Water Resources Management 12: 31–50, 1998.
c 1998 Kluwer Academic Publishers. Printed in the Netherlands.
31
A Review on Monthly Water Balance Models for
Water Resources Investigations
C.-Y. XU1 and V. P. SINGH2
1
Hydrology Division, Uppsala University, Norbyvägen 18B, 75236 Uppsala, Sweden.
Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge,
Louisiana 70803–6405, U.S.A.
2
(Received: 7 May 1996; in final form: 1 November 1996
Abstract. Research on the development and application of monthly water balance models has been
carried out since the 1940s. A good body of experience has been gained for many models, and a review
of these models is needed. Beginning with the development of monthly water balance models from
the earliest times, this paper discusses the relevance of various aspects of the practical application of
such models. Monthly water balance models were introduced originally to evaluate the importance
of different hydrologic parameters under a variety of hydrologic conditions. Present applications
of water balance models are directed along three main lines: reconstruction of the hydrology of
catchments, assessment of climatic impact changes, and evaluation of the seasonal and geographical
patterns of water supply and irrigation demand.
Key words: monthly water balance models, review, water resources investigation, climatic impact
assessment.
1. Introduction
Water balance models have been developed at various time scales (e.g. hourly, daily,
monthly and yearly) and to varying degrees of complexity. Monthly water balance
models were first developed in the 1940s by Thornthwaite (1948) and later revised
by Thornthwaite and Mather (1955, 1957). These models have since been adopted,
modified, and applied to a wide spectrum of hydrological problems (e.g. Gabos and
Gasparri, 1983; Alley, 1984, 1985; Vandewiele et al., 1992; Xu and Vandewiele,
1995). Recently, they have been employed to explore the impact of climatic change
(e.g. Schaake and Liu, 1989; Arnell, 1992; Xu and Halldin, 1996). They also
have been utilized for long-range streamflow forecasting (e.g. Alley, 1985; Xu and
Vandewiele, 1995). Although such applications may use hourly or daily models,
these models are however more data intensive and have more parameters than do
the corresponding monthly models.
With increasing use of monthly water balance models to address a range of
hydrological problems, a considerable amount of effort is being devoted to the
development of such models and techniques for their parameter estimation. A
variety of models and parameter estimation algorithms have been considered,
ranging from relatively complex conceptual models with 10 to 15 parameters for
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C.-Y. XU AND V. P. SINGH
arid regions in Africa (e.g. Pitman, 1973) to very simple models with 2 to 5
parameters for humid regions in temperate zones (e.g. Vandewiele et al., 1992;
Makhlouf and Michel, 1994; and Xu et al., 1996a). Although a great deal of
experience has been gained for many models, there is a continuing need to upgrade
the models and test them against practical requirements. Additionally, model users
must be given the opportunity to become familiar with the models and to develop a
good working knowledge of their sensitivity, and strengths and weaknesses. There
is, therefore, a need to take stock of such models and review them. Such a review
does not appear to have been reported.
The objective of this study is to discuss the current state of the art and the
assumptions and limitations of monthly water balance models by introducing a set
of representative models that have been used world wide. This eventually leads
to the development of guidelines for the model selection process. It is, however,
not the intention of this paper to discuss all the models that have appeared in the
literature.
The paper is organized as follows. After this brief introduction, Section 2 begins
with the development of monthly water balance models from the earliest times,
and discusses practical applications of such models for water resources investigations. The models reviewed here are first grouped by their principal objectives,
and a subgrouping based on the input data requirements is provided. Beginning
with a discussion of the models with precipitation as input, which are useful tools
in those regions where other meterological variables are not available, we consider the models with precipitation, temperature and/or potential evaporation as
input. Most of the available monthly water balance models belong in this group.
Thereafter, we review the monthly water balance models that use daily input data
which can provide a better estimation of actual evapotranspiration. Monthly water
balance models used for seasonally snow-covered regions are discussed in Section
3. Monthly water balance models used for applications other than to investigate
the importance of different hydrological vriables in diverse watersheds are also
discussed in this Section. These are exemplified by applications of monthly water
balance models for climate-change inpact assessment, for river flow forecasting,
for water project design and operation, etc. A discussion of the present status of
monthly water balance models and the possible future developments is presented
in Section 4. The spatial scale included in the review ranges from a few km2 to
several thousand km2 . The paper is concluded in Section 5.
2. Monthly Water Balance Models for Water Resources Investigations
The principal and traditional use of monthly water balance models has been to
investigate the importance of different hydrologic variables in diverse watersheds.
Many monthly water balance models have been developed for this purpose.
