The mathematical basis for this method had been described by Buryak et al. (1991) and Buryak et al. (1994). A recent detailed discussion can be found in Doroshkevich et al. (1996).
First let us give the salient points of the core-sampling method.
, and the linear density of sheets, 
.
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 cylinders around straight lines, but they can be easily replaced by cones for observational surveys. Thus, the core-sampling method can be directly used to analyze pencil beam surveys as well as almost two-dimensional samples like the slices of the deep Las Campanas survey or real three-dimensional surveys. Moreover, in case of two- and three-dimensional samples it allows measurement of structure 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 distribution of structure elements along the core is essential for our method, because it enables transformation of the point distribution into a distribution of structure elements. Once this transformation is accomplished, the appropriateness of the assumed functional form can be tested and the values of the functional parameters can be determined.
As stated above, we assume a 1D Poissonian law for the distribution of structure elements (not galaxies) along the axis of cylinder. It is a simple distribution, and we shall see that it is also a reasonable assumption. Indeed, the Poissonian distribution arises naturally for some theoretical models (White 1979; Buryak et al. 1991) when the mean separation of structure element exceeds the correlation length. The validity of this assumption, however, cannot be tested a priori, and a possible difference between the assumed and the actual distributions depends on the sample in question, thus limiting the precision of the final results. This assumption was valid for all cases in previous investigations. Here we are testing the method with Voronoi tessellations, for which our assumption has to be tested as well.
The first step of core-sampling analysis is the preparation of a
sample with suitable parameters. This sample depends both on the
radius of the cylinder 
and the linking length 
of the
culling procedure. The sample is the basis for the further analysis.
The diameter of the cylinder must be a few times smaller than the expected size of the cells in order to avoid a masking effect due to possible overlapping of projected elements. On the other hand, the cylinder must be wide enough to allow during further analysis the sequential reduction of the radius by a factor of at least 2, preferably more, because the radius is that diagnostic parameter which allows one to distinguish between filaments and sheets. The number of particles has to be large enough to guarantee stability and reliability of results even for the smallest radius used for analysis. In practice one observes that 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 reduction, which is performed sequentially during the analysis. Sample reduction provides an additional method of distinguishing between different populations of filaments and sheets according to the mean density or, in other words, according to the multiplicity of the structure elements. It has to be performed before any further steps of analysis.
To do the sample reduction, a 1D cluster analysis of each field is
performed with decreasing linking length 
of the culling
procedure. All clusters with fewer members than a constant threshold
multiplicity 
are rejected. Then the remaining number of
particles in the sample 
depends on the current linking length
and the threshold 
. For each linking length
a certain fraction 
of points remains
(
is the total number of particles). It is convenient to use the
fraction f together with the threshold multiplicity 
, as
parameters characterizing the reduced sample.
This procedure reduces the full sample, rejecting more and more points associated with poorer or sparse clumps. Only the tighter clumps are retained for further analysis. This approach emphasizes the local density within the structure elements. It allows the rejection of the low-density haloes from the structure elements and sparse structure elements as a whole.
The next step of analysis is the detection of the mean separation of
structure elements. At first, the axes of all cylinders are randomly
combined to one line by identifying the last point of one axis with
the first of the next. At this line a 1D cluster analysis with
increasing linking length R is performed. Assuming a Poisson
distribution, the number of clusters 
as a function of R is
given by the relation
For a true Poissonian sample, these values are related to the length of
the sample, 
, defined by the nearest and the farthest points in
an average field, via the relationship
Thus, the difference between the linear density of structure elements
and their mean separation 
obtained from
Eqs. (1 (click here)) and (2 (click here)) can be considered a measure of the
systematic error due to the deviation of the actual from the assumed
(Poissonian) distribution of structure elements along the analyzed
cylinder. To decrease this error we use an automatic procedure to find the
optimal interval for R for the fit to Eq. (1 (click here)). The upper
limit to this interval is fixed by the input of a minimum number
, whereas the lower limit can vary and is defined by
the condition
where 
is a desirable precision of fitting parameters.
In general, a reasonable precision (
) can be achieved
in the analysis.
If a desirable precision cannot be achieved, the reason is usually that the distribution used is far from Poissonian and the sample under consideration must be changed (e.g., by changing the cylinder radius).
The final step of analysis is the discrimination of filaments and
sheet-like structure elements and determination of the parameters
and 
for both populations. To this end, we use
a simple geometrical model for the structure elements (Buryak et al.
1994). According to this model, in each narrow
cylinder the structure can be considered as a system of randomly
distributed lines (filaments) and planes (sheets) which contain all
points. In this model, the filaments are considered straight lines,
and the sheets are considered flat planes. Of course, this approach
is limited, and it cannot be used as an accurate description of the
true matter distribution on large scales; one can best characterize it
as an intermediate step between the local description of the matter
distribution with density and velocity fields and the global
description obtained by the topology or Minkowski techniques.
