# Calibration of the Logarithmic-Periodic Dipole Antenna (LPDA) Radio Stations
at the Pierre Auger Observatory using an Octocopter

A. Aab  P. Abreu  M. Aglietta  I. Al Samarai  I.F.M. Albuquerque  I. Allekotte
A. Almela  J. Alvarez Castillo  J. Alvarez-Muñiz  G.A. Anastasi  L.
Anchordoqui  B. Andrada  S. Andringa  C. Aramo  F. Arqueros  N. Arsene  H.
Asorey  P. Assis  J. Aublin  G. Avila  A.M. Badescu  A. Balaceanu  F. Barbato
R.J. Barreira Luz  J.J. Beatty  K.H. Becker  J.A. Bellido  C. Berat  M.E.
Bertaina  X. Bertou  P.L. Biermann  P. Billoir  J. Biteau  S.G. Blaess  A.
Blanco  J. Blazek  C. Bleve  M. Boháčová  D. Boncioli  C. Bonifazi  N. Borodai
A.M. Botti  J. Brack  I. Brancus  T. Bretz  A. Bridgeman  F.L. Briechle  P.
Buchholz  A. Bueno  S. Buitink  M. Buscemi  K.S. Caballero-Mora  L. Caccianiga
A. Cancio  F. Canfora  L. Caramete  R. Caruso  A. Castellina  G. Cataldi  L.
Cazon  A.G. Chavez  J.A. Chinellato  J. Chudoba  R.W. Clay  A. Cobos  R.
Colalillo  A. Coleman  L. Collica  M.R. Coluccia  R. Conceição  G. Consolati
F. Contreras  M.J. Cooper  S. Coutu  C.E. Covault  J. Cronin  S. D’Amico  B.
Daniel  S. Dasso  K. Daumiller  B.R. Dawson  R.M. de Almeida  S.J. de Jong  G.
De Mauro  J.R.T. de Mello Neto  I. De Mitri  J. de Oliveira  V. de Souza  J.
Debatin  O. Deligny  C. Di Giulio  A. Di Matteo  M.L. Díaz Castro  F. Diogo
C. Dobrigkeit  J.C. D’Olivo  Q. Dorosti  R.C. dos Anjos  M.T. Dova  A.
Dundovic  J. Ebr  R. Engel  M. Erdmann  M. Erfani  C.O. Escobar  J. Espadanal
A. Etchegoyen  H. Falcke  G. Farrar  A.C. Fauth  N. Fazzini  F. Fenu  B. Fick
J.M. Figueira  A. Filipčič  O. Fratu  M.M. Freire  T. Fujii  A. Fuster  R.
Gaior  B. García  D. Garcia-Pinto  F. Gaté  H. Gemmeke  A. Gherghel-Lascu
P.L. Ghia  U. Giaccari  M. Giammarchi  M. Giller  D. Głas  C. Glaser  G. Golup
M. Gómez Berisso  P.F. Gómez Vitale  N. González  A. Gorgi  P. Gorham  A.F.
Grillo  T.D. Grubb  F. Guarino  G.P. Guedes  M.R. Hampel  P. Hansen  D. Harari
T.A. Harrison  J.L. Harton  A. Haungs  T. Hebbeker  D. Heck  P. Heimann  A.E.
Herve  G.C. Hill  C. Hojvat  E. Holt  P. Homola  J.R. Hörandel  P. Horvath  M.
Hrabovský  T. Huege  J. Hulsman  A. Insolia  P.G. Isar  I. Jandt  S. Jansen
J.A. Johnsen  M. Josebachuili  A. Kääpä  O. Kambeitz  K.H. Kampert  I. Katkov
B. Keilhauer  N. Kemmerich  E. Kemp  J. Kemp  R.M. Kieckhafer  H.O. Klages  M.
Kleifges  J. Kleinfeller  R. Krause  N. Krohm  D. Kuempel  G. Kukec Mezek  N.
Kunka  A. Kuotb Awad  D. LaHurd  M. Lauscher  R. Legumina  M.A. Leigui de
Oliveira  A. Letessier-Selvon  I. Lhenry-Yvon  K. Link  D. Lo Presti  L. Lopes
R. López  A. López Casado  Q. Luce  A. Lucero  M. Malacari  M. Mallamaci  D.
Mandat  P. Mantsch  A.G. Mariazzi  I.C. Mariş  G. Marsella  D. Martello  H.
Martinez  O. Martínez Bravo  J.J. Masías Meza  H.J. Mathes  S. Mathys  J.
Matthews  J.A.J. Matthews  G. Matthiae  E. Mayotte  P.O. Mazur  C. Medina  G.
Medina-Tanco  D. Melo  A. Menshikov  K.-D. Merenda  M.I. Micheletti  L.
Middendorf  I.A. Minaya  L. Miramonti  B. Mitrica  D. Mockler  S. Mollerach
F. Montanet  C. Morello  M. Mostafá  A.L. Müller  G. Müller  M.A. Muller  S.
Müller  R. Mussa  I. Naranjo  L. Nellen  P.H. Nguyen  M. Niculescu-Oglinzanu
M. Niechciol  L. Niemietz  T. Niggemann  D. Nitz  D. Nosek  V. Novotny  H.
Nožka  L.A. Núñez  L. Ochilo  F. Oikonomou  A. Olinto  M. Palatka  J. Pallotta
P. Papenbreer  G. Parente  A. Parra  T. Paul  M. Pech  F. Pedreira  J. Pȩkala
R. Pelayo  J. Peña-Rodriguez  L. A. S. Pereira  M. Perlín  L. Perrone  C.
Peters  S. Petrera  J. Phuntsok  R. Piegaia  T. Pierog  P. Pieroni  M. Pimenta
V. Pirronello  M. Platino  M. Plum  C. Porowski  R.R. Prado  P. Privitera  M.
Prouza  E.J. Quel  S. Querchfeld  S. Quinn  R. Ramos-Pollan  J. Rautenberg  D.
Ravignani  B. Revenu  J. Ridky  M. Risse  P. Ristori  V. Rizi  W. Rodrigues de
Carvalho  G. Rodriguez Fernandez  J. Rodriguez Rojo  D. Rogozin  M.J.
Roncoroni  M. Roth  E. Roulet  A.C. Rovero  P. Ruehl  S.J. Saffi  A. Saftoiu
F. Salamida  H. Salazar  A. Saleh  F. Salesa Greus  G. Salina  F. Sánchez  P.
Sanchez-Lucas  E.M. Santos  E. Santos  F. Sarazin  R. Sarmento  C.A. Sarmiento
R. Sato  M. Schauer  V. Scherini  H. Schieler  M. Schimp  D. Schmidt  O.
Scholten  P. Schovánek  F.G. Schröder  A. Schulz  J. Schumacher  S.J. Sciutto
A. Segreto  M. Settimo  A. Shadkam  R.C. Shellard  G. Sigl  G. Silli  O. Sima
A. Śmiałkowski  R. Šmída  G.R. Snow  P. Sommers  S. Sonntag  J. Sorokin  R.
Squartini  D. Stanca  S. Stanič  J. Stasielak  P. Stassi  F. Strafella  F.
Suarez  M. Suarez Durán  T. Sudholz  T. Suomijärvi  A.D. Supanitsky  J. Swain
Z. Szadkowski  A. Taboada  O.A. Taborda  A. Tapia  V.M. Theodoro  C.
Timmermans  C.J. Todero Peixoto  L. Tomankova  B. Tomé  G. Torralba Elipe  P.
Travnicek  M. Trini  R. Ulrich  M. Unger  M. Urban  J.F. Valdés Galicia  I.
Valiño  L. Valore  G. van Aar  P. van Bodegom  A.M. van den Berg  A. van Vliet
E. Varela  B. Vargas Cárdenas  G. Varner  R.A. Vázquez  D. Veberič  I.D.
Vergara Quispe  V. Verzi  J. Vicha  L. Villaseñor  S. Vorobiov  H. Wahlberg
O. Wainberg  D. Walz  A.A. Watson  M. Weber  A. Weindl  L. Wiencke  H.
Wilczyński  T. Winchen  M. Wirtz  D. Wittkowski  B. Wundheiler  L. Yang  D.
Yelos  A. Yushkov  E. Zas  D. Zavrtanik  M. Zavrtanik  A. Zepeda  B.
Zimmermann  M. Ziolkowski  Z. Zong  and F. Zuccarello

###### Abstract

An in-situ calibration of a logarithmic periodic dipole antenna with a
frequency coverage of $30\,\rm{MHz}$ to $80\,\rm{MHz}$ is performed. Such
antennas are part of a radio station system used for detection of cosmic ray
induced air showers at the Engineering Radio Array of the Pierre Auger
Observatory, the so-called Auger Engineering Radio Array (AERA). The
directional and frequency characteristics of the broadband antenna are
investigated using a remotely piloted aircraft carrying a small transmitting
antenna. The antenna sensitivity is described by the vector effective length
relating the measured voltage with the electric-field components perpendicular
to the incoming signal direction. The horizontal and meridional components are
determined with an overall uncertainty of $7.4^{+0.9}_{-0.3}\,\rm{\%}$ and
$10.3^{+2.8}_{-1.7}\,\rm{\%}$ respectively. The measurement is used to correct
a simulated response of the frequency and directional response of the antenna.
In addition, the influence of the ground conductivity and permittivity on the
antenna response is simulated. Both have a negligible influence given the
ground conditions measured at the detector site. The overall uncertainties of
the vector effective length components result in an uncertainty of
$8.8^{+2.1}_{-1.3}\,\rm{\%}$ in the square root of the energy fluence for
incoming signal directions with zenith angles smaller than
$60\rm{{}^{\circ}}$.