A REVIEW OF MONTHLY WATER BALANCE MODELS
33
2.1. MODELS USING PRECIPITATION (RAINFALL) AS INPUT
In general precipitation constitutes the largest component in a water balance equation. The derivation of a relationship between rainfall over a catchment and the
resulting flow in a river is a fundamental problem in hydrology. In most countries,
there are usually plenty of rainfall records, but the streamflow data are often limited and are rarely available for the specific river under investigation. The need
to evaluate river discharges from rainfall has therefore stimulated a great deal of
research. A number of monthly water balance models have been developed using
only precipitation (rainfall) as input. Examples of this kind of model are given
below.
Snyder (1963) developed the Tennessee Valley Authority (TVA) model for prediction of monthly water yield. The model partitions runoff into three components:
(1) immediate runoff, calculated as some portion of precipitation during the current
month; (2) delayed runoff, calculated using a linear reservoir concept; and (3) time
function, assumed to have no interaction with other components. Kuczera (1982)
used a modified version of Fiering’s (1967) model for describing the hydrology of
a catchment near Ocala, Florida. Monthly precipitation, pt , used as the sole input,
was divided into three parts by using two parameters a and b; the expected evapotranspiration bpt , the expected aquifer recharge apt (delayed by one time interval),
and the expected direct runoff (1–a–b)pt . A third parameter, c, was used to model
the base flow which is a fraction c of water storage mt 1 at the beginning of the
month: cmt 1 . A time-variant model was reported by Gabos and Gasparri (1983).
It is a single linear-reservoir model. The model uses two parameters to divide precipitation into three parts, i.e. direct runoff, infiltration, and evapotranspiration in
a similar way as in the preceding model, and another two parameters are used for
regulation of the underground reservoir. The first two parameters are allowed to
vary with time. Tuffuor and Labadie (1973) used a nonlinear time-variant model
for modeling monthly or seasonal data for the Todzie River in Ghana. It has 3 n
parameters to be determined, and 1 n 12 is the number of seasons defined per
year.
A common feature of these models is that evapotranspiration is calculated as
a fraction of the precipitation (rainfall) and the rest of the precipitation (rainfall)
is considered empirically as either infiltration and/or direct runoff. The respects in
which the models differ are that the fraction is either time invariant or time variant,
and the equations are either linear or nonlinear. Estimation of evapotranspiration as
a fraction of rainfall is, clearly, not reliable on a monthly time scale, since it is not
unusual for evapotranspiration to be greater than precipitation, especially during
those months that follow immediately the end of the rainy season, and the fact that
rainfall is highly variable in most parts of the world. Nevertheless, these models
can be used as approximate tools for water resources planning in those regions
where no other meteorological data are available.
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C.-Y. XU AND V. P. SINGH
2.2. MONTHLY MODELS USING RAINFALL AND TEMPERATURE AS INPUT
Four water balance models previously reported in the literature were discussed
by Alley (1984) and Vandewiele et al. (1992) in their comparative studies. The
first model is the Thornthwaite and Mather’s (1955) T-model. It is a model with
two storages: ‘soil moisture index’ mt and ‘water surplus’ vt . The model has two
parameters: soil moisture capacity a1 and storage constant a2 for vt . The second
model is Alley’s (1984) T-model. This model is a modification of the preceding
model in that a fraction a3 of the precipitation is immediately transformed into
direct runoff. The rest of the precipitation then enters the system as before. The
third model is Thomas’s (1981) abcd-model. There are also two storages in this
model: groundwater storage and soil moisture storage. The fourth model is Palmer’s
(1965) and Alley’s (1984) P-model. Palmer (1965) used a water balance model to
develop an index of meteorological drought. The model is unusual in that it uses a
‘root constant’ concept in calculating evapotranspiration.
The common feature of these four models is that the temperature was used
as a driving force to estimate potential evapotranspiration by the Thornthwaite
approach, which together with monthly rainfall was used as input data for the models. These models differ in their treatment of the relationship between actual and
potential evapotranspiration and of soil moisture accounting and aquifer recharge.
Alley (1984) reported that all of these four models did well in simulating annual
flows but less well in simulating monthly flows. The main weaknesses of these
models are: (1) state variables simulated by different models may be quite different
and the result is often an unrealistic simulation of soil moisture storage; (2) very
high correlations have been found between parameters; and (3) in some cases
parameter values have the tendency during optimization to overstep the physical
contraints. A comparative study performed by Vandewiele et al. (1992) confirmed
the above findings of Alley (1984).
2.3. MODELS USING MONTHLY RAINFALL AND POTENTIAL EVAPORATION AS INPUT
Monthly areal precipitation and potential evapotranspiration have been used as
the sole inputs to most monthly rainfall-runoff models. These models have been
developed in a wide range of climatic regions for an extensive range of applications
and vary considerably in their complexity.
