Hydrological Sciences -Journal- des Sciences Hydrologiques,40,5, October 1995
Quantification of the stability of river flow
regimes
IRINA KRASOVSKAIA
Hydroconsult AB, Lergravsvâgen 33, S-263S2 Hôganâs, Sweden
Abstract An important characteristic of a river flow regime type is the
time of year when high and low flows are likely to occur. How likely is
it, however, to observe an identified seasonal pattern each individual
year? Stability is an often neglected property of a flow regime, though
shifts in the seasonal behaviour of flows affect both environmental and
economic activities. An approach to characterize objectively the stability
of a flow regime type, based on the concept of entropy, is presented. The
stabilities of river flow maxima and minima are studied separately to
investigate their respective contributions to the stability character of a
particular regime type. A quantitative "instability index" permits a study
of the development of a flow regime's stability in time, especially important in the context of a possible climate change. The method is presented
using the example of a quantitative flow regime classification developed
for Scandinavia and western Europe.
Quantification de la stabilité du régime des rivières
Résumé Une importante caractéristique du régime des rivières est la
période de l'année où il est probable que se produisent les crues ou les
étiages. Cependant quelle est la probabilité d'observer le modèle saisonnier identifié au cours d'une année particulière? La stabilité est une
propriété du régime souvent négligée, alors que des variations du comportement saisonnier de l'écoulement affectent à la fois l'environnement
et les activités économiques. Nous proposons une approche fondée sur
le concept d'entropie qui permet de caractériser objectivement la stabilité
d'un régime. La stabilité des maxima et minima d'écoulement est étudiée
séparément afin de rechercher leur contributions respectives à la stabilité
caractéristique d'un régime particulier. Un "index de instabilité"
quantitatif permet d'étudier l'évolution de la stabilité du régime au cours
du temps, ce qui est particulièrement important dans la perspective
d'éventuelles modifications climatiques. La méthode est présentée à partir
d'une classification quantitative du régime des rivières réalisée pour la
Scandinavie et l'Europe de l'ouest.
INTRODUCTION
A river flow regime describes the average seasonal behaviour of flow dependent on its genetic sources and on basin climate and physiography. Flow
regime classifications are made on the basis of a thorough analysis of a number
of available annual hydrographs and the results are usually presented in the
form of maps with characteristic hydrographs (e.g. Arnell et al., 1993) or are
Open for discussion until 1 April 1996
587
588
Irina Krasovskaia
interpolated by some method (e.g. Arnell et al., 1993; Krasovskaia &
Gottschalk, 1992; Krasovskaia et al., 1994).
Many different flow regime classification systems exist, both local and
global (e.g. Lvovich, 1938; Pardé, 1955; Haines et al., 1988). As stated by
Pardé (1955), seasonal variations of flow are characterized by their "stable
rhythm or irregularity from year to year". A flow regime classification identifies the times of the year when high and low flows are likely to occur. How
likely is it, however, to observe an identified seasonal pattern each individual
year? How "rhythmic" or "irregular" is it? Stability is an important property
of a flow regime, one which is often neglected or described only qualitatively.
Shifts in the seasonal behaviour of flow affect both environmental and economic activities. There are classifications using predefined quantitative percentages of the occurrence of flow within a certain season (Lvovich, 1938), but
these percentages are arbitrary and very approximate.
In several studies involving the stability of river flow regimes
(Krasovskaia & Gottschalk, 1992, 1993; Krasovskaia et al., 1994) it has been
shown that a given flow series can demonstrate different flow regimes from
year to year (as a result of variability in climatic conditions) and that flow
regimes classified on the basis of long term monthly means sometimes do not
coincide with the flow regime that occurs most frequently when each year is
classified separately. The deviation can occur in as much as 45% of cases
(Krasovskaia et al., 1994). Qualitative analyses have shown that certain regime
types are more unstable than others. Thus, flow regime stability is a property
directly linked to flow regime type. However, using qualitative methods, it is
difficult to compare flow regimes with respect to their stability and to quantify
the probability of observing a predetermined flow regime type each individual
year for a particular series. It is also difficult to study the development of the
stability of a particular flow regime type in time, especially important in the
context of a possible climate change.
The objective of this study is to develop a unique quantitative measure
of the stability of river flow which can be directly determined from observed
series and used for regime classification. The instability index developed is
demonstrated using an automatic quantitative flow regime classification.