##  1 Introduction

When ultrahigh-energy cosmic rays (UHECRs) hit the Earth, they collide with
air nuclei and create a particle cascade of millions of secondary particles, a
so-called air shower. The atmosphere acts thereby as a giant calorimeter of
${\sim}11$ hadronic interaction lengths. Instrumentation of such a giant
detector volume is challenging in every respect, especially concerning
readout, calibration and monitoring. Well-established solutions are stochastic
measurements of the remaining secondary particles at ground level and direct
detection of fluorescence light emitted from air molecules excited by the
particle cascade. Both techniques are successfully applied in the Pierre Auger
Observatory in Argentina, covering $3000\,\rm{km^{2}}$ with $1660$ water-
Cherenkov detectors and $27$ telescopes for detection of fluorescence light
[1].
In recent years, measurement of radio emission from air showers in the
megahertz (MHz) regime has become a complementary detection technique [2, 3,
4, 5, 6, 7, 8]. For this, the Pierre Auger Observatory was extended by $153$
radio stations, the so-called Auger Engineering Radio Array (AERA). These
antenna stations at ground level provide information on the radio signal and
are used to reconstruct the electric field generated by an air shower.
Two mechanisms contribute to coherent radio emission from air showers, namely
the geomagnetic effect induced by charged particle motion in the Earth’s
magnetic field [2, 9, 10, 11, 12, 13] and the time varying negative charge
excess in the shower front. The charge excess is due to the knock-out of
electrons from air molecules and annihilation of positrons in the shower front
[14, 15, 16, 17, 18]. The radio emission can be calculated from first
principles using classical electrodynamics [19, 20, 21, 22]. The emission
primarily originates from the well-understood electromagnetic part of the air
shower. Thus, the theoretical aspect of radio measurements is on solid grounds
[7].
As the atmosphere is transparent to radio waves, the radio technique has a
high potential for precision measurements in cosmic-ray physics. Correlation
of the strength of the radio signal with the primary cosmic-ray energy has
meanwhile been demonstrated by several observatories [23, 24, 25, 26, 27].
Furthermore, the radiation energy, i.e., the energy contained in the radio
signal has been determined [27]. It was shown that the radio energy resolution
is competitive with the results of particle measurements at ground level.
Furthermore using above-mentioned first-principle calculations, a novel way of
a stand-alone absolute energy calibration of the atmospheric calorimeter
appears feasible [26].
In all these considerations, the antenna to detect the electric field and a
thorough description of its characteristics is of central importance. Precise
knowledge of the directional antenna characteristics is essential to
reconstruct the electric field and therefore enables high quality measurements
of the cosmic-ray properties. For a complete description of the antenna
characteristics an absolute antenna calibration needs to be performed. The
uncertainties of the absolute calibration directly impact the energy scale for
air shower measurements from radio detectors. Therefore, a central challenge
of the absolute antenna calibration is to reduce the uncertainties of the
antenna characteristics to the level of $10\,\rm{\%}$ which is a significant
improvement in comparison with the uncertainties obtained in calibration
campaigns at other radio detectors [28, 29, 30].
In this work, the reconstruction quality of the electric-field signal from the
measured voltage trace, which includes the directional characteristics of the
antenna and dispersion of the signal owing to the antenna size, is
investigated. All information are described with the vector effective length
$\vec{H}$, a complex measure that relates the measured voltage to the incoming
electric field. One antenna of the subset of $24$ radio stations equipped with
logarithmic periodic dipole antennas (LPDAs) is investigated here exemplarily.
This antenna is representative of all the LPDAs which are mechanically and
electrically identical at the percent level [31]. While the low-noise
amplifier attached to the antenna was included in the signal chain during the
calibration, amplifiers and the subsequent electronics of all radio stations
have been characterized individually. The LPDA antennas have the advantage of
low sensitivity to radio waves reflecting from the ground which makes them
largely independent of potentially changing ground conditions.
The LPDA antennas have been studied before and a first absolute calibration of
one signal polarization was performed in $2012$ giving an overall systematic
uncertainty of $12.5\,\rm{\%}$ [32]. In comparison to the first absolute
calibration of AERA, in this paper a new absolute calibration is presented
using a new setup enabling a much more dense sampling of the arrival
directions, more field polarization measurements, and an extended control of
systematic effects including the full repetition of calibration series. To
ensure far-field measurements, instead of the previously used balloon, a drone
was employed, carrying a separate signal generator and a calibrated
transmitting antenna.
This work is structured as follows. Firstly, a calculation of the absolute
value of the vector effective length $|\vec{H}|$ of the LPDA is presented.
Then, the LPDA antenna and the calibration setup are specified. In the next
section the calibration strategy is presented using one example flight where
$|\vec{H}|$ is measured on site at the Pierre Auger Observatory at one of the
radio stations. The main section contains detailed comparisons of all the
measurements with the calculated vector effective length and the determination
of the uncertainties in the current understanding of the antenna. Finally, the
influence of the calibration results are discussed in applications before
presenting the conclusions.

##  2 Antenna Response Pattern

This section gives a theoretical overview of the antenna response pattern. The
vector effective length (VEL) is introduced as a measure of the directional
dependent antenna sensitivity. Furthermore, it is explained how the VEL is
obtained for an uncalibrated antenna. For more details refer to [32].

###  2.1 The Vector Effective Length (VEL)

Electromagnetic fields induce a voltage at the antenna output. The antenna
signal depends on the incoming field $\vec{E}(t)$, the contributing
frequencies $f$, as well as on the incoming direction with the azimuthal angle
$\Phi$ and the zenith angle $\Theta$ to the antenna. The relation between the
Fourier-transformed electric field $\vec{\mathcal{E}}(f)$ and the Fourier
transformed observed voltage $\mathcal{U}$ for $\Phi,\Theta,f$ is referred to
as the antenna response pattern and is expressed in terms of the VEL
$\vec{H}$:

$\mathcal{U}(\Phi,\Theta,f)=\vec{H}(\Phi,\Theta,f)\cdot\mathcal{\vec{E}}(f)$
(2.1)

The VEL $\vec{H}$ is oriented in the plane perpendicular to the arrival
direction of the signal and can be expressed as a superposition of a
horizontal component $H_{\phi}$ and a component $H_{\theta}$ oriented
perpendicular to $H_{\phi}$ which is called meridional component:

$\vec{H}=H_{\phi}\vec{e}_{\phi}+H_{\theta}\vec{e}_{\theta}.$ (2.2)

The VEL is a complex quantity $H_{k}=|H_{k}|e^{i\alpha_{k}}$ with
$k=\phi,~{}\theta$ and accounts for the frequency-dependent electrical losses
within the antenna as well as reflection effects which arise in the case of
differences between the antenna and read-out system impedances. Both effects
lead to dispersion of the signal shape.
The antenna response pattern is often expressed in terms of the antenna gain
based on the directional dependence of the received power. With the quadratic
relation between voltage and power, the antenna gain and the absolute value of
the VEL are related by:

$|H_{k}(\Phi,\Theta,f)|^{2}=\frac{c^{2}Z_{R}}{f^{2}4\pi
Z_{0}}G_{k}(\Phi,\Theta,f).$ (2.3)

Here, $f$ is the signal frequency, $c$ is the vacuum speed of light,
$Z_{R}=50\,\rm{\Omega}$ is the read-out impedance, $Z_{0}\approx
120\,\pi\,\Omega$ is the impedance of free space, the index $k=\phi$ or
$\theta$ indicates the polarization, and $\Phi$ and $\Theta$ denote the
azimuth and zenith angle of the arrival direction.

###  2.2 Calculating the Absolute Value of the VEL from a Transmission
Measurement

The antenna characteristics of an uncalibrated antenna under test (AUT) is
determined by measuring the antenna response of the AUT in a transmission
setup using a calibrated transmission antenna. The relation between
transmitted and received power is described by the Friis equation [33]
considering the free-space path loss in vacuum as well as the signal
frequency:

$\frac{P_{r}(\Phi,\Theta,f)}{P_{t}(f)}=G_{t}(f)G_{r}(\Phi,\Theta,f)\left(\frac{c}{f4\pi
R}\right)^{2},$ (2.4)

with the received power $P_{r}$ at the AUT, the transmitted power $P_{t}$
induced on the transmission antenna, the known antenna gain $G_{t}$ of the
calibrated transmission antenna, the unknown antenna gain $G_{r}$ of the AUT,
the distance $R$ between both antennas and the signal frequency $f$.
By considering Eq. (2.3) and Eq. (2.4) the VEL of the AUT in a transmission
setup is then determined by

$|H_{k}(\Phi,\Theta,f)|=\sqrt{\frac{4\pi
Z_{R}}{Z_{0}}}R\sqrt{\frac{P_{r,k}(\Phi,\Theta,f)}{P_{t}(f)G_{t}(f)}}$ (2.5)

###  2.3 Calculating the Absolute Value of the Antenna VEL with separate
Amplifier from a Transmission Simulation

In this work, the NEC-2 [34] simulation code is used to simulate the response
pattern of the passive part of the AUT. With the passive part of the AUT the
antenna without a then following low-noise amplifier stage is meant. These
simulations provide information about the received voltage directly at the
antenna footpoint (AF) which is the location where the signals of all dipoles
are collected and converted to the then following $50\,\Omega$ system of the
read-out system. In the case of an amplifier (AMP) connected to the AF, the
voltage at the output of the AMP is the parameter of interest. The AMP is
connected to the AF using a transmission line (TL). Both, the AMP and the TL,
then constitute the active part of the AUT. In the simulation, mismatch and
reflection effects between the AF, the TL and the AMP, which arise if the
impedances $Z_{j}$ ($j=\rm{AF},\rm{TL},\rm{AMP}$) of two connected components
differ from each other, have to be considered separately. Moreover, the
attenuation of the TL with a cable length $l_{TL}$ as well as the AMP itself
described by the AMP S-parameters have to be taken into account. The
transformation of the received voltage at the AF to the received voltage at
the AMP output is described by the transfer function $\rho$:

$\rho=\frac{1}{\sqrt{r}}\frac{Z_{TL}}{Z_{TL}+Z_{AF}/r}\left(1+\Gamma_{AMP}\right)\frac{e^{(\gamma+i\frac{2\pi
f}{c_{n}})l_{TL}}}{e^{2(\gamma+i\frac{2\pi
f}{c_{n}})l_{TL}}-\Gamma_{AMP}\Gamma_{AF}}\frac{S21}{1+S11}$ (2.6)

with $\Gamma_{AMP}=\frac{Z_{AMP}-Z_{TL}}{Z_{AMP}+Z_{TL}}$ and
$\Gamma_{AF}=\frac{Z_{AF}/r-Z_{TL}}{Z_{AF}/r+Z_{TL}}$. Furthermore, $\gamma$
denotes the attenuation loss along the transmission line, $f$ is the frequency
of the signal, $c_{n}$ denotes the transfer rate inside the TL, and $r$ is the
transfer factor from an impedance transformer at the AF which transforms the
balanced signal of the antenna to an unbalanced signal of a TL. For more
details refer to [32].

##  3 Logarithmic Periodic Dipole Antenna (LPDA)

In this section, the Logarithmic Periodic Dipole Antenna (LPDA) which is used
in a subset of the radio stations of AERA is presented. An LPDA consists of
several $\lambda/2$-dipoles of different lengths which are combined to one
single antenna with the largest dipole located at the bottom and the shortest
dipole at the top of the LPDA. The sensitive frequency range is defined by the
length of the smallest $l_{min}$ and largest $l_{max}$ dipole. The ratio of
the distance between two dipoles and their size is described by $\sigma$ and
the ratio of the dipole length between two neighboring dipoles is denoted by
$\tau$. The four design parameters of the LPDAs used at AERA are $\tau=0.875$,
$\sigma=0.038$, $l_{min}=1470\,\rm{mm}$ and $l_{max}=4250\,\rm{mm}$. These
values were chosen to cover the frequency range from around $30\,\rm{MHz}$ to
$80\,\rm{MHz}$ and to combine a high antenna sensitivity in a broad field of
view using a limited number of dipoles and reasonable dimensions. They lead to
a LPDA with nine separate dipoles. For more details refer to [32]. A full
drawing of the LPDA used at AERA including all sizes is shown in Fig. 1.

