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Lockerbie Wildlife Trust (LWT)Scottish Charity Number - SC005538
Lockerbie Wildlife Trust and Eskrigg Reserve
Lockerbie Wildlife Trust manages Eskrigg Reserve and runs an annual programme of talks, visits and workshops relating to wildlife and conservation for members, schools and other interested parties.
Eskrigg Reserve and the adjacent woodland walks offer visitors the opportunity to relax in peaceful surroundings and observe or photograph the resident fungi, flora and fauna. Some may prefer to be a bit more energetic and explore a range of habitats, walk their dogs or try out the mountain bike trail.
The Reserve is open all year round. Why not come along and see for yourself or book in for any of the organised activities?
If you would like to support
Lockerbie Wildlife Trust
just click on the button below:
Star Species
Objectives & History
Willow Tit
Tawny Owl
Adder's Tongue Fern
Objects of the Trust
To encourage wildlife conservation.
To manage such conservation sites as the Trust accepts responsibility for.
To promote the education and understanding of all age groups regarding local wildlife and its management.
To encourage the use of Trust properties for educative recreation.
To do all such other things as are incidental or conducive to the attainment of the above objects or any of them.
Lockerbie Wildlife Trust was established in September 1988 and was granted charitable status in November 1989.
Scottish Charity Number - SC005538
The History of Lockerbie Wildlife Trust and Eskrigg Reserve
The Eskrigg Reserve is on the site of an old curling pond. It occupies 7 acres (2.83 hectares) and contains a mixture of habitats including mixed woodland, grassland, heathland, willow carr, sedge bed, reed bed, stream and pond each with its own associated flora and fauna. There is coniferous woodland to the north and west with a rich deciduous understory, permanent grassland to the east and marshland to the south. Notable animal species include Red Squirrel, Willow Tit, Little Grebe, Kingfisher and Tawny Owl. You will also find the rare Adder's Tongue Fern there.
In the beginning, 1986 - 1989
Jim Rae learned of the old curling pond from a pupil in 1986, when he was Principal Teacher of Biology at the local secondary school in Lockerbie. After visiting the site, he was impressed by the range of habitats in such a small area, but noticed that one of the habitats, a monoculture of reed-grass, covered more than three acres and supported a very limited fauna. The pond had been drained and the natural succession of vegetation over many years had eliminated all open water. Jim quickly realised that, by digging out much of the reed-grass and repairing the sluice gate, a pond could be re-established thereby increasing the biodiversity and, at the same time, creating a resource for the local community, including his Natural Resources class at the school.
He explained his ideas to Sir Rupert Buchanan-Jardine, owner of Castle Milk Estate, then applied for, and was granted in January 1987, a joint one-year lease of the curling pond with the then shooting tenant Henry Strath. Jim spent the following year seeking advice and assistance from many experts in the field of conservation. He arranged a survey of the area to determine future water levels. With the assistance of Mary Martin (Regional Plant Recorder) and Elizabeth Kungu (expert on mosses), he identified as many plants and animals as possible and drew up a potential management plan.
In August 1987, Jim started a Conservation Class at Lockerbie Academy and together they discussed and worked on the management plan. They also organised a sponsored cycle run to raise the funds needed for digging out the pond.
In September 1987, a public meeting was held at Lockerbie Academy at which Jim Rae explained the draft plans to interested members of the local community. His plans received overwhelming approval and that evening a Local Support Group was set up. At the first meeting of the support group, in October 1987, the members decided to form the Lockerbie Wildlife Trust and to name the old curling site Eskrigg Reserve.
Jim Rae took on the sole lease of the pond site for a year in 1988 and, throughout the spring of that year, continued his research, obtained estimates for digging out the pond and, in discussion with members of the support group, determined the objects of the Lockerbie Wildlife Trust. In May 1988, a local firm (Dalgliesh Brothers) dug out the pond area creating islands and banks, the sluice was repaired, an overflow pipe (donated by Express Cheese Factory) was installed and the pond area flooded. Within a month there was good growth on the islands and a wide variety of wildlife in the pond. In the summer of '88, eight weeks of dry weather resulted in the pond drying up. This helped to determine the future management policy for the site. In June, pupils from Lockerbie Academy erected two bird-watching hides.
In September 1988, the inaugural meeting of the Lockerbie Wildlife Trust was held and a programme of talks, visits, workshops and fund-raising events was organised for the year ahead. Over the next year, Academy pupils, former pupils, Scouts, Cubs and students from the Barony College helped Trust members with clearing, construction and maintenance work.
In January 1989, the lease for the pond site was transferred to Lockerbie Wildlife Trust for an initial term of ten years, subject to review after five. The pond site then became officially recognised as Eskrigg Reserve. Throughout '89 Jim Rae consulted widely and drafted a Constitution for the Trust.
Jim continued to manage Eskrigg Reserve for the Lockerbie Wildlife Trust, during his spare time, over the next 20 years. He received a lot of support from the local community and members of the Trust. The reserve and adjacent woodland was an ideal site for biological fieldwork with his science classes. He organised many different wildlife workshops and events, particularly for local youngsters. A number of social events, including a wedding, woodland fair and a concert, were also held at the reserve.
In spring 2008, the first Dumfries and Galloway Wildlife Festival was organised. Although the Easter activities at Eskrigg were not on the official festival programme, they continued to prove popular with local children. In the summer of 2008, Jim organised the first of the now annual Wildlife Weekends at Eskrigg. Over one weekend in the summer holidays, the Trust runs a range of workshops for both children and adults. These include pond dipping, small mammal trapping, moth trapping, wild food foray and bird or bat box workshops. Fungal forays and den building were included in the autumn activities.
IIn August 2009, Jim retired from teaching at Lockerbie Academy in order to oversee the building of the Eskrigg Centre. This timber building has wheelchair access. It contains a viewing area with four windows overlooking the pond and another four facing the trees and wildlife feeders on either side making it ideal for wildlife observation and photography. Also included is a comfortable reading area with access to the latest wildlife magazines and leaflets. By arrangement with the Reserve Manager, there is secure area that can be opened up to convert the hide into a small classroom, lecture hall or research laboratory as required. Sir Rupert Buchanan-Jardine opened the Eskrigg Centre in April 2010.
Since then, Jim and the Lockerbie Wildlife Trust have contributed to the Dumfries and Galloway Wildlife Festival, assisted with the Wild Seasons initiative to educate people about the wonderful wildlife in our area and helped promote Dumfries and Galloway as a prime venue for nature tourism. From 2012, Eskrigg activities were listed in the D & G Wild Seasons programme.
In 2013, Lockerbie Wildlife Trust, assisted by Lockerbie and District Rotary Club, marked the 25th Anniversary of Eskrigg Reserve by replacing one of the original bird hides with the new Red Squirrel Hide. This also has disabled access and provides visitors with an excellent opportunity to view and photograph the Red Squirrels and other wildlife.
In 2014, Lockerbie Wildlife Trust started its 'Access For All' project. Castle Milk Estate, Dumfries and Galloway Ranger Service and Lockerbie Wildlife Trust worked together to provide a car park (Phase 1) and hardcore path to the Reserve (Phase 2) allowing safer access for all. These were in place by the end of August.
During 2015, Lockerbie Wildlife Trust proceeded with Phase 3 of the 'Access For All' project and raised funds to purchase and install a NatSol Compost Toilet with disabled access. The underground vault was lowered into place and the timber building was erected above it at the end of April 2016. During May and June, the toilet building was painted and fitted out. A concrete ramp was put in place and the toilet was ready for use at the beginning of July 2016.
Meanwhile, Phase 4 of the 'Access For All' project* had already begun - the resurfacing of the Woodland Walks next to the Reserve with gravel in order to facilitate access to the woods. This work will continue over the next few years until there is disabled access to at least one mile of woodland walks.
* report available for download
Membership of Lockerbie Wildlife Trust is by subscription, renewable annually in September.
At present, each member receives a membership card giving details of
a) The Winter Talks Programme (normally from October to March)
b) The Main Fund-raising Events and
c) The Planned Maintenance Days (normally the last Sunday of each month)
Copies of the monthly Eskrigg Nature Reserve Monthly News Bulletins are available on the website and a copy will be sent to every member who has supplied an email address. Copies are also available for viewing in Eskrigg Centre, Lockerbie Library and the Dryfesdale Lodge Visitors' Centre. Members can enter the Winter Talks for half the normal entry fee and will receive priority booking for seats on any trips. Entry to the A.G.M. (last Wednesday in September) is free.
Trust Meetings and Winter Talks Programme
For details click here
These usually take place in the early summer (May/June) or early autumn (September/October). Transport depends on the number of people interested - either shared private cars, ATI minibus or hired coach. Each trip is costed independently and the costs shared by those involved. For details of planned trips see 'Activities'.
To become a member
The Trust's financial year begins on the 1st of September and ends on the 31st August the following year. New members may join at any time of the year and will be due to pay an initial sum which is equal to 1/12th of the annual subscription for each month of the financial year remaining. Thereafter they should pay the full membership subscription each September.
Click on 'Membership Application Form', print out a copy, complete the form and then send it, together with the appropriate membership fee to the Membership Secretary as detailed on the form.
Personal data that we store
If you decide to contact us in any way, we may hold records of your name, phone number, email address and other information that you have opted to provide. This also includes donations you made with PayPal.
If you have opted in for our Monthly News Bulletin, then we will send you monthly updates. We may also contact you to keep you updated with oncoming events like talks, activities, fundraises or trips. We respect your personal data and will never send it to third parties without prior consent.
Where we store your data
How you have interacted with us will help determine what data is stored where.
If you have opted in to our email communication, or if you have communicated with us by email, then there may be records of your communications in the corresponding members' email systems.
