ASTRONOMY & ASTROPHYSICS JUNE II 1997, PAGE 495

SUPPLEMENT SERIES

Astron. Astrophys. Suppl. Ser. 123, 495-506 (1997)

The accuracy of parameters determined with thecore-sampling method: Application to Voronoi tessellationsA.G. Doroshkevich1, S. Gottlober2, and S. Madsen3

1 Theoretical Astrophysics Center, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark2 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany3 Copenhagen University, Astronomical Observatory, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark

Received May 14; accepted October 10, 1996

Abstract. The large-scale matter distribution representsa complex network of structure elements such as voids,clusters, filaments, and sheets. This network is spannedby a point distribution. The global properties of the pointprocess can be measured by different statistical methods,which, however, do not describe directly the structureelements.

The morphology of structure elements is an impor-tant property of the point distribution. Here we applythe core-sampling method to various Voronoi tessella-tions. Using the core-sampling method we identify one-and two-dimensional structure elements (filaments andsheets) in these Voronoi tessellations and reconstruct theirmean separation along random straight lines. We com-pare the results of the core-sampling method with the apriori known structure elements of the Voronoi tessella-tions under consideration and find good agreement be-tween the expected and found structure parameters, evenin the presence of substantial noise. We conclude that thecore-sampling method is a potentially powerful tool to in-vestigate the distribution of such structure elements likefilaments and walls of galaxies.

Key words: large-scale structure of Universe — methods:statistical

1. Introduction

The first deep galaxy surveys which have become avail-able have shown that the galaxies are not at all homo-geneously distributed on large scales. Large voids, smallfilaments, massive clusters, and walls of galaxies can beobserved. How these structures have evolved from the ini-tially nearly homogeneous universe is one of the central

Send offprint requests to: S. Gottlober, sgottloeber@aip.de

problems in modern cosmology. Many theoretical mod-els of structure formation have been suggested. Powerfulmathematical statistical methods are necessary in orderto describe this structure, to obtain its quantitative char-acteristics, to compare the theoretical models and simu-lations with observational catalogues, and, eventually, tofind the correct theory of structure formation.

The standard approach for testing models is to definea point process which can be characterized by statisti-cal methods. This could be the distribution of galaxiesof a specific type in deep surveys or clusters of galaxies.In order to compare models of structure formation, thedifferent distribution of dark matter particles in N-bodysimulations could be tested as well.

The most widely used statistics are the n-point cor-relation functions, the counts-in-cells method, and thevoid probability function. The geometrical and topologi-cal properties of the point distribution can be investigatedby the percolation technique (Zeldovich et al. 1982), theminimal spanning tree (Barrow et al. 1985), the genus ofthe smoothed density field (Gott et al. 1986, 1989), andthe Minkowski functionals (Mecke et al. 1994).

All these different statistics measure the global prop-erties of a point process. However, when looking at a pointprocess we also see different structures. The morphologyof these structures is an important property of the pointdistribution. (Note also the warning of Barrow & Bhavsar(1987) that we tend to see by eye structures which donot exist). Therefore, it is important to develop statisticswhich distinguish between different structure elements ina point distribution and find their typical scales (if anyexists).

In this direction a first attempt was made by Vishniac(1986). He used the moments of a point distribution ina window to measure the amount of filaments in two-dimensional galaxy distributions. Recently, this methodwas generalized to three-dimensional point distributions(Luo & Vishniac 1995). Both the number of filaments andwalls and typical scales of the structure can be found.

496 A.G. Doroshkevich et al.: Core-sampling

Fig. 1. Top: a slice of 8 Mpc thickness, bottom: radial particle distribution in the four cylinders shown at the top part of thefigure (the dashed line indicates the minimum core radius used in the analysis), left: only walls, right: walls with a backgroundof randomly distributed particles

Minkowski functionals are also efficient discriminators foridealized one-, two- or three-dimensional structure ele-ments (Schmalzing et al. 1996). However, using Minkowskifunctionals it is difficult to extract such structure elementsfrom a superposition of all possible elements includingPoisson noise.