Figure 1: Drawing of the Logarithmic Periodic Dipole Antenna (LPDA), units are
millimeter.

Each radio station at AERA consists of two perpendicular polarized antennas
which are aligned to magnetic north with a precision better than
$1\rm{{}^{\circ}}$. The dipoles are connected to a waveguide with the
footpoint at the top of the antenna. The footpoint is connected by an RG58
[35] coaxial transmission line to a low-noise amplifier (LNA) which amplifies
the signal typically by $(18.1\pm 0.2)\,\rm{dB}$. The LNA of the radio station
and used in the calibration setup amplifies the signal by $18.1\,\rm{dB}$. The
amplification is nearly constant in the frequency range $30\,\rm{MHz}$ to
$80\,\rm{MHz}$ and variates at the level of $0.5\,\rm{dB}$. For more technical
details about the LNA refer to [36].

##  4 Calibration Setup

The antenna VEL of the LPDA is determined by transmitting a defined signal
from a calibrated signal source from different arrival directions and
measuring the LPDA response. The signal source consists of a signal generator,
producing known signals, and a calibrating transmitting antenna with known
emission characteristics. The transmission measurement needs to be done in the
far-field region, which is fulfilled to a reasonable approximation at a
distance of $R>2\lambda=20\,\rm{m}$ for the LPDA frequency range of
$30\,\rm{MHz}$ to $80\,\rm{MHz}$.
In a first calibration campaign [32] a large weather balloon was used to lift
the transmitting antenna and a cable to the signal source placed on ground. As
a vector network analyzer was used to provide the source and to measure the
AUT output this transmission measurement allowed to determine both, the VEL
magnitude and phase. This setup has the disadvantages that it requires calm
weather conditions and the cost per flight including the balloon and gas are
high. Moreover, the cable potentially impacts the measurements if not properly
shielded. In this first calibration campaign only the horizontal VEL was
investigated. A new calibration campaign was necessary and a new setup was
developed.
Now, a signal generator as well as a transmission antenna were both mounted
beneath a flying drone, a so-called remotely piloted aircraft (RPA), to
position the calibration source. Hence, the cable from ground to the
transmitting antenna is not needed anymore. Furthermore, the RPA is much less
dependent on wind, and thus it is easier to perform the measurement compared
to the balloon-based calibration. The new calibration is performed with a
higher repetition rate and with more positions per measurement.
During the measurement, the RPA flies straight up to a height of more than
$20\,\rm{m}$ and then towards the AUT until it is directly above it. Finally,
it flies back and lands again at the starting position. A sketch of the setup
is shown at the top of Fig. 2.

Figure 2: (top) LPDA calibration setup. The calibration signal is produced by
a signal generator and radiated by a transmitting antenna. Both the signal
generator and the transmitting antenna are attached underneath a flying drone,
a so-called RPA, to realize far-field conditions during the measurement. On
arrival of the signal at the LPDA, the antenna response is measured using a
spectrum analyzer. The orientation of the RPA is described by the yaw (twist
of front measured from north in the mathematically negative direction), and
the tilt by the pitch and the roll angles. (bottom) Sketch of the expected
(blue arrow) and measured (red arrow) electric field polarization at the LPDA
emitted by the transmitting antenna from the nominal (blue) and measured (red)
position. The real transmitting antenna position is shifted from the nominal
position, e.g., due to GPS accuracy. This misplacement changes the electric-
field strength and polarization measured at the LPDA and, therefore,
influences the measurement.

The RPA used here was an octocopter obtained from the company MikroKopter
[37]. Such an octocopter also has been used for the fluorescence detector [38]
and CROME [39] calibrations. The horizontal octocopter position is measured by
GPS and a barometer provides information about the height above ground. Both
are autonomously recorded nearly each second which enables measurements of the
VEL with good resolution in zenith angle $\Theta$. To obtain further
improvements of the octocopter position determination an optical method using
two cameras taking pictures of the flight was developed. The cameras are
placed on orthogonal axes with a distance of around $100\,\rm{m}$ to the AUT.
Canon Ixus 132 cameras [40] with a resolution of 16 MegaPixel are utilized.
They are set to an autonomous mode where they take pictures every three
seconds. From these pictures the full flight path of the octocopter can be
reconstructed. The method is explained in detail in [41, 42]. Beside the
octocopter position, information about rotation angles (yaw, pitch, roll as
defined in Fig. 2) are recorded during the flight which are later used to
determine the orientation of the transmission antenna with respect to the AUT.
The position of the LPDA station was measured by a differential GPS (DGPS)
(Hiper V system [43]) and is therefore known with centimeter accuracy.
The reference spectrum generator, model RSG1000 produced by the company TESEQ
[44], is used as the signal generator. It continuously produces a frequency
comb spectrum between $5\,\rm{MHz}$ and $1000\,\rm{MHz}$ with a spacing of
$5\,\rm{MHz}$. This signal is further amplified in order to accomplish power
well above background for the measurement using the LPDA. The output signal
injected into the transmission antenna has been measured twice in the lab
using a FSH4 spectrum analyzer from the company Rohde&Schwarz [45] and using
an Agilent N9030A ESA spectrum analyzer [46] both with a readout impedance of
$50\,\rm{\Omega}$.
In an effort to maintain the strict $2.5\,\rm{kg}$ octocopter payload limit, a
small biconical antenna from Schwarzbeck (model BBOC 9217 [47]) is mounted
$0.7\,\rm{m}$ beneath the octocopter. This antenna has been calibrated by the
manufacturer in the frequency range from $30\,\rm{MHz}$ to $1000\,\rm{MHz}$
with an accuracy of $0.5\,\rm{dB}$. This response pattern and its uncertainty
comprise all mismatch effects when connecting a $50\,\rm{\Omega}$ signal
source to such a transmitting antenna. The power received at the LPDA during
the calibration procedure is measured using the same FSH4 spectrum analyzer as
above.
The different VEL components mentioned in Eq. (2.2) are determined by
performing multiple flights in which the orientation of the transmitting
antenna is varied with respect to the AUT. Sketches of the antenna
orientations during the flights are shown on the left side of Fig. 3. The
horizontal component $|H_{\phi}|$ of the LPDA is measured in the LPDA main
axis perpendicular to the LPDA orientation. Then, both antennas are aligned in
parallel for the whole flight. The meridional component $|H_{\theta}|$ is
split into two subcomponents: The other horizontally but perpendicular to
$\vec{e}_{\phi}$ oriented component $|H_{\theta,\mathrm{hor}}|$ and the
vertical component $|H_{\theta,\mathrm{vert}}|$. As the orientation of the
transmission antennas is the main difference between both measurements, the
phase $\alpha_{k}$ with $k=(\theta,\mathrm{hor}),(\theta,\mathrm{vert})$ is
the same. Then, these two subcomponents are combined to the meridional
component $|H_{\theta}|$:

$|H_{\theta}|=\cos(\Theta)|H_{\theta,\mathrm{hor}}|+\sin(\Theta)|H_{\theta,\mathrm{vert}}|.$
(4.1)

Both meridional subcomponents are measured in the axis perpendicular to the
LPDA main axis. Therefore, the transmission antenna needs to be rotated by
$90\rm{{}^{\circ}}$ and the flight path needs to start at the
$90\rm{{}^{\circ}}$ rotated position in comparison to the measurement of
$|H_{\phi}|$. For the case of the $|H_{\theta,\mathrm{vert}}|$ measurement the
transmitting antenna is vertically aligned.
As the receiving power is measured directly at the output of the LPDA
amplifier, all matching effects from connecting a transmission line to the
LPDA footpoint and the LPDA LNA are taken into account. The VEL is calculated
using Eq. (2.5).

##  5 Calibration Strategy

To explain the LPDA calibration strategy a measurement of each of the three
VEL components is presented. Several flights at different days with different
environmental conditions were performed and finally combined to give an
average LPDA VEL. Here, one of the measurements of each VEL component is
presented to show the reconstruction procedure as well as the statistical
precision of the measurements. Furthermore, the corrections taking into
account cable damping, background measurements, misalignments of the
transmitting antenna and shift of the octocopter position are discussed in
detail. Afterwards, an overview of the measurement uncertainties is given.

###  5.1 Example Measurement

In the right diagrams of Fig. 3 the measured VEL components $|H_{\phi}|$,
$|H_{\theta,\mathrm{hor}}|$ and $|H_{\theta,\mathrm{vert}}|$ at the output of
the LPDA LNA as a function of the zenith angle $\Theta$ at $55\,\rm{MHz}$ are
shown. In the left drawings the respective antenna orientations are visible.
The antenna response pattern reveals the following features. For the VEL
component $|H_{\phi}|$, the LPDA is most sensitive in the zenith direction.
The pattern shows a side lobe at around $65\rm{{}^{\circ}}$. For
$|H_{\theta,\mathrm{hor}}|$ the most sensitive direction is the zenith while
at larger zenith angles the sensitivity is strongly reduced. At the zenith the
components $|H_{\phi}|$ and $|H_{\theta,\mathrm{hor}}|$ are equal which is
expected as the antenna orientations are identical. The fluctuations in
$|H_{\theta,\mathrm{hor}}|$ are larger than those in $|H_{\phi}|$ due to the
larger dependencies on the octocopter rotations. When flying towards the
antenna, any acceleration causes a rotation around the pitch angle (Fig. 2)
which does not influence $|H_{\phi}|$. However, for both meridional
subcomponents the pitch angle already changes the transmitting antenna
orientation (Fig. 3). Therefore, it influences both measurements. In
comparison to the other components $|H_{\theta,\mathrm{vert}}|$ is much
smaller. Therefore, the LPDA is marginally sensitive to such a signal
polarization especially at vertical incoming directions. All these results are
frequency dependent.

Figure 3: (left) NEC-2 realization of the setup to simulate the three VEL
components (from top to bottom) $|H_{\phi}|$, $|H_{\theta,\mathrm{hor}}|$ and
$|H_{\theta,\mathrm{vert}}|$. The meridional component $|H_{\theta}|$ is a
combination of $|H_{\theta,\mathrm{hor}}|$ and $|H_{\theta,\mathrm{vert}}|$.
The distance between transmitting and receiving antenna is reduced and the
transmitting antenna is scaled by a factor of $3$ to make both antennas
visible. For clarity, the LPDA and the transmitting antenna (assumed as a
simple dipole) orientations are sketched in the lower right corner of each
picture in the XY-plane as well as in the XZ-plane. (right) Measured VEL as
function of the zenith angle (red dots) of three example flights for the three
VEL components at $55\,\rm{MHz}$.

