Hydrological Sciences Journal
ISSN: 0262-6667 (Print) 2150-3435 (Online) Journal homepage: http://www.tandfonline.com/loi/thsj20
The waters of the Earth
J. C. I. DOOGE
To cite this article: J. C. I. DOOGE (1984) The waters of the Earth, Hydrological Sciences Journal,
29:2, 149-176, DOI: 10.1080/02626668409490931
To link to this article: http://dx.doi.org/10.1080/02626668409490931
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Hydrological Sciences - Journal - des Sciences Hydrologiques, 29, 2, 6/1984
The waters of the Earth
J. C. I. DOOGE
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Civil Engineering
College,
Earlsfort
Department,
University
Terrace, Dublin 2,
Ireland
ABSTRACT
Hydrology is considered in the context of
the solar system, the water-related cycles of geophysics,
the human environment, and international scientific
cooperation. Global water balances and dynamic
hydrological models are briefly described. The existence
of different levels of description of hydrological
phenomena is referred to and recent attempts to formulate
hydrological laws at basin scale briefly reviewed.
Les eaux de la
Terre
RESUME
L'hydrologie est examinée dans le contexte du
système solaire, des cycles en rapport avec l'eau dans
le domaine de la géophysique, de l'environnement humain
et de la coopération scientifique internationale. Les
bilans hydriques globaux et les modèles dynamiques
hydrologiques sont passés en revue succinctement. On
note l'existence d'une description des phénomènes
hydrologiques à trois niveaux différents, et on fait la
critique des essais récents pour formuler des lois
hydrologiques à l'échelle des bassins.
THE CONTEXT OF HYDROLOGY
Water and
geophysics
The nature of the hydrosphere and hence of the hydrology of any
planet depends critically upon its surface temperature. The planet
Earth has a particularly vigorous hydrology because its surface
temperature is in the neighbourhood of the triple point of water.
As a result of this circumstance, water in the Earth's hydrosphere
occurs in all three phases and is readily transformed from one
phase to another.
It would appear from the abundance of noble gases in the Earth's
atmosphere that either the Earth originated without a primary
atmosphere or that the primary atmosphere was largely lost by the
dissociation and thermal escape to space (Rubey, 1951). It is
generally accepted that our present atmosphere and our present
hydrosphere arose from outgassing of the solid Earth (Prinn, 1982).
Thus we are indebted for the hydrosphere which we study to chemical
and physical processes in the Earth's interior which are the
concern of two of our colleague associations in IUGG. The present
atmosphere of the Earth is over one hundred times greater (as
*Text of the Union Lecture delivered
of IUGG, Hamburg, August 1983.
at the 18th General
Assembly
149
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150 J.C.I. Dooge
measured by the surface pressure) than that of Mars but almost one
hundred times less than that of Venus. The effect of the Earth's
atmosphere is to reduce the effective temperature due to the increase
in albedo but to increase the surface temperature due to the
atmospheric greenhouse effect. In the case of Venus these two
effects are much more marked resulting in a surface temperature of
over 700 K. This is due to the runaway greenhouse effect described
by Rasool & de Bergh (1970) which is illustrated on Fig.l (Goody &
Walker, 1972). It is clear from Fig.l that hydrologists on Mars
would all belong to the International Commission on Snow and Ice
and hydrologists on Venus would all belong to the International
Association for Meteorology and Atmospheric Physics.
10~6
!0~5
\0'a
!0~ 3
\0~2
Î0"1
!
Vapour pressure of water (bars)
Fig. 1
Evolution of surface temperatures.
It is interesting to note that if the Earth were closer to the
sun by about 5% a runaway greenhouse effect could have occurred and
the bulk of the Earth's hydrosphere would be in vapour form (Rasool
& de Bergh, 1970; Kondratyev & Hunt, 1982). The present average
surface temperature of 15°C is close enough to the triple point
of water for water vapour, liquid water and frozen water all to
occur in substantial amounts. The occurrence of these three forms
of water, their movement from one storage location to another, and
their transformation from one form to another give rise to problems
of considerable scientific interest and to problems of considerable
practical importance.
Water
and man
The struggle of man to control any element of his environment may
be divided into four phases: observation, understanding, prediction
and control. Even in regard to a single aspect of the environment
such as the control of water, these four phases of human endeavour
are, as suggested in Fig.2, iterative rather than strictly
consecutive. Because of the pervading influence of water (physically,
biologically, economically, socially), the concerns of the
hydrologist have always transcended the boundaries that would be
appropriate if hydrology were to be considered only as a branch of
geophysics. This is reflected in the statutes of IAHS in which the
main objective of the Association is given as "to promote the study
of hydrology as an aspect of the earth sciences and of water
resources" and goes on to include in the more detailed statement
Waters of the Earth
151
observation
'1
''
''
science J > — * - understanding
"
^—«^engineering
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prediction
,
"
i „ i
Fig. 2
Man and his environment.
of its objectives "the examination of the hydrological aspects for
the use and management of water resources and their change under
the influence of man's activities".
It is interesting to record that the earliest hydrological
measurements were made for very practical purposes. The 4000 year
old nilometer on the Isle of Elephantine near Assuan was fixed on
the wall of a covered corridor accessible only to the priests
(Borchardt, 1906). The level of taxation for the year depended on
the water level of the Nile. The first reference to a raingauge
occurs in a book by Kautilya (400 BC) on politics and administration
because the amount of annual rainfall was the basis of the land tax
(CBI, 1951). The relation of the hydrological cycle to the other
water-related cycles and to some important socio-economic factors
is shown in Fig.3 (Golubev, 1983). The relationship of the
hydrological cycle to the cycle of erosion and sedimentation is a
close one and this is reflected by the fact that one of the seven
international commissions of IAHS is the International Commission
on Continental Erosion. The amount of soil erosion at present is
water resources use
population j
growth
cycle
erosion/sedimentation
A
hydrological
cycle
economic
development
biochemical
cycles
change of geosystems
Fig. 3
Water-related cycles.
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152 J.C.I, Dooge
probably more than five times what it was before the emergence of
agriculture on a worldwide basis and may double in the future as
more and more land is converted for the harvesting of crops.
Similarly, the hydrological cycle plays a key role in the biogeochemical cycles of the more common elements which have been the
subject of close study by the ICSU Scientific Committee on Problems
of the Environment (SCOPE, 1981). All of these cycles have been
changed considerably because of human activity. The pressures of
population increase and of economic development on both the
quantitative and the qualitative aspects of the hydrological cycle
have been quite considerable (Pereira, 1972; Dooge et al.,
1973).
Development
of modern
hydrology
It is sometimes convenient to divide the history of a people or the
history of an art form into three periods of time and to designate
these as the heroic age, the classical age, and the modern age. If
this were to be done in the case of hydrology, the transition from
the heroic age to the classical age would probably be placed about
50 years ago and the transition from the classical age to the
modern age about 20 years ago. The development of new measuring
techniques and the availability of computers has had enormous
influence on hydrological research and practice in the past two
decades. High technology techniques for measurement which have
been applied in hydrology include: radar measurement of rainfall,
ultrasonic measurement of streamflow, gamma radiation measurement
of snow cover, new techniques of spectrometry and chromatography
in water quality, tracing by isotopes in both the unsaturated and
saturated zones and in ice cores, radio-echo sounding of the polar
ice masses, remote sensing of hydrological phenomena, and many
more. Over the same period there has been both an extension of
hydrological networks, particularly with respect to the measurement
of sediment transport and water quality, and also a better understanding of optimum methods of network design and coordination.
The availability of telemetering systems and of high-speed computers
has been availed of in the processing of hydrological data of all
types.
Hydrological understanding has also advanced considerably over
the past two decades. Better and more representative measurements
have been matched by a growth in physical understanding and
analytical insight. Linearized theories of the individual hydrological processes have been developed to the stage where they
constitute a comprehensive and consistent system of analysis. A
substantial start has been made in building on this foundation
techniques of nonlinear analysis which are realistic without being
unduly complex. In the field of glaciology the application of the
methods of physics and of continuum mechanics to the flow of
glaciers and to the flow of water within glaciers has improved
greatly the understanding of glacier movement and glacier runoff.