Using the core-sampling method (Buryak et al. 1991;Buryak et al. 1994) one can find structure elements andtheir typical separation. The method was designed tofind one-dimensional filaments and two-dimensional wallswithin observational surveys of galaxies (Buryak et al.1994; Doroshkevich et al. 1996) and in simulated sam-ples (Doroshkevich et al. 1997). Here we want to applythe method to an idealized model mimicking many fea-tures of the observed patterns in the galaxy distribution.The model employed here is based on the concept of aVoronoi tessellation (Goldwirth et al. 1995). Such a testallows us to compare the input and output structure pa-rameters and, thus, to test how powerfully this method

can discriminate structure elements, determine their dis-tribution and typical parameters.

The basic idea of the core-sampling method is to re-duce the analysis of a three-dimensional galaxy distribu-tion to the investigation of the distribution of structureelements along random straight lines (the “core”). Suchan approach allows us to avoid any discussions and de-scriptions of the very complicated multiconnected struc-ture as a whole as well as any conventional definition of a“structure element” and “void” in the three-dimensionalspace. The core-sampling method has to deal only with aone-dimensional point distribution. It allows us to definetwo typical populations of structure elements, namely, fila-ments and sheets, and provides us with two objective char-acteristics of the spatial distribution of these elements,namely the surface density of filaments and the linear den-sity of sheets. The surface density of filaments σf is simplythe mean number of filaments intersecting a unit area ofarbitrary orientation, while the linear density of sheets

A.G. Doroshkevich et al.: Core-sampling 497

Fig. 2. A fit to the equation ln(NR) = ln(N0) −R/R0 (left: only walls, right: with a background)

Fig. 3. A fit to the Eqs. (5) and (6) from which N0/D0) (dots) and 1/R0 (triangles) are obtained, (left: only walls, right: witha background)

498 A.G. Doroshkevich et al.: Core-sampling

Fig. 4. Mean separation of walls for three multiplicity thresholds (left: only walls, right: with background particles)

Fig. 5. Mean surface density of apparent filaments for three multiplicity thresholds (left: only walls, right: with backgroundparticles)

A.G. Doroshkevich et al.: Core-sampling 499

Fig. 6. Top: a slice of 4 Mpc thickness, bottom: radial particle distribution in the four cylinders shown at the top part of thefigure (the dashed line indicates the minimum core radius used in the analysis), left: only filaments, right: filaments with abackground of randomly distributed particles

σs is simply the mean number density of sheets crossingan arbitrary straight line. These characteristics provide alocal description of a random three-dimensional networkstructure. In Sect. 2 we will briefly outline the generalproperties of this method

The three-dimensional Voronoi tessellation consists ofa number of Voronoi cells enclosed by the planes bisect-ing the lines between the nuclei of neighbouring cells (vande Weygaert 1991). The intersections of these planes arelines and points. Therefore, the three-dimensional Voronoitessellations consists of three structure elements: the wallsaround the voids, the edges (intersection of walls) and thenodes. Within the three-dimensional Voronoi tessellationparticles may be distributed on each of these structure el-ements or distributed around them according to a givendistribution function. Moreover, the Voronoi tessellationcan be superimposed by randomly distributed particles.Thus it is a unique tool for testing algorithms purportingto describe the structure elements of a point distribution.

In Sect. 3 we will briefly describe Voronoi tessellationsand the creation of synthetic samples. We apply the core-sampling method to these synthetic samples and comparethe results with the input structure element. We discussthe accordance between the expected and found structureparameters and conclude that the core-sampling method isa powerful tool to investigate the distribution of structureelements.

2. The core-sampling method

2.1. Basis of core-sampling method

The mathematical basis for this method had been de-scribed by Buryak et al. (1991) and Buryak et al. (1994).A recent detailed discussion can be found in Doroshkevichet al. (1996).

First let us give the salient points of the core-samplingmethod.

