Snowmelt runoff forecasts -theoretical problems
E.G. Popov
HydrometeoroZogicaZ Research Centre of the USSR, Moscow, USSR
ABSTRACT: This report is concerned with the theoretical concepts of
long-range forecasting of seasonal snowmelt runoff, the role of separate factors, and practical forecasting techniques.
RESUME: Ce rapport est consacré aux fondements théoriques de
prévisions du temps 2 longue échéance du ruissellement de fonte
saisonnier, au rôle des facteurs différents et aux méthodes pratiques
des prévisions.
INTRODUCTION
In northern countries and in mountainous areas where winter precipitation is accumulated in the form of snow, melting during the
warm season is one of the main sources ot river feed. Spring floods
resulting from a relatively brief period of snowmelt are characteristic features of the regime of lowland rivers. In mountain
catchment areas, the melting season spreads over several months,
causing as lengthy a period of high water. The need for prediction
of spring flood crest stages arose long ago and defined, to a great
extent, the requirements for observations and investigations at an
early stage in the development of hydrology.
First attempts at long-range forecasting of spring flood crest
stages go back as far as the early twenties. The possibilities for
such forecasts were limited, however, and forecasting techniques were
crude due to lack of knowledge and observational data. To that time,
also, belong the first attempts of water supply forecasting for mountain rivers for thc purposes of irrigation.
Since then, hydrology has made great progress. Requirements of
the economy stimulated development of the study of the urocesses of
snowmelt and rainfall runoff formation and, in particular, of the
theory and practical procedures for runoff forecasting. Requirements
for long-range forecasts of seasonal flow are increasing from year to
year with the development of irrigation, construction of hydroelectric power plants, and reservoirs for streamflow control. Forecasts for such characteristics of spring floods as volume, time distribution of flow, and peak discharges are most important. The
warnings of crest stages for non-controlled rivers also remain very
important.
In accordance with the program of this Symposium, the purpose of
this paper is to review the existing theoretical concepts of snowmelt
runoff formation, the principles and possibilities for 1 ong-range
flood flow forecasting, and the prospects of its further development.
The processes of snowmelt runoff formation in lowland basins
differ considerably from those in mountainous terrain. The feasibility of hydromctcorological observations and measurements in the
lowlands also differs'from that for those in the mountains. Specific
829
features of runoff formation processes and lack of observational data
fully explain why the principles of snowmelt flow forecasting are at
present less developed for mountain rivers than for lowland rivers.
FACTORS OF SNOWMELT RUNOFF AND POSSIBILITIES FOR ITS FORECASTING
The volume of snowmelt runoff in any basin depends on three main
factors: (1) water equivalent of snow cover, ( 2 ) water absorbability
of a river basin, and (3) evaporation of meltwater during the flood
period. All other factors being equal, the volume of seasonal runoff
increases with the increase of the water storage accumulated in the
form of snowpack, and, vice versa, decreases with the increase of
water absorbability of a basin and evaporation. The first two factors are subject to considerable variations from year to year. The
variability of these factors defines, in the final analysis, variations in the seasonal snowmelt runoff. Unlike the first two factors,
meltwater losses by evaporation are comparatively small when the
melting period is brief, and not so changeable from year to year.
Thus, in many cases this factor does not play as important a role in
runoff variations as do the first two factors.
Of the three factors above, only the water equivalent o€ snow
cover can be estimated with greater or lesser accuracy on the basis
of direct measurements. Evaporation losses can be practically estimated only ayproximately using meteorological data. The ability of
a river basin to absorb a greater or smaller amount of meltwater depends on a number of factors and cannot be measured directly.
The amount of water absorbed in a given year by a basin can be
determined only from the water balance equation when water equivalent
of snow cover (S), total runoff (R), and water losses by evaporation
(E) are known:
A
= S-R-E
(11
It is obvious that the accuracy of determining A from equation (1)
depends on the accuracy of estimating the other variables.