Much of the increased understanding of hydrological processes in
the last two decades has been a recognition of their complexity and
of the inter-relationship between factors which had previously
been separated for analytical purposes. However, it can be said
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Waters of the Earth
153
in general that over the past two decades the advances in the
understanding of hydrological processes do not appear to have been
as rapid as the advances in the measurement and processing of data
or in the simulation of hydrological variables based on mathematical
models. This may partly be due to the rapid advance in understanding
over the immediately preceeding decades but it does present a
challenge to hydrologists today if progress is to be maintained
evenly on all fronts.
The great advances in the power of hydrological forecasting in
real time and of hydrological prediction for design purposes can
be attributed to the availability of high-speed computers and to
developments in mathematical modelling that took full advantage of
this facility. In surface water hydrology, the main advances have
been in the application of systems techniques and of statistical
methods to problems of hydrological analysis. In groundwater
hydrology, the main advances have been in the application of finite
difference methods to groundwater problems by the use of both
analogue and digital computers. In the field of continental erosion,
an important advance has been the synthesis of the vast body of data
for purposes such as the development of a universal soil loss
equation to predict soil losses from agricultural land under a wide
range of conditions.
Like every other scientific discipline, hydrology faces a host
of unsolved problems. There is still a need for improved methods
for the measurement of precipitation, of evaporation and of soil
moisture storage. There is a need to develop an adequate theory
for the relationship between storm rainfall and flood runoff and
for a linkage of this relationship with physiographic basin
characteristics. There is a need to develop methods for the analysis
of the coupled flow of heat and of water within an unsaturated soil
or within an ice sheet. Above all there is a need to integrate the
various approaches to hydrological problems. There is a need for
an integration of the separate approaches to the various parts of
the hydrological cycle. There is a need for the integration of the
approaches used in hydrology and used in cognate sciences. Finally,
there is an urgent need for the integration of the methods used by
research hydrologists and the methods used in applied hydrology.
International
cooperation
in
hydrology
As in all other branches of geophysics, international cooperation
is of great significance in hydrology. The first important move
towards such cooperation in hydrology was the formation by the
General Assembly of IUGG in Rome in 1922 of an International Branch
of Scientific Hydrology; this name was soon changed to the International Association of Scientific Hydrology. It has been variously
reported that the use of the term "scientific hydrology" in these
titles was to distinguish the members and their interest in
hydrology from (a) water diviners and (b) promoters of natural
mineral waters. At the Moscow General Assembly (1971) the name of
the Association was changed to the International Association of
Hydrological Sciences.
The Association developed very vigorously in the years after the
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154 J.C.I. Dooge
Second World War under the guidance of Professor Leon Tison who
first became associated with the work of IASH about 1930 and was
Secretary General of the Association from 1939 until forced to
retire due to ill health in 1970. The Association has published a
quarterly journal since 1956 and also publishes the proceedings of
symposia which reflect the development of all branches of hydrology.
The latest issues in this series of some 150 volumes are the five
Association Symposia for this General Assembly, two of which were
pre-published and three which will be post-published in 1984.
Inter-governmental cooperation in hydrology has developed strongly
in recent years under the auspices of UNESCO and WMO. As a result
of the interest fostered by the International Geophysical Year of
1957 and following the UN Conference on Water for Peace in 1967,
the International Hydrological Decade (IHD) was instituted by
UNESCO and promoted by the member states . The purpose of the
Decade (1965-1974) was to promote international cooperation in the
field of hydrological research and education as a means for
achieving a full assessment of the world's water resources and a
more rational use of them. A continuation of the work of the IHD
was ensured by the establishment of a long-term International
Hydrological Programme which is now in its second phase. In 1971,
under a unified Operational Hydrology Programme, WMO grouped its
activities on the development of hydrological networks, the standardization of observations, the improvement of data collection and
processing, and the supply of data for design and hydrological forecasting. The IHP/OHP are closely coordinated and IAHS plays an active
role both in programme development and in project implementation.
Cooperation among nongovernmental organizations in the field of
water research has also been formalized to some degree. In 1964
the International Council of Scientific Unions (ICSU) established
a scientific committee on water research (COWAR) to act in the
name of ICSU in relation to the IHD. The role of this Committee
was later extended to include the formulation and promotion of
programmes of research in water resources and to establish contact
with all governmental and nongovernmental organizations concerned
with problems of water in order to ensure the coordination of
research in this field. Meanwhile, IAHS signed an agreement in
1973 with the International Association for Hydraulic Research (IAHR)
and with the International Commission for Irrigation and Drainage
(ICID) establishing an informal Presidential Council to coordinate
and complement each other's activities for the common benefit of
all three organizations. This agreement was later adhered to by
the International Association of Hydrogeologists (IAH), the
International Association for Water Pollution Research (IAWPR), the
International Congress on Large Dams (ICOLD) and the Permanent
Association for International Navigational Congresses (PIANO. In
1976, both these liaison bodies were reconstituted as a joint
committee for the coordination of water research (COWAR) of the
International Council of Scientific Unions and the Union of
International Engineering Associations (UATI). The new COWAR seeks
to combine the bridging of the physical and biological sciences
which was present in the original COWAR with the bridging of
scientific research and engineering practice which was present in
the informal Presidential Council.
Waters of the Earth
155
THE GUOBAL WATER CYCLE
of the hydrological
cycle
Hydrology is concerned with the occurrence and movement of the
water on our planet. The forms in which this water may occur are
shown in Fig.4 which illustrates the modern concept of the global
hydrological cycle. The rectangles of the figure denote various
forms of water storage: in the atmosphere, on the surface of the
ground, in the unsaturated soil moisture zone, in the groundwater
reservoir below the water table, in the channel network draining
the basin, or in the oceans. The arrows in the diagram denote the
Ri
Atmosph ere
* •
P
Fi
T
—J
Surface
F
i
i
Soil
i
i
i
i
80
Q
-M
.
i »
R
\'
Groundwater
__lk —
%
2LLithosphère
Fig. 4
4î
3
>E
!
i
i
i
%
JRb
l
Channel Network
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Nature
__|
t tF
i
Atmos.
"p
i'E
"I
I
I
I
RO,
Ocean
1
Lithos.
Block diagram of hydrological cycle.
various hydrological processes responsible for the transfer of
water from one form of storage to another. Thus the precipitable
water (W) in the atmosphere may be transformed by precipitation (P)
to water stored on the surface of the ground. In the reverse
direction water may be transferred from the surface of the ground
by evaporation (E) or from the unsaturated soil by transpiration
through vegetation and subsequent evaporation from the leaf surface
(ET). Some of the water on the surface of the ground will
infiltrate through the surface into the unsaturated soil (F) but
some of it may find its way as overland flow (Q 0 ) into the channel
network. During precipitation, if the field moisture deficit of
the soil which has arisen since the previous precipitation is
substantially satisfied then there will be either recharge (R) to
the groundwater or else lateral interflow (Q^) through the soil
into the channel network. The groundwater storage is depleted by
groundwater outflow (Q e ) which enters the channel network and
supplies the streamflow during dry periods. During prolonged
droughts, soil moisture may be replenished by capillary rise (C)
from groundwater to the unsaturated zone and subsequent loss to the
atmosphere by évapotranspiration. Overland flow ( Q 0 ) , lateral
interflow (Q-^) and groundwater outflow (Q g ) are all combined and
modified in the channel network to form the runoff (RO) from the
area for which the balance is being calculated. These various
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156 J.C.I. Dooge
hydrological processes form the subject matter of physical hydrology.
This hydrological cycle was not established on a quantitative
basis until modern times. In many Greek writings, and particularly
in those of Plato, we find references to the gathering of water in
great underground reservoirs from which the rivers flow. This
concept is an interesting reflection of the wide distribution of
karstic conditions in Greece. The debate as to whether rivers
originated from rainfall or from some subterranean process of
condensation waxed and waned for another two thousand years. The
controversy was only finally settled by the rise of quantitative
hydrology and the estimation of water balances on individual basins.
Quantitative hydrology is usually taken as dating from the
publication of an anonymous work on "The origin of springs" in Paris
in 1674 which compared rainfall and streamflow in the upper reaches
of the Seine basin. This work is now attributed to Pierre Perrault
(Dooge, 1959; Tixeront , 1974).