###  5.2 Corrections

For the raw VEL determined according to Eq. (2.5) corrections for the
experimental conditions have to be applied. The VEL is averaged in zenith
angle intervals of $5\rm{{}^{\circ}}$. This is motivated by the observed
variations in the repeated measurements which were recorded on different days
(see e.g. below Fig. 8). All corrections to the VEL are expressed relative to
the measured raw VEL at a zenith angle of $(42.5\pm 2.5)\rm{{}^{\circ}}$ and a
frequency of $55\,\rm{MHz}$ and are listed in Tab. 1. The corrections are
partly zenith angle and/or frequency dependent.

corrections |  $\Delta|H_{\phi}|$ [%] |  $\Delta|H_{\theta,\mathrm{hor}}|$ [%] |  $\Delta|H_{\theta,\mathrm{vert}}|$ [%]
---|---|---|---
background noise | $-0.1$ | $-0.5$ | $-0.9$
cable attenuations | $+44.4$ | $+44.4$ | $+53.2$
background noise + cable attenuation | $+44.3$ | $+43.7$ | $+51.8$
octocopter influence | $+0.6$ | $+0.6$ | $-0.2$
octocopter misalignment and misplacement | $+0.3$ | – | –
height at take off and landing | $+1.8$ | $+15.8$ | $+5.8$
height barometric formula | $-5.2$ | $-10.2$ | $-2.5$
combined height | $-3.6$ | $-5.4$ | $+1.3$
shift to optical method | $-14.5$ | $-4.8$ | $+0.2$
combined height + shift to optical method | $-14.6$ | $-5.5$ | $-0.3$
all | $+24.6$ | $+36.4$ | $+51.1$
Table 1: VEL corrections taking into account different kinds of corrections
for the three measured VEL components $|H_{\phi}|$,
$|H_{\theta,\mathrm{hor}}|$ and $|H_{\theta,\mathrm{vert}}|$ of the example
flights at a zenith angle of $(42.5\pm 2.5)\rm{{}^{\circ}}$ and a frequency of
$55\,\rm{MHz}$ with $\Delta|H_{k}|$ = $\frac{|H_{k}|-|H_{k,0}|}{|H_{k,0}|}$
and $k=\phi,(\theta,\mathrm{hor}),(\theta,\mathrm{vert})$.

The following paragraphs describe the corrections of the raw VEL at the LPDA
LNA output from the measurement.

####  5.2.1 Background Noise

During the calibration background noise is also recorded. In a separate
measurement the frequency spectrum of the background has been determined and
is then subtracted from the calibration signal spectrum. Typically, the
background noise is several orders of magnitude below the signal strength.
This is even the case for the component $|H_{\theta,\mathrm{vert}}|$ with
lowest LPDA sensitivity. For large zenith angles close to $90\rm{{}^{\circ}}$
and in the case of the component $|H_{\theta,\mathrm{vert}}|$ also for small
zenith angles directly above the radio station, however, the background noise
and the signal can be of the same order of magnitude. In this case, the
calibration signal spectrum constitutes an upper limit of the LPDA
sensitivity. If more than $50\,\rm{\%}$ of the events in a zenith angle bin of
$5\rm{{}^{\circ}}$ are affected, no background is subtracted but half of the
measured total signal is used for calculating the VEL and a $100\%$ systematic
uncertainty on the VEL is assigned.

####  5.2.2 Cable Attenuation

To avoid crosstalk in the LPDA read-out system, the read-out system was placed
at a distance of about $25\,\rm{m}$ from the LPDA. The RG58 coaxial cable
[35], used to connect the LPDA to the read-out system, has a frequency-
dependent ohmic resistance that attenuates the receiving power by a frequency-
dependent factor $\delta$. To obtain the VEL at the LNA output the cable
attenuation is corrected from lab measurements using the FSH4.

####  5.2.3 Octocopter Influence

During the LPDA VEL measurement the transmitting antenna is mounted underneath
the octocopter which contains conductive elements and is powered electrically.
Therefore, the octocopter itself may change the antenna response pattern of
the transmitting antenna with respect to the zenith angle. To find a
compromise between signal reflections at the octocopter and stability during
take off, flight and landing, the distance between transmitting antenna and
octocopter has been chosen to be $0.7\,\rm{m}$. The influence has been
investigated by simulating the antenna response pattern of the transmitting
antenna with and without mounting underneath an octocopter. It is found that
the average gain of the transmission antenna changes by $0.05\,\rm{dB}$ [48].
At a zenith angle of $(42.5\pm 2.5)\rm{{}^{\circ}}$ and a frequency of
$55\,\rm{MHz}$ the octocopter influences the transmitting antenna VEL with
$0.6\,\rm{\%}$.

####  5.2.4 Octocopter Misalignments and Misplacements

Misalignments and misplacements of the octocopter during the calibration
flight have a direct impact on the transmitting antenna position and
orientation changing the signal polarization at the position of the AUT. For
this investigation the orientation of the transmission antenna is assumed to
correspond to a dipole, which holds to a good approximation. The electric
field $\vec{E}_{t}$ emitted from a dipole antenna with orientation
$\hat{A}_{t}$ in the direction $\hat{n}$ in the far-field region is
proportional to $\vec{E}_{t}\sim(\hat{n}\times\hat{A}_{t})\times\hat{n}$, and
the amplitude is given by $|\vec{E}_{t}|=\sin(\alpha)$. Here, $\alpha$
describes the smallest angle between the transmitting antenna alignment
$\hat{A}_{t}$ and the direction from the transmitting antenna to the AUT
denoted as $\hat{n}$ (see lower sketch of Fig. 2). The orientation of the
transmitting antenna $\hat{A}_{t}$ is calculated by an intrinsic rotation of
the initial orientation of the transmitting antenna rotating first by the yaw
angle $G$, then by the pitch angle $P$ and finally, by the roll angle $R$. The
AUT sensitivity $\eta$ to the emitted electric field is then calculated by the
absolute value of the scalar product of the electric field and the AUT
orientation $\hat{A}_{r}$:
$\eta=|\vec{E}_{t}\cdot\hat{A}_{r}|=\sin(\alpha)\cos(\beta)$ with $\beta$
describing the smallest angle between $\vec{E}_{t}$ and $\hat{A}_{r}$ (see
lower sketch of Fig. 2). Finally, the correction factor $\epsilon$ of the
power measured at the AUT is determined by the square of the quotient of the
nominal and the real value of $\eta$. In case of the horizontal component
$|H_{\phi}|$ the VEL is systematically shifted to larger values for all zenith
angles and frequencies due to the octocopter misalignment and misplacement.
The correction factor $\epsilon$ is used to determine the horizontal VEL
$|H_{\phi}|$. In both meridional subcomponents the VEL becomes small at large
zenith angles and strongly dependent on the antenna alignments. Therefore, in
the meridional subcomponents $|H_{\theta,\mathrm{hor}}|$ and
$|H_{\theta,\mathrm{vert}}|$ the effects of the octocopter misalignment and
misplacement are included in the systematic uncertainties.

####  5.2.5 Octocopter Flight Height

The octocopter flight height is determined by a barometer measuring the change
of air pressure $\Delta p$ during the flight. The octocopter software assumes
a linear dependency of $\Delta p$ and the octocopter flight height over ground
$h_{raw}$. Two corrections have been applied to the raw flight height.
Firstly, it was observed that the flight height differs at take off and
landing. Therefore, a linear time dependent correction is applied which
constrains the flight height over ground at take off and landing to zero.
Secondly, AERA is located at a height of about $1550\,\rm{m}$ above sea level.
Therefore, such a linear relation between $\Delta p$ and $h_{raw}$ used by the
octocopter software is not precise enough. A more realistic calculation
considering an exponential model of the barometric formula [49] as well as the
height and latitude dependent gravitation is used to determine the more
precise octocopter height $h_{octo}$. An inverse quadratic relation between
gravitation and the height above sea level with a value at sea level of
$g(0)=9.797\,\rm{\frac{m}{s^{2}}}$ at the latitude of AERA is taken into
account. The raw octocopter height as well as the height after all corrections
of the $|H_{\phi}|$ example flight are shown on the left side of Fig. 4 in
comparison to the octocopter height determined with the optical method. Both
methods agree at the level of $1.1\,\rm{\%}$ in the median. The quotient of
the octocopter height measured by the camera method and by the full corrected
barometer method is shown in the histogram on the right side of Fig. 4. The
optical method is used to correct for the small difference.

Figure 4: (left) Corrections for the measured octocopter height with the raw
data denoted by the green rectangles. The black diamonds refer to the height
after linear correction for the start and end positions. The blue circular
symbols show the corrections for the linear barometric formula used in the
octocopter electronics. The octocopter height determined by the optical method
is denoted by the red dots. All measurements are shown as a function of the
flight time. (right) Histogram of the quotient of the full corrected barometer
height and measured height from the optical method.

####  5.2.6 Octocopter Position Shift from Optical Method Position
Reconstruction

While the octocopter position measured by the built-in sensors (air pressure,
GPS) is recorded nearly each second, the cameras used in the optical method
take pictures of the flight every $3\,\rm{s}$. Furthermore, it turned out that
the fluctuations of the built-in sensors are smaller in comparison to the
optical method. Nevertheless, the systematic uncertainties of the octocopter
position reconstruction using the optical method are still much smaller. The
uncertainties are described in detail in the following subsection. To combine
both advantages of high statistics and small uncertainties, the octocopter
position measured by the built-in sensors is taken and then shifted towards
the position measured with the optical method. Therefore, the octocopter
position in the XY-plane is shifted by the median distance and the octocopter
height measured by the barometer is shifted by the median factor between both
methods. For the $|H_{\phi}|$ example flight the octocopter XY-position
measured by GPS is shifted by $0.83\,\rm{m}$ to the west and $3.22\,\rm{m}$ to
the south. The full corrected flight height measured by the barometer is
shifted by $1.1\,\rm{\%}$.

