University Computing Centre, Edvard Kardelj University, Vojkova 69, 61000 Ljubljana, YUGOSLAVIA

This paper is devoted to a model for approximation of cave space, where the body of a cave is divided into galleries and gallery into sections, each spreading between two adjacent cross-sections. Every section is illustrated by a multilateral prism, having both cross-sections (approximated by multangles) as on upper and a lower base. The model has been tested on two fossile ponors of the Planinsko polje near Postojna: Skednena jama, volume 8.878 m3, length 225 m (error < 5%) and Mačkovica, 38.770 m3 and 650 m (error < 2%).

L'objêt principal de cet ouvrage est le modèle pour la présentation de l'espace d'une grotte
où le corps de la grotte est divisé en parties, dont chacune s'étend entre deux conjointes
sections trans-versales. Chacune partie est illustrée par un prisme multilatéral ayant pour
base inférieure et supérieure les deux sections transversales (approximées par multiangles).

Le modèle était vérifié sur deux grottes, situées au bord septentrional de
Planinsko polje près de Postojna: Skednena jama, 8878 m3 en volume et 225 m en longueur (erreur < 5%),
et Mačkovica, 38770 m3 et 650 m (erreur < 2%).

The work has been made possible by two institutions. Members of the Društvo za raziskovanje
jam Ljubljana (Ljubljana Cave Research Society) have helped me with great enthusiasm;
the most valuable contribution was given by:

Jože Stražišar,

Marina Brancelj, Jaka Jakofčič, Vanja Janežič, Marko Krevs, Uroš Kunaver, Andrej Pokorn,
Joerg Prestor and Ina Šuklje.

It is Edvard Kardelj University, Computing Centre, where all the results have been
computed. Its creative environment has enabled the project to develop and to come to a
successful end.

Caves as such are three-dimensional structures and it is known for a long time that the
intensity of cave-generating processes is at best illustrated by their volume. It would
therefore seem quite obvious to have the volume as the first and the most important
numerical parameter which describes the cave. It is, however, not so; and there are
many good reasons. First of them is the number of points needed. Even the task to get
the exact position of a few dozen points of cave body, in order to make a sketch of it,
to obtain its length and depth, is in most cases untrivial. And to get a representation,
suitable for any serious cave space analysis, not a few dozen, but a few hundred points
are necessary. Another good reason was the absence of a suitable method, which would,
with reasonable amount of effort, allow to compute the volume from the representation
just mentioned.

The challenge is but great and the solution to the problem was only
to be expected. When mr. Corbel published his list of 22 top caves, as far as the
volume of the rock, in which the cave extends, is concerned (Corbel, 1965), he did
not forget to justify the somewhat unusual method by the phrase: "à défaut du cubage exact".
The next such list, available to the author, was the one published recently by
Dubljanski (Dubljanski et al., 1980); here the volume is presented in a modest way,
unveiled only after such parameters as length and depth. But, nevertheless, it is there,
and it is an estimate for the actual volume of cave body. The method, using two formulas,
for horizontal and vertical caverns, respectively, is not described as accurate,
but "of sufficient precision for solving of all the speleological, geological
and hydrogeological tasks, connected to field research" (10-20%).

Author's interest in the problem began some ten years ago; it is a sad thing if a country that had eight caves on the list of the world's top ten only three generations ago, has to look how its largest cave, Škocjanske jame, has disappeared from the list of top hundred, and how its longest one, Postojnska jama, keeps falling towards the bottom of the same list. The paper "On Numerical Valuation of Karst Objects" was presented at the Sixth Yugoslav Speleological Congress (Jakopin, 1972); to reevaluate these objects by volume was stressed as a difficult, but very interesting task, coming in the future. No method was available at the time but the work started. In 1973 it became clear that a cave shall have to be divided into galleries and each gallery into a set of adjacent sections, each spreading between the two consecutive cross-sections. Every cross-section can be approximated by a multangle, but how to approximate a section?

An analytical model was chosen first, as it is more elegant and less expensive
to compute. The idea was to compute the area of both cross-sections (multangles),
delimiting the section and to choose as the base of it the cross-section having
the larger area. The section was then approximated by a truncated cone, having as
a base a circle with the same area and centre of weight as the basic cross-section;
the upper surface was in general an ellipse, having the same centre of weight and
the same area as the second cross-section. Both basic surfaces of the truncated
cone were, of course, also situated in the same plane as the basic and the second
cross-section, respectively. In 1974 the skeleton of the model was derived,
the extra-long equations solved and in a small cave, Skednena jama, near Laze
(10 km NE from Postojna), 51 cross-sections were measured, composed of 305 points.
Only usual cave-surveying equipment was used, combined with three wooden sticks:
1 m, 3 m and 5 m of length. A candle was attached to the top of the last so that
the height of inaccessible ceiling points could be estimated.

Yet the model was
not implemented, the computer program not written. It was felt that the model
would do only as far as the volume of the gallery is concerned, but not in general.
In the case of low and wide galleries the boundary area would be much underestimated.
The second weak point was the poor qraphical representation of the cave, which
results from such a model. In 1977 the idea of analytical solution to the problem
was definitely abandoned.

The new model for approximation of cave space illustrated every section by a
multilateral prism, having the delimiting cross-sections as both basic surfaces.
Manipulation with such an irregular structure is not particularly elegant, it requires
extensive computing and cannot be done without a machine; but the resulting model
is very realistic and can approximate the cave with great precision. It also allows
better definition of cave gallery length: it can now be defined as the sum of the
lengths of all its sections, where the length of a section is the distance between
the two centres of weight of delimiting cross-sections.

