In particular I wish to thank Dr. Arne Forsman for experienced advice and ...... lyng,layers when fresh snow was fallingon snow that had already started to melt.
DEVELOPMENTAND APPLICATIONOF A CONCEPTUALRUNOFF MODEL FOR SCANDINAVIAN CATCHMENTS UTVECKLINGOCH TILIIMPNINGAV EN BEGREPPSMXSSIGAVRINNINGSMODELLFÖR SKANDINAVISKANEDERBÖRDSOMRÅDEN by Sten Bergström SMHI Rapporter HYDROLOGIOCH OCEANOGRAFI Nr RHO 7 (1976)
SVERIGESMETEOROLOGISKAOCH HYDROLOGISKAINSTITUT NorrkOping1976
ERRATA
"Developmentand Applicationof a ConceptualRunoff Model for ScandinavianCatchments" by Sten Bergström
Page
Location
IV
Line 3
1
11
12
3
11
15
4 5
Line 5 and 17 Line 18
Original
Should be
Fc
Fc
Linsley and Crawford result
Crawfordand Linsley
Clarke (1972)
Clarke (1973)
results
11
/1
23
11
11
stationery
Stationary
38
11
13
Steindalsvatn
Steinslandsvatn
45
11
21
C -value ul
C -value u2
50
11
18
starts.
starts,
60
Ti
3
fig. 7.3
60
11
8
(Ssm)
fig. 7.13 S (7-111 )
64
11
14
70
I?
8
70
11
72
t?
82 82 92
Fc value of s sm
value of sact
lower zones
lower zone
23
on
of
18
Roche (1970)
Roche (1971)
table 7.13 (Pw) 12.0 line 8 table 7.13 (Pw) 10.0 line 9 line 13 look
10.0 12.0
looks
103
11
29
1962-75
1963-75
126
It
25
cathcments
catchments
132
11
35
1975
1971
SUMMARY
The experiencesof conceptualrunoff modellingat the Swedish Meteorological and HydrologicalInstituteare summarizedin the present work. The basic philosophyand the methodologywhen developingthe HBV-modelare discussed.The structureof the model is describedwith a discussionof its physical relevanceand examples of alternatives.The sensitivityof the model to changes in parametervalues is studiedthrough mappings of the response surfacesof a sum of squares criterionof fit. Applications to a variety of catchmentsin Sweden and Norway are presentedand the performanceof the model is verifiedby a numerical criterionof fit, plottings of computedhydrographsand recorded ones, scatter diagramsof peak flows and flow duration curves. Examples of both short range and long range hydrologicalforecastingare given. A general conclusionis that the HBV-modelcan be used for the reconstruction of the dischargein catchmentsof the presentedtype, if it is properly calibrated.The model can also be used for hydrologicalforecasting, if combinedwith meteorologicalforecastsor recorded climatic series.
SAMMANFATTNING
Erfarenheternaav begreppsmåssigaavrinningsmodellervid SverigesMeteorologiska och HydrologiskaInstitut summeras i föreliggandearbete. Den grundläggandefilosofinoch metodikenvid utvecklingenav HBV-modellen diskuteras.Modellstrukturenbeskrivesmed en presentationav den fysikaliska bakgrundenoch exempel på alternativ.Modellenskånslighet fOr störningarav parametervårdenstuderasgenom kartlåggningav ett minsta kvadratkriteriumsresponsytor.Tillämpningarav modellenpå ett antal avrinningsområdeni Sverige och Norge redovisasoch simuleringenutvårderas med ett numeriskt anpassningskriterium, uppritningarav den beråknade och observeradehydrografen,jåmförelsermellan simlerade och beräknade flödestoppar samt varaktighetsdiagram. Exempel ges på hydrologiskalång- och korttidsprognoser. En allmån slutsatsår att HBV-modellenår anvåndbarför rekonstruktionav
vattenföringsserieri den typ av områden, där den tillämpats,under förutsåttning att den kalibreraspå ett riktigt sätt. Modellen kan också anvåndas för hydrologiskaprognosermed hjälp av meteorologiskaprognosereller observeradeklimatserier.
ACKNOWLEDEMENTS
The staff of the Swedish Meteorological and Hydrological Institute is gratefully acknowledged for valuable assistance in all phases of this work. In particular I wish to thank Dr. Arne Forsman for experienced advice and encouragements, Mr Stig Jönsson, partner on this project since September l974, Mr Lars-Erik Eggertsson for valuable help with computer problems and Mrs Vera Kuylenstierna for typing the manuscript. I also want to thank professor Gunnar Lindh at the Department of Water Resources Engineering, Lund Institute of Technology, who convinced me about the value in a summary of my work on conceptual modelling. During the four years of work on conceptual runoff modelling valuable criticism and suggestions in the form of meetings or by more informal contacts, have been provided, both on a national and on an international basis. I am grateful to all those who participated in this process. Parts of this work have been financed by the Swedish Natural Sience Research Council and Ångermanälvens Vattenregleringsföretag which is gratefully acknowledged. Finally I want to express my gratitude to all those who have participated in the lengthy process of data collection and processing. Reliable records of meteorological and hydrological data are indispensable when developing conceptual runoff models.
LIST OF CONTENTS
Page
LIST OF CONTENTS LIST OF SYMBOLS
III
1
INTRODUCTION
1
1.1
The work at the Swedish Meteorologicaland HydrologicalInstitute (SMHI)
2
2
DEFINITIONS
14
3
STRATEGY
6
4
TEST CATCHMENTS
9
5
METHODOLOGY
12
5.1
Verificationcriteria
12
5.1.1
Numericalcriteria
13
5.1.2
The accumulateddifference
16
5.1.3
Scatter diagramsof peak flows
16
5.1.4
Flow duration curves
16
5.2
Split sample test
17
5.3
Mappings of the error function
22
5.4
Automatic calibration
24
5.5
Subjectivecalibration
26
6
DATA BASE
29
6.1
Precipitation
30
6.2
Temperatures
30
6.3
Potentialevaporation
30
6.4
Runoff
32
6.5
Consequencesof errors in data
33
7 7.1
THE MODEL STRUCTURE
34
Snow accumulationand ablation
40
7.1.1
The form of precipitation
41
7.1.2
Snow fall and evaporationcorrections
42
7.1.3
Wind corrections on snow accumulation
44
7.1.4
Temperatureindex methods for snowmeltcomputations
146
II
7.1.5
Alternativemeltfunctions
49
7.1.6
Water retentionin the snowpack
50
7.1.7
Refreezingof liquid water
51
7.1.8
The effect of frozen ground
52
7.1.9
Statisticaldistributionof the degree-dayfactor
52
7.1.10
Area-elevationdistributionof the snowroutine
54
7.2
The soil moisture zone
57
7.2.1
A simple reservoirapproach
58
7.2.2
Soil moisture accountingin the HBV-model
59
7.2.3
An alternativesoil moisture routine
65
7.2.4
Distributionaccordingto the area-elevationcurve
67
7.3
The response functionof the HBV-model
69
7.3.1
The single linear reservoir
70
7.3.2
The upper zone
72
7.3.3
The lower zone
75
7.3.4
The transformationfunction
77
7.3.5
Parametervalues of the response function
81
7.4
Computationaldetails
83
8
APPLICATIONS
86
8.1
Reconstructionof the hydrograph
86
8.1.1
R2-values
88
8.1.2
Scatter diagrams of peak flows
89
8.1.3
Flow duration curves
92
8.2
Hydrologicalforecasting
95
8.2.1
Updating
97
8.2.2
Short range forecasting
99
8.2.3
Long range forecasting
103
8.2.4
Operationalsystems
106
9
CONCLUSIONS
107
APPENDIX 1: PLOTTINGSOF COMPUTEDAND RECORDED HYDROGRAPHS
109
APPENDIX 2: SOME ASPECTS ON THE INTERPRETATIONOF THE RESPONSE SURFACES
126
LIST OF REFERENCES
128
LIST OF SYMBOLS
at
accumulated difference at time t.
Bmax
maximum base in the transformation function. actual base in the transformation function.
Co
degree-day melt factor.
C . o inf
range in distribution of the degree-day melt factor.
O o max
maximum in the distribution of the degree-day melt factor.
o
minimum in the distribution of the degree-day melt factor.
C1
general parameter.
02
general parameter.
Cel
parameter in the snow evaporation equation.
0e2
parameter in the snow evaporation equation.
Ceff
parameter relating melt rate to accumulated snowmelt.
o min
ml
parameter accounting for windspeed in the snowmelt equation.
0m2
parameter accounting for windspeed in the snowmelt equation.
Cperc
percolation capacity.