If you are a member of the Lockerbie Wildlife Trust, a record of your membership data is stored with the membership secretary, both physically and electronically.
Membership data is only accessible to the following eligible LWT Management Committee members:
The Membership Secretary, the Chairperson, the Vice Chairperson, the Treasurer, the Reserve Manager, the Events Secretary and, at the discretion of the Management Committee members, the IT Manager.
This will also apply to Gift Aid* forms and donations you made through PayPal.
*We will also record your name and address in a spreadsheet provided by HMRC.
Your rights to your data
You are entitled to request access to the data we have stored on you. You are also entitled to request corrections to your data as well as deletions. Please get in touch if you would like to do so.
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Eskrigg Reserve
Eskrigg Reserve is on the site of an old curling pond. It occupies seven acres and contains a mixture of habitats including mixed woodland, grassland, heathland, willow carr, sedge bed, reed bed, stream and pond each with its own associated flora and fauna.
There is coniferous woodland to the north and west with a rich deciduous understory, permanent grassland to the east and marshland to the south.
You can help look after the reserve if you:
•	keep to footpaths
•	keep your dog on a lead while within the Nature Reserve
•	clean up after your dog
•	take your litter home
•	respect all wildlife habitats, plants, animals and other visitors
•	record any wildlife sightings at Eskrigg Centre
Monthly Bulletins
Trust Meetings & Winter Talks Programme
The Reserve is located 1.6 miles (2.57 km) south-west of Lockerbie Town Centre, on the east side of South Turnmuir Plantation.
(OS map ref. N.Y. 126805)
Getting to the Reserve from Lockerbie
Take the A709, Lockerbie to Dumfries road.
Just before leaving Lockerbie, turn left into Vallance Drive and then immediately right on to Eskrigg Farm Road.
Follow the farm road for 400m and the Reserve Car Park is on the right.
Please observe the 10mph speed limit.
Getting to the Reserve from Lochmaben or Dumfries
Take the A709 to Lockerbie.
As you enter the town, take the very first turning on the right into Vallance Drive and then immediately right on to Eskrigg Farm Road. Follow the farm road for 400m and the Reserve Car Park is on the right.
Reserve Car Park to Reserve
There is a hard-core, all-user path, leading 320m down a gentle slope from the Car Park to the Reserve. The path is fenced off from the fields on either side and is suitable for wheelchairs, mobility scooters, pushchairs and prams. There are 2 small gates near the beginning of the path that may need to be opened and closed when the farmer requires the fields on either side to be linked. When he doesn't, the gates will be open.
Access for those with mobility problems
It is possible, by prior arrangement with the Reserve Manager, to take a car or minibus down the forest road opposite the Dryfesdale Lodge Visitors' Centre to a small parking area within 100m of Eskrigg Centre. The single bar gate at the entrance to the forest road is normally kept locked and parking is not allowed in front of the gate. Access will not be possible when there are forest operations taking place in the woods.
Two mobility scooters are now available for use at Eskrigg Reserve by anyone with mobility problems. Please contact the Reserve Manager at least 24 hours before the scooter is needed so that it can be fully charged and delivered to the Reserve Car Park when required.
Access by Bus or on Foot
For those travelling by bus, there are bus stops on the Dumfries Road at the end of Vallance Drive. From the bus stop it is a ten-minute walk to the Car Park and another ten-minute walk from the Car Park to the Reserve. The Reserve is roughly forty-five minutes from Lockerbie Town Centre.
Map & leaflet of Reserve and Woodland Walks
Click on the symbols to view and download the map (pdf)
A hardcore path runs through the reserve offering views of most habitats and with interpretation boards at regular intervals.
•	A picnic area with tables and a shelter.
•	One small bird-watching hide (without wheelchair access) at the end of a boardwalk, overlooking the pond.
•	One medium-sized red squirrel/bird-watching hide (with wheelchair access) with views over the pond and wildlife feeding areas.
•	Eskrigg Centre – a large, high-quality hide/visitor centre with interpretation and teaching/research facilities and wheelchair access.
•	A toilet with disabled access is available during events and by arrangement.
•	School group visits and guided tours can be arranged.
Eskrigg Centre
Eskrigg Centre is open all year
beginning of March - end of October
from 8.00am - 5.00pm
beginning of November - end of February
other times by arrangement
View of access ramp into Centre
Front of Centre
View of seating area inside Centre
View of windows from inside
Red Squirrel HIde
Entrance to Red Squirrel Hide
Front and side of Red Squirrel Hide
Eskrigg Reserve Activities
Attendance at any of the Eskrigg Reserve activities is by prior booking only. To book you can send Jim an email or you can text or call him on 07739 987 009.
There is no charge for any of the activities but donations to help with the running of Eskrigg Centre and the Reserve are always welcome.
>> Children must be accompanied by an adult at all events <<
Meet at Eskrigg Centre unless otherwise stated.
Open or Close
Wildlife at Eskrigg Reserve and Woodland Walks
All wildlife can be classified into groups that have features in common. A group of living things that have so many common features that they can reproduce and have fertile young is called a species. Most species at the Reserve can be put into the two major groups: PLANTS and ANIMALS. All the other organisms can be grouped under a third heading: the PROTISTS. Each of these groups can be sub-divided as follows:
Other Protists
(including Cyanobacteria or blue-green algae)
(including Amoebae, Ciliates & Flagellates)
Slime Moulds
(including Cellular & Plasmodial forms)
(including unicellular, filamentous & colonial)
(including crusty, leafy & shrubby forms)
Spore Producing
- Mosses and Liverworts
- Ferns and Horsetails
Seed Producing
> Conifers
> Flowering Plants
- Deciduous Trees & Shrubs
- Climbers
- Herbaceous Plants
(without backbones)
- Arthropods
(Hard outer skeleton & jointed legs)
> Insects
- Butterflies & Moths
- Damselflies & Dragonflies
- Other Insects
> Other Arthropods
- Crustaceans
- Myriapods
- Arachnids
- Other Invertebrates
- Coelenterates
- Platyhelminths
- Annelids
- Molluscs (Many with shell, no segments)
(with backbones)
> Cold Blooded
(Aquatic - fins, gills & scaly skin)
(Breed in water, smooth, moist skin)
(Dry, scaly skin, lay eggs with soft shells)
> Warm Blooded
(wings, eggs with hard shells)
(Hair or fur, live young fed on milk)
The LWT Committee Members for 2018 / 2019
David Hughes (Chairman)
Freda Seddon (Vice-Chairman)
Jim Rae (Reserve Manager)
Kathryn Hughes (Treasurer)
Kath Davidson (Events Secretary)
John Reid (Membership Secretary)
Alan Elkington (Minutes Secretary)
Norah Muirhead
George Trudt
Sybille Spägele Hon. IT Manager
Date of next Committee Meeting:
Tuesday, 24th April 2018
Venue: Kings Arms Hotel, Lockerbie at 7.00 pm
Lockerbie Wildlife Trust, Reserve Manager: Jim Rae
Mobile (daytime): 07739 987 009
Home Tel. (evening): 01576 203 314 Email
Annual Reports of the Lockerbie Wildlife Trust
Annual Report for 2017-2018
(opens in a new window and requires Adobe © Acrobat Reader)
Date of next AGM:
Venue: Town Hall, Lockerbie at 7.30 pm
Trust Meetings and Winter Talks Programme 2018 / 2019
These will be held in the Lockerbie Town Hall, Community Rooms, at 7.30 pm unless otherwise stated.
Entry to meetings: All welcome - Adults £2.00 (Members, Senior Citizens and School children £1.00).
Entry to A.G.M. and Members' Night is free.
Title of meeting / talk and speaker
LWT AGM
Fund Raisers 2018 / 2019
Autumn Coffee Morning
Lockerbie Town Hall
(Community Rooms)
10.00 am - 12.00 noon
© 1988 - 2019 | contact
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Research into the design, implementation and maintenance of software, enabling development of software that is dependable, efficient and maintainable. Includes topics such as requirements engineering, software design, software quality (including reliability, safety, security and usability), software testing and analysis, software adaptation and evolution, software process and automation, and empirical software engineering.
We aim to grow this area as a proportion of the EPSRC portfolio. This strategy recognises the high quality of this research area in the UK and the key underpinning role it has to play in several challenges of national importance.
There is a substantial level of high-quality software engineering research in the UK, demonstrated by strong representation of UK researchers at international conferences such as ICSE (International Conference on Software Engineering), FSE (Foundations of Software Engineering) and ASE (Automated Software Engineering). EPSRC recognises that software engineering supports many other areas of ICT and beyond. Although impacts arising from this research area are significant, they may sometimes be partially 'hidden' behind an associated application area.
Researchers in this area can continue to contribute to challenges posed by the world moving to complex, open systems and environments (e.g. autonomous systems and an Internet of Things), particularly around a need for research into new techniques for software to reliably adapt and learn. This will require work across the portfolio with researchers in areas such as Artificial Intelligence, Verification and Correctness, and Pervasive and Ubiquitous Technologies. Engagement with the academic and user community has identified that further software engineering research is essential to future advances in cyber security (Evidence source 1).
Researchers in this area have significant impact on skills uptake in industry through undergraduate teaching and involvement in standards boards. Many disciplines rely on good-quality software. The SSI and Research Software Engineer Fellowships are encouraging translation of novel software engineering techniques into practical use across computational science and a range of disciplines, demonstrated by the co-funding of SSI by three Research Councils (EPSRC, BBSRC and ESRC) (Evidence source 2,3).
In terms of research training, there is a balanced portfolio between studentships and cohort-based training (Evidence source 4). There is high demand from industry for students trained in this area, demonstrated by a significant number of Industrial Cooperative Awards in Science & Engineering (CASE).
Although there is a range of EPSRC funding in this area, including programme grants and fellowships, there has been a reduction in the number of large groups and there are fewer UK leaders than in the past. The demands placed on researchers by the challenges described above therefore exceed their capacity to deliver (Evidence source 4,1).
Development of reliable, robust and usable tools will require advances in software engineering.
Software engineering is expected to be a key contributor because it can enable uptake of intelligent technologies in complex systems.
Advances in software engineering will be necessary to anticipate and adapt to acute threats in systems.
Community engagement (individual input, group feedback, team visits and evidence gathering) and input from the ICT Strategic Advisory Team, the UK Computing Research Council Executive Committee and Research Excellence Framework (REF) 2014 panellists (sub-panel 11).
SSI, Measuring the Success of the Software Sustainability Institute (PDF), (2014).
SSI, Research Software Sustainability (PDF), (2016).


