and not linked to the antenna
dimensions. When the antenna size D is electrically large, D > λ, the common definition of the
far-field condition is : r > 2D2
λ XXXXXXXXXXXXXX
The space surrounding an antenna is usually subdivided into three regions: reactive near-field,
radiating near field (Fresnel) and far-field (Fraunhofer) regions. These regions are so designated
to identify the field structure in each. Apres repetition de calcul, je crois que D n’est pas liee a
l’antenne recepteur mais a l’antenne emettrice pour cela je crois qu’on est en train de travailler
vraiment dans une zone tres limite entre la region de far field et la region de near field ce qui
complique encore de plus la recontruction des front spheriques des gerbes radio et des signaux
Il est clair qu’il existe 3 choses distinctes pour pouvoir reconstruire la position de la source:
savoir si le front de l’onde est vraiment courbe et savoir si le reseau d’antennes est capable de
resoudre cette courbure et dans le cas ou les deux premieres conditions sont validees il faut trouver
un algorithme qui permet de reconstruire la courbure de l’onde.
XXXXXXXXXXXXXXXXXXXX The near and far field are spatial regions of the radio emis-
sion around their sources. A far field emission is occured when its source is located at a distance
greater than rlim = 2.D2
λ , where D is the antenna array characteristic scale for CODALEMA III
array, D = 2 km and λ is the radio emission wavelength. In this case, the wavefront can be assim-
ilated to a plan. For a distance lower than rlim = 2.D2
λ , we talk about a near-field emission then the
wavefront has a curved shape. For CODALEMA experiment, the dimension of the array is about 2
km and the used band is [23, 80] MHz, then the previous statical sources are in a near-field region
so emit a spherical wave 11. This calculation justifies our use of a spherical model to adjust the
radio wavefront.
Note that the sun can be considered as a perfect plane wave source which motivated the use
of its emission during solar flare periods to calibrate the CODALEMA antennas array angular
– 14 –

Figure 7. Representation of rlim as a function of the frequency f in the detection band [23, 80] MHz.
resolution [37].
2.4 Discussion
XXXXXXXXXXXXXXXX Times delays are measured with respect to a plane perpendicular to
shower axis that starts in phase with the primary particle and moves parallel to the axis at speed
c, XXXXXXXXXXXXXXX we will show in the next section that ∆ttheo
i and ∆texp
i are used for
showing the deviation from the plane wave model.
the above approaches XXXXXXXXXXXXXXXX Let us ﬁrst consider the case of two an-
tennas triggering on a wave propagating spherically from a point source. Their trigger times dif-
ference is noted ∆tsph. If we perform the reconstruction of the signal direction of origin assuming
a plane wave hypothesis, the resulting trigger time difference ∆tplan will be wrong by a quantity
ε = ∆tplan −∆tsph given at leading order in d/R by ε = d3
8cR2 cos(θ)sin2(θ) where c is the velocity
of light, R the distance to the source and θ the angle between the source direction and d, the vec-
tor joining the two antennas. Taking d = 250 m (the maximum extension of the CODALEMA III
array), we ﬁnd that ε < 10 ns (our estimated experimental resolution on the trigger time measure-
ment) for R > 500m. According to this calculation, the discrimination between spherical and plane
wavefronts is hardly possible for sources further than 500 m with the trends setup geometry, yield
similar results. Note howeverthat with larger setups, the distance up to which wavefronts curvature
radii can be reconstructed will increase. XXXXXXXXXXXXXXXX
• Pour discuter la courbure du front d’onde champ proche/champ loitain.
• Discuter les formules de Balanis
• discuter l’effet des mecanismes d’emission sur la forme de l’onde par exemple le mecanisme
coulombien booste ne donne jamais une onde spherique ou parabolique alors on peut rejeter
ce mecanisme alors on peut rejeter ce mecanisme.
– 15 –

3. A new hyperbolic model to fit the EAS radio wavefront curvature
Good expression: Finding the minimum is a challenge for some algorithms since it has a
shallow minimum inside a deeply curved valley. When the vertical axis is log-scaled in other
words. With these considerations in mind, write a function file for the nonlinear constraint.
As explicitly mentionned above, we have demonstrated that the wave front is slightly curved.
This curvature is due to the fact that the source of the radio signal is space-localized. Now, we
propose to reconstruct the emission center position. Our reconstruction is not based upon adjusting
the wavefront shape which has a complicated geometry dependent on the shower developpement
but based on fitting the difference between its real and a hypothetical plane wavefront by a hyper-
bola and this is correct for 3 basic geometrical considerations. Then, modelling of this difference
requires four hypothesis:
• The lateral spread is ignored.
• The emission region is situated at a large distance Rc compared to distances between antennas
and shower axis (Rc >> d).
• Radio waves are supposed to travel at the speed of light.
• Antenna and shower core coordinates need to be changed into the shower coordinate system
by 2 angular rotation.
We can write this difference as follows:
∆ = MG−MO,
= (d2
+R2
c)
1
2 −Rc,
= Rc(((
d
Rc
)2
+1)
1
2 −1),
≈ Rc((
1
2
(
d
Rc
)2
+1)−1),
≈
1
2
d2
Rc
,
Developing more the four hypothesis assumed at this section: Let’s start with the first hypothesis,
one can be considered the air shower particles responsible for the radio emission are concentrated
in a region of space close to the shower axis. The coherence property of the signal leeds to a lateral
spatial extension variate between 3 m to 13 m order the chosen frequency band. For the longitudinal
thickness of the region, it is known after the work of Linsley [?] that the particles swarm has a few
meters of longitudinal thickness. It is clear now that most electrons/positrons are concentrated
in a small symmetric cylindrically volume with negligible dimensions compared to the distances
between the emission center and the array of antennas which explains the above approximation
Rc >> d. Finally, the last hypothesis was necessary to generalize the reconstruction model to all
showers with different zenith angles. Yet, the difference ∆ is a parabolic function of the distance
d. In term of arrival times, ∆ is expressed by the time delay between the instant tpred
i predicted
– 16 –

Figure 8. Sketch of a simpliﬁed relation between wavefront shape and curvature radius
by the hypothetical passage of the plane wave front on antenna i and the instant tmax
i measured
experimentaly by the slightly curved wave front on the same antenna (see Appendix A). In order to
ensure identical treatment for all showers despite of their zenith angles θ. The coordinates of the
antennas (xi,yi,zi = 0) and times (tmax
i ,tpred
i ) must be expressed in a new frame called the shower
frame deﬁned by two rotation involves both the azimuthal and zenithal angles (φ,θ) as used in [?].
This correspondence is then written for an antenna i as follows:
c(tmax
i −tpred
i ) = a+
1
2Rc
(dr
i )2
,
where dr
i the distance between antenna i and the shower axis in the shower frame,
dr
i = (xr
i −xr
c)2 +(yr
i −yr
c)2 +(zr
i −zr
c)2,
The 3D rotation matrix. Tout d’abord, considerons la matrice rotation Rz(φ) de l’angle φ par
rapport a l’axe (oz).
Rz(φ) =



cos(φ) sin(φ) 0
−sin(φ) cos(φ) 0
0 0 1



Maintenant, il faut determiner la matrice rotation par rapport a l’axe (oy) avec l’angle θ.
Ry(θ) =



cos(θ) 0 sin(θ)
0 1 0
−sin(θ) 0 cos(θ)



– 17 –

Ry(θ)∗Rz(φ) =



cos(θ) 0 sin(θ)
0 1 0
−sin(θ) 0 cos(θ)


