GEOL. CROAT. 50/1 79 - 88 5 Figs. 3 Tabs. ZAGREB 1997 

Frequency Distribution Curves as an Indicator of Evolutionary Trends in 
Geomorphological Systems: A Case Study from the Northwestern Part 

of Hrvatsko Zagorje (Croatia) 

Zoran PEH 

Key words: Geomorphological systems, 'I'hc rmocly­
nami cs or nOllcquilibrium systems, Frequency dis­
tribut ion curves, Horizontal dissection, Vertical dis­
sectio n, HYPsoll1ctric iJ1lcgral, Drainage densit y, 
Slope, Erosional levels, Hrva tsko Zagorje, Croat ia. 

Abstract 
Frequency (lislribulion curves can be indicative of the dynamics 

and CVO!lIliol1<ITY development of gcomorpho logic.tl systems. Normal­
ly distriblllCd geomorphological data arc a reliable indicator of th e 
equi librium tluctualions prevalent within such systems. Thi s occurs, 
for example, when i\ delicate balance is established between varioll s 
lypes of landform s and geomorphological proces ses . Landforms Illat 
ex ist in thi s geomorphological steady state can be properly ca ll ed 
"equilibrium" structures . They show no pcrmanen t trends of either 
growth or destruction. On the contrary, a positively skewed frequency 
distribution curvc indicutes a type of non-equilibrium cond iti ons and 
Iloll -line:lr rel:ltionships w ithi n geomorphological systems, which 
Illay be the resu lt of major cx ternal energy inputs (particu larly from 
tcc tonic uplift ). Thi s. in turn. gi ves rise to new and more complex 
higher-order landfonns. These represent structures of growth . or evo­
lut ion. Alle rnati vely, a negati vely skewed freq uency distri bution 
curve, also be ing a consequence of non-equilibrium conditions, ind i­
cates a lack o f free energy in Ihe system. which induces destflletion of 
[he existing landforms. Such landforms represent strUClllres of degen­
eration. The rlu vial landscape i n the northwestern pari of Hrvatsko 
Zagorj e, portrayed in terms of an open system, has been used as the 
geomorphologica l framework for this study, the primmy purpose o f 
which was (0 apply frequency di~tribution curves in the interprelll ti oll 
of the processes influenc ing the creation and destructi on of the 1'\I1d­
sCilpe. I [ori zo lltal and vertical dissection, slopc and other morphomct­
ric clwracteris tics rcnecllhc spcc ific dynamics of the geomorphologi­
cal processes, which relate a variety of landforms to th eir geologic 
framework and climate in thc study arca. 

1. INTRODUCTION 

A diligent search for solid proof of how the proccss­
es shaping the face of our planet actually work, resulted 
in the acceptance and application of new concepts in 
geo logy. These originate from the domain of funda ­
mentnl natural sc iences, particularly mathematics and 
phys ics . However, despite the fact that geology, nnci 
geomorphology in particular, has taken a great leap in 

tnstitutc of Geology. Sachsova 2, HR-l 0000 Zagreb, Croatia . 

the last few decad es owing to newly promoted ideas, 
many questions, especially those concerning the prob­
lems of evolution of geomorphological sys tcms, remain 
unanswered. Very often, emphasis is placed on the fact 
tha i all natura l systems, including geomorpho logical 
ones, are too complex for analy tical treatment beyond a 
rout ine search for the empirical relati ons hips among 
assessed data (PATEE, 1973; KARCZ. 1980). Never­
theless, many recent ideas may be directly applied, par­
ticularly with regard to the problem of struc tural stabili­
ty and related fluctuation phenomcna (osci llations of 
system parameters), that are considered to be the main 
cause of pcrmanent and irrevers ibl e change in natural 
sys tems. Stability and cquilibrium , re fl ected in the geo­
morphological steady-state concept, present the general­
ly accept cd axioms around which the analysis of vari­
ous geological and geomorphological systems has 
revolved in the past few decades (Ki\RCZ, 1980). 

One should emphasize. though, thai evolution of the 
natura l systems, including landscapes. docs not proceed 
via the equilibrium condi tions that prescribe Ihe stable 
llux of matter and cnergy. On the contrary, evolution is 
induced by a shock occurring as a system's response to 
strong external impul ses. Evolutionary processes, trig­
gered as the system reaches the far-from- equilibrium, 
non-linear regime, eventually inc rease its structural 
comp lexity (NICOLlS & PRIGOGINE, 1977; PRI­
GOGlNE. 1978; KARCZ, 1980; PRIGOGINE & 
STENGERS. 1985 , and others) . In geomorphological 
sys tems these changes have bee n exemplified by 
drainage network expansion, Ihe grow ing intens ity of 
vertical dissection and Ihe general appearance o f ne w, 
hi gher-order features in a landscape. A re liable indica­
lOr of such changes a re frequency distribution curvcs of 
Ihe system parameters. Their analysis, therefore, can be 
of great imporLanee in understandi ng geomorphological, 
and geological processes in a givcn area. 

The aim of thi s investi ga ti o n is to pl ace greater 
emphasis on thc potential applicat ion of frequency di s­
tribution curves in the analys is and appraisal or various 
geomorphic phenomena and the ir dynamics in a fluvial 
landscape (the northwestern part of I-Irvat sko Zagorje). 
Previous studies in this area, carricd out to exam ine the 
dynamic interplay of various fluvial landform s (PEl-I, 
1990), as weI! as to e lucidate the re lationship between 
the driving forc es and resistant geological framework 



'0 

Slovenia 

Croatia 

Fig. 1 Location map of the study area. 