Croute
parameter in the transformation function.
Csf
snow fall correction factor.
Cul
Cwh
parameter accounting for windspeed in the snow accumulation routine. parameter accounting for windspeed in the snow accumulation routine. water holding capacity of snow.
Ea
actual evaporation.
0u2
potential evaporation. ea
vapor pressure in the atmosphere.
es
surface vapor pressure.
F2
sum of squares criterion of f t.
Fo2
initial variance.
Fabs
alternative criterion of fit.
IV
Fdiff
alternativecriterionof fit.
F2 mv
alternativecriterionof fit.
Fc
maximum soil moisture storagein the model.
Fcmax
max.valueof Fc in a distributedroutine.
Fc . min
min.value of Fc in a distributedroutine.
f (x)
frequencydistributionfunctionof x. elevationin the catchment. general variable in time or space. storage dischargeparameter.
Ko
storage dischargeparameter in the upper zone.
K1
storage dischargeparameter in the upper zone.
K2
storage dischargeparameter in the lower zone.
Lp
limit for potentialevaporation.
uz
limit for slow drainageof the upper zone. snowmelt. precipitation.
corr lapse lapse Pw
rainfall correctionfactor. area-elevationcorrectionof precipitation. average area-elevationcorrectionof precipitation. part of the model representinglakes, rivers and other wet areas. runoff.
Qo Q1 Q2 Qc
runoff generatedfrom the upper zone. runoff generatedfrom the upper zone. runoff generatedfrom the lower zone. computed runoff.
Qg
total generatedrunoff.
Q.n
inflow from an upstream situatedreservoir.
loc r Qr
local inflow to a reservoir. recorded runoff. mean of recorded runoff.
total runoff from a reservoir.
tot R2
criterion of fit. alternative storage discharge description. storage.
Sact
active storage in the soil moisture zone.
Sb
bottom storage under the snowpack of the model.
Sls
storage in the lower zone of the model.
Sr
storage in a reservoir in a catchment.
Ss
storage of snow in the model. soil moisture storage in the model.
sm sm
min
uz
minimum soil moisture storage for a given period of time. storage in the upper zone of the model.
sact
relative active soil moisture storage for a given period of time. temperature.
To
general temperature correction.
T
area-elevation correction of temperature.
lapse
variable in time. to
initial time. windspeed. average windspeed.
Wc
convective heat flux.
Wg
heat flux from the ground.
W1
latent heat flux.
Wlw
net long wave radiation.
Wm
heat equivalent of the snowmelt.
Wp
contribution of heat from precipitation.
sw Wt
absorbed short wave radiation. change in the energy content of the snowloack. area of contributing zone in the distributed soil moisture routine.
VI
parameter in the distributedsoil moisture routine. parameter in the lumped soil moisture routine. period of time.
Appendix 1 ACC. DIFF. EVP
accumulateddifferencebetween the computed and the observed hydrographs. computed actual evaporation.
MELT
yield from the snow routine. precipitation.
SM
computed soil moisture storage.
SNOWCOV
computed snow covered area.
SP
computed average snowpack.
TEMP
temperature. computedand recorded discharge.
Appendix 2 F2
sum of squares criterionof fit. number of values for the computationof F2.
R2
criterionof fit. mean error.
1. INTRODUCTION
The processes in nature governingthe hydrologicalcycle have long attracted the attentionof those working on the rationaluse of existing water resources. Particularlythe processesconvertingthe driving forces, precipitationand evaporationinto runoff have been subjectto great efforts by scientistsand engineers. In recent years the advent of electroniccomputershas made it possible to organize and analyze data and to carry out computationsin a way previously inconceivable.This possibilityhas resulted in the development of mathematicalhydrologicalmodels which in a conceptualway are capable of simulatingthe most significantcomponentsof the runoff generating processes.Among the pioneers in this field were Linsley and Crawford (1966)when presentingthe Stanfordwatershedmodel, a model which today has found a lot of applicationsall over the world. Also well known is the SSARR model (Schermerhornand Kuehl, 1968) developedfor river regulations in the Columbiabasin in the north-westernof U.S.A. The SSARR model is also
in operationaluse for forecastingpurposes in
Sweden (Danielssonand Wretborn,1975) and in Poland (Bobinski,Piwecki and Zelazinski,1975). In the Scandinaviancountriesthe first models were presentedby Nyberg (1972),Nielsen and Hansen (1973) and Grip (1973).The first model from the Swedish Meteorologicaland HydrologicalInstitute (SMHI)was presented by Bergström (1972A).
The increasinginterest in hydrologicalmodels caused the World Meteorological Organizationto initiatethe project on "Intercomparisonof Conceptual Models Used in OperationalHydrologicalForecasting"in 1968 with ten participatingmodels from differentparts of the world. The final report from this project appeared in 1975 (WMO, 1975 B). Mathemtical runoff models can be applied to a vast field of water resource problems. Some of the most common applicationsare summarized in the followingthree points.
2
Simulationof natural discharge. Operationalforecasting. Predictionof effects of future physical changes in a catchment. Simulationof natural dischargemeans that the model is used tc simulate runoff from meteorologicalinput data availablein the catchmentor in its neighbourhood.The performanceof the model is generallyverified against a recorded runoff series. The model can be used to extend runoff records bymeans of long records of meteorologicalobservations.It can also be used to tell artificialfrom natural variations in a catchmentwhere human influenceor other changes in the hydrologicalregimearesuspected.A tempting, but rather difficult,applicaticnis the estimationof runoff in ungauged catchments.To do this with any certaintyrequires long experience with the model so that its componentscan be related to the physiographic characteristicsof the catchment.
In operationalforecasting,the model is first fitted and tested in order to verify its capabilityof runoff simulationfrom meteorologicaldata. Then meteorologicalforecastsor recorded climatic series can be used to forecast dischargein rivers. The most difficultpoint is to predict the effects of future physical changes in a catchment,as it requires not only an accurate runoff model but also that its componentswith certaintycan be related to the characteristicsof the catchments,such as degree of urbanizationor percentageof clear cuttings in a forest. If so, the model can be used to study the possible effect on the hydrologicalregime of a proposed activity in the catchment.
1.1. The work at the Swedish Meteorolo ical and Hrdrclo ical Institute (SMHI)
Since 1972 work on the developmentof an operationalhydrolowicalrunoff model has been in prowress at the SMHZ The efforts have been concentrated on the simulationof natural dischargeand operationalforecasting.The objective is to develop a modelwhichis applicableto most Swedish catchments, This
means that it must not require better data coveragethan can general-
ly be satisfiedalso in rather remote areas. Furthermore the intentionis to make the model so flexiblethat it can be transferredfrom catchmentto catchmentwithout too much modificationof its basic structure.
3
Initiallya simplemodel for the snow-freeseason was developed,which showed encouraging results when applied to a few small basins. Since then the model has been tested in several catchmentsranging in size from )4 to neafly 4 000 km2. The model has been modified slightlyas catchment size and other characteristicshave changed,but the basic structureis still the same. It has been named the HBV-model,from the water balance section (HBV), where
it was developed,followedby a number to identi
fy differentversions. The present work intendsto give some insiRht into the philosophy and methodology in the developmentof the HBV-model,to discuss the structureand to show some practicalapplications.As developmentand application are closely related, examples from differentcatchmentswill be used as illustrationsin case they have had significantimpact on the methodologyor structureof the model. A more systematicpresentationof the
result is given in a separatechapter, No. 8, and some graphs of
computed and observed dischargeare presented in appendix 1.
4
2. DEFINITIMS
In the following text some expressions that might not be self-explanatory will be used. Therefore it was felt appropriate to give some clarifications. For more complete definitions of the terms in hydrological modelling, reference is made to, for example, Clarke (1972).
A system has a very broad meaning. It can be defined in a simple way as by von Bertalanffy(1968,page 55): "A system can be defined as a set of elements standing in interrelations".Dooge (1973,page 4) gave a more specifieddefinition:"A system is any structure,device, scheeme, or procedure,real or abstract,that interrelatesin a given time reference, an input, cause, or stimulus,of matter, energy, or information, and an output, effect, or response,of information,energy, or matter".
A linear system is defined as one where the principleof superpositionholds, which means
that if yl(t), y2(t) are the outputs correspondingto
the inputs xl(t), x2(t), the outputs correspondingto xl(t) + x2(t) will be yl(t) + y2(t). A model can be defined accordingto Clanke (1972) as "a simplifiedrepresentationof a complex system".This simple definitionapplies to the use of the term model in this text. A mathenaticalmodel is defined as a set of mathematicalexpressions and logical statementscombined in order to simulatethe behaviourof a given system. A deterministicmodel is a model where two equal sets of inputs will always yield the same output, if run through the model under identicalconditions. The model has no componentcontrolledby chance. A stochasticmodel has some componentof random character.Identicalinputs may result in unequal outputs, if run through the model under identical conditions.
