## LANDSCAPE AS A SPATIALLY ORGANIZED SYSTEM AND THE GEORELIEF AS A SUBSYSTEM OF LANDSCAPE - THE INFLUENCE OF GEORELIEF ON SPATIAL DIFFERENTIATION OF LANDSCAPE PROCESSES

#### Jozef KRCHO

Dean of the Faculty of Natural Sciences, Comenius University Bratislava

CONCLUSIONS This study discusses following topies: position of the georelief in the landscape and its influence on the spatial differetation of landscape-ecological processes, landscape as aspatially organized autoregulating system and the georelief as its spatial subsystem, expression of the set of its morphometric quantities characterising georelief in its arbitrary point, geometric forms the georelief on the basis of morphometric parameters, moddeling of georelief by Conplex Digital Model as integral part of GIS.

#### INTRODUCTION

Landscape is a planet-wide, complex, spatially organized whole possessing its own self control mechanisms and its own circulation of mater, energy and information. The man (human society) interacts with the natural parl of the landscape and the human activity is thus embeldded in the landscape. One of the landscape. One of the most important factors affecting the spatial differentiation of processes in the landscape is the georelief.

Georelief is one of dominant components in landscape. It influences by its geometry upon not only spatial differentiation of ecological processes in lanscape but spatial differentation various kinds of human activities in space as well.

The relief of the Earth (georelief) with it geometric characters is from one side the resultant of processes running in landscape and from second side these processes are influenced and spatially differentiated by its own geometry.

Georelief is noticed in each scientific discipline in its point of view. This point of view is given by object and subject of study of relevant scientific discipline as well as by the aim of work and realisation result. So e.g. genetic forms of georelief, their development, dynamics considering to characters of geologic underlying rock and considering to all processes which model the surface of underlying rock, there is studied by geomorphology. These processes are a part of total processes in landscape. Geomorphometry studies the geometry of relief of the Earth which can by studied from many points of view however its own substance there is the geometry of georelief.

Georelief is interesting as firm but dynamic interface between atmosphere and hydrospere resp. on the one side and pedosphere with part of biosphere and lithosphere on the other side. This firm but dynamic interface is always considered in certain scale and on the certain discintive level. From theoretico-methodological point of view relief is interesting as dynamic surface characterized in each arbitrary point (exactly in differently small surface element dp) by the set of quantitative indicese and quantities resp.(morphometric parameter). Such morphometric quantities are interesting which express geometric characters of the relief including its geometric forms in relation to other landscape elements as spatially organised dynamic system with own specific circulation of subslantes, energy and information.

The geometry of this way considered surface is interesting which as a result of modelling processes there is from one side "a respond" of physical, chemical and other characters of geologic underlying rock (lithospheric upper layers) to these processes, from other side it influences upon spatial differentiation of all processes in landscape. However it is expressed simplier. An important fact wanted to be pointed out that one of the basic tasks of geomorphometry there is the observation of the Earth relief geometry as a part landscape and the formulation such morphometric quantities characterizing relief in each its arbitrary position which is characterized by structural geometric characters of relief relevant simultaneously even from point of view of landscape and processes running in it.

These morphometric quantities from one side:

- give information enclosed in relief geometry about its development so they are "memory quantities",
- from the other side they are input variable quantities form exact expression of georelief influence upon processes in landscape, upon their spatial differentiation.

Firstly let landscape characterize from point of view of our requirements so that relief as a part of landscape can be appropriately expressed. In literature a landscape is characterised as substance - energetic spatially organized complex with own specific substances and energy circulation in which according the paper NEEF, E (1967) there exists interrelations and interconnections.

It can be briefly mentioned that the problems of landscape (in broad sense of word of physico-geographical sphere) as spatially organized complex with own specific substances and energy circulation which is spatialy differentiated into particular spatial landscape units have been worked out mainly in German and Russian (later Soviet) so called "landschaft school". From large amount of papers concerning this sphere, from our point of view papers can be mentioned as follows: ARMAND, D. L. (1975), CAROL, H-NEEF, E. (1957), BARSCH, H. (1971), BOBEK, H.- SCHMITHUSEN, J. (1949), DRDOŠ, J. (1973), GERASIMOV, I. P. (1983), HASSE, G. (1964), ISAČENKO, A. G. (1965,1981), LESER, H. (1974, 1976), NEEF, E. (1963,1967), SCHMITHUSEN, J. (1948,1968,1970), SOČAVA (1978).

In Anglo-Saxon literature these problems have been formulated and developed in ecology with biological accent. There is newer approach in broade inter-disciplinary context formulated and developed in so called environmentalism which has been gradually constituted as independent broaddrawn inter-disciplinary scientific division. This term in broad sense begins to extend and develop even in countries of German and Russian "landschaft schools". However, in environmentalism, the part which develops landscape ecology and understands landscape as patially organized complex is more orless identical with so called complex physical geography of German, Russian and Soviet "landschaft schools". On the whole nowadays understood environmentalism has broader inter-disciplinary range, even if it has big penetriation into mentioned "landschaft school". In German geographical literature of "landscaft school" even the term "geoecology" has been introduced.

The problem of landscape as spatially organized complex in scientific literature has been transmissed even from point of view of systems theory.

On various level of complexity from landscape point of view the theory of geosystems has been worked out ALEXANDROVA, T. D. - PREOBRAŽENSKO, V.S. (1978), ARMAND, A. D. (1966, 1972, 1973, 1975), AURAD, K. D. (1979, 1982, 1983), BENNET, R. J. - CHORLEY, R. J.(1978), DEMEK, J. (1974,1977), GOCHMAN, V. M.-MINC, A. A. - PREOBRAŽENSKIJ, V. S.(1971), CHORLEY, R J.-KENNEDY, B. A. (1971), JUDIN, E. G.(1978), KLUG, H.-LANG, R. (1983), LIPEC, J. G. (1980), SCHMIDT, G. (1974), ÚJVARI, I. (1979).