An example of the models for generating monthly river flows for the South
African catchments is the model developed by Pitman (1973, 1978). This model
became most popular in African countries. It uses 12 parameters to compute actual
evapotranspiration, interception, surface runoff, soil moisture and runoff from upper
and lower zones. Roberts (1979) developed a monthly model based on an hourly
model developed by Krzystofowicz and Diskin (1978). The model has two storages:
interception and soil moisture storage, and uses essentially eight parameters. The
unusual part of this model in comparison to most other models is that the amount
of ‘base flow’ is not a function of the soil moisture level but of the total runoff,
A REVIEW OF MONTHLY WATER BALANCE MODELS
35
reducing the dependence of the model results on the soil moisture levels, which
are seldom accurately reproduced (Roberts, 1979). Hughes (1982) modified the
daily model developed by Roberts (1978) and converted it to a monthly model. The
model retains a similar structure as the preceding one, but a particular characteristic
of the model is that the single soil moisture storage is represented by a container
with various shapes. Salas et al. (1986) presented a conceptual simulation model
with the objective to simulate the various hydrological processes occurring in a
watershed at a monthly or seasonal time scale. This model is space distributed on
monthly time scale and assumes that the watershed can be divided into a number of
sub-watersheds. The model has a total of 11 parameters, some of them are obtained
by optimization and others are determined by judgement and/or trial and error.
The above-mentioned four models belong to the deterministic type and have
relatively complicated structure as far as monthly flow simulation is concerned. A
number of simpler monthly water balance models have been reported recently.
Vandewiele et al. (1992) and Xu (1992) proposed a series of models which
are variants of the monthly water balance model developed by Van der Beken
and Byloos (1977). The number of parameters used in the description of the
hydrological phenomena in the catchments is in most cases three, sometimes four.
The models were successfully applied to nearly 100 catchemnts from Belgium,
China and Burma. Makhlouf and Michel (1994) reported a two-parameter monthly
water balance model for French watersheds, GR2M, which originated from a daily
rainfall-runoff model of Edijatno and Michel (1989).
2.4. MONTHLY MODELS USING DAILY INPUT DATA
Haan (1972) developed a model for simulating monthly streamflow by using daily
time step. The estimated average potential evapotranspiration and daily rainfall are
used as input. The model has two storages and uses four parameters: the maximum possible infiltration rate, the maximum possible seepage rate, the maximum
capacity of that part of the soil’s moisture-holding capacity which is less readily
available for evapotranspiration, and the constant defining the fraction of seepage
that becomes runoff. The model was developed for small catchments since no
lag time is considered from rainfall incidence to the appearance of runoff at the
gauging station. Kuczera (1983b) developed a water balance model from that used
by Langford et al. (1978) for the Slip Creek catchment, which computes monthly
runoff by using a daily time step. The model has two storages: a quick response
storage contributes to quick response flow and the soil storage contributes to base
flow. The seepage loss function was introduced into the model because studies of
annual water balance and stream chemistry suggest that not all runoff produced by
the catchment is observed. In all, the model contains nine parameters. McMahon
and Mein modified the Boughton model (1973) by including a baseflow routine
with a double recession characteristic, and used the model to estimate monthly
flows for the Thomson River at the Narrows. The inputs for the model are daily
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catchment rainfall and monthly evaporation. The model has three storages and uses
10 parameters. It can also be used to simulate daily runoff when some years of
daily flow records are available for calibration.
It is believed that the use of daily rainfall as input improves the estimation
of such processes as infiltration, evapotranspiration, interception, and depression
storage. On the other hand, the use of daily data increases the amount of work and
may limit research to fewer catchments instead of water balance computations over
large geographical units.
2.5. SUMMARY AND COMPARISON OF MODELS DISCUSSED IN SECTIONS 2.1 TO 2.4
Four different types of monthly water balance models have been reviewed with
respect to their input data requirements, assumptions, uses and limitations. A
summery is shown in Table I.
3. Other Applications of Monthly Water Balance Models
3.1. SNOWMELT SIMULATION
Snowfall and snowmelt play a significant role in the hydrologic regime in many
parts of the world. Snow has received attention as a water resource, primarily in
the northern part of North America, Europe, and Asia. For hydrologic purposes,
the water content is more important than depth, unless one is interested in the
insulating properties of snow as in soil-freezing studies.
The problem of snowmelt runoff modeling associated with climatic and physiographic conditions are functions of data availability, regional characteristics, modeling approach, and model application. Many of these problems are common to all
models and regions, whereas others are unique to specific models or regions. The
more universal problems are generally associated with data constraints, whereas
the more unique problems are associated with model formulation and the climatic
and physiographic conditions of a region.
The use of the energy balance technique results in a model which may be very
close to being correct, but which may be unwieldy to use, except in very specialized,
highly instrumented situations. The energy balance approach uses a form of the
energy balance equation for a snowpack that can be written as (e.g. U.S. Army
Corps of Engineers, 1956; Gray and O’Neill, 1974):
Hm = Hsn + H1n + Hc + He +Hg + Hp + Hq ;
(1)
where Hm = energy available for snowmelt, Hsn = net shortwave radiation, H1n
= net longwave radiation, Hc = convective heat flux, He = latent heat flux, Hg =
conduction of heat from the ground, Hp = heat content of rain drops, and Hq =
change in energy content of the snowpack.