∗



cos(φ) sin(φ) 0
0 0 1



The 3D rotation matrix used is as follows :



xr
i
yr
i
zr
i


 =



cos(φ).cos(θ) cos(θ).sin(φ) sin(θ)
−cos(φ).sin(θ) −sin(θ).sin(φ) cos(θ)






xi
yi
zi



The development of calculation gives the following system of equations.



xr
i = cos(θ).(cos(φ).xi +sin(φ).yi)+sin(θ).zi(1)
yr
i = −sin(φ).xi +cos(φ).yi(2)
zr
i = −sin(θ).(cos(φ).xi +sin(φ).yi)+cos(θ).zi(3)
The same transformation is performed to the shower core coordinates (xc,yc,zc). The term time
will not be affected by the transformation since the difference will remove the same added term
zr
i
c .
Giving the χ2 function:
χ2
=
N
∑
i=1
(c(tmax
i −tpred
i )−a−
(xr
i −xr
c)2 +(yr
i −yr
c)2 +(zr
i −zr
c)2
2Rc
)2
This estimator has five free parameters the constant a, the radius of curvature Rc and (xr
c,yr
c,zr
c)
expressed in the shower frame. The nonlinear terms force us to use a numerical method for the χ2
minimization. Both the matlab Curvefitting toolbox and Optimization toolbox have been used and
give the same results. We found that the more appropriate algorithm for the resolution of the min-
imization problem was the Levenberg-Marquardt designed for non-linear problems. Data analysis
and events selection Criteria Selection strategy Our strategy for estimating the radius of curvature
demanded the selection of only those events in which we are sure of their quality and their pa-
rameters reconstructed by other models in order to facilitate comparison between different models.
For this we have chosen a selection with cuts similar to those used to fit the lateral distribution
function. The data used in this paper were collected by the CODALEMA experiment during over
than 3 years between november 2006 and january 2010. We find a yield of 196526 events detected
by the scintillator array after selections we use 450 internal events.
Thus the key ingredients for selecting our set of events are the following:
• Selection of radio events candidate by choosing events were detected in coincidence between
scintillator and antennas array. je parle ici de l’arbre la fenetre en temps et la fenetre angu-
laire the following criteria must be met: a time coincidence with +/-100 ns and an angular
difference smaller than 20 degree in the arrival directions reconstructed from both the parti-
cle and radio arrays. je peux parler ici du taux du trigger et de taux d’evets fisiks par jour
comme c’est indique dans ma presentation au SF2A
• Selection of internal events to be sure that shower core was situated inside the two array with
a very good estimation of energy (Fenergy=1).
– 18 –

• Multiplicity 5 because our model has 5 free parameters
• Only tagged antennas by event. This cut is applied to eliminate the antennas that have a low
signal to noise ratio in order to improve reconstruction.
This last cut does not remove any event although it improves their quality by getting rid of not
tagged antennas.
Figure 9. Results of the fit with the hyperbolic model.
Events Samples
Table shows the numbers of collected events and their types. We report here the efficiency of
samples.
Type Number Efficiency
Trigger SD 196526 100%
Coincidences (SD and antennas) 2030 1.03%
Internal events 450 22.17%
Verification and Confirmation of Results
Numerical minimization of the χ2 function gives the shower core position (xr
c,yr
c,zr
c) expressed
in the shower coordinate system. For using coordinates its need to be transformed by an inverse
transformation that involves the inverse rotation matrix (see Appendix D) to the ground frame.
Our approach for the validation of the model is based on the comparison of these reconstructed
parameters with other models and with confirmed physical values.
Consistent shower core elevation
The CODALEMA experiment is situated on a flat land of geographical altitude of 134 meters.
Given the lateral extension of the antenna array. We can be considered with a good approximation
that antennas have an altitude equal to zero meter in the ground local reference. The figure 10
shows a histogram of the shower core altitudes for selected events. We can conclude that elevations
are consistent with the geometric configuration of the antenna array. Then the model give a correct
zc consistent with zero. j’ajoute une etude statistique pour les evets qui ont un z vraiment egale a 0
et les z qui sont a peu pres different quantification avec des pourcentages
– 19 –

Figure 10. histogram of shower core elevation for selected events
Confirmation of the radio core east shifting signature of charge excess mechanism
We can consider that the real test of validation of our experimental reconstruction is whether
it predicts the systematic shift between the radio core and the particle radio. This shifting is an
evidence of a negative charge excess in the electromagnetic component during the shower devel-
opement. This effect was predicted by Askaryan [?] in the sixties of the last century. According to
[?], this negative charge excess acts as a monopoly that moves with the speed of light and which
contributes to the emission by coherent radio signal. The processes responsible for this negative
charge excess are:
• Compton recoil electrons ejected into shower by photons with energy less than 20 MeV.
• δ-ray process which consist of electrons ejected from external atomic orbital under the in-
fluence of electromagnetic cascade.
• Fast annihilation of positrons in flight.
Further explanations are compiled in the Allan review [26]. This effect has several signatures. it
appears in the polarization of the electric field on the ground as shown in [?] also in the systematic
shift between radio shower core and particle shower core seen in data with [?] and [?] and explained
by simulations in [?]. The reconstruction model used in these papers assume that the lateral density
profile (LDF) of the radio shower follow a decreasing exponential as mentionned by Allan in [26].
Then, the electric field has this formula
E = E0.exp(−
((x−xc)2 +(y−yc)2 −((x−xc).cos(φ).sin(θ)+(y−yc).sin(φ).sin(θ))2)
d0
)
with xld f
c , yld f
c were coordinates of the radio shower core by the LDF model. The radio core were
expressed in particle core frame with the next geometrical transformation
S = rr −rp
with rr and rp are vectors respectively for radio and particle shower cores and S the vector which
represent the systematic shift.
– 20 –

Figure 12 demonstrates a comparaison between the east-west projection of the systematic shift
SEW measured by PM and LDF models. Obtained curves are fitted by a gaussian. According to
our statistical approach, it can be concluded that the radio shower cores are shifted towards the east
with respect to the particle shower cores. This shift is a physical effect verified by both methods.
We remember that the two methods are completely independent. PM method is based on the
distribution of arrival times and the LDF method is based on the amplitudes of the radio signal on
the antennas. One can interpret the difference in the mean shift value between the two models by
the signal to noise ratio is different for the two methods. LDF model is based on the radio signal
amplitudes on the antennas. CODALEMA antennas are occupied by a low noise amplifier (LNA)
are very sensitive to the signals detected. Knowing that the noise level of the galactic background
is worth?? and the value of a signal typically developed by a shower with an energy of 1017 in the
range of ???? µV/m. This sensibility can expect a ratio of the order ???
Figure 11. A comparison between the spherical model, the conical model and the hyperbolical model.
Results of the Curvature Radius reconstruction
J’insere l’histogramme des rayons de courbure avec une explication du pic vers 4 km et du
queue de la distribution les Rc tres grands qui sont peut etre les evenements qui ont un centre
d’emission tres loin qui donne d’une onde plane ou bien de defauts de reconstruction ou bien le
modele arrive a ces limites il y a la these autrichienne qui montre un histogramme des Rc dans
Auger reconstruit avec la methode particule je peux prendre l’interpretation qui se trouve dans
cette these. The shower front curvature radius at the core also represents the apparent distance
– 21 –