###  5.3 Uncertainties

In this subsection the statistical and systematic uncertainties are discussed
using the $|H_{\phi}|$ example flight at a middle frequency of
$f=55\,\rm{MHz}$ and a zenith angle bin of $(\Theta=42.5\pm 2.5)\,\rm^{\circ}$
as mentioned above. This zenith angle is chosen as most events at AERA are
reconstructed coming from this direction. While some systematic uncertainties
are stable between flights, e.g., measurement of the power injected to the
transmitting antenna or the transmitting antenna response pattern, others are
flight dependent, e.g., the octocopter position and the measurement of the
receiving power at the AUT. The VEL relative uncertainties are listed in Tab.
2. These individual uncertainties are described in detail in the following
subsections. The constant systematic uncertainties add quadratically to
$6.3\,\rm{\%}$ and the flight dependent systematic uncertainty is
$6.9\,\rm{\%}$.

source of uncertainty / % | systematic | statistical
---|---|---
flight dependent uncertainties | 6.9 | 2.7
transmitting antenna XY-position | $1.5$ | $1.0$
transmitting antenna height | $0.1$ | $0.6$
transmitting antenna tilt | $<0.1$ | $<0.1$
size of antenna under test | $1.4$ | -
uniformity of ground | $<0.1$ | -
RSG1000 output power | $2.9$ | $2.3$
influence of octocopter | $<0.1$ | -
electric-field twist | $0.4$ | $0.2$
LNA temperature drift | $1.0$ | $0.6$
receiving power | $5.8$ | -
background | $0.4$ | -
global uncertainties | 6.3 |  $<$0.1
injected power | $2.5$ | $<0.1$
transmitting antenna gain | $5.8$ | -
cable attenuation | $0.5$ | $<0.1$
all / % | 9.3 | 4.7
Table 2: Uncertainties of the horizontal VEL $|H_{\phi}|$ of the example
flight at $55\,\rm{MHz}$ and $(42.5\pm 2.5)\,\rm^{\circ}$. While the overall
systematic uncertainty is the quadratic sum of each single systematic
uncertainty, the overall statistical uncertainty is described by the observed
signal fluctuation during the measurement. The statistical uncertainty of each
source of uncertainty describes the expected uncertainty, e.g., from the
manufacturer’s information.

####  5.3.1 Transmitting Antenna Position

The systematic uncertainty of the position reconstruction of the optical
method was determined by comparing the reconstructed octocopter position with
the position measured by a DGPS which gives an accurate position
determination. The combined mass of the transmission antenna and the
additional DGPS exceeds the maximal payload capacity of the octocopter.
Therefore, a separate flight with DGPS but without transmitting antenna and
source generator was performed. The octocopter positions measured with the
optical method and the DGPS are compared in Fig. 5.

Figure 5: Comparison of the octocopter position measured with the optical
method and with an additional DGPS mounted at the octocopter during one
flight. (upper left) Raw position data measured with DGPS (lines) and the
optical method (dots) as function of the flight time. The distance between the
reconstructed octocopter position measured by optical method and DGPS in X and
Y direction are shown in the (upper right) and (lower left) figure. The
difference of the octocopter height measured by the barometer and DGPS is
shown in the (lower right) figure. The systematic uncertainty in the XY-plane
of the octocopter position is calculated by the quadratic sum of both median
values (red dashed lines) in X and Y direction. Similarly, the median of the
height difference of both measurement setups is taken as systematic
uncertainty of the octocopter height.

The systematic uncertainty of the octocopter position in the XY-plane is
calculated using the quadratic sum of both median values (red dashed lines) in
the X and Y direction which is smaller than $1\,\rm{m}$. Equally, the
systematic uncertainty of the octocopter height is $\sigma_{h}=0.06\,\rm{m}$.
The influence on the VEL is determined by shifting the reconstructed
octocopter position by these uncertainties and redoing the VEL calculation
given in Eq. (2.5) of each zenith angle bin separately for the XY-plane and
the height. The VEL systematic uncertainty is given by half the difference of
the upper and lower shift of the VEL. The systematic uncertainty on the VEL at
a zenith angle of
$\Theta=42.5\rm{{}^{\circ}}(2.5\rm{{}^{\circ}},72.5\rm{{}^{\circ}})\pm
2.5\rm{{}^{\circ}}$ due to the octocopter’s XY-position is $1.5\,\rm{\%}$
($0.2\,\rm{\%}$, $2.9\,\rm{\%}$) and due to the octocopter’s height is
$0.1\,\rm{\%}$ ($0.2\,\rm{\%}$, $<0.1\,\rm{\%}$).
The statistical uncertainty of the octocopter’s built-in sensors is determined
in the following way. The flight height measured by the barometer has to be
corrected as described in section 5.2.5 which causes further uncertainties
during the flight. The statistical uncertainty of the octocopter height
measured with the barometer is then determined by comparing the measured
height with the height measured by the DGPS (lower right panel of Fig. 6). The
statistical uncertainty are found to be $\sigma=0.33\,\rm{m}$ which results in
a $0.6\,\rm{\%}$ uncertainty in the VEL. The horizontal position uncertainties
are determined in a measurement where the octocopter remains stationary on the
ground. The measurement is presented in Fig. 6. The diagrams show a
statistical uncertainty of
$\sigma=\sqrt{0.48^{2}+0.39^{2}}\,\rm{m}=0.6\,\rm{m}$ in the XY-plane which
results in a $1.0\,\rm{\%}$ uncertainty in the VEL. All these uncertainties
are smaller than those of the optical method described by the widths of the
distributions shown in Fig. 5 where the octocopter positions measured with
DGPS and the camera method are compared.

Figure 6: The statistical uncertainties of the octocopter position
reconstruction using the built-in sensors. The uncertainty of the horizontal
position is determined in a measurement while the octocopter is on ground and
does not move. (upper left) Measured octocopter GPS-position with respect to
the average position at $(0,0)$. Color coded is the time. (upper right)
Histogram of the distance between measured and average position in Y
direction. (lower left) Histogram of the distance between measured and average
position in X direction. (lower right) The statistical uncertainty of the
octocopter height measured with the barometer is determined by comparing the
measured flight height with the height measured using a DGPS. Then,
uncertainties arising from the height corrections are taken into account. The
histogram of the octocopter height difference over ground measured with the
barometer compared to the DGPS measurement is shown.

The transmission antenna is mounted at a distance of $s_{Ant}=0.7\,\rm{m}$
beneath the octocopter. Hence, a tilt of the octocopter, described by the
pitch and the roll angle, changes the position in the XY-plane of the
transmission antenna as well as its height over ground. In the case of the
example flight, the average pitch (roll) angle of the octocopter is
$-0.6\rm{{}^{\circ}}$ ($0.9\rm{{}^{\circ}}$) which lead to a systematic
uncertainty smaller than $0.1\,\rm{\%}$ at $55\,\rm{MHz}$ and $(42.5\pm
2.5)\,\rm^{\circ}$.

####  5.3.2 Size of AUT

The size of the LPDA in the z-direction is $1.7\,\rm{m}$. The interaction
point of the signal at each frequency is set to the center of the LPDA.
Therefore, there is a systematic uncertainty in the height interval between
transmitting antenna and AUT which is conservatively estimated to be
$0.85\,\rm{m}$. For the example flight, this systematic results in a VEL
systematic uncertainty of $1.4\,\rm{\%}$ at $55\,\rm{MHz}$ and $(42.5\pm
2.5)\,\rm^{\circ}$.

####  5.3.3 Uniformity of Ground Height

The ground height above sea level at the octocopter starting position and at
the LPDA is measured by DGPS. The ground is not completely flat but varies at
the level of a few $\rm{cm}$ over a distance of $5\,\rm{m}$ which is
incorporated as additional uncertainty on the height. The resulting influence
on the VEL is less than $0.1\,\rm{\%}$.

####  5.3.4 Emitted Signal towards the Antenna Under Test

The uncertainty of the emitted signal contains effects from the power output
of the RSG1000, the injected power into the transmission antenna, the
transmission response pattern, the influence of the octocopter on the pattern
as well as the misalignment and misplacement of the transmitting antenna which
changes the emitted power transmitted towards the AUT and lead to a twist of
the signal polarization at the AUT.
The manufacturer of the RSG1000 states a signal stability of $0.2\,\rm{dB}$
measured at a constant temperature of $20\,\rm{{}^{\circ}}$ which results in a
statistical uncertainty of $2.3\,\rm{\%}$ in the VEL. The calibration
measurements were performed at temperatures between $15\,\rm{{}^{\circ}C}$ and
$25\,\rm{{}^{\circ}C}$. Here, the manufacturer denotes a systematic
uncertainty of $0.25\,\rm{dB}$ due to temperature shifts which results in
$2.9\,\rm{\%}$ in the VEL.
The injected power from the RSG1000 to the transmission antenna is measured
twice in the lab using the FSH4 spectrum analyzer averaged over $100$ samples
and a Agilent N9030A ESA spectrum analyzer averaged over $1000$ samples. The
systematic uncertainty of the FSH4 measurement is $0.5\,\rm{dB}$ and the
systematic uncertainty of the Agilent N9030A ESA measurement is
$0.24\,\rm{dB}$. Both are combined yielding a total systematic uncertainty of
$0.22\,\rm{dB}$ in the VEL. As there is a quadratic relation between injected
power and the VEL (refer to Eq. (2.5)) the systematic uncertainty on the VEL
is $2.5\,\rm{\%}$. The statistical uncertainties of these measurements are
small due to the number of samples and can be neglected.
The antenna manufacturer specifies a systematic uncertainty of the
transmitting antenna pattern of $0.5\,\rm{dB}$ which results in a systematic
uncertainty on the VEL of $5.8\,\rm{\%}$. The influence of the octocopter on
the transmission antenna pattern investigated with simulations is small [48]
and, therefore, a systematic uncertainty due to the octocopter influence on
the transmission antenna pattern can be neglected.
Misalignment and misplacement of the transmitting antenna lead to a twist of
the signal polarization and furthermore, altered the signal strength at the
AUT. The AUT sensitivity to an electric field is given by
$\eta=\sin(\alpha)\cos(\beta)$ with the angles $\alpha$ and $\beta$ as
described in section 5.2.4. Both angles, and therefore $\eta$, depend on the
octocopter rotation angles as well as on the octocopter position. The angle
$\beta$ linearly depends on $\alpha$ and on the AUT orientation which is known
with a precision of $1\rm{{}^{\circ}}$. The uncertainty of all three
octocopter rotation angles is estimated to be $1\rm{{}^{\circ}}$. In the case
of the horizontal VEL the uncertainty of $\alpha$ is described by the
quadratic sum of two octocopter rotation angles and the angle which arises
from the octocopter position uncertainties as well as the size of the AUT. For
the example flight, the resulting influence on the VEL is $0.4\,\rm{\%}$ at
$55\,\rm{MHz}$ and $(42.5\pm 2.5)\,\rm^{\circ}$. In contrast, both meridional
subcomponents are not corrected for the octocopter misalignment and
misplacement. Here, the octocopter misalignment and misplacement is completely
included in the systematic uncertainty. Therefore, the systematic uncertainty
of the VEL due to an octocopter misalignment and misplacement is larger for
both meridional subcomponents than in the case of the horizontal component.
The systematic uncertainty on the VEL is calculated in the same way but using
the nominal values of $\alpha$ and $\beta$ in each zenith angle bin of
$5\rm{{}^{\circ}}$ instead. As $\beta$ linearly depends on $\alpha$, only a
further uncertainty on $\alpha$ given by the difference between the measured
median values and nominal values of $\alpha$ is needed, quadratically added
and then propagated to the systematic uncertainty on the VEL. In case of both
meridional subcomponents, both angles $\alpha$ and $\beta$ depend on the
zenith angle. Hence, this systematic uncertainty is strongly zenith angle
dependent for both meridional subcomponents.
The uncertainties of the injected power to the transmitting antenna and the
transmitting antenna pattern limit the overall calibration accuracy. In
comparison to other calibration campaigns at LOFAR or Tunka-Rex, a RSG1000
were used as signal source as well but a different transmitting antenna. Both
RSG1000 signal sources differ on a percent level only. However, the
manufacturer of the transmitting antenna used at LOFAR and Tunka-Rex states a
systematic uncertainty of the transmitting antenna pattern of $1.25\,\rm{dB}$
[50]. Hence, the AERA calibration has a significantly smaller systematic
uncertainty due to the more precise calibration of the transmitting antenna.