DATA USED
The study is based on 49 Scandinavian series of monthly flows with a common
observation period of 66 years (1922-1987). These data were complemented by
49 mean monthly flow series for western Europe from the FRIEND data base
(Roald et al., 1993). The lengths of the latter data series varied between 35 and
87 years, covering approximately the same period as the Scandinavian data. It
was not possible to find a common period of the same length for all the stations. Most of the basins have areas of less than 2000 km2.
Quantification of the stability of river flow regimes
589
AUTOMATIC RIVER FLOW REGIME CLASSIFICATION
A flow regime classification scheme used in the study was developed for
Scandinavia (Krasovskaia & Gottschalk, 1992) and is based on the principles
formulated by the Scandinavian Working Group on Flow Regimes (Gottschalk
etal., 1979), shown in Table 1. This rough classification, based on mean
monthly values, was later generalized to include river flow regimes of western
Europe and applied for flow regime classification of a total of 1385 monthly
flow series taken from the FRIEND data base (Krasovskaia et al., 1994).
Table 1 Scandinavianflowregime classification (from Krasovskaia et al., 1994)
High water
HI
H2
H3
Dominant snowmelt high water
Transition to secondary high water
Dominant rain high water
LI
L2
L3
Dominant low water in winter
Transition zone, two low water periods Dominant summer low
in different seasons
water
Low water
This automatic computer-adapted classification distinguishes three main
sources of flow: snowmelt, rain and mixed snowmelt and rain. It defines
exactly the time of year when high/low water is observed in a certain regime
type. The identified flow regime types for the FRIEND-area of Scandinavia and
western Europe are summarized in Table 2 together with the criteria for classification. Three discriminating periods were used for high water: April-August,
September-November and December-March. For low water two periods were
used: January-April and May-December. Six main flow regime types have been
identified for Scandinavia and eight for western Europe, with a number of
transition types. (A complete description of the regime types can be found in
Krasovskaia et al., 1994). Figure 1 offers examples of the main flow regime
types. The mountain regime is illustrated by two examples: one for the glacial
and one for the nival mountain regime type.
The flow regime classification utilized, unlike many traditional ones, is
based on strictly defined discriminating criteria and is computer-adapted, which
is a clear advantage when large amounts of data both in space and time are
involved for flow regime characterization in a changing environment. The strict
description of the discriminating criteria also permits a strict quantitative
assessment of the problem of the stability of the flow regimes.
590
lrina
Krasovskaia
Table 2 Flow regime types for the "FRIEND" area of western Europe (from Krasovskaia
etal, 1994)
Index
High water
Low water
North-Scandinavian
H1L1
Maxl
Max2
Max3
IV-VIII
IV-VIII
IV-VIII
Mini
Min2
I-IV
I-IV
Mountain nival
Mountain glacial (>500 m a.m.s.l)
H1ML1
H1MGL1
Max
Max
IV-VI
VII-VIII
Min
I-IV
Northern Inland
H2L1
Maxl
Max2
Max3
IV-VIII
IX-XI
IX-XI
Mini
I-IV
Southern Inland
H2L2
Maxl
Max2
Max3
IV-VIII
X-XI
X-XI
Mini
Min2
I-IV
V-XII
Baltic
H2L3
Maxl
Max2
Max3
IV-VIII
IX-XI
IX-XI
Mini
Min2
VI-VIII
VI-VIII
Atlantic
H3L3
Maxl
Max2
IX-III
ix-m
Mini
Min2
VI-IX
VI-IX
Maxl
Max2
I-II
XI-XII
Mini
Min2
VIII-IX
VIII-IX
Regime type
Main flow regime types:
South-European
H3SEL3
Maritime/Azonal Max/Min < 10%
H0L0
Transition flow regime types:
North-Scandinavian Inland
H1L2
Baltic Inland
H1L3
South European Inland
H3SEL2
Mountain transition
H1ML2, ]H1MGL2, H1ML3, H1MGL3
Maxl, Max2 and Max3 denote the highest, second and third highest observed flow and Mini and Min2
the lowest and second lowest observed flow.
Roman numerals are used for month numbers.