A black box model is developedwithout any considerationsof the physical processes in the catchment.The model is merely based on analysisof input and output. A conce tual model is based on some considerationsof the physicalprocesses in the catchment.In a hydrologicalmodel the use of a lower zone representingground water and an upper zone representingquicker runoff, for example,will give the model a conceptualstatus. A.routine, functionor procedure is a part of a model, for example the simulationof snowmeltor'the soil moisture accountingprocedure. A lumped model or routimis one where the catchmentis regarded as one unit without any considerationof the.geographicalor statisticaldistribution of its properties. A distributedmodel or routine is accountingfor the statisticalor geographicaldistributionof propertieswithin the catchment. Ambiguoususe of the terms parameter and variable sometimescauses confusion when discussingmathematicalmodels. The definitionsbelow have, however,become practice in hydrologicalmodelling.They have been suggested by Clarke (1972) among others. A parameteris a constant in the mathematicalexpressionsor logical statementsof the mathematicalmodel. It remains constant in time. Alternativelythe terms coefficient,constantor factor will be used in this text. A variable is a quantitywhich varies in time. It can be a series of input to or output from the model but also a descriptionof the conditions in the differentcomponentsof the model. In this text the parameterswillbe classifiedfurther as free or confined dependingon their use in the model. A free arameter is found empiricallyduring the calibrationof the model. Alternativelythe term em irical coefficientwill be used. A confined arameter is estimated from the map or other informationabout .t.11,e_catchment. It is kept constant during the calibration-procedure.
6
3. STRATEGY
The runoff modellingproblem can be approachedfrom two extreme standpoints. One is the analysis of time series of input and output and constructionof the model without any notice of the known or unknown physical propertiesof the catchment.This is sometimescalled the "black box" approach.The other is representedby the school advocatingfor, if not a complete,then at least a thorough understandingof the physical processesbefore any model can be established. The most famous example of the black box approach in hydrology is probatly the instantaneousunit hydrograph (IUH) as describedby Dooge (1973) among others. The elegantmathematicalsolutionsare attractive,but the determination of the excess water that will form the runoff is a very weak point. This problem is sometimesovercome by an antecedentprecipitationindex (API),meanirg that the length of the dry period precedingprecipitationis a factor reflected in the response.A model of the latter type is not, however, to be regarded as a black box model any longer as the effect of a soil moisture deficit is accounted for implicitly. The black box approach has its main drawback in its incapabilityto account for the general knowledgewhich we have about the catchmentand the different physical processes. It is therefore stronglyfelt that the applicationof pure black box methods is a waste of information. On the other hand the school advocatingthat a runoff model can be developed only with a thorough knowledgeof the physical laws and mechanisms of the runoff generatingprocess, will run into unsurmountableproblems due to shortageof information.Detailed knowledge of the physical laws at one spot in the catchmentwill be more or less impossibleto extrapolateto areal behaviour because
of limited knowledge
of
the distributionof the physical
characteristicsover the catchnent. Most runoff models are compromises between these two extreme standpoints, and so is the HBV-model.It is built up on a frameworkjustifiedby physical considerationsbut parts of the model have more the characterof a black box
7
approach as it includesparameterswhichhave to be calibratedthrough the analysis of input and output. When developingthe HBV-model,itwas immediatelyrealized that a detailed descriptionof all componentsof the hydrologicalcycle would lead to a model of a complexitythat cannot be justifiedby the objective to simulate runoff. Thereforeit was decided to concentrateon the most significant parts of the runoff generatingprocesses.In order to find these significantparts,workhas been carried out accordingto a scheme which can be siunmarized in the following six points (Bergström197)4): Assume a very simple model based on field measurementsand intuition. Test the model against recorded data. Modify the model and test it again, make sure that the modifications are improvingthe model. Study the stabilityand interactionsof the parameters. Test the model in catchmentsof different size, characterand geographiclocation. Carry out field investigationsof the processesicl- are particularly importantfor the model structure. Emphasis is put on the restrictionof the number of free parameters. Thereforeeach parameter has to be used as efficientlyas possible, meaning that a large spectrumof possibilitieshas to be covered by alternativevalues of one parameter. The method has a lot in common with the ideas presentedby Nash and Sutcliffe (1970).They suggestedthat one should start with a simplemodel and then elaborateit further,always with a glance at the efficiencyof the model and the stabilityof its parameters.Other authors supportingthis approach are von Bertalanffy(1968,page 187), who stated that "oversimplificationsprogressivelycorrectedin subsequentdevelopmentare the most potent or indeed the only means towards conceptualmastery of nature", and Kalinin (1971,page 2)44),who wrote that "when a problem goes beyond a certain boundry of mathematicalcomplexity,a simplerapparatus is used in its solution".Fiering (1975),finally, stronglyemphazises the need for more understandingof the physical laws before more complex
8
models are developed.On the question of free parameters,Ibbit(1974) stated that "The economicsof model operationnecessitatethat fitting problems be minimized and this impliesthat the number of parametersshould be as low as possible".
9
4. TEST CATCHMENTS
The model has so far been run by the SMHI during 96 years or seasons in 11 catchmentsas presented in table 4.1 and fig. 4.1. The catchmentscover a large spectrumof physiographicsettingsand geographicallocations.The soil is mostly morain or of a pervious type. It remains an interesting future task to model a catchmentdominatedby clay.
Some of the catchmentsin table 4.1 are subbasinsto RepresentativeBasins establishedwithin the frameworkof the InternationalHydrologicalDecade. Lilla Tivsjön is a part of the Kassjöånbasin, Nolsjn a part of Velen and Solmyrena part of the Lapptråsketbasin. The Filefjellbasin is one of three NorwegianRepresentativeBasins. Apart from being used as test catchmentsthe RepresentativeBasins have been valuable sources of information when developingthe model structure,as they have improvedour general understandingof the hydrologicalprocesses.Characteristicsof all Nordic RepresentativeBasins are presentedby Falkenmark (1972). The applicationsof the model to the Kultsjön,Malgomaj and Ströms Vattudal catchmentswere made for the river regulationcompany of the river Ångermanälven(ÅngermanålvensVattenregleringsföretag). These catchments will be subject.tooperationalforecastsstarting in the spring of 1976. The applicationto the Steinslandsvatncatchmentwas made at the Norwegian Water Resourcesand ElectricityBoard and was about to be finishedwhen this was written. Worth mentioning is also a modificationof the HBV-model made by HoumcAler (1976)and applied to the 46.6 km2 large Giber å basin outside Aarhus in Denmark. The results from the differentcatchmentswill be incorporatedin the followingdiscussionon the methodologyand the model structure.A more systematic presentationof the results is made in chapter 8 and in appendix1. For more details concerningcatchmentcharacteristicsand applications,the reader is referredto the referencesto table 4.1.
10
Table 4.1 Test catchmentsfor the HBV-model
Catchment
Area (km2)
L. Tivsjön 1) 2) 3) 12.7
Altitude range (m)
Lakes (%)
200
2.7
Predominant vegetation cover below timberline Coniferous forest
Swamp (%)
Area above timberline (%)
8.0
0
Nolsjön 2) 3)
18.2
55
1.5
Stabby 3) 4)
6.4
37
o
2.3
o
Stormyra 2) 3)
4.0
67
o
1.6
0
Solmyren 3)
27.5
130
33.1
0
Gimdalsbyn 2178 5)
300
13
t,
4
0
T,
6
51
0.5
u
v
14.2
Kultsjön 7)
1109
1040
6
Malgomaj 7)
1862
1240
8
18
7
Malgomaj + Kultsjön 7) 2971
1240
7
14
23
Stråms Vattudal 6) 7) 3851
1015
lo
5
13
3.7
86
2
88
Filefjell 8)
154
900
Steinslands216 vatn
1115
8.8 6
BergstrOm (1972 A) Bergstråm (1973) Bergströmand Forsman (1973) Bergstråm (1972 B) Bergstrm (1975) Bergstråmand Jönsson (1975) Bergströmand Jönsson (1976 A) Bergströmand Jönsson (1976 B)
11
Deciduous forest
11
30°
250
2o°
15°
35°
1 300 km
200
100
435°
SOLMYREN
\ KULTSJdN \'''',.., — 1109Km2
o
27.5Km2 6s°
STROMS VATTUDAL ;',;:-
MALGOMAJ '.!..
3851Km2
1862 Km2 . .,. . -
\,
)
.,
\\\
1.