500 A.G. Doroshkevich et al.: Core-sampling

Fig. 7. Mean separation of apparent walls for two multiplicity thresholds (top: radius of filaments 0.2 Mpc, bottom: radius offilaments 0.1 Mpc, left: only filaments, right: with background particles)

1. The distribution of structure elements along a randomstraight line is assumed to be Poissonian. Thus, a 1Dcluster analysis is utilized to discriminate the structureelements among the sample of points and to find thenumber and the mean separation of structure elements.

2. For the following cluster analysis the fields are then allorganized into an “equivalent single field” by combin-ing the separate 1D distributions one after the otheralong a line, with the first point of a field placed ontop of the last point of the preceding field.

3. The dependence of the number and mean separationof structure elements on the diameter of the core al-lows a rough discrimination between the filament andsheet-like populations of the structure elements andyields the fundamental characteristics of the structure,namely, the surface density of filaments, σf , and thelinear density of sheets, σs.

4. The sample of points under consideration is reducedby rejecting poorer and sparser structure elements. Inthis manner, the mean characteristics can be found asa function of the threshold richness.

Let us emphasize here that in practice the one-dimensional analysis is very convenient in many respects.We are using here and in numerical simulations cylin-ders around straight lines, but they can be easily re-placed by cones for observational surveys. Thus, the core-sampling method can be directly used to analyze pen-cil beam surveys as well as almost two-dimensional sam-ples like the slices of the deep Las Campanas survey orreal three-dimensional surveys. Moreover, in case of two-and three-dimensional samples it allows measurement ofstructure parameters in different directions, for instance,both along the line of sight and the transverse direction.Thus, redshift space distortions can be extracted(Doroshkevich et al. 1996).

The assumption of a functional form for the distri-bution of structure elements along the core is essential forour method, because it enables transformation of the pointdistribution into a distribution of structure elements. Oncethis transformation is accomplished, the appropriatenessof the assumed functional form can be tested and the val-ues of the functional parameters can be determined.

A.G. Doroshkevich et al.: Core-sampling 501

Fig. 8. Mean surface density of filaments for two multiplicity thresholds (top: radius of filaments 0.2 Mpc, bottom: radius offilaments 0.1 Mpc, left: only filaments, right: with background particles)

As stated above, we assume a 1D Poissonian law forthe distribution of structure elements (not galaxies) alongthe axis of cylinder. It is a simple distribution, and weshall see that it is also a reasonable assumption. Indeed,the Poissonian distribution arises naturally for some theo-retical models (White 1979; Buryak et al. 1991) when themean separation of structure element exceeds the corre-lation length. The validity of this assumption, however,cannot be tested a priori, and a possible difference be-tween the assumed and the actual distributions dependson the sample in question, thus limiting the precision ofthe final results. This assumption was valid for all casesin previous investigations. Here we are testing the methodwith Voronoi tessellations, for which our assumption hasto be tested as well.

2.2. Sample preparation

The first step of core-sampling analysis is the prepara-tion of a sample with suitable parameters. This sampledepends both on the radius of the cylinder Rc and thelinking length lc of the culling procedure. The sample isthe basis for the further analysis.

The diameter of the cylinder must be a few timessmaller than the expected size of the cells in order to avoida masking effect due to possible overlapping of projectedelements. On the other hand, the cylinder must be wideenough to allow during further analysis the sequential re-duction of the radius by a factor of at least 2, prefer-ably more, because the radius is that diagnostic parame-ter which allows one to distinguish between filaments andsheets. The number of particles has to be large enough toguarantee stability and reliability of results even for thesmallest radius used for analysis. In practice one observesthat this can be achieved if a structure element contains,in the mean, at least 5, or preferably more, particles.

The final step of the sample preparation is sample re-duction, which is performed sequentially during the anal-ysis. Sample reduction provides an additional method ofdistinguishing between different populations of filamentsand sheets according to the mean density or, in otherwords, according to the multiplicity of the structure el-ements. It has to be performed before any further steps ofanalysis.

502 A.G. Doroshkevich et al.: Core-sampling

To do the sample reduction, a 1D cluster analysis ofeach field is performed with decreasing linking length lcof the culling procedure. All clusters with fewer mem-bers than a constant threshold multiplicity µthr are re-jected. Then the remaining number of particles in the sam-ple Nr depends on the current linking length lc and thethreshold µthr. For each linking length lc a certain fractionf = Nr/Na of points remains (Na is the total number ofparticles). It is convenient to use the fraction f togetherwith the threshold multiplicity µthr, as parameters char-acterizing the reduced sample.