The variables necessary for a complete heat budget computation according
to Equation (1) include total solar radiation, albedo, longwave radiation balance
A REVIEW OF MONTHLY WATER BALANCE MODELS
37
Table I. A comparison of models discussed in Sections 2.1 through 2.4
Common features
(1) They describe conceptually land-based hydrologic process or
processes which are spatially averaged or lumped.
(2) Some of their parameters (if not all) are estimated by fitting
to observed hydrologic data such as rainfall and streamflow.
(3) They are specific-purpose models concerned primarily with
streamflow simulation.
(4) They have a relatively simple structure and a small number of
parameters compared with other short-period models.
(5) The basis of such models is the equation of continuity, that is
the water balance equation.
Different features
Despite the lengthy period over which these models have been
developed, it is seen that they differ more in matters of detail than in
broad concept. The models differ mainly in the following respects:
(1) Their requirement of input data.
(2) Their treatment of soil moisture accounting and aquifer
recharge. The number of storages considered varies from one
(e.g. models in Section 2.1) to three (e.g. Salas et al., 1986, in
Section 2.3).
(3) Their representation of the number of hydrologic processes.
In the models discussed in Section 2.1, only evapotranspiration, fast flow and slow flow are considered in a very simple
manner. In some of models mentioned in Section 2.3, interception, surface runoff, infiltration, evapotranspiration, deep
percolation, base flow and ground water flow are calculated.
Applications
and limitations
(1) In view of the availability of universally more reliable models,
the models in Section 2.1 cannot be recommended when other
meteorological data besides precipitation are available.
(2) The models in Section 2.2 can be used in reproducing annual
and seasonal flows when only precipitation and temperature
are available, but the state variable simulated by these models
may be unrealistic, as mentioned by Alley (1984) and confirmed later by Vandewiele et al. (1992).
(3) In view of the treatment of hydrologic processes and the way
that input information is utilized, models in Section 2.3 and
2.4 are more reliable. They give not only better estimates of
monthly flow, but also more reliable estimates of other water
balance components, such as, actual evapotranspiration, surface and base flows and soil moisture content, etc. Moreover,
application of such models to ungauged catchments by relating
model parameters with physical characteristics of catchments
is possible (see Section 3.4 for details).
(effective radiation), air temperature, air humidity, wind speed, temperature gradients in the soil and in snow, and precipitation. In addition to these variables some
physical parameters governing heat exchange with the atmosphere, heat transfer
within the snowpack, liquid water content in the snow, and drainage of the snowpack, would have to be estimated. Limitations on the availability of some of these
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data on the techniques to extrapolate point measurements to areal mean values have
restricted most applications of Equation (1) to snowmelt studies at a point or on
small plots (Leavesley, 1989).
Due to the above-mentioned difficulties in water balance modeling, emphasis
has been put on determining snowmelt by use of air temperature or a temperature
index. The temperature index approach is more empirical and is expressed in the
general form as
M = Cm (Ta –Tb );
(2)
Mt = 0.06[tt;max )2 – (tt;max * tt;min )];
(3)
where M = snow melt (mm); Cm = melt factor (mm/ C); Ta = air temperature ( C);
and Tb = base temperature ( C).
For most cases, Tb is assumed to be a constant; in some cases, it is also considered as a model parameter. In the literature, there are many different versions
of Equation (2) in which Ta is replaced by maximum daily temperature, minimum
daily temperature, etc. (e.g. Woo, 1972; Lang, 1984; Power, 1986; Martinec and
Range, 1986; and Moussavi, 1988).
Moussavi et al. (1989, 1990) developed and compared different structures of a
monthly water yield model for seasonally snow-covered mountainous watersheds
in Iran. In the recommended form of the model, snowmelt was calculated as
where Mt = the total melt depth (mm) over month t; tt;max = the average daily
maximum air temperature ( C) in month t; and tt;min = the average daily minimum
air temperature ( C) in month t.
Xu et al. (1996a) applied a monthly water balance model to 11 seasonally
snow-covered catchments in central Sweden. The uncommon feature of the model
is that the calculation of snowmelt is considered as a fraction of the snow storage
similar to the one used by Vandewiele and Ni-Lar-Win (1993). The fraction is an
increasing function of monthly air temperature as
mt = spt
n
1
1
e
Ta a2 )=(a1 a2 )]2
[(
o
;
(4)
where spt 1 is the snow storage at the beginning of the month t, and a1 and a2 are
model parameters with the constraint a1 a2 .
3.2. CLIMATIC CHANGE IMPACT ASSESSMENT
In the last 10 years, monthly water balance models have been used to explore the
impact of climatic change (e.g. Gleick, 1986, 1987; Schaake and Liu, 1989; Arnell,
1992). Gleick (1986) reviewed various approaches for evaluating the regional
hydrologic impacts of global climatic change and presented a series of criteria
for choosing among the different methods. He concluded that the use of monthly
water balance models appears to offer significant advantages over other methods
in accuracy, flexibility, and ease of use. Several case studies were reported.