And how the volume of a
section is computed? The method goes as follows: areas of both cross-sections
(approximated by multangles), are computed first. The cross-section with larger
area is chosen as the basis. The prism that approximates a section is then defined
by connecting all the corners of the basic cross-section with proper corners of the
second cross-section. It is then cut into two parts by the plane, that is parallel
to the basic cross-section and which touches the nearest corner of the second
cross-section (nearest to the base). One of the two parts may be empty - the
second one if both cross-sections are parallel and the first one if they have
some point in common. The volume of each part is then determined by an iterative
process: first approximation for the volume of each part is the sum of areas of
the lower base (So) and the upper base (Sh), divided by two and multiplied by
its height (h):

At the next step, the part is cut at half its height into two slices of equal thickness (h/2) and the next approximation is computed as the sum of the two volumes:

At step n, the prism is divided into 2^{n} slices and the resultinq volume given by:

Such an iteration is usually continued until the relative difference between the two
consecutive approximations: |(V_{n} - V_{n-1})/V_{n}|, falls
under prescribed level of accuracy (0.00001 = five decimal digits, for example). In
our case it turned out that is is more suitable to stop the iteration when the
abso1ute difference is small enough, (|V_{n} - V_{n-1}| < 0.005 m^{3}).
Volume of the section is then obtained as the sum of the volumes of both parts.
Number of slices necessary varies from section to section but, as it showed up later,
usually ranges from 16 - 128 for the first part and from 32 - 4096 for the second,
more complicated one. The area of the sections coat, which adds into the area of
the body surface, can be computed directly.

All that remained was to implement
the idea. In 1978 several circumstances, not exactly favorable for the author, gave
necessary push to the project. Old measurements, made in 1974 in Skednena jama,
were used as the testing ground. The cave, in fossile ponor of the Planinsko polje,
is composed of simple main gallery to which several small side tunnels are connected.
In the beginning of 1979 the model was implemented and the results which followed,
were these: volume 8878 m^{3}, area of the body surface 6455 m^{2},
length 225 m, depth 31 m, 7 galleries, 44 sections, 51 cross-sections, all derived
from 305 measured points (6 per section). The estimated error, due mostly to primitive
technique for the measuring of inaccessible (ceiling) points, was below 5%.

A better technique was necessary if one was to attack some larger cave. Therefore two
equal prototypes of a special inclinometer were constructed and completed. Both consist
of a wooden rod (1.2 m long), to which a school inclinometer (0.5 m long) is attached.
The rod is furnished on the top with a narrow, battery operated light beam. Inclinometer's
pointer is equipped with a water level, for greater accuracy and comfort. To get a proper
representation of a vertical cross-section (which are most common) the ground points
are measured first, in usual way (at best all from the same central point). Then each
chosen ceiling point is determined by illuminating it from two measured ground points
(narrow light beams just mentioned) and putting down the angles on both inclinometers.
The equipment described was tested in another fossile ponor of the Planinsko polje,
cave Mačkovica (pronounced muchcovytsa), which is also situated near the village Laze.
Eastern Gallery and the Great Hall were measured: the largest cross-section was 55.7 m wide,
24.4 m high, had an area of 516 m^{2} and was represented by 14 points
(6 inaccessible). The results were published in spring 1979, toqether with an article
"On Some Terms Concerning Cave Space" (Jakopin, 1979), that has explained author's
view on several widely used, but often differently defined terms (as an example:
depth was defined as difference in above sea level altitude of two such points of
cave boundary, that one has the biggest and the other the smallest altitude).

Author's plans to move with the work from an amateur to professional
environment have in spring 1979 definitely failed and so the project discontinued
for a while. But, strange as it may seem, it has shown up later, that the project
even benefited from this. In the fall of 1980 the work started again with full power,
the supporting software being switched from one large scale computer (CDC CYBER 172)
to another (DEC 1091). The set of routines has been much improved, it is now more
convenient to use and some important features have been added. To define a section,
for example, all the corners of one delimiting cross-section have to be connected to
proper corners of the second cross-section. Up to now this had to be done explicitly,
which is a time consuming and not very pleasant task. It is now typically determined
by an algorithm.

In January 1981 four more excursions were made into Mačkovica cave. The
measuring of vertical cross-sections has been brought to the level of routine
(the last excursion achieved a record of 40 cross-sections) and the survey is
practically completed. 725 points were measured and 709 vere included in a model
of the cave body (105 inaccessible). The cave was divided into 10 galleries,
composed of 106 sections, which ware delimited by 116 vertical cross-sections
(again 6 points per sectlon). The net results, which can be seen in more detail on
Figure 1, were these: volume 38.770 m^{3}, area of the body surface
20.821 mm^{2}, length 650 m and depth 57 m (the error was estimated as
less than 2%).

Mačkovica is not a large cave but may be it has opened the way up. For the real giants.

- Corbel, J., 1965: Notes sur les plus grandes grottes du monde, Proceedings of the 4th International Congress of Speleology in Yugoslavia, Ljubljana, 1971, vol. 6, p. 19 - 24.
- Dubljanski, V.N., Iljuhin, V.V., Gobanov, J.E., 1980: Nekateri problemi morfometrije kraških votlin, Naše jame, Ljubljana, 1980, vol. 21, p. 75 - 84.
- Jakopin, P., 1972: O numeričnem vrednotenju kraških objektov, Povzetki predavanj 6. kongresa speleologov Jugoslavije, Postojna, 1972, p. 41 - 42.
- Jakopin, P., 1979: O nekaterih pojmih v zvezi z jamskim prostorom, Glas podzemlja,
Ljubljana, 1979, p. 17 - 18.

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