#### LANDSCAPE AS A SPATIALLY ORGANIZED SYSTEM SG(P,T) AND THE POSITION OF THE GEORELIEF IN THE LANDSCAPE AS ITS SUBSYSTEM SRF(P,T).

The landscape is conceived of as a spatially organized matter-energic and information system SG(P,T), whose real geometric space is delimited with respect to the reference surface of the Earth and which is studied in the coordinate system <O, , ,h>, where O is the origin of the coordinate system being identical with the centre of the reference surface of the Earth, - is the geographic latitude, - is the geographic longitude and ±h is the altitude measured in the direction of the normal to the reference surface, starting from the reference surface of the Earth outside (+h) and inside (-h) (Fig.l).

We express therefore geographic sphere as spatially organized system SG(P,T) in a form of an ordered pair (couples)

SG(P,T) = (GG(P,T), RG(P,T)) (1)

where

GG(P,T) = [gi(P,T)]11i=1 (2)

is a set of spatially located elements gi(P,T), of whieh this system consists, and RG(P,T) is a set of spatial relations rij(P,T) between bouth the elements gi(P,T) GG(P,T)and the elements gi(P,T) and enviromnent (surroundning) go of the system SG(P,T). (Fig l).

Under the environment (surroundnig) go we understand, according to works of KRCHO, J. (1968, 1974, 1981, 1983, 1990, 1991) the inside if the Earth E under Lower frontiere of the geographic sphere HD and high stratums of the athmosphere over the upper frontiere of the geographic sphere HD with a neighbouring cosmic space (Fig.l).

Set (2) can be, at the lowest resolution level, divided in two subsets

GAG(P,T) = [gf(P,T)]6f=1 [ef(P,T)]6f=1 (3)

GFG(P,T) = [g6+k]5k=1 [ak(P,T)]5k=1 (4)

In the subset subset GAG(P,T) is gf(P,T) ef(P,T) for f=1,2,...,6

and in the subset GFG(P,T) is gG+K(P,T) ak(P,T) for k=1,2,...,5, where

GG(P,T) = GAG(P,T) GFG(P,T) ; GAG(P,T) GFG(P,T) 0

Subset (3) is a subset of elements representing basic components of the socio-economic (human) sphere at the lowest resolution level i. e.

g1(P,T) e1(P,T) : forestry sphere
g2(P,T) e2(P,T) : agricultural sphere
g3(P,T) e3(P,T) : industrial sphere
g4(P,T) e4(P,T) : transport and communication sphere
g5(P,T) e5(P,T) : settlements sphere
g6(P,T) e6(P,T) : biosphere (fito and zoosphere)

Subset (4) is a subset of elements representing basic components of the physico-geographic sphere at the lowest resolution level i. e.

g6+1(P,T) a1(P,T) : athmosphere
g6+2(P,T) a2(P,T) : hydrosphere
g6+3(P,T) a3(P,T) : lithosphere
g6+4(P,T) a4(P,T) : pedosphere
g6+5(P,T) a5(P,T) : biosphere

By the symbol P = , , h is expressed, in given coordinate system <O, , ,h>, an absolute position of individual elements in a given space,by the symbol P( , , h ) is expressed relative position of two elements always, i. e. ei(P,T), ej(P,T), ai(P,T) , aj(P,T) , where

ij = j - i; ij = j - i; hij = hj - hi .

By the symbol T is expressed a time parameter.

By the set is expressed a structure of the system (1); according to the works KRCHO, J. (1968, 1974, 1076, 1981, 1990) it is possible to

(5)

The elements of this matrix are submatrices, where
ROA(P,T) : submatrix of relations between environment (surrounding) go and elements of the set GAG(P,T) ; this submatrix expresses the influence of the environment (surrounding) go on socio-economic sphere,
ROF(P,T) : submatrix of relations between environment (surrounding) go and elements of the set GFG(P,T) ; this submatrix expresses the influence of the environment (surrounding) go on natural part of the landscape,
RAO(P,T) : submatrix of relations between elements of the set GAG(P,T) and environment (surrounding) go , this submatrix expresses the influence of elements of socio - economic sphere on the environment (surrounding) go ,
RFO (P,T): submatrix of ralations between the set GFG(P,T) and the environment (surrounding) go, this submatrix expresses the influence of the natural part of the landscape on the environment (surrounding) go
(RAG) (P,T): submatrix of relations between elements of the subset GAG(P,T) ; this submatrix expresses the structure within the subsystem SAG(P,T) which will be a subject of further interst,
(RFG) (P,T): submatrix of relations between elements of the subset GFG(P,T) ; this submatrix expresses the structure within the subsystem SFG(P,T) , which will be a subject of further interest,
RAF(P,T) : submatrix of relations between elements of GAG(P,T) and elements of GFG(P,T) ; this submatrix expresses the influence of elements of socio-economic sphere on elements of natural part of the landscape,
RAF(P,T) : submatrix of ralations between elements of GFG(P,T) and elements of GAG(P,T) ; this submatrix expresses the influence of natural part of the landscape on elements of socio-economic sphere.

All this can be ec onomically expressed in the form of an oriented graph (Fig 2).

In the system SG(P,T) (l) it is possible, according to the works KRCHO, J. (l968, 1974, 1981, 1990, 1991) to express two spatialy organized autonomous subsystem i.e.

SAG(P,T) = (GAG(P,T), RAG(P,T)) (6)

SFG(P,T) = (GFG(P,T), RFG(P,T)) (7)

which are in a interaction. Subsystem SAG(P,T) (6) is a subsystem of the socio-economic sphere, subsystem SFG(P,T) (7) is a subsystem of the physico-geographic sphere, GAG(P,T), GFG(P,T) are elements (3), (4), RAG(P,T) is z matrix of the structure of the subsystem SAG(P,T) ; RFG(P,T) is a matrix of the structure of the subsystem SFG(P,T).