Figure 12. Comparison between results from the RLDF model based on electric field amplitudes distribution
(red) and the hyperbolic model based on arrivals time distribution (blue). The fit of the observable SEW shows
a spatial shift towards the East for both models. The adjustement takes account the statistical errors.
Figure 13. pieds de gerbe avec 3 méthodes ici je dois mettre les courbes bi-dim pour la comparaison des
pieds de gerbes
to the initial cosmic ray interaction with atmospheric nucleus with the atmosphere. the dist of the
apparent fisrt interaction height Rcostheta shows a distinct peak at 7 km which is the height at
which most air shower signals seems to originate
Comme une explication possible du queue de la distri des Rc qui presente des Rc tres grands on peut
expliquer ca par la multiplicite des evets cad moins l’event a touches d’antennes moins la recon-
struction est bonne ou bien precise un autre argument a passer avec l argument de l’eloignement
du centre d’emission
il faut aussi montrer la courbe Rc en fct de theta ou bien en fonction du cos(theta) pour discuter
le fait que Rc augmente avec l’angle zenithal je pense qu’il faut ajuster avec une loi de forme R =
cte1 + cte2*(theta)n pour comparer apres entre d0 = cte1 + cte2*(theta)n l’idee est de tirer une
similarite entre les deux observables physiques R et d0 et theta
– 22 –

Figure 14. pieds de gerbe avec 3 méthodes ici je dois mettre les courbes bi-dim pour la comparaison des
pieds de gerbes
Figure 15. Histogram of the radius of curvature for 1010 events show a peak at about 4 km.
4. A spherical model to ﬁt the anthropic radio wavefront signals
4.1 Compilation of experimental results from self-triggered radiodetection experiments
Well as the autonomous radiodetection technique is not yet mature, the current experimental efforts
focuse on the engineering work to test its feasibility such as the CODALEMA III experiment at
the Nançay radio astronomy facility in France [22], the AERA experiment at the Pierre Auger
Observatory in Argentina [29] and the TREND experiment at the XinJiang 21 cm array (21CMA)
radio telescope in China [18].
For all these experiments, given a set of arrival time on antennas, the used technique to ex-
tract the radius of curvature is based on condensing data in a model that contains the following
propagation term:
ti = ts +
(xi −xs)2 +(yi −ys)2 +(zi −zs)2
c
where (xi,yi,zi,ti) is the position et time of the reception of the antenna i, (xs,ys,zs,ts).
– 23 –

In frequentist approach, the merit function (χ2) or the objective function is conventionally
arranged so that small values represent close agreement with the solution. The Figure 16 shows
a typical radio sources reconstruction obtained with the CODALEMA experiment [17], by using
a spherical wavefront model. This observation uses an array formed by 34 autonomous stations
equiped with two butterflies antennas capable of measuring simultaneously the electric field’s two
horizontal polarizations (North-South and East-West) [27]. For each station, the wavefront record-
ing is subjected to one trigger based on a voltage threshold analyzed by a comparator in the band
[45 − 55 MHz]. The distribution of arrival time are dated by a GPS that allows a temporal res-
olution of σt = 5 ns [22]. For this analysis, only events with a multiplicity of at least 4 stations
in coincidence were selected. The non-linear least-squares function describing the development of
the spherical wavefront and taking into account the timing error (including 4 free parameters) is as
follows:
χ2
=
N
∑
i=1
(x0 −xi)2 +(y0 −yi)2 +(z0 −zi)2 −c2(t0 −ti)2
σti
2
Figure 16. Typical result of reconstruction of two entropic emitters at ground, observed with the stand-
alone stations of CODALEMA, through standard minimization algorithms. Despite the spreading of the
reconstructed positions, these two transmitters are, in reality, two stationary point sources.
Then, we used the Levenberg-Marquardt algorithm to solve this non-linear minimization prob-
lem [30]. The reconstruction results show unexpected behavior. Indeed for a fixed emission source
on the ground, whereas one would expect to observe a well localized gaussian distribution, the
reconstructed points are distributed on a half-line pointing towards the antenna array centre and
whose direction is oriented towards the real source position. The comparison between the recon-
structed positions and the Nançay site map shows that the two half-lines pointing to an electrical
transformer (the southwest sector source) and an electric gate at the entrance of a house (the north-
west source). These sources are external to the antenna array. The topology of the points distri-
butions suggests that the reconstruction method leads to a large uncertainties in the reconstructed
positions (bias and width distributions). It may also be noted that the arrival directions estimated
by the planar fit (θ plan, φ plan) or derived from this spherical fit (θsphe, φsphe) are very close.
– 24 –

Figure 17. Typical result of reconstruction of two entropic emitters at ground, observed with the stand-
alone stations of AERA experiment, through standard minimization algorithms. Despite the spreading of
the reconstructed positions, these many transmitters are, in reality, stationary point sources.
Such patterns are also observed in other radiodetection experiments, such as the AERA exper-
iment [19]. The model used is weighted by the time errors on the antennas and it is given by the
following formula:
χ2
=
N
∑
i=1
((τi −τ0)−(ti −t0))2
(σi)2
+
(1−γ)2
(σγ)2
The propagation takes place with a speed of v = γc. When γ = 1, the factor (1−γ)2
(σγ )2 represents
the contribution of deviation from the speed of light. The factor τ0 is the antennas arrival time
average [19]. The chi2 function minimization is performed with two algorithms: Simplex and
Migrad within the ROOT-CERN software. The reconstruction results for the anthropogenic radio-
sources in the AERA site is given in several conferences [28] and publications [29]. Although a
satisfactory agreement exists with the directions of the known noisy sources located in the analyzed
region that covers an area of 13∗13 km2, the distributions of points have an elongated shape and are
behaving similarly to those observed in the CODALEMA III experiment but the interpretation of
observations remains difficult. And the longitudinal extension of the reconstructed sources argues
again for an effect of a solutions degeneracy in the localization problem. On the other hand, it
is interesting to note that the distances of the observed sources are significantly larger than those
observed in the CODALEMA experiment. In this case, the wavefront curvature is significantly
greater and it is possible to imagine that the reconstruction error can become more important.
Moreover the TREND experiment has published the calibration results of antenna array with
an the localization of an intentionally positioned source on the ground in the middle of the detector
array which a truck with a running engine [18]. The minimization is performed using a Levenberg-
Marquardt algorithm with the following function:
χ2
=
N
∑
i=1
ti −t0 −
||Xi −X0||
c
2
– 25 –

Figure 18. Typical result of reconstruction of entropic source located inside the antennas array, observed
with the stand-alone stations of TREND experiment in China, through standard minimization algorithms.
Despite the spreading of the reconstructed positions, these two transmitters are, in reality, two stationary
point sources.
The algorithm reproduces reasonably the truck position inside the array. However, given the ob-
servation conditions, the best quality of the reconstruction could also result from the proximity of
the transients source, which indicates a strong spherical wavefront curvature. It appears that the
position of the source relative to the antenna array has an important role in the algorithms conver-
gence. One can also be noted that similar experimental observations were seen by the LUNASKA
experiment in Australia but results are not yet published.
Finally, we will give a summary of these experimental observations from different radiodetec-
tion experiments. These results show that :
- if the emission source is not located inside the detectors array, then the source is reconstructed
with a large error and a solutions degeneration appears.
- The reconstruction procedure (the minimization algorithm and/or the non-linear chi2) de-
termines less correctly the source distance Rs = x2
s +y2
s +z2
s than its zenith angle given by
θ = arcsin( ( xs
Rs
)2 +( ys
Rs
)2) or its azimut angle given by φ = arctan(ys/xs).
- An additional statistical analysis on the reconstructed points distribution, that spread over a
– 26 –