This procedure reduces the full sample, rejecting moreand more points associated with poorer or sparse clumps.Only the tighter clumps are retained for further analy-sis. This approach emphasizes the local density withinthe structure elements. It allows the rejection of the low-density haloes from the structure elements and sparsestructure elements as a whole.

2.3. Detection of the mean separation of structureelements

The next step of analysis is the detection of the meanseparation of structure elements. At first, the axes of allcylinders are randomly combined to one line by identify-ing the last point of one axis with the first of the next.At this line a 1D cluster analysis with increasing linkinglength R is performed. Assuming a Poisson distribution,the number of clusters NR as a function of R is given bythe relation

ln(NR) = ln(N0)− R/R0. (1)

For a true Poissonian sample, these values are relatedto the length of the sample,D0, defined by the nearest andthe farthest points in an average field, via the relationship

N0/D0 = 1/R0. (2)

Thus, the difference between the linear density ofstructure elements N0/D0 and their mean separation R0

obtained from Eqs. (1) and (2) can be considered a mea-sure of the systematic error due to the deviation of the ac-tual from the assumed (Poissonian) distribution of struc-ture elements along the analyzed cylinder. To decreasethis error we use an automatic procedure to find the op-timal interval for R for the fit to Eq. (1). The upper limitto this interval is fixed by the input of a minimum num-ber NR ≥ Nmin, whereas the lower limit can vary and isdefined by the condition

1− ε ≤ R0N0/D0 ≤ 1 + ε (3)

where ε is a desirable precision of fitting parameters. Ingeneral, a reasonable precision (≈ 10%) can be achievedin the analysis.

If a desirable precision cannot be achieved, the reasonis usually that the distribution used is far from Poissonianand the sample under consideration must be changed (e.g.,by changing the cylinder radius).

2.4. Identification of structure elements

The final step of analysis is the discrimination of filamentsand sheet-like structure elements and determination of theparameters σf and σs for both populations. To this end,we use a simple geometrical model for the structure el-ements (Buryak et al. 1994). According to this model,in each narrow cylinder the structure can be consideredas a system of randomly distributed lines (filaments) andplanes (sheets) which contain all points. In this model,the filaments are considered straight lines, and the sheetsare considered flat planes. Of course, this approach is lim-ited, and it cannot be used as an accurate description ofthe true matter distribution on large scales; one can bestcharacterize it as an intermediate step between the localdescription of the matter distribution with density andvelocity fields and the global description obtained by thetopology or Minkowski techniques.

As it was mentioned above, we characterize the ran-dom distribution of straight lines (filaments) by their sur-face density, σf , i.e., the mean number of lines intersectingan unit area of arbitrary orientation. The random distri-bution of planes can be characterized by the linear density,σs, i.e., the mean number density of planes (sheets) cross-ing an arbitrary straight line. Equivalently, we can use the

values Ds = σ−1s and Df = σ

−1/2f as typical measures of

the mean separation of structure elements. To character-ize the structure as a whole we can estimate the meanseparation of structure elements — filaments and sheetscombined — by < Dfs >, which can be thought of as thediameter of a sphere containing, on average, two structureelements,

< Dfs >=2

σs +√σ2

s + πσf

· (4)

The main characteristics of the structure can be ob-tained by fits to the radial dependence of the linear den-sity of clusters, N0/D0, and to the radial dependence ofthe mean separation of structure elements, R0,

N0/D0 = σs + π Rcσf , (5)

and

1/R0 = σs + π Rcσf , (6)

over the range of variation of core radius Rc. Here D0 isthe mean depth of field.

Clearly, these parameters also depend on the samplebeing studied. In our analysis, Eqs. (5) and (6) were fit-ted by a maximum likelihood technique, and the resultingmean values of σf and σs were accepted as the final esti-mates of the structure parameters. The difference of theestimates from Eqs. (5) and (6) was included into the er-rors of the final values.