Subsystem SFG(P,T) (7) - physico-geographic sphere is a spatially organized and spatially complex with a specifie cireulation of matter, energy and information and with a specifie selfregulation mechanism.

Subsystem SAG(P,T) (6) - Human-geographic (socio-economic) sphere is also spatially differentiated and spatially highly organized complex with a specific selfregulation mechanism. Looking at the problem spatially, socio-economic sphere occur in a geometric space of physico-geographic sphere (in natural part of landscape).

Interaction of socio-economic and physico-geographic sphere produces spatially organized cultural landscape. Man's activity can produce spatially distributed disturbances in the natural landscape; it can also disturb selfregulation processe. Physico-geographic sphere attemps to compensate disturbances produced by man's activity.

Analysing the georelief we are interested in the following subsystem SFG(P,T)(7), which can also be studied as an autonomous, planet-wide and spatially organised system.

The natural part of the landscape, considered as a subsystem (7) is a spatially organized whole consisting of basic, spatially differentiated components:

a1(P,T), a2(P,T), a3(P,T), a4(P,T), a5(P,T),

all being in interaction.

The natural part of the landscape (7) is, with respect to the reference surface of the Earth, spatialy delimited by the region of mutual spatial intersection and by the interaction of individual spheres ak(P,T)(k = 1,2,...,5).Vetical delimination of the system SFG(P,T) (7) is given in the direction of normals to the reference surface of the Earth, i.e.by HH hmax from above nad by HD hmin from below (Fig.l).

Interaction of individual spheres ak(P,T) GFG(P,T) as elements of the system SFG(P,T) (7) produces processes, specifiec only for the region of intersection, which are characterized by the circulation of matter, energy and information.

To be able to express spatial differentiation of subsystems SAG(P,T) , SFG(P,T) (6), (7) we raise the resolution level in such a way that each of the elements ef(P,T), ak(P,T) of the sets GAG(P,T) , GFG(P,T) (3), (4) becomes an ordered subset of distinguished elements.

Gef = [efi]mfi=1 Gak = [akj]nkj=1 (8)

for each f = 1,2,...,6 for each k = 1,2,...,5

The elements

efi(P,T) Gef (P,T)

represented for individual f = l, 2, 3, 4, 5, 6 components of individual original, however now interally distinguished components of socio-economic sphere, and elements

akj (P,T) Gak(P,T)

represent form individual k = l, 2, 3, 4, 5 components of individual, however, now internally distinguished components (geospheres) of physico-geographical sphere.

The change (elevation) of the resolution level can lead to the situation under which the individual spheres ak (P,T) of GFG(P,T) are considered as ordered sets

ak (P,T) = Gak(P,T) = [akj]nkj=1 for each k = 1, 2, ..., 5

where individual elements akj present for k = l, 2, 3, 4, 5, and j = l, 2, ...,nk, ( n1 n2 ... n5) individual components of the geosphere ak (P,T). Then the set GFG(P,T) can be expressed as follows:

(9)

Since between elements

akj (P,T) Gak(P,T)

for each individual k = 1, 2, ..., 5, a set of relation can be observed, it is possible the individual spheres ak(P,T) to be analysed as spatially organized subsystems

Sak(P,T) = (Gak(P,T), Rak(P,T)) (10)

of the system SFG(P,T). The structure of the system SFG(P,T) is then given by the set of distinguishable relations RFG(P,T) between interacting subsystems (10).

The system SFG(P,T) is in each position P = , , h of the delimited space and each T in some whereby the general state of each element akj (P,T) , general state ZFG(P,T) , which is characterized by general state Zak(P,T) of the subsystems Sak(P,T)(10) and their elements akj (P,T) , whereby the general state of each element akj (P,T) , general states Zak(P,T) and hence also the general state ZFG(P,T), are determined by state variables z1, z2, ...,zn. These variable can presemnt physical, physio-chemical, chemical, mathematical and mathematico-statical quantities. States of each subsystem Sak(P,T) of system SFG(P,T) and processes in lanscape which are originated from mutual interaction of subsystems Sak(P,T) expressed by the set RFG(P,T) can be expressed by state quantities Z1, Z2, ..., Zn. The course of prosesses in landscape in each time moment T and in each position P = (, , h) is characterized by these state quantities. The size of particular state quantities Zi (i = 1, 2, ..., n) is changed either in course of time T or in dependance on position P = (, , h). There are values of particular state quantities Zi for particular i = l, 2,..., n within certain intervals

[ZD(P,T)]i Zi(P,T) [ZH(P,T)]i (11)

where [ZD(P,T)]i is Lower value of i-order state quantity which it reaches in position P = (, , h) and [ZH(P,T)]i is upper value. Presented record (11) expresses that the value [ZD(P,T)]i , [ZH(P,T)]i for each one i = l, 2, ...., n depends either on position P = (, , h) or the time T .

The state variables Z1, Z2, ..., Zn are functionally interrelated and their values are functions of the position P = (, , h) and the time T . On this basis it is possible the system SFG(P,T) (7) to be expressed by the equation system

Z1 = 1(Z2, Z3, ...Zn) Z1= Gi ( ,, h, T)
Z2 = 2(Z1, Z3, ...Zn) i = l, 2, ...., n
Zn = n(Z1, Z2, ...Zn) (12)

The system SFG(P,T) is spatially differentiated into individual spatial subsystems SFG,n (P,T), (n = 1, 2, ...) which in the landscape present individual landscape spatial units - the geosystems (Fig.3). In this spatial differentiation the important role has the georelief which presents a firm, but dynamical interface of the surface parts of the lithosphere Sa3(P,T) and the pedosphere Sa4(P,T) with the atmosphere Sa1(P,T) and the hydrophere Sa2(P,T).