half-line joining the true source to the array, allows the extraction of the emission distance (mean
value and standard deviation) when the number of realizations is large.
- By cons, an UHECR event is a unique realization of physical observables (arrival time and
maximum amplitude distributions) then, the application of these statistical methods to the identifi-
cation of a pointlike source (Xmax position) becomes more difficult.
All these observations pushed us to analyze more in details the various parameters of the
problem and the different minimization algorithms. To test our hypotheses, we had recourse to
extensive numerical simulation to the reality of experimental observations with respect to these
parameters.
4.2 Simulation studies of the localization of emission source with minimization algorithms
To improve the understanding of the experimental reconstruction results, we test many minimiza-
tion algorithms performances with simulated data. The test antennas array used is formed by 5
antennas where this number is imposed by the number of free parameters in the reconstruction
model, then the antennas positions −→ri = (xi,yi,zi) are fixed (see Fig. 19) (this corresponds to a
multiplicity of antennas similar to that sought at the detection threshold in current setups).
Figure 19. Left, scheme of the antenna array used for the simulations. The antenna location is took from
a uniform distribution of 1 m width. Right, the diagram shows our adopted strategy steps during the re-
construction phase. The planar fit is used as a first step for the estimation of the signal arrival direction
(θ plan, φ plan). In the second step, the spherical fit uses these parameters as initial conditions to reduce the
explored phase space.
A source S with a spatial position −→rs = (xs,ys,zs) is set at the desired value. Assuming ts
the unknown instant of the wave emission from S, c the wave velocity in the medium considered
constant during the propagation, and assuming that the emitted wave is spherical, the reception
time ti on each antenna i ∈ {1,...,N} can written:
ti = ts +
(xi −xs)2
+(yi −ys)2
+(zi −zs)2
c
+G(0,σt)
– 27 –

where G(0,σt) is the Gaussian probability density function centered to t = 0 and of standard devi-
ation σt. This latter parameter stand for the the global time resolution, which depends as well on
technological specifications of the apparatus than on analysis methods.
The theoretical predictions are compared to the reconstructions given by the different algo-
rithms. The latter are setup in two steps. First, a planar adjustment is made, in order to pres-tress
the region of the zenith angle θ and azimuth angle φ of the source arrival direction. It specifies
a target region in this subset of the phase space, reducing the computing time of the search of the
minimum of the objective function of the spherical emission. Reconstruction of the source location
is achieved, choosing an objective-function that measures the agreement between the data and the
model of the form, by calculating the difference between data and a theoretical model (in frequen-
tist statistics, the objective-function is conventionally arranged so that small values represent close
agreement):
f(rs,t∗
s ) =
1
2
N
∑
i=1
−→rs −−→ri
2
−(t∗
s −t∗
i )2
2
(4.1)
The partial terms −→rs −−→ri
2
− (t∗
s −t∗
i )2
represents the difference between the square of the
radius calculated using coordinates and the square of the radius calculated using wave propagation
time for each of the N antennas. The functional f can be interpreted as the sum of squared errors.
Intuitively the source positions −→rs at the instant ts is one that minimizes this error.
In the context of this paper, we did not use genetic algorithms or multivariate analysis meth-
ods but we focused on three minimization algorithms, used extensively in statistical data analysis
software of high energy physics [31, 32]: Simplex, Line-Search and Levenberg-Marquardt (see
table 1). They can be found in many scientific libraries as the Optimization Toolbox in Matlab, the
MPFIT in IDL and the library Minuit in Root that uses 2 algorithms Migrad and Simplex which
are based respectively on a variable-metric linear search method with calculation of the objective
function first derivative and a simple search method. For the present study, we have used with their
default parameters.
We tested three time resolutions with times values took within 3σt.
• σt = 0ns plays the role of the perfect theoretical detection and serves as reference;
• σt = 3ns reflects the optimum performances expected in the current state of the art;
• σt = 10ns stands for the timing resolution estimate of an experiment like CODALEMA [22].
For every source distance and temporal resolution, one million events were generated. Antenna
location was taken in a uniform distribution of 1m width. A blind search was simulated using
uniform distribution of the initial rs values from 0.1km to 20km. Typical results obtained with our
simulations are presented in Figures 20 and 21. The summary of the reconstructed parameters is
given in table 2.
One can summarize the simulation results:
• when the temporal resolution increases: we have a reconstruction quality degradation, a
spread in the distribution of points and an appearance of bias,
– 28 –

Table 1. Summary of the different algorithms and methods used to minimize the objective-function. The sec-
ond row indicates framework functions corresponding to each algorithm; third recalls the framework names.
The key information used for optimization are recalled down, noting that a differentiable optimization algo-
rithm (ie. non-probabilistic and non-heuristic) consists of building a sequence of points in the phase space
as follows: xk+1 = xk +tk.dk, and that it is ranked based on its calculation method of tk and dk parameters
([31, 32, 33]).
Minimization algo-
rithms
Levenberg-Marquardt Simplex Line-Search
Libraries lsqnonlin - MPFIT fminsearch - SIMPLEX MIGRAD - lsqcurveﬁt
Software Optimization Toolbox
Matlab - IDL
Optimization Toolbox
Matlab - MINUIT-ROOT
Optimization Toolbox
Matlab - MINUIT-ROOT
Method Principles Gauss-Newton method
combined with trust
region method
Direct search method Compute the step-size by
optimizing the merit func-
tion f(x+t.d)
Used information Compute gradient (∇ f)k
and an approximate hes-
sian (∇2 f)k
No use of numerical or
analytical gradients
f(x + t.d, d) where d
is a direction descent
computed with gradi-
ent/hessian
Advantages / Dis-
advantages
Stabilize ill-conditioned
Hessian matrix / time
consuming and local
minimum trap
No reliable information
about parameter errors
and correlations
Need initialization with
another method, give
the optimal step size for
the optimization algo-
rithm then reduce the
complexity
• but the temporal resolution is not the only factor, in fact, if the source is located outside the
antennas array a bad reconstruction is obtained and if the source is located inside the array a
good reconstruction is obtained,
• the localization is sensitive to the minimization algorithms (simplex and LVM),
• we have an initial conditions dependence: when the distance is close to the middle of the
blind search interval [0,20] km used as initial condition for the minimization algorithms as
indicated in the bottom Fig. 20,
• we have a multiple solutions (degeneration).
As is well known, a problem is said to be well-posed in the sense of Hadamard when it veriﬁes
these conditions:
• Existence of a solution (for all admissible data),
• Uniqueness of this solution,
• Continuous dependence of solution on the data ie the solution has not a strong dependence
in the problem different parameters for example initial conditions, boundary conditions and
data errors.
– 29 –

Figure 20. Results of the reconstruction of a source with a radius of curvature equal to 1 and 10 km with
the LVM algorithm. For Rtrue = 1km, the effect of the blind search leads to non-convergence of the LVM
algorithm, when initialization values are greater than Rtrue = 1km.
Then, whatever the simulations samples (versus any source distances, arrival directions, time
resolutions), (also with several detector conﬁgurations) and the three minimization algorithms,
large spreads were generally observed for the source locations reconstructed. This suggests that
the objective-function presents local minima. Moreover, the results depend strongly on initial
conditions. Thus, when subjected to these limitations, the radio source localization problem is
liable to be an ill-posed problem in the sense of Hadamard [35].
Otherwise, the continuous dependence with respect to data means that small perturbations
in the data induce small changes in the problem solution. More precisely, one can quantify this
property with the calculation of the condition number used to measure the solution sensitivity to
error in data. A high condition number (»1) indicates a poorly conditioned problem against a
condition number close to 1 indicates that the problem is well conditioned. This method is used in
other ﬁelds such as radar detection and geophysics to study the localization problem sensitivity as
mentioned in [36]. Then, the condition number is given by the calculation of the Hessian matrix
– 30 –