A.G. Doroshkevich et al.: Core-sampling 503

2.5. Methodological remarks

For the idealized model considered above, the core-sampling method would be expected to reproduce wellthe characteristics of the sample under consideration.However, in reality, several factors may distort the final re-sults. For example, filaments may be apparently detectedas sheets, if they pass through the center of the core. Alsothe noise of randomly distributed particles must be takeninto account. Therefore, a specific technique must be usedto obtain stable results.

The analysis of the Las Campanas Redshift Survey(Doroshkevich et al. 1996) has shown that the stability ofcore-sampling with respect to various accidental or sys-tematic variations of density is very high. During the se-quential random rejection of particles, the final estimatesof mean separation of structure elements increased onlyas f−1/3. In general, the precision and stability of thecore-sampling method depends on the density contrast instructure elements relative to the mean number densityof points. In case of pure Poissonian point distributionthe “structure” parameters have been found with errorsof about 50% (Buryak et al. 1994). Similar errors werefound for the Soneira-Peebles model. On the contrary, forthe observed Las Campanas Redshift Survey errors werefound to be about 10% only.

1. Sample preparation.During the first step of analysis, when the sample isprepared, one has to check whether the resulting 1Dcluster distribution is Poissonian as assumed. One hasto ensure this requirement by the variation of the ra-dius of cylinder and the range of fit in Eq. (1). Duringthe further analysis the range of core radii used for thefit of Eqs. (5) and (6) is important to get the two val-ues in agreement. These questions have to be solvedbefore the final analysis. This means that during eachof the intermediate steps leading to Figs. 1 to 3, the va-lidity of all assumptions must be checked. Otherwise,the final result will be meaningless.

2. The surface density of filaments.The richness of a filament in the core depends both onthe actual properties of the point distribution analyzedand the geometry, i.e. the position and orientation ofthe filament relative to the core. Only the first piece ofinformation is of interest; the geometry can be charac-terized analytically and used to improve the stabilityof the method.The surface density of filaments σf(f, µthr) dependson the fraction of particles retained in the sample andthe multiplicity threshold. The main characteristic offilamentary structure is the full surface density of fila-ments, σf(1, 1). However, at this point the noise is usu-ally high and it is desirable to improve the estimate ofthis value.Basing on the geometrical model described above onecan calculate the distribution function for the length of

intersection of identical filaments with the core. Thus,calculating the dependence of both the fraction f andthe surface density σf on this length, we find in linearapproximation

σf(f) ≈ σf(1)(1−

√2

3(1− f)1/2 + ...). (7)

This relation estimates the fraction of poor filamentsfor the ideal case without noise. The linear fit of Eq. (7)over some range of f allows one also to improve the es-timate of σf(1) for a moderate noisy sample. However,in the case of very noise samples with a significantfraction of filaments consisting only of one point, itcan be only used for small linking lengths, lc, of theculling procedure. Indeed, if lc ≤ R0 the main partof noise particles will be rejected (see Fig. 8, right).Evidently, for smaller values of f , the function σf(f)resembles the actual properties of the filaments (see,e.g. Doroshkevich et al. 1996).

3. The linear density of sheets.The richness of the elements of the sheet-like popu-lation both in observational data and simulations isusually quite high. Thus, the linear density of sheetscan be estimated for a wide range of parameters f andµthr, providing more reliable results. However, even inthis case the final results are sometimes distorted be-cause during reduction the fields are shortened as a re-sult of the appearance of empty edges. To avoid thesedistortions, a correction procedure (see Doroshkevichet al. (1996) was used.The next problem is a filament population masquerad-ing as sheets. These apparent sheets are detected bythe program if filaments cross the cylinder near itsaxes. Obviously, this effect will be proportional to thethickness of the filaments. The surface density σa

s ofthese apparent sheets is σa

s = πσfrf , where rf is theradius of the filament (Buryak et al. 1994). Usuallythe influence of this effect is negligible, but it canbecome important for thick filaments. Moreover, ap-parent sheets can be generated by the branch points(knots) of filamentary structure or other clumps in thepoint distribution if their size is comparable to the di-ameter of the core. The contribution of these knotscan be estimated as σa

s = πnclmr2clm, where nclm and

rclm are respectively the volume density and the effec-tive radius of the clump. For a random network struc-ture, the volume density of the knots is approximately

nclm ≈ σ3/2f .

