RAPPORT D’ACTIVITEIll-posedness formulation of the emission source localization in the radio-detection arrays of extensive air showers initiated by Ultra High Energy Cos-mic RaysAHMED REBAI1 FOR THE CODALEMA COLLABORATION,1 Subatech IN2P3-CNRS/Universite de Nantes/Ecole des Mines de Nantes, Nantes, France.

ahmed.rebai@subatech.in2p3.fr

Abstract: In the field of radio detection in astroparticle physics, many studies have shown the strong dependenceof the solution of the radio-transient sources localization problem (the radio-shower time of arrival on antennas)such solutions are purely numerical artifacts. Based on a detailed analysis of some already published results ofradio-detection experiments like : CODALEMA 3 in France, AERA in Argentina and TREND in China, wedemonstrate the ill-posed character of this problem in the sens of Hadamard. Two approaches have been used asthe existence of solutions degeneration and the bad conditioning of the mathematical formulation problem. Acomparison between experimental results and simulations have been made, to highlight the mathematical studies.Many properties of the non-linear least square function are discussed such as the configuration of the set ofsolutions and the bias.

Keywords: UHECR; radio-detection; antennas; non-convex analysis; optimization; ill-posed problem

1 IntroductionThe determination of the ultra-high energy cosmic rays(UHECR) nature is an old fundamental problem in cosmicrays studies. New promising approaches could emerge fromthe use of the radio-detection method which uses, throughantennas, the radio emission that accompanies the exten-sive air shower emission (EAS). Several experiences likeCODALEMA [1] in France and LOPES [2] in Germanyshown the feasibility and the potential of the method. Manyparameters have been reconstructed as the arrival direction,the shower core at ground, the electric field lateral distribu-tion function and the primary particle energy. However, thetemporal radio wavefront characteristics remain still poorlydetermined. In fact, the arrival time distribution defined bythe filtred radio signal maximum amplitude. On the oth-er hand, the migration of small scale radio experiments tolarge scale experiments with huge surfaces of several ten-s of 1000 km2 using autonomous antennas, is challenging.This technique is subjected limitations in regard to UHE-CR recognition, due to noises induced by human activities(high voltage power lines, electric transformers, cars, trainsand planes) or by stormy weather conditions (lightning).Figure 1 shows a typical reconstruction of sources obtainedwith the CODALEMA experiment [?], by using a spheri-cal wave minimization. Such results are also observed inmany others radio experiments [?, ?]. In general, one ofthe striking results is that the radio emission sources arereconstructed with great inaccuracy, although they are fixedand although the huge measured events number.

The frequentist approach uses an objective functionminimization that depends the wavefront shape, employingthe arrival times and antennas positions.

2 Reconstruction with common algorithmsThe performances of different algorithms has been testedusing the simplest test array of antennas. Within the con-straints imposed by the number of free parameters used for

Fig. 1: Typical result of reconstruction of two entropicemitters at ground, observed with the stand-alone stations ofCODALEMA, through standard minimization algorithms.Despite the spreading of the reconstructed positions, thesetwo transmitters are, in reality, two stationary point sources.

reconstruction, we choose an array of 5 antennas for whichthe antennas positions −→ri = (xi,yi,zi) are fixed (see Fig. 4)(this corresponds to a multiplicity of antennas similar tothat sought at the detection threshold in current setups).

A source S with a spatial position −→rs = (xs,ys,zs) is setat the desired value. Assuming ts the unknown instant of thewave emission from S, c the wave velocity in the mediumconsidered constant during the propagation, and assumingthat the emitted wave is spherical, the reception time ti oneach antenna i ∈ {1, . . . ,N} can written:

ti = ts +

√(xi− xs)

2 +(yi− ys)2 +(zi− zs)

2

c+G(0,σt)

where G(0,σt) is the Gaussian probability density functioncentered to t = 0 and of standard deviation σt . This latterparameter stand for the the global time resolution, which

Rebai A. Ill-posedness formulation of the emission source localization in the radio-detection arraysRAPPORT D’ACTIVITE

Fig. 2: ypical result of the reconstruction of a source ob-served with AERA (Auger Engineering Radio Array) exper-iment for an entropic emitter at ground, the minimizationhas been implemented usign Minuit with the Simplex andMigrad algorithm and shown in [10].