Figure 21. Results of the reconstruction of a source with a radius of curvature equal to 1 and 10 km with the
Simplex algorithm.
conditioning (see the next section for the Hessian matrix) and it has the following expression:
κ(H) = ||H−1
∗H|| =
λmax(H)
λmin(H)
Where, λ are the Hessian matrix eigenvalues and many norms have been used to calculate κ(H)
(the Euclidean norm, the maximum absolute row sum norm and the Frobenius norm). The Fig. 4.2
indicates large values (> 104), when a well-posed problem should induce values close to 1. We can
conclude that the problem is ill-conditioned, in addition to its ill-posedness formulation.
Lastly, it obvious that a further mathematical study is necessary to understand this spherical
minimization for that we have undertaken to calculte the main features of this objective-function.
– 31 –

Reconstruction Results
σt (ns) Rtrue(m) Algorithms Rmean(m) |Bias|(m) σR(m)
0
1000
Levenberg-Marquardt (10071) 1002 (9071) 2 (5763) 102
Simplex 1198 198 1477
3000
Simplex 3134 134 3437
10000
Levenberg-Marquardt 9999 1 56
Simplex 10466 466 5817
3
1000
Simplex 1199 199 1486
3000
Simplex 3132 132 3485
10000
Simplex 8194 1806 6154
10
1000
Simplex 1189 189 1507
3000
Simplex 2760 240 3703
10000
Simplex 3675 6325 4048
Table 2. Summary of parameters reconstructed with different algorithms for several distances of source
and several timing resolutions. On the Levenberg-Marquardt, the results in parentheses are those taking
into account the ﬂat portion of the resulting distribution (see Fig. 20). They are typical of initialization
values which are starting too far from the actual source distance. The Line-Search method was ultimately
rejected for this quantitative study, because results too dependent on the starting algorithm ﬁxing the initial
conditions.
– 32 –

Figure 22. Condition numbers obtained using the formulaCond(Q) = Q . Q−1 with Q the Hessian matrix
(see next section) as a function of the source distance and for different timing resolutions. The large values
of conditioning suggest that we face an ill-posed problem.
– 33 –

rs, ri: position of the source, position of the ith antenna
ts, ti: emission time of the signal, signal arrival time at the ith antenna
t∗
s , t∗
i : reduced time variables (ie. t∗ = c.t)
σt
i : time resolution on the ith antenna
Xs, Xi spacio-temporal position of the source, of the ith antenna
∇ f, ∇2 f: first and second derivative of the objective function f
M = I4 −2E44 =





1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 −1





: second order tensor related to the Minkowski metric
Q, Li: quadratic and linear form
< .|. >: inner product
XT : transpose of a vector or a matrix X
Table 3. List of notations
5. Ill-posed formulation of the emission source localization for the spherical emission
To estimate the source position Xs = (rs,ts) using the sequence of arrival ti, the natural method is
to formulate an unconstrained optimization problem of type a non-linear least square [33], starting
from eq. 1. which can rewrite
1 (see the notations listed in table 5):
f (Xs) =
1
2
N
∑
i=1
2
2 −(t∗
s −t∗
i )2
2
=
1
2
N
∑
i=1
f2
i (Xs) (5.1)
Several properties of the objective-function f were studied: the coercive property to indicate
the existence of at least one minima, the non-convexity to indicate the existence of several local
minima, and the jacobian to locate the critical points. (Bias study, which corresponds to a sys-
tematic shift of the estimator, is postponed to another contribution). In mathematical terms, this
analysis amounts to:
• Estimate the limits of f to make evidence of critical points; obviously, the objective function
f is positive, regular and coercive. Indeed, f tends to +∞ when X → ±∞, because it is a
polynomial and contains positive square terms. So, f admits at least a minimum.
• Verify the second optimality condition: the convexity property of a function on a domain for
a sufficiently regular function is equivalent to positive-definiteness character of its Hessian
matrix.
• Solve the first optimality condition: ∇ f(Xs) = 0 (jacobian) to find the critical points.
1In practice of the minimization, it is usual to take into account errors on the measured parameters by putting them
in the objective function denominator. In our theoretical study, it is assumed that the arrival times errors are the same for
all the antennas (σt = constant ∀ i). The present studied functional is generic and does not include errors, but as will see
later, introduction of a multiplicative constant doesn’t change the results of our study.
– 34 –

5.1 Convexity property of the non-linear χ2 function
Using fi(Xs) = (Xs −Xi)T .M.(Xs −Xi) where M designates the Minkowski matrix and given ∇ fi(Xs) =
2.M(Xs −Xi), the f gradient function can written (see appendix 1):
1
2
∇ f(Xs) = (∑ fi(Xs))M.Xs −M.(∑ fi(Xs)Xi)
The Hessian matrix, which is the f second derivative can written:
∇2
f(Xs) = ∑∇ fi(Xs).∇fT
i +∑ fi.∇2
fi
that becomes, replacing ∇ fi by its expression:
∇2
f(Xs) = (∑ fi(Xs)).M +2M.[N.Xs.XT
s +∑XiXT
i −Xs(∑Xi)T
−(∑Xi)XT
s ].M
Using a Taylor series expansion to order 2 (see appendix 1), an expanded form of the Hessian
matrix, equivalent to the previous formula of the f second derivative, is:
1
2
Q(Xs,Xi) =





∑i Ki +2∑i (xs −xi)2
2∑i (xs −xi)(ys −yi) 2∑i (xs −xi)(zs −zi) 2∑i (xs −xi) t∗
i −t∗
s
∗ ∑i Ki +2∑i (ys −yi)2
2∑i (ys −yi)(zs −zi) 2∑i (ys −yi) t∗
i −t∗
s
∗ ∗ ∑i Ki +2∑i (zs −zi)2
2∑i (zs −zi) t∗
i −t∗
s
∗ ∗ ∗ −∑i Ki +2∑i t∗
i −t∗
s
2





(5.2)
This latter allowed us to study the convexity of f (see appendix 1). Indeed, because its math-
ematical form is not appropriate for a direct use of the convexity definition, we have preferred
to use the property of semi-positive-definiteness of the Hessian matrix. Our calculus lead to the
conclusion that:
• Using the criterion of Sylvester [34] and the analysis of the principal minors of the Hessian
matrix , we find that f is not convex on small domains, and thus is likely to exhibit several
local minima, according to Xs and Xi. It is these minima, which induce convergence problems
to the correct solution for the common minimization algorithms.
5.2 Critical points
The study of the first optimality condition (Jacobian = 0) gives the following system ∇f(Xs) = 0
and allows finding the critical points and their phase-space distributions. Taking into account the
following expression:
1
2∇ f( ¯Xs) = (∑ fi( ¯Xs))M. ¯Xs −M.(∑ fi( ¯Xs)Xi)
we get the relation:
Xs =
N
∑
i=1
fi(Xs)
∑j fj(Xs)
Xi (5.3)
This formula looks like the traditional relationship of a barycenter. Thus, we interpret it in terms of
the antennas positions barycenter and its weights. The weight function fi expressing the space-time
distance error between the position exact and calculated, the predominant direction will be the one
presenting the greatest error between its exact and calculated position. The antennas of greatest
weight will be those the closest to the source.
– 35 –