The first quantitative comparison between streamflow and
evaporation was made for the Mediterranean by the astronomer Halley
(1691). The first catchment water balance involving the estimation
of all of the elements of the cycle was not made until over one
hundred years later when John Dalton, better known as a chemist,
estimated the three elements of rainfall, streamflow and evaporation
for England and Wales (Dalton, 1802). For another hundred years,
isolated shrinking pockets of resistance to the modern concept of
the hydrological cycle still persisted.
Global
water
balances
The results of modern studies on global water balances are to be
found in the text by Kalinin (1968), in the papers presented at the
Symposium on World Water Balance held in Reading (IAHS/UNESCO, 1972),
in the monograph of the IHD National Committee for the USSR
(Korzun et al.,
1974), in the monograph by Baumgartner & Reichel
(1975), and in papers scattered throughout hydrological journals.
The estimates of the components of the water balance for the Earth as
a whole and for the individual continents still differ from one
another though the overall pattern is similar.
Table 1
Storage and replenishment of global water
Type of storage
Amount
(106 km 3 )
Flux
(103 km 3 )
Total
Oceans
Inactive groundwater
Frozen water
Active groundwater
Soil water
Atmosphere
Rivers
Biological water
1460
1370
56
29
4
65 X 10"3
14 X 10"3
1.2 X 10~3
0.7 X 10^3
E = 520
E = 449
R = 1.8
R = 13
E + R = 85
P=520
R=36
Mean
residence
time
2800 years
3100 years
16000 years
300 years
280 days
9 days
12 days
7 days
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Waters of the Earth
157
For the purpose of discussion here the figures will be taken
from the textbook by Kalinin (1968) who made notable contributions
in a number of areas of modern hydrology and was active in the
affairs of IAHS. The distribution of water among the various forms
of storage is shown in Table 1 (Kalinin, 1968). The figure of
greatest uncertainty here is that for inactive groundwater. The
total groundwater included in Table 1 represents an estimate of the
water down to a level of 5 km. Hydrologists concerned with global
water balances could benefit through information from their
colleagues in IAVCEI in dealing with the problem of the amount of
water in the Earth's mantle and the rate of exchange between this
inactive water and the rest of the hydrosphere.
It is clear that well over 90% of the available fresh water is
accounted for by ice caps and glaciers, and that over 90% of the
active fresh water is accounted for by groundwater. When the
estimated fluxes between these forms of water storages are used to
calculate the mean residence times, the complete disparity in time
scale between the waters of the oceans and the ice caps on the one
hand and the water of the atmosphere and the land surface on the
other becomes apparent.
Many of the modern estimates of global or regional fluxes of
precipitation, evaporation and runoff are presented as closed
balances thus ignoring the effects of changes in the mean sea level
or in the ice caps which could be quite large in comparison with
the components of the water balance of more immediate concern in
hydrology. Even when these variations on a large scale and storage
elements are taken into account there are still unexplained
discrepancies. This problem has recently been discussed by Meier
(1983) who considers that there is an appreciable geophysical
enigma to be resolved which may involve tectonic redistributions as
well as a re-evaluation of the assumptions and estimates of
glaciologists and oceanographers.
Water related
cycles
Since the hydrological cycle is linked with the cycle for erosion
and sedimentation and with the biogeochemical cycles, there are
other aspects concerning the global view of the Earth's water that
also call for attention and research. Table 2 shows the contrast
Table 2
Runoff and sediment yield
Continent
Annual
surface
runoff
(103km3)
Annual
sediment
yield
(109 t)
Specific
yield
(t m~3)
Africa
Asia
Australia
Europe
North America
South America
4.1
13.2
2.3
3.0
6.7
11.2
0.48
14.53
0.21
0.30
1.78
1.09
0.12
1.10
0 09
0.10
0.27
OJO
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158 J.C.I. Dooge
between the distribution among the continents of the estimated
annual surface runoff and the estimated annual discharge of sediment
(Holdgate et al., 1982). The effect of vegetation and of population
pressure as well as soil type is reflected in the contrast between
the specific yield for Asia and that for the other continents. The
nitrogen cycle and the phosphorus cycle have been affected by human
activity. The amount of nitrogonous fertiliser applied annually
throughout the world and the amount of nitrogen world harvest are
of the same order of magnitude as the natural components of the
biogeochemical cycle of nitrogen. In the case of phosphorus the
amount of phosphorus used in fertilisers is now of the same order of
magnitude as the total amount of phosphorus stored in the oceans but
is still insufficient to replace the amount of phosphorus removed
during harvesting. The interrelation of these biogeochemical cycles
with the hydrological cycle is a close and a complex one.
The need for complete understanding of the global and regional
aspects of the hydrological cycle is evidently of great significance
in studies of climatic change and variation. In the field of
palaeohydrology this effect is very marked. Figure 5 (Schumm, 1977)
shows a hypothetical series of curves showing the relationship
between relative sediment yield and precipitation at various stages
during the past 400 million years. Curve 1 shows the conditions in
0
10
20
30
40
50
MEAN ANNUAL PRECIPITATION
(inches)
Fig. 5
Hypothetical relationship between relative sediment yield and precipitation
during past 400 million years (Schumm, 1977).
Precambrian and early Palaeozoic time (400 million years ago) when
there was no effective vegetation cover. Curve 2 shows the
estimated relationship during the later Palaeozoic and early
Mesozoic times (between 400 million and 135 million years ago)
following the appearance of primitive vegetation on coastal plains
and in humid valleys. Curve 3 shows the estimated relationship
following the emergence of flowering plants and conifers during the
Mesozoic and early Cenozoic periods (between 125 million and 25
million years ago). Curve 4 shows the relationship following the
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Waters of the Earth
159
appearance of grasses during the late Cenozoic time which gave rise
to erosion and runoff conditions similar to those of the present
but without the effect of man's influence. Without an accurate
estimate of the elements of the hydrological cycle it would not be
possible to estimate the biogeochemical cycle either of the common
elements or of particular pollutants. Hydrology can make a real
contribution to these studies. A recent attempt to close the
balance for the carbon cycle involved a year-long study of the major
rivers of the world (Degens, 1982).
It has been pointed out by O'Kane (1982) that the mass balances
used in environmental studies can never be exhaustive exact balances
and hence should be considered as statistical mass balances, a
concept which has not been considered previously in the literature.
Such an approach recognizes the existence of the closing errors due
to errors of statistical sampling, errors of measurement, and errors
due to incomplete specification of the elements of the mass balance.
O'Kane argues that, under these conditions, probability sampling
rather than systematic sampling should be used and that the key to
a mass balance is the statistical testing of the hypotheses that
the residual lack of balance is due to chance alone.
MODELS OF WATER MOVEMENT
Range of hydrological
models
When we turn from consideration of global or regional water balance
on an annual or long-term time scale to the dynamic behaviour on a
catchment scale, we encounter a bewildering variety of hydrological
models. This wide range of models is reflected in the proceedings
of the symposia organized by IAHS at Warsaw on "Mathematical Models
in Hydrology" (IAHS, 1974), at Bratislava on "The Application of
Mathematical Models in Hydrology" (IAHS, 1975), and at Baden on
"Mathematical Models of Water Quality Systems" (IAHS, 1978) and at
many other international symposia. It is difficult for the research
hydrologist and even more for the non-hydrological research worker
or for the applied hydrologist to find a way through the jungle
which has resulted from the luxurious growth of mathematical models
in hydrology in the past 15 years. Since there is no universal
model which is appropriate for the solution of all problems, the
choice of the applied hydrologist in any given situation is a most
difficult one.
There are a number of headings on which the classification and
description of hydrological models can be based (Dooge, 1981). Here
we will discuss only the spatial extent of the model, the nature of
the data, the type of structure assumed and the degree of simplification involved. There is often a strong link between the first two
criteria. Thus models aimed at physical understanding can rarely
be applied over a wide area.
Classification
based on nature
of
data
Hydrological models can be classified (on the basis of the nature of
the input data appropriate to them) into probabilistic models,
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160 J.C.I. Dooge
stochastic models and deterministic models. In probabilistic models,
the analysis is based completely on the historical values -of a
single hydrological variable such as storm rainfall or the maximum
annual flow or other variable and these data are assumed to be
independent. The frequency distributions most often used in
hydrology are the lognormal distribution, the Pearson type III
(gamma) distribution, and the Gumbel extreme value distribution.