Fig. 3: Reconstruction results in the TREND experimentwith the same algorithm in the case of an radio emitterlocated on ground inside the antennas array. The algorithmprovide a good estimation of the emitter position.

Fig. 4: Scheme of the antenna array used for the simulations.The antenna location is took from a uniform distribution of1 m width.

depends as well on technological specifications of theapparatus than on analysis methods.

The theoretical predictions are compared to the recon-structions given by the different algorithms. The latter aresetup in two steps. First, a planar adjustment is made, inorder to pres-tress the region of the zenith angle θ and az-imuth angle φ of the source arrival direction. It specifiesa target region in this subset of the phase space, reducing

the computing time of the search of the minimum of theobjective function of the spherical emission. Reconstructionof the source location is achieved, choosing an objective-function that measures the agreement between the data andthe model of the form, by calculating the difference be-tween data and a theoretical model (in frequentist statistics,the objective-function is conventionally arranged so thatsmall values represent close agreement):

f (~rs, t∗s ) =12

N

∑i=1

[‖−→rs −−→ri ‖2− (t∗s − t∗i )

2]2

(1)

The partial terms ‖−→rs −−→ri ‖2− (t∗s − t∗i )2 represents the

difference between the square of the radius calculated usingcoordinates and the square of the radius calculated usingwave propagation time for each of the N antennas. Thefunctional f can be interpreted as the sum of squared errors.Intuitively the source positions −→rs at the instant ts is onethat minimizes this error.

In the context of this paper, we did not use genetic al-gorithms or multivariate analysis methods but we focusedon three minimization algorithms, used extensively in sta-tistical data analysis software of high energy physics[?, ?]:Simplex, Line-Search and Levenberg-Marquardt (see table1). They can be found in many scientific libraries as theOptimization Toolbox in Matlab, the MPFIT in IDL and thelibrary Minuit in Root that uses 2 algorithms Migrad andSimplex which are based respectively on a variable-metriclinear search method with calculation of the objective func-tion first derivative and a simple search method. For thepresent study, we have used with their default parameters.

We tested three time resolutions with times values tookwithin 3σt .

• σt = 0ns plays the role of the perfect theoreticaldetection and serves as reference;

• σt = 3ns reflects the optimum performances expectedin the current state of the art;

• σt = 10ns stands for the timing resolution estimateof an experiment like CODALEMA [?].

For every source distance and temporal resolution, onemillion events were generated. Antenna location was takenin a uniform distribution of 1m width. A blind search wassimulated using uniform distribution of the initial rs valuesfrom 0.1km to 20km. Typical results obtained with oursimulations are presented in Figures 5 and 6. The summaryof the reconstructed parameters is given in table ??.

Whatever the simulations samples (versus any source dis-tances, arrival directions, time resolutions), (also with sever-al detector configurations) and the three minimization algo-rithms, large spreads were generally observed for the sourcelocations reconstructed. This suggests that the objective-function presents local minima. Moreover, the results de-pend strongly on initial conditions. All these phenomenamay indicate that we are facing an ill-posed problem. In-deed, condition number calculations [?] (see Fig. 2), whichmeasures the sensitivity of the solution to errors in the data(as the distance of the source, the timing resolution, etc.),indicate large values (> 104), when a well-posed problemshould induce values close to 1.

To understand the observed source reconstruction pat-terns, we have undertaken to study the main features of thisobjective-function.

Rebai A. Ill-posedness formulation of the emission source localization in the radio-detection arraysRAPPORT D’ACTIVITE

Table 1: Summary of the different algorithms and methods used to minimize the objective-function. The second rowindicates framework functions corresponding to each algorithm; third recalls the framework names. The key informationused for optimization are recalled down, noting that a differentiable optimization algorithm (ie. non-probabilistic and non-heuristic) consists of building a sequence of points in the phase space as follows: xk+1 = xk + tk.dk, and that it is rankedbased on its calculation method of tk and dk parameters (see [11, 12, 13]).