APPLICATION OF THE CONCEPT OF ENTROPY FOR
QUANTIFICATION OF THE STABILITY OF RIVER FLOW
REGIMES
To describe the uncertainty in the assigning of a certain regime type to each
series one can use the measure of uncertainty of an experiment as defined by
information theory. An appearance of a certain pattern of flow out of n types
during individual years in a series can be regarded as coming from events
Eu ..., En. Each of these events can be characterized by a probability of
appearance:
Pi
= P(E,)
i = !,...,«,
where "£Pi = 1
1=1
(1)
Quantification of the stability of river flow regimes
NORTH-SCANDINAVIAN (H1L1)
Ôvertjuktan Sweden
591
MOUNTAIN GLACIAL (H1MGL1)
Isère at Val d'Isère Fran
MOUNTAIN NIVAL (H1ML1)
NORTHERN INLAND (H2L1)
Ybbs at Opponitz Austria
Fora at Hoggars bru Norway
SOUTHERN INLAND (H2L2)
BALTIC (H2L3)
Aurajoki at Hypôlnsn, Finland
Harnmarbyàn at Hammarby Sweden
ATLANTIC (H3L3)
AZONAL (HOLO)
Bekhamp Brook at Bardfield bridge UK
Riss at Untersalmetingen Germany
SOUTH-EUROPEAN (H3SEL3)
BALTIC INLAND (H1L3)
La Rulies at Tintigny Belgium
Prosna at Boguslaw Poland
Fig. 1 Examples of the main flow regime patterns for northern and western
Europe (From Krasovskaia et al., 1994)
Irina Krasovskaia
592
A measure of the uncertainty of an experiment H, called the entropy of the
experiment, can then be applied (Shannon & Weaver, 1941):
H= -ZPilnfp;)
(2)
From this expression it follows (Pugachev, 1962) that the entropy of the
experiment, which is a continuous non-negative function of the probabilities
px, ..., pn, equals 0 only if one of these probabilities equals 1 and all the rest
are 0 (as p -» 0, limpln(p) = 0). In other words, an event with the probability
of 1 is not random and there is no uncertainty in the experiment. This means
that the series will have the same flow pattern for all the years in the observation period.
It can also be proved that the entropy value H reaches its maximum when
all the possible values of the events Eh ..., En are equally probable:
px = ... = pn = \ln. In this case the entropy has the value of ln(n).
The higher the entropy value of an experiment, the smaller is the probability of observing the assigned flow pattern for the series in each individual
year. Flow patterns with low entropy values will be called stable and those with
high entropy values unstable. Thus entropy may be employed as an "instability
index" of a flow regime type, a concept developed further below.
Entropy has the property of additivity (Pugachev, 1962), which gives it
a clear advantage as a measure of stability of a flow regime (which reflects the
stability of maxima and minima) over, for example, the percentage of cases
that fit an assigned regime type (i.e. probability).
Equation (2) for the entropy of an experiment has its parallels in the
definitions of entropy in thermodynamics and in various "stability" and
"diversity" indices in ecology (such as Shannon's index and MacArthur's
stability index (MacArthur, 1955) often used to describe stability of a system.
There are also examples of the use of entropy in hydrology. Leopold &
Langbein (1962) used entropy to find the most probable description of a longitudinal river profile. Amorocho & Espildora (1973), using marginal and
conditional entropy, characterized the uncertainty of hydrological data and
developed a criterion for an objective evaluation of the goodness of a model.
This approach was extended by Chapman (1986) who suggested the use of proportional class intervals for the calculation of entropy. Sonuga (1972, 1976)
used the maximization of entropy, subject to certain constraints, to develop a
simplified probability distribution for a hydrological frequency analysis, useful
in areas with scarce hydrological data. A maximum entropy approach was also
used by Dalezios & Tyraskis (1989) for spectral estimation in regional precipitation analysis. Yang & Burn (1994) developed an approach for data collection
network design employing entropy as a measure of the information flow between gauging stations. Culling (1988) used K-entropy (Kolmogorov-entropy)
as a measure of the irregularity of a landscape surface.
The concept of entropy has also recently been introduced in geostatistics;
Christakos (1990) used entropy as a measure of the uncertainty of a prior
Quantification of the stability of river flow regimes
593
distribution model. Rossi & Posa (1992) used entropy as an alternative measure
to characterize the bivariate distribution of a stationary spatial process and
introduced a "relative entropy" estimator to measure the departure of experimental and theoretical distributions.
Here, a statistical interpretation of entropy as a measure of the uncertainty of an experiment will be used to describe quantitatively the inherent
property of a river flow regime, namely its stability, characterizing the
uncertainty in assigning a certain regime type to a series.