(PEH, 1992), dealt with a single li"agment on a hierar­
chical scale of system relationships. Conclusions drawn 
from this research have adequately described the spatial 
changes caused by climate and tectonic movements 
during the particular geologic period defined by the rel­
ative age (order) of drainage basins. However, exclu­
sive study of the spatial relationships among geomor­
phological parameters thus defined, has not established 
temporal trends of landscape evolution with any degree 
of certainty. Only after several adjacent, hierarchically 
ordered geomorphological systems had been analyzed 
toget her , could a more explicit refe rence to the time 
component involved in the investigation of the dynam­
ics of geomorphologic processes be made (PEH, 1994). 
It seems possible that the probability distribution curve, 
illustrating a specific geomorphologic process , can be 
exceptionally helpful as a very simple indicator of evo­
lution. Therefore, when applied with reference to a par­
ticular geolog ical framework, this method can supply 
the necessary information about the origin of the land­
scape and its destruction. 

The primary objective of this study is to highlight 
utilization of the frequency distribution curves, with 
examples of how the drainage network, slopes and oth­
er geomorphic parameters develop within hierarchically 
de fin ed geomorphological systems in the NW part of 
Hrva tsko Zagorje (Fig. I). However, the parameters 
lIsed refer to the recent period in development of the 

Geologia Croatic<l 50/1 

study area, which is defined by the relative age of the 
drainage basins ranging from the fifth to the third order. 
Data used in the study derive from the mathematically 
treated morphometric and geological variables. They 
are presented in the form of deviations (fluctuations) 
from standardized means and displayed in appropriate 
tables and diagrams (PEH, 1994). 

2. THERMODYNAMIC BACKGROUND 

Earlier investigations of the fluvial geomorphologi­
cal phenomena, were mostly founded on the concept of 
energy distribution on the surface of our planet being 
both stable and uniform in time and space. The steady­
state approach assumcd that system parameters (variety 
of landforms presented mainly in the form of morpho­
metric data) uniformly oscillate around an equilibrium 
position as they adjust (with minimum internal change) 
to external forces, allowing a system as unity to remain 
in a stable equilibrium (CHORLEY, 1962; LEOPOLD 
& LANGBEIN, 1962; HOWARD, 1965; SCHEIDEG­
GER & LANGBEIN, 1966; KING, 1967 and others). 
The fact that permanent changes of morphometric para­
meters do not allow the system to reach stability, even 
during the long term (SCHUMM & LICHTY , 1965; 
CHORLEY & KENNEDY, 1971; SCHUMM, 1973; 
BULL, 1975, and others) had largely been ignored 
while, at best, a number of temporary steady states can 
be form ed which represent transient phases in the land­
scape development. The basic idea underlying more 
recent geomorphological research (COATES & 
VITEK, 1980; KARCZ, 1980; RITTER, 1988) is 
founded on the premise that only a particular Iype of 
d isequilibrium, driven by high iluctuations within the 
geomorphological system, can generate new landforms. 

In the process of adjustment, the mechanisms of 
self-regulation and self-stabilizing negative feedback 
formulated in Prigogine's rule of minimum entropy 
production (PRIGOGINE, 1978), and the Lc Chatelier­
Brown principle of linear reaction, play the key role . 
The stability zone in the neighbourhood of equilibrium 
is usually called the thermodynamic branch (GLANS­
DORFF & PRIGOGINE, 1971; NICOLlS & PRI­
GOGINE, 1977). This is a domain of a linear, station­
ary regime , where the probability distributions of the 
system components obey the Gaussian law (NICOLlS 
& PRIGOGINE, 1977; KEIZER, 1987). Landforms 
described in terms of different morphometric variables 
do not greatly deviate from the average, which is valid 
for a system as a whole. Their main feature, defined in 
the language of thermodynamics, is aptly expressed in 
the term of "equilibrium structures". 

However, strong external energy input s can upset a 
system and drive it completely out of equilibrium. 
These generate fluc tuations with sufficiently high 
potential energy 10 trigger irreversible loss of structural 
stability, which manifests itself largely through the 
mechanism of positive feedback. Instead of being 



Pch: Frequcncy Dislribution Curves as an Indic:lIor of Evolutionary Trends in Geomorphological Systems ... 81 

a 

- 11 o +11 

b 

Types of distribution curves 

The above considerations lead to the conclusion that 
the instability of landforms (s tructural instability when 
cons idered in a the rmodynamic sense) is caused by 
great fluc tuations which serve as the basic evolutionary 
impulse in geomo rpholog ical systems. In this respect, 
insight into laws govern ing the behaviour o f the proba~ 
bility di stribution of flu ctuations is of the g reatest 
importance for the study of their developmental charac~ 
teristics. In a system which is characterized by struClur~ 

al stability within the limit s of the thermodynamic 
branch, flu ctuations will be dist inguished by a qu ite dif~ 
rerent pro bability dis tribu tion with regard to another 
system wi th an unstable response in the neighbourhood 
of, or beyond the thermodynamic threshold. In the first 
case , when a systcm is dominated by an eq uilibrium 
reg ime, the frequency di stribution of Jluctuati ons w ill 
ex hibit a normal (Gauss ian) shape of probability eurve 
(Fig. 2a). Tn the latter, however, when the balance of a 
system is upset, and relationships approach the non~lin~ 
car domain, the frequency distribution of flu ctuations 
will assume positive skewness (Fig. 2b). Thi s type of 
asy mmetry is a direct effect of the presence of a very 
small number of Ouctuations carrying a hi gh energy 
potential. Thcy have the leas t probability of occurrence 