Minimization algorithms Levenberg-Marquardt Simplex Line-SearchLibraries lsqnonlin - MPFIT fminsearch - SIMPLEX MIGRAD - lsqcurvefitSoftware Optimization Toolbox Mat-

lab - IDLOptimization Toolbox Mat-lab - MINUIT-ROOT

Optimization Toolbox Mat-lab - MINUIT-ROOT

Method Principles Gauss-Newton methodcombined with trust regionmethod

Direct search method Compute the step-size by op-timizing the merit functionf (x+ t.d)

Used information Compute gradient (∇ f )kand an approximate hessian(∇2 f )k

No use of numerical or ana-lytical gradients

f (x + t.d, d) where d is adirection descent computedwith gradient/hessian

Advantages / Disadvantages Stabilize ill-conditioned Hes-sian matrix / time consumingand local minimum trap

No reliable informationabout parameter errors andcorrelations

Need initialization with an-other method, give the op-timal step size for the opti-mization algorithm then re-duce the complexity

Fig. 5: Results of the reconstruction of a source with aradius of curvature equal to 1 and 10 km with the LVMalgorithm. For Rtrue = 1km, the effect of the blind searchleads to non-convergence of the LVM algorithm, wheninitialization values are greater than Rtrue = 1km.

Several properties of the objective-function f were stud-ied: the coercive property to indicate the existence of at leastone minima, the non-convexity to indicate the existence ofseveral local minima, and the jacobian to locate the critical

Fig. 6: Results of the reconstruction of a source with aradius of curvature equal to 1 and 10 km with the Simplexalgorithm.

points. (Bias study, which corresponds to a systematic shiftof the estimator, is postponed to another contribution). Inmathematical terms, this analysis amounts to:

• Estimate the limits of f to make evidence of criticalpoints; obviously, the objective function f is positive,

Rebai A. Ill-posedness formulation of the emission source localization in the radio-detection arraysRAPPORT D’ACTIVITE

Fig. 7: Condition numbers obtained using the formulaCond(Q) = ‖Q‖.‖Q−1‖with Q the Hessian matrix (see nex-t section) as a function of the source distance and for dif-ferent timing resolutions. The large values of conditioningsuggest that we face an ill-posed problem.

regular and coercive. Indeed, f tends to +∞ when‖X‖→±∞, because it is a polynomial and containspositive square terms. So, f admits at least a mini-mum.

• Verify the second optimality condition: the convexityproperty of a function on a domain for a sufficientlyregular function is equivalent to positive-definitenesscharacter of its Hessian matrix.

• Solve the first optimality condition: ∇ f (Xs) = 0(jacobian) to find the critical points.

2.1 Convexity propertyUsing fi(Xs) = (Xs−Xi)

T .M.(Xs−Xi) where M designatesthe Minkowski matrix and given ∇ fi(Xs) = 2.M(Xs−Xi),the f gradient function can written:

12

∇ f (Xs) = (∑ fi(Xs))M.Xs−M.(∑ fi(Xs)Xi)

The Hessian matrix, which is the f second derivative canwritten:

∇2 f (Xs) = ∑∇ fi(Xs).∇ f T

i +∑ fi.∇2 fi

that becomes, replacing ∇ fi by its expression:

∇2 f (Xs) = (∑ fi(Xs)).M+2M.[N.Xs.XT

s +

∑XiXTi −Xs(∑Xi)

T − (∑Xi)XTs ].M

Using a Taylor series expansion to order 2, an expandedform of the Hessian matrix, equivalent to the previousformula of the f second derivative, is:

This latter allowed us to study the convexity of f . Indeed,because its mathematical form is not appropriate for a directuse of the convexity definition, we have preferred to usethe property of semi-positive-definiteness of the Hessianmatrix. Our calculus lead to the conclusion that:

• Using the criterion of Sylvester [?] and the analysisof the principal minors of the Hessian matrix , we findthat f is not convex on small domains, and thus islikely to exhibit several local minima, according to Xsand Xi. It is these minima, which induce convergenceproblems to the correct solution for the commonminimization algorithms.

2.2 Critical pointsThe study of the first optimality condition (Jacobian =0) gives the following system ∇ f (Xs) = 0 and allowsfinding the critical points and their phase-space distributions.Taking into account the following expression:

12 ∇ f (Xs) = (∑ fi(Xs))M.Xs−M.(∑ fi(Xs)Xi)

we get the relation:

X s =N

∑i=1

fi(X s)

∑ j f j(X s)Xi (2)

This formula looks like the traditional relationship of abarycenter. Thus, we interpret it in terms of the antennaspositions barycenter and its weights. The weight functionfi expressing the space-time distance error between theposition exact and calculated, the predominant direction willbe the one presenting the greatest error between its exactand calculated position. The antennas of greatest weightwill be those the closest to the source.

