In the field of radiodetection in astroparticle physics, the Codalema experiment is devoted to the detection of ultra high energy cosmic rays by the radio method. The main objective is to study the features of the radio signal induced by the development of extensive air showers (EAS) generated by cosmic rays in the energy range of 10 PeV-1 EeV. After a brief presentation of the recent results of UHECR, a description the CODALEMA II and III experiments characteristics is reported. Next, a study of the response in energy of the radio-detection method is presented. The analysis of the CODALEMA II experiment data shows that a strong correlation can be demonstrated between the primary energy and the electric field amplitude on the axis shower. Its sensitivity to the shower characteristics suggests that energy resolution of less than 20% can be achieved. It suggests also that, not only the geomagnetic emission, but also another contribution proportional to all charged particles number in the shower, could play a significant role in the radio emission measured by the antennas (as Askaryan charge-excess radiation or a Cherenkov like coherence effect). Finally, the transition from small-scale prototype experiments, triggered by particle detectors, to large-scale antenna array experiments based on standalone detection, has emerged new problems. These problems are related to the localization, recognition and the suppression of the noisy background sources induced by human activities (such as high voltage power lines, electric transformers, cars, trains and planes) or by stormy weather conditions (such as lightning). In this talk, we focus on the localization problem which belongs to a class of more general problems usually termed as inverse problems. 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 already published results of radio-detection experiments like : CODALEMA 3 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. Two approaches have been used as the existence of solutions degeneration and the bad conditioning of the mathematical formulation of the problem. A comparison between the experimental results and the simulations have been made, to support the mathematical studies. Many properties of the non-linear least square function are discussed such as the configuration of the set of solutions and the bias.

1. Why radiodetection of UHECR still matters ?
Lessons learned from the CODALEMA II & III experiments
+ Measurement of the primary particle energy
+ Localization of the radio emitting sources
+ Towards the identification of the primary particle
Dr. Ahmed REBAI
Thursday, February 20, 2014
Das Karlsruher Institut für Technologie
Laboratoire de physique subatomique
et des technologies associées Nantes

2. This work has been made a part under a grant
from Region Pays de la Loire France and
CNRS/IN2P3 (Centre national de la recherche
scientifique).
I would also like to thank Dr. Andreas Haungs
and Dr. Tim Huege for the invitation
and Dr. Sabine Bucher for her administrative
collaboration
1

3. Introduction
Recent Results in Ultra High Energy Cosmic
Rays physics and their interpretation
Radiodetection of UHECR
The CODALEMA Experiment
●The measurement of the energy of the primary
particle
Localization of the radio-emission source
2

4. Ultra High Energy Cosmic Rays puzzles
Many regions :
- Low energies
- Knees
- ankle
Many origins :
- solar
- galactic
-extragalactic and ?
Many techniques :
 Direct: satellites, balloons
 Indirect: ground base arrays (fluorescence,
particle detectors and antennas)
Open questions:
Origin ?
Nature ?
Limit ?
Power law:
Flux ~ E-2.7
Gaisser T. K. et al. Front. Phys. China. 8 (2013)
Transition
???
Knee
GZKorSOMETHINGELSE?
3

5. Current results on the Ultra High Energy Cosmic
Rays Physics
4

6. UHECR Origins
Question: How to reach
100 EeV (1020
eV)?
Cosmological origin (top-down
mechanism)

Massive particles decay (M.c2
~ 1024
eV)
 Signature: photons/neutrinos
 Excluded by the Pierre Auger
Experiment (The Astrophysical
Journal Letters, 755:L4 (7pp), 2012
August 10)
Astrophysic origin (bottom-up mechanism)
 Accélération de Fermi des particules chargées (Fermi: Phys. Rev. 75, 1169,
1949) => Limite ~ 1018
eV
 Proximité d'objets astrophysiques
=> Now, need to find sources
5

7. Sky maps @ Ep
>55 EeV
UHECR sources
South hemisphere : Auger North hemisphere : TA/Hires
TA/Hires : directional correlation 44%
(Astrophys.J. 757 (2012) 26)
Auger :increase of statistics and
decrease of statistical correlation (61%
to 33%)
(Science 318 (2007) no. 5852, 938–943)
Propagation: effect of the
Intergalactic magnetic fields?
6