-6 

C owing to the second law of thermodynamics which prc~ 

sc ribes that equil ibri um in the neighbourhood o f the 
mean system trajectory exemplifies the most probable 
system state. The grcaler the magnitude of fluctuations 
(that is, their potential energy), the smaller the probabil~ 
it y of the ir occ urrence, which accounts for a posi ti ve 

Fig. 2 Typical rrobabilily distri buti on Cllrve of nu cl uatiolls: a) 
Gauss ian normal probability di s tri but ion curv c; b) pos itively 
skewed probabilit y distribution c urv e: c) negative ly skewed 
probability distribution curvc. On abscissa cr stands ror the mea­
surc or nucluation.~ . 

damped, such flu ctuations arc amplified, leading a sys~ 
tem into the domain o f a qualitatively diffcrent regime 
of processes, with a dist incti ve ly different development 
nature. Changes at the boundaries of the thel11lOdynam ~ 

ic branch, which cause the onset of the loss of stability, 
are charactcrized by the occurrence of n OJl~ lillear rela~ 

tionshi ps among system parameters, by the shi ft to a 
non~Gaussian , skewed frequency distribution of f1uctu~ 

at iolls (D1M ENTBERG, 1989). Landforms o riginating 
in such thcrmodynamic cond itions depart considerably 
from the mean which is charactcristic for a system as a 
whole. Their main properties can be conveniently sum-
marized as being the epitome of "diss ipative structures" 
owing to the strong and continuous dissipation of encr~ 
gy th ro ughout the geomorpholog ical realm (KARCZ, 
1980). They are new, self~organi zed landforms which 
orig inatcd as a resu lt of " order through flu ct uations" 
(G LANS DO RFF & PRIGOGI NE, 1971 ; NICOllS & 
PRIGOGINE, 1977; PRIGOGINE & STENGERS , 
1985; KNORI NG & DEC, 1989). 

skew of their distribution becoming more and morc 
pronounced (DIMENTBERG , 1989). 

There is another type or geomorphological process 
which is also characteri zed by a skewed frequency dis­
tr ibut ion curve. Tn this case, q uite a different set of 
processes is observed, wh ich are dist inguished by a lack 
of energy. Thi s sta te is charac terized by a re la tively 
sm all number of flu c tuations with exceptiona ll y high 
negative amp litudes. General ly, this is the case when 
many of the flu ct uati ons are clustered at or nea r an 
upper limit of some natural process (KOCH & LI NK, 
197 1), as di stinguished from the system processes 
which lend to maintai n the structural stability of ex i s t ~ 

ing steady states (normal distribution of flu c tua ti ons), 
or processes that g ive rise to new, more organi zed 
structures (positively s kewed di stributi on of nue tua~ 

tions). The emphasis here is on the processes of gradual 
decay of preexisting structures which are surrounded by 
condition s of decreas in g pot ential ene rgy, necessary 
ei ther to maintain the stationary state or to spur evo l u~ 

ti onary change, thus produc ing a negati vely skewed 
d istribution. 

3. A CASE STUDY 

Geomorphol og ical processes arc di stinguished not 
only by their ample di vers ity, but also by their specific 



82 Geologia Cro:nic<l 50/1 

min max skewness kurtosis KS- KS+ type of 
distribution 

PROCESSES CONCERNING DRAINAGE NETWORK 

horizontal dissection -1.49 4.79 1.42 2.95 0.09 0.11 
drainage density -2.84 2.90 -0.30 0.42 -0.06 0 .04 

=> 

+ 

PROCESSES CONCERNING VERTICAL DIFFERENTIATION 

vertical dissection -1.55 3.11 0.75 1.60 0.05 0.08 + 
slope -1.78 2.15 -0.11 -0.99 0.08 0 .07 <= 
erosional levels 

hypsometric integral -3.00 3.64 0 .54 1.96 0.06 0.09 + 

Table I Distribution of l1uctualions in the investigated geomorphological system or the 3'" order (N= [80). Critical values of K-$ lest ror 180 
objects: KS+. KS- = 0.0909 (0.=0.05): ~ positive skewness; ¢:::: negative skewness: + nonnal di stri butioll. 

spatial dynamics. The system dynam ics produce a 
who le array o f landforms, particularl y in the fluvial 
landscape. Inves ti gations have shown (e .g. STRA-
1-ILER, 1952; MELTON , 1957; BRUS H, 196 1; CAR­
SON, 1971 ; RITTER , 1978; HART, 1986, and others) 
that the nature or geomorphological processes is affect­
ed no t on ly by the magni tude, durat ion and direction of 
driving forces, but equally, if not to a grcater extent, by 
the resist ive forces of the underlying geological frame­
work. In the mutual interaction o f the driving and res is­
tant forces, energy and material entering a sys tem is 
Iransfeneel c ilhcr th tO ugh spec ifi c composil c lanel­
forms, or landscape subsystems, which sim ultaneously 
undergo different stages of development. The study of 
the relat io nship between typi ca l geomorphological 
processes and their respective landforms in the NW part 
of I-lrva tsko Zagorje (PEl-I, 1994), has demonstrated 
tha t a geomorphological system, observed as a structur­
al unity over certain period of time, can, if only tem­
porari ly, ex ist in a state of balance with these processes. 
Nevertheless, it s sepa rate part s, or subsyste ms, can 
ex hibit diffe rent stages of non-equilibrium state, the 
ultimate effect of wh ich would either be evol uti onary 
growth or degeneration. Tectonism and lithological fae-

tors are of extraordinary significance owing to their role 
as regulatory agents moderating the influence of exter­
nal factors, as they tend to absorb the effects of external 
disturbance (driv ing forces, particularly tectonism) and 
to cstablish a steady state between form and processes 
(CHORLEY & KENNEDY, 197 1). 