As it was mentioned above, we characterize the random distribution of
straight lines (filaments) by their surface density, 
, i.e.,
the mean number of lines intersecting an unit area of arbitrary
orientation. The random distribution of planes can be characterized
by the linear density, 
, i.e., the mean number density of
planes (sheets) crossing an arbitrary straight line. Equivalently, we
can use the values 
and 
as typical measures of the mean separation of structure elements. To
characterize the structure as a whole we can estimate the mean
separation of structure elements -- filaments and sheets combined --
by 
, which can be thought of as the diameter of a sphere
containing, on average, two structure elements,
The main characteristics of the structure can be obtained by
fits to the radial dependence of the linear density of clusters,
, and to the radial dependence of the mean separation of
structure elements, 
,
and
over the range of variation of core radius 
. Here 
is the
mean depth of field.
Clearly, these parameters also depend on the sample being studied. In
our analysis, Eqs. (5 (click here)) and (6 (click here)) were fitted by a
maximum likelihood technique, and the resulting mean values of
and 
were accepted as the final estimates of
the structure parameters. The difference of the estimates from
Eqs. (5 (click here)) and (6 (click here)) was included into the errors of the
final values.
For the idealized model considered above, the core-sampling method would be expected to reproduce well the characteristics of the sample under consideration. However, in reality, several factors may distort the final results. For example, filaments may be apparently detected as sheets, if they pass through the center of the core. Also the noise of randomly distributed particles must be taken into account. Therefore, a specific technique must be used to obtain stable results.
The analysis of the Las Campanas Redshift Survey (Doroshkevich et
al. 1996) has shown that the stability of core-sampling with respect
to various accidental or systematic variations of density is very
high. During the sequential random rejection of particles, the final
estimates of mean separation of structure elements increased only as
. In general, the precision and stability of the
core-sampling method depends on the density contrast in structure
elements relative to the mean number density of points. In case of pure
Poissonian point distribution the "structure" parameters have been
found with errors of about 50% (Buryak et al.
1994). Similar errors were found for the Soneira-Peebles model. On
the contrary, for the observed Las Campanas Redshift Survey errors were
found to be about 10% only.
During the first step of analysis, when the sample is prepared, one has to check whether the resulting 1D cluster distribution is Poissonian as assumed. One has to ensure this requirement by the variation of the radius of cylinder and the range of fit in Eq. (1 (click here)). During the further analysis the range of core radii used for the fit of Eqs. (5 (click here)) and (6 (click here)) is important to get the two values in agreement. These questions have to be solved before the final analysis. This means that during each of the intermediate steps leading to Figs. 1 (click here) to 3 (click here), the validity of all assumptions must be checked. Otherwise, the final result will be meaningless.
The richness of a filament in the core depends both on the actual properties of the point distribution analyzed and the geometry, i.e. the position and orientation of the filament relative to the core. Only the first piece of information is of interest; the geometry can be characterized analytically and used to improve the stability of the method.
The surface density of filaments 
depends
on the fraction of particles retained in the sample and the multiplicity
threshold. The main characteristic of filamentary structure is the full
surface density of filaments, 
. However, at this point
the noise is usually high and it is desirable to improve the estimate
of this value.
Basing on the geometrical model described above one can calculate the
distribution function for the length of intersection of identical
filaments with the core. Thus, calculating the dependence of both the
fraction f and the surface density 
on this length, we
find in linear approximation
This relation estimates the fraction of poor filaments for the ideal
case without noise. The linear fit of Eq. (7 (click here)) over some range
of f allows one also to improve the estimate of 
for a
moderate noisy sample. However, in the case of very noise samples with
a significant fraction of filaments consisting only of one point, it
can be only used for small linking lengths, 
, of the culling
procedure. Indeed, if 
the main part of noise particles
will be rejected (see Fig. 8 (click here), right). Evidently, for smaller
values of f, the function 
resembles the actual
properties of the filaments (see, e.g. Doroshkevich et al.
1996).
The richness of the elements of the sheet-like population both in
observational data and simulations is usually quite high. Thus, the
linear density of sheets can be estimated for a wide range of
parameters f and 
, providing more reliable
results. However, even in this case the final results are sometimes
distorted because during reduction the fields are shortened as a result
of the appearance of empty edges. To avoid these distortions, a
correction procedure (see Doroshkevich et al. (1996) was used.
The next problem is a filament population masquerading as sheets.
These apparent sheets are detected by the program if filaments cross
the cylinder near its axes. Obviously, this effect will be proportional
to the thickness of the filaments. The surface density 
of
these apparent sheets is 
, where 
is the radius of the filament (Buryak et al.
1994). Usually the influence of this effect is
negligible, but it can become important for thick filaments. Moreover,
apparent sheets can be generated by the branch points (knots) of filamentary
structure or other clumps in the point distribution if their size is
comparable to the diameter of the core. The contribution of these knots can
be estimated as 
,
where 
and 
are respectively the volume density
and the effective radius of the clump. For a random network structure, the
volume density of the knots is approximately 
.