Factors that define the water absorbability of river basins will
be discussed later. Here, I would just like to note that they include both relatively constant physical characteristics of basins,
such as topography, soils, vegetation, and some others, and variable
factors, such as soil moisture content, temperature, and depth of
freezing. These factors depend, in their turn, on the antecedent
meteorological conditions. It is these factors that ultimately define the variations in the water absorbability of river basins from
year to year.
Theoretically, as it appears from the water balance equation,
the possibility of long-range forecasting of snowmelt runoff depends
on the feasibility of estimating well in advance the water absorbability of river basins.
In other words, as the water storage in the snowpack is obtained
from measurements, the problem of a long-range forecast for the seasonal smowmelt runoff involves estimating the amount of meltwater to
be absorbed by soil, retained on the surface of the catchment area,
and evaporated during the period of snowmelt. As to the period
covered by such forecasts, it is equal to the difference in time between the date of issuing the forecast plus the duration of flood in
a given basin. From the water balance equation, it also follows that
with the same accuracy of the areal estimation of the snow cover
830
water equivalent, the less variability there is in meltwater losses
from year to year, the better conditions are for long-range forecasting. To determine the values and variability of the meltwater
losses in any given river basin, it is necessary to have the historical data on snow cover and streamflow for a sufficient number of
years.
Under natural conditions snow cover is seldom the only source of
river feed. In many regions, precipitation occurring during and
after the snowmelt period contributes significantly to seasonal runoff. The problem of long-range forecasting is complicated by this
fact both in theoretical and practical aspects. All other conditions
being equal, the reliability of seasonal runoff forecasts decreases,
as the contribution of liquid precipitation increases,because its
amount is unknown at the moment of the prediction and can practically
be taken into account only statistically.
As is known from relevant studies, the volume of snowmelt flow
depends on rate and/or duration of snowmelt to a lesser extent than
such parameters as time distribution and peak discharge. Nevertheless, both parameters are related to the volume of runoff. It is the
existence of such a relationship that in principle makes possible a
long-range prediction of those elements.
WATER ABSORPTION AND LOSSES
Absorption of meltwater in a river basin consists of infiltration and surface retention. The ratio between infiltration and
surface retention can change within wide limits depending on soil
permeability and topography of river basins. ïhe subdivision of
water losses into infiltration and surface retention is theoretically
important for the modelling of runoff and its prediction. Nonuniform areal distribution of surface retention capacity and infiltration rate is the important factor on which depends the nonsimultaneous beginning of overland runoff and variability of the
active area.
InfiZtratim
Infiltration is the descending movement of water through the
pores of the soil. The infiltration rate is defined as the amount of
water infiltrated into the soil during a unit of time and is expressed in millimetres per minute or millimetres per hour. Total infiltration which is calculated as the product of infiltration rate
and time, depends on the duration of snowmelt.
Infiltration of water into frozen soil is a very complicated
process. Besides the gravitational and molecular forces, thermophysical processes of changes in the aggregative state of soil moisture, and hence of soil porosity, have a pronounced effect on the
infiltration rate. Although a number of aspects of infiltration into
the frozen soil are not yet sufficiently investigated, the experience
gained in field and laboratory studies gives not only an idea of the
order of the magnitude of the infiltration rate for different types
of soil, but also of the factors on which the soil permeability to
water depends. The most important of these factors are: soil moisture content, temperature, and depth of freezing. The structure and
type of soil are also highly important. The clogging of soil occurring under the action of autumn rainfalls plays an essential role in
changing the permeability of some types of soil.
831
Experiments show that the permeability of frozen soils changes
from year to year within wide limits. A well-moistened frozen soil
at n temperature of 2-3" below zero becomes practically impermeable
to meltwater. Conversely, when it is dry it remains highly permeable. Even impermeable frozen soil can retain from 5 to 15 mm of
meltwater due to the presence of free non-capillary pores in its
upper layer [l].
These losses, however, should be attributed to
surface retention.
Approximate estimates for steppe basins with chernozem soils
show that average infiltration rate for the melting period ranges
from 0.5-1.0 to 12-14 mm per day depending on the antecedent soil
moisture content [3].