8. UHECR propagation : GZK effect
During the 90s => disagreement
between AGASA (Japan)-Hires1 (USA)
experiments
2008: GZK cut confirmed
by TA/Hires (Phys, Rev. Lett. 101
(2008) 061101)
by Auger in 2010 (Phys. Lett. B 685
(2010) 239–246)
Interaction of UHECR with the Cosmic Microwave Background CMB
Auger
=>limit the observable universe at 100 Mpc
=> depends on the primary nature
=> @3.119
eV : Max energy of acceleration or propagation ?
7

9. UHECR Nature
Auger : Heavy composition favoured
(Phys.Rev.Lett.104:091101,2010)
TA/Hires : lightening of the
composition in function of the energy
(Phys.Rev.Lett.104:161101,2010)
Xmax
: depth of the shower
maximum development =>
related to the nature of the
primary
Difficulties on measurements and interpretations
and strongly increasing cross-section
8

10. Proton interaction cross section @57TeV
(Phys.Rev.Lett. 109 (2012) 062002)
Constrained the hadronic models (QGSJET, Sibyll, Epos) used in particle physics
9

11. UHECR detection methods
Particle detection on the
ground :
 Cerenkov detectors
 Scintillators
Detection of the
fluorescence light
Advantages Disadvantages
Ground based
detectors
Duty cycle near to 100% Dependence on hadronic models
Deployed large surfaces > 1000 km2
Fluorescence
telescopes
Low dependence on
hadronic models
Large volume detection
Low duty cycle near to 10%
10

12. Spectrum remains understood @UHE
Ill-defined chemical composition @UHE
Unknown astrophysical origin @UHE
Low statistics at extreme energies
Radio-detection of Extensive Air Shower:
A complementary detection method in evaluation
11

13. 13
γγ
ee e e
γ γee
Electromagnetic cascadePions cascadeNucleons cascade
γe γe γe
nπ°
2n(Κ±
π±
...hadrons)
Near shower axis
Hadrons
π±
desintegration
µ µ µµ
~90% of γ (>50 keV) ~9% electrons (>250 keV)
~1% µ (>1 GeV) small hadron fraction
Sol
z First interaction
Xmax Nmax
Extensive Air Shower 12

14. Radio-detection of EAS
9% electrons/positrons
Radio emission mechanisms
Geomagnetic effect => deviation of electrons/positrons under
the geomagnetic field effect => bipolar emission, transverse
current, synchrotron emission => linear polarisation
Askaryan Negative charge-excess radiation => temporal
variation of the negative charge excess => monopolar emission
=> radial field
+ Cherenkov-like coherence effect? (2010)
Forme du signal radio
Distance
shower-antenna
Radio signal shape
13

15. Unipolar pulse models :
● REAS1, REAS2, ReAIRES
Bipolar pulse models:
● MGMR, REAS3, SELFAS2
Radio emission theoretical models
Comprehension of the radio signal => Intense theoretical efforts => several
models available :
● Microscopic : use of CORSIKA and AIRES codes
● Macroscopic : simplistic assumptions on the EAS phenomenology
+The total electric field
14

16. MHz radio-detection experiments
EASIER
16
AUGER
TREND
AERA
LOFAR
LOPES
15

17. CODALEMA experiment @Nançay
• Radio-astronomy environment
• Electromagnetic quiet
environment
• Far from big cities => Non-existence of
strong transmitter
• But no possibility for end-to-end
calibration.
~1 Km
~2 Km
16

18. CODALEMA Actual Setup
Array of 24 short antennas
21 polar. E-W
Array of 17 scintillator
Experiment Trigger
Energy Estimator
Shower core location
Arrival direction
Decametric array
18 groups of 8 log-
periodic phased
antennas
30 self-
triggered
Antennas
2 polar.
(E-W + N-S)
Objective : 60
on 1.5 km24 detector arrays
17

19. CODALEMA II : method of detection
Slave trigger mode
recording of the radio sky
state
recording of the
radio sky state
18

20. Radio transient selection
Recording radio waveforms in 0-200 MHz:
Sampling frequency 1Gs / s on 2520 Points
19

21. Filtered radio transients
Event = 2 physical quantities/antenna (transient maximum amplitude and time of
maximum transient )
Corrections :
+Cables delays
+Attenuation
+Antennas gain
Digital filtering in
the [20-83] MHz
band
20