3. Analysis of Voronoi-tessellations

3.1. Construction of Voronoi tessellations

We have used the code developed by R. Van de Weygaert(1991, 1994, see also Goldwirth et al. 1995) in order to con-struct Voronoi tessellations. Although the code has beendeveloped to allow for much more sophisticated cases, we

504 A.G. Doroshkevich et al.: Core-sampling

limit ourselves to the simplest case where a given num-ber of points is distributed in walls or filaments of a giventhickness. To mimic noise we have added an Poisson dis-tribution of particles in the whole box.

Using the same nuclei of the Voronoi tessellation, thefilaments (defined as the edges of the walls) would have ap-proximately the same separation as the walls (contrary towhat we expect in nature). Therefore, we did not combinea filamentary structure with a wall-like structure in onerealization, but made a second realization for filamentsalone. Note also that the filaments of a Voronoi tessel-lation are not distributed like filaments of real galaxies.Nevertheless the points of these Voronoi tessellation formone-dimensional structure elements with a priori knownproperties which should be recovered by the core samplingmethod.

The mean diameter of the Voronoi cells and the sizeof the box are free input parameters of the code. For thecore-sampling analysis it is only important that the meannumber of structure elements along a random line crossingthe box be large enough. We have chosen 6 cells and 20cells for the analysis of walls and filaments, respectively.The analysis itself can be performed in units of the boxlength. For the graphical representation we have chosenformally a box size of 300 Mpc. We have chosen 200 000and 400 000 points to represent the walls and the fila-ments, respectively. Due to the formal definition of the boxsize these numbers are not directly comparable with thenumber of galaxies in the corresponding volume. However,in that case the mean number density of points is of theorder 10−2 Mpc−3, i.e of the same order as the galaxy den-sity. Also the the mean separation of structure elements isof the same order as found for galaxies, so that we can in-deed draw conclusions from our test to the situation foundin observational surveys. The noise level chosen (50% forwalls and 25% for filaments) is in these realisations an ap-proximate upper limit for which the known properties canbe recovered with high accuracy.

3.2. Wall distribution

For the first test, we chose a mean size of the cells of50 Mpc. Our first realization consists of 63 randomly dis-tributed cells in box of 300 Mpc length. 200 000 pointsare distributed only in walls around the cells. We chose aGaussian density profile of the walls with a thickness of2 Mpc. On the left hand side of Fig. 1, a slice of 8 Mpcthickness of this realization is shown. In order to test thestability of the core-sampling method, we made a secondrealization with 100 000 additional points which were ran-domly distributed (Fig. 1, right).

As explained in Sect. 2.1, the starting point of thecore-sampling analysis is the construction of randomly dis-tributed cylinders in the simulation. For illustration, weput four of these cylinders into each of the slices shownin Fig. 1. The diameter of the cylinders is equal to the

thickness of the slice (8 Mpc). In the lower part of the fig-ures, the distribution of points in these cylinders is plottedvs. the radius of the cylinder. This radius will be used inthe following as one diagnostic parameter as described inSect. 2. Indeed, for sheet-like structures like the walls inour Voronoi tessellations, the mean number of clusters inthe 1D cluster analysis is independent of the core radius.One can clearly see from Fig. 1 that in both cases almostall walls will be identified as walls.

Next, all particles Na in a given cylinder of radius Rc

are projected onto the axis of the cylinder. From Fig. 1it is clear that there is also noise besides real structuresin the resulting one-dimensional point distribution. Thedegree of noise depends on the chosen parameters of thegenerated sheets and on the parameters of the core. Thenoise particles in underdense regions can be removed us-ing the reduction procedure described in Sect. 2.1. Thereduced sample at which the further analysis will be per-formed is characterized by three parameters: the fractionf , the threshold µthr, and the core radius Rc.