ANALYSIS OF THE STABILITY OF FLOW REGIMES
The approach described above was first applied to the mean monthly flow data
sets for Scandinavia, classified according to the scheme presented earlier. In
order to investigate the stabilities of maxima and minima, their instability
indices were calculated separately. Thus, n represents the number of periods
used for the discrimination of a particular regime type.
The instability index reaches its maximum value, as was shown earlier,
when all the events are equally probable. The probability p1 that the maxima
during high water will be observed within one of the three discriminating
periods will then be 1/3. Similarly, for low water, the probability that the
minima will be observed within one of the two discriminating periods will be
1/2. Using the additivity of entropy, the total instability index of a flow regime
type will then be the sum of the indices for maxima and minima. From
equation (2) it follows that the maximum instability index value for a flow
regime with three maxima and two minima is 4.682 and for those with two
maxima it equals 3.584. For the case with three maxima and one minimum, the
maximum value of the instability index will be 3.990.
The maximum value of the instability index means that maxima and
minima can occur with equal chances in any of the discriminating periods used
for classification. In this case the flow regime is naturally characterized as very
unstable. On the other hand, the lower the value of the instability index, the
greater is the probability that the identified regime type will be the most
frequent one in the series. Table 3 (column 3) presents the values of the
instability index obtained (the average across the samples) and in parentheses
shows the percentage of the maximum value for the respective regime type.
To illustrate the chance of observing the assigned regime type during
each individual year in a series with a certain instability index, one can use the
following procedure. First, give/?,- in equation (2) the values of the percentage
of the years in the series with the assigned regime type and those with a
different regime. Repeat the procedure for all maxima and minima involved in
the definition and sum the results. The values obtained will give the corresponding values of instability indices for one of many possible combinations,
namely when all maxima and minima involved in the regime definition show
the same probability of occurrence within a discriminating period. Table 4
594
Irina Krasovskaia
Table 3 Stability indices for flow regime types
Regime type
Instability index of max/min
Instability index of flow regime types
Scandinavia
W. Europe
Scandinavia
W. Europe
Scandinavia + W.
Europe
0.287
0.394
0.552
0.590
0.639
2.372
(=51%)
2.462
( = 52%)
2.409
( = 51%)
3.644
(=78%)
3.591
( = 77%)
1.816
(=51%)
1.903
(=53%)
H1L1
Maxl
Max2
Max3
Mini
Min2
0.287
0.458
0.709
0.442
0.476
H1L2
Maxl
Max2
Max3
Mini
Min2
0.879
0.903
1.026
0.652
0.660
H1L3
Maxl
Max2
Max3
Mini
Min2
0.914
0.962
0.989
0.316
0.355
H2L1
Maxl
Max2
Max3
Mini
0.478
0.810
0.962
0.604
2.854
(=71%)
H2L2
Maxl
Max2
Max3
Mini
Min2
0.882
0.959
1.034
0.666
0.682
4.223
( = 90%)
H2L3
Maxl
Max2
Max3
Mini
Min2
0.998
1.028
1.046
0.504
0.475
4.041
( = 86%)
H3L3
Maxl
Max2
Mini
Min2
0.912
0.996
0.342
0.344
4.142
( = 88%)
0.832
0.837
0.943
0.546
0.486
0.652
0.819
0.146
0.199
3.536
(=76%)
2.594
( = 72%)
The values in parentheses show the instability index as a percentage of the maximum possible value for a
respective regime type.
illustrates the results of this procedure for the percentages from 50 to 95 % of
the years.
It is seen from Table 3 (column 3) that different regime types demonstrate different values of the instability index. The North-Scandinavian flow
regime (H1L1) is the most stable one according to the definition of stable
regimes. From Table 4 it is seen that if a series is assigned to the NorthScandinavian regime, 80-85% of years in the series will have flows with this
particular regime type.
The Atlantic (H3L3) and the Northern Inland (H2L1) flow regime types
show instability index values of approximately 70% of the maximum possible.
This would roughly correspond to about 65% and 50% chance, respectively,
of finding these flow regime types among the regimes of the individual years
Quantification of the stability of river flow regimes
595
Table 4 Comparison of the instability index to the probability that each
year in the series will have the assigned regime
% of cases with the assigned Instability index
regime type
Instability index
(% max value)
95
0.990
21 (22,20)
90
1.625
35 (36,33)
85
2.110
45 (47,42)
80
2.505
53 (56,50)
75
2.805
60 (63,56)
70
3.055
65 (68,61)
65
3.235
69 (72,65)
60
3.360
72 (75,67)
55
3.440
73 (77,69)
50
3.465
74 (77,70)
The values in parentheses are, respectively, for regime types with 2 maxima and 2
minima, and 3 maxima and 1 minimum, in the definition.
of the series that have been assigned the Atlantic or Northern Inland regimes.