STORMYRA 4.0 Km2
_ss°N
Fig.
4.1.
Test
catchments
for
the
applications
of
the
HBV-model.
12
5. METRODOLOGY
In this chapter some of the most importantmatheraticaltools and methods used when developingthe HBV-modelwill be discussed.Emphasiswill be Dut on the points which have given rise to the most vivid discussionsamong modellers in the Scandinaviancountries.An attempt will alsc be made tc reflect reevaluationsof some of the methods in the light of experiencefrom the applicationsof the model.
5.1. Verificationcriteria A verificationcriterion is a rule after which the performanceof the model is judged. As this work is aimed towards the developmentof runoff models, this criterionis based on the agreementbetween a computed and a recorded hydrograph. Several types of criteriacan be found in the litterature,rangingfrom single values of goodness of fit to scatter diagramsand comparisonsbetween computed and recorded flow duration curves (see for example WO, 1975 B). As a matter of fact we have almost endless Dossibilitiesof describingthe agreementbetween two graphs. The problem is to find the criterionthat is most in agreementwith our intentionsregardingthe model. The choice of criterionis an extremelycriticalpoint as it effects the optimum parametervalues and thus the performanceof the calibratedmodel. In the followingfour sections some criterionsused in the applications of the HEV-modelwill be discussed.They are all giving some representation of the agreementbetween the otserved and the computed hydrographs, but as they all have their drawbacksand limitations,a rigorousvisual inspectionof the hydrographsis still indispensablewhen analysingthe performanceof the model.
13
5.1.1. Numericalcriteria A numericalcriteriongives the agreementbetween the computedand the recorded hydrographsexpressedas one single value. One of the most simple and most popular criteria is the•sum of squares of the residuals, expressedas:
F2 = E t=0 wherer(t) Q(t)
(5.1)
r
= observed discharge at time, t; ” = computed
= total period of time. If the initial variance is expressedas: —
= E (Qr(t)-r)2 t=0 wherer
F02 (5.2)
= arithmeticmean of the observed hydrographover the time "C
the proportionof the initialvariance accountedfor by the model can be expressedas:
R2
2 F o
F2
Fo2
This criterionwas
defined by Nash and Sutcliffe (1970) as the effic±-
ency of the model. The same criterionwas proposed by Ibbit (197)4).
R2 and F2 are essentiallythe same criterion.They will yield the sa=e general shape of the error function and the same optimum parameter values. The advantageof the R2-criterionis its characterof a relative measure, facilitatingcomparisonsbetween models. The value of R2 will range from minus infinityto plus one, where plus one is representinga complete agreementbetween the two hydrographs.
The applicationof the HBV-modelwith the R2-criterionhas thrown some doubt on its representativenesswhen comparingmodels tested in different catchmentsor during differentperiods of time. If the initial variance is low as in fig. 5.1, small errors will cause low R2-values,while at high
initial variance the situationis the opposite,as seen in fig. 5.2.
O (t15) —w--
400
OBSERVED HYDROGRAPH COMPUTED HYDROGRAPH
R2 =- 0.64
200
MAY
APR
JUN
JUL
AUG
SEP
OCT
NOV
Fig. 5.1. Low R2 value as a result of low initialvariance (Stabby,1959). —
OBSERVED HYDROGRAPI4 —x
0(L/s) 600
COMPUTED
HYDROGRAPH
R2= 0.84
400
200
0
APR
M AY
JUN
JUL
AUG
P
OCT
NOv
Fig. 5.2. High R2-value as a result of high initialvariance (L. Tivsjön, 1968).
In spite of these uncertaintiesthe R2-value has generallybeen a good indicationof the overall fit of the model as long as the comparisonis restrictedto identicalperiods of time and to one catchment.This means that the best fit accordingto visual inspectionand the R2-value mostly coincide.Occasionally,as shown by Bergströmand Jönsson (1976 B), the R2-criterioncan be misleading especiallyat values near optimum. This is due to the fact that the residualsare of differentorigin during different seasons.Large deviationsdue to input errors, for example during snowmelt,can mask small errors in another sequenceof the hydrograph.
When applyingthe HBV-modelto the reservoirsin the river Ångermanälven,the R2-valuesfairly well reflectedthe general conclusionsconcerningthe performanceof the model accordingto visual inspection.The problem
15
with variabilityof the initialvariancewas overcomeby the long periods used for calibrationand test (see table 8.2). Bergström (1973) investigatedsome modified forms of (5.1) in the Stormyra basin accordingto:
Fabs =
IQ (t) t=0 r
(5.4)
Qc(t)!
Fdiff = El( Qr(t) - Q (t-1)) - (Qc(t) - Q (t-1) t=0
(5.5)
2 2 ,, E (E (t+i) - E Qc(t+i))2 t=0 i=0 r i=0
(5.6)
1
F2mv
T
Two parameters in the soil moisture zone of the HBV-model, f3 and L (see chapter 7.2.2), were fitted for snowfreeperiods with the different criteria.As can be seen from table 5.1 the choice of criterion strongly effected the optimum parametervalues.
Table 5.1. Optimum parameters in the Stormyrabasin with different numericalverificationcriteria.Fc = 50 (see chapter 7.2.2.) F abs L /Fc
Fdiff L /Fc
F2 mv R L /Fc
1963
8.0
0.8
2.0
0.2
4.o
0.6
4.o
0.6
1964
2.0
0.2
8.0
0.2
4.0
0.2
4.0
0.2
1965
8.0
1.0
16.0
0.4
4.0
1.0
4.o
1.0
1966
4.o
0.4
16.o
0.6
8.0
0.6
8.0
0.6
1967
4.0
1.2
16.0
0.4
32.0
1.2
32.0
1.2
1968
4.0
0.8
4.0
0.2
4.0
1.0
4.0
1.0
1969
8.0
1.0
8.0
0.8
4.0
1.0
4.o
1.0
Average 5.43 0.77 10.00 0.40
8.57 0.80
8.57 0.80
The average parametervalues in table 5.1 are computed for this comparison only. When fitting a model for more than one season, optimum parameters should be sought from the joint error functionand not as a mean of the optimum parametersfor each individualperiod.
5.1.2. The accumulateddifference
A graph showingthe accumulateddifferencebetween the observedand computed hydrographshas been a valuable help when analysingthe results of the model.
( 5.7 )
at = where at
= accumulateddifferenceat time t,
Qc(i) = computed dischargeat time (i) = observed dischargeat tie i.
If at is plotted along the time axis of the runoff, it is a convenientway of keeping track of errors in the simulatedvolumes over long periods of time. The usefulnessof the accumulateddifferencewas recognizedby WMO (1975 E) when recommendingits use as an alternativeto double mass plots of computed and recorded discharges.
5.1.3. Scatter diagrams of peak flows If the modelling of high flows is of particular interest, plottingsof recorded
peak dischargeagainst computedpeak dischargecan be studied.Regarding
scatter diagrams the problem with visual inspectionof a graphicalrepresentation remains,butitcan be overcome,if some numericalcriterionof goodness of fit is adopted. WMO (1975 B) applied the method to maximum monthly flows, but in the work on
the HBV-modeleach peak has been studied sepa-
rately, neglectingminor timing errors. Scatter diagrams have not been plotted regularly for all catchments.A few exampleswillbe given in chapter 8.1.2.
5.1.4. Flow duration curves
In the recommendationson verifcation criteriathe WMO (1975 B) emphasizes the use of flow duration curvesas theyconvey
a maximum of information.Such
curves were computed when applying the HBV-modelto four catchments,aswill be shown in chapter 8.1.3. With flow duration curves, as with scatterdiagrams,the analysis of a gra-
17
phical representation remains a problem. In this case the curve represents the accumulated relative frequency of discharge. Because of the relatively few events with high flows, the curve must be interpreted carefully. The applications of the HBV-model indicate that the low 2lows easily can be overemphasized. Therefore a combination of scatter diagrams of maximum flows and flow duration curves is preferable if a representation of flow statistics is sought.
5.2. Split sample test
The test of a model must be carried out in two steps, if there are free parameters to optimize.
Calibration of the model. The free parameters are adjusted until an acceptable agreement between the observed and the computed hydrographs is obtained. Test on independent data. The calibrated model is run for a period that was not used for calibration and thus did not effect the final values of the free parameters.
It is often argued that this procedure is a waste of information, and that as long a record as possible should be used to derive parameter values. This is certainly a valid point, but without the test on independent data our knowledge of the behaviour of the model is very limited. We only know that the given model successfully manages to interpolate a curve between a series of points. This can
be achieved with several
mathematical expressions, provided the number of constants (free parameters) is
sufficient. The performance of the model is thus better re-
flected by the independent test period.