The analysis of geomorphological sys tems of differ­
ent hierarchical levels (ranked aner the order or relati ve 
age of the main valleys) in the NW part o f Hrvatsko 
Zagorje (Fig. I) shows that the study area is composed 
of severa l very homogeneou s landscape units, or sub­
systems, being quite distinctive on the local and reg ion­
a l scales of system relat ionships. These subsystems 
comprise: horizontal dissection, drainage density, vert i­
cal dissection, slope, erosional levels and the hypsomet­
ric integra l. They deve lop as a result of two basic types 
of land forming processes - the processes of horizontal 
and ve rti ca l differen ti ation. Appa rent var iat ions ca n 
arise frolll differences in magnitude involving the spa­
tial and temporal sca les on which the relationships in 
the geomorphological systems o f thi rd, fourth and fi fth 
order arc observed (PEH, 1994). It is logical, therefore, 
to expect the nature of flu ctuations, lypical for particu­
lar subsystems, to depend on both the type of geomor-

min max skewness kurtosis KS- KS+ type of 
distribution 

horizontal dissection 

drainage density 

vertical dissection 

slope 

erosional levels 

hypsometric integral 

PROCESSES CONCERNING DRAINAGE NETWORK 

-1.30 4.30 1.41 2.50 0. 11 

PROCESSES CONCERNING VERTICAL DIFFERENTIATION 

-1.58 2.99 1.03 0.94 0.06 
-2.37 1.75 -0.46 -0.38 0.10 
-2.11 3.21 0.38 0.53 0.04 
-1.94 4.10 1.06 3.56 0.04 

0.16 

0. 10 
0.05 
0.05 
0.05 

= 

=> 
<= 

+ 
+ 

Ta ble 2 Distribut ion of fluctuat ions in the investigated geomorphological system of the 4th order (N= 160). Critical values of K-S [cst for t 60 
objects: KS+, KS- = 0.0964 (<<=0.05): => posi ti ve skew ness: ¢::: negative skewness; + normal dis tribution. 



Pch: Fn."<llIcncy Di\l ribluiOIl CurvCS;ls an Indic:uor of Evolutionary Trends in Cicomolphologico'll Systems ... 83 

min max skewness kurtosis KS- KS+ type of 

distribution 

PROCESSES CONCERNING DRAINAGE NETWORK 

horizontal dissection -1.29 2.74 0.61 -0.50 0.10 0.13 => 
drainage density -2.72 2.99 0.15 0.91 0.06 0.08 + 

PROCESSES CONCERNING VERTICAL DIFFERENTIATION 

vertical dissection -1 .68 2.83 1.21 1.78 0.06 0.13 => 
slope -2.70 1.55 -1.19 1.57 0.13 0.08 <= 
erosional levels -3.55 1.90 -0.99 2.73 0.13 0.07 <= 
hypsometric integral -1.97 2.36 -0.32 0.20 0.08 0.05 + 

Table 3 Di .~ tribu l ion of fluctua tions in thc invc\tigalcd gcomorphological systc m of thc 51h order (N= I02). Crit ical values of K-S lest fo r 102 
objec ts: KS+, KS- = 0. 1208 (u=O.05); ~ po.~ iti vc skewness; <= negative skewness: + norma! distribution. 

phologie process and thc scale inferred. If the distribu­
tional characteristics of fluctuation s arc to be properly 
stud ied , it is necessary to tcst the normality o r their fre­
quency dist ri bution wi th respect to each spec ifi c sub­
system (Tab les I, 2 and 3). Thi s norm ality tes t can 
allow conclusions to be made about the dynamics of the 
most promin ent geo morphological process . Di stribu ­
tional c haracte ri sti cs o/" nuctuations can be tes ted by 
nonparam et ri c stati stical tests incl uding the Kol ­
mogorov-Sm irnov (KS) tcst where the c ri ti cal values 
(KSK) can be take n from the appropriate tables (e.g. 
DAVIS, 1986). 