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\section{Introduction}

Hundred years after the discovery of cosmic rays \cite{Hess1912} the origin of the highest energy particles is still unclear and requires further measurements. The interesting energy range above $10^{17}\,$eV where the transition from galactic to extragalactic cosmic rays is presumed \cite{Haungs2003} cannot be accessed by direct measurements since the flux of cosmic rays above $\sim 10^{14}\,$eV is too low. Instead, cosmic rays are measured indirectly by detecting air showers of secondary particles. Established techniques for air shower measurements are the detection of the secondary particles at ground, and the measurement of atmospheric Cherenkov and fluorescence light emitted by air showers. The aim of all measurements is to reconstruct the properties of the primary cosmic ray particles from the air-shower observables, i.e.~their arrival direction, energy and mass. In particular, the latter two techniques give more precise measurements of the energy since they provide a calorimetric measurement of the air showers. However, they have the disadvantage that they can be used only during dark, moonless nights \cite{Bluemer2009}.

An alternative instrument for air shower detection is given by digital radio antenna arrays, which also can provide a measurement of the shower energy \cite{HuegeUlrichEngel2008, PalmieriICRC2013} and feature a duty-cycle of almost $100\,\%$ like particle detector arrays \cite{Allan1971, Melissas2010}. Current efforts like LOPES \cite{FalckeNature2005, HuegeARENA2010, SchroederLOPES_ARENA2012}, CODALEMA \cite{ArdouinBelletoileCharrier2005, RavelARENA2010}, at ANITA \cite{Hoover2010ANITA}, the Pierre Auger Observatory \cite{FuchsRICAP2011}, or at Tunka \cite{SchroederTunkaRex_ARENA2012} still focus on engineering work, i.e.~to prove the applicability of the radio technique to large scale observatories, and to show that a precision similar to the one of the fluorescence and air-Cherenkov techniques can be achieved.

To make the radio technique competitive, it is crucial to understand the radio emission mechanism in sufficient detail. From previous work (e.g., Ref.~\cite{Allan1971}) we know that the radio emission of air showers originates mainly from the geomagnetic deflection of electrons and positrons \cite{KahnLerche1966, FalckeGorham2003}, but also other effects play a role, in particular the Askaryan effect \cite{Askaryan1962}, i.e.~the variation of the net charge excess over the shower development. Recent models and simulation codes also take into account the refractive index of the air, which affects the coherence conditions for all radio emission mechanisms \cite{deVries2011, LudwigICRC2011, AlvarezZHAires2012}, as was already discussed more than 40 years ago in Ref.~\cite{AllanICRC1971}. Cherenkov emission due to constant, not-accelerated charges moving with superluminal velocity in a refractive medium, however, is generally not included in recent simulation codes because its contribution is considered negligible \cite{James2011}. The best way to test our understanding of the overall emission is to compare experimentally measured quantities, like the lateral distribution of the radio signal, with simulations based on certain models.

The lateral distribution of the radio signal is the variation of the radio amplitude $\epsilon$ with the distance to air shower axis $d$. It has already been studied with LOPES \cite{ApelArteagaAsch2010, ApelLOPES_MTD2012} and other experiments (e.g., CODALEMA \cite{RavelARENA2010}), and already it was compared to simulations \cite{SchroederECRS2010}, however, only for single events or with a limited statistics. These limited studies had been sufficient to show that earlier models, not yet including the refractive index of the air, could not fully reproduce the measurements. Now, we present the results of a systematic, per-event comparison of radio lateral distributions measured with LOPES to two recent, publicly available Monte Carlo simulation codes: REAS 3.11 \cite{LudwigHuege2010} and CoREAS (v 1.0) \cite{HuegeCoREAS_ARENA2012}, respectively. Since LOPES features an absolute amplitude calibration \cite{NehlsHakenjosArts2007}, and neither REAS nor CoREAS have free parameters to tune the absolute scale, the presented comparison is not only qualitatively, but also quantitatively meaningful.

There are other simulation codes available, e.g., ZHAireS \cite{AlvarezZHAires2012}, EVA \cite{Werner2012}, MGMR \cite{ScholtenMGMR2008}, SELFAS \cite{Marin2012} and the model described in Ref.~\cite{KalmykovKonstantinov2011}. Although they use different approaches and techniques, they basically simulate the same physics for the radio emission of air showers, i.e., the geomagnetic and the Askaryan effect, and some codes also include the refractive index of the air. Since we have chosen on purpose two simulation codes with a fixed, non-tunable absolute scale, other codes can be indirectly compared to LOPES when comparing them with REAS 3.11 or CoREAS, respectively. Such model-to-model comparisons have already been performed, e.g., in Refs.~\cite{HuegeARENAmodels2010, HuegeARENAtheory2012}. In future, the next-generations experiments LOFAR \cite{BuitinkLOFAR_ICRC2013} and AERA \cite{SchroederAERA_ICRC2013} can be used to test simulations in much more detail or, respectively, at much larger axis distances.

\section{Measurements}
For the reconstruction of the lateral distribution we use LOPES measurements of the years 2005-2009. During this period, LOPES consisted of 30 absolute calibrated, inverted v-shaped dipole antennas. Until the end of 2006, all antennas were east-west aligned, since for most shower geometries the radio signal is preferentially east-west polarized due to the geomagnetic radio emission. Afterwards, half of the antennas were aligned in the north-south direction. In this paper we focus on the lateral distribution measured with the east-west aligned antennas, since the statistics are larger and the signal-to-noise ratio is generally higher. At the end of the paper, we also present first results for lateral distributions measured with the north-south aligned antennas.

The LOPES antennas are located within the KASCADE-Grande experiment \cite{ApelArteagaBadea2010} at the Karlsruhe Institute of Technology, Germany, and form an array with a lateral extension of about $200\,$m. Whenever KASCADE-Grande detects a high-energy shower ($10^{16}-10^{18}\,$eV), it triggers LOPES which then digitally measures the radio emission. The effective bandwidth of LOPES is $43-74\,$MHz, the trace length about $0.8\,$ms, and the sampling rate $80\,$MHz. This means that the LOPES setup fulfills the Nyquist sampling theorem, i.e.~the full information of the electrical field strength in the measurement bandwidth is contained in the data. Thus, the electrical field strength between the sampled data points is retrieved by up-sampling.

\begin{figure*}[t]
  \centering
  \includegraphics[width=0.92\columnwidth]{2005_11_25_02_11_53_428-all.eps}
  \hskip 0.13\columnwidth
  \includegraphics[width=0.92\columnwidth]{2005_11_25_02_11_53_428-CC.eps}
  \vskip 0.2 cm
  \includegraphics[width=0.92\columnwidth]{2005_11_25_02_11_53_428-06.eps}
  \hskip 0.13\columnwidth
  \includegraphics[width=0.92\columnwidth]{lateral-1132884713.eps}
  \caption{Example event measured with LOPES in 2005, when all antennas were aligned in the east-west direction: electrical field strength traces of all antennas (a), smoothed cross-correlation (CC) beam and power beam (b), electrical field strength trace in an individual antenna with a Hilbert envelope used to measure the maximum amplitude (c), lateral distribution (d).}
   \label{fig_exampleEvent}
 \end{figure*}

\subsection{Analysis Procedure}

The KASCADE-Grande reconstruction provides the shower direction and the shower core (= intersection of the air shower axis with the ground), as well as the primary energy, and the measured numbers of muons and electromagnetic particles. A subset of these parameters (arrival direction, core position, energy, muon number) is used as input for the REAS 3.11 and CoREAS simulations, and another subset (shower direction, core position) as input for the LOPES analysis, whereby the shower direction is optimized during the LOPES analysis using the radio measurements.\footnote{The LOPES analysis software is available as open-source at \lq http://usg.lofar.org\rq.}

An essential part of the LOPES analysis is cross-correlation beamforming. First, the electrical field strength traces of each antenna are shifted in time corresponding to the arrival direction of the air shower (Fig.~\ref{fig_exampleEvent} a), and, second, a cross-correlation between the different traces is calculated (CC beam, Fig.~\ref{fig_exampleEvent} b) as well as a power beam measuring the total power from the air shower arrival direction. Details of this procedure can be found in Refs.~\cite{HornefferThesis2006, HornefferICRC2007}. The present analysis of the lateral distribution uses the cross-correlation and power beam only for two purposes: a selection of events with a clear radio signal, and a determination of the exact time of the radio pulse, which is an important input to determine the radio amplitude in antennas with low signal-to-noise ratio.

The radio signal in each individual antenna is determined with a Hilbert envelope of the up-sampled trace (Fig.~\ref{fig_exampleEvent} c), where the maximum of the Hilbert envelope is a measure for the maximum instantaneous amplitude of the radio signal. Since the time of the radio pulse is known from the preceding cross-correlation beamforming, a measurement of the radio amplitude at this time is possible in each individual antenna. For this, we take the maximum of the Hilbert envelope closest to the time of the CC beam maximum as input for the lateral distribution (Fig.~\ref{fig_exampleEvent} d). To convert the measured radio pulse amplitudes into the absolute electrical field strength, we use simulations of our antenna gain pattern and calibration measurements of the complete electronics chain including the antenna. However, for a correct conversion of the measured amplitudes to the east-west or, respectively, the north-south polarization component of the field strength vector, a measurement with at least two differently aligned antennas at the same location would be required. Since this is only available for about half of the LOPES events and a few antenna positions, we generally use a simplified conversion formula, which is exact in the limit that the radio signal is purely east-west polarized. For the simulations, we determine the pulse amplitude by digitally filtering the east-west polarization component of the simulated radio signal to the effective bandwidth of LOPES (cf. section \ref{sec_simulations}).

We point out that a different analysis procedure (e.g., using the integrated pulse power instead of the amplitude) might result in a different lateral distribution. The lateral distribution likely depends on the used frequency band, too. Theoretical models predict that the pulse shape and thus also the shape of the frequency spectrum of the radio emission depends on the distance to the air shower axis \cite{HuegeARENAmodels2010}. A direct test of our results will still be possible when results of measurements with a larger effective bandwidth are digitally filtered to the effective bandwidth of LOPES ($43-74\,$MHz). To facilitate a comparison with other experiments, we have normalized the electrical field strength dividing by the effective bandwidth of LOPES, though this will allow only a comparison at first order, since the frequency spectrum is not flat \cite{NiglFrequencySpectrum2008, GrebeSpectralSlope_ARENA2012}.