22. Reconstruction of physical observables
Hypothesis : a plane wave front with
equation :
u.x+v.y+w.z+cte = 0
(u,v,w) normal vector coordinates
Electric field profile
Arrivals direction : θ zenith angle ϕ azimuth angle
Allan model: exponential function with 4
parameters
Ε = ε0
exp(-d/d0
(xc
,yc
))
=> ε0
electric field on the shower axis
=> d0
radio shower lateral distance
=> (xc
,yc
) shower core on the ground
Event = 2 physical quantities
- Maximum amplitude of the transient
- Time of maximum transient
21

23. CODALEMA Results
CODALEMA model |vXB|EW
=>Est-West Polarisation of the electric field
=> Signal amplitude ~ |vXB|EW
D. Ardouin et al. Astro.ph 31 2009
 Deficit of events near the magnetic axis
Evidence of a geomagnetic effect in the electric field generation
mechanisms
22
B

24. The primary particle energy measurement with ε0
radio observable
+ A. Rebai et al. ArXiv:1210.1739, Oct. 2012 (submitted to Astro.Ph)
+ ARENA2012, AIP Conf. Proc. 1535, 99-104 (2013)
ε0
Primary particle
energy Ep
Correction factors:
+ Geomagnetic emission?
+ Askaryan charge-excess radiation?
+ Cherenkov-like coherence effect?
23

25. Study of correlation between Ep
and ε0
E0 ~ Ep^alpha avec alpha ~
1.0=> dépendance linéaire
Corrélation dépend :
Erreurs sur Ep
Erreurs sur e0
Fit function :
3 assumptions :
* Linear-linear fit
* Gaussian error
* Independence relation between ε0
and Ep
Goodness of fit study => Standard deviation
of the distribution of Ep
and E0
residual
Existence of outlier events
σ(Ep
)/Ep
~ 30%
σ(ε0
)/ε0
~ 22% (Monte-Carlo)
(Only statistical errors No
systematics for this study)
24

26. 1st
correction factor : geomagnetic emission
Geomagnetic effect :
ε0
~ Ep
.|(vXB)EW
|
ε'0
~ Ep
.|(v'XB)EW
|
=> ε0
→ ε0
/|(vXB)EW
|
Overestimation of the energy of the
events near to the geomagnetic axis
But no effect in Ep
=> The existence of a second contribution
25

27. Additional mechanism
An simple assumption :
Contribution proportional to the energy
(i.e. total charge produced in the shower)
ε0
~ Ep
.|(vXB)EW
| + Ep
.c
=> ε0
→ ε0
/ ( |(vXB)EW
| + c )
0 < |(vXB)EW
| < 1
c > 0
 Best resolution for |(vXB)EW
| close to 1 and c=0
=> Geomagnetic effect dominance
 For small |(vXB)EW
| => improvement in
resolution when c increase => Ep
.c dominates
 70 events per window
Qualitycriteria
26

28. Additional mechanism
Can we combine this new contribution to an
existing electric field emission mechanism ?
 Shower axis perpendicular component
 Shower axis parallel component
ε0
~ Ep
.|(vXB)EW
|+Ep
.c.|sin(θ).sin(ϕ)|
Resolution degradation=> we reject the hypothesis
Interpretations with the current data set of 315 events:
ε0
~ Ep
.|(vXB)EW
|+Ep
.c
1st
term depends on the geomagnetic effect
2nd
term depends on the shower total charge => Askaryan charge-excess
mechanism ?
ε0
~ Ep
.|(vXB)EW
|(1+c/|(vXB)EW
|+d/|(vXB)EW
|2
+ ….)
Analogy with a magnetic field deflection created by a dipole =>
deflection of charged particles increases with |(vXB)EW
| => distance
between the particles increases => s there a limit imposed by the
coherence of the emission??
27

29. Summary of the analysis
Our interpretation with only 315 events :
ε0
~ Ep
.|(vXB)EW
| + Ep
.c
1st
term depends in the geomagnetic emission
2nd term depends in the shower charge => Askaryan negative charge-excess
mechanism ?
ε0
~ Ep
.|(vXB)EW
|.(1+c/|(vXB)EW
|)
28

30. Energy resolution
Monte Carlo: Construction (E0
, EP
)
distributions for fixed (∆E0
, ∆EP
)
=> Construction of the abacus σ(EP
-
E0
)/EP
=> σ(E0
) ~ 20%
=> Adopting a better parametrization RLDF (Gaussian)
=> Improving the analysis chain + including systematic errors
Radio energy spectrum after calibrationRadio energy
“Particle” energy
29