The procedu re of how the frequency distributions 
compare to a normal eurve is shown in Tables 1-3, for 
each separately investi gated geomorphologic system. 
Ana lys is of frequ ency distributions y ield s some very 
sign ificant result s ror the interpretati on o f the develop­
ment of the in ves ti gated geomorpholog ical sys tems. Jt 
is obvious from Tabl es 1-3 that the indi vidual subsys­
tems have follo wed different paths of deve lopmen t in 
the various geomorphological sys tems. Morover, pro­
ceeding [rom lower 10 highcr lcvel rela ti onsh ips, a reg­
ular transformation of the shape or the dis tribution 
curve is apparent. Thi s change manifes ts it se lf by a 
decrcase of potential energy cont ai ned in the fluctua­
tions, which cause a shift toward the sinistral, negative 
skewness w ith regard 10 spec ific geomorpholog ical 
processes. The faclO rs governing the mig ra tion of the 
mean system state trajectory (x:::: 0) toward the negative 
values (the probability oC the appea rance of negative 
flu ctuation s dec reases) can be numerous, but for the 
most part they ar ise from the effect o f seria l autocorre­
lation which becomes conspicuous as the higher-order 
geomorphological sys tems increase their structural 
complex ity (CHORLEY & KENNEDY , 1971). Name­
ly, int rica te and mutua lly perplexed re laxa tion pat hs, 
paved by serial autocorrelation due to the presence of 
the same morphornetri c variables, especial ly in the sys­
tcms of a higher rank, can considerably damp the effect 
of individual inputs o f ex ternal forces causing their 
activi ty to fa ll under some significant energy threshold 

(CII ORLEY & KENNE DY , 197 1). In addition, it Illust 
be emphasized that the increase or a system rank (both 
enl argement of the phase space and expans ion o f it s 
physical borders) unavoidably leads a system to a state 
or hi gher entropy (WOOD & FR ASER , 1977). Thus, 
one ca n log ically ex pec t a system of the fifth -order 
drai nage basins (0 have hi gher entropy with respect to 
one o r the fourth -order, and the laller with respect to the 
one o r the third-order. 

Exami nation of Tables 1-3 and Figs. 3-5 reveals that 
only the processes concern ing the development o r hori­
zontal di ssection cause da ta to assume a marked ly posi­
tive skew of distribution, irrespective or the hierarchical 
o rder under considerat ion. Such a trend can be 
ex plained by the continuolls reorganization or the sub­
system of horizontal dissecti on which manifests it self at 
all hi e rarch ica l leve ls . When a drainage basin with 
expanding drainage ne twork of a g ivcn order LI exceeds 
some lim iti ng size, there is a jump in the reorganization 
o f dra inage texture. Th is produces a new drainage basi n 
or higher order u+1 (CHORLEY & KENNEDY, 197 1). 
In accordance with the earli er cons ideration about sta­
bility, the new drainage basin of order u+l ex hi bits the 
features or dissipative structures wh ich are di stin ctly 
dirferent in re lation to the rest of the drainage basins of 
order u to wh ich it prev ious ly belonged. ;\ positi vely 
skewed frequency distribution curve (a lso histogram of 
Iluctuati ons), which is indicative of the processes con­
cerning development of horizontal dissection , serves as 
a s ure indicator of the drai nage network evo lution in 
loca l (third-order drai nage bas in system), and reg ional 
(fifth -order drainage basin system) boundari es or sys­
tem re la tions hips. 11 is a lso character isti c of th e hi gh 
energy potent ial of the c li matic ractor (insol ati on, pre­
cipitation) in the study area, which operates approx i­
mately wi th the same intensity, regard less of the space­
time sca le defined by the hi erarchical order of the 
inves ti ga ted geomorphological system (Fig. 3). 

'rhe processes of vertical dissection are also di stin ­
gui shed by an expressly positively skewed distribution, 
wit h the exception of the thirel -oreler drainage bas in sys-



100 

80 
m z 

~ 60 .! .. 

" :3 
~ 

~ 

0 40 

'" w on 

" => 20 z 

0 
-3 -2 -1 0 

100 

80 . ---. 
m z 

~ 
=> 60 ~ 

~ 
~ 
~ 

is 40 

'" w on 

" => 20 z 

0 
-2 -1 0 

100 

• HORIZONTAL DISSECTION 

l' 00Dffi 

_ expe<;ted normal 

2 3 5 

... HORIZONTAL DISSECTION 
4'" ORDER 

......... ~. 

_ expected normal 

2 3 4 

Gcolo~ia Croatica 5011 

a 

6 

b 

5 

80 , ...... HORIZONTAL DISSECTION ; .. m z 511 ORDER 

! 60 

~ 
~ C 
~ 

40 0 

'" W on _ oxpected normal 

" => 20 z 

0 
-2 -1 0 2 

tem, where the subsystem of vertical dissection is found 
to be in stable equilibrium. Thi s stability, irrespecti ve 
of the s lightly prominent positive asymmetlY of the fre­
quency diSiribu tion curves (Table I), can be ex plained 
as a resul t of the compl ex response to external vari ­
ables. The characte ri sti c properties of vertical di ssec­
tion such as, for instance, the third-order drainage basin 
rclicf, arc combined wit h other altitudinal aspects of 
drainage basins - valley-side slope, for example - which 
inc reases the multiplicity of possib le linkages among 
variables. Thi s allows many alternatives for potential 
adjustments among the landscape properties. Additional 
relaxation paths occur so that a stationary state can be 
achieved relatively easily and quickly, even under great 

Fig. 3 Frequency distribution histogram of fluc-
3 tuations for the horizontal dissection, 3_5 '11 

order, in lhe area of inve.~t igation. 

variations in the external regime incruding both endoge­
netic and exogenelic forces (C HORLEY & KEN­
NEDY , 197 1), In the fOUrlh - and firth-order drainage 
basins (Tables 2 and 3), where the ve rLical dissection is 
dominantl y characterized by the hi ghest point on a 
watershed , and the total re li ef ins id e the basin (PEH, 
1990, 1992), such a possibility is considerably reduced. 
On thi s account the subsystem of vertica l dissection, as 
ari integral part of the fourth- and fifth -order drainage 
basin system, adjusts to externa l impulses through the 
reorganization of its structural properties, and develop­
ment of dissipative structures in the zones where the 
structural stability has been irreversibly lost. Since hori­
zo ntal and verti cal di ssec tion represent the domi nant 