Further, we use the iterative fitting procedure de-scribed in Sect. 2.2 to find the best fit to Eq. (1). Accordingto our assumption, the resulting clusters must be Poissondistributed for the some range of linking length. However,this assumption must be tested for the samples under con-sideration. As an example we show in Fig. 2 the fit toEq. (1) for four sets of parameters (f = 1, 0.5, r = 4 Mpc,2 Mpc and µthr = 2: left for walls and right for walls withrandomly distributed points).

Using Eqs. (5) and (6), where Rc is the diagnostic pa-rameter, we determine now the surface density of filamentsand the linear density of walls for each pair of parametersf and µthr. This procedure is illustrated by Fig. 3. for fourpairs of parameters. (Note that each curve of Fig. 2 corre-sponds to one point in Fig. 3). From Fig. 3 we extract thelinear density of walls as the zero point of the curve andthe surface density of filaments as the gradient. The fitsto Eqs. (5) and (6) were performed over 9 equally spacedvalues of Rc in the interval 2 Mpc ≤ Rc ≤ 4 Mpc.

As the final result of the analysis, we present in Fig. 4the mean distance of the walls Ds = σ−1

s and in Fig. 5 thesurface density of filaments σf depending on the thresholdµthr. (Note, that each curve of Fig. 3 provides one pointin Fig. 4 and one point in Fig. 5.) The final values of σs

and σf are listed in Table 1.

Table 1. Test of the wall distribution

µthr σs σf σnoises σnoise

f10−2Mpc−1 10−2Mpc−2 10−2Mpc−1 10−2Mpc−2

2 2.1± 0.29 0.43± 0.21 1.8± 0.20 2.7± 0.164 2.0± 0.13 0.43± 0.19 2.5± 0.29 2.7± 0.166 1.8± 0.16 0.48± 0.21 2.1± 0.17 2.7± 0.16

A.G. Doroshkevich et al.: Core-sampling 505

The noise of apparent filaments is an objective char-acteristic of the sample under consideration. The popu-lation of filaments dominates for all thresholds µthr thatresemble the small density contrast in the walls, since thenumber of particles concentrated into sheets exceeds thenoise particles only by a factor of two. In this respect oursynthetic models differ from the Las Campanas RedshiftSurvey, where for high threshold multiplicity the filamentpopulation had disappeared, and only the richest sheet-like elements survived.

3.3. Filament distribution

As a second test we have investigated a pure filamen-tary structure and filaments superimposed by randomlydistributed particles. To this end we have constructed aVoronoi tessellation with 400 000 particles which are dis-tributed at the edges of the cells, and a second one with anadditional 100 000 randomly distributed particles. For thistest we chose a mean size of the cells of 15 Mpc. In Fig. 6we show a slice of 4 Mpc thickness and four cores of ra-dius 2 Mpc. Further steps in the analysis are as above.We made two realisations of Voronoi tessellations con-taining only filamentary structure: In the first realisationwe chose a radius of filaments of 0.2 Mpc, in the second0.1 Mpc.

For these synthetic models, we omit the figures corre-sponding to Figs. 2-3, but present only the final results, i.e.the density of walls σs(f) and filaments σf(f) vs. (1−f)1/2

for two thresholds µthr = 2, 4 and the two different radiiof filaments. (see Figs. 7-8). The final structure parame-ters are given in Table 2.

Let us note that in this case the mean edge separationalong a random straight line is not identical to the meansize of the cell but is of the order of the mean length ofthe filaments. The mean distance of particles along the fil-aments in our sample is of the order of 1 Mpc. Therefore,we expect that the set of particles forms a broken net-work structure which can be characterized by the surfacedensity of filaments σf(f). With a higher mean densitywe could create also a complete network structure which,however, in reality does not exist.

Since all filaments were created using the same pro-cedure their parameters can vary only statistically. Wecan use Eq. (7) to determine σf(1) as is shown in Fig. 8(dashed line). In case of µthr = 4 the points correspondingto a large fraction f do not follow the line predicted byEq. (7). Rejecting these points, a linear fit would predicta somewhat smaller σf . We expect this result, because weare now measuring only the rare high-density filaments.