####  5.3.5 Received Signal at the Antenna Under Test

Within the uncertainty of the received signal all uncertainty effects of the
received power at the AUT including the full signal chain from the LPDA to the
spectrum analyzer as well as the LNA and cables are considered. In the
following a drift of the LPDA LNA gain due to temperature fluctuations, the
uncertainty of the received power using the FSH4 and the influence of
background noise as well as the uncertainty of the cable attenuation
measurements are discussed.
The LPDA LNA gain depends on the temperature. The gain temperature drift was
measured in the laboratory and was determined to $0.01\,\rm{dB/K}$ using the
FSH4 in the vector network analyzer mode [48]. The calibration measurements
were performed at temperatures between $15\,\rm{{}^{\circ}C}$ and
$25\,\rm{{}^{\circ}C}$ which results in a systematic uncertainty of
$1\,\rm{\%}$ in the VEL due to temperature drifts of the LNA. The measurements
of the LPDA LNA gain due to temperature fluctuations using the FSH4 show
fluctuations of the LNA gain at the level of $0.1\,\rm{dB}$ which results in
an expected statistical uncertainty of $0.6\,\rm{\%}$ in the VEL.
The event power is measured using the FSH4 spectrum analyzer. The manufacturer
states a systematic uncertainty of $0.5\,\rm{dB}$. The systematic uncertainty
in the VEL is then $5.8\,\rm{\%}$. Also the background noise is measured using
the FSH4 in spectrum analyzer mode. The systematic uncertainty of the VEL
considering event power (P) and background noise (B) is
$\sqrt{\frac{P^{2}+B^{2}}{P^{2}-B^{2}}}\frac{0.5}{2}\,\rm{dB}$. If the
background noise is of the same order of magnitude as the measured event power
for more than $50\,\rm{\%}$ of events in a $5\rm{{}^{\circ}}$ zenith angle
bin, the systematic uncertainty for this zenith angle bin is set to
$100\,\rm{\%}$. For the example flight, the systematic due to background noise
results in an additional VEL systematic uncertainty of $0.4\,\rm{\%}$ at
$55\,\rm{MHz}$ and $(42.5\pm 2.5)\,\rm^{\circ}$. A further background
influence on the measured signal at the LPDA due to the communication between
the remote control and the octocopter is not expected, as they communicate at
$2.4\,\rm{GHz}$ and the LPDA is sensitive in the frequency range from
$30\,\rm{MHz}$ to $80\,\rm{MHz}$.
The attenuation of the cable is measured with the FSH4 in network analyzer
mode transmitting a signal with a power of $0\,\rm{dBm}$ and averaged over
$100$ samples. Therefore, the statistical uncertainty can be neglected. The
manufacturer states a systematic uncertainty of $0.04\,\rm{dB}$ for
transmission measurements with a transmission higher than $-20\,\rm{dB}$ which
applies in case of the cables. This results in a systematic uncertainty of
$0.5\,\rm{\%}$ in the VEL.

###  5.4 Simulation of the Experimental Setup

The calibration measurement is simulated using the NEC-2 simulation code.
Here, the AUT, the transmission antenna and realistic ground properties are
taken into account. At standard ground conditions the ground conductivity is
set to be $0.0014\,\rm{S/m}$ which was measured at the AERA site. Values of
the conductivity of dry sand, which is the typical ground consistency at AERA,
are reported here [13, 51]. Measurements of the ground permittivity at the
AERA site yield values between $2$ and $10$ depending on the soil wetness
[48]. The standard ground permittivity is set to be $5.5$ in the simulation.
The distance between both antennas is set to be $30.3\,\rm{m}$. The VEL is
calculated using Eq. (2.5) modified with Eq. (2.6) considering the
manufacturer information for the response pattern of the transmitting antenna
as well as the transfer function from the AUT output to the system consisting
of the transmission line from the LPDA footpoint to the LNA and the LNA
itself. To investigate the simulation stability several simulations with
varying antenna separations and changing ground conditions were performed
[48]. Antenna separations ranging from $25\,\rm{m}$ to $50\,\rm{m}$ were
simulated and did not change the resulting VEL of the LPDA. Hence, the
simulation confirms that the measurement is being done in the far-field
region. Furthermore, the influence of different ground conditions is
investigated. Conductivity and permittivity impact the signal reflections on
ground. The LPDA VEL is simulated using ground conductivities ranging from
$0.0005\,\rm{\frac{S}{m}}$ to $0.005\,\rm{\frac{S}{m}}$ and using ground
permittivities ranging from $2$ to $10$. Within the given ranges the
conductivity and permittivity independently influence the signal reflection
properties of the ground. In Fig. 7 the simulations of the horizontal and
meridional VEL for these different ground conditions as function of the zenith
angle at $55\,\rm{MHz}$ are shown.

Figure 7: Simulations of the VEL for different ground conditions. A variation
in conductivity is shown in the upper diagrams whereas a variation in
permittivity is shown in the lower diagrams. In the (left) diagrams the
horizontal VEL $|H_{\phi}|$ and in the (right) diagrams the meridional VEL
$|H_{\theta}|$ as function of the zenith angle $\Theta$ at $55\,\rm{MHz}$ are
shown.

Different ground conductivities do not change the LPDA response pattern. In
contrast the influence of the ground permittivity on the antenna response is
slightly higher. In the case of an applied ground permittivity of $2$ and of
$10$, the influence on the horizontal VEL is at the level of $1\,\rm{\%}$
averaged over all frequencies and zenith angles with a scatter of less than
$6\,\rm{\%}$. The influence of the ground permittivity on the electric-field
reconstruction is discussed in section 7.2.
Simulations of an electronic box beneath the LPDA show influences on the
horizontal antenna VEL smaller than $0.3\,\rm{\%}$ which is negligible
compared to the influence of the ground permittivity [48].

##  6 Measurement of the LPDA Vector Effective Length

In this section, the reproducibility and the combination of all measurements
performed on different days and under different environmental conditions are
discussed. Furthermore, the combined results of the LPDA VEL are compared to
the values obtained from the NEC-2 simulation.

###  6.1 Horizontal Vector Effective Length

Here, the results of the measurements of the horizontal VEL $|H_{\phi}|$ are
presented. In total, five independent measurements were performed to determine
$|H_{\phi}|$ as a function of the zenith angle $\Theta$. The horizontal VEL
$|H_{\phi}|$ in zenith angle intervals of $5\rm{{}^{\circ}}$ for three
different measurements at $35\,\rm{MHz}$, $55\,\rm{MHz}$ and $75\,\rm{MHz}$ is
shown on the left side of Fig. 8. The constant systematic uncertainties of
each flight are denoted by the light colored band and the flight dependent
systematic uncertainties are indicated by the dark colored band. Compared to
the average $\overline{\mathrm{VEL}}$ from $5$ measurements the median value
of the ratio $\sigma/\overline{\mathrm{VEL}}$ is $6\,\rm{\%}$ which is well
compatible with the estimated uncertainties presented in Tab. 2. At the right
side of Fig. 8 all performed measurements to determine $|H_{\phi}|$ are
combined in zenith angle intervals of $5\rm{{}^{\circ}}$, weighted by the
quadratic sum of the systematic and the statistical uncertainties of each
flight. The gray band describes the constant systematic uncertainties whereas
the statistical and flight-dependent systematic uncertainties are combined
within the error bars.

Figure 8: (left) Mean horizontal VEL $|H_{\phi}|$ (dots) and standard
deviation (error bars) of three different measurements and (right) the overall
combinations in comparison to the simulation (green curve) as a function of
the zenith angle in $5\rm{{}^{\circ}}$ bins at (from top to bottom)
$35\,\rm{MHz}$, $55\,\rm{MHz}$ and $75\,\rm{MHz}$. The colored bands in the
left diagrams describe the constant (light color) and flight-dependent (dark
color) systematic uncertainties of each flight. The gray band in the right
diagrams describes the constant systematic uncertainties whereas the
statistical and flight-dependent systematic uncertainties are combined within
the error bars.

The constant systematic uncertainty of the combined horizontal VEL is
$6.3\,\rm{\%}$ and the uncertainties considering flight dependent systematic
and statistical uncertainties for the combined horizontal VEL result in
$4.7\,\rm{\%}$ at a zenith angle of $(42.5\pm 2.5)\rm{{}^{\circ}}$ and a
frequency of $55\,\rm{MHz}$. The overall uncertainty of the determined LPDA
VEL in the horizontal polarization adds quadratically to $7.9\,\rm{\%}$. The
overall uncertainty of all other arrival directions and frequencies are shown
on the left side of Fig. 9. On the right side of Fig. 9 a histogram of all
overall uncertainties for all frequencies and all zenith angles up to
$85\rm{{}^{\circ}}$ is shown. For larger zenith angles the LPDA loses
sensitivity and the systematic uncertainty exceeds $20\,\rm{\%}$. Therefore,
angles beyond $85\rm{{}^{\circ}}$ are not considered in the following
discussion. Taking all intervals of the frequencies and zenith angles with
equal weight the median overall uncertainty including statistical and
systematic uncertainties is $7.4^{+0.9}_{-0.3}\,\rm{\%}$.

Figure 9: (left) Overall uncertainty of the horizontal VEL $|H_{\phi}|$
including statistical and systematic uncertainties for all frequencies as a
function of the zenith angle $\Theta$ up to $85\rm{{}^{\circ}}$ in
$5\rm{{}^{\circ}}$ bins. (right) Histogram of all overall uncertainties for
all frequencies and all zenith angle bins previously mentioned. The median
(average value $\mu$) is marked as red dashed line (red line).