These two regime types can be considered to be moderately unstable.
The Southern Inland (H2L2), Baltic (H2L3), North-Scandinavian Inland
(H1L2) and Baltic Inland (H1L3) flow regime types have very high values of
instability indices, 80-90% of the maximum value. In other words, according
to Table 4, this corresponds to a less than 50% chance of finding these types
among the regimes of the individual years in the series that have been assigned
these flow regimes. This is a typical case when it is practically impossible to
tell which regime type is the most frequent one. The regimes of this group all
belong to a group with a transition from snowmelt dominated types to rain
dominated ones, which are extremely sensitive to small fluctuations in climate.
The same approach has been used to determine the instability indices for
49 flow series for western Europe. The following regime types were identified:
North-Scandinavian (H1L1), Baltic Inland (H1L3) and Atlantic (H3L3). The
values of the instability indices obtained, given in the fourth column of
Table 3, proved to be very similar to those for the Scandinavian series. For
H3L3 the value is lower, which perhaps can be explained by a much larger
number of the European series that demonstrated this regime type and a more
central role of rain in flow formation. It is seen from Table 4 that if a series
is assigned this regime, about 80% of years in it will have flows with this
particular regime type. This regime can be considered to be stable. The fifth
column of Table 3 presents the weighted mean values of the instability indices
for both Scandinavia and western Europe.
The total instability index of a flow regime is a sum of instability indices
of maxima and minima used for its identification. The contribution of the
stability character of maxima and minima to the total stability character of a
flow regime can be very different. Figure 2 illustrates the stability of different
596
Irina Krasovskaia
100
I
£
Ê
X
£
REGIME TYPE
H1L1
H1L2
H1L3
H2L1
H2L2
H2L3
H3L3
MAXIMA
H
42.0
85.0
83.0
68.2
87.2
92.9
69,0
MINIMA
HH
74.0
94.6
61.0
87.0
97.2
70.6
28.0
Fig. 2 Instability index for maxima and minima of different flow regime
types (% of the maximum possible entropy).
regime types by showing the instability indices of maxima and minima (as
percentage of the maximum possible) separately. The values of the instability
indices of maxima and minima for each particular regime are given in the
second column of Table 3.
For very unstable regime types, such as North-Scandinavian Inland
(H1L2) and Southern Inland (H2L2), both maxima and minima are very unstable. For the other two unstable regime types, Baltic Inland (H1L3) and
Baltic (H2L3), the difference in the stability character of maxima and minima
is striking. For both of them it is the maxima that give the regime its unstable
character. The minima are more stable, especially for H1L3. For the moderately unstable Northern Inland regime (H2L1) it is the minimum which is more
unstable, while the maxima are moderately stable.
Turning to the more stable regime types, it can be seen that their stability
depends on stable maxima for the North Scandinavian type (HILI) and stable
minima for the Atlantic type (H3L3). A conclusion can be drawn that, in this
case, there is a chance of observing low flow in some season other than winter
for the flow regime type H1L1, while for the flow regime type H3L3 the
maximum flow can be caused by rain (or snowmelt?) in some season other than
winter.
Table 5 summarizes the stability characteristics of different flow regime
types for the combined data sets for Scandinavia and western Europe.
Quantification
of the stability
of river flow regimes
597
Table 5 A summary of stability characteristics of different flow regime types
Flow regime type
Instability index
( % of max: value)
Stability character of regime
As a whole
Maxima
Minima
North-Scandinavian
H1L1
51
stable
stable
moderately
stable
North-Scandinavian Inland
H1L2
88
unstable
unstable
unstable
Baltic Inland
H1L3
77
unstable
unstable
stable
Northern Inland
H2L1
71
moderately
unstable
moderately
stable
unstable
Southern Inland
H2L2
90
unstable
unstable
unstable
Baltic
H2L3
86
unstable
unstable
moderately
stable
Atlantic
H3L3
53
stable
moderately
stable
stable
CONCLUSIONS
The suggested approach for the calculation of the instability indices for flow
regime types permits quantification of the inherent property of a flow regime,
namely, its stability. It allows one to compare regimes according to their
stability and to reveal if it is maxima or minima which bring the specific
stability features to a particular regime type. The instability index provides as
important a characteristic of a flow regime as the average seasonal behaviour
of the flow. Finally, a quantitative instability index permits the study of the
changes of flow regimes in time, for example, by comparing the indices for
different time periods. The approach is general and can be applied to other
flow regime classifications with quantitatively formulated discriminating
criteria.