Surface Re tention
Before reaching the channel network, meltwater should fill a
great number of various cavities along its way including large noncapillary pores of the upper layer of soil, a variety of depressions
of microrelief, and larger depressions. The losses of water from the
filling of these cavities are inevitable and do not depend on the
rate or duration of snowmelt. The volume of these losses depends on
topography, type of soil, and the area of non-outflow lakes and
marshes.
The order of magnitude of surface retention can be illustrated
by the following figures. For the steppe basins of the European part
of the USSR, the surface retention makes up approximately 20-25 mm,
and in the northern forest zone it increases to 60-70 mm due to great
porosity of the upper soil layer, debris layer, and non-outflow
marshes capacity. The non-uniform areal distribution of retention
capacity over the catchment areas plays an important role in the surface runoff formation. As was already noted, this non-uniformity
governs the change in the active area.
The capacity of a single depression (c) and the depth of water
needed to fill it (D) are related as follows:
c
= $D
(2)
where I#I denotes the relative area of a depression catchment. Variables c and D are expressed here in millimetres, ie., in the form of
a layer of water over the whole area of the basin.
Equation (2) shows that if the depth of meltwater is less than
or equal to D (M 5 D) the area 4 cannot yield runoff.
Even a relatively small surface capacity can produce a great effect on the active area if infiltration occurs. In this case, the
amount of meltwater needed for filling up the surface capacity is expressed by the following equation:
M =
D
1 - 1
(31
m
where i and m are the rates of infiltration and snowmelt respectively.
Assume, for example, that the surfacc capacity of half the area
of a certain basin (9 =.0.5) is only 5 mm and is reached at the water
= 0.9, this capacity is filled up at the
layer D = 10 mm. With
meltwater depth o f 100 mm. If the water equivalent of the snow cover
is less or equal to this value (M 5 100 mm), there will be no runoff?
from this area even though the rate of snowmelt exceeds that of
,
infiltration.
I
&
832
m o Types of Water Absorption
From the theoretical point of view, two types of water absorption are to be distinguished in accordance with the degree of soil
permeability to water:
(1) The capacity type, where the soil in a river basin is impermeable, or its permeability is so high that the infiltration rate is
equal to the rate of snowmelt. Runoff is possible only from areas
where the water equivalent of the snow cover exceeds the capacity of
surface depressions or of surface depressions and soil together.
This type of water absorption is characteristic for loose soils with
shallow bedding of the impermeable layer, for example, for forest
podzol and for basins with impermeable soil due to high moisture content and deep freezing. In this case, the rate of snowmelt has no
effect on the total runoff.
(2) Capacity-infiltration type, where surface runoff is formed by
the excess of the rate of snowmelt over the rate of infiltration and
is possible in areas where the difference between the water equivalent of the snow cover and the total infiltration exceeds the surface
capacity. The relation between the rates of infiltration and snowmelt should affect considerably the total runoff.
In reality, hardly any fairly large river basins exist where
only one type of water absorption occurs. A combination of both
types is most probable, although, certainly, there can be basins
where one or the other type predominates.
INTEGRAL EQUATIONS OF TOTAL RUNOFF
A theoretical grounding and a mathematical description of the
general relationship between total runoff and the water equivalent of
the snow cover are of great importance for analytical and practical
purposes. As they are necessary for the development of practical
forecasting techniques and procedures, they also make it possible to
evaluate the role of individual factors, such as the depression storage, the rates of snowmelt and infiltration, and finally to estimate
the probable accuracy of forecasting. An exact mathematical description of such a relationship presents great difficulties. The main
difficulties consist of lack of knowledge about infiltration, great
irregularities in snow accumulation, snowmelt and water absorption
processes, and in the essential role that chance plays in these processes. These difficulties allow only an approximate approach to the
mathematical description using simplified models of a river basin.
The following integral equations of runoff based on the multicapacity concept were derived by the author [3, 61 for lowland
basins.
(1) For the capacity type of water absorption, the equation of the
total snowmelt runoff (in mm) is of the form
In this equation S is the water equivalent of the snow cover in
mm, @(D,u) is a differential distribution function of the active area
where D is the depth of the water needed for meeting the capacity and
u is a variable that characterizes the change in the capacity depending on the antecedent soil moisture conditions in a basin.
