To contact the author use ahmed.rebai2@gmail.com Radio signal from extensive air showers EAS studied by the CODALEMA experiment have been detected by means of the classic short fat antennas array working in a slave trigger mode by a particle scintillator array. It is shown that the radio shower wavefront is curved with respect to the plane wavefront hypothesis. Then a new fitting model (parabolic model) is proposed to fit the radio signal time delay distributions in an event-by-event basis. This model take into account this wavefront property and several shower geometry parameters such as: the existence of an apparent localised radio-emission source located at a distance Rc from the antenna array of and the radio shower core on the ground. Comparison of the outputs from this model and other reconstruction models used in the same experiment show: 1)- That the radio shower core is shifted from the particle shower core in a statistic analysis approach. 2)- The capability of the radiodetection method to reconstruct the curvature radius with a statistical error less than 50 g.cm−2 . Finally a preliminary study of the primary particle nature has been performed based on a comparison between data and Xmax distribution from Aires Monte-Carlo simulations for the same set of events.

1. RAPPORT D’ACTIVIT ´E
Towards the identification of the primary particle nature by the radiodetection
method with the CODALEMA experiment
AHMED REBAI1 FOR THE CODALEMA COLLABORATION,
1 Subatech IN2P3-CNRS/Universit´e de Nantes/ ´Ecole des Mines de Nantes, Nantes, France.
ahmed.rebai@subatech.in2p3.fr
Abstract: Radio signal from extensive air showers EAS studied by the CODALEMA experiment have been
detected by means of the classic short fat antennas array working in a slave trigger mode by a particle scintillator
array. It is shown that the radio shower wavefront is curved with respect to the plane wavefront hypothesis. Then a
new fitting model (parabolic model) is proposed to fit the radio signal time delay distributions in an event-by-event
basis. This model take into account this wavefront property and several shower geometry parameters such as: the
existence of an apparent localised radio-emission source located at a distance Rc from the antenna array of and the
radio shower core on the ground. Comparison of the outputs from this model and other reconstruction models
used in the same experiment show: 1)- That the radio shower core is shifted from the particle shower core in a
statistic analysis approach. 2)- The capability of the radiodetection method to reconstruct the curvature radius
with a statistical error less than 50 g.cm−2 . Finally a preliminary study of the primary particle nature has been
performed based on a comparison between data and Xmax distribution from Aires Monte-Carlo simulations for
the same set of events.
Keywords: UHECR, radio-detection, inverse problem, ill-posed problem, regularization.
1 Introduction
Since the last decade, radiodetection of the ultra high ener-
gy cosmic rays has arised again as a complementary detec-
tion technique to ground-based particle detector arrays and
fluorescence telescopes. The lastest results from CODALE-
MA and LOPES experiments have shown the potential and
feasibility of this technique in terms of sensitivity to the
shower longitudinal development , the detection duty cycle
was near to 100% and the low cost of detectors. CODALE-
MA at Nanc¸ay has shown a north south asymmetry signa-
ture of a geomagnetic effect in the radio signal production
mechanism [2]. On the other hand LOPES has shown the
possibility of reconstruction of the radio lateral distribution
function [3] that allows to have an observable linked to the
shower developement and correlated with the primary parti-
cle energy [4] and [5]. The radio emission center is a very
important observable since it is related to two properties of
the primary particle its energy and its chemical composition
through the shower maximum developpement Xmax. In this
paper, we discuss a new reconstruction method of the radio
signal wave front radius of curvature. We use a parabolic
model (PM) that fit the distribution of time residuals rela-
tive to plane wave hypothesis. We show the origin of the
reconstruction model and the results from the CODALEMA
data.
2 Experimental observations
In this paper, we focus on the problem of localization of
the emitting source which belong to a more general class of
problems called inverse problems. Based on a detailed anal-
ysis of experimental results of many experiments around
the world such as CODALEMA III in France, AERA in
Argentina, TREND in China and LUNASKA in Australia,
we demonstrate that this problem is ill-posed in the sense
of Hadamard (mathematician). This demonstration is divid-
Fig. 1: Set up of CODALEMA experience showing the
disposition of the particle detectors array (red) and the
dipole antenna array (yellow) used for this study.
ed into two parts which are: the existence of degeneration
of solutions (non-positive definition Hessian matrix of the
objective function + Sylvester criteria) and the bad condi-
tioning of the problem.
3 Experimental situation
Since 2002, CODALEMA experience [6], hosted on the
radio observatory site at Nanc¸ay with geographical coor-
dinates (47.3◦N, 2.1◦E and 137 m above sea level), aim-
s 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). It consists of an
array of 24 active dipole antennas spread over a surface of
about 1
4 km2, an array of 17 particle scintillator detectors
and a 144 conic logarithmic antennas from the Nanc¸ay deca-
metric array (see fig. 1). Triggering the scintillator data ac-
quisition system is defined by the passage in coincidence of
secondary particles, created in extensive atmospheric show-