Peh: Frequency DiMributiOIl Curves as an Indicator of Evo]utiollmy Trends in GeomOlvhological Systcms ... 85 

100 

HYPSOMETRIC INTEGRAL 

80 
w z 
0 

~ 60 
() 

" ~ ~ a 
~ 

0 40 

"' w 
rn 

" OJ 20 z 

0 
-4 -3 -2 -1 0 2 3 4 5 

100 

80 
w HYPSOMETRIC INTEGRAL 
z 
0 

~ 
OJ 60 r 
() 
OJ 
~ 
~ 

~ 

0 40 

"' w 
rn 

" " 20 z 

0 
-3 

100 

80 
w z 
0 

~ 

" 60 
() 

'3 
~ 

~ 

0 40 

"' w 
rn 

" OJ 
20 z 

0 
-3 

-2 -1 0 

_ expectod nonnul 

_ "q><>ctoo normal 

2 3 

HYPSOMETRIC INTEGRAL 
5th OROER 

geomorphological processes in the landscape of the 
study area, it is logical to assume that they will mostly 
reflect the evolutionary trends of the obscrved geomor­
phological systems. 

Another group of geomorphological processes is 
distinguished by a normal frequency distribution curve 
of lluctuations, indicating structural stabilit y of the per­
taining subsystems, which is a consequence of their 
ability to maintain thei r structural attributes in the exist­
ing regime of external forces. Such is the case of the 
subsystem of hypsometric integral (Fig. 4) and the sub­
system of drainage density being integral parts of the 
fourth- and fifth -order drainage basin systems. System 
componenls fluctuate within the limits of the them10dy-

b 

5 

c 

Fig. 4 Frcquency distribution hislogram of Iluc-
3 luations for the hypsometr ic integral, 3_5'1> 

order, in the area of Investigation. 

namic branch, sustaining the existing stat ionary state. In 
other words, potential energy carried by the fluctuations 
is neithcr too high to spur cvolutionary changes, nor too 
low to bring about degeneration of the present struc­
tures. The normal frequency distribution curve of l1uc­
tllations in the subsystem of drainage density supports 
the general concensus of opinion that drainage density 
in its present extent retlects a balance between the input 
and output of the hydrological cycle. Its pattem disclos­
es a number of mutual adjustments providing for the 
maximum and most efficient surface runoff in the 
drainage basin (CHORLEY & KENNEDY, 1971)_ 
Conversely, the subsystem of the hypsometric integral 
(Fig. 4) reflects equilibrium tendencies in the processes 



X6 

' 00 

SLOPE 
80 

3'11 ORDER ~ z 
0 

~ 
co 60 ~ 
0 
=> 
~ 
~ 

~ 
0 40 
~ 
w 
rn 

" => 20 z 

0 
-3 -2 -1 0 

100 

SLOPE 
80 

4lt1 ORDER ~ 
z 
0 

~ 
=> 60 ~ 
u 
co 
~ 
~ 

~ 
0 40 
~ 

_ ~ednoml"l w 
rn 

" => 20 z 

0 
-3 -2 -, 0 

' 00 

80 SLOPE 
~ z 

5th ORDER 0 

~ co 60 ~ 
0 
3 
~ 

~ 

0 40 
~ w 
rn _ expected normal 

" '" 20 z 

0 
-4 -2 

of denudation, erosion, transportation and aggradation. 
In lower-order drainage basins the balance is es tab­
lished owing to the negative feedback between total 
relief and hypsometric integral (that is, among morpho­
metric propert ies only). In the higher-order drainage 
basins lithology plays the role of moderator in the 
process of mutual adjustment of the two morphometr ic 

variables - channel gradient and valley-side slope 
(PEH, 1992)_ 

Processes involved in shaping the slope subsystem 
(Fig. 5) and the subsystem of erosional levels afe distin­

gui shed by the progressive modification of the frequen­
cy distribution curve which shifts [rom normality 

towards negative skeweness as the order increases. The 

2 

Geologia Cromica 5011 

a 

b 

2 

c 

Fig. 5 Frequency distribuLion histogram of tluc-
2 Illations for the slope, 3_5'10 order, in the area 

of investigation. 

slope subsystem in the third-order drainage basin sys­
tem is characterized by the normal distribution of fluc­
tuations. In the fou rth- and fifth-order drainage basin 
system the distribution is more negatively skewed (Fig. 

5). The subsystem of erosional levels, which dissociates 
from the subsystem of vertical dissection in the fourth­
order drainage basin system, is at this hierarchical level 
characterized by normal distribution of fluctuations, but 

becomes negatively skewed in the fifth -order drainage 
basin system. It can be stated that in this case only the 
m inority of system components in these basin subsys­
tems oscillates in the proximity of the lower limit of the 
thermodynamic branch (that is , the probability of 

exceptionally high negat ive fluctuations is very small). 