8 33
The first integral in equation (4) represents the active area
ie., the relative area from which, with given S, runoff is possible.
This integral distribution function should meet the following
conditi on :
S-tDmax
= i$(D,u)dD
O(S,u)
+
1
(53
The second integral in equation (4) represents the amount of
water (in mm) retained in the active area, which equals the retention
capacity of this area. This integral function should meet the fol1owing condition:
m)
is the retention capacity of the whole basin in mm.
where
Equation (4) shows that for the capacity type of water absorption, total snowmelt runoff is a function of snow cover water equivalent and the antecedent moisture conditions, if the change of the
latter from year to year results in essential changing of the retention capacity of the basin. This equation may be presented in
another form:
R =
S
I@
S,u)dS
17)
(2) For the capacity-infiltration type of water absorption, the runoff equation is more complicated.
In this equation I denotes the total infiltration in mm calculated in the general case from
T
I
= li(u,t)dt
(9)
The excess of the water equivalent of the snow cover over the
total infiltration can be presented in the following form:
s-
I
=
i
m
(1- -)S
(10)
where i and m denote the rates of infiltration and snowmelt,
respectively .
Equations (4) and (8) are not limited by any assumptions regarding the form of retention capacity distribution.
i
In the general case the relationships R = f(S,u) and R = f(Sare nonlinear.
Equation (4) can be transformed so that the areal distribution
of snowpack over the basin could be taken into account.
3
In this equation, the first integral denotes a relative water
equivalent of the snow cover on an active area (y= S/S), and the
834
second integral denotes a relative value of the capacity on the same
area (E= C/C).
P RACI'I CAL FORE CASTING TECHNIQUES
The integration of equation (8), using simplest one-parameter
distributions of retention capacity, makes it possible to obtain the
following simple formulae
R = (1- -I)S
S
- C(u)
(12)
or
These equations are widely used in long-range forecasting of spring
flood runoff for lowland rivers to obtain empirical water balance
relationships of the following type R = f(x,u), where R denotes the
total flood runoff excepting the base flow, and x = S i- P is the sum
of the water equivalent of the snow cover and the precipitation for
+he period of flood formation.
For river basins where the capacity type of water absorption
prevails, infiltration can be assumed equal to zero (I = O). The
parameter C(u) is obtained empirically for a given basin using historical data. For a basin in which the capacity-infiltration type
of water absorption prevails, not only parameter C(u) but also the
values of total infiltration are obtained empirically.
The relationship of the relative infiltration I/x to the antecedent moistening index (u) and the depth of soil freezing (1) is
plotted I/x = f(u,l).
The difference between observed precipitation and computed evapo
transpiration over a relatively long (2-4 months) period, before the
formation of snow cover is most often used as the antecedent moistening index. In the humid forest zone the runoff caused by late
rainfalls is also used as an index to the basin water absorbability.
In order to use equation (11) in forecasting practices, it is
necessary to know functions of areal distribution of snow cover and
those of surface capacities, and to possess methods for estimating
the inactive area. One possible method for estimating such an area
was suggested by V.D. Komarov El].
For this purpose, he used the
interrelated distributions of snow cover and depth of frost in the
soil. For the area from which practically all meltwater is absorbed,
Komarov recommends using the section in which the depth of soil
freezing is less than 20 cm.
The possibilities for obtaining empirical water balance relationships and their reliability as a tool of runoff forecasting depend on the availability and accuracy of observational data. Unfortunately, in many cases lack of data, or insufficient accuracy, limit
the possibilities of elaboration of the reliable forecasting methods.
As follows from the analysis of the integral runoff equations
above,it is necessary to know the values of water losses for sufficiently long historical period. Because they are computed from the
water balance equation, to successfully develop forecasting techniques, completely accurate data on the water equivalent of the snow
835
cover and on the runoff itself are required. The following formula
may be used for the estimation of this accuracy.