Stochastic models are used when the output cannot be assumed to
consist of statistically independent data. The time-dependency or
persistence in the time series is measured by the autocovariance
function, and the most commonly used stochastic hydrological models
simulate the autocovariance functions as well as the mean and
variance of the hydrological time series. The use of stochastic
models is essential in the planning of the size of reservoirs since
a stochastic model that allows for the effects of persistence will
give a larger range of reservoir level than a probabilistic model
which neglects the persistence. Once a particular type of stochastic
model has been chosen and its parameters determined from the
historical record, the resulting stochastic process can be used to
generate a synthetic hydrological time series to serve as a basis
for the design of the reservoir or other hydraulic project.
In the case of deterministic models, the available data of both
input and output are used in order to choose an appropriate model
for the hydrological process and to choose appropriate values of
the parameters of this model. Dependence between input and output
reflects the assumption of causality and may be represented by a
wide range of models from simple regression to complex conceptual
models.
The modelling of the behaviour of a basin as indicated by a
record of precipitation and runoff may require an overall model
involving probabilistic, stochastic and deterministic elements.
Such an overall model can be considered as consisting of three
distinct parts, one belonging to each of the three classes of model
discussed above. If a linear model is used for forecasting
purposes then the deterministic part will give an estimate of the
expected value of the output at a given time. The stochastic part
will simulate the portion of the persistence indicated by the
recorded output which is not accounted for by the deterministic
component, and will also contribute an estimate of the probable
variation of the output from the expected value. The probabilistic
time-independent component of the model would contribute to the
remainder of the variation which is not accounted for in the
stochastic part.
Classification
based on knowledge
of system
structure
Dynamic hydrological models intended to represent physical
processes or total basin response may be divided into (a) black-box
models which link the input and output of a basin module or of a
total basin, (b) regression models linking hydrological variables,
(c) conceptual models of hydrological processes and (d) models based
on the application to hydrological processes of the principles of
mathematical physics. Such a procedure of classification is
tantamount to distinguishing between models on the basis of the
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Waters of the Earth
161
assumed degree of knowledge of the connection between the input and
the output. In the four cases listed above, this knowledge is
respectively assumed to be (a) a simple causality of unknown form,
(b) a regular statistical correspondence, (c) a simulation consistent
with physical principles, (d) a more complex physical theory capable
of improving the understanding of the phenomena involved.
In the black-box approach, an attempt is made to characterize
mathematically the operation of the hydrological system on the basis
of recorded inputs to and outputs from that system without any knowledge of the physical laws involved or of the physical characteristics
of the actual basin under study. A black-box approach has been used
as the basis for the prediction of direct storm runoff for the past
50 years. It is perhaps significant that such an approach was first
adopted by practical hydrologists and only after 25 years was an
adequate theory of the method developed.
At the other end of the spectrum we have dynamic hydrological
models which are based on the principles of continuum mechanics and
the applications of such physical principles as the conservation of
mass and the conservation of linear momentum. The prediction of
hydrological processes through the classical methods of mathematical
physics soon runs into difficulties. The physics itself is complex
and consequently the solution is difficult even for homogeneous
conditions and for simple boundary conditions. If the nonhomogeneous
nature of the basin under study can be quantified, then a solution
may be possible with the aid of large scale computer simulation but
even in this case serious difficulties have to be overcome.
Even the most complete mathematical model has ultimately to be
calibrated by applying it to an historical event in order to
optimize coefficients involving quantities such as friction factors
or sorptivities or hydraulic conductivities which represent a
parameterization at the scale of interest of those spatially nonhomogeneous variables that are part of the basic physical equations
formulation for conditions at a point.
The choice between an a priori
model based on physical knowledge
and an a posteriori
model based on measurement is common to all
types of physical system and is contrasted in Fig.6. In any
physical system, both approaches are liable to result in errors of
prediction because of errors in the data (process noise in Fig.6)
and approximations used in the analysis.
An intermediate approach frequently used in hydrology is the use
of a conceptual model in which a certain structure is assumed for
the system operation and a number of undefined parameters are
optimized on the basis of historical records. A conceptual model
may be defined as a simple arrangement of a relatively small number
of elements each of which is in itself a simple simulation of a
physical process. Thus, for example, the transformation of storm
rainfall to direct flood runoff by a basin can be represented by an
arrangement of linear channels which simulate pure translation and
linear reservoirs which simulate a concentrated storage effect.
Linearity,
lumping,
time-invariance
No matter what type of model is used, simplifying assumptions are
usually required in order to effect an analysis of the hydrological
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162 J.C.I. Dooge
system and to predict its behaviour. This can be illustrated for
the case of the deterministic input-output model. As indicated in
Fig.6, a deterministic model based on physical laws can be
simplified by linearization which reduces a set of nonlinear partial
differential equations to a set of linear partial differential
equations. By neglecting spatial variations these linear partial
differential equations can be reduced to a set of linear ordinary
differential equations. If the further assumption is made that the
system operation does not vary in time then these equations finally
reduce to a set of linear ordinary differential equations with
constant coefficients and the mathematical difficulties are
substantially reduced. In the case of black-box analysis, a similar
series of simplifications can be made, firstly by reducing the
concept of a general system to that of a linear system, then to
modelling
errors
linearization
errors
1
physical
partial
laws
differential
equations
(non-linear}
lumping
errors
ordinary
partial
differential reduction differential
equations
equations
(linear)
structure
a prion )
structural knowledge _
measurement knowledge
model
( o posteriori )
parameters
X
measurements
measurement
data
measurement
noise
Fig. 6
data
handling
e.g.
quantized
samples
e.g.
quantization
states
parameter
estimators
state
estimators
I
estimation errors,e.g. due to
truncation, model order,
data processing, etc.
Choice between an a priori and a posteriori model (Eykhoff, 1974).
that of a linear lumped system, and then to that of a linear lumped
time-invariant system. In this case, the general but intractable
mathematical specification that the output is obtained by an
unspecified transformation of input to output is reduced to the
mathematical specification that the output is given by the
convolution of the input and the impulse response of the system.
A consistent and comprehensive theory of linear hydrological
systems has been available for 10 years or more and is widely used
today in hydrological practice (Dooge, 1973). Recently, research
has been directed towards the relaxation of the restrictive
assumptions of linearity, lumping and time-invariance. The linear
black-box model can be generalized to the nonlinear case by the
use of Volterra series (Amorocho, 1973) or by the use of simple
nonlinear models (Natale & Todini, 1973; Napiorkowski, 1984). The
lumped models have been generalized to allow for spatial variability
and attempts made to cope with the increased complexity involved.
Waters of the Earth
163
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Relaxation of the assumption of lumping was applied to deterministic
models (Roche, 1965) and to stochastic models (Amorocho, 1979; Ord
& Rees, 1979; Clarke, 1979). Nash & Barsi (1983) have shown that
the modelling of total runoff as a linear time-invariant deviation
from the long-term seasonal trend can represent the total basin
response as well as most of the more complex nonlinear models of
total basin response.
Choosing
a mathematical
model
Hydrological practice would be improved if models were objectively
chosen on the basis of making the best use of the information
available and following some systematic procedure of selection and
verification. Some guidance can be obtained by comparing the time
interval constituting the record with the estimated memory length
of the hydrological system. In studies involving floods with a
large return period, this interval is clearly much longer than any
memory length which is significant for the conversion of rainfall to
runoff. Accordingly it would be appropriate to apply a timeindependent probabilistic model either directly to the flow record
or indirectly by combining the frequency study of the storm rainfall
with the use of a unit hydrograph or other deterministic operator.
It is possible that such an approach would be as reliable and less
time-consuming than the determination of a suitable stochastic
model for the simulation of the flow and the subsequent generation
of an extremely long record of synthetic flows in order to estimate
the extreme flood event.
At the other end of the time scale, the forecasting in real time
of downstream outflow when information is available either of
upstream inflow or of precipitation would appear to suggest the use
of a deterministic model whose memory time would be expressed in
hours or days for most basins. For the intermediate case where the
designer is concerned with the effects of monthly and annual flows,
the problem can probably best be tackled either by fitting a
stochastic model directly to the recorded flows and generating a
sequence of synthetic flows or by fitting the stochastic model to
the precipitation and generating a synthetic record of precipitation
which would be converted to flows by some deterministic model.