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{eventMap.eps}
\caption{Map of the LOPES array co-located with the KASCADE-Grande experiment, and shower cores of the selected LOPES events. Due to quality cuts of the KASCADE-Grande reconstruction, only events with a core location within the fiducial area of the KASCADE array or the Grande extension are accepted which gives rise to the distinct shapes of the core locations.} \label{fig_eventMap}
\end{figure}

\subsection{Event selection}

For the present study, we preselected 3968 LOPES events using the KASCADE-Grande measurement of the same events, requiring a minimum reconstructed energy of $10^{17}\,$eV and a zenith angle $\theta \le 45^\circ$. Then, we selected only events which show a clear signal in the CC beam, i.e.~the fraction of correlated power in the antennas (= height of CC beam / height of power beam) must be larger than $80\,\%$. In addition, the signal-to-noise ratio of the CC beam must be larger than $14$ (normalized with a factor $\sqrt{N_\textrm{ant}/30}$, since not all 30 LOPES antennas are available in each event). The signal-to-noise ratio is calculated as the height of a Gaussian fit to the CC beam smoothed by block-averaging, divided by the RMS of the smoothed CC beam in a part of the trace before the radio pulse. Furthermore, we exclude events for which we measured a high atmospheric electrical field at ground ($E_\mathrm{atm} > 3000\,$V/m), since this can significantly affect the radio emission of air showers \cite{BuitinkApelAsch2006, LopesThunderstormARS2011}.

This way we attained 528 events. From these events, we exclude in total 15 events for different reasons: in 3 cases there have been technical problems with one of the performed simulations, in one cases the fit to the measured lateral distribution did not converge for technical reasons in the fit algorithms, in 14 cases the fit failed because at least one antenna measured a too low amplitude. Thus, 513 events remain for the lateral distribution analysis of this paper (see Fig.~\ref{fig_eventMap}). This is only a fraction of the preselected events, since LOPES is not fully efficient at the chosen energy threshold. For example, the efficiency depends on the angle between the Earth's magnetic field and the air shower axis (= geomagnetic angle $\alpha$), since the signal height depends on the geomagnetic Lorentz force. Furthermore, preselected events closer to the LOPES antennas are more likely to pass the quality cuts than events at larger axis distances, since the radio amplitude decreases with increasing distance. However, since we have done the simulations for exactly the selected air showers, neither full efficiency nor a uniform distribution of the events is required to test the simulations with LOPES measurements.

\subsection{Lateral distribution function (LDF)}
According to experimental and theoretical studies, the 'true' lateral distribution function (LDF) is quite complex, cf.~Refs.~\cite{AllanICRC1971, deVries2010, FuchsPhD, NellesLOFAR_ARENA2012}: in the LOPES distance and frequency range, the LDF flattens towards the shower core, in some cases shows a bump at an axis distance of about $120\,$m and then shows an approximately exponential decrease. On top of that, the interference between the dominant geomagnetic and the weaker Askaryan effect leads to a small asymmetry depending on the observer azimuth relative to the shower core. According to Ref.~\cite{SchoorlemmerPhD}, the relative strength of the Askaryan effect compared to the geomagnetic effect is about $11.5\,\%$ at the location of the Pierre Auger Observatory, where the geomagnetic field is only half as strong as at the LOPES site. Thus, the relative size of this effect should be about $5-6\,\%$ at LOPES, which is small compared to the typical measurement uncertainty of the amplitude in an individual LOPES antenna. Consequently, the asymmetry of the lateral distribution is neglected when defining the LDF for the present analysis. Finding a function describing all these features correctly would be an interesting study on its own, but is not the focus of this paper.

Instead, we have decided to use a simplified 2-parameter function which still is sufficient to test whether the simulations can correctly describe the absolute amplitude scale and the general shape of the measured lateral distributions. Using a more complex LDF with additional parameters describing the features mentioned above would drastically reduce the statistics of usable events, because only for a small fraction of the events all fit parameters could be determined with sufficient precision. E.g., if there were one additional parameter to describe the flattening of the lateral distribution near the shower axis, the fit would be poorly confined for all events which have no antenna close to the shower axis. Thus, we have decided to use an exponential LDF with an amplitude and a slope parameter, as already in Refs.~\cite{ApelArteagaAsch2010, ApelLOPES_MTD2012} -- in particular since a power law as alternative 2-parameter LDF has been shown to be inappropriate for LOPES events \cite{ApelArteagaAsch2010}.

Even with the simple exponential LDF we can test whether the simulations reproduce the more complex features of the true lateral distribution in two different ways. First, we compare individual events and see generally a good agreement in the shape. Second, we compare the dependencies of the slope parameter on the shower inclination and the mean axis distance where the event is observed. The fact that there is a dependence reflects that the true LDF is indeed not a pure exponential. By comparing the measured dependence with the simulated one, we can consequently test (at least in first order) whether simulations would fail to reproduce the general shape of the true LDF. I.e., the complex nature of the true lateral distribution is reflected in the dependencies of the LDF on the shower parameters and can be tested this way. For this reason, testing simulations against measurements does not require a complex LDF describing all the detailed features of the lateral distribution. Instead, it is useful to choose a simple LDF which gives a sufficient description of each individual event. This allows for large statistics of events and a relatively low uncertainty of the fit parameters at the same time. For other purposes, the decision for a specific LDF might be different, though.

Following this goal, we slightly modified the exponential LDF used in Ref.~\cite{ApelArteagaAsch2010}: First, we use the amplitude at an axis distance of $100\,$m as fit parameter instead of the amplitude at the shower core, since $100\,$m is close to the mean axis distance of about $120\,$m and close to the distance where least fluctuations due to the primary particle type are expected for LOPES ($70 - 90\,$m) \cite{PalmieriLOPES_ARENA2012}. This reduces the uncertainty of the amplitude fit parameter while still describing the same physics. Second, we switched from the original slope parameter $R_0$ (distance over which the amplitude decreases by a factor 1/e) to its reciprocal $\eta=1/R_0$. There are two reasons for this change: First, some events show flat lateral distributions with an original slope parameter $R_0$ much larger than the lateral extension of LOPES, which is about $200\,$m. For such flat events, exact values of $R_0$ can differ a lot while the absolute values of $\eta$ are similar. Second, there are a few events with rising lateral distributions, i.e.~with a negative $\eta$, which for technical reasons would be difficult to fit using $R_0$. Thus, $\eta=1/R_0$ is the physically more sensible parameter. Therefore, we use the following fit function for the measured and simulated lateral distributions:

\begin{equation}
\label{eq_ldfEXP}
\epsilon(d) = \epsilon_{100} \cdot \exp{(- \eta \cdot (d-100\,\mathrm{m}))}
\end{equation}

with the amplitude (electrical field strength) $\epsilon(d)$ at a distance to the air shower axis $d$, the amplitude at $100\,$m as amplitude fit parameter $\epsilon_{100}$, and the slope fit parameter $\eta$.

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{dispLOPES_RMSvsEpsKG_EW.eps}
\caption{Spread of the amplitudes in individual antennas around the exponential LDF fit for LOPES events with shower core in the KASCADE array and the Grande extension, respectively (cf. Fig.~\ref{fig_eventMap}). Most Grande events have larger uncertainties in $\epsilon_{100}$ than KASCADE events, since their shower core is at a larger distance to the LOPES antennas.}
\label{fig_dispersionFromLDFfitLOPES}
\end{figure}

We checked whether a uniform exponential LDF is indeed a sufficient approximation for individual events, although the \lq real\rq~lateral distribution is more complex. Therefore, we studied the spread of the amplitudes measured in individual antennas around the exponential LDF fit. If the exponential LDF were a perfect description of the measurements, we would expect a spread only due to measurement uncertainties. In this case the spread is expected to decrease with rising signal-to-noise ratio, and thus with rising amplitude of the event. The spread indeed decreases with increasing amplitude (Fig.~\ref{fig_dispersionFromLDFfitLOPES}), however, it does not completely vanish since the exponential LDF is only a simplification. Still, for most events the spread seems to be dominated by noise, and the remaining spread at high amplitudes is small. Therefore, a uniform exponential LDF is justified for the purpose of comparing simulations to LOPES measurements, and the uncertainties introduced by this simplification will be small compared to other uncertainties.

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{noiseBehavior.eps}
\caption{True amplitude of test pulses (before adding noise) as function of the test pulse amplitude after adding noise to the pulses. All amplitudes are normalized to the noise level. Thus, the amplitude after adding noise is the signal-to-noise ratio. The profile reflects the mean change of the amplitude due to noise, and the standard deviation (vertical bars) is used as a measure for the uncertainty of an amplitude measurement at a certain signal-to-noise ratio.} \label{fig_noiseDependency}
\end{figure}

\subsection{Uncertainties}

Numerous systematic effects have been studied to check whether they influence the amplitude or slope parameter of the lateral distribution. The three main sources of uncertainties are the high ambient radio background at the site of the LOPES experiment, the calibration uncertainty of the absolute amplitude scale, and the energy uncertainty of the KASCADE-Grande reconstruction used as input for the REAS 3.11 and CoREAS simulations, respectively. All other studied effects can either be avoided or are negligible as summarized in the following subsections. However, there remain two uncertainties which are difficult to quantify: first, the simplification in the conversion from measured pulse amplitudes to electrical field strengths, since the used conversion formula is exact only for east-west polarized signals; second, the simplified treatment of the simulations by just applying a rectangular band filter to the effective LOPES bandwidth instead of a full detector simulation which is not available, yet. Nevertheless, we tested that our main conclusions do not depend on these two simplifications: first, all statements on the compatibility between the measurements and the simulations are also true when only events with arrival directions are selected for which the simplified treatment gives approximately the true east-west polarization component; second, with a preliminary version of a full detector simulation, we have tested that ignoring the frequency dependence of the experimental hardware properties will only have a slight systematic effect on the slope of the lateral distributions. This might be important when studying the composition of the primary cosmic rays, but it is not important when testing if simulations are compatible to measurements. A summary of all considered uncertainties can be found in table \ref{tab_summaryOfUncertainties}.