The distinction between fitting and calibration is particularly important when testing snowmelt models, because of the few events available for parameter estimation every year. This was exemplified clearly when 2, applying the HBV-model to the Torpshammar catchment (4 230 km ). The model was first calibrated to a close fit as shown in fig. 5.3. The independent period in fig. 5.4 revealed, however, that the calibration period had been too short to obtain proper parameter values. The close agreement in fig. 5.3 had little to do with modelling because of the
large number of parametersand the few degrees of freedom for each estimate (i.e. few hydrologicalevents).
TORPSHAMMAR
1967 —1969
—RECORDED
200
•••
••COMPUTE
100
0
NDJIF.144114J1
ASONO
IFINAIN)1NS
Fig. 5.3. Calibrationperiod in the Torpshammarbasin (1967-69).R2 = 0.95.
200'
TORPSHAMMAR
(Js)
1969-1971
NECONDe CONPUTED
100
0
NI)JrNA141.1A
SONO.IFNAN.I.IAS
Fig. 5.4. Independenttest period in the Torpshammarbasin (1969-71).R2 = 0.50.
If fig. 5.3 and 5.4 are analysed closer, it is evident that the climaticvariability is one of the main sources of the unstable parametervalues. In order to get a longer record the work in the Torpshammarcatchmentwas abandonedand interest was concentratedto its upper half, the Gimdalsbyncatchment.A four year period was fitted, as shown in fig. 5.5. The independenttest period still caused problems, as can be seen in fig. 5.6, due to the fact that some of the springfloods were of much higher order of magnitude than those used for calibration.
19
GIMDALSEWN1961-1965 Q(rn//s) COMPUTED RECORDED
100
50
ONDJFMAMJJ
ASONDJFMAMJJ
ASONDJ
FMAMJJ
ASONDJ
FMAMJJ
AS
Fig. 5.5. Calibrationperiod in the Gimdalsbyncatchment.R2 = 0.83.
GIMDALSE3YN1965-1969 Q(m)/s) COMPUTED RECORDEO
100
50
ONDJFMT-A-5,-,1JJ
ASONDJFMAMJJ
ASONDJ
FMAMJJ
ASONDJFMAMJ
J AS
Fig. 5.6 Independenttest period in the Gimdalsbyncatchment.R2 = 0.81. The recalibrationof the model for the entire eight year period was not successfuluntil the model structurehad been modified (Bergström,1975). An
independenttest for anotherfouryear period gave results comparable
to those during calibration,and the model was accepted (see appendix 1, fig. A 16). The results from this Drocedure,Dresented as R2-values, are given in table 5.2.
Table 5.2. R2-values in Gimdalsbyn.Independenttest periods underlined. Ori inal model
Modifiedmodel
1961 - 1965
0.83
0.86
1965 - 1969
0.81
0.91
1969 - 1973
0.86
It must be stressed,however, that althoughthe test period behaved fairly
20
well with the modified model, the performancewhen simulatingeven larger or smaller events is still uncertain. The more free parameters in the model, the more the problem with proper parameter values will become pronounced.Kuchment (1972) exemplifiedthis when analysingmodels with a differentnumber of free parameters (table
5.3).The results showed that an increasingnumber of parametersalways improvedthe fit of the calibrationperiod, while the independentperiod attainedthe best agreementwith only a few parameters.
Table 5.3. The effect on the sum of squares criterionwhen gradually increasingthe complexityof the model (Kuchment,1972). Number of parameters
Year
3
4
6
10
Optimization
1959 a 1960 1965 1964 1965
b b a a
23.7 15.9 41.1 7.1 54.6
17.4 10.6 38.7 5.8 52.4
13.6 9.0 36.6 9.0 34.7
13.2 6.6 26.6 2.5 28.8
142.4
124.9
102.1
77.7
(Independentdata) Control 1962 a 1962 b 1959 b 1964 b 1960 a
E tot.
43.7
49.2
50.9 99.6
41.3 82.0
44.8
32.1
19.6
32.0
30.4
23.3
30.6
26.9
269.4
227.9
411.8
352.8
259.8 361.9
267.8 345.5
48.3 68.4
42.1 57.2
92.9
109.6
Split-sampletesting has been acceptedby most modellers,butunfortunately work still appears which only shows the fitted period. Dawdy, Lichty and Bergmann (1972) support the split-sampletheory when stating: "Accuracy should be measured in terms of prediction,rather than in terms of fitting.Accuracy of fitting indicatesonly how well the model can reproduce a set of data from adjustedmodel parameters.Accuracy of prediction indicateshow well the model can reproducea set of data that was not used to derive the parametervalues. Therefore,predictioninvolvesan independent test of accuracy of the model." In the WMO project on "Intercompari-
21
son of ConceptualModels Used in OperationalForecasting"(WMO, 1975 B) split-sampletesting was consideredthe best way of comparingthe performance of the participatingmodels.
The optimal length of the calibrationperiod is a delicate questionwhich has to be consideredwhen fittingthe model. The period must include sufficient information,in terms of hydrologicalevents, for the estimation of the parametersthat are valid under all conditionsin the basin. A successful independentperiod is no guaranteefor proper parametervalues, if both the calibrationand the test periods are unrepresentativeof the hydrologicalregime. With a conceptualmodel the hydrologistis, however, in a better position than without any model at all, as an extreme event will mostly give an extreme response even if the exact behaviouris represented poorly. Fig. 5.7 is an example where the highest peak during the independenttest period (2 193 l/s) was more than four times as large as the maximum flow during calibration(498 l/s). The model underestimated the peak (1 680 l/s) but overestimatedthe total volumes. In general can be said that the model performedwell taking into considerationthe relatively small peaks during the calibrationperiod. The number of years or seasons used when calibratingand testing the HBVmodel in different catchmentsare summarizedin table 5.4, which clearly reflects the need for long periods when working with snowmeltmodels. Table 5.4. Years or seasons in the calibrationand test periods. Calibration
Test
Snowroutine
Lilla Tivsjön
1
2
no
Nolsj5n
2
no
Stabby Stormyra
3 4
3 6
no
Solmyren
1
3 2
no
Gimdalsbyn
8 8 8
4
yes
5 4
yes yes
4
yes
5 3
yes
Kultsjön Malgomaj Malgomaj + Kultsjön Ströms Vattudal Filefjell
8 8 4
no
yes
22
NOLSJdN NEDRE 1967
ACC. DIFF.(1 /s) 6 000
4 000
2000
0
-2000
0,(t/s)
OBSERVED HYDROGRAPH
1200
COMPUTEDHYDRO6RAPH
1000
800
600
400
20 0
0 P(rnm) 40
20
0
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
Fig. 5.7. Independenttest period in the Nolsjön basin. (The highest level during calibrationis indicatedby a broken line.)
5.3.
Ma in s of the errorfunction
Mappings of the error functionhave proved very useful when developingthe HBV-model.The principle is to evaluate the verificationcriterion in a grid, defined by differentparametervalues. The topographyof the response surface is visualized through isolinesand subject to analysis.The method was suggested by Nash and Sutcliffe (1970)and exemplifiedby O'Connell,Nash and Farrell (1970) and by Mandeville,O'Connell,Sutcliffeand Nash (1970) in a series
2
of papers on conceptualmodels. Further exampleswere given by Plinston (1972)and Dickinson and Douglas (1972). The method has been used when calibratingthe HBV-modelunder snowfree conditions (Bergstr8m,1972 B) as all but two parameterscould be estimated from the map, from analysisof the hydrographsor by considerationof the characteristicsof the catchment.A more importantapplication,however, is the investigationof the general behaviourof the model as regards sensitivityto its parametersor their interdependence. When applying a model to a new catchmenta general knowledgeof the error functiontopography simplifiesthe estimationof parametervalues. If the stationerypoint is situatedon a large plateau as in fig. 5.8.A, we can expect to reach a good fit without much adjustment,as the model is rather tolerant to deviationsfrom optimum parametervalues. If the optimum is well defined by steep slopes as in fig. 5.8.B, large deviationsfrom optimum parameters cannot be accepted.If, finally,the response surface has the shape of a long flat valley, as in fig, 5.8.C, we can concentrateour adjustmentsto one parameterkeeping the other at a reasonablevalue.
2
2
cl
2
cl
cl
Fig. 5.8. Some alternativeshapes of the response surface of the error function.C1 and C2 are free parameters. Fig. 5.8.C. representsa situationthat is often encountered.If sensitivity analyses are carried out with all but one parameter constant,the result will be a false impressionof sensitivityto both parametersC1 and C2 as the analysesrepresent diagonalcuttingsthrough the valley. If the shape of the error functionis of type C, it can be worth considering simplificationsof the model by the exclusionof one parameter.