In practice (see appendix 2), because the analytical development of this optimality condition
in a three-dimensional formulation is not practical, especially considering the nonlinear terms, we
chose to study particular cases. We considered the case of a linear antennas array (1D) for which
the optimality condition is easier to express with an emission source located in the same plane.
This approach allows us to understand the origin of the observed degeneration which appears from
the wave equation invariance by translation and by time reversal (known reversibility of the wave
equation in theory of partial differential equations) and provides us a intuition of the overall solu-
tion. It also enlightens the importance of the position of the actual source relative to the antennas
array (the latter point is linked to the convex hull of the antenna array and is the object of the next
section). Our study led to the following interpretations:
• The iso-barycenter of the antenna array (of the lit antennas for a given event) plays an impor-
tant role in explaining the observed numerical degeneration. The nature of the critical points
set determines the convergence of algorithms and therefore the reconstruction result.
• There are strong indications, in agreement with the experimental results and our calculations
(for 1D geometry), that the critical points are distributed on a line connecting the barycenter
of the lit antennas and the actual source location. We used this observation to construct an
alternative method of locating the source (section 4).
• According to the source position relative to the antenna array, the reconstruction can lead to
an ill-posed or well-posed problem, in the sense of J. Hadamard.
5.3 The antennas array convex hull concept
In the previous section we pointed that to face a well-posed problem (no degeneration in solution
set), it was necessary to add constraints reflecting the propagation law in the medium, the causality
constraints, and a condition linking the source location and the antenna array, the latter inducing
the concept of convex hull of the array of antennas. From appendix 2, we also saw that analyti-
cally the critical points evidence could become very complex from the mathematical point of view.
Therefore, we chose again an intuitive approach to characterize the convex hull, by exploring math-
ematically the case of a linear array with an emission source located in the same plane. This is the
subject of the appendix 3.
The results extend to a 2D antenna array, illuminated by a source located anywhere at ground,
arguing that it is possible to separate the array into sub-arrays arranged linearly. The superposition
of all the convex segments of the sub-arrays leads then to conceptualize a final convex surface, built
by all the peripheral antennas illuminated (see Figure 23).
The generalization of these results to real practical experience (with a source located any-
where in the sky) was guided by our experimental observations (performed through minimization
algorithms) that provide a first idea of what happens. For this, we chose to directly calculate nu-
merically the objective function for both general topologies: a source inside the antenna array (ie.
and at ground level) and an external source to the antenna array (in the sky ). As can be deduced
from the results (see Figs. 24 and 25), for a surface antenna array, the convex hull is the surface
defined by the antennas illuminated. (An extrapolation of reasoning to a 3D array (such as Ice
Cube, ANTARES,...) should lead, this time, to the convex volume of the setup).
– 36 –

Figure 23. Scheme of the reconstruction problem of spherical waves for our testing array of antennas (2D),
with a source located at ground. For this conﬁguration, the convex hull becomes the surface depicted in red.
The result is the same for a source in the sky.
Our results suggest the following interpretations:
• If the source is in the convex hull of the detector, the solution is unique. In contrast, the
location of the source outside the convex hull of the detector, causes degeneration of solutions
(multiple local minimums) regarding to the constrained optimization problem. The source
position, outside or inside the array, affects the convergence of reconstruction algorithms.
6. Conclusion
Experimental results indicated that the common methods of minimization of spherical wavefronts
could induce a mis-localisation of the emission sources. In the current form of our objective func-
tion, a ﬁrst elementary mathematical study indicates that the source localization method may lead
to ill-posed problems, according to the actual source position. However, further developments are
without any doubt still necessary, maybe based on advanced statistical theories, like Tikhonovs
regularization by adding further information as the signal amplitude or the functional of the radio
lateral distribution, this could be achieved by trying a generalized objective function which includes
these parameters or the exploitation of the lit antennas convex-hull concept for introducing a new
generation of 3D antennas array. In addition, the interactions with other disciplines which face
this problem could also provide tracks of work (especially regarding earth sciences which focus on
technics of petroleum prospecting in geophysics or aircraft radar detection).
– 37 –

Figure 24. Plots of the objective-function versus R and versus the phase space (R, t), in the case of our
testing array (2D), for a source on the ground and located inside the convex surface of the antenna array.
This conﬁguration leads to a single solution. In this case the problem is well-posed.
7. Appendix 1
7.1 Symbolic calculus
Keeping the same notation as in table 5, the objective function can be written:
f (Xs) =
1
2
N
∑
i=1
f2
i (Xs)
with fi (Xs) = (Xs −Xi)T
·M ·(Xs −Xi) = −→rs −−→ri
2
−(t∗
s −t∗
i )2
.
The formula ∇ f (Xs) = ∑ fi (Xs)·∇fi (Xs) is derived from the formula of a product derivation.
Using the bi-linearity of the inner product, we show that ∇fi (Xs) = 2M · (Xs −Xi). By injecting
this formula into the formula of ∇f , we obtain the following formula:
∇f (Xs) =∑ fi (Xs)·∇fi (Xs)
=∑ fi (Xs)·2M ·(Xs −Xi)
– 38 –

Figure 25. Plots of the objective-function versus R and versus the phase space (R, t), in the case of our
testing array (2D) for a source outside the convex hull. This conﬁguration leads to multiple local minima.
All minima are located on the line joining the antenna barycenter to the true source. In this case the problem
is ill-posed.
It then leads to the following form:
1
2
∇ f (Xs) = ∑ fi (Xs) M ·Xs −M · ∑ fi (Xs)Xi
With the same method, the second derivative matrix (Hessian matrix) is given by the following
formula :
∇2
f (Xs) = ∑∇ fi (Xs)·∇ fi (Xs)T
+∑ fi (Xs)·∇2
fi (Xs)
By injecting in the previous formula the following formula of the second derivatives ∇2 fi (Xs) = 2M
and by using the relation (AB)T
= BT AT , we get the following formula:
∇2
f (Xs) =∑∇fi (Xs)·∇fi (Xs)T
+∑ fi (Xs)·∇2
fi (Xs)
=∑2M ·(Xs −Xi)·(2M ·(Xs −Xi))T
+∑ fi (Xs)·2M
=4M · ∑ XsXT
s −XsXT
i −XiXT
s +XiXT
i ·M +2 ∑ fi (X) ·M
=4M · NXsXT
s +∑XiXT
i −Xs ∑Xi
T
− ∑Xi XT
s ·M +2 ∑ fi (Xs) ·M
– 39 –

Both relationships correspond to the end-calculus forms given in the 3rd section. These forms are
easy to handle for symbolic calculus but not convenient for explicit calculation used for studying
the convexity.
7.2 Explicit calculus using the Taylor expansion
An explicit form for the objective function first and second differential can be obtained using a Tay-
lor expansion. Indeed, the function f is an element of C∞ R4,R 2 and is therefore differentiable
in the sense of FrÃl’chet. Let Xs = (−→rs ,t∗
s )
T
be a fixed vector of R4 and
−→
ε =
−→
h ,t∗
T
another
vector of R4. In order to simplify the calculus, we use the following notations : Ki = −→rs −−→ri
2
2 −
(t∗
s −t∗
i )2
a constant term when setting the vector Xs; Li
−→
ε = −→rs −−→ri |
−→
h − (t∗
s −t∗
i ) ·t∗ the
linear form; and Q
−→
h ,t∗ =
−→
h
2
2
−t∗2 the quadratic form. The Taylor expansion leads to:
f Xs +
−→
ε =
1
2 ∑
i
−→rs +
−→
h −−→ri
2
2
−(t∗
0 +t∗
−t∗
i )2
2
=
1
2 ∑
i
−→rs +
−→
h −−→ri | −→rs +
−→
h −−→ri −(t∗
s +t∗
−t∗
i )2
2
=
1
2 ∑
i
2
2 +
−→
h
2
2
+2 −→rs −−→ri |
−→
h −(t∗
s −t∗
i )2
−t∗2
−2t∗
(t∗
s −t∗
i )
2
Using the multinomial expansion, the function f can then be approximated by the second-
order Taylor expansion following:
f −→rs +
−→
h ,t∗
s +t∗
≈
1
2 ∑
i
K2
i +2∑
i
Ki ·Li
−→
h ,t∗
+2∑
i
L2
i
−→
h ,t∗
+ ∑
i
Ki ·Q
−→
h ,t∗
We identify from this formula:
the constant term 1
2 ∑
i
K2
i ;
the linear term which is ∇f (Xs)T
·
−→
ε = 2·∑
i
Ki
t∗
i −t∗
s
T
·
−→
ε (the f first differential in
(−→rs ,t∗
s ) );
and the quadratic form at the point Xs:
1
2
Q(Xs,Xi) =