The noise strongly increases the full surface densityof filaments σf(1) (Fig. 8, right). The rejection of (10 −15%) of particles depresses the noise impact and leads usback to noiseless data. Since most or all observed galaxiesare incorporated into structure elements the noise impactis not expected to be severe in catalogues. Following the

rejection procedure we can obtain for all realizations thesame result σf(1) = 0.0054 which corresponds to 13.6 Mpcmean separation (see Fig. 8).

Figure 6 (bottom) indicates that some of the filamentsmasquerade as sheets. We have discussed this problem al-ready in Sect. 2.4. In this case the error bars stronglyincrease as a result of the sparse sample statistics of el-ements which the core-sampling method wrongly detectsas sheets (see Fig. 7).

Table 2. Test of the filament distribution

µthr σs σf σnoises σnoise

f10−3Mpc−1 10−2Mpc−2 10−3Mpc−1 10−2Mpc−2

r = 0.2 Mpc2 4.4± 0.8 0.57± 0.02 5.3± 1.0 0.92± 0.054 7.3± 0.8 0.30± 0.02 6.5± 0.6 0.50± 0.04

r = 0.1 Mpc2 4.5± 0.9 0.58± 0.01 3.5± 1.2 0.90± 0.054 6.1± 0.8 0.31± 0.01 5.5± 1.0 0.49± 0.04

4. Conclusions

We have constructed a series of Voronoi tessellations withdifferent input parameters. Our realisations consist eitheronly of structure elements or of structure elements andnoise. We have chosen a number of points comparable withthe situation in observational samples and simulations oflarge scale structure formation, i.e. our realisations cor-respond to models with all galaxies within structure ele-ments and models with additional randomly distributedgalaxies.

In the first two tessellations, only walls were con-structed. Using the core-sampling method, we detectedwalls with a mean separation of 50 Mpc (see Table 1 andFig. 2) in these synthetic models. This corresponds exactlyto the input parameter (the mean void size) of the Voronoitesselation. The method provided an accurate reconstruc-tion despite the additional noise which was added to thetessellation (Fig. 4, right). On the other hand, the scalesof the spurious filaments in the tessellation (Fig. 5) aresensitive to the addition of noise. The specific dependencefound appears to be a property of the particular syntheticmodel generated.

The next four tessellations are all characterized by amean cell size of 15 Mpc. The cells are surrounded only byfilaments. Two tessellations were made with filaments ofradius 0.2 Mpc, two more with 0.1 Mpc. Independent ofthe radii of filaments, we were able to determine the samemean distance of filaments (13.6 Mpc); however, this esti-mate was slightly reduced by adding randomly distributedpoints as background (cp. Fig. 8 , dashed lines). The de-tected spurious walls have distances of the order of the

506 A.G. Doroshkevich et al.: Core-sampling

box size and very large errors so that they can be easilyruled out.

Our results demonstrate clearly that the core-samplingmethod allows one not only to determine the structure el-ements, but also to measure their characteristic mean sep-aration or density almost independently of the influence ofa substantial noise component. From the good agreementbetween the expected and recovered structure parameters,we conclude that the core-sampling method is a powerfultool for further investigations of observational surveys andthe determination of structure elements such as filamentsand walls of galaxies. Recently, the core-sampling methodhas been successfully applied both to numerical simula-tions (Doroshkevich et al. 1997) and the Las Campanasgalaxy sample (Doroshkevich et al. 1996).

Acknowledgements. We would like to thank Rien Van deWeygaert for allowing us to use his code for generating Voronoitessellations and Ron Kates for valuable discussion of themanuscript. This paper was supported in part by DanmarkGrundforskningsfond through its support for the establishmentof the Theoretical Astrophysics Center. A.G.D. was partly sup-ported by the INTAS grant 93-0068 and by the Center forCosmoparticle Physics “Cosmion” in the framework of theproject “Cosmoparticle Physics”. S.G. wishes to express grat-itude for the hospitality of the TAC Copenhagen.

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