2
Examples of this application will be given in chapter
T.
It is important to realize a few points when interpreting the topography of the error function. The error function topography is only showing the response of the verification criterion and is only valid as long as this criterion is properly describing the performance of the model. Only a few parameters can be studied simultaneously. The conclusions are therefore valid for this particular set of parameters only. Consequently it is important to choose parameters that are likely to interact or show some other interesting qualities.
It is important not to judge the relative importance of the parameters from the density of the isolines only. The structure of the model must be considered as well in order to draw the correct conclusions. An important parameter controlling the level of baseflow, for example, will effect the sum of squares error much less than will a parameter for the timing of high peaks. Examples of mappings in three dimensions (two parameters) and four dimensions (three parameters) will be given in the discussion of the model structure in chapter
7. In all applications F2, according to eq. 5.1, will
be used as verification criterion. Systematic error function studies by the HBV-model can be found in the work by Bergström and Forsman (1973), Bergstr.5m (1975) and Bergström and Jönsson (1976 B). Some guidance to the interpretation of the response surfaces in this text is given in appendix 2.
5.4. Automatic calibration The need for rapid estimates of parameters in complex mathematical expressions has led to the development of a large number of algorithms for automatic optimization, or minimization, of an objective function. Hydrological modelling is a field, where these methods are often applied for calibration purposes. One of the most popular "hill climbing" techniques was developed by Rosenbrock (1960). The search for a minimum (or maximum) of the objective function is made in a very
efficient way by means of a transformation of the coor-
dinate axes defined by the parameters in question. The procedure is illust-
25
rated schematiCallyin three dimensions (two parameters)in fig. 5.9.
C2
STA T
cl Fig. 5.9. The search techniquedevelopedby Rosenbrock (1960)with 01 and 02 as free parameters. Ibbitt and O'Donnell (1971) found this method very efficientin a comparative test of differentoptimizationprocedures. The Rosenbrockmethod was adopted in the work on the HBV-modelwhen the introductionof a snowmeltroutine increasedthe number of free parameters. It proved to be able to fit the model rapidly but there were many restrictionson its usewhich gave rise to more scepticismthan enthusiasm. The use of automaticmethods demands a verificationcriterion which is objectivelygiving the best fit expressedas one single value. This problem was discussedfurther in chapter 5.1. The lack of informationabout the error functiontopographywas soon felt very unsatisfactory.The optimizationthrough visual inspectionand successivetrials gives a much better feeling for the sensitivityof the model to its parameters.This is particularly importantwhen developinga model. Restrictionsmust be put on the parameterstoprevent their values from violatingthe conceptsbehind the model structure.With these restrictionsthe restrictingvalues will not infrequently be found as optimum values. The optimizationproceduremight end up in a local instead of an absolute optimum, a problem which increaseswith the number of free parameters.
26
5. The calibrationof snow melt models requires long calibration
periods especiallyin highly danped catchments.Due to the limited size and capacity of the SAAB D22 ocn;u_ter used-in the work
on the
Rosenbrockmethod the calibrationperiod had to be split into parts. The computer time needed for each calibrationwas still considerable, and we had introduceda new problem, trying to average the optimum parametersfrom the differentcalibrationperiods. The existenceoflocal stationarypoints far fror_the absolute optimum was exemplifiedwhen applying the Rosenbrockmethod to the Gimdalsbynbasin. The snowmeltparameterswere given such poor initial estimatesthat the springfloods were completelyout of phase. The automaticmethod, instead of adjusting this, simply gave the parameters such values that the springfloods were suppressed. Generally it can be said that the confidencein automaticcomputer based procedures for calibrationwithout any human interactionhas decreased during the work on the HBV-model.This is in agreementwith experience elsewhere.Burnash, Ferral and McGuire (1973) suggest the use of a manmachine interactiontechniquewith initialmanual calibrationand possibility for a final "polishing" of the parameters with an automaticalgorithm. WMO (1975 B) in the report from the project on "Intercomparisonof Conceptual Models Used in OperationalHydrologicalForecasting"makes the following statement:"As far as possible,a combinationof manual and automatic procedures should be used in model calibration." Sugawara,Ozaki, Watanabe and Katsuyama (1974) state that the calibration of a model "somewhatresemblesthe driving of a motor car on the street. It is very difficultand probably impossibleto make an automaticmachine that can drive a car on the street,but many men can drive easily".
5.5. Sub'ectivecalibration
"In hydrologicresearch,there is no excuse for avoidable aubjectivity. Nevertheless,the hydrologicliteratureis full of models justifiedby a single illustrationwhich shows that the predicted outputscloselyresembles the actual output. Such 'optimizationby eye' is incapableof being
integratedinto a general bcdy cf scientifickncledge and is unworzhy of the name of scientifichydrclogy.
The above statementwas male by :ocge (1973,page 150). Theoreticallyit is a valid statement,practically,however, it is not easy to comply. The problem is thaz if we use an tobjective"verificationcriterion,we conceal instead of avoid subjectivity.The choice of criterionis a subjective procedure in itself. Furthermore, the risk is that we blindly believe in the criterionwithout any glance at the hydrographcomputedby the model and without sufficientknowledge
of
the representativenessof the crite-
rion of fit. Visual inspectionof the computedand observed hydrographshas regained confidenceand is ncw used in combinationwith the graph of the accumulated differencewhen calibratingthe HEV-nodel.The R2-value (eq. 5.3) is computedas a complement.A multicolorplotter simplifiesthe analysis considerablyas not only the hydrographsbut also the input variables and the conditionsin the differentcomponentsof the model can be visualized simultaneously.Examplesof plottingsare shown in appendix 1 (fig. A 17- A 21). The time scale is often expandedcomparedto these figures in order to increase the resolution. The calibrationis generallycarried out in four steps with some interaction between them. The confined parametersare determinedfrom the map and from other informationabout the catchment. First estimatesare given to the free parameters.These can be based on experiencefrom other catchmentsor analysis of the hydrograph.An initial test run is made. Parameters,whichmainly effect the volumes,areadjusted after inspectionof the accumulateddifference. The remainingfree parametersare adjusted after visual compari-: son between the computed and observedhydrographs. Point 3 and 4 are repeated after each test run until an acceptableagreement is obtainedbetween the computedand observed hydrographs.What is i_tacceptable" will not be determineduntil
severalunsuccessfulattempts
to adjust the model further have been made. A detailed discussionof this
proceduretogetherwith a completeexample from the Str8= Vattudal catchment was given by Bergstr5mand Jönsson (1975). The number of runs,needed to calibratethe model with this trial and error technique,and the degree of modificationof the structurewhen applying the model to differentcatchmentsare shown in table 5.5. Table 5.5. Number of runs and degree of modificationwhen calibratingthe HEV-model. Catchment
De ree of modification
No. of runs
Gimdalsbyn
moderate IT
33 38
Str5ms Vattudal
no
18
Malgomaj
no
16
Filefjell
low
Kultsjn
9
As the catchmentsare presented in the order in which they were tested, table 5.5 reveals that the experienceof the hydrologistis a factor stronglyeffectingthe efforts needed for the calibrationof hydrological models.
29
6. DATA BASE The complexityof a model is limited by our knowledgeof the physical processes and the quality and quantityof availabledata. If the model is to be used for forecasts,the possibilitiy to forecasta certain variable can also be a limiting factor. When calibratinga model the need for long, horogeneousseries will exclude all instrumentationthat does not have the status of a permanent station.Detailed studies of
short duration can be used to back up the
model structurebut will not be of mucl-help for operationalpurposes. The main input and output variables in this work are precipitation,temperature,potential evaporationand discharge.Other meteorologicalvariables have been used in some special investigationsin attempts to improve the model. The number of precipitationand temperaturestationsused in each basin is shown in table 6.1. In the Swedish catchmentsdata were provided by the SMHI, while for the Filefjelland Steinslandsvatncatchments data from the NorwegianWater Resourcesand ElectricityBoard were used. Table 6.1. The number of temperatureand precipitationstationsin the applicationsof the HBV-model. Catchment
Area (km2)
Preci itation
Ter erature
L. TivsjOn
12.7
1
Nolsjön
18.2
1
6.4 4.0
1 1
Solnyren
27.5
1
Gimdalsbyn
2178
3
1
Kultsjön
1109
3
1
Malgomaj
1862
3
1
Malgonaj + Kultsj5n
2971
2
Ströms Vattudal
3851
5 4
Filefjell
154
4
1
Steinslandsvatn
216
1
1
Stormyra Stabby
2
30
6.1. Precipitation Daily totals from standardgauges were used in all test cåtchments.Areal means were computedby Thiessenpolygons in the Gimdalsbyncatchment,while the topographyand representativenessof eacb gauge were consideredin the other catchments. Changes in
precipitationdue to elevationdifferenceswere accounted
for by a parameter,Plapse' in all large catchments (> 100 km2) with the exception of Gimdalsbyn,where the altitude range is moderate. This will be discussed further in chapter 7.1.10. In the other catchmentsuncorrected precipitationvalueswereused for the snowfreeperiods. A snowfall correctionfactor, Csf'Was used in the snow accumulationprocedure as will be discussed in chapter 7.1.2.