The spatial differentiation, or course of the surface parts the lithosphere as a subsystem Sa3(P,T) including pedosphere Sa4(P,T) is a result of the interaction of all susbystems Sak(P,T) in the system SaFG(P,T)during the time T .

Let us present a brief definition of the georelief as a subsystem SaRF(P,T)in the system SaFG(P,T).

Georelief is, at some resolution level, a firm, but dynamical interface between lithosphere Sa3(P,T) or pedosphere Sa4(P,T) with biosphere Sa5(P,T) on the one side (hand) and the atmosphere Sa1(P,T) or hydrosphere Sa2(P,T) on the other side (hand). This firm, but dynamical interface, when analysed its spatial course in the coordinate system <O,, , h> has attributes of the surface, created by the set of points on the normals N to the reference surface within the interval (HH, HD).

A complete definition of the georelief was given in the papers of KRCHO, J. (1973, 1987, 1989, 1990); there also aditional remarks can be found, which specify the land relief, the relief of the sea bottom, bottom of lakes and the glacial relief. It is clear, already from the brief definition of the georelief, that the value of the altitude h is in each point Ai(i, i, hir) of the georelief and or each time point T an output of the preceding gradual change of the general states Zak(P,T) of the subsystems Sak(P,T) ( k = 1,2,3,4,5) determined by state variables Z1, Z2, ..., Zn . The spatial distribution of altitudes hir of points Ai(i, i, hir) creating the georelief, is thus in functional relation to state variables Zn (n = 1,2, ...) whereby the values of the individual state variables both interrelate and functions of the position , , h and the time T ; hence, they are determined by the equation system (12). The value of the altitude k as a function of state variables Z1, Z2, ..., Zn , position , , h and time T can be expressed in the following general form:

h = fRF (Z2, Z3, ...Zn)
Z1 = 1(Z2, Z3, ...Zn) Z1= gi ( ,, h, T)
Z2 = 2(Z1, Z3, ...Zn)
Zn = n(Z1, Z2, ...Zn) (13)

Equation system (13) puts down the spatial distribution of altitudes of georelief and its development in the time T with respect to the modelling processes. Equation system (13) enables to express an ordered set of morphometrical parameters

GRF(P,T) = [ h(P,T), N(P,T), AN(P,T), (P,T), (KN)t(P,T), Kr(P,T), ...] (14)

as dependent variables upon state variables Z1, Z2, ..., Zn, position , , h and the time T , where

h(P,T) denotes the relative height in the direction of orthogonal lines,
N(P,T) the slope (inclination) of the georelief in the direction of orthogonal lines,
AN(P,T) the orientation (exposition) of the georelief with respect to the cardinal points,
(P,T) the normal curvature of the georelief in the direction of orthogonal lines (not identical with extreme normal curvature of Dupen indikatrix),
(KN)t(P,T) the normal curvature of the georelief in the direction of tangent to the contour line,
Kr(P,T) the horizontal curvature of the georelief (curvature in the direction of contour lines), etc. Each individual morphometrical parameter from the set (14) is determined by the functional relationship derived from equation system (13).

Georelief in the spirit of the preceding definition, as a dynamic surface, can thus be considered as a specific, dynamic, spatially, organized, inmaterial subsystem

SRF (P,T) = [GRF (P,T), RRF (P,T)] (15)

being determined both by the equation system (13) and by the set of functions associated with the set of morphometric parameters GRF (P,T) (14), derived from the equation system (13).

Georelief is, on the one hand, an output of processes resulting from interaction of the subsystems Sak (P,T), on the other hand, however, it alone affects these subsystems. Its altitude h and hence its individual elements in the ordered set GRF (P,T) (14) change in each point Ai(i, i, hir) in the time T in dependence upon state variables and upon the dynamics of the general state of subsystems Sak (P,T) of the system SFG (P,T).

Morphometric quantities (14) are dynamic quantities which are changed in course of time T due to modelling processes. Starting point form their derivation on dynamic terrain surface there are first and second differentials dh, d2h on the function

h = fRF (Z2, Z3, ...Zn)

in equation system (13), where

(16)

whereby dZi( i = 1,2, n) on the basic of relations (13) are expressed by relations

(17)

for i = 1,2,..., n i.e. for i = 1 j = 2,3,...,n
for i = 2 j = 1,3,...,n
- -
- -
for i = n j = 1,2,...,n -1

Because state quantities Z1, Z2, ..., Zn are simultaneously the functions of position , , h and time T , for their differential due to

Z1= gi ( ,, h, T)

for particular i = 1,2,...,n there is valid

(18)

Therefore the differentials dZi in relations (17) can be expressed due to (13) in the form

dZi = Didh + Mid+ Nid+ LidT (19)

who

(19')

In this way there are the differentials dZi( i = 1,2, n) in relation (16) of particular variable quantities Zi expressed due to the position , , h and due to the time T .