\begin{table*}
\centering
\caption{Summary of uncertainties (relative values) which are considered for the comparison of measured and simulated lateral distributions. Some effects are thought to have a negligible effect on the slope parameter $\eta$, which is indicated by '-'.} \label{tab_summaryOfUncertainties}
\vspace{0.1 cm}
\begin{tabular}{lcc}
 & ~~~ uncertainty on $\epsilon_{100}$ ~~~  & ~~~ uncertainty on $\eta$ ~~~ \\
\hline
\textbf{Effects affecting the individual event}   & &\\
Noise                                                & \multicolumn{2}{c}{determined for each event by LDF fit}\\
Energy uncertainty per event                          & $20\,\%$& - \\
Amplitude calibration (environmental effects)~~~~~~~~& $5\,\%$ & - \\
\hline
\textbf{Effects affecting the absolute amplitude scale}              & &\\
Amplitude calibration                                & $35\,\%$ & - \\
Energy scale uncertainty (KASCADE-Grande)            & $20\,\%$ & - \\
Noise: average corrected systematic effect           & $4-5\,\%$ & $7-8\,\%$ \\
Pulse distortion due to dispersion                   & $< 5\,\%$ & $< 2\,\%$ \\
Up-Sampling                                          & $< 2\,\%$ & $< 2\,\%$ \\
\hline
\end{tabular}
\end{table*}

\subsubsection{Noise}
In the noisy environment of a large research center understanding the influence of noise on the measurements is important to avoid any biases on the reconstruction of lateral distributions \cite{SchroederARENA2010}. Noise generates an uncertainty for amplitude measurements which is reflected in the statistical uncertainties of the lateral distribution fit (minimizing $\chi^2$-fit done with ROOT 5.30.06 \cite{ROOT}). Understanding the influence of noise on radio measurements in more detail requires some effort: Radio noise can in principle interfere constructively or destructively with the radio signal and thus increase or decrease the true signal. This effect has been studied by adding measured noise in the analysis software to measured test pulses. Depending on the signal-to-noise ratio, an increase or decrease is not equally likely (see Fig.~\ref{fig_noiseDependency}). For details on the test pulse study and on the method how the noise level is determined in a consistent way see Ref.~\cite{SchroederPhD}. Correcting for the noise influence is important, since noise systematically flattens lateral distributions, because antennas at larger distances typically have a lower signal-to-noise ratio than antennas close to the shower axis. The resulting effect on $\eta$ is an average decrease due to noise by about $7-8\,\%$, and the effect on $\epsilon_{100}$ is an average increase by about $4-5\,\%$. To avoid these biases, every single amplitude measurement in each individual antenna is corrected for the mean effect of noise observed at the corresponding signal-to-noise ratio (profile in Fig.~\ref{fig_noiseDependency}).

\subsubsection{Amplitude calibration}
The uncertainties of the absolute amplitude calibration are discussed in detail in Ref.~\cite{NehlsHakenjosArts2007}. However, all uncertainties in Ref.~\cite{NehlsHakenjosArts2007} are given for the gain factor which is proportional to the square of the amplitude. The systematic calibration uncertainty originates from different sources, which contribute to this analysis in different ways. The largest uncertainty is the gain uncertainty of the calibration antenna. It is given by the manufacturer as $2.5\,$dB which corresponds to a scale uncertainty of about $35\,\%$ for the radio amplitude. The contribution of the LOPES antenna model to the scale uncertainty is significantly smaller than $35\,\%$ and thus neglected for this analysis. Although a scale uncertainty is not important when comparing several LOPES events with each other (as in Ref.~\cite{ApelArteagaAsch2010}), it is crucial when comparing LOPES events with other experiments or simulations. In particular, the scale uncertainty limits the testing power whether a theoretical model can predict the absolute amplitude correctly. Even for a perfect simulation, the measured and simulated amplitude (i.e.~$\epsilon_{100}$) could deviate by up to $35\,\%$, though by the same amount and in the same direction for all events.

In addition, there are two other types of calibration uncertainties, mainly due to the unknown influence of environmental variations. First, the gain of all antennas can change simultaneously in the same direction, since all antennas are subject to more-or-less the same environmental conditions at the same time. This results in a per-event uncertainty of $\epsilon_{100}$ which can be treated in a similar way as a statistical uncertainty and thus is quadratically added to the statistical uncertainty of the LDF fit. Second, since not all antennas react exactly equally to environmental changes, there is an additional uncertainty for the amplitude measurement in each individual antenna which has to be quadratically added to the uncertainty due to noise. Both types of uncertainties are almost an order of magnitude smaller than the scale uncertainty. However, an underestimation of either of the two types of uncertainties might lead to a wrong rejection of the theoretical model. Thus, we conservatively estimate both type of amplitude uncertainties to $5\,\%$ which is slightly more than the total environmental uncertainty stated in Ref.~\cite{NehlsHakenjosArts2007} ($9\,\%$ for the gain corresponding to $4.5\,\%$ for the amplitude).

\subsubsection{Energy uncertainty}
For the KASCADE-Grande energy reconstruction we use formulas based on the measurement of the muon and electron numbers of the KASCADE array \cite{Glasstetter2006} and the numbers of muons and charged particles for KASCADE-Grande \cite{Wommer2007, ApelGrandeEnergySpectrum2012}. The uncertainties of this energy reconstruction formulas have been studied with Monte Carlo simulations. For the events of the LOPES selection, the typical uncertainty is about $20\,\%$ for individual events. Since the radio amplitude scales almost linearly with the energy, this means a corresponding systematic uncertainty of $\epsilon_{100}$ for the simulations, which we add quadratically to the statistical uncertainty when comparing $\epsilon_{100}$ of LOPES and the simulations event-by-event. In addition, the energy reconstruction of KASCADE, respectively, KASCADE-Grande has a scale uncertainty estimated to about $20\,\%$. The effect of this scale uncertainty is similar to the amplitude calibration scale uncertainty, i.e., all simulated amplitudes could be simultaneously by up-to $20\,\%$ too high or too low. For the slope parameter $\eta$, the energy uncertainty is neglected. First, the energy dependence of $\eta$ is expected to be very weak and, second, only the mean $\eta$ is compared between measurements and simulations, since a comparison of $\eta$ for individual events is hampered by shower-to-shower fluctuations.

\subsubsection{Antenna altitude and shower inclination}
All LOPES antennas are placed within $1\,$m at the same altitude. Therefore, any possible effects of the altitude on the measured lateral distributions can be neglected to very good approximation. However, for inclined showers the antennas have a different altitude in shower coordinates, i.e.~relative to a plane perpendicular to the shower axis. Thus, the air shower radio signal can have different propagation lengths to antennas at an equal distance $d$ to the shower axis. The order-of-magnitude of this effect can roughly be estimated by assuming a simple point source in the sky with a spherical wavefront. Although, the real radio wavefront of air showers is approximately conical \cite{SchroederICRC2011}, already the simple spherical approximation reveals that the effect of the shower inclination is negligible for LOPES. When assuming a distance of $5\,$km (typical radius when fitting a spherical wavefront during the beamforming analysis) and a zenith angle $\theta = 30^\circ$, the different antenna heights in shower coordinates lead to a maximum difference in the radio amplitude of $(200\,\mathrm{m} \cdot \sin{30^\circ}) / 5\,\mathrm{km} \approx 2\,\%$, since the lateral extension of LOPES is about $200\,$m. The typical effect ought to be even smaller. Thus, the systematic uncertainties due to the shower inclination can be neglected for LOPES, but might be important for radio arrays with larger lateral extensions.

\subsubsection{Pulse distortion (filter dispersion)}
Pulse distortion by the dispersion of the LOPES bandpass filter can increase or decrease the pulse amplitude depending on the pulse shape \cite{SchroederAschBaehren2010}. However, the pulse shape of measured air shower radio pulses is known only roughly, since the high ambient noise level at LOPES and the limited effective bandwidth ($43-74\,$MHz) do not allow precise pulse shape measurements. Thus, we try to decrease systematic uncertainties by digitally correcting the measured radio pulses for the known pulse distortion of the LOPES bandpass filters. All other LOPES components (e.g., the antennas) contribute less to pulse distortion, and we can take the correctable effect of the bandpass filter dispersion as an upper limit for the remaining pulse distortion of all other LOPES components. It has been studied by reconstructing the lateral distribution with and without digitally correcting for the dispersion of the bandpass filter: The remaining uncertainty of $\epsilon_{100}$ is smaller than $5\,\%$ and the uncertainty of $\eta$ is smaller than $2\,\%$.

\subsubsection{Up-Sampling}
Since the LOPES hardware fulfills the Nyquist sampling theorem by sampling in the second Nyquist domain ($80\,$MHz sampling rate and an effective bandwidth of $43-74\,$MHz), up-sampling is applied. Up-sampling is the correct interpolation between the sampled data points of the time series. To test the effect of up-sampling on the reconstruction of the lateral distributions, the same events have been analyzed with different up-sampling rates: up-sampling is mostly important for the cross-correlation beamforming which precedes the reconstruction of the lateral distribution. The mean effect on the lateral distribution itself is smaller than $2\,\%$ for both $\eta$ and $\epsilon_{100}$. In principle, the effect can be made even smaller by using a higher up-sampling factor. However, this would demand additional computing recourses, and is not necessary, since the total uncertainty on $\eta$ and $\epsilon_{100}$ is dominated by other effects.

\begin{figure*}[t]
 \centering
 \includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000084-1162144854p.eps}
  \hskip 0.13\columnwidth
 \includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000084-1162144854p_Coreas.eps}
  \vskip 0.2 cm
 \includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000084-1162144854Fe.eps}
  \hskip 0.13\columnwidth
 \includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000084-1162144854Fe_Coreas.eps}
  \caption{Example event for the comparison of LOPES lateral distributions to REAS 3.11 simulations (left), and CoREAS simulations (right), for protons (top) and iron nuclei (bottom) as primary particles: amplitudes at the individual antennas and an exponential fit. The stated uncertainties of $\epsilon_{100}$ and $\eta$ are the statistical fit uncertainties. The systematic uncertainties are given as bands: a $35\,\%$ scale uncertainty for the LOPES measurements and a systematic uncertainty of typically $20\,\%$ for each simulated event due to the energy uncertainty of KASCADE-Grande. For reference, also the energy $E$, the zenith angle $\theta$ and the azimuth angle $\phi$ are given.}
 \label{fig_REAS3exampleLateralDistributions}
\end{figure*}

\section{Simulations}
\label{sec_simulations}
For the comparison of the lateral distribution of LOPES measurements with the lateral distribution predicted by simulations, REAS (v 3.11) and CoREAS (v 1.0) were selected to model the radio emission from air showers. Both are publicly available Monte Carlo simulation codes developed in the frame of LOPES. In principle, both simulate the same physics, i.e., they calculate the radio emission from air showers by superposing the radiation of the individual particles of the air shower. Both, the established REAS simulations and the new CoREAS simulations, are based on the endpoint formalism, i.e. no specific assumptions on the emission mechanism have to be made \cite{LudwigHuege2010, James2011}.