6.2. Temperatures Temperaturesmeasured in standardthermometershelters1.5 - 2.0 m above the ground were used in all the
catchments.Daily means are generally
computed,fromthree daily readings and maximum and minimum observations, by empiricalcoefficientsderived at the SMHI (SMhI, 1966). In case data were missing, means of daily maximum and minimum temperatureshave been used. In most catchmentsonly one station was
available. If two
stationswere at hand their artihmeticmeans are representingthe areal temperature. The temperaturevalues are subjectsto two correctionfactors. One is a thresholdvalue, To'whichcan
be said to correct for poor representative-
ness of the station,aswill be discussedin chapters 7.1.1 and 7.1.4. The other is a lapse rate, Tlapse accountingfor the temperaturegradient in catchmentswith considerableelevationrange. Tlapse is discussed further in chapter 7.1.10.
6.3. Potentialeva oration
Different ways of computingthe potential evaporationhave been used in the differentcatchments,dependingmainly on the availabledata. Wallen (1966) computed long term means of the potential evaporationwith Pen-
31
man's formula att differentlocationsin Sweden. These monthly values have been used with a correctionfor monthly mean temperaturesin L. Tivsj5n and Nolsjön. The same values vere used without any temperaturecorNalgomaj and Ströms Vattudal. rection in Stormyra,Stabby, Kultsj'ån, Monthly totals of the potentialevaporationwere computed by Penman's formula, specificallyfor the catchmentsof Solmyren (Persson, 1972), Gimdalsbynand Filefjell. Differentvalues of the potential evaporationwere tested when applying the model to the Stormyrabasin outside Stockholm (Bergstr5m,1973). The conclusionfrom this investigationwas that the results with daily values computedby Penman's formula are slightlybetter than those with monthly averages and still better than those with the values computedby Wallen. The choice between these methods had some effect on the optimum parameter values in the soil moisture zone. When using mean monthly values of potentialevaporation,a subjectiveinterpolationwas carried out in the smallerbasins to avoid unwanted effects due to discontinuitiesin the histogramformed seasonal curve. As can be seen in the graphicalrepresentationof evaporationin appendix 1, fig. A 17 - A 21, this techniquehas not been used in the Kultsj3n,Malgomaj, Stråms Vattudal or Filefjellcatchments.In Kultsjön comparativetest runs were made with a smoothedevaporationcurve accordingto fig. 6.1.
E (mrn/day) 3 2
F
M
MJJA
Fig. 6.1. The smoothedcurve for potentialevaporationin the Kultsjön catchment. The effect on the discharge,when using the smoothedcurve instead of the histogram,wasnot detectableby eye in the computedhydrograph.The results expressedas R2-valuesare shown in table 6.2.
32
Table 6.2. The effect, expressedas R2-values,of a smoothedcurve of potential evaporationin the Kultsjön catchment.
Histo ram 1962 - 1966
1970 - 1974
0.7977 0.8402
Smoothed
0.7975 0.8399
The results are interesting,as they show that random errors in the computations of potential evaporationare of minor importanceas long as the volumes are correct. This conclusionwassupportedby Parmele (1972)when studyingthe effect of errors in potential evaporationon the output from hydrologicalmodels.
6.4. Runoff Runoff has been measured by means of dischargeweirs in the small basins including Filefjell. In Gimdalsbynobservationsof lake levels were transformed to outflow by a stage-dischargerelation for the outlet of lake Idsjön. In Ku1tsjOn3Malgomaj and Ströms Vattudal the model was used to simulatethe local inflow to reservoirswhichare parts of a rather complex system of hydroelectricpower stations.Thereforethe recorded hydrographhad to be computed accordingto: Q
loc
= Qtot + ASr - Qin,
(6.1)
where Q10C
= local inflow to the reservoir,
= total outflow as reported by the regulationcompany, Qtot = inflow from an above situatedreservoiras reported by the regulaQ. tion company, LSr
= change in storage in the reservoirfrom observationsof its level.
As all terms in eq. 6.1 are subjectsto errors, the resultinghydrographis rather uncertain.Particularlycritical are the observationsof the level of the lake, or reservoir,as one reading will effect the storagecomponent, ASr, of two consecutivedays. These uncertaintiesare causing fluctuationsin the hydrographas can be seen in appendix 1 (fig. Al7 - A 20). In Ströms Vattudal the runoff values were averaged for three days in an attempt to overcome the problem, but when working on the other catchmentsthe use of a
33
multicolor plotter made it possible to analyse the hydrographs in spite of the fluctuations.
6.5. Conse uences of errors in data
The errors in the data will effect the modelling procedure differently depending on their type and origin. S stematic errors may be caused by poor measurements, poor representativeness of a station, or by uncertainties in the computations when estimating a variable. They can, to some degree, be accounted for by the free parameters when calibrating the model. These implicit corrections will effect the optimum parameters and thus complicate the generalizations of these. Random errors may have the same origin as systematic errors. They cannot, however, be accounted for implicitly by the model. They will therefore effect the performance of reconstruction. Random, uncorrelated, errors in the input variables will cause persistance in the residuals between the computed and the observed hydrographs due to the different storages in the model structure. In operational hydrological forecasting these persistent errors can be eliminated by means of an updating procedure, as will be discussed in chapter 8.2.1. Inhomo eneities in a data series is a type of systematic errors, which may be caused by, for example, a changed location of a meteorological station or a new method for the estimation of runoff during the period subject to analysis. Due to the implicit corrections by the free parameters inhomogeneities will cause unstable parameter values. Bergstr3m and J3nsson (1975) showed an example, where a slight modification of the procedure for the computation of local inflow to the StrOms Vattudal reservoir was detected when calibrating the HBV-model.
3)4
T. THF NCDEL STRUCTURE
The simulationof runoff by
the HPV-modelis made in three steps.
Snow accumulationand ablation. Soil moisture accounting. Generationof runoff and transformationof the hydrograph. The basic philosophyhas been to regard these steps as relativelyindependent. First of all snow accumulationand melt is considered.Neltwater and rainfall is then fed into a soil moisture accountingroutine, which is the most essentialpart for the determinationof runoff volumes. Finally the volumes are given an appropriateshape in the response function including rcuting of the,waterthrough a series of reservoirsand damping of the hydrograph by neans of a transformationfunction. A survey of the most importantstructures,which have been attempted,is shown in fig. 7.1. As the intentionis to illustrateonly the general linkage of components,the subroutinesare describedmore in detail in the following chapters.Either these subroutinescan be lumped or tey can account for the statisticalor geographicaldistributionof characteristicsof the catchment. Although the mcdels in fig. 7.1 look different,they are all developments from the first very simple HBV-1 structureand will also degenerateto this version with appropriateparameter settings.A list of parametersin the HEV-3 version is given in table 7.1. Each parameter is subjectto a more detailed descriptionin the chapter referred to in the table. The symbols used in table 7.1 can be criticized.fornot being a very consequent system of notation.The author is aware of this but has tried not to diverge too much from symbclsused in other publicationson the HEV-model.Table 7.2 shows where the differentmodels have been applied so far. "Enowroutine"in the column of comment;means that a
snowroutinehas been incorporated
and that the model has been run throughoutthe year. "Distributedsnowroutine" indicatesthat a distributionof this snowroutineaccordingto the areaelevationcurve was made.