On the basis (19) the relation (16) for differential dh after modification can be expressed in the form

(20)

In this relation (20) from point of view of its following simplification particular components are signed according , , h,T by symbols

(21)

then relation for first diffrential dh (20) can be expressed in the form

(22)

Second differential d2h is expressed on basis of relation (20) so

(23)

If relation (22) is modificated due to dh into the form

(24)

and info the form

(24')

The differential equation of dynamic terrain surface which geometry is resultant of state quantities Z1, Z2, ..., Zn as functions of position P = ( , , h) and their changes in time T will acquired in common form . For determined value T = const. there is (22), (23), (24) and (24') the component Ft= 0, so in this case the equations (24) and (24') express the state of georelief as a result of modelling processes in determined time T . The realtions (22),(23) for T = const. will aquire the form

(25)

(25')

The relations (22),(23) form T = const and h = const. will aquire the form

(26)

(26')

The equations (24), (24') resp. are rather complicated therefore from point of view on the aim of paper they have only theoretico-methodological meaning, mainly their initial form (22). It is only starting point for derivation of morphometric quantities N(P,T), AN(P,T) in the set (14). On the basis (22) relation for vector gradient grad h wich |grad h| = tgN and from which can be acquired that

N = arctg(|grad h|) (27)

whereby N(P,T) is be the component of the set (14). Vector grad h forms gradient field on referential surface of the Earth whereby gradient grad h is oriented to the side dh/dn > 0 in direction of orthogonal trajectories to the isolines field of altitudes h , i.e. to the contour lines field on referential surface. Referential surface of the Earth (referential global surface or surface of referential ellipsoid) is simultaneously scalar basis either vector field h or the other fields, so called structural fields of georelief formed by componenets of the set (14).

The relation for grad h for T = const. from the relation (25) in this way that the derivation dh/dn direction of tangents to orthogonal trajectories of isolines field of altitudes (to the contour lines field) i. e. in direction of normal n to contour lines on referential surface of the Earth which has the form after arrangement is expressed

(27')

and which is adjusted in coordinates system <O, , , h> , into the form of scalar product of two vector .

(28)

In relation (28) the vector grad h is expressed by the relation

(29)

and its unitary vector n0 is expressed by the relation

(30)

The vectors u, v are considered as unitary vectors oriented in direction of parallels and in direction of meridians . From the relation (29) is resulted that

(31)

For direction of vector grad h determined by angle which is simultaneously in coordinates system <O, , , h> the angle of relief orientation to cardinal points, the relation is aquired

(32)

In following procedure, due to simplification of problem, which does not influence upon the correctness of procedure and correctness of solution there will be supposed that state quantities Zi (i = 1, 2, ..., n) are only the functions of posotion , (i.e. h = const., T = const.). So in the relations (17), (18) for particular i = 1, 2, ..., n will be Di = 0 and Li = 0, so the relations (22), (23) will have the form (26) (26').

The dynamics of the change of h-values, as well individual morphometric parameters in GRF(P,T) (14) is but in lanrger spatial dimension much slower than the dynamics of the subsystems Sa1(P,T), Sa2(P,T); it depends upon the attributes of the subsystem Sa3(P,T) . From point of view of the morphometric analysis of georelief as subsystem of landscape for different time interval (TPK)i <Tp, Tk>i (TPK)i = 5, 10, 20, 30, 50, 100, ... years which depends on the scale of map 1: Mi( i = 1,2,...) and its discintive level where M1 = 1 000, M2 = 2 000, M3 = 5 000, M4 = 10 000, M5 = 25 000, M6 = 50 000 the parameter of time can be eliminated T = const. and only the influence of relief geometry upon ecological proceses in landscape can be studied however not on the contrary. Modelling the influence of the georelief on the spatial differentiation of proceses in the landscape makes it thas possible the time parameter T , considered in the form of shorter time intervals (TPK)i= 5,10,20,... years, to be taken in equation system (13) as a constant, viz. starting with some time point Tpand finishing with the time point Tk , i. e. the interval (TPK)i <Tp, Tk> ((TPK)i = (Tp)i - (Tk)i i = 1, 2, ... )The length of interval (TPK)idepends upon the scale of the map its resolution level.

In the influence of relief upon ecological processes in landscape is only studied then the chandge of relief for given time interval (TPK)i in selected scale 1: Mi will be considered as null i. e. relief will be considered as constant and so static system. There is the substance of this consideration that the change od relief for selected scale 1: Mi in the length of its relevant time Ti is below its distinctive level therefore during this interval it does not manifest. Moreover its morphometric quantities (morphometric parameters) as input variables their physical meaning is simultaneously saved. Relief influences by them upon processes and their spatial differentiation in landscape whereby in selected scale 1: Mi during time (TPK)i is considered as non-variable (constant). It means that during this time the influence of processes upon relief is not considered.

In this case (for T = const.) even in the relation (22) (23) there will be FT = 0, so they will have the form (26),(26´). From (26) on the reference global surface (on reference surface of the Earth) for vector grad h and its unitary vector n0 the relation is acquired

(33)

There is resulted from the relation (33) that absolute value of vector grad h i. e.

(33´)

The spatial distribution of the altitudes h is in this case dependent only on the position , , i. e. the altitudes are now only the function of the position , . Thaking into account this theoretico-methodic fact, spatial distribution of altitudes h due to referential surface of the Earth for T = const. in gravitational field of the Earth in coordinates system <O, , , h> can be simply expressed in form of function

h = f (, ) (34)

Georelief is thus, within the given interval (TPK)i <Tp, Tk> , considered as a static surface expressed by the function (34).

This function in resulting form briefly expresses fact that altitudes h are the function of position P = (, ) on referential surface of the Earth whereby they are considered in direction of its normals N as tangens to gravitation power curves. The differential of altitude dh of the function (34) is the shortest connecting line of two altitudes levels in the direction of gravitation. The function (34) neither its first and second differentials

(34´)

where

however in comparison with previous equations it does not contain the fact that altitudes h are result of modelling processe in landscape characterised in each time moment T by state quantities Z1, Z2, ..., Zn . Besides these state quantities are the function, of position P = (, , h) and time T expressed in the form Z1= gi ( ,, h, T) (13). The function (34), as a matter of fact, in coordinates system only states the fact that altitudes of the Earth georelief as well their differentials (34') measured in the direction of normals to referential surface are in gravitational field of the Eath the function of position , and nothing more. However it does not express their cause and development in time. It is the basis for description of relief geometry considered in gravitational field of the Earth what is in given time. The function (34) is considered as common static function, geometrically theoretically exactly described the Earth georelief for given time moment T . Structural characters of georelief as static surface which the set morphometric parameters influence upon spatial differentiation of processes in landscape but not on the country will be derived.