In practice, because the analytical development of thisoptimality condition in a three-dimensional formulation isnot practical, especially considering the nonlinear terms,we chose to study particular cases. We considered the caseof a linear antennas array (1D) for which the optimalitycondition is easier to express with an emission sourcelocated in the same plane. This approach allows us tounderstand the origin of the observed degeneration whichappears from the wave equation invariance by translationand by time reversal (known reversibility of the waveequation in theory of partial differential equations) andprovides us a intuition of the overall solution. It alsoenlightens the importance of the position of the actualsource relative to the antennas array (the latter point islinked to the convex hull of the antenna array and is theobject of the next section). Our study led to the followinginterpretations:

• The iso-barycenter of the antenna array (of the lit an-tennas for a given event) plays an important role inexplaining the observed numerical degeneration. Thenature of the critical points set determines the conver-gence of algorithms and therefore the reconstructionresult.

• There are strong indications, in agreement with theexperimental results and our calculations (for 1Dgeometry), that the critical points are distributed on aline connecting the barycenter of the lit antennas andthe actual source location. We used this observationto construct an alternative method of locating thesource (section 4).

• According to the source position relative to the anten-na array, the reconstruction can lead to an ill-posedor well-posed problem, in the sense of J. Hadamard.

Rebai A. Ill-posedness formulation of the emission source localization in the radio-detection arraysRAPPORT D’ACTIVITE

Fig. 8: Scheme of the reconstruction problem of sphericalwaves for our testing array of antennas (2D), with a sourcelocated at ground. For this configuration, the convex hullbecomes the surface depicted in red. The result is the samefor a source in the sky.

2.3 Convex hullIn the previous section we pointed that to face a well-posed problem (no degeneration in solution set), it wasnecessary to add constraints reflecting the propagation lawin the medium, the causality constraints, and a conditionlinking the source location and the antenna array, the latterinducing the concept of convex hull of the array of antennas.We also saw that analytically the critical points evidencecould become very complex from the mathematical pointof view. Therefore, we chose again an intuitive approach tocharacterize the convex hull, by exploring mathematicallythe case of a linear array with an emission source located inthe same plane.

The results extend to a 2D antenna array, illuminatedby a source located anywhere at ground, arguing that itis possible to separate the array into sub-arrays arrangedlinearly. The superposition of all the convex segments ofthe sub-arrays leads then to conceptualize a final convexsurface, built by all the peripheral antennas illuminated (seeFigure 8).

The generalization of these results to real practical expe-rience (with a source located anywhere in the sky) was guid-ed by our experimental observations (performed throughminimization algorithms) that provide a first idea of whathappens. For this, we chose to directly calculate numeri-cally the objective function for both general topologies: asource inside the antenna array (ie. and at ground level) andan external source to the antenna array (in the sky ). Ascan be deduced from the results (see Figs. 9 and 10), for asurface antenna array, the convex hull is the surface definedby the antennas illuminated. (An extrapolation of reasoningto a 3D network (such as Ice Cube, ANTARES,...) shouldlead, this time, to the convex volume of the setup).

Our results suggest the following interpretations:

• If the source is in the convex hull of the detector, thesolution is unique. In contrast, the location of thesource outside the convex hull of the detector, causesdegeneration of solutions (multiple local minimums)regarding to the constrained optimization problem.The source position, outside or inside the array, af-fects the convergence of reconstruction algorithms.

Fig. 9: Plots of the objective-function versus R and versusthe phase space (R, t), in the case of our testing array (2D),for a source on the ground and located inside the convexsurface of the antenna array. This configuration leads to asingle solution. In this case the problem is well-posed.

Fig. 10: Plots of the objective-function versus R and versusthe phase space (R, t), in the case of our testing array (2D)for a source outside the convex hull. This configurationleads to multiple local minima. All minima are located onthe line joining the antenna barycenter to the true source. Inthis case the problem is ill-posed.

3 ConclusionExperimental results indicated that the common methods ofminimization of spherical wavefronts could induce a mis-localisation of the emission sources. In the current formof our objective function, a first elementary mathematicalstudy indicates that the source localization method maylead to ill-posed problems, according to the actual sourceposition.

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