Once the class of model has been chosen, it remains to choose an
individual model within that class. In doing so, standard methods
can be used but should not be applied blindly. Thus, the chisquared test or the Kolmogorov-Smirnov test can be applied to
evaluate the fit of a probabilistic model but there is the
disadvantage that the less rare events that contribute heavily to
the criterion figure are those events least likely to satisfy the
assumption of independence. Stochastic models can be compared with
one another or with catchment data on an auto-correlogram which
reveals the auto-regressive structure (Kisiel, 1969; Yevjevich, 1972).
Dimensionless moments can be used to characterize the shape of unit
hydrographs (Nash, 1959) and also used to compare conceptual models
with catchment data, to compare conceptual models with one another
and to compare conceptual models with models based on continuum
mechanics (Dooge, 1973).
164
J.C.I. Dooge
L£VELS OF HYDROLOGICAL DESCRIPTION
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Time scales
and space
scales
Wheeler (1973, 1980) pictures the development of physics as a
staircase. Each step symbolizes the discovery of a new law; each
riser marks the attainment of conditions so extreme as to transcend
that law. In a sense, the laws are not violated - they are
transcended. Equally we would expect that there would be no unique
set of physical laws for describing the movement of water. Each
specialist in one of the physical sciences has his own concept of
what water is and how it behaves (Dooge, 1983). The hydrologist is
therefore in a dilemma as to what concepts and models to apply to
the movement of water at the scales of interest to him.
We have already seen that the special circumstances of our
position in the solar system and the closeness of our geophysical
temperatures to the triple point of water are responsible for the
vigour of the Earth's hydrological cycle. A comparison of the
melting points and boiling points of the hydrides of the elements in
the same group as oxygen reveals that the values for water are
highly anomalous. The physical chemist can explain these anomalies
on the basis that the water molecule is polar, i.e. highly nonisotropic, and that the numerous anomalous properties of water are
due to the resulting hydrogen bonding. When we come to continuum
mechanics , the complexity of the fourth order tensor involved in
the general relationship between stress and rate of strain is wiped
away by the sweeping assumption of isotropy in order to produce the
two-parameter constitutive equation of a Newtonian fluid and hence
the Navier-Stokes equations. Thus we have an apparent contradiction
as we move from the molecular level of description to the continuum
level. As we move further to geophysical fluid dynamics (whether
in relation to the atmosphere, the continental waters or the
oceans), we are faced with the problem of describing turbulent flow
in all its complexity. In the case of such turbulent flow the
Navier-Stokes equations do not cease to be true at the continuum
scale but they cease to be useful for the purpose of describing or
predicting such flow. Accordingly we have recourse to Reynolds
averaging of the various terms of the Navier-Stokes equation and
the parameterization of some of the average terms in the original
equation as functions of the mean variables in the new level of
description.
The difference between the concepts of water and models of water
behaviour used by the different sciences depends greatly on the
question of scale (Dooge, 1983; Klemes, 1983). An attempt is made
in Table 3 to give an approximate estimate of the significant
length scales and significant time scales for various approaches to
the study of water. This reveals the enormous gap between, on the
one hand, the space and time scales of clusters of water molecules
continually forming, breaking down and reforming with a half life
about 10
seconds, and on the other, the space and time scales
of a general circulation climate model in which the space scale is
over 100 km and the time scale of interest is 10 to 15 days. This
poses sharply the question whether the laws of hydrology should be
formulated on the basis of physical equations which can be verified
Waters of the Earth
165
in the laboratory (or on small plots) or whether an alternative
set of laws should be sought on the basin scale. The difficulty
of parameterization from point processes to a meaningful hydrological
scale is emphasized by the fact that the parameters governing the
key behaviour of unsaturated soil can vary by one or two orders of
magnitude in a 10 ha field (Dooge, 1982).
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Table 3
Significant length and time scales
Water molecule
Water cluster
Continuum point
Turbulent flow
Experimental plot
Basin module
Sub-basin
Basin
General circulation model
Length (m)
Time (s)
10~10
10"8
10 5
10~2
10
1Û2
103
10 4 -10 5
10s
10~13
10_u
10
10 2 -10 3
103
10 4
106
In attempting to tackle this problem we can learn not only from
experiences in the physical sciences but from experiences in the
social sciences as well (Whyte et al.,
1969; Pattee, 1973). Simon
(1962) in his essay on "The architecture of complexity" speaks of
complex systems as follows:
"In such systems, the whole is more than the sum of the
parts, not in an ultimate, metaphysical sense, but in
the important pragmatic sense that, given the properties
of the parts and the laws of their interaction, it is not
a trivial matter to infer properties of the whole."
Simon goes on to suggest that the existence of hierarchies in
physical , biological and social systems may be largely due to the
fact that these systems are the end product of an evolutionary
process. Since our hydrological basins are certainly the product
of the evolutionary processes of geomorphology, the arguments of
Simon would suggest that we should be able to find laws of regularity
at the basin scale as well as at the local scale.
Micro-models
of hydrological
processes
Physical hydrology is concerned with the application of such
physical principles as the conservation of mass and the conservation
of linear momentum to the various hydrological processes of
precipitation, infiltration, percolation, groundwater outflow,
channel flow etc. which were shown earlier on Fig.4. The equations
of continuity based on the conservation of mass are all linear but
the dynamic equations based on the conservation of linear momentum
are all nonlinear. These internal descriptions, i.e. local
descriptions at a point, can be illustrated for the case of the
flow in the unsaturated zone. The other hydrological processes
which represent a flux of water from one form of storage to another
166 j.c.l. Dooge
can be similarly treated.
For the case of one-dimensional vertical flow in unsaturated
soil we can write the equation of continuity for either percolation
downwards or capillary rise upwards:
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3w
^
3c
+ JÏ
„
= 0
(1)
where z is the elevation above a fixed datum, w(z,t) is the face
velocity measured vertically upwards and c(z,t) is the moisture
content on a volume basis.
Since the movement is very gradual, the acceleration terms of the
Navier-Stokes equation for laminar flow are neglected and the
remaining terms parameterized twice, firstly by integrating across
each individual pore and then by integrating across the whole
horizontal section of the soil column (Bear, 1972). This gives us
the Darcy equation for unsaturated flow:
w(z,t) = -K(c) — {<f)<z(t)}
(2)
dz
where K(c) is the unsaturated hydraulic conductivity of the soil in
the vertical direction and <f>(z,t) is the total moisture potential
which includes the effects of hydrodynamic pressure, gravity,
osmotic pressure, etc. A combination of equations (1) and (2) gives
us the single equation for the unsteady vertical movement of water
in an unsaturated soil:
ft =£[ K < C > £<•<*•*»]
which is
contains
a second
content.
flow can
3c
3t
(3)
commonly known as the Richards' equation. This equation
two dependent variables and hence must be supplemented by
equation relating the moisture potential to the water
If hysteresis is ignored, the equation for unsaturated
be written in terms of the moisture content c(z,t) as:
3 r_ , „ 3c -i
3z *3zJ
3E
3k
...
where D(c) is the soil diffusivity defined from the single soil
moisture characteristic linking soil suction (S) and moisture content
(c):
D(c) = K(c) •£- (S(c)}
dz
(5)
This is still a nonlinear partial differential equation and can only
be solved analytically for further simplifying assumptions.
During the period of high infiltration following a dry spell,
the capillary potential is far larger than all the other elements
of the total moisture potential. If these other elements including
gravity are neglected then it is possible to show that, for any
form of soil diffusivity relationship, D(c), the volume of
infiltration, F(t), will vary in time according to the relationship:
Waters of the Earth
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F(t) = A.t
2
+ Bt
167
(6)
where the parameters A (usually known as the sorptivity) and B
depend on the form of the soil diffusivity and on the initial
conditions. In order to obtain this solution for equation (4) it
is necessary to make the following simplifying assumptions: (a) the
soil profile is semi-infinite, i.e. the water table is at an
infinite depth below the soil surface; (b) the initial condition is
one of constant initial moisture content which is equivalent to
assuming that there is a constant initial rate of infiltration
equal to the hydraulic conductivity corresponding to this initial
moisture content; (c) the moisture content at the surface changes
instantly from the initial value to the saturation value. The
family of simplified solutions represented by equation (6) is in
fact a set of self-similar solutions of the problem since the
developing moisture profiles of the soil column will all be similar
to one another.