A REVIEW OF MONTHLY WATER BALANCE MODELS
39
Gleick (1987) developed and tested a monthly water balance model for climatic
impact assessment for the Sacramento basin. The model works well under conditions of stationary climate and includes various capabilities to imcorporate changes
in climatic variables. The results suggest that the application of such models may
provide considerably more information on regional hydrologic effects of climate
change than is currently available. Schaake and Liu (1989) developed and used
simple monthly water balance models to understand the relationship between climate and water resources for more than 50 catchments from eastern China and
throughout the south-eastern U.S.A. Arnall (1992) used Thornthwaite and Mather’s (1955) T model, as summarized in Section 2.2, together with a number of
realistic climate change scenarios to examine the factors controlling the effects of
climate change on 15 catchments in the U.K., which represented a wide range of
climatic and geological conditions. In favour of using conceptual water balance
models rather than physically based models or black-box models, he stated that the
detailed realism of a physically based model posed a different set of complications.
First, the physically based models require high resolution, in both space and time,
of climatic input data that may not be available; and, second, it is possible that model parameters may need to change as climate evolves: soil structure may change,
for example, as summers become drier, and more importantly, the distribution and
composition of catchment vegetation will probably alter. There are at present too
many unknowns for detailed physically based models to be used in climate impact
studies. On the other hand, parameters of black-box empirical models that simply
reflect the current relationship between climate and hydrological response may not
remain valid in changed climatic circumstances, and such models therefore are not
suitable for this kind of study.
3.3. FLOW FORECASTING AND WATER PROJECT DESIGN
Two examples of using simple monthly water balance models for river flow forecasting and simulation needed in design and control of water resources systems
were reported by Alley (1985) and Xu and Vandewiele (1995). In water supply
projects (irrigation, drinking, and hydroelectricity), problems can be solved with
use of monthly models, as easily calibrated and easily obtainable input variables
can be acquired by the engineer in charge of water resources projects. Forecasting
even on a monthly scale can be used for real time control, e.g. in irrigation and
hydropower generation. Another direct application of the monthly water balance
models to water project design, as demonstrated by Xu and Vandewiele (1995), is
as follows. On a given river a dam is to be built in order to meet a given water
demand. The problem is to decide on the capacity of the dam. A useful aid to this
decision is the knowledge of the return period of water shortages as a function of
dam capacity. A fundamental requirement for the study is a long monthly runoff
series. Because the observed runoff series is usually short, say, less than 20 years,
a long synthetic flow record has to be generated. This can be done in several ways.
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C.-Y. XU AND V. P. SINGH
One way of doing this is to use water balance models to simulate a longer runoff
series based on the much longer rainfall series.
3.4. FLOW RECORD GENERATION IN UNGAUGED CATCHMENTS
One of the main objectives in the development of a conceptual water balance model
is to provide a model that can be used on ungauged catchments to generate a record
of runoff for planning and design purposes. To be successful in this approach, the
number of parameters has to be small, since ‘keeping the number of parameters
as low as possible increases the information content per parameter and therefore
allows both a more accurate determination of the parameter and a more reliable
correlation of the values obtained with catchment characteristics’ (Dooge, 1977).
Monthly water balance models usually have a simple structure and a small number of parameters which have led to some successful studies in the field. Jarboe
and Haan (1974) used a multiple linear regression method to relate each of four
parameters of Haan’s model and measurable catchment characteristics. Magette et
al. (1976) used Jones’s (1976) procedure to fit a subset of six parameters of the
Kentucky watershed model, and were able to obtain acceptable multiple regression
equations using indices of 15 watershed characteristics. Gabos and Gasparri (1983)
expressed four model parameters as functions of some measurable factors, such
as catchment area, average slope, catchment permeability, etc. Vandewiele et al.
(1991) successfully related three parameters of a monthly water balance model to
the lithological characteristics of Belgian catchments. Servat and Dezette (1993)
related catchment characteristics to parameters of two monthly water balance models; one model had seven parameters and the other had three parameters. Better
results were obtained for the three-parameter model. More recently, Vandewiele
and Elias (1995) used two techniques for obtaining parameter values of a monthly
water balance model for ungauged basins in Belgium. It was found that the Kriging technique was significantly better than the method using the parameter values
of a few neighboring basins. Xu et al. (1996b) developed relationships between
parameters of a monthly water balance model and the land-use data for Swedish
catchments. Regression equations were used to calculate model parameters from
catchment characteristics. Simulation of annual runoff based on these parameter
values showed an average absolute error of prediction for nine catchments to be
less than 1% with a maximum error of 20% for one catchment.