2. Rebai A. Towards the primary particle identification by the CODALEMA experiment
RAPPORT D’ACTIVIT ´E
er, through each of the 5 central particle detectors. Trig-
ger detection threshold energy is 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. Radio events that are detected by
dipole antenna array in coincidence with atmospheric show-
er events are identified during offline analysis [6] [2]. After
this analysis phase, a data set containing the parameters of
the shower 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 distri-
bution, radio signal amplitude) and the observables of the
reconstructed shower by the use of radio data alone (arrival
time, direction arrival, 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 variables the nature of
the primary (estimate of Xmax).
4 Experimental motivations
In the first approximation the radio signal front is assumed
to be a plane perpendicular to the shower axis. Then, the
primary particle direction of motion can be determined
directly by triangulation using the time of flight between
different antennas. According to this hypothesis if we take
the first tagged antenna, in each event, as a reference for
arrival time and we plot the theoretical time delay ∆ttheo as a
function of the experimental time delay ∆texp (see Appendix
A for time delay calculation methods). We should observe
an alignment with the plane wave best line fit. But when this
test is performed on data we see that points deviate from
this line (see fig. 2) despite the 10 ns experimental timing
uncertainty (See Appendix B for understanding the origin
of such uncertainties). This deviation from planarity is not a
systematic experimetal bias time measurements on antennas
so it can be explained by the fact that the wave front has
not a plane form (shape) but another one and the signal
generation region in the shower was located at a distance
Rc from the ground with respect to the arrival direction. 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 (See Appendix C).
The figure 2 shows a simulation where we have used the
same parameters of the event (see fig. 2) and the emission
center is distant of 3, 5 and 10 km from the ground. We can
conclude two important effects: the simulations reproduced
the data in the context that the wavefront shape is different
from a plan and the emission center 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.
5 Theoretical foundation of the
reconstruction model
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.
We now propose to reconstruct the emission center position.
Our reconstruction is not based upon adjusting the wave-
Fig. 2: 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 systematically from the line which shows that
the wavefront is not a plan
front shape which has a complicated geometry dependent
on the shower developpement but based on fitting the differ-
ence between real and a hypothetical plane wavefront by a
parabola this is correct for basic geometrical consideration
3. Modeling 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) (see fig. 3).
• 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

3. Rebai A. Towards the primary particle identification by the CODALEMA experiment
RAPPORT D’ACTIVIT ´E
Fig. 3: Sketch of a simplified relation between wavefront
shape and curvature radius
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 [15] 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 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 [14]. 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 used is as follows :


xr
i
yr
i
zr
i

 =


cos(φ).cos(θ) cos(θ).sin(φ) sin(θ)
−sin(φ) cos(φ) 0
−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)
Fig. 4:
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 minimization
problem was the Levenberg-Marquardt designed for non-
linear problems.
6 Data analysis and events selection
Criteria
6.1 Selection strategy
Our strategy for estimating the radius of curvature demand-
ed the selection of only those events in which we are sure
of their quality and their parameters reconstructed by other
models in order to facilitate comparison between different
models. For this we have chosen a selection with cuts simi-
lar to those used to fit the lateral distribution function [7].
The data used in this paper were collected by the CODALE-
MA 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.
6.2 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%

4. Rebai A. Towards the primary particle identification by the CODALEMA experiment
RAPPORT D’ACTIVIT ´E
Fig. 5: histogram of shower core elevation for selected
events
7 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.
7.1 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 5
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.
7.2 Confirmation of the radio core east shifting
signature of charge excess mechanism
We can consider that the real test of validation of our exper-
imental reconstruction is whether it predicts the systemat-
ic 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 develope-
ment. This effect was predicted by Askaryan [11] in the
sixties of the last century. According to [11], 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 influence of
electromagnetic cascade.
• Fast annihilation of positrons in flight.
Further explanations are compiled in the Allan review
[12]. This effect has several signatures. it appears in the
polarization of the electric field on the ground as shown in
[13] also in the systematic shift between radio shower core
Fig. 6:
and particle shower core seen in data with [8] and [9] and
explained by simulations in [10]. 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 [12]. 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))1/2
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 systemat-
ic shift. Figure 6 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 remem-
ber 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 sig-
nal 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. CO-
DALEMA 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 100 µV/m. This sen-
sibility can expect a ratio of the order 1000.