Mesoscale
models of basin
modules
When we attempt to move from the study of hydrological processes on
the microscale to the behaviour of components or modules in a basin,
considerable difficulty arises. As mentioned already, the effect
of soil properties can vary by one or two orders of magnitude in a
field as small as 10 ha. Because the underlying phenomena are
nonlinear, it is not possible to use the simple averaging process
in order to obtain representative values of parameters for use in
the same equation at the larger basin scale.
In modelling individual components of the hydrological cycle,
recourse is often made to conceptual models. The approach based
on conceptual models has not been used in the case of a soil
moisture component to the same extent as for other components, but
it is equally valid. It is known that following a dry period the
capacity of soil for infiltration is high but that during
precipitation this high rate will decline and ultimately reach a
constant value. If the rate of excess infiltration, f e (t), i.e.
the rate over and above the ultimate constant value, f c , is taken
as inversely proportional to some power of the volume of excess
infiltration then we have:
f e (t) = f(t) - fc = { F ( t ) I
f
t }g
<7)
Since the rate of infiltration is the derivative of the volume of
infiltration this conceptual model can readily be shown to result
in the volume of infiltration given by:
F(t) = {(c + 1) a t } 1 / ( C + 1 ) + fc.t
(8)
For c = 1 this is clearly of the same form as equation (6). The
value of the constants A = /2a and B = fc in the conceptual model
could either be based on some assumption in relation to D(c) and
K(c) or be derived by parameter optimization from records of actual
168 J.C.I. Dooge
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basin behaviour.
In this connection it is interesting to note that the theoretical
values for the parameters A and B in equation (6) do not varywidely for different assumptions in regard to the relationship of
soil diffusivity to moisture content. If we make the very simple
assumption that the soil above the wetting front is completely
saturated while the soil below this abrupt front is at its initial
constant value, (c 0 ) , then the value of the sorptivity in equation
(6) is given by:
A = [2(c s a t - c o ) K s a t S0]i
(9)
where S 0 is the soil suction corresponding to the initial constant
moisture content, c Q . In this particular model the diffusivity
takes the form of a delta function, being zero everywhere except at
the wetting front itself where it takes on an infinite value. If
we make the completely contrasting assumption that diffusivity and
conductivity are constant at all values of the moisture content
then the constant saturation at the surface will result in diffusion
throughout the soil column but with moisture content always below
saturation. For this case the sorptivity will be given by:
A = [(4/TT)(csat - c 0 ) K s a t S 0 ] z
(10)
A comparison of equations (9) and (10) indicates that the variation
in A due to differing simplifying assumptions is about 50% and will
be completely swamped by the spatial variability of an order of
magnitude in soil properties that occurs in practice over short
distances.
Links
between
descriptive
levels
It would appear that, in moving from a smaller scale description to
a larger scale description, there are a number of ways of linking
the two forms of description. If the phenomena described
at the smaller scale were truly linear, then averaging from an
infinitesimal volume to a finite volume would give the values of
the parameters at the larger scale in terms of the values of the
parameters on the smaller scale. Alternatively, a set of equations
or a conceptual model can be formulated at the larger scale which
is compatible with the physical principles governing the smaller
scale, and the parameters of the equations or of the conceptual
model can be determined from measurements made on the larger scale.
Finally, an attempt can be made to formulate a completely separate
set of laws for a larger scale on the basis of observations of that
scale and without reference to the lower scale description.
There are many examples in science of these different approaches.
One classical example is Einstein's work on Brownian motion
(Einstein, 1905). Another example is the derivation of the macroscopic form of the gas laws by a statistical treatment of the
kinetic behaviour on a microscale. In these examples and a number
of others we have a deterministic law on a microscale replaced by
statistical laws at intermediate scale and these in turn replaced by
a new and different deterministic law at the macroscale. It must
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Waters of the Earth
169
be accepted that the laws of the macroscale may be different from
the laws of the microscale and that there may be apparent contradictions between them. As was mentioned already, the physical
behaviour of water at equilibrium can only be explained on the basis
of the non-isotropic nature of the water molecule, whereas the laws
of water movement developed in continuum mechanics depend
fundamentally on the assumption that at the scale of interest water
behaves isotropically. In attempting to move from the scale of
continuum mechanics to the basin scale, we must be prepared for
similar disparities and apparent contradictions. The alternation
of deterministic and stochastic models of different scales suggests
that we might accept as a representation of the different scales in
water flow the lithograph by M.C.Escher of the Waterfall shown on
page 11 of Hofstadter (1979).
A comparison of the bulk parameters of a conceptual model with
the point parameters of models based on some special solutions in
continuum mechanics may give valuable information for the estimation
of bulk parameters and the use of conceptual models. In the last
section it was shown that the sorptivity is proportional to the
square root of the hydraulic conductivity ( K s a t ) , the square root of
the initial soil suction (S 0 ) and the square root of the initial
moisture deficit (c s a t - c ) . Similar comparison between macroscale
parameters and microscale parameters can be derived for linearized
models of groundwater flow (Kraijenhoff, 1958) and of open channel
flow (Dooge, 1973; Dooge et al.,
1982).
TOWARDS MACROSCALE MODELS OF BASIN RESPONSE
Empirical
laws of drainage
composition
If hydrological laws on the basin scale are to be discovered, then
they must depend on some uniformities or regularities other than
those on which the laws of fluid mechanics are based. The best
hope of finding a basis for these macroscale laws would appear to
be the regularities resulting from geomorphological equilibrium in
the formation of the drainage channel network and in the maintenance
of cross-sectional areas and slopes at various points in that
channel network. Such an approach may be said to have begun with
the paper by Horton (1945) entitled "Erosional development of
streams and their drainage basins: hydrophysical approach to
quantitative morphology." This work was extended by Strahler (1952)
and Shreve (1966). Smart (1972) has published a review of this
work on quantitative geomorphology.
The basic unit for the quantitative analysis of any drainage
network is the set of all channels above the given point in the
network. This point is known as the outlet and the points farthest
upstream in the channel network are known as sources. The point at
which two channels combine to form one is called a junction and it
is assumed that multiple junctions do not occur. An exterior link
is a segment of the channel network between a source and the first
downstream junction. An interior link is a segment of channel
network between two successive junctions between the outlet and
the first upstream junction. The magnitude of a link is the number
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170 J.C.I. Dooge
of sources upstream and the magnitude of a channel network is the
magnitude of its outlet link.
The first step towards developing empirical laws of drainage
composition is to classify the channels in the network on the basis
of their order. In the Strahler (1952) method of ordering, channels
that originate at a source are defined to be first order streams;
when two streams of a given order join, the order of the downstream
link is taken as one higher than the order of the two upstream links;
and when two streams of different order join, the downstream link
has the higher of the orders of the two combining streams. Horton's
laws of drainage composition state that, for a given channel network,
the number of streams of successive orders and the mean lengths of
successive orders can both be approximately represented by geometric
progressions. Thus for the stream numbers we would have:
Nw-1
_ï_± =
RB
(11)
"w
where Rg is known as the bifurcation ratio and is usually between 3
and 5. The law of stream lengths states that:
Lw
7= RL
(12)
^w-1
where R L is the stream ratio which is usually between 1.5 and 3.5.
These results indicate a degree of regularity in natural channel
networks which is somewhat surprising. Schumm (1956) proposed a
further geometric law for drainage areas:
Aw
Vi
(13)
= RA
where the area ratio, R^, normally falls in the range 3 to 6. A
given basin area is examined for Horton stability by plotting the
numbers of streams of a given order against the appropriate order.
An example of such a Horton plotting is shown on Fig.7 for a basin
of order 6 and magnitude 1181 (Morisawa, 1962).
Random theory
of network
formation
The next important step in relation to Horton's laws (which were
formulated on a deterministic basis) was the work of Shreve who
introduced the idea of a random population of channel networks and
studied the properties of this population. Shreve (1966) defined
a topologically random population as one in which all topological
distinct channel networks with a given number of sources are
equally likely. He developed formulae for the relative probability
of different sets of stream numbers in such a population and showed
that the most probable networks conform to Horton's law of stream
numbers. Thus it can be shown that for five sources there are 14
topologically distinct channel networks. Eight of these are second
order (five first order streams and one second order stream) and
six of them are third order streams. It can be shown that in
random samples of streams from an infinite topologically random
Waters of the Earth
—
1000
1
1
V
DADDY'S CREEK TENN
MORISAWA (1962)
N.