The green curve in Fig. 8 marks the simulation of $|H_{\phi}|$. The agreement
between the combined measurements and the simulation of $|H_{\phi}|$ is
illustrated in the diagram of their ratio versus zenith angle $\Theta$ and
frequency $f$ in the upper left panel of Fig. 10. In the upper right panel of
Fig. 10 all ratios are filled into a histogram with entries weighted by
$\sin(\Theta)$ in consideration of the decrease in field of view at small
zenith angles. The combined measurement and the simulation agree to within
$1\,\rm{\%}$ in the median. The fluctuation described by the $68\,\rm{\%}$
quantile is at the level of $12\,\rm{\%}$. The two lower panels of Fig. 10
show the median ratio as a function of the frequency (left) and as a function
of the zenith angle (right). In both cases, the red error bars mark the
$68\,\rm{\%}$ quantile of the distributions.

Figure 10: Comparison of the combined horizontal VEL $|H_{\phi}|$ with the
simulation. (top left) Ratio of the combination of all measurements and
simulation for all frequencies as a function of the zenith angle $\Theta$ up
to $84\rm{{}^{\circ}}$ in $3\rm{{}^{\circ}}$ bins. (top right) Histogram of
all ratios of the combination of all measurements and simulation for all
frequencies and all zenith angle bins previously mentioned weighted with
$\sin(\Theta)$. The median value is marked as the red dashed line. (bottom
left) Median (red dots) and the $68\,\rm{\%}$ quantile (red error bars) of the
zenith angle weighted ratio distribution as a function of the frequency.
(bottom right) Median (red dots) and the $68\,\rm{\%}$ quantile (red error
bars) of the ratio distribution as a function of $\Theta$. The gray band
indicates the constant systematic uncertainty of the measurement and the red
dashed lines mark the overall zenith angle weighted average in both lower
diagrams.

###  6.2 Meridional Vector Effective Length

In this subsection, the results of the meridional VEL $|H_{\theta}|$ are
discussed. For both subcomponents $|H_{\theta,\mathrm{hor}}|$ and
$|H_{\theta,\mathrm{vert}}|$ three independent measurements were taken and
averaged. The averaged components are combined to determine $|H_{\theta}|$ as
a function of the zenith angle $\Theta$ using Eq.(4.1). In Fig. 11 all
performed measurements of $|H_{\theta}|$ are combined in zenith angle
intervals of $5\rm{{}^{\circ}}$, weighted by the quadratic sum of the
systematic and the statistical uncertainties of each flight. The gray band
describes the constant systematic uncertainties whereas the statistical and
flight-dependent systematic uncertainties are combined within the red error
bars.

Figure 11: Combination of all measurements of the meridional VEL
$|H_{\theta}|$ (red dots) as a function of the zenith angle $\Theta$ in
comparison to the simulation (green curve) for three different frequencies
(from top to bottom) $35\,\rm{MHz}$, $55\,\rm{MHz}$ and $75\,\rm{MHz}$. The
gray band describes the constant systematic uncertainties whereas the
statistical and flight-dependent systematic uncertainties are combined within
the error bars.

The constant systematic uncertainty of the combined VEL is $6.3\,\rm{\%}$. The
uncertainties considering flight dependent systematic and statistical
uncertainties of the combined VEL result in $11.2\,\rm{\%}$ at a zenith angle
of $(42.5\pm 2.5)\rm{{}^{\circ}}$ and a frequency of $55\,\rm{MHz}$. The
overall uncertainty of the determined LPDA VEL in the meridional polarization
adds quadratically to $12.9\,\rm{\%}$. The overall uncertainty of all other
arrival directions and frequencies are shown on the left side of Fig. 12. On
the right side of Fig. 12, a histogram of all overall uncertainties for all
frequencies and all zenith angles up to $65\rm{{}^{\circ}}$ is shown. For
larger zenith angles the LPDA loses sensitivity and the systematic uncertainty
exceeds $20\,\rm{\%}$. Therefore, these angles are not considered in the
following discussion. Taking all intervals of the frequencies and zenith
angles with equal weight the median overall uncertainty including statistical
and systematic uncertainties is $10.3^{+2.8}_{-1.7}\,\rm{\%}$. This is larger
than the uncertainty of the horizontal component $|H_{\phi}|$. The reasons are
that firstly, the meridional component $|H_{\theta}|$ is a combination of two
measurements of $|H_{\theta,\mathrm{hor}}|$ and $|H_{\theta,\mathrm{vert}}|$
whereas $|H_{\phi}|$ is directly measured. Secondly, the number of
measurements is smaller than in the case of $|H_{\phi}|$ and thirdly, the
horizontal component is corrected for the octocopter misplacement and
misalignment in comparison to the meridional subcomponents where this effect
is included in the systematic uncertainties.

Figure 12: (left) Overall uncertainty of the meridional VEL $|H_{\theta}|$
including statistical and systematic uncertainties for all frequencies as a
function of the zenith angle $\Theta$ up to $65\rm{{}^{\circ}}$ in
$5\rm{{}^{\circ}}$ bins. (right) Histogram of all overall uncertainties for
all frequencies and all $\Theta$ up to $65\rm{{}^{\circ}}$. The median
(average value $\mu$) is marked as red dashed line (red line).

The green curve in Fig. 11 indicates the simulation of $|H_{\theta}|$. The
agreement between the combination of all measurements and the simulations of
$|H_{\theta}|$ is illustrated by the diagram of their ratio versus zenith
angle $\Theta$ and frequency $f$ shown in the upper left panel of Fig. 13. In
the upper right panel all ratios for all zenith angles and frequencies are
filled into a histogram with entries weighted by $\sin(\Theta)$ in
consideration of the decrease in field of view at small zenith angles. The
combined measurement and the simulation agree to within $5\,\rm{\%}$ in the
median. The fluctuation described by the $68\,\rm{\%}$ quantile is at the
level of $26\,\rm{\%}$. The two lower panels of Fig. 13 show the median ratio
as a function of the frequency (left) and as a function of the zenith angle
(right). In both cases, the red error bars mark the $68\,\rm{\%}$ quantile of
the distributions.

Figure 13: Comparison of the combined meridional VEL $|H_{\theta}|$ with the
simulation. (top left) Ratio of combination of all measurements and simulation
for all frequencies as a function of the zenith angle $\Theta$ up to
$63\rm{{}^{\circ}}$ in $3\rm{{}^{\circ}}$ bins. (top right) Histogram of all
ratios of the combination of all measurements and simulation for all
frequencies and all zenith angle bins previously mentioned weighted with
$\sin(\Theta)$. The median value is marked as the red dashed line. (bottom
left) Median (red dots) and the $68\,\rm{\%}$ quantile (red error bars) of the
zenith angle weighted correction factor distribution as a function of the
frequency. (bottom right) Median (red dots) and the $68\,\rm{\%}$ quantile
(red error bars) of the ratio distribution as a function of $\Theta$. The gray
band indicates the constant systematic uncertainty of the measurement and the
red dashed lines mark the overall zenith angle weighted average in both lower
diagrams.

###  6.3 Interpolation to all Arrival Directions and Frequencies

The horizontal and meridional VEL magnitudes are measured in the LPDA axis
with highest sensitivity to the respective VEL component (refer to section 4)
and in frequency bins of $5\,\rm{MHz}$. To achieve an overall LPDA calibration
for all incoming directions and frequencies the measurement is combined with
simulations. The LPDA VEL pattern is simulated using the NEC-2 simulation
code. In contrast to the previous simulations presented in this work, only the
LPDA with the following amplifier stage but without the transmitting antenna
is taken into account. This original simulation of the LPDA pattern is then
combined with the results from the calibration campaign. The calibrated LPDA
VEL pattern is obtained by multiplying the full pattern of the simulated VEL
with the ratio of measured to simulated VEL magnitudes shown in Figs. 10 and
13. The ratios are linearly interpolated between the measurements at each
zenith angle and each frequency bin. The respective frequency and zenith angle
dependent ratios are applied at all azimuth angles. With this method, the
magnitude of the VEL is normalized to the calibration measurements, while the
VEL phase comes entirely from the original simulation.

##  7 Influence on Cosmic-Ray Signal Reconstruction

In this section the influence of the modified LPDA pattern on the cosmic-ray
signal reconstruction is discussed. In the first part of this section the
influence of the differences between simulated and measured VEL on the
electric field as well as on the radiation energy for one event with a
specific arrival direction are presented. In the second part the influence of
the uncertainty of both components of the VEL on the electric-field is
discussed.

###  7.1 Influence of Modified Pattern on one Example Event

To reconstruct the electric field of a measured air shower induced radio
signal the Auger software framework $\overline{\textrm{Off}}$
$\underline{\textrm{line}}$ [52] is used. To show the influence of the
improved VEL, an air shower measured in $9$ stations at AERA with a zenith
angle of $30\rm{{}^{\circ}}$ and an azimuth angle of $14\rm{{}^{\circ}}$ south
of east is presented as an example. The energy of the primary cosmic ray is
reconstructed to $1.1\times 10^{18}\,\rm{eV}$ using information from the
surface detector. In Fig. 14 the electric field reconstructed at the station
with highest signal-to-noise ratio (SNR) is shown once using the simulated
antenna response with and once without the corrections owing to the
measurements of the VEL magnitude in both components. For clarity only one
polarization component of the electric field is shown.

Figure 14: (top) Reconstructed electric-field trace at the station with
highest SNR in the east-west polarization of a signal measured at AERA with a
zenith angle of $30\rm{{}^{\circ}}$ and an azimuth angle of
$14\rm{{}^{\circ}}$ south of east using the simulated LPDA pattern (blue line)
and using the modified pattern considering the correction factors between
measurement and simulation (green line). The residual between both
reconstructed traces as function of the time is shown in the (lower) diagram.
The measured energy fluence in the east-west polarization changes from
$100\frac{\rm{eV}}{\rm{m^{2}}}$ to $112\frac{\rm{eV}}{\rm{m^{2}}}$.

The general shape of the electric-field trace is the same for both
reconstructions. The trace of the modified LPDA pattern exhibits an up to
$7\,\rm{\%}$ larger amplitude. The measured energy fluence that scales with
the amplitude squared in the east-west polarization at this station with
highest SNR changes from $100\frac{\rm{eV}}{\rm{m^{2}}}$ to
$112\frac{\rm{eV}}{\rm{m^{2}}}$. The total energy fluence of all polarizations
changes from $141\frac{\rm{eV}}{\rm{m^{2}}}$ using the simulated antenna
response pattern to $156\frac{\rm{eV}}{\rm{m^{2}}}$ using the modified antenna
response pattern which is an effect at the level of $9\,\rm{\%}$. The
reconstructed radiation energy of the full event changes from $7.96\,\rm{MeV}$
to $8.54\,\rm{MeV}$. The ratio of these radiation energies is $0.93$.