The subsystem SRF (P,T) (15) becomes thus a static spatially organized subsystem

SRF (P) = [GRF (P), RRF (P)] (35)

in which the values of its individual elements from the set

GRF(P) = [ h(P), N(P), AN(P), (P), (KN)(P), Kr(P), ...] (35´)

are only the function of the position , .

In analysing the interaction of static subsystem SRF (P) (35) with the systems Sak (P,T) we explore for each T in the given time interval Tp T Tk only the influence of the georelief on the spatial differentiation of the states of the subsystems Sak (P,T) in the system SFG (P,T) , but not more the influence of these subsystems on the georelief as a subsystem SRF (P) (35). Components of the set GRF(P) (35') there are static component expressing geometric structure of georelief as static surface.

Note that the system SFG (P,T) as well as its subsystem SRF (P) are explored, expressed and interpreted in the map (in abstract projection space) of the scale 1: Mi( i = 1,2,...) which is expressed in the Cartesian coordinate system <O, , , h> . The positon in the map is, with respect to the real space <O, , , h> determined by the projection equations.

#### FORMALIZED EXPRESSION OF GEORELIEF IN ABSTRACT (CARTOGRAPHIC) PROJECTION SPACE IN CARTESIAN COORDINATES SYSTEM <O, , , h>

So far basic consideration have been formulated due to the real geometric space of geographical sphere (landscape) which has been determined in coordinates system. <O, , , h> due to referential surface of the Earth. In this space either landscape considered as system SFG (P,T) , (1), its susbystems SFG (P,T), SAG (P,T) (6),(7), or as relief of the Earth (georelief) expressed as the subsystem SRF (P,T)(13), (15) have been expressed. The same way in this space dynamics of processes in landscape by the system of equations (12) and relief considered as dynamic surface expressed by the system equation (13) have been expressed.

Georelief as well as whole subsystem SFG (P,T), with its subsystems Sak (P,T) are modelled in abstract (cartographic) projection space determined in Cartesian coordinates system <O, , , h> in which real space of geographic sphere (landscape) will be projected. Referential surface of the Earth (referential global surface or referential surface of rotative ellipsoid) due to which real space of the Earth was defined, it will be projected into projection level (x,y) of abstract (cartographic) space (info map). Projection plane (x,y) (map) of projection space has exactly defined mathematical characters determined by projection equations. Projection of whole defined real space of the Earth in mathematical sense is given by unambiquous operation of projection determined by projection (transformation) equations

x = f1( , );y = f2( , ); z = f3(h)
= F1( x,y) ; = F2( x,y); h = F3(z) (36)

In this projection space even processes expressed primarily in real space of the Earth by the equation system (12), (13). Therefore the equation system (13) expressed in the Cartesian coordinate system <O, , , h> , will take of the following form:

z = fRF (Z1, Z2, ...Zn)
Z1 = 1(Z2, Z3, ...Zn) Zi= gi ( x, y, z, T)
Z2 = 2(Z1, Z3, ...Zn)
Zn = n(Z1, Z2, ...Zn) (37)

where x, y, z are determined by equations (36). Therefore in projection space even mentioned relations will be changed by relevant way whereby firstly differentials dZifrom the equations (37) are presented

(38)

Consequently the relation (20) will have the form

(39)

Second defferential d2z is expressed on basis of relation (39) so

(40)

The relations (39), (40) form z = const. and T = const. aquire the form

(41) , (42)

Note briefly that the morphometric parameters of the georelief

(43)

P' =(x,y) from the set GRF(P) (35') are in the map, when the time parameter is not taken into account, determined by the relations

(44)

(45)

(46)

(47)

(48)

The set (43) is the projection (transformation) of the set (35') from real space of the Earth into abstract (cartographic) space one withou parameter T , i.e.

GRF(P) : --> GRF(P') (49)

The symbol P = , expresses in (50) the position of real space determined considering referential surface of the Earth in coordinates system <O, , , h> and the symbol P' = (x,y) expresses relevant position in projection plane (x,y) i.e. on map whereby coordinates are determined by projection (transformation) equations (36).

Because in real space of the Earth, from point of view of paper aim , the georelief as static surface in gravitational field ef the Earth has been expressed by common functional relation (34) in which altitudes h are only the function of position , in projection (cartographic) space the relation

z = z (x,y) (50)

will correspond with it. The function (49) express only the fact that there are projected altitudes z (z = f3(h))in projection plane (x, y) i.e. in the map only the function of position x,y and nothing more.

The function (50) is a one-valued map (transformation) of the function (34) which is expressed in the form

h = f( , ): --> z = z (x,y) (51)

and the morphometric parameters of the georelief (43) are in the map determined by the relations

(52)

(53)

(54)

(55)

(56)

In these relations zx, zy, zxx, zyy, zxy denote the partial derivateof the function (50). Morphometric parameters , (KN)t , Kr take on the values

<0, =0, >0,
(KN)t<0, (KN)t=0, (KN)t>0,
Kr<0, Kr<0, Kr<0,

The geometric form of the georelief are expressed by means of the paremeter values , Kr follous from the equations (55) and (58) that (KN)t has an identical spatial distribution as Kr , for which it is valid

The isoline Kr = 0 is thus as to spatial course, identical with the isoline (KN)t = 0.

Note that the normal curvalure of the georelief in the direction of orthogonal lines and the normal curvature of the georelief (KN)t in the direction of tangent t to the contour line are not identical with principal surface curvature defined in differential geometry in the direction of axes of the Dupin's indikatrix. The curvatures , (KN)t ,and Kr morphometric parameters, where derived for the needs of the morphometric analysis of the georelief conceived of as a subsystem of the Landscape KRCHO, J. (1973, 1987, 1989, 1990).