1
1
1
/
»
A/
Ri=4.7l
100
171
).
R B = 4-10
RL=2-18
10
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1
1
1
3
2
l
4
S
i
6
ORDER—»
Fig. 7
Example of a Horton plotting fora basin of order 6 and magnitude 1181
(Morisawa, 1962).
network the expected value of the number of streams of successive
orders approaches a geometric series with the ratio of 4 as size of
the sample increases (Gupta & Waymire, 1983).
The random model can be used to predict individual stream length
ratios (A w ) in contrast to the overall bifurcation ratio (Rg) shown
on Fig.7. This theory is based on the assumption that the interior
link lengths are independent random variables drawn from a common
distribution. Figure 8 shows the comparison of prediction and
observation for a seventh order drainage basin with 5156 sources
(Smart, 1972). The predicted values follow the trends of the
observed ones whereas Horton's deterministic law of stream lengths
would be represented by a horizontal line corresponding to a value
of R L = 2.36.
3
4
5
ORDER Ul
Fig. 8 Observed and predicted results of stream length ratios vs. bifurcation ratio for
a seventh order basin with 5156 sources (Smart, 1972).
Geomorphic
unit
hydrograph
The consideration that regularity in the shapes of unit hydrographs
should reflect the laws of drainage composition led to the formulation of the question: what is the runoff from a pulse input of
effective precipitation to a Horton or a Shreve network? This
problem was tackled by Rodriguez-Iturbe & Valdez (1979) who treated
the progression of a drop of water from one stream link to another
as a state transition for which a transition probability matrix
172 J.C.I. Dooge
exists. The waiting time distributions were taken to be exponential
and a series of computer runs was made to derive the unit hydrograph
for networks with different values of Rg, R L and R^. As a result of
this numerical experimentation, relationships between the properties
of the unit hydrograph and the properties of the basin were derived.
One such expression gave:
• 55
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q
P
t =0.58
P
(14)
R
l A.
where q is the peak of the unit hydrograph and t p is the time to
peak.
This work has been extended by Gupta et al.
(1980) and Gupta &
Waymire (1983) who showed that the unit hydrograph can be determined
on the basis of the basin characteristics with only one unknown
parameter to be determined from historical records. This approach
shows great promise for future advances in basin hydrology.
Climate,
vegetation
and
hydrology
An alternative to working from a smaller scale upwards towards the
scale of interest is to start with a larger scale on a long-term
equilibrium and attempt to work downwards to the space scales and
time scales of interest. Such an approach is made by Eagleson who
examined the average annual water balance of basins on the basis of
simplified assumptions of the hydrological processes (Eagleson,
1978). This one-dimensional water balance relates five surface
parameters: the vegetation canopy density, the species-dependent
plant water use coefficient, the effective porosity of the soil,
the saturated hydraulic conductivity of the soil and the disconnectedness index of the soil. Eagleson argued that under conditions of
limited water, there will be a tendency to limit water demand
through the adjustment of the canopy density and the plant species
so that the soil moisture is maximized. This gives the curve for
vegetal equilibrium which appears towards the left of Fig.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
VEGETAL CANOPY DENSITY
Fig. 9
Relationships between potential transpiration efficiency and vegetal canopy
density (Eagleson, 1982).
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Waters of the Earth
173
(Eagleson, 1982). In climates where the limitation is one of
energy rather than of water there will be a tendency to maximize
the biomass for any given species, and this condition is shown by
the curve in the upper part of Fig.9. A short-term equilibrium is
sought in terms of a minimum water demand stress and a long-term
equilibrium is sought in the soil-vegetation system such that the
density of the minimum stress canopy is maximized. For a number
of basins which have been studied, these hypotheses as shown in Fig.9
seem to be well verified (Eagleson & Tellers, 1982).
The material reviewed above offers hope for the development of a
consistent theory of macroscale hydrology. Such a theory would
probably owe more to the concepts of Hofstadter (1979) and the
mathematics of Mandelbrot (1982) than to our current concepts and
techniques in either physical hydrology or systems hydrology.
REFERENCES
Amorocho, J. (1973) Nonlinear hydrologie analysis. In: Advances
in
Hydroscience,
vol. 9 (ed. by V.T.Chow), 203-251. Academic Press,
New York.
Amorocho, J. (1979) Spatially distributed variables in hydrologie
modelling. In: The Mathematics of Hydrology and Water
Resources
(ed. by E.H.Lloyd, T.O'Donnell & J.C.Wilkinson), 87-94. Academic
Press, London.
Baumgartner, A. & Reichel, E. (1975) World Water Balance.
Olderbourg, Munich.
Bear, J. (1972) Dynamics of Fluids in Porous Media.
Elsevier,
Amsterdam.
Borchardt, L. (1906) Nilmesser und Nilstandsmarken.
Preussische
Akademie der Wissenschaften
Philosophisch-Historische
Abhandungen
nicht zur Akademie Gehoriger Gelehrter
no.1,
Berlin.
CBI (1951) Irrigation
in India through the Ages.
Central Board of
Irrigation, New Delhi.
Clarke, R.T. (1979) Multivariate synthetic hydrology: a theoretical
viewpoint. In: The Mathematics
of Hydrology and Water
Resources
(ed. by E.H.Lloyd, T.O'Donnell & J.C.Wilkinson), 119-138.
Academic Press, London.
Dalton, J. (1802) Experiments and observations to determine whether
the quantity of rainfall and dew is equal to the quantity of
water carried off by the rivers and raised by evaporation, with
an enquiry into the origin of springs. Mem. Proc. Lit.
Phil.
Soc. Manchester
5, part 2, 346-372.
Degens, E.T. (editor) (1982) Transport of carbon and minerals in
major world rivers. Mitt.
Geologisch-Palaontologischen
Institut
der Universitat
Hamburg, Hefte 52.
Dooge, J.C.I. (1959) Un bilan hydrologique aux XVIIe siècle. La
Houille Blanche no.6.
Dooge, J.C.I. (1973) Linear theory of hydrologie systems.
Tech.
Bull. no. 1468, US Agricultural
Research Service
Washington.
Dooge, J.C.I. (1981) Model structure and classification. In:
Logistics
and Benefits
of Using Mathematical
Models of
Hydrologie
and Water Resource
Systems
(ed. by A.J.Askew, F.Greco
& J.Kindler), 1-25. Pergamon Press, Oxford, UK.
Downloaded by [University of Minnesota Libraries, Twin Cities] at 13:41 20 February 2016
174 J.C.I. Dooge
Dooge, J.C.I. (1982) The parametrization of hydrologie processes.
In: Land Surface
Processes
in Atmospheric
General
Circulation
Models (ed. by P.S.Eagleson), 243-288. Cambridge University Press,
Cambridge, UK.
Dooge, J.C.I. (1983) On the study of water. Hydrol.
Sci.
J. 28 (1),
23-48.
Dooge, J.C.I. , Costin, A.B. & Finkel, L.H.J. (1973) Man's
Influence
on the Hydrological
Cycle.
Irrigation and Drainage Paper no. 17,
FAO, Rome.
Dooge, J.C.I., Strupczewski, W.G. & Napiorkowski, J.J. (1982)
Hydrodynamic derivation of storage parameters of the Muskingum
model. J. Hydrol.
54, 371-387.
Eagleson, P.S. (1978) Climate, soil and vegetation. Wat. Resour.
Res.
14 (5), 705-776.
Eagleson, P.S. (1982) Ecological optimality in water-limited natural
soil-vegetation systems. 1. Theory and hypothesis. Wat.
Resour.
Res.
18 (2), 325-340.
Eagleson, P.S. & Tellers, T.T. (1982) Ecological optimality
in water-limited natural soil-vegetation systems. 2.
Tests and applications. Wat. Resour.
Res.
18 (2),
341-354.
Einstein, H.A. (1905) Investigations
on the Theory of the
Brownian
Movement.
Translated into English 1926. Methuen.