###  7.2 Uncertainty of the Cosmic-Ray Signal Reconstruction

In this subsection the systematic uncertainty of the cosmic-ray signal
reconstruction that results from the overall uncertainty of the antenna VEL
magnitude and from the uncertainty due to different ground permittivities is
determined. In the first case, the VEL magnitude is shifted up and down by one
standard deviation of the overall uncertainty. The VEL phase remains
unchanged. In the case of the uncertainty due to different ground
permittivities the antenna pattern with a ground permittivity of $2$ and of
$10$ are used (see Fig. 7). The respective VEL is denoted as
$H^{\mathrm{down}}$ and $H^{\mathrm{up}}$. The antenna response is applied to
a simulated electric-field pulse using once $H^{\mathrm{up}}$ and once
$H^{\mathrm{down}}$, to obtain the corresponding voltage traces
$\mathcal{U}^{\mathrm{up}}$ and $\mathcal{U}^{\mathrm{down}}$ according to Eq.
(2.1). Then, the original VEL is used to reconstruct the electric-field pulse
once from $\mathcal{U}^{\mathrm{up}}$ and once from
$\mathcal{U}^{\mathrm{down}}$. From the difference of the two resulting
electric-field pulses, the systematic uncertainty of the amplitude or the
energy fluence can be determined. Both uncertainties resulting from the
antenna VEL magnitude uncertainty and resulting from different ground
permittivities, are then combined quadratically.
An additional uncertainty on the electric-field trace can arise due to an
uncertainty on the VEL phase. An uncertainty in the VEL phase leads to a
signal distortion of the radio pulse resulting in an increased signal pulse
width and a smaller electric-field amplitude or vice versa. However, the
energy fluence of the RD pulse which is given by the integral over the
electric-field trace remains constant. Hence, a VEL phase uncertainty
propagates to an additional uncertainty in the electric-field amplitude
whereas the energy fluence does not change due to a VEL phase uncertainty.
Therefore, the systematic uncertainty of the energy fluence due to the VEL
uncertainty is discussed in the following.
The radio pulse is approximated with a bandpass-limited Dirac pulse and the
polarization is adjusted according to the dominant geomagnetic emission
process. As the uncertainty of the VEL and the polarization of the electric-
field pulse depend on the incoming signal direction, different directions in
bins of $10\rm{{}^{\circ}}$ in the azimuth angle and in bins of
$5\rm{{}^{\circ}}$ in the zenith angle are simulated. Due to the changing
polarization also the relative influences of the $|H_{\phi}|$ and
$|H_{\theta}|$ components change with direction. The resulting systematic
uncertainty of the energy fluence is presented in Fig. 15. The square root of
the energy fluence is shown because the energy fluence scales quadratically
with the electric-field amplitude and the cosmic-ray energy. Hence, the
systematic uncertainty of the square root of the energy fluence is the
relevant uncertainty in most analyses.

Figure 15: (left) Systematic uncertainty of the square root of the energy
fluence for all arrival directions taking into account a signal polarization
due to the dominant geomagnetic emission process. The square root of the
energy fluence is shown because the energy fluence scales quadratically with
the electric-field amplitude and the cosmic-ray energy. Hence, the
uncertainties of the square root of the energy fluence is the relevant
uncertainty in most analyses. (right) Histogram of the systematic uncertainty
of the square root of the energy fluence of signals with zenith angles smaller
than $80\rm{{}^{\circ}}$ (blue) and of signals with zenith angles smaller than
$60\rm{{}^{\circ}}$ (green).

For most regions the systematic uncertainty is at the level of $10\,\rm{\%}$.
The uncertainty increases only at large zenith angles
($\theta>60\rm{{}^{\circ}}$) due to the increased uncertainty of
$|H_{\theta}|$. An azimuthal pattern appears at $90\rm{{}^{\circ}}$ and
$270\rm{{}^{\circ}}$. At these azimuth angles the uncertainty is smaller
because the electric-field pulse is polarized in the $\vec{e}_{\phi}$
component and only the more precise $|H_{\phi}|$ component contributes. For
incoming signal directions with zenith angles smaller than $60\rm{{}^{\circ}}$
the systematic uncertainty of the square root of the energy fluence owing to
the antenna calibration and different ground permittivities is at most
$14.2\,\rm{\%}$ with a median of $8.8^{+2.1}_{-1.3}\,\rm{\%}$.

##  8 Conclusion

In this work, the results of an absolute antenna calibration are presented
performed on a radio station equipped with a logarithmic periodic dipole
antenna (LPDA). The station belongs to the AERA field of radio stations at the
site of the Pierre Auger Observatory. The calibrated LPDA is representative of
all the LPDAs which are mechanically and electrically identical at the percent
level.
The radio stations are used to reconstruct the electric field emitted by
cosmic particle induced air showers which gives, e.g., a precise measure of
the energy contained in the electromagnetic shower. The accuracy of the
reconstructed shower energy is limited by the uncertainty in the absolute
antenna calibration such that reduction of the uncertainties was most
desirable.
The frequency and directional dependent sensitivity of the LPDA has been
probed by an octocopter carrying a calibrated radio source with dedicated
polarization of the emitted radio signals. The measured LPDA response has been
quantified using the formalism of the vector effective length and decomposed
in terms of a horizontal and a meridional component.
All experimental components involved in the calibration campaign were
quantified with respect to their uncertainties. Special emphasis was put on
the precision in the position reconstruction of the source which was supported
by a newly developed optical system with two cameras used in conjunction with
on-board measurements of inclination, GPS, and barometric height. To ensure
reproducible results, all calibration measurements were repeated by several
flights on different days under different environmental conditions.
The combination of all measurements gives an overall accuracy for the
horizontal component of the vector effective length of
$7.4^{+0.9}_{-0.3}\,\rm{\%}$, and for the meridional component of
$10.3^{+2.8}_{-1.7}\,\rm{\%}$. Note that for air showers with zenith angles
below $60\rm{{}^{\circ}}$ the horizontal component gives the dominant
contribution. The obtained accuracy is to be compared with a previous balloon
based measurement probing a smaller phase space of the horizontal component
with a systematic uncertainty of $12.5\,\rm{\%}$.
The measurements of the new calibration campaign enable thorough comparisons
with simulations of the calibration setup including ground condition
dependencies using the NEC-2 program. The measurements were used to correct
the simulated pattern at multiple points in the phase space described by
arrival direction, frequency and polarization of the waves. While the median
of all correction factors are close to unity at standard ground conditions,
corrections of the simulated vector effective length vary with an rms of
$12\,\rm{\%}$ for the horizontal component, and with rms of $26\,\rm{\%}$ for
the meridional component. The simulations have been further used to confirm
that the measurements have been done in the far-field region. Additionally,
the LPDA sensitivity to different ground conditions has been investigated
showing that the LPDA is insensitive to different ground conductivities and
the sensitivity to different permittivity is only at the level of
$1\,\rm{\%}$.
The effect of the correction factors on the simulated vector effective length
has been demonstrated in the reconstruction of one example radio event
measured with AERA.
Finally, the uncertainty of the two VEL components are propagated onto the
square root of the energy fluence that is obtained by unfolding the antenna
response from the measured voltage traces. For incoming directions up to
$60\rm{{}^{\circ}}$, the expected systematic uncertainty in the square root of
the energy fluence due to the LPDA calibration is $8.8^{+2.1}_{-1.3}\,\rm{\%}$
in the median.

## Acknowledgments

The successful installation, commissioning, and operation of the Pierre Auger
Observatory would not have been possible without the strong commitment and
effort from the technical and administrative staff in Malargüe. We are very
grateful to the following agencies and organizations for financial support:

Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de
Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de
Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in
gratitude for their continuing cooperation over land access; Australia – the
Australian Research Council; Brazil – Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq); Financiadora de Estudos e Projetos (FINEP);
Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ); São Paulo
Research Foundation (FAPESP) Grants No. 2010/07359-6 and No. 1999/05404-3;
Ministério de Ciência e Tecnologia (MCT); Czech Republic – Grant No. MSMT CR
LG15014, LO1305 and LM2015038 and the Czech Science Foundation Grant No.
14-17501S; France – Centre de Calcul IN2P3/CNRS; Centre National de la
Recherche Scientifique (CNRS); Conseil Régional Ile-de-France; Département
Physique Nucléaire et Corpusculaire (PNC-IN2P3/CNRS); Département Sciences de
l’Univers (SDU-INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No. LABEX
ANR-10-LABX-63 within the Investissements d’Avenir Programme Grant No.
ANR-11-IDEX-0004-02; Germany – Bundesministerium für Bildung und Forschung
(BMBF); Deutsche Forschungsgemeinschaft (DFG); Finanzministerium Baden-
Württemberg; Helmholtz Alliance for Astroparticle Physics (HAP); Helmholtz-
Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium für Innovation,
Wissenschaft und Forschung des Landes Nordrhein-Westfalen; Ministerium für
Wissenschaft, Forschung und Kunst des Landes Baden-Württemberg; Italy –
Istituto Nazionale di Fisica Nucleare (INFN); Istituto Nazionale di
Astrofisica (INAF); Ministero dell’Istruzione, dell’Universitá e della Ricerca
(MIUR); CETEMPS Center of Excellence; Ministero degli Affari Esteri (MAE);
Mexico – Consejo Nacional de Ciencia y Tecnología (CONACYT) No. 167733;
Universidad Nacional Autónoma de México (UNAM); PAPIIT DGAPA-UNAM; The
Netherlands – Ministerie van Onderwijs, Cultuur en Wetenschap; Nederlandse
Organisatie voor Wetenschappelijk Onderzoek (NWO); Stichting voor Fundamenteel
Onderzoek der Materie (FOM); Poland – National Centre for Research and
Development, Grants No. ERA-NET-ASPERA/01/11 and No. ERA-NET-ASPERA/02/11;
National Science Centre, Grants No. 2013/08/M/ST9/00322, No.
2013/08/M/ST9/00728 and No. HARMONIA 5 – 2013/10/M/ST9/00062; Portugal –
Portuguese national funds and FEDER funds within Programa Operacional Factores
de Competitividade through Fundação para a Ciência e a Tecnologia (COMPETE);
Romania – Romanian Authority for Scientific Research ANCS; CNDI-UEFISCDI
partnership projects Grants No. 20/2012 and No.194/2012 and PN 16 42 01 02;
Slovenia – Slovenian Research Agency; Spain – Comunidad de Madrid; Fondo
Europeo de Desarrollo Regional (FEDER) funds; Ministerio de Economía y
Competitividad; Xunta de Galicia; European Community 7th Framework Program
Grant No. FP7-PEOPLE-2012-IEF-328826; USA – Department of Energy, Contracts
No. DE-AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No.
DE-SC0011689; National Science Foundation, Grant No. 0450696; The Grainger
Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin
American Network; European Union 7th Framework Program, Grant No.
PIRSES-2009-GA-246806; European Union’s Horizon 2020 research and innovation
programme (Grant No. 646623); and UNESCO.

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