∑i Ki +2∑i (xs −xi)2
2∑i (xs −xi)(ys −yi) 2∑i (xs −xi)(zs −zi) 2∑i (xs −xi) t∗
i −t∗
s
∗ ∑i Ki +2∑i (ys −yi)2
2∑i (ys −yi)(zs −zi) 2∑i (ys −yi) t∗
i −t∗
s
∗ ∗ ∑i Ki +2∑i (zs −zi)2
2∑i (zs −zi) t∗
i −t∗
s
∗ ∗ ∗ −∑i Ki +2∑i t∗
i −t∗
s
2





which is the f Hessian matrix in (−→rs ,t∗
s ), or the second differential of f also denoted ∇2 f (Xs,Xi).
The use of ∗ indicates that the coefficients above and below the diagonal are equal (Schwarz
Lemma). The quadratic form represented by this matrix gives us the local second-order proper-
ties for the function f. To show that a critical point is a local minimum, it will suffice to verify that
the Hessian matrix is definite positive in the vicinity of this point.
2The function is also an element of the algebra R[X1,...,X4]
– 40 –

7.3 Study of the convexity property
The convex analysis occupies a capital place in the problems of minimization. Indeed, an important
theorem yet intuitive stated that if a convex function has a local minimum, it is automatically global.
We will shows that the function f is not convex in R4, i.e. that the Hessian matrix in non-positive
define.
Let ∇2 f (X) the Hessian matrix, and let’s suppose d a vector, since the function f is twice
differentiable, using the Sylvester’s criterion [34] to characterize the convexity of f , we can write
the following equivalence:
f is convex ⇔ Hessian is positive semi-definite ⇔ All Hessian principal minors are just nonnegative
f is convex ⇔ ∀d, ∀X, dT
·∇2
f (X)·d 0
So if we can find an element X and d such as dT ·∇2 f (X)·d < 0, f will be non-positive definite.
For this, it is sufficient to find a single negative principal minor to demonstrate the Hessian matrix
is non-positive definite. The objective function f will present then several local minimums and will
be thus locally non-convex.
So let Q the explicit expression of the Hessian and let us choose dT = (0001) then:
dT
·∇2
f(X)·d = (0001)·Q(Xs,Xi)·





0
0
0
1





= −∑
i
Ki +2∑
i
(t∗
i −t∗
s )2
which is represent the principal minor of order 4 of the Hessian.
For a family of fixed positions antennas and for a signal source with coordinates Xs such as
ys = zs = t∗
s = 0, the negativity condition of the principal minor of order 4 can then written:
∑
i
(xs −xi)2
> ∑
i
−y2
i −z2
i +3t∗2
i
Now the left term tends to infinity when the source tends to infinity3. It is written in terms of limits,
lim
|xs|→+∞
∑
i
(xs −xi)2
= +∞ ⇔ ∀A > 0, ∃η > 0 |xs| > η ⇒ ∑
i
(xs −xi)2
> A
Taking a value ∑
i
−y2
i −z2
i +3t2∗
i of the constant A, it exist a real η and therefore a xs such that
∑
i
(xs −xi)2
> ∑
i
−y2
i −z2
i +3t∗2
i . We deduce that the function is not convex in the vicinity of this
point. It suffices to take dT = (0001) and xs = η +1.
Q is the explicit expression of the Hessian and we take dT = (0001) alors:
dT
·∇2
f(X)·d = (0001)·Q(Xs,Xi)·





0
0
0
1





3We say that the function is coercive
– 41 –

= −∑
i
Ki +2∑
i
(t∗
i −t∗
s )2
it represents the Hessian principal minor of order 4.
We can also show this inequality without passing to the limit using the fact that a second
degree polynomial in xs tends to infinity as xs tends to infinity, the inequality is satisfied for a least
one value of xs
∑
i
(xs −xi)2
> ∑
i
−y2
i −z2
i +3t∗2
i
so
x2
s −
2∑i xi
N
xs +
1
N ∑
i
x2
i +y2
i +z2
i −3t∗
i
2
> 0
8. Appendix 2
8.1 Degeneration line for a linear antenna array
According to experimental data analysis and to our simulations (see Fig. 16 and 25), the results of
the common minimization algorithms appear to fall on a half-line in the phase space (x,y,z) which
we shall call the degeneration line, which is linked to the existence of local minima. We present the
mathematical development in the case of a linear array using an analysis-synthesis method. Then
we try to generalize results to the higher dimension cases.
Let suppose Xs = (xs,t∗
s ) a critical point of f, ie. ∇f(Xs) = 0, for a linear array, the minimiza-
tion problem with constraints can written:



arg min f(xs,t∗
s ) =
1
2
N
∑
i=1
((xs −xi)2
−(t∗
s −t∗
i )2
)2
1 i N
Propagation constraint : |xs −xi| = |t∗
s −t∗
i |
Causality constraint : t∗
s < mini(t∗
i )
and Let suppose L =
L
L
so that Xs − L is also a a solution of the minimization problem, ie.
∇ f(Xs −L) = 0)
The Jacobian of f is written as:
∇f (xs,t∗
) = 2



∑
i
(xs −xi) (xs −xi)2
−(t∗
s −t∗
i )2
∑
i
(t∗
i −t∗
s ) (xs −xi)2
−(t∗
s −t∗
i )2



If Xs being a critical point, this leads to two equations:



∑i (xs −xi) (xs −xi)2
−(t∗
s −t∗
i )2
= 0 (1)
∑i (t∗
i −t∗
s ) (xs −xi)2
−(t∗
s −t∗
i )2
= 0 (2)
Assuming that Xs −L being also a critical point, this leads to two equations:



∑i (xs −xi −L) (xs −xi −L)2
−(t∗
s −t∗
i −L)2
= 0 (3)
∑i (t∗
i −t∗
s +L) (xs −xi −L)2
−(t∗
s −t∗
i −L)2
= 0 (4)
– 42 –