Eykhoff, P. (1974)
Systems
Identification.
Wiley.
Golubev, G.N. (1983) Economic activity, water resources in the
environment: a challenge for hydrology. Hydrol.
Sci.
J. 28 (1),
57-75.
Goody, R.N. & Walker, J.C.G. (1972) Atmospheres.
Prentice-hall,
Englewood Cliffs, New Jersey, USA.
Gupta, V.K., Waymire, E. & Wang, C.T. (1980) A representation of an
instantaneous unit hydrograph from geomorphology. Wat.
Resour.
Res.
16 (5), 855-862.
Gupta, V.K. & Waymire, E. (1983) On the formulation of an analytical
approach to hydrologie response and similarity at the basin
scale. J. Hydrol.
65, 95-123.
Halley, E. (1691) On the circulation of the vapours of the sea and
the origin of springs. Phil.
Trans,
no. 192 (January/February
1691), vol. 17, 468-473.
Hofstadter, D.R. (1979) Godel, Escher,
Bach: An Eternal
Golden
Braid.
Penguin.
Holdgate, M.W., Kassas, M. & White, G.F. (1982) The World
Environment
1972-1982.
Tycooly, Dublin.
Horton, R.E. (1945) Erosional development of streams and their
drainage basins: hydrolophysical approach to quantitative
morphology. Geol. Soc. Am. Bull.
56, 275-370.
IAHS/UNESCO (1972) Symposium on World Water Balance
(Proc.
Reading Symp. 1970), vols. I, II and III. IAHS Pubis 92,
93 and 94.
IAHS (1974) Mathematical
Models in Hydrology
(Proc. Warsaw Symp.,
July 1971), vols 1, 2 and 3. IAHS Pubis 100, 101 and 102.
IAHS (1975) Application
of Mathematical
Models in Hydrology
and
Water Resources
(Proc. Bratislava Symp., September 1975).
IAHS Publ. no. 115.
IAHS (1978) Modelling
the Water Quality
of the Hydrological
Cycle
Downloaded by [University of Minnesota Libraries, Twin Cities] at 13:41 20 February 2016
Waters of the Earth
175
(Proc. Baden Symp., September 1978). IAHS Publ. no. 125.
Kalinin, G.P. (1968) Problemy
Global'noi
Gidrologii.
Gidrometrologischeskoe Izdatel*stvo, Leningrad. Translated into English as
Global Hydrology.
Israel Programme for Scientific Translations,
Jerusalem, 1971.
Kisiel, C.C. (1969) Time series analysis of hydrologie data. In:
Advances
in Hydroscience,
vol. 5, 1-119. Academic Press, New
York.
Klemes, V. (1983) Conceptualization and scale in hydrology. J.
Hydrol.
65, 1-23.
Kondratyev, K.Y. & Hunt, G.E. (1982) Weather and Climate
on
Planets.
Pergamon Press, Oxford.
Korzun, V.I. et al.
(1974) Mirovoi
Vodnyi Balans i Vodnye
Resursy
Zemlii.
Me.iduvedomstvennyi Komitet SSSR po Mejdunarodnomu
Gidrologicheskomu Desiatiletiu. Gidrometeoizdat, Leningrad.
Translated into English as World Water Balance
and Water
Resources
of the Earth.
Studies and Reports in Hydrology no. 25, UNESCO,
Paris, 1978.
Mandelbrot, B.B. (1982) The Fractal Geometry of Nature. W.H.Freeman,
San Francisco.
Meier, M.F. (1983) Snow and ice in a changing hydrological world.
Hydrol.
Sci.
J. 28 (1), 3-22.
Morisawa, M.E. (1962) Quantitative geomorphology of some watersheds
in the Appalachians. Geol. Soc. Am. Bull.
73, 1025-1046.
Napiorkowski, J.J. (1984) The optimisation of a third order surface
runoff model. In: Scientific
Procedures
Applied
to the
Planning,
Design and Management of Water Resources
Systems
(Proc. Hamburg
Symp., August 1983). IAHS Publ. no. 147 (in press).
Nash, J.E. (1959) Systematic determination of unit hydrograph
parameters. J. Geophys.
Res. 64 (1), 111-115.
Nash, J.E. & Barsi, B.I. (1983) A hybrid model for flow forecasting
on large catchments. J. Hydrol.
65, 125-137.
Natale, L. & Todini, E. (1973) Black-box identification of a flood
wave propagation linear model. In: Proceedings
of
Istanbul
Congress
vol. 5, 165-168. IAHR.
O'Kane, J.P. (1982) Statistical mass balances for estuaries.
Workshop on Estuarine Processes: An Application to the Tagus
Estuary (Lisbon, December 1982).
Ord, K. & Rees, M. (1979) Spatial processes: recent developments
with applications to hydrology. In: The Mathmatics
of
Hydrology
and Water Resources
(ed. by E.H.Lloyd, T.O'Donnel & J.C.Wilkinson),
95-118. Academic Press, London.
Pattee, H.H. (1973) Hierarchy
Theory.
Braziller.
Pereira, H.C. (1972) A guide to policies for the safe development of
land and water resources. In: Status
and Trends of Research
in
Hydrology.
UNESCO, Paris.
Perrault, P. (1674) De l'Origine
des Fontaines.
Paris. Translated
with commentary by A.Larocque. Hafner, New York, 1967.
Prinn, R.G. (1982) Origin and evolution of planetary atmospheres:
an introduction to the problem. Planet.
Space Sci.
30 (8).
Rasool, S.I. & de Bergh, C. (1970) The runaway greenhouse and the
accumulation of CO2 in the Venus atmosphere. Nature
226, 1037-1039.
Roche, M. (1965) Point de vue matriciel sur un opérateur linéaire
de transformation pluies-débits. Cah. 0RST0M no.
2.
Downloaded by [University of Minnesota Libraries, Twin Cities] at 13:41 20 February 2016
176 J.C.I. Dooge
Rodrïguez-Iturbe, I. & Valdez, J.B. (1979) The geomorphic structure
of hydrologie response. Wat. Resour.
Res. 15 (6), 1409-1420.
Rubey, W.W. (1951) Geological history of sea water. Bull.
Geol.
Soc. Am. Reprinted as pages 1-63 of The Origin
and Evolution
of
Atmospheres
and Oceans (ed. by P.J.Brancazio & A.E.W.Cameron).
Wiley, 1964.
Schumm, S.A. (1956) Evolution of drainage systems and slopes in
badlands at Perth Amboy, New Jersey. Geol. Soc. Am. Bull.
67,
597-646.
Schumm, S.A. (1977) The Fluvial
System.
Wiley.
SCOPE (1981) Some Perspectives
of the Major Biogeochemical
Cycles.
(SCOPE Publ. no. 17). Wiley, Chichester.
Shreve, R.L. (1966) Statistical laws of stream numbers. J. Geol.
75, 179-186.
Simon, H.A. (1962) The architecture of complexity. Proc. Am.
Phil.
Soc. 106, 467-482.
Smart, J.S. (1972) Channel networks. In: Advances
in
Hydroscience,
vol. 8, 305-346. Academic Press, New York.
Strahler, A.N. (1952) Hypsometric analysis of erosional topography.
Geol. Soc. Am. Bull.
63, 1117-1142.
Tixeront, T.J. (1974) L'hydrologie en France au dix-septième siècle.
In: Three Centuries
of Scientific
Hydrology,
24-39. UNESC0-WM0IAHS, Paris.
Wheeler, J.A. (1973) From relativity to mutability. In: The
Physicist's
Conception
of Nature
(ed. by J.Mehra), 202-247.
Reidel, Dordrecht.
Wheeler, J.A. (1980) Beyond the black hole. In: Some
Strangeness
in the Proportion
(ed. by H.Woolf) , 341-375. Addison-Wesley.
Whyte, L., Wilson, A. & Wilson, R. (1969) Hierarchial
Structures.
Elsevier, Amsterdam.
Wittfogel, K.A. (1956) Hydraulic civilizations. In: Man's Role
in
Changing
the Face of the Earth
(ed. by W.M.Thomas). Wenner-Gren
Foundation, Chicago.
Yevjevich, V. (1972) Stochastic
Processes
in Hydrology.
Water
Resources Publications, Fort Collins, Colorado, USA.