In this formula ß = sx/crx = S R / ~ Rdenotes the relative standard error
of areal estimates of snow cover water equivalent and precipitation
and that of the estimation of snowmelt flood flow; CI = SL/OL is the
desirable relative standard error of the estimation of water losses;
ox, UR, U L are the standard deviations of the snow cover water equivalent, runoff, and water losses, respectively; y x is
~ the correlation
coefficient between x and R.
The accuracy of snowmelt runoff forecasts depends not only on
the accuracy of water equivalent of snow cover estimation but also on
the representativeness of the indices of water absorbability of river
basins and on variability of precipitation during the forecast period. The latter are assumed to be normal as no reliable methods for
long-range quantitative forecasts of precipitation are as yet available.
The possibilities for long-range forecasting of time distribution, peak discharges, and crest stages of snowmelt floods are more
limited than the possibilities for total runoff forecasting. The
reason is that those elements depend to a considerable extent on
weather conditions that remain unknown at the moment of preparing
the long-range forecast.
As the first approximation, the unit hydrograph obtained empirically for normal duration of snowmelt may be used for the prediction
of time distribution of flood flow using the forecasted value of
total runoff. The unit graph method makes it possible to determine
peak discharges and crest stages as well. The correlation between
these two parameters and the total runoff can also be used for their
prediction. Long-range forecasts of seasonal flow and lowland rivers
are usually issued before the beginning of snowmelt when snow accumulation approaches its maximum. In all cases where the available historical data make it possible to carry out statistical estimation of
forecasting errors, forecasts should be issued with an indication of
probable errors [3, 61.
SEASONAL SNOWMELT FLOW FORECASTS FOR MOUNTAIN RIVERS
Considerable differences in altitude, and therefore in climate,
soil, and vegetative conditions, are specific features of highland
river basins. This is why one of the most important characteristics
of a mountain basin is its area-altitude distribution.
Because of a great time difference between accumulation of seasonal snowpack and its melting, the forecast of seasonal flow may be
prepared several months in advance. The possibilities €or forecasting
are most favourable for those areas where rainfall during the melting period is scarce with respect to accumulated winter precipitation,
and the total heat inflow is sufficient every year for complete melting of the seasonal snowpack. Conditions for runoff in highland
river basins are such that losses of meltwater cannot vary greatly
from year to year. Under such conditions, there should exist a
relationship between the accumulated winter precipitation and seasonal
runoff. In principle, such a relationship can be established empir8 36
ically for any basin if observational data are available over a
number of years. In practice, however, the problem of determining
such relationships is far from easy.
The main difficulty consists in estimating true values of the
water equivalent of snowpack. Feasurcments of precipitation and snow
cover in high mountain basins do not, as a rule, make it possible to
determine the true amount of accumulated water and may serve only for
obtaining some indices of this value. For this reason, relationships
between seasonal flow and snow accumulation index are statistical in
nature and, in most cases, cannot be used for water balance analysis.
The representativeness of the snow accumulation indices is a decisive
factor governing the possibility of establishing seasonal runoff
relations.
At least two additional factors have an influence on runoff and,
consequently, on the degree of correlation between runoff and the
snow accumulation index. These factors are: the amount of precipitation for the melting season and the antecedent water storage in a
basin. Both these factors should be taken into account in the development of seasonal runoff relations and in the preparation of forecasts. Methods of obtaining different indices of snow accumulation
are discussed in detail in the WMO "Guide to Hydrometeorological
Practices". With no opportunity to dwell on them in this paper we
will only note that to obtain the most representative index, the
area-altitude and the precipitation-altitude distribut ions should be
taken into account if possible.
For hilly forested catchment areas and basins located on plateaux, the theory discussed above in connection with lowland rivers
is applicable if adequate data are available.
HYDRO GRAPH COMPUTATIONS
Analysis and computation of snowmelt flow hydrographs embrace a
wide range of complicated problems that cannot be discussed in detail
within the limits of this paper. In this connection, just a few
points concerning the main problems of hydrograph synthesis are
touched upon.