4. Discussion and Prospects
4.1. PURPOSE
Although a major factor in determining the form of a model for hydrologic prediction is the availability of resources (which may be expressed in terms of time,
availability of data, computing facilities, and human resources of training, experience and creativity, etc.), the primary factor to be considered is the purpose
A REVIEW OF MONTHLY WATER BALANCE MODELS
41
for which the model is required. Discussions on the relative methods of different
approaches to modeling are often resolved when the primary purpose of each is
stated explicitly.
The principal uses of monthly water balance models can be summarized as:
(1) synthesis of long-term records of basins, (2) generation of runoff records for
ungauged basins, (3) provision of hydrological data as inputs for validation of
deterministic general circulation models, (4) forecasting of the yield within one
or two months for real time control of water resources systems, (5) derivation of
climatic and hydrological regional classifications, and (6) forecasting of possible
hydrologic effects of changes in land use and climate change. Although these
objectives may be derived from hourly or daily models, the use of monthly water
balance models is preferred for the following reasons: (a) short-term models are
more data intensive and in many cases those data are not available, and (b) shortterm models are usually far more complicated; some models took years to develop
and thousand of man-hours to use.
4.2. CONCEPTS AND STRUCTURE
Hydrologic models are frequently classified into three types (e.g. Singh, 1989,
1995), the empirical models (also called in the literature black-box models), the
conceptual models (grey-box models) and theoretical models (sometimes called
white-box models). The block-box models relate outputs to inputs through a structure which may be wholly statistical or partly mathematical and does not aid in
physical understanding, as in the application of linear and nonlinear systems theory. Hydrologic models here are considered as comceptual if the form of model
equations is suggested by consideration of the physical processes acting upon the
inputs and outputs in a highly simplified form. The theoretical models have a
logical structure similar to the real-world system and may be helpful in changed
circumstances.
Monthly conceptual water balance models purport to simulate selected hydrological processes usually by conceptualizing the catchment as an assemblage of
interconnected storages, through which water passes from input as rainfall to output
as streamflow at the catchment outlet; the controlling equations satisfy the water
balance requirement.
In recent years there has been an increasing trend toward the development of
physically based models (Beven, 1989). However, it is apparent that there are
problems associated with the application of physically based models (e.g. Hughes,
1989). It would appear that future developments in hydrological modeling techniques face a dilemma. Modeling studies can contribute to the understanding of
hydrological processes at various scales, but only if uncertainties related to the
quality of the model input information can be overcome. The ability of the user to
provide such models with the information that they need has not kept pace with
recent model developments (e.g. the SHE model of Abbott et al., 1986). Any indi-
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C.-Y. XU AND V. P. SINGH
vidual model user is therefore faced with a choice of using either a sophisticated
model with less than perfect input data or a less complex model, based upon a
simpler conceptualization of ‘known reality’, for which the data requirements are
less stringent. A specific model application can be defined by the nature of the
catchment response characteristics and prevailing climate inputs as well as by the
type of model output required and the availability of information with which to
evaluate parameter values and the required input data.
4.3. METHODS OF COMPUTATION OF THE MAIN WATER BALANCE COMPONENTS
4.3.1. Precipitation
In general, precipitation is the major input for water balance models. The accuracy of measurement and computation of precipitation from a network of stations
determines to a considerable extent the reliability of water balance computations.
In discussing computer models, Linsley (1967) stated that with adequate amounts
of the proper kinds of hydrologic data, streamflow hydrographs can be reproduced
which are as accurate as the input supplied. The input with the greatest variability
is rainfall. Therefore, the accuracy of streamflow simulation depends primarily on
how well the variability can be defined in a specific case. No reliable water balance
computation is possible with insufficient knowledge of the spatial rainfall patterns.
Areal rainfall may be incorrect both randomly and systematically. Recently, Xu
and Vandewiele (1992, 1994) examined the influence of rainfall errors on the performance of monthly water balance models for Belgian and Chinese catchments.
It showed that: (1) random errors in rainfall data affect the model performance
adversely, and significant effects on model performance are expected when the
random errors go up to 10% and 15% for Chinese and Belgian catchments, respectively; (2) systematic errors in rainfall data have a serious effect on the model
parameter values.
A variety of general interpolation methods is available for areal estimation (e.g.
Rainbird, 1976). For selecting the best-suited method for computation of mean
precipitation over an area, we have to take into account the following controlling factors (Dyck, 1983): requirements of the water balance (spatial and temporal
resolution), spatial and temporal distribution of precipitation, density and distribution of the precipitation network, variability of precipitation events (structure of
precipitation distribution), available data, and possibilities of practical realization.
Remote-sening and satellite data can be used for catchment modeling (e.g.
Singh, 1995). Raders are employed for rainfall measurement. Satellite data can
be used for estimation of area and intensity of rainfall. An application combining
remote-sensing techniques based on satellite imagery and ground-based radar with
a network of recording raingauges for calibration holds promise in the future.