31. Localization of radio emission sources
➢ Motivations
➢ Experimental observations
➢ Mathematical framework
➢ Ill-posedness formulation
➢ Convex hull concept
30
A. Rebai et al. arXiv:1208.3539
+ ARENA2012, AIP Conf. Proc. 1535, 99-104 (2013)

32. Towards a self-radio trigger
In 3 years : 2030 eventsIn 4 days : 107
events
Antenna trigger Antennas triggered by scintillators
Noise sources appearance
Objective : avoid to trigger the antenna array by another array
Transition from prototype experiments triggered by a particle detector arrays to self-
triggered antenna arrays deployed on large surfaces
31

33. Study of the radio interference (RFI)
Need to accurately locate interference sources to remove it =>
spherical emission assumption
Anthropogenic sources:
Aircraft, power lines, transformers,
electric motors ...
Natural sources: Atmospheric
storm discharges...
This is the crucial problem to be fixed in order to make
radiodetection auto-triggering mode (radio trigger)!
32

34. ● The near field and far field are regions of the radio emission around the source
● Near field => spherical wavefront (airplanes, electric transformers …)
● Far field => planar wavefront (the sun during solar flare periods cf. J. Lamblin for the CODALEMA
collab.. Radiodetection of astronomical phenomena in the cosmic ray dedicated CODALEMA
experiment. In Proceeding of the 30th
ICRC 2008)
Why a spherical wavefront hypothesis ?
source
Near field Far field
2*D2
/λ
33
Near to Lofar array
EDF electric transformer
Electric gate of a farm

35. RFI localization
►Correct localization expected for a
spherical reconstruction:
● immobile sources
● Large number of detected events
+ Source/array position effect
►► localization problem ?
►►numerical simulation
CODALEMA III
TREND
AERA
34

36. Model and simulation of the spherical wave
Test array Spherical Propagation
1-Source at distance Rs
2- Arrival times distribution Computing
3-Introduction of errors
4-Generation of 1000 events
5-Reconstruction of the emission centre by
minimizing an objective function with Simplex
and Levenberg Marquardt (LVM) algorithms
35

37. Time resolution effect
Simplex algorithm: direct search (no
gradient calculation)
Effect of temporal error :
● σt
=0 ns : good localization with a
statistical estimator
● σt
=3, 10 ns : Degradation of
reconstruction: spread of the Rs
distribution
σt
=0 ns
σt
=3 ns
σt
=10 ns
Elongated distribution points : but θ, ϕ =>
good estimation !!!
36

38. LVM algorithm
Sensibility in initial conditions :
 Rini
>> Rs
=> False results

Small modification in Rini
=>
unpredictable results
=> Need to refine the analysis: apply other
selection criteria
Initial conditions effect37

39. Simulation conclusions
When the temporal resolution increases: :
 Reconstruction degradation=> distribution points spread
 Bias appearance
==> Need for a detailed study of this spherical minimization
But the temporal resolution is not the only factor
● Source position relative to the array :
✔ Source outside the array => bad reconstruction
✗ Source inside the array => good reconstruction
 Localization sensitive to the minimization algorithms (simplex and
Levenberg-marquardt, linear search)
 Initial conditions dependence
 Multiple solutions (degeneration)
38

40. The followed approach
The minimization algorithms based on f descent direction
search to reach the minimum (local or global)
● Solution = function minimum found
● Global minimum => convex function => unicity of the
solution
● Local minimum => non-convexe function=>
degeneration of the solution
● Need to watch the first differential (minima)
and second (convexity)
Study the convexity of f:
Jacobian and Hessian
● Coercivity of the objective function
● Sylvester criterion
● Ill-posed problem in the sense of Hadamard
● Ill-conditioned problem
Classification of the localization
problem
in a more general framework
39

41. Convexity of the objective function
Reasoning on the principal Hessian minor of order 4:
Sylvester Criterion : f is convex ⇔ All Hessian minors are positive
=> a negative minor => f is not convex => existence of several minimum
We choose
symbolic calculus:
With M is the Minkowski matrix
40