In principle, any radiation originating from the acceleration, creation, annihilation or disappearance of air-shower particles can be described. In practice, the air showers are simulated with CORSIKA \cite{HeckKnappCapdevielle1998} using the hadronic interaction models QGSJetII.03 \cite{OstapchenkoQGSjetII2006} and FLUKA \cite{BattistoniFLUKA2007, FerrariFLUKA2005}. Other hadronic interaction models might result in a slightly different radio emission, depending on how much the electromagnetic air-shower component changes. Since only the relativistic air-shower electrons and positrons have a significant effect on the simulated radio emission \cite{LudwigPhD}, other particles are neglected. Moreover, acceleration by electric fields is neglected, since experimental results indicate that atmospheric electric fields do not have any significant effect on the radio emission during normal weather conditions \cite{BuitinkApelAsch2006, LopesThunderstormARS2011}. Finally, REAS 3.11 and CoREAS both consider a realistic, height-dependent refractive index of the atmosphere, which changes the coherence conditions for any radio emission generated by the air shower.

The difference between REAS 3.11 and CoREAS is that CoREAS is directly implemented in CORSIKA, while REAS 3.11 is a separate software which makes use of output generated by CORSIKA. For REAS, the information on the electrons and positrons is taken from histograms, thus losing information on the individual particles. Consequently, CoREAS is expected to provide a more accurate simulation of the same physics. Nevertheless, we see in the performed comparisons that REAS 3.11 gives a better description of the LOPES data, and thus compare to both simulation codes.

For the comparison between the simulations and LOPES data as shown in this article, each event was simulated for two different types of the primary particle of the air shower, namely protons and iron nuclei (see Fig.~\ref{fig_REAS3exampleLateralDistributions} for example lateral distributions). For this, a simulated air shower reproducing the observables measured by KASCADE is chosen, in particular the geometry, the energy and the muon number. Furthermore, before comparing the output of the simulations with the data obtained with LOPES, a frequency filter needs to be applied to the simulations since they are performed for an unlimited frequency band and LOPES is measuring in the effective band of $43-74\,$MHz. The filtering of both the REAS and the CoREAS simulations to a finite observing bandwidth is performed with REASPlot (a helper application included in the REAS package) using an idealized rectangle filter.

\begin{figure*}[t]
\centering
\includegraphics[width=0.65\textwidth]{simchain_muon.eps}
\caption{Flow chart of the simulation chain for the production of the REAS 3.11 and CoREAS simulations used in this article. In the ellipses are the input parameters determining the air shower physics. In the hexagons are parameters or information from the programs themselves for the following simulation program.} \label{fig_simchain_muon}
\end{figure*}

\subsection{Selection of CORSIKA shower by muon number}
\label{sec_simSelection}
For the purpose of testing simulation codes against data, it would be ideal if one could reproduce the experimentally observed shower with all its properties. For the geometry and the energy this is relatively easy, since these properties can be selected as input parameters for the shower simulations, in our case CORSIKA. However, due to shower-to-shower fluctuations, each simulated shower (with the same energy and geometry) is different. This leads to systematic uncertainties when comparing simulations and measurements event-by-event, and this is the reason why we give quantitative conclusions based on histograms and average values.

To select one of the numerous air-shower simulations which are possible for each event, we developed a new methodology taking into account the measurements of the KASCADE particle detector array, i.e., in addition to the shower geometry and energy, in particular the muon number. A selection of a specific shower for the radio simulation is necessary, since only a few air-shower and REAS 3.11 simulations per measured cosmic-ray event are processable due to limited computing times. Of the many possible simulated showers for the geometry and energy given by KASCADE-Grande, we decided to choose a shower with a muon number similar to the measurement, instead of a random shower.

The methodology we developed for the shower selection is summarized in Fig.~\ref{fig_simchain_muon}. It makes use of CONEX \cite{BergmannCONEX2007, PierogICRC2011}, an air-shower simulation code incorporated in the used version of CORSIKA. CONEX saves computing time compared to a standard CORSIKA simulation, since it uses a Monte Carlo simulation only for the first few air-shower interactions, and then makes use of cascade-equations to calculate the air shower. Thus, with CONEX is is possible to simulate several 100 showers within reasonable computing time, and then select one of these showers for the complete CORSIKA simulation.

The method for shower selection might be especially useful for multi-hybrid observatories, it can also be used to select showers by $X_\textrm{max}$ instead of by muon number. Therefore, we give a detailed description of the method in Appendix B. It can be used to improve the method and transfer it to other experiments.

If a measurement of additional, complementary shower parameters were available (e.g., $X_\textrm{max}$ and muon number), then in principle it should be possible to beat shower-to-shower fluctuations by selecting a simulated shower which reproduces all measured shower parameters. It is clear, that in the present analysis we can only partially reach this goal, because KASCADE-Grande features only a measurement of the muon number, but no measurement of $X_\textrm{max}$. In addition, there are systematic uncertainties, since there are no hadronic models available which completely reproduce all measured properties of air-shower muons. For these reasons, the shower selection by muon number has on average only a small effect on the impact of shower-to-shower fluctuations, at least in the way we performed it (see Appendix B for details). In particular, the average values and the standard deviations of the slope parameter $\eta$ and the amplitude parameter $\epsilon_{100}$ change only marginally when the showers are selected by muon number instead of randomly. Still, for principle reasons we want to use all observables provided by KASCADE-Grande. Consequently, we decided to use the muon selection, although it has no significant effect on the conclusions of the analysis. In any case, such a selection criterion might be useful for future studies at other experiments.

\begin{figure*}[p!]
  \centering
  \includegraphics[width=0.77\columnwidth]{epsLOPESvsREAS3_11_p_EW.eps}
  \hskip 0.25\columnwidth
  \includegraphics[width=0.77\columnwidth]{epsLOPESvsCOREAS_p_EW.eps}

  \includegraphics[width=0.77\columnwidth]{epsLOPESvsREAS3_11_Fe_EW.eps}
  \hskip 0.25\columnwidth
  \includegraphics[width=0.77\columnwidth]{epsLOPESvsCOREAS_Fe_EW.eps}
  \caption{Per event comparison of the amplitude parameter $\epsilon_{100}$ between LOPES measurements and REAS 3.11 (left) and CoREAS (right), respectively, simulations for protons (top) and iron nuclei (bottom) as primary particles. The dashed lines around the solid line indicate the scale uncertainty of $\pm 35\,\%$ due to the absolute amplitude calibration of LOPES. The dotted lines around the long-dashed line indicate the $\pm 20$\,\% scale uncertainty of the KASCADE-Grande energy reconstruction. These lines have been shifted by the scale mismatch between the simulations and the measurements (see Fig.~\ref{fig_spread} for the scale factors).}
   \label{fig_amplitudeComparison}
\end{figure*}

\begin{figure*}[p!]
\centering
  \includegraphics[width=0.92\columnwidth]{histEpsdevLopesVsREAS3_11_p_Fe_EW.eps}
  \hskip 0.12\columnwidth
  \includegraphics[width=0.92\columnwidth]{histEpsdevLopesVsCOREAS_p_Fe_EW.eps}
\caption{Relative deviation of $\epsilon_{100}$. For REAS 3.11 the mean deviation is significantly smaller than the calibration scale uncertainty of $35\,\%$, which, however, is not the case for CoREAS.} \label{fig_histEpsDev}
\end{figure*}

With this, the computing time is optimized while taking into account all shower information available from the KASCADE measurements and selecting a suitable air shower and its radio emission for the comparison. The selection criterion based on the number of muons measured by KASCADE has the advantage that the specific air shower represents the measured air shower with a higher probability than a typical or randomly chosen air shower. However, even when fixing the muon number, there are still shower-to-shower fluctuations. To completely avoid the effect of shower-to-shower fluctuations, one would have to fix at least another shower parameter, e.g., the atmospheric depth of the shower maximum, $X_\textrm{max}$. This is not possible for LOPES, since KASCADE features no $X_\textrm{max}$ measurement. Consequently, the simulation of a single event is not expected to exactly reproduce the measurement of that individual event. Nevertheless, the simulations are expected to reproduce the measurements on average and the statistics of simulated and measured events are sufficiently large to test this.

\begin{figure*}
\centering
\includegraphics[width=0.92\columnwidth]{histEtaLopesVsREAS3_11_p_Fe_EW.eps}
\hskip 0.12\columnwidth
\includegraphics[width=0.92\columnwidth]{histEtaLopesVsCOREAS_p_Fe_EW.eps}
\caption{Comparison of the slope parameter $\eta$ between LOPES measurements and REAS 3.11, respectively, CoREAS simulations for protons and iron nuclei as primary particles.} \label{fig_slopeComparison}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.92\columnwidth]{histEtaErLopesVsREAS3_11_p_Fe_EW.eps}
\hskip 0.12\columnwidth
\includegraphics[width=0.92\columnwidth]{histEtaErLopesVsCOREAS_p_Fe_EW.eps}
\caption{Statistical uncertainty of the slope parameter $\eta$. The uncertainties of the LOPES measurements are generally larger than for the simulations, since the measurements are affected by noise, but the simulations do only contain numerical noise.} \label{fig_slopeUncertainty}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.92\columnwidth]{histRelEpsdevREAS3_11_p_Fe_EW.eps}
\hskip 0.12\columnwidth
\includegraphics[width=0.92\columnwidth]{histRelEpsdevCOREAS_p_Fe_EW.eps}
\caption{Deviation between the measured $\epsilon_{100}$ and the scaled simulated $\epsilon_{100}$ divided by the total uncertainty of the individual events, and a fitted Gaussian for each histogram. The scaling factor $f_\mathrm{scale}$ has been chosen such that the fitted Gaussian is centered around 0. Apart from $f_\mathrm{scale}$, also the resulting standard deviations of the fitted Gaussians are given. In case of a perfect agreement between simulations and measurements, a perfect estimation and no correlation of the individual uncertainties $\sigma_\mathrm{LOPES}$ and $\sigma_\mathrm{sim}$, we would expect a standard deviation of 1. The main reason for the difference between the proton and iron simulations is that the uncertainties of $\epsilon_{100}$ are slightly larger for the proton simulations.} \label{fig_spread}
\end{figure*}