The georelief given by the function (50) without the time parameter T and its height field is depicted in form coputer map in Fig. 6 from the Zemianska Olca region. The spatial distribution of the individual morphometric parameters N, AN, , Kr and total general geometric forms of relief from the region Zemianska O3/4ča and region Šamorín is depicted in Fig. 7, Fig. 8, Fig. 9, Fig. 11, Fig. 12 and in Fig. 13, Fig. 14.

In Fig. 7 the spatial distribution of slopes N in the direction if orthogonal lines is from region Zemianska Olca depicted by isolines of equal slope of the georelief N in the direction orthogonal lines. For purposes of illustration the slopes N are divided into intervals.

In Fig. 8 from region Zemianska Olca depict the spatial course of the normal curvature of the georelief . In regions where <0 the georelief exhibits the convex normal forms in the direction of orthogonal lines; they are denoted by NnFX. Converselly, where >0 the georelief exhibits the concave normal forms denoted by NnFK. Convex normal forms NnFX (>0) are separated from concave normal forms NnFK <0 by an isoline of the zero normal curvature ( =0 ) . The individual letters in NnFX and NnFK are to be interpreted as follows: N denotes normal form F,n denotes form in the direction of tangent to the orthogonal line, X denotes convex form and K denotes concave form.

Fig. 9, thaken from the same region Zemianska Olca, depict the spatial course of the horizontal curvature Kr, i. e. curvature in the direction of contour lines. In these parts of the georelief where Kr > 0 the georelief exibits convex horizontal forms denoted by KrFX; conversely, where Kr < 0 the georelief exibits the concave horizontal forms denoted by KrFK. Convex horizontal forms KrFX (Kr > 0)and concave horizontal forms KrFK (Kr < 0)are each from other separated by an isoline of the zero horizontal curvature Kr = 0.

Fig. 11 (computer map of the slope of relief in the direction of orthogen direction of orthogonal lines) depicit the spatial distribution of slopes N in the direction of orthogonal lines foom region of Šamorín.

Fig. 12 (computer map of the orientation of the georelief with respect to the cardinal points) depiet the spatial distribution of values of the orientation AN of georelief with respect to cardinal point from region Šamorín.

On the basis of the normal curvature , normal curvature (KN)t , or horizontal curvature Kr and their signs (±) it is possible to state and quantitatively characterize general geometric forms of the georelief:

(>0, Kr > 0) convex-convex forms FXX,
(<0, Kr > 0) concave-convex forms FKX
(<0, Kr < 0) concave-concave forms FKK
(>0, Kr < 0) convex-concave forms FXK

(>0, Kr = 0) convex-linear forms FXL
(=0, Kr > 0) linear-convex forms FLX
(<0, Kr = 0) concave-linear forms FKL
(=0, Kr < 0) linear-concave forms FLK
(=0, Kr = 0) linear-linear forms FLL

These total geometric georelief forms (57), (58) present a set

F = (FXX, FKX, FKK, FXK, FXL, FLX, FKL, FLK, FLL, ) (59)

Detailed determination of the geometric forms of the georelief is very important both for understanding of slope modelling and from the general lanscape-ecological point of view.

The influence of individual total gemetric forms (57), (58) manifesting itself through , Kr ,find, for instance, its expresion in the hydrothermic regiment of water run-off regiment of water from soils and hence also in the soil erosion. In the convex normal forms NnFx (>0) erosion processes take place, whereas in the concave normal forms NnFK (<0) accumulation processes. Moreover, in any point Ai (xi ,yi ,zi )of the georelief folloving items are relevant:

- size of i, (Kr)i together with the size of j, (Kr)j in its neighbouring points A'j (xj ,yj ,zj ) which create the corresponding geometric form,
- size of (N )i the slope together with the size of (N )j in its neighbouring points A'j (xj ,yj ,zj ),
- change of the slope N / n providing the change of points,
- direction of the vector grad z determined by the angle AN,
- change of the direction AN by the element ANin the direction of contour lines, i. e. AN / s,
- length of the hill-side the direction of orthogonal line. In the given point A'i (xi ,yi ,zi ) of the georelief are thus important both the vector grad zi zwith its absolute value | grad z | and

i. e. the values of the derivate of the vector grad zi and its absolute value | grad zi | in the point A'i.

This is expressed in graphs on the profile led in the direction of orthogonal trajectory to contour lines at the Fig. 5. Unfolded slope curve into plane in local coordinates system <O,z,n> where O is the beginning of cordinates system at the beginning of slope curve, z are altitudes n is unfolded lenght of orthogonal trajectory.

The values | grad z | aequire the extreme | grad z |E in inflex point If of slope curve which separates its concave forms from convex ones. The curve | grad z | reaches the inflex points in the such position of profile where profil of slope curves reaches extreme normal curature E either maximum (E)max or minimum (E)min. The values ,

have extrenes in inflex points | grad z | , i.e. in such positions in which the profil of slope curve reaches the extreme values (E)maxor minima (E)min .

The normal curvature by which, according to its sign (±), normal forms NnF in the direction of orthogonal lines are characterized ( NnFX for >0, NnFL for =0, NnFK for <0 ) accelerates, balances or decelerates the run-off regimen and other processes of slope modelling, viz. according to its size and sign. In the normal forms NnF manifests itself as follows:

- convex normal forms NnFX , for which in each point Ai (xi ,yi ,zi ) it is valid that >0, accelente the processes of slope modelling,
- linear normal forms NnFL, for which in each point Ai (xi ,yi ,zi ) it is valid that =0, balance the processes of slope modelling,
- concave normal forms NnFK , for which in each point Ai (xi ,yi ,zi ) it is valid that <0 ecelarate the processes of slope modelling.