A REVIEW OF MONTHLY WATER BALANCE MODELS
43
4.3.2. Evapotranspiration
In general, evapotranspiration is the second largest quantity in a hydrological water
balance. Accurate spatial and temporal predictions of ET are required for water
balance models. Complex models for determining evapotranspiration have been
developed. The best example of such models is the Penman-Monteith method
(Monteith, 1965). These plot or site models with high temporal resolution were
developed for agricultural crops and forest stands and were established on soil
physical, plant physiological and climatic parameters (Halldin, 1979). But even
one-dimensional dynamic continuous simulation models of water state and flow in
the soil-plant-atmosphere system are too complex for water balance computation of
drainage basins. Therefore, the calculation of evapotranspiration for water balance
studies has to relay on simplified treatments.
In water balance models, most investigators have found it necessary to derive
‘actual’ evapotranspiration as a function of potential evapotranspiration and the
dryness of the soil (e.g. Palmer, 1965; Dyck, 1983; Xu, 1992):
ET = PET f(SM)
(5)
where ET and PET are actual and potential evapotranspiration, respectively; f(SM)
is a function of soil moisture, SM. Different forms of the function f(SM) have been
studied by Roberts (1978), Dyck (1983), and Xu (1992) among others. In fact,
this has been the most popular method of computing evapotranspiration for most
conceptual hydrologic models. With regard to this method of evapotranspiration
calculation the following two concepts are dominant (e.g. Dyck, 1983): (i) determination of site evapotranspiration of a soil-plant system and generalization of the site
data for basins; (ii) determination of areal evapotranspiration using the concept of
complementary relationship between areal and potential evapotranspiration. Many
studies have been carried out utilizing the potential evapotranspiration of certain
crops in the water balance calculation on a catchment level (e.g. Dyck, 1983; Oliver, 1983). Interpolation and aggregation methods are being applied to present both
weather and land cover data at the relevant space and time scales for the hydrology
of a runoff basin. Remote-sensing methods are going to play an ever-increasing
role in the assessment of water balance which use models of varying complexity
to estimate the evaporation from the surface (Oliver, 1983).
4.3.3. Runoff
Streamflow records provide a measure of the response of a catchment to the time
variable input and internal hydrological processes. Although model output may be
a single or multiple output, the ability to predict stream discharge remains the most
important objective of most models.
It is important for water balance studies to know different runoff components
and their regimes. The number of runoff components to be analyzed depends on
the characteristics of the basin and the objective of the separation, including the
44
C.-Y. XU AND V. P. SINGH
time base to be considered. For water balance modeling four runoff components,
as proposed by Dyck (1983), may be identified.
surface flow
fast interflow
slow interflow
baseflow
fast components;
slow components;
The following storage concepts can be applied to model those components:
single linear reservoir
single logarithmic reservoir
single nonlinear reservoir
S
S
S
=
=
=
K1 Q ;
K2 ln Q ;
K3 Q m ;
where S = storage; Q = reservoir outflow (discharge); Ki = storage constant; and
m = exponent. In several cases the concept of a single linear reservoir may be
sufficient.
4.4. MODEL EVALUATION AND PARAMETERS ESTIMATION
When a model has been developed or selected for use in predicting hydrologic outputs for a particular practical problem, it is then necessary to assess its applicability
and potential accuracy for the problem at hand, and to determine the values of the
model parameters or constants for the catchment under consideration. In general,
several levels of evaluation are necessary before a model should be applied to
estimate the output from a catchment (Pilgrim, 1975). These are: (i) rational examination of the model structure, (ii) estimation of parameter values, (iii) testing the
fitted model to verify its accuracy, and (iv) estimation of its range of applicability.
Conceptually, these evaluations are done in sequence. Estimation of the parameter values and model tests are emphasized in this paper, although it is important
to recognize that all four evaluations are of equal importance, and neglect of any
one can lead to serious errors.
Many types of techniques are employed for estimation of parameters of different
hydrological models (e.g. Pilgrim, 1975). Of these, automatic optimization using
search techniques has been the most common method in calibration of monthly
water balance models. This is partly because most monthly water balance models
have a simpler structure and a smaller number of parameters, which surmount some
of the practical difficulties encountered with optimization methods. Moreover,
automatic optimization techniques yield a reproducible and unique parameter set,
which is one of the conditions when the relationship between parameter values and
physical characteristics is to be established.
A REVIEW OF MONTHLY WATER BALANCE MODELS
45
4.4.1. Residual Analysis
Before the evaluation of parameter uncertainty can be considered, the catchment
model must be posed in a statistical framework. Typically, the catchment model is
formulated as (e.g. Clarke, 1973; Soorooshian and Dracup, 1980)
qt = q̂(xt ; ) + "t , t = 1, 2, ..., n;
(6)
where for time interval t, qt is the actual catchment response (such as runoff)
predicted by the function q̂(.) given xt , a vector of inputs (such as rainfall and
evaporation), and , a parameter vector about which inference is sought. The
difference between the observed and predicted response is given by the error "t .