\section{Results}

In this section we compare the LOPES measurements with the REAS 3.11 and CoREAS simulations. With the exception of the absolute amplitude, REAS 3.11 and CoREAS show a very similar behavior, and can both describe the measured lateral distributions. A comparison for the amplitude parameter $\epsilon_{100}$ is shown in Figs.~\ref{fig_amplitudeComparison} and \ref{fig_histEpsDev}. REAS 3.11 generally gives slightly lower amplitudes than measured -- independently whether protons or iron nuclei are used as primary particles for the simulations, but we expect a constant shift of the measured versus the simulated amplitudes of up to $\pm 35\,\%$ due to the calibration scale uncertainty of LOPES. Thus the amplitudes of the REAS 3.11 simulations are fully compatible with the LOPES measurements. However, the amplitudes simulated with CoREAS for the same events are significantly lower, and hardly compatible with the measurements, even when taking into account the calibration scale uncertainty and the energy scale uncertainty of KASCADE-Grande, respectively, KASCADE. This difference between REAS 3.11 and CoREAS is not understood yet. Maybe the information loss due to the histogramming of the particles for the REAS simulations is responsible for this effect. Moreover, we recognize that the agreement between LOPES and REAS 3.11 amplitudes is slightly better for small values of $\epsilon_{100}$, i.e.~for events with lower energy. The reason might be that the energy reconstruction of the original KASCADE experiment has not been developed for energies above $10^{17}\,$eV. Hence, the energy and, consequently the radio amplitude, of some high energy events might be slightly underestimated.

Moreover, we have studied whether the spread visible in Fig.~\ref{fig_amplitudeComparison} is compatible with the expectations due to the uncertainties of the individual data points. A simple $\chi^2$-test, however, would be insufficient because it would be dominated by the mismatch in the scale: since the CoREAS simulations have generally smaller amplitude parameters $\epsilon_{100}$ than REAS, they also have smaller absolute uncertainties, which would lead to a larger $\chi^2$, but does not necessarily mean a worse agreement regarding the spread. Therefore, we have first rescaled both the simulated $\epsilon_{100}$ values and their uncertainties to get rid of the scale mismatch. The $\chi^2$-spread around the shifted lines in Fig.~\ref{fig_amplitudeComparison}, is dominated by the outliers and yields reduced $\chi^2$ values between $1.7$ and $1.9$ for the different cases. This means that we have more outliers in the comparison than expected by a standard Gaussian interpretation of our uncertainties. The reason for these outliers will be further investigated.

Consequently, another statistical test less sensitive to outliers is applied, to judge on the compatibility of the individual deviations in the amplitude comparison (Fig.~\ref{fig_amplitudeComparison}). Fig.~\ref{fig_spread} displays histograms of the individual deviations between the measured and simulated amplitudes $\epsilon_{100}$ divided by the individual uncertainties of each event and taking into account the scale mismatch. The width (= standard deviation of a fitted Gaussian) is slightly smaller than 1. Thus, we either have slightly overestimated the individual uncertainties or there is a correlation between the uncertainties of simulations and measurements. A possible explanation is that a part of the statistical fit uncertainty does not originate from statistical uncertainties of the amplitudes in individual antennas, but from 'true' deviations around the fit, e.g., due to the weak geometrical asymmetry of the radio lateral distribution caused by the interference of the geomagnetic and the Askaryan effect. If these true deviations are reproduced by the simulations correctly, this would indeed lead to some degree of correlation between the fit uncertainties of the simulations and measurements. Consequently, with the exception of a few events with a very large deviation, the spread in Fig.~\ref{fig_amplitudeComparison} is compatible with our expectations, and does not express any incompatibility between simulations and measurements.

The slope parameter $\eta$ is compared on average (Fig.~\ref{fig_slopeComparison}), but not per individual event, since it depends on the atmospheric depth of the shower maximum \cite{ApelLOPES_MTD2012}, and there is no way to reproduce the (unknown) shower maximum of the measured events with the individual simulations. Experimental observation indicates that the composition in the relevant energy range is relatively heavy \cite{ApelIronKnee2011}, and that the average mass almost certainly is in-between the masses of protons and iron nuclei. Thus, we expect that the measured distribution of $\eta$ lies in-between the simulations done for the extreme cases of a pure proton and a pure iron primary composition, if the simulations are compatible with the measurements. This is the case for both the REAS 3.11 and the CoREAS simulations.

The qualitative difference between proton and iron simulations is reasonable, and confirms earlier results that the slope of the lateral distribution is sensitive to the composition of the primary cosmic rays \cite{HuegeUlrichEngel2008, deVries2010, PalmieriLOPES_ARENA2012}. Thus, protons on average interact deeper in the air and have a steeper radio lateral distribution than iron nuclei. However, quantitative results for the reconstruction of the primary mass with radio lateral distributions have to be taken very carefully, since they depend on the used hadronic interaction models for the CORSIKA simulations. Moreover, the reconstruction of the composition will likely be affected by the simplified application of the detector properties for the reconstruction of the simulated radio amplitude. However, comparing the difference of the mean $\eta$ between the proton and iron simulations (approx.~$1.7\,$km\textsuperscript{-1}, Fig.~\ref{fig_slopeComparison}, and also Fig.~\ref{fig_corEta}) with the typical uncertainty of $\eta$ (Fig.~\ref{fig_slopeUncertainty}) provides an idea of the precision a possible mass reconstruction could have. For the LOPES measurements, which are affected by the high level of ambient radio background at Karlsruhe, the mean uncertainty of $\eta$ is in the same order as the typical difference between proton and iron induced showers. For the simulations, which do only contain numerical noise, the mean uncertainty of $\eta$ is significantly smaller than the difference between proton and iron induced showers. This shows that the primary mass can in principle be reconstructed with radio measurements, provided that the experiment is located in a region with low radio background or dimensioned for higher primary energies.

\begin{figure*}[t!]
\centering
\includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000164-1240619537Fe.eps}
\hskip 0.13\columnwidth
\includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000164-1240619537Fe_Coreas.eps}
\caption{Lateral distribution with a negative slope parameter in the LOPES measurements which in contrast to earlier versions of REAS not considering the refractive index of the atmosphere, is now reproduced by both REAS 3.11 and CoREAS.} \label{fig_risingLDF}
\end{figure*}

\begin{figure*}
  \centering
  \includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000018-1146508536p.eps}
  \hskip 0.13\columnwidth
  \includegraphics[width=0.92\columnwidth]{lateral-EW-SIM000018-1146508536p_Coreas.eps}
  \caption{Example event for which a flattening towards the shower axis is observed.}
   \label{fig_flatEvents}
\end{figure*}

Also with respect to the shape of the lateral distribution both the REAS 3.11 and the CoREAS simulations are in much better agreement with the measurements than previously tested simulations which did not take into account the refractive index of the air. Already in Ref.~\cite{ApelArteagaAsch2010} we observed that the lateral distribution of some events flattens towards the shower core. With the improved fitting function, we discovered that in a few cases the slope parameter $\eta$ is even negative, i.e.~the lateral distribution rises (see Fig.~\ref{fig_risingLDF} for a rising lateral distribution, and Fig.~\ref{fig_flatEvents} for a flattening lateral distribution). First indications for rising lateral distributions have already been described in Ref. \cite{Hazen1969}, and a first theoretical explanation has been given by Ref.~\cite{AllanICRC1971}, which we now confirm. Both the flattening and the rising lateral distributions can be caused by the refractive index of the air which changes the coherence conditions \cite{deVries2011}. Only at the Cherenkov angle, the radio emission from all stages of the shower development arrives at approximately the same time, which results in a higher radio amplitude at an axis distance of about $120\,$m. Also in first measurements reported by LOFAR \cite{LOFAR_ICRC2011}, indications for such a bump can be seen. It can be interpreted as result of a Cherenkov-like beaming. However, one must carefully keep in mind that the main origin of the radio emission is not the polarization of the ambient medium by constantly moving charges as for normal Cherenkov light detectable at near-optical frequencies, but still the geo-magnetic deflection of electrons and positrons in the air shower, and the time-variation of the charge excess. Depending on the shower geometry, this bump can lead to rising lateral distributions -- in particular, when most of the antennas have an axis distance below $120\,$m. Concluding, both REAS 3.11 and CoREAS seem to reproduce the shape of measured lateral distributions correctly, at least in the frequency band and distance range of LOPES.