(1) The main processes that form the complex basin outflow hydrograph are:
(a) areal snowmelt under the action of heat exchange including
the change of the snow-covered area;
(b) retention and yield of meltwater by the snow cover including
its tempora 1 accumulat ion ;
(c) absorption and retention of water by the basin including the
change of the active area;
(d) water yield and inflow into the channel network;
(e) transformation of the inflow into the outflow hydrograph.
All these processes are closely interrelated and should be taken into
consideration in the development of methods of hydrograph computatioii.
(2) A wide range of literature describes these processes. However,
they are still imperfectly investigated, and methods suggested by
different authors for the computations of snowmelt and snowmelt flow
hydrographs, which are sometimes insufficiently reliable, remain as
yet approximate.
(3) The future of the computations of snowmelt flow hydrographs consists in the development of a theoretical model realized by computers.
The development of models of this kind is becoming more and more
widespread. At present, however, more attention is sometimes given
837
to mathematical and technical problems of modelling and not to the
physical essence of the model itself and the physical sense of its
parameters. "here is no doubt, however, that future models will be
based on the fullest possible description of physical processes and
their synthesis.
CONCLUDING REMARKS AND ACKNOWLEDGMENTS
In conclusion, I would like to stress once again that the key
problem in snowmelt runoff forecasting is one of estimating water
absorption. Scientists' efforts should be aimed at a deeper insight
into the physics of infiltration and development of its theory and,
on the basis of this theory, the elaboration of practical methods of
determining reliable indices of water absorbability of river basins.
Finally, I wish to express my deep gratitude to those responsible for organizing this Symposium and for my opportunity to
participate.
RE FEREN CES
KOMAROV, V.D. (1959). Vesennii stok ravninnyh rek Evropeiskoi
chasti SSSR. (Spring flow of lowland rivers of the European
part of the 1JSSR.) Gidrometeoizdat, Moscow.
POPOV, E. G. (1957). Gidrologicheskie prognozy.
forecasts.) Gidrometeoizdat, Leningrad.
(Hydrological
POPOV, E.G. (1963'). Voprosy teorii i praktiki prognozov
rechnogo stoka. (Theory and practice of river runoff forecasts.] Gidrometeoizdat, Moscow.
LINSLEY, R.K., KOHLER, M.A., and PAULHUS, J.L.H. (1949).
Applied hydrology. New 'fori, Toronto, London.
LINSLEY, R.K., KOHLER M.A., and PAULHUS, J.L.H. (1958).
Hydrology for engineers. New York, Toronto, London.
WMO. (1970). Guide to hydrometeorological practices. WMO No.
168-'IT-82, Geneva.
This paper was presented by Professor J. Nemec
DI SCUSSION
-
J. Murtinec (Switzerland) Frequent reference was made in the
paper to long-range forecasts. This term is sometimes reserved for
year to year forecasts that may be based on the sunspot cycle. Would
it not be better to use the term "seasonal" or "medium-range forecasts11. Perhaps, in the future WMO would standardize the terminology.
-
J. Nemec (World Meteorological Organization)
(After discussion
with Professor V.D. Komarov) - Professor Komarov says the method of
forecasting explained in Professor Popov's paper is applicable to
short-term forecasts as well as long-term ones. The problem remains
as to the definition of short-term and long-term forecasts. We are
in agreement with the suggestions that a more explicit definition is
required.
838
-
K.S. Davar (Canada) As indicated in Professor Popov’s paper,
and the comments of Professor Nemec, improvement in our understanding
of snowmelt-runoff processes and derivation o€ more reliable simulation models require improvements in our data collection programs.
What do you suggest as the technique(s) to be used for actual measurements of snowmelt, to yield reliable inputs to models such as we
may have for rainfall? Also, as snowmelt runoff occurs through a
continuum of snowpack and ground, the processes of storage and transmission occur in the snowpack, in soil, and in the ground. What suggestions do you have for a data acquisition program to obtain this
information?
-
J. Ueemec (World Meteorological Organization)
(After discussion
with Professor V.D. Komarov) - Professor Komarov has no particular
comments nor does he have a proposal for an international program.
On my own behalf, I agree with you and I was attempting, in my
comments, to point out those processes that still require extensive
observational and experimental work.
839