REFERENCES
Amorocho, J. & Espildora, B. (1973) Entropy in the assessment of uncertainty in hydrologie systems and
models. Wat. Resour. Res. 9(6), 1511-1522.
Arnell, N. W., Krasovskaia, I. & Gottschalk, L. (1993) River flow regimes in Europe. In: Flow Regimes
from International Experimental and Network Data (FRIEND), ed. A. Gustard, vol. I, Hydrological
Studies, 112-121. Institute of Hydrology, Wallingford, UK.
Chapman, T. G. (1986) Entropy as a measure of hydrologie data uncertainty and model performance. J.
Hydrol. 85, 111-126.
Christakos, G. (1990) A Bayesian/maximum-entropy view to the spatial estimation problem. Math. Geol.
22, 763-777.
Culling, W. E. H. (1988) Dimension and entropy in the soil-covered landscape. Earth Surf. Proc.
Landforms 13, 619-648.
Dalezios, N. R. & Tyraskis, P. A. (1989) Maximum entropy spectra for regional precipitation analysis and
forecasting. / . Hydrol. 109, 25-42.
598
Irina
Krasovskaia
Gottschalk, L., Jensen, J. L., Lundquist, D,, Solantie, R. & Tollan, A. (1979) Hydrological regions in
Nordic countries, Nordic Hydrol. 10, 273-276.
Haines, A. T., Finlayson, B. L. & McMahon, T. A. (1988) A global classification of river regimes. Appl.
Geogr. 8, 255-272.
Krasovskaia, I. & Gottschalk, L. (1992) Stability of river flow regimes. Nordic Hydrol. 23, 137-154.
Krasovskaia, I. & Gottschalk, L. (1993) Frequency of extremes and its relation to climate fluctuations.
Nordic Hydrol. 24, 1-12.
Krasovskaia, I., Arnell, N. W. & Gottschalk, L. (1994) Flow regimes in northern and western Europe:
development and application of procedures for classifying flow regimes. In: FRIEND: Flow
Regimes from International Experimental and Network Data, ed. P. Seuna, A. Gustard, N. W.
Arnell & G. A. Cole (Proc. 2nd Int. FRIEND Conf., UNESCO, Braunschweig, Germany, October
1993), 185-193. IAHS Publ. no. 221.
Leopold, L. B. & Langbein, W. B. (1962) The concept of entropy in landscape evolution. USGS Prof. Pap.
500-A.
Lvovich, M. I. (1938) Opyt Klassifikatsii Rek SSSR (Experience of classification of rivers of the USSR).
Trudy GGI6, Leningrad (in Russian).
MacArthur, R. H. (1955) Fluctuations of animal populations and a measure of community stability. Am.
Zool. 10, 17-25.
Pardé, M.(1955) Fleuves et Rivières. Colin, Paris.
Pugachev, V. S. (1962) Teorija sluchainyh funktsij (Theory of random functions). Fizmatgiz, Moskva (in
Russian).
Roald, L. A., Wesselink, A. J., Arnell, N. W., Dixon, J. M., Rees, G. & Andrews, A. J. (1993) European
water archive. In: flow regimes from international experimental and network data (FRIEND), ed.
A. Gustard, vol. 1, Hydrological Studies, 7-20. Institute of Hydrology, Wallingford, UK.
Rossi, M. & Posa, D. (1992) A non-parametric bivariate entropy. Math. Geol, 24, 539-553.
Shannon, C. E. & Weaver, W. (1941) The Mathematical Theory of Communication. University of Illinois
Press, Urbana, USA.
Sonuga, J. O. (1972) Principle of maximum entropy in hydrologie frequency analysis. / . Hydrol. 17,
177-191.
Sonuga, J. O. (1976) Entropy principle applied to the rainfall-runoff process. J. Hydrol. 30, 81-94.
Yang, Y. & Burn, D. (1994) An entropy approach to data collection network design. / . Hydrol. 157,
307-324.
Received 8 November 1994; accepted 7 March 1995