5. Rebai A. Towards the primary particle identification by the CODALEMA experiment
RAPPORT D’ACTIVIT ´E
Fig. 7:
Fig. 8:
Fig. 9: Histogram of the radius of curvature for 1010 events
show a peak at about 4 km.
Fig. 10: Correlation between Rc et θ. The mean value of Rc
increase with the zenithal angle.
8 Results of the Curvature Radius
reconstruction
9 Towards a primary particle nature
identification with the radio method
9.1 Atmospheric density profile
The earth’s atmosphere acts like a layer of matter with
1000g.cm−2 of thickness. The earth’s atmosphere acts like
a volume of detection where the primary particle deposits
its energy as huge number of secondary. Then, any attempt
for the determination of chemical composition of UHECR
passes through the fine understanding of the atmosphere
density variation as function of the altitude above the
sea level exactly at the experiment site in France. For
these reasons, the atmospheric density profile is a highly
required knowledge for converting the reconstructed radius
of curvature into the shower maximum Xmax using this
formula:
Xradio
max =
fLinsley(Rc.cos(θ)))
cos(θ))
where fLinsley is a function following the Linsley’s param-
eterization which divides the atmosphere into five layers
and give a realistic approximation. So we have compiled
data from the US standard atmosphere cited in Aires us-
er manual [16] and from the middle europe atmosphere in
7 months implemented in Corsika package [17]. We have
used the Linsley’s parameterization [16]. Our compilation
shows that both atmospheres have very similar character-
istics (see fig. 11). The same figure shows that the error in
the Xmax estimation due to the atmosphere collected data
is the order of σatm = 45g.cm−2 which represents a first

6. Rebai A. Towards the primary particle identification by the CODALEMA experiment
RAPPORT D’ACTIVIT ´E
Fig. 11: (Above) Compilation of data that represent the ver-
tical atmospheric depth according to the altitude above the
sea level with respect to the Linsley’s parameterization. At-
mospheric data are collected from the US standard atmo-
sphere (dash black curve) [16] and the middle Europe at-
mosphere with measurements at 7 different months (col-
ored solid curve) [17]. (Below) Comparison between the
same data taking as reference the US standard atmosphere
XEU −XUS.
source of systematic error on the our chemical composition
estimation.
9.2 Composition results:Preliminary
il est tres important d’interpreter les courbes d’identification
avec les simulations et les resultats theoriques je dois ap-
prendre comment mettre les droites theoriques SIBYL et
QSJET. je dois aussi lire la these de Frank Schrodder.
10 Conclusions
11 Appendix A. Calculation method for
theoretical and experimental time delays
11.1 Theoretical time delay calculation
We assume here that the signal propagation is carried out
with a constant speed which is the speed of light in the
vacuum c and in the hypothesis that the radio signal wave
front is a plan perpendicular to the arrival direction. The
plan equation can be written as:
u.x+v.y+w.z+d = 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.
Fig. 12: For 38 events, we use two reconstruction methods
spheric and parabolic.
Fig. 13:
Now we take the first tagged antenna (fta) as reference to
calculate the constant d. 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 this formula:
di =
|u.xi +v.yi +w.zi −(u.xfta +v.yfta +w.zfta)|
√
u2 +v2 +w2
The arrival time of the plan on each antenna is given by
simple division of the distance di by c then:
tpred
i = tfta +
di
c
.
this formula allows to reproduce the plane wave propaga-
tion from the first tagged antenna until other antennas. The-
oretical delay is then written:
∆ttheo
i = tpred
i −tfta =
di
c
.
11.2 Experimental time delay calculation
The filtered signals maximum in each antenna enable the
determination of the experimental arrival time the real time
noted tmax
i . Experimental delay is then written:
∆texp
i = tmax
i −tmax
fta =
di
c
.
∆ttheo
i and ∆texp
i are used in the begining of this paper for
showing the deviation from the plane wave model.