Pch: Frequcncy Distribution Curves as ,lTl Indicator or Evolution;)ry Trends in Geomorphologic;)1 Systems ... 87 

It suggests that the potential energy in specific parts of 
a system is so low that existing structures (landforms) 
undergo spontaneous degenerat ion in the places where 
its magnitudc falls below some critical value necessary 
for maintaining the stationary state. A reason for this 
can be found either in the unfavourable mechanical 
properties of the underlying bedrock (high permeability 
and low cohesion of rock and soil material), or in vary­
ing rates of uplift that affect some major tectonic blocks 
in the area of invcstigation. If the fonner is case, the 
vallcy-sidc slopes tend to form lower angles due to 
ineffective st ream downcutting (caused chiefly by rela­
tively high infiltration capacity), which is most conspic­
uously reflected in higher-order and larger drainage 
basins (fourth- and, particularly, fifth-order). In the lat­
ter, denudation processes give way to aggradation 
processes when the uplift only slightly exceeds or evcn 
tends to be lowcr than the rales of denudation. As an 
example, in the subsystem of erosional levels onc of the 
key variables - thc main-valley mouth (which is also the 
mouth of a referring drainagc basin) - represents an 
overt indicator of uplift and subsidence that can be 
helpful in disclosing "act ive" and " inactive" gcologic 
structures in a landscape. Obviously, due 10 their rela­
tive lag in respect of the general uplift typical for the 
study area (PRELOGOYIC, 1975), some tectonic 
blocks have considerably lower potential energy with 
respect to the surrounding area, inducing the processes 
of aggradation and development of "inactive" geologic 
structures. Both cxamples show that the energy expen­
diture in these subsystems occurs mostly at the lowcr 
hierarchical levels (PEH, 1992, 1994). 

4. CONCLUSIONS 

Examples presented in this study show that varioLls 
landforms in the fluvial landscape, such as the north­
western part of Hrvatsko Zagorje, can pursue different 
evolutionary schcmes which can be discerned from the 
corresponding frequency dist ributions of rIuetuations. 
The thermodynamic theory of non-equilibrium states 
and related fluctuations as the source of instability is 
the kcy clement in the interpretation of evolutionary 
trends in gcomorphological systems and their subsys­
tems. This results from utilizing new insights in funda­
menta l physics in the geological sciences. 

It has been shown in this study that thrce types of 
frequency distribution curves exist which, in a thermo­
dynamic sensc, can be indicative of equil ibrium and 
non-equilibrium relationships between driving and 
resisting forces in land forming processes. They show 
how landforms try to adjust themselves with minimum 
internal change to the external driving forces. The 
example of the processes responsible for the develop­
ment of the subsystem of drainage density and subsys­
tem of hypsometric integral is presented. 

Non-normal frequency distribution of systcm para­
meters indicatcs non-equilibrium conditions which give 

rise to non -linear relationships between process and 
form and, accordingly, to structural instability within 
specific subsystems. Such a case is demonstrated by 
both the positive and negative skew of distribut ion fre­
quency curves. Positivc skew is indicativc of structural 
changc which is the response to strong energy inputs 
from the system surroundings. The loss of structural 
stability and emergence of new and more complex 
landforms is the final result which spurs landscape evo­
lution. An example of this was shown by thc processes 
accounting for drainage network expansion (horizontal 
dissection) and, to some extent, by the proccsses of val­
ley incision (vertical dissection). Negative skew, on the 
contrary, can suggest structural changes distinguished 
by the lack of energy in a system, which is insufficient 
to reinstatc thc original equilibrium conditions and, 
thus, to sustain existing landforms. This case predicts 
the downward path to landscape degeneration, that is, 
its involution. Such a case is exemplified by the process­
es affecting valley-slope angles and erosional levels. 

Although the relevant space and time scales have 
been taken into consideration, this st udy has been put 
into a frame which, in a geomorphological and geologi­
cal sense, could not be str ictly viewed upon as regional. 
However, the results point to distinct evolutionary laws 
that rule the development of a fluvial landscape in the 
temperate climate zones. Further investigations should 
therefore be directed toward the spatial evaluation of 
thermodynamically specified geomorphol ogical para­
meters with assistance of the relevant maps ("process 
maps", PEH, 1994). Attention should also be paid to 
the concept of geomorphological thresholds and bound­
ary values delimiting landform stability. This would 
allow individual landforms and their evo lutionary (or, 
contrari ly, degenerative) changes to be directly related 
to geological struclurcs, lithological characteristics and 
tectonic activity. 

Acknowledgments 

During preparation of this paper the author benefit­
ted greatly from critical comments by Prof. Eduard 
PRELOGOYTC (Faculty of Mining, Geology and 
Petrolcum Engineering, University of Zagreb, Croatia) 
which were cxt remely valuable. His help is grateful ly 
acknowledged. Also, the author is greatly indebted to 
Prof. Simon PIRC (Institute of Geology, Universi ty of 
Ljubljana, Slovenia) for his thoughtful reviews and use­
ful comments on earlier drafts of the manuscript. 
Thanks are also due to all my colleagues who supported 
me in this erfort. 

5. REFERENCES 

BRUSH, M.LJr. (1961): Drainage basins, channels and 
now characteristics of selected streams in Central 
Pennsylvania.- U.S. Geo!. Survey Prof. Paper, 282-
F,145-180. 



BULL, W.B. (1975): Landforms that do not tend 
toward steady state.- In: MELHORN, W.N. & FLE­
MAL, R.C. (cds.): Theories of landform develop­
ment. Allen & Unwin , London -Boston-Sydney, 
259-263. 

CARSON, M.A. (1971 ): The mechanics of erosion.­
Pi on, London, 174 p. 