The horizontal curcave Kr by which, according to its sign (±), are determined horizontal forms

KrF ( KrFX for Kr > 0 , KrFL for Kr = 0 , KrFK for Kr < 0 )

has in forming the run-off regimen and processes of slope modelling a following role:

- convex horizontal forms KrFX, for which, in each point Ai (xi ,yi ,zi ) it is valid that Kr > 0, disperse the run-off regimen and the processes of slope modelling,
- linear horizontal forms KrFL, for which, in each point Ai (xi ,yi ,zi ) it is valid that Kr = 0, balance the run-off regimen and the processes of slope modelling,
- concave horizontal forms KrFK, for which, in each point Ai (xi ,yi ,zi ) it is valid that Kr < 0, concentrate the run-off regimen and the processes of slope modelling.

On the basis of morphometric analysis (designed by KRCHO, J. 1964 a, b, 1973, 1983) this problem was tacked , from the landscape-ecological point of view, in the papers of RUŽIČKA, M., MIKLÓS, L. et al. (1976, 1982), MIKLÓS, L. (1986), MIKLÓS, L., HRNČIAROVÁ, KOZOVA, M. (1984) and others. From the point of viw of slope modelling this problem, again on the basis of papers KRCHO, J. (1964 a, b, 1973, 1983), was tacked by MINÁR, J.(1986), and from the point of view of run-off regimen by REHÁK (1988), MIKLÓS, KOZOVÁ, RUŽIČKA et al. (1986), MIKLÓS (1986), MIKLÓS, HRNČIAROVÁ, KOZOVÁ (1984).

The influence of the total forms of the georelief (57), (58), (59) on the course of slope modelling is by means of the course of size and direction of the vector grad z depicted in Fig.4. The influence of the forms can be expressed as follows:

FXX (>0, Kr > 0) Forms acceleratin and dispersing the run-off regimen,
FKX (<0, Kr > 0) Forms decelerating and dispersing the run-off regimen,
FKK (<0, Kr <0) Forms decelerating and concentrating the run-off regimen,
FXK (>0, Kr < 0) Forms accelerating and concentrating the run-off regimen,
FLX (=0, Kr > 0) Forms balancing and dispersing the run-off regimen,
FXL (>0, Kr = 0) Forms accerating and balancing the run-off regimen,
FKL (<0, Kr = 0) Forms decelerating and balancing the run-off regimen,
FLK (=0, Kr < 0) Forms balancing and concentrating the run-off regimen

The spatial distribution of the total (general) geometric froms of the georelief from same regions Šamorín and Zemianska Olča is depicted in Fig.13 and in Fig.14.

The influence of georelief morphometric parameters on ecological processes in landscape in manifested in mutual dependence of the set GRF(P) parameters which are not mutually independent. Mutual relation of morphometric parameters is differently manifested in mountains regions, hilly countries regions and lovland regions.

The influence of georelief morphometric parameters on ecological processes in landcape in lowland regions of georelief in sence of the paper KRCHO, J. (1973) was illustrated in the paper MIKLÓS, MIKLISOVÁ (1987) in which so called micro-basins of georelief and their role in landscape were formulated.

In lowland areas of georelief the influence of georelief on landscape-ecological processes is manifested by means of total geometric forms of georelief. Considerable differentiation of hydrothermic regime of soils and whole geochemic regime and dynamic odf soils is related to these quantities.

#### CONCLUSION

In the paper there was an effort to present geometry and geometric structure of georelief from point of view of broader relations of processes running in landscape. Landcape and its spatial organisation were formulated as spatially organized and spatialy differentiated systems further to citaded and others papers. Simultaneously the importance of spatial human activity in landscape and its influence on spatial processes differentiation in natural part of landscape was briefly pointed out.

Simultaneously there was an effort to point out the importance of relief role in landscape and its influence

- on spatial processes differentiation in natural part of landscape,
- spatial differntiation of human activity in landscape.

There was an effort to point out the manin spheres of whole problem and in this sence to point out the position of georelief in landscape. The main subject of interest (in context with presented problems) was geometry of georelief from point of views as follows:

- geometry of relief as "respond" of lithospere characters to processes in landscape from one side and from second one as "memory" of relief with respect to processes so far occured,
- geometric structure of georelief as subsystem of landscape in gravitational field of the Earth,
- the expression of morphometric quantities of relief GRF(P,T), GRF(P) as well a morphometric quantities GRF(P') where P = (, , h), P' = (x, y, z)in each arbitry localized areal element P of georelief, correct mathematical characterization of these quantities as well as their mutual relation,
- the expresion of morphometric quantities (morphometric parameters) influence on processes in landscape.

Fruther the problem of cartographic modelling of georelief by digital models was briefly presented. Simultaneously there was an effort to present only basis groups of problems.

The complex digital model of georelief integrates theoretical conception of georelief considered as spatial landscape subsystem SRF(P).

It enables detailed calculation and cartographic modelling of spatial differentiation of values of various morphometric parameters from GRF(P) set.

On the basis of exact morphometric analysis calculation and cartographic modelling of the following morphometric quantities with the output in the form of computer maps was made by means of complex digital georelief model of the larger area of Gabčíkovo hydroelectric system:

- the slope of the georelief in the direction of the orthogonal lines to the contour lines
- orientation of the georelief with respect to the cardinal points
- the normal curvatures of the georelief in the direction of slope curves

Kr - horizontal curvature of georelief
NnF- normal forms of georelief (NnFx ,NnFk),
KrF- horizontal forms of georelief (KrFx, KrFk),
F - complete forms of georelief

These morphometric parameters are input quantities for the calculation of the georelief influence on spatial differentiation of hydrothermic and geochemical soil regime and other landscape components.

The whole complex of outputs included in the database is illustrated by graphical outputs shown in Fig. 6, 7, 8, 9, 10, 11, 12, 13, 14.

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