\begin{figure*}[p!]
  \centering
  \includegraphics[width=0.92\columnwidth]{energyVsEpsProfilesREAS3_11_EW.eps}
  \hskip 0.13\columnwidth
  \includegraphics[width=0.92\columnwidth]{energyVsEpsProfilesCOREAS_EW.eps}
  \caption{Profiles of the amplitude parameter $\epsilon_{100}$ of measured and simulated lateral distributions versus the primary energy $E$. The horizontal bars represent the bin width; the vertical bars denote the statistical error calculated as standard deviation divided by the square root of the number of events, where the bins without vertical bar contain only one event. While the general shift between the measurements and the REAS 3.11 simulations can easily be explained by the calibration scale uncertainty, the amplitude discrepancy between CoREAS and LOPES is significantly larger than the known scale uncertainties.}
   \label{fig_corEps}
\end{figure*}

\begin{figure*}[p!]
  \centering
  \includegraphics[width=0.92\columnwidth]{zenithVsEtaProfilesREAS3_11_EW.eps}
  \hskip 0.13\columnwidth
  \includegraphics[width=0.92\columnwidth]{zenithVsEtaProfilesCOREAS_EW.eps}
  \vskip 0.2 cm
  \includegraphics[width=0.92\columnwidth]{rMeanVsEtaProfilesREAS3_11_EW.eps}
  \hskip 0.13\columnwidth
  \includegraphics[width=0.92\columnwidth]{rMeanVsEtaProfilesCOREAS_EW.eps}
  \caption{Profiles of the slope parameter $\eta$ of measured and simulated lateral distributions versus the zenith angle $\theta$ and the mean axis distance of the events. Although, there seems to be a slightly different dependence of $\eta$ with the mean axis distance, we cannot exclude that simulations and measurements are compatible with each other. Our event selection and reconstruction might be biased for distant events, i.e., events whose core is not contained in the LOPES array.}
   \label{fig_corEta}
\end{figure*}

\subsection{Correlations with shower parameters}

In the previously published analysis focusing on the experimental results \cite{ApelArteagaAsch2010}, dependencies of the amplitude and slope parameter with other shower parameters were studied. With the improved analysis and increased statistics we confirm the dependencies reported in this previous paper for the LOPES measurements. Furthermore, we have tested whether REAS 3.11 and CoREAS can reproduce the characteristics of the measurement, and find a general agreement between simulations and measurements. In particular, the simulations reproduce the dependencies on the arrival direction, which is implicitly taken into account in our study, since each simulated event is made using the geometry of the measured shower. For instance, due to the geomagnetic effect, the amplitude parameter $\epsilon_{100}$ depends both in the measurements and in the simulations in the same way on the azimuth and zenith angle.

Moreover, the amplitude parameter $\epsilon_{100}$ is clearly correlated with the primary energy (Fig.~\ref{fig_corEps}), and shows only a weak dependence on the composition of the primary cosmic rays. This confirms that the primary energy can be estimated by measuring the lateral distribution of the radio emission (and the arrival direction), as shown already in Ref.~\cite{HuegeUlrichEngel2008}, and confirmed at CODALEMA \cite{RebaiCODALEMAenergy2012} and LOPES \cite{PalmieriLOPES_ARENA2012}. Using REAS 2 and REAS 3 simulations not yet including the refractive index of the air, Refs.~\cite{HuegeUlrichEngel2008} and \cite{PalmieriLOPES_ARENA2012} give an optimal distance to measure the radio amplitude for energy reconstruction, which is the distance for which the mass of the primary particle has a minimal influence on the radio amplitude. In newer studies \cite{PalmieriICRC2013}, we have shown that this typical distance range changes only slightly to $70-100\,$m, when CoREAS simulations including the refractive index are used. This distance is by chance close to the typical antenna distance of $100\,$m which we use for the fit parameter $\epsilon_{100}$. In Refs. \cite{SchroederLOPES_ARENA2012, PalmieriLOPES_ARENA2012}, we demonstrated by comparing the LOPES to the KASCADE-Grande energy reconstruction that a statistical precision for the energy of better than $20\,\%$ can be achieved.

Also for the slope parameter $\eta$ we see the same correlations to other shower parameters in the LOPES measurements and in the simulations. $\eta$ is clearly correlated with the zenith angle $\theta$ and the mean distance of the antennas to the shower axis (Fig.~\ref{fig_corEta}): the more inclined the shower and the smaller the average distance from the antennas to the shower axis, the flatter is the radio lateral distribution. Both effects have to be taken into account if one aims for a reconstruction of the type and mass of the primary cosmic ray particles \cite{ApelLOPES_MTD2012}. An interesting aspect of the correlation of $\eta$ with the mean axis distance is that for near events, the measurements seem to be closer to the proton assumption, and for distant events closer to the iron assumption. However, this does not necessarily mean that the simulations would be incompatible to the measurements. Since the shower core of the distant events is outside of the LOPES array, we likely have a higher detection chance for events with flatter lateral distribution. Thus, the effect can probably be explained by an experimental selection bias, and consequently does not indicate a discrepancy between the simulations and the measurements. Consequently, for any analyses aiming for a reconstruction of the cosmic-ray composition, only events with shower cores contained inside of an antenna array should be used.

\section{Conclusions}

We performed a comparison of radio lateral distributions measured with LOPES and simulated on an event-by-event basis with REAS 3.11 and CoREAS, respectively. For this purpose, and in the limited distance and frequency range of LOPES, a simple, uniform exponential lateral-distribution function is sufficient. The \lq true\rq~lateral distribution is more complex and depends on the shower geometry: in many cases the lateral distribution is flattening towards the shower core and steepening towards larger distances. In some cases the lateral distribution seems to have a bump at about $120\,$m, which even can lead to a rising lateral distribution up to roughly this distance. Consequently, other antenna arrays (e.g., LOFAR \cite{NellesLOFAR_ARENA2012}, AERA \cite{MelissasAERA_ARENA2012}, Tunka-Rex \cite{SchroederTunkaRex_ARENA2012}) covering a larger bandwidth and a larger distance range in individual events might have to use more complicated lateral distribution functions.

The qualitative and quantitative results of the comparison between the simulations and measurements reflect a remarkable progress in the understanding of the radio emission generated by air-showers: In contrast to previously tested simulations, REAS 3.11 can reproduce all tested features of the lateral distribution, including the absolute amplitude, the shape, and correlations with other shower observables within the uncertainties. Thus, we conclude that REAS 3.11 can correctly reproduce the radio amplitude in the effective frequency band of LOPES ($43-74\,$MHz), and in the distance range of LOPES $d_\textrm{axis}\lesssim300\,$m.

Also CoREAS can reproduce the shape of the lateral distribution and the correlations with other shower parameters. The absolute amplitude of the radio signal predicted by CoREAS is lower by a factor of approximately two than for REAS 3.11. Thus, the absolute amplitude predicted by CoREAS is only marginally compatible with the LOPES measurements. The reason for the difference in the amplitude scale is not known yet. It is interesting since technically CoREAS is the more accurate model: it does not loose information of the CORSIKA air-shower simulation by histogramming, as REAS 3.11 does. Of course, we can not completely exclude that the mismatch is due to some kind of yet undiscovered mistake, e.g., an unknown error in either the simulations themselves, the analysis procedure, or an underestimation of the scale uncertainties, but we are not aware of any unsolved issue which could explain this difference. Perhaps, there still remains an important aspect of the air shower and its radio emission to be understood. In the future, we can also test other radio simulation codes against LOPES data to study this issue. Furthermore, LOFAR, a much denser antenna array operating approximately in the same frequency and distance range as LOPES, will be able to test the structure of the lateral distribution in greater detail \cite{BuitinkLOFAR_ICRC2013}. Complementary, the sparser antenna array AERA will be able to study the lateral distribution at large distances, since it features an extension of several km \cite{SchroederAERA_ICRC2013}.

With the exception of the absolute amplitude, no significant difference is observed by us between REAS 3.11 and CoREAS for the bandwidth and distance range of LOPES. However, a purely theoretical comparison between REAS 3.11 and CoREAS indicates that differences can be expected at high frequencies and very small axis distances \cite{HuegeARENAtheory2012}. Moreover, the simulated radio emission depends on the hadronic interaction models used for the underlying air-shower simulations. Recently, different hadronic interaction models converged, but still, all available models have some deficits describing air showers at ultra-high energies. Thus, a \lq correct\rq~hadronic interaction model could shift the absolute scale of the radio amplitude, for two reasons: first, the underlying energy scale of KASCADE-Grande depends on the hadronic interaction models; second, for a given energy, the radio emission depends on the electromagnetic air-shower component. In case that these systematic effects act by chance in the same direction as the calibration scale uncertainty of LOPES, we cannot exclude that the measured amplitudes might become compatible with the CoREAS simulations.

Both REAS 3.11 and CoREAS reproduce now rising lateral distributions which we measure in a few LOPES events. We interpret this as effect of the refractive index of air, since previous versions of REAS not including the refractive index could not reproduce this feature. Consequently, in addition to the dominant geomagnetic emission and the Askaryan effect, the refractive index of air apparently plays an important role for the radio emission of air showers. This is because the refractive index changes the coherence conditions for the radio emission which leads to a Cherenkov-like time compression of the radio signal \cite{deVries2011} and, consequently, to an amplification of the radio emission at the Cherenkov angle causing the bump mentioned above. Still, one should keep in mind one essential difference to the emission of Cherenkov light by air-showers at near-optical frequencies: While this 'standard' Cherenkov light is caused by constant charges moving with approximately constant, superluminal speed, the dominant origin of the radio emission is the acceleration and variation of charges in the air shower. The Cherenkov emission due to the constant movement of the charges is considered negligible at radio frequencies \cite{James2011}, and not included in the tested simulation codes.

Finally, we confirm with the measurements and simulations that the radio lateral distribution is sensitive to the energy and mass of the primary cosmic-ray particles. The amplitude is clearly correlated with the energy, and the slope depends on the type and mass of the primary particle, which is expected since we already know that the slope of measured lateral distributions is sensitive to the longitudinal shower development \cite{ApelLOPES_MTD2012}. At LOPES, this method seems to be limited by the high level of ambient radio background. Future experiments at locations with lower background have to show whether the precision of other established detection techniques for air showers like air-fluorescence or air-Cherenkov light detection can be achieved.

\section*{Acknowledgments}
LOPES and KASCADE-Grande have been supported by the German Federal Ministry of Education and Research. KASCADE-Grande is partly supported by the MIUR and INAF of Italy, the Polish Ministry of Science and Higher Education and by the Romanian Authority for Scientific Research UEFISCDI (PNII-IDEI grant 271/2011). This research has been supported by grant number VH-NG-413 of the Helmholtz Association.  The present study is supported by the 'Helmholtz Alliance for Astroparticle Physics - HAP' funded by the Initiative and Networking Fund of the Helmholtz Association, Germany. The authors thank Steffen Nehls for fruitful discussions and his previous work at LOPES.

redpajama_set_name: RedPajamaArXiv


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