CHORLEY , R.1. (1962): Geomorphology and general 
systems theory. - U.S. Geol. Survey Prof. Paper, 
500-B, 10 p. 

CHORLEY, R.1. & KENNEDY, B.A. (1971): Physical 
geography: a systems approach. - Prcntice- Hall 
Intcrnational Inc., London , 370 p. 

COATES, D.R. & VITEK , J.D. (eds.) (1980): Thresh­
olds in geomorphology.- Allen & Unwin, London­
BoslOn-Sydney, 485 p. 

DA VI S, LC. (1986): Statistics and data analysis in 
geology. - 2"" edition , John Wil ey & Sons, New 
York, 646 p. 

D1MENTBERG , M.F. (1989): Slucajnije proeessi v 
dinamiceskih sistemah s pcrcmcnnimi parametra­
mi. - Nauka, Moskva, 175 p. 

GLANSDORFF, P. & PRIGOGINE, I. (1971): Ther­
modynam ic theory of structure, stability and Ouctu­
ations.- John Wiley Interscicnce, New York, 306 p. 

HART, M.G. ( 1986): Geomorphology pure and 
applied.- Allen & Unwin, London, 45 p. 

HOWARD, A.D. (1965): Geomorphological systems­
equilibrium and dynamies .- Am. Jour. Sc i. , 302-
3 12. 

KARCZ, I. ( 1980): Thermodynamic approach to geo­
morphic thresholds.- In: COATES, D.R. & VITEK, 
J.D. (cds.): Thresholds in geomorphology. Allen & 
Unwin, London-Boston-Sydney, 209-226. 

KEI ZER , J. ( 1987): Statisticeskaja tennodinamika ner­
avnovesnih processov.- Mir, Moskva, 607 p. 

KI NG, R.I-!. ( 1967): The concept of general systems 
theory as app licd to geomorphology.- Albertan Geo­
grapher, 3, 29-34. 

KNORI NG, L.D. & DEC, V.N. (1989): Geologu 0 

m3temalike .- Nedra (Leningradskoe oldeleni e ), 
Leningrad, 208 p. 

KOCH, G. S. & LINK, R.F. (1971): Statistical analysis 
o f gcological dala. - Dover Publications Inc ., New 
York , 438 p. 

LEOPOLD, L.B & LANGBEIN, W.B. ( 1962): The 
concept of entropy in landscape evoJution. - US 
Geol. Survcy Prof. Paper, SOO-A, 20 p. 

Geolog ia Cro:tlic:t 50/1 

MELTON, M.A. (1957): An analys is of the relat ions 
among e lemen ts of climate , surface prope rties and 
geomorphology.- Columbia Univ., Dept. Geol. , 
Omce of Nav. Rcs. , Project NR 389-402 , 102 p. 

NICOLlS, G. & PRIGOGINE, I. ( 1977): Sc lf-organiza­
tion in non-equilibrium sy stems.- Wiley-Inter­
science, New York, 491 p. 

PATTEE, H.H. (1973): Hierarchy theory, the challenge 
or complex systcms.- Braziller, New York, 156 p. 

PEB, Z. (1990): App lication of factor analysis in 
dynamic geomorphology.- Geo l. vjcsnik, 43, 59-67. 

PEB, Z. (1992): Factor model of the geomorphological 
system of a part of Northweste rn Croatia. - Geologia 
Croatica, 45, 163-1 72. 

PEH , Z. ( 1994): Faktorski model kao izraz dinamike 
geoJoskih procesa l1a primje ru sjevc rozapadnog 
dijela Hrvalskog zagorja.- Unpublished PhD. T he­
sis, University of Zagreb, 162 p. 

PRELOGOVIC, E. (1975 ): Neot ekton ska karta SR 
I-Irvatske.- Geo!. vjesnik, 28 , 97- 108. 

PRIGOG INE, I. (1978): Time , structure and f1 uctua­
tions. - Science, 20 1, 777-785. 

PRIGOGINE, I. & STENGERS, I. (1985): Order out of 
chaos.- Flamingo, London , 349 p. 

RITTER , D.F. (1978): Process geomorphology.- V.c. 
Brown Publishers, Dubuque, Iowa, 579 p. 

RITTER , D.F. (1988): Landscape anal ys is and the 
search for geomorphic unity.- Geo!. Soc . Am. Bull. , 
100, 160~171. 

SCHEIDEGGER, A.E. & LANGBEIN, W.B. (1966): 
Probability concepts in geomorphology. - U.S. Geol. 
Survey Prof. Paper, 500-C, 14 p. 

SCHUMM, S.A . (l973) : Geomorphi c thresholds and 
complex response of drainage system s.- In: MORI­
SA WA, M. (cd.): Fluvial geomorphology. Allen & 
Unwin, London-Boston-Sydney, 299-3 10. 

SCHUMM, S.A & LICHTY, R.W. (1965): Time, space 
and causality in geomorphology.- Am. Jour. Sci., 
263 , 110-119. 

STRAHLER, A.N. (1952): Dynamic basis of geomor­
phology.- Geo!. Soc. Am. Bull. , 63, 923-938. 

WOOD , B.1. & FRASER, D.G. ( 1977) : Elementary 
thermodynamics for geo!ogislS.- Oxford University 
Press, Oxford, 303 p. 

Manuscript received March 15, 1995. 
Revised manuscript accepted April 28, 1997.