7. Rebai A. Towards the primary particle identification by the CODALEMA experiment
RAPPORT D’ACTIVIT ´E
12 Appendix B. Time uncertainty
calculation
Timing errors are due to the method of digital filtering
by the filter. Codalema uses the 23-83 Mhz band clean
from the parasites transmitters. This filtering gives signals
that oscillate with periods ranging between 12 and 43 ns
and since we are interested in the filtred signal positive
maximum.
13 Appendix C. Simulations of wave front
propagation with spheric shape
Simulated events generation is based on purely geometric
considerations (In this study, we did not use complete
simulation given by REAS3 or SELFAS2 but its can be
used for future more realistic tests). To generate a simulated
event, we fix the radius of curvature Rc, azimuthal and
zenith angles values (φ, θ) and coordinates of the shower
core (xc,yc,zc), we calculate the coordinates of the the
emission center with:



x0 = R.cos(φ).sin(θ)+xc(1)
y0 = R.sin(φ).sin(θ)+yc(2)
z0 = R.cos(θ)+zc(3)
These coordinates allow us to determine the distance
between a given antenna and the center of emission.
d = (x0 −xi)2 +(y0 −yi)2 +(z0 −zi)2. We calculate the
wave arrival time at each antenna by the following for-
mula: ti = t0 + d/c. Since the antenna has a time reso-
lution and non-zero error that affects off-line analysis,
one must take into account in our simulation so we fluc-
tuates over time on the antenna of a normal distribu-
tion with this formula: ti = t0 + d
c .gauss(0,1).σtime with
gauss(0,1) =
1
σ
√
2π
exp(−0.5λ2)
References
[1] Pierre Auger Collaboration, Astropart. Phys. 31 (2009)
399, Pierre Auger Collaboration, Phys. Rev. D 79 (2009)
102001, Pierre Auger Collaboration, Astropart. Phys. 29
(2008) 188, The High Resolution Fly’s Eye
Collaboration, Phys. Lett. B 619 (2005) 271, Pierre
Auger Collaboration ICRC 09, arXiv:0906.2347 [astro-
ph], Pierre Auger Collaboration ICRC 09,
arXiv:0906.2319 [astro-ph], M. Lemoine et E. Waxman,
JCAP 0911:009 (2009).
[2] D. Ardouin et al. Astroparticle Phys, 31, 192-200,
2009.
[3] W.D. Apel et al. Astroparticle Phys, 32, 294-303, 2010.
[4] P. Lautridou, Contribution to the RICAP conference,
Roma, 2011.
[5] A. Haungs, Contribution to the RICAP conference,
Roma, 2011.
[6] D. Ardouin et al. Astroparticle Phys, 26, 341-350,
2006.
[7] O. Ravel et al. Nucl. Instrum. methods A,
doi:10.1016/j.nima.2010.12.057.
[8] A. Lecacheux, A. Bell´etoile for the CODALEMA
collaboration, Proc. of the 31st ICRC 2009.
[9] A. Bell´etoile et al. Submitted to Phys Rev D.
[10] V. Marin for the CODALEMA collaboration Proc. of
the 32 ICRC 2011.
[11] G. Askaryan J. Exp. Theor. Phy 21 658 1962.
[12] H. Allan Progress in Elementary Particles and
Cosmic Ray Physics, Amesterdam, 10, 171 1971.
[13] H. Schoorlemmer, Results from polarization studies
of radio signals induced by cosmic rays at the Pierre
Auger Observatory, Nuclear Instruments and Methods in
Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment, In
Press, Corrected Proof”, 2010 DOI:
10.1016/j.nima.2010.11.145.
[14] H. Hedayati Kh. and A. Anvari and M. Bahmanabadi
and J. Samimi and M. Khakian Ghomi, A new simple
method for the analysis of extensive air showers,
Astroparticle Physics, 34, 9, 699 - 704, 2011.
[15] J Linsley, Thickness of the particle swarm in cosmic-
ray air showers, Journal of Physics G: Nuclear Physics,
12, 1, 511986.
[16] S.J. Sciutto, Manual and Reference Guide, version
2.8.4a www.fisica.unlp.edu.ar/auger/aires.
[17] D. Heck et al., Report FZKA6019
Forschungszentrum Karlsruhe, 1998