By developing the equation (3) and by using the equation (1), then:
(3) ⇒ ∑
i
(xs −xi) (xs −xi)2
−(t∗
s −t∗
i )2
−2L[(xs −xi)−(t∗
s −t∗
i )] −L∑
i
(xs −xi)2
−(t∗
s −t∗
i )2
... +2L2
∑
i
[(xs −xi)−(t∗
s −t∗
i )] = 0
⇒ −L∑
i
(xs −xi)2
+L2
∑
i
[(xs −xi)−(t∗
s −t∗
i )]−L∑
i
(xs −xi)2
−(t∗
s −t∗
i )2
... +L∑
i
(xs −xi)(t∗
s −t∗
i ) = 0
The set of constraints requires that the term ∑
i
(xs −xi)2
− (t∗
s −t∗
i )2
is null. We get the simplified
equation:
L∑
i
(xs −xi)−(t∗
s −t∗
i ) = ∑
i
(xs −xi)((xs −xi)−(t∗
s −t∗
i ))
In the cases where xs −xi < 0 for all i, the set of constraints is equivalent to (xs −xi)−(t∗
s −t∗
i ) = 0.
Thus, if one assumes that xs −xi < 0 for all i, i.e that the source is outside the array convex hull (a
segment), we find that previous implications are equivalences and thus that equation (3) is verified.
Operating in the same manner for the equation (4), we obtain the following equations:
(4) ⇒ ∑
i
(t∗
i −t∗
s ) (xs −xi)2
−(t∗
s −t∗
i )2
−2L[(xs −xi)−(t∗
s −t∗
i )] +L∑
i
(xs −xi)2
−(t∗
s −t∗
i )2
...−2L2
∑
i
[(xs −xi)−(t∗
s −t∗
i )] = 0
⇒ −2L∑
i
(t∗
i −t∗
s )(xs −xi)−2L∑
i
(t∗
i −t∗
s )2
+L∑
i
(xs −xi)2
−(t∗
s −t∗
i )2
...−2L2
∑
i
(xs −xi)−(t∗
s −t∗
i ) = 0
Using the set of constraints as above, we obtain the following equation:
L∑
i
(xs −xi)−(t∗
s −t∗
i ) = ∑
i
(t∗
i −t∗
s )((xs −xi)−(t∗
s −t∗
i ))
The same analysis as above gives us the condition that the source is out of the antennas convex hull.
This degeneration is an important point because it determines the convergence of minimization
algorithms. In this case the problem of the reconstruction is ill-posed.
The generalization of the previous calculation to higher dimensions is more delicate, insofar
as there are infinitely many directions in which the source can move. The idea now is to translate
the source, from its position −→rs , simultaneously in all directions −→rs − −→ri and with the same dis-
tances. We define the unit vector on the direction source-antenna. It will be noted: −→ei =
−→rs −−→ri 2
.
The translation spatial direction thus defined, is given by the vector
−→
L = ∑
i
−→ei = ∑
i
−→ri −−→rs 2
=
−∑
i
−→ri
−→rs −−→ri 2
+ ∑
i
1
−→rs −−→ri 2
−→rs . Considering the reduced temporal variables, the wave required
– 43 –

delay to traverse the distance induced by the translation
−→
L . Let V the vector of coordinates
V =
−→
L ,
−→
L
T
. We write the first order optimality condition for the vector of R4: Xs −V:
∇f (Xs −V) = (∑ fi (Xs −V))·M ·(Xs −V)−M ·(∑ fi (Xs −V)·Xi)
By introducing the condition ∇ f (Xs) = 0 which implies that:
∑ fi (Xs) ·M ·Xs −M · ∑ fi (Xs)Xi = 0
we obtain:
∇ f (Xs −V) =N VT
·M ·V ·M · Xs −V −
1
N ∑
i
Xi
2M ∑(Xs −Xi)T
M ·V ·
−→
Xi −2 ∑(Xs −Xi)T
M ·V ·M ·(Xs −V)
− ∑ fi (Xs) ·M ·V
According to the imposed form of the vector
−→
V , then:
VT
·M ·V =
−→
L
−→
L
2
M
−→
L
−→
L
2
T
= 0
It remains then the following expression:
∇ f (Xs −V) =2M ∑(Xs −Xi)T
M ·V ·Xi −2 ∑(Xs −Xi)T
M ·V ·M ·(Xs −V)
− ∑ fi (Xs) ·M ·V
The resolution of this equation should lead to an analytical expression for the topology of critical
points. We failed to develop it, but we can already see that the explicit development leads to cross
terms that will make simplifications difficult. Therefore, we have tried again an intuitive approach
based on the numerical simulations presented section 3.3.
9. Appendix 3
9.1 Convex hull for a linear antenna array
Let us consider the sub-array of the 3 aligned upper antennas presented in Fig. 19). The figure 26
shows the physical principle of the reconstruction of the source.
Three situations must be considered:
• the source located inside the array;
• the source located outside the array but on the detector axis;
• the source located outside this main axis. The latter corresponds to the typical problems
encountered with of the man-made emitters located on the ground.
– 44 –

Figure 26. Scheme of the reconstruction problem of spherical waves for a 1D array of antennas. For this
configuration, the convex hull is the segment shown in red.
The first situation leads to 2 half-lines cutting each other in a single point: the solution is unique
(Figure 27) and the localization problem is well-posed. The source is unique and inside the line
segment linking the nearest antennas to the source. This segment correspond to the convex hull
within this geometry. We can also note that only the two antennas flanking the source then play a
role in its localization. The problem writes:



arg min f(xs,t∗
s ) =
1
2
N
∑
i=1
((xs −xi)2
−(t∗
s −t∗
i )2
)2
Propagation constraint : |xs −xi| = |t∗
s −t∗
i |
s < mini(t∗
i )
About the source on-axis, but outside the convex hull, the arrival times between the antennas,
are no longer related to the source position, but to their locations. Whatever their positions, the
time differences remain constant (for equally spaced antennas). It becomes impossible to distin-
guish between two different shifted sources by any length. The only relevant information lies in the
direction of propagation of the wave (see figure 28). This result appears by a degeneration of solu-
tions because all points located on the half-line starting from the first tagged antenna are solutions
of the problem which is ill-posed. The source is outside the convex hull of the antenna array.
On the configuration where source located outside this antenna axis (problem in two dimen-
sions), the solving starts with:



arg min f(xs,ys,t∗
s ) =
1
2
N
∑
i=1
((xs −xi)2
+y2
s −(t∗
s −t∗
i )2
)2
Propagation constraint : (xs −xi)2
+y2
s = (t∗
s −t∗
i )2
s < mini(t∗
i )
The constraint set reduces the problem of characterization of critical points to the search of the
half-cones intersections induced by each antenna, in the 3 dimensional phase space (x, y, t) and
– 45 –

Figure 27. Phase space representation in the case of a linear array of three antennas (shown as green squares
located at x1 = −200 m, x2 = 0 m, x3 = 200 m). The source is located at xs = 60m when the instant of the
emission is taken as the time origin (ts = 0s). Because the source is outside this sub-array, the constraints on
the positions of antenna 1 and 2 lead to the same equation ts = 60−xs (black line). Equation for the antenna
3 (blue line) leads to ts = −60 + xs. The causality conditions restrict the initial lines to two half-lines (red
lines). The source location (black star) is at the intercept of the both half-lines.
Figure 28. Same as ﬁgure 27 but for an on-axis source outside the linear array of three antennas. The whole
constraints lead the same equation t∗
s = 60 − xs. All points belonging to the lower half-line are solutions
of the source localization problem (red dashed line), which becomes, in this case, ill-posed, and creates the
degenerations.
which presents a great similarity of constraints with the light cone used in special relativity (Fig.
31). Intersection of the half-cones, two to two, induces multiple critical points which are local
minima.
10. Appendix 4: Systematics due to atmospheric earth
Since the earth atmosphere is acting like a shield for these extreme energy particles, where UHECR
interact and initiate the extensive air shower, it is necessary to modeling it with accuracy to reduce
– 46 –

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