42. Distribution of f minima
Search of minima:
=> Importance of the barycentre of
the spatio-temporal variations of
tagged antennas
 Existence of a privileged
direction of barycentre-source
● Importance of the closest
antennas to the source
Analogy with the barycentre formula:
Difficulties in resolution => use of an empirical method
symbolic calculus:
=>
41

43. 1D array case
3 hypothesis :
● Source inside the array
● Source outside the array
● Source outside the array but off-line
Source interne
Particular role for the segment => convex hull
Role of the nearest antenna to the source
Source externe
42

44. 1D array case (external source)
Source outside the segment and off-line
Constraints => light cones
Unconstrained Minimization results =>
Solutions lie on a half-line
43

45. 2D array case (internal source)
Convex hull role
Real case : source inside the antenna
array
44

46. 2D array case (external source)
Presence of local minima
Distribution on a line
Role of the convex hull
Existence of privileged half-line
Real case : source on the ground and
outside the array convex hull (RFI
sources)
45

47. 2D array case : source in the sky
Good estimation of the arrivals direction
source
Antenna array on the ground
46

48. Ill-conditioned problem47
Condition number : measure how sensitive a function is to changes or errors in the input
κ(H)=||H-1
||*||H|| ~ λmax
(H)/λmin
(H) (eignevalues of H)
(F. Delprat-Jannaud and P. Lailly, Ill-Posed and Well-Posed Formulation of the Reflection Travel Time
Tomography Problem, J. of Geophysical Research, vol. 98, No. B4, p. 6589, April 10, 1993.)
Low condition number (~1) => well-conditioned problem
High condition number (>>1) => ill-conditioned problem

49. Classification of the localization problem
(1902) A physics problem is ill-posed if:
1 – no solution
or
2 – has many solutions
or
3 – the solution has a strong dependence
In the different parameters of the problem (initial conditions,
boundary conditions, data errors)
(2) et (3) => localization problem is ill-posed in the case of a
source external to the tagged antenna convex hull
localization problem is well posed in the case of an internal
source to the convex hull of antenna
Jacque Hadamard
48

50. Best method to circumvent the problem is not yet found
R(estimated)=4000 m
R(estimated)=9700 m
Direct search attempt
1 - Quantification of the phase space in a cubic grid: step ~ 50 m in space, time
step ~ 10 ns
2 - Using the planar fit to determine the search directions in the phase space
3 - Calculate the value of f on a grid
4 - Search absolute minimum
49

51. Best method to circumvent the problem is not yet found
Direct search attempt
But in the case of a source in the sky => Bias
50
Source on the ground
PhD of Diego Torres

52. And now EAS ?
Observation of a time shift relative to the
plane wavefront assumption
But only one realization per event ►
Problem for a statistical estimation of
the source position
Hypothesis : the signal maximum amplitude linked
To the development region of the shower (Xmax
?)
51
Dimensions of the charged particles pancake:
+ Longitudinal spreading ~ few meters (J. Linsley, "Thickness of the
particle swarm in cosmic-ray air showers," Journal Phys G vol. 12,
No. 1., P. 51, 1986)
+ Lateral spreading: limited by the effect of coherence => band [23-
83] MHz => ~ 3 - ~ 13 m
Apparent point source localized in space => spherical emission
Data

53. 3D detectors in the water and in the ice
Volume detector array (not a new idea) : Askaryan during the 70s and DUMAND
array during 90s.
Slide from ARENA 2010 presentation “H. Ralf”
52

54. 3D detectors in the water and the ice53

55. Exposure of high altitude antennas (mountains, ballons, satellite) P. Motloch, N.
Hollon, P. Privitera, On the prospects of ultra-high energy cosmic rays detection by
high altitude antennas (arXiv:1309.0561) accepted in Astro. Part.
3D convex hull: Towards 3D antenna array
Our idea
A. Rebai and Ramzi Boussaid Formulation of the emission sources localization problem in the case
of a selftriggered radio-detection experiment: Between Ill-posedness and Regularization (to be submitted)
Anita
Forte satellite
54

56. 55

57. Conclusions
+ Energy analysis was improved
indication of the presence of several radio emission mechanisms
Estimation of the energy resolution of ~ 20%
+ Prespective: More accurate LDF (Gaussian) => resolution enhancement
+ RFI observations and simulations => difficulties in interpretations
Study the objective function
Role of the convex hull
Role of the half-line that binds the antennas barycentre and the source
We need to work with mathematicians (multidisciplinary approach)
56