Typically, catchment model parameters are estimated by ordinary least squares
(OLS), which involves solving the minimization problem:
SSQ = min
X
qt q^(xt ; ) 2 :
(7)
Provided the errors "t are (1) uncorrelated and (2) have constant variance with zero
mean. OLS inference furnishes valid statements of parameter uncertainty.
However, as Clarke (1973) and Soorooshian and Dracup (1980) note, these are
particularly strong assumptions, which often are not satisfied. Hence there is a
need for an inference procedure capable of handling violation of both OLS error
assumptions. Specifically, it must be capable of exhibiting heteroscedasticity, or
nonstationary variances, and autocorrelation. Residual analysis checks whether the
residuals, "t , behave as is required by the model hypotheses, especially whether
they are independent, homoscedastic and whether they have zero expectation.
There are various ways of doing so (e.g. Kuczera, 1983a; Vandewiele et al., 1992).
Customarily, independence can be checked by computing the autocorrelation of
residuals with time lag k. Homoscedasticity can be checked by the residuals versus
time diagram and the residuals versus computed discharge diagram, etc. (e.g.
Vandewiele et al., 1992; Xu, 1996).
Moreover, a variety of criteria have been used for checking whether the model
gives a computed output that is a sufficiently close reproduction of the observed
output. Clearly, to test a good fit between observed and computed output certain
statistical parameters have to be determined. Model efficiency as defined by Nash
and Sutcliffe (1970) may be used in this aspect.
4.4.2. Parameter Analysis
Three different techniques can be applied in order to evaluate the parameter significance and sensitivity (e.g. Xu, 1996).
1. Evaluation of the parameter values during the optimization. Automatic optimiation procedures are mathematical search algorithms that seek to minimize
differences between selected features of modeled and observed streamflows by
46
C.-Y. XU AND V. P. SINGH
systematic trial alterations in the values of the model parameters. These trial alterations are called ‘literations’. The objective function, i.e. the quantitative measure
of the fit of modeled runoff to the observed runoff, is calculated after each parameter search iteration. Successful iterations are those which cause a reduction in the
value of the objective function. During the search only the parameter set associated
with the current least objective function value is retained, which, at the end of a
search, is regarded as the optimal parameter set. The end of a search is usually
decided by: (1) a convergence test of the rate of reduction of the objective function
value; (2) a predetermined number of iterations; and (3) a computer run-time limitation. The stabilization of the parameter values can be studied with the graphs of
the parameter values versus the number of iterations. If the search is ended under
conditions (2) or (3) it does not guarantee that the parameters are stabilized.
2. Checking if global minimum is obtained. Note that stabilized parameter values
do not necessarily mean that a global optimum has been found. In order to check
whether the minimization is performed properly, graphs of the sum of squares
(SSQ) versus parameter values at the neighbourhood of the optimal value can be
plotted.
3. Detailed analysis of the variance–covariance matrix. The correlation matrix
of the parameters has to be checked. If the correlation coefficient between two
parameters is very near to –1 or 1, this means that perhaps a model can be found
with a smaller number of parameters and with the same explanatory power, or
that perhaps the parameters have to be built into the model in a different way, so
that their explanatory effects are more dissociated, and optimization is easier. To
answer the question of whether all parameters are really necessary, one can test the
hypothesis that parameters are significantly different from zero. This can be done
by checking whether the zero value belongs to the 95% confidence interval.
5. Conclusions
Monthly water balance models being used all over the world are reviewed and the
present status of development and applications of such models discussed. In the 40
years since the development of water balance methods, considerable experience has
been gained in identifying the best ways to apply water balance models to answer
questions about water availability, watershed characteristics, and water resources
management. The water balance models have proven to be a valuable tool not
only for assessing the hydrologic characteristics of diverse watersheds but also for
evaluating the hydrologic consequences of climatic change.
There are two practical reasons, among others, for using monthly models.
First, for the purposes of planning water resources and predicting the effects of
climatic change, the monthly variation of discharges may be sufficient. Second,
monthly hydroclimatological data are most readily available. Monthly precipita-
A REVIEW OF MONTHLY WATER BALANCE MODELS
47
tion, temperature and/or evaporation seem to be sufficient, and sometimes even
only precipitation data suffice.
It appears that three to five parameters may be sufficient to reproduce most of
the information in a hydrological record on a monthly scale in humid regions. It
may be worth while to use models with a relatively complex structure in arid and
semi-arid regions, such as in African catchments.
Because most monthly water balance models require fewer parameters to
explain hydrological phenomena, the information contained per parameter is then
increased, which permits a more accurate determination of parameters and more
reliable correlations between parameter values and catchment characteristics. Consequently, applicability to ungauged catchments is another important advantage of
such models.
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