35
Table 7.1 Parametersin the HEV-3 model. Chapter
Correctionson in ut variables = correctionfactor on rainfall P corr correcticn = precipitation-elevation P lapse Tlapse = temperature-elevation correction
7.1.10
To
7.1.1, 7.1.4
7.1.10 7.1.10
= general temperaturecorrection
Parametersin the snowroutine Csf
= snow fall correctionfactor
7.1.2
Co
= degree-daymelt factor
7.1.4
Cwh
= water holding capacity
Sb
= bottom storageunder snowpack
Parametersin the soil moisture routine Fc
= maximum soil moisture storage
7.2.2
L P ' 13
= linit for potential evaporation
7.2.2
= empiricalcoefficient
7.2.2
Parametersin the res onse function = storage dischargeconstant in the upper zone
Ko K1
=
K2
=
11
/T
11
t/
ti
I/
"
lower
r(
L = limit for slow drainage iiithe upper zone uz Cperc = percolationcapacity into the lower zone = part of the lower zone representinglakes and other wet areas
pw
Bmax -= maximum base in the transformationfunction e
relating the base in the transforrout = parameter mation functionto the generatedflow
7.3.2 7.3.2 7.3.3 7.3.2 7.3.3 7.3.3 7.3.4 7.3.4
RAI N, EVAPORATION SOIL MOISTUREZONE
TIME LAG
Ld‘krEFT 2-6NT E,-.1-; Q_=K•S -g 2 lz
HBV-1
PRECIPITATION
SNOWROUTINE RAIN, SNOWMELT, EVAPORATION SOIL MOISTURE ZONE
- Q1= Kl*Suz UPPERZONE,
„ `42=^2.Stz
HBV-2
TRANSF. FUNCT1ON
37
P RECIPITATION
SNOW ROUTINE RAIN , SNOWMELT, EVAPORATION SOIL MOISTURE ZONE
UPPER ZONE,
Q0=K0
Q•
LOWER
j lz
=K2-Siz
uz
Luz)
uz 1S TRANSF. FU NCTION
HBV-3
PRECIPITATION
S NOW ROUT1NE RAIN SNOWMELT, EVAPORATION SOIL MO1STURE ZONE
P0C 90= .S uz.)
=K .Suz
—— s Suz 2
121 T RANSF. FUNCTI 0 N
L Q =K•S 2 2 lz
HBV-4 Fig.
7.1.
Schematic
representation
of
different
versions
of the
HBV-model.
38
Table T.2. Model structuresin the differentcatchments.
Catchment
Model
L. Tivsjön
FEV-1
Nolsjn
HBV-2
Stalby
HEV-2
Stormyra
HBV-2
Solmyren
HBV-2
Gimdalsbyn
HEV-2
Snowroutine
KultsjOn
HBV-2
Distributedsnowroutine
Ströms Vattudal
HBV-2
ti
11
Malgomaj
HEV-2
It
I.
Filefjell
HBV-3
11
IT
Steindalsvatn
HBV-3
!?
11
Comments
The soil moisture accountingproceduresare essentiallythe same in all versions of the model. The subsequentincrease in the number of runoff components is justifiedby experiencefrom the differentcatchments.Two componentshave generallybeen sufficientin the Swedish catchments,but when applying the model to the Filefjellalpine basin three components were detected.Two ways of modelling this were attempted,representedby the HEV-3 and HBV-4 models. HBV-3 was consideredsuperiorboth according to visual inspectionand in terms of the R2-criterionof fit. This comparison was discussedin detail by Bergstri5m and Jånsson (1976 E).
All versions of the HBV-modelare lacking componentsfor direct surface runoff, as the water is
controlledby the conditionsin the soil mois-
ture zone before any runoff can be generated.The position of the upper zone below the soil moisture zone is more an indicationof the order in which computationsare carried out than an attempt to describethe natural system. It would probably be more correct to integratethe soil moisture zone and the upper zone, as the upper zone representsthe superficialdrainagewhich might occur within the topmost layers of the soil. For computationalreasons it is, however, convenientto make this separationinto two storages, each one with its own budget. The soil moisture zone as a buffer controllingthe total volumes has been verified by a lot of applications of the model. Fig. 7.2 shows an example where a surface runoff componenton top of the soil moisture•zone
39
would certainlyyield poor results. After a long dry summer 46 mm of precipitationin August did not cause any runoff, which was modelled well by the HBV-2 model.
OBSERVED HYDROGRAPH
GL(t/s) 600
COMPUTED HYDROGRAPH
400
200
0 P (mm) 40
20
0
OCT
NOV
Fig. 7.2. Simulationof-the response to a rainstormafter a long dry spell. Nolsjön (1969). One explanaticn of the good performance of this model, without a direct surface flow component,might be the highly pervious soils in the test catchmentsand the fact that rainfall intensitiesare mostly moderate. One must, however, bear in mind that the soil moisture zone is accounting for all losses, includinginterceptionin vegetation,and thus it is ntorean index of the total wetness of the catchmentthan a detaileddescription of the conditionsin the soil. Thereforedetailed explanationscan be questionable.
The capabilityof the model to reconstructa given hydrographis the rule, by
which its performanceis judged. The fact that a model is doing this
properly must not, however, lead us to the conclusionthat all components in the model are in agreementwith the real hydrologicalsystem.Axelsson (1975) showed that he could obtain almost identicalresults,aswith the HBV-2 model when applying a model with two parallell soil moisture zones and two parallell reservoirsto the Stormyrabasin. To take the capabilityof reconstructionas an evidence of physical relevancewould thus lead to two contradictorystatements.This phenomenumwas recognizedby Amorocho and Hart (196)-t) as the problem of nonuniquenessof the processesof synthesis.
7.1. Snow accumulationand ablation One glance at an annual hydrographis sufficientto realize the significance of snow accumulationand melt for formationof runoff in Sweden (see for example fig. 5.3). To model these processesmeans that a few central problemsmust be considered. Determinationof the form of precipitation. Correctionsfor the aerodynamiceffect around the precipitation gauges and poor representativenessof the meteorologicalstations. Determinationof evaporationlosses.
4• Computationof snowmelt. Estimationof retention and refreezingof liquid water in the snowpack. Correctionof precipitationand temperatureaccordingto the altitude differencesin the catchment. A physicallycorrect way to approach the snow modellingproblem would be from the total heat budget of the snowpack,as discussedby,for example, U.S. Corps of Engineers (1956) and (1960),Forsman (1963) and Kuz'min (1972). The general equationwould be of the form: Wsw + Wlw + Wc + Wl + Wg + Wp
+ Wt + Wm = 0
(7.1)
where: = absorbed short wave radiation, sw Wlw = net long wave radiation,
W
Wc
= convectiveheat flux,
W1
= latent heat flux (condensationand evaporation),
W
= heat flux from the ground,
W
= contributionof heat from precipitation,
Wt = change in the energy content of the snowpack, m
= heat equivalentof the snowmelt.
Among the variablesnecessary for a completeheat budget computationacccrding to eq. 7.1 can be mentioned: total solar radiation, albedo, longwave radiationbalance (effectiveradiation),
)41
air temperature, air humidity, wind speed, temperaturegradients in the soil and ir the snow, precipitation. In addition to these variables some physical parametersgoverningheat exchange with the atmosphere,heat transferwithin the snowpack,liquid water content in the sncw and drainage of the snowpack,wouldhave to be estimated. Consideringthe data that are generallyavailable in a catchmentdetailed heat budget computations,suchas eq. 7.1,are hardly warranted.The great inhomogenitiesin the areal distributionof the snow cover is complicating the picture furter. Rather crude index methods are thereforeoften preferred in operationalmodels. When developingthe snowroutinefor the HBV-modelrecordingsof air temperature and precipitationhave been the mair input variables.The air temperatureis used as an index and is thus not only representingthe convectivecomponentin eq. 7.1, but also other effects such as radiation and condensation.The routine was initiallydeveloped for the Gimdalsbyn catchment (BergstrOm,1975) but has since then been modified in the light of experiencefrom other applications.This will be discussed in the remaining part of chapter 7.1, in which some alternativemethods and special investigationswill be presentedas well.
7.1.1. The form of reci itation The problem with determinationof the form of precipitationis usually solved in a rather simple manner. The air temperatureis accepted as a determiningfactor meaning that snow accumulationstarts as soon as the temperatureis lower than a certain thresholdvalue, T. This method has been used by the U.S. Corps of Engineers (1956) and
Anderson (1973)
among others. According to an investigationmade by the U.S. Corps of Engineers,the To-valuemay vary beti,Teen - 1.7 °C and + 4.4 °C when studying hourly values. An investigationof daily values made at the Lilla Tivsjön climate station in Sweden is shown in fig. 7.3 (Bergström, 1975).
RAIN RAIN+ SNOW
.
.
SNOW
-4
• ••
:
• .
..:.:.:
-5
••
-3
-2
-1
.t.0
1
2
3
4
5 TEMP (°C)
Fig. 7.3. The observer'snote on the form of precipitationrelated to mean daily temperatures.Each point representsone day with precipitation. The variabilityof To indicatedby these investigationsis of couree inherent in the crudenessof the method. The risk for erroneous floods when using the To as a thresholdvalue,is greatest when there is no snow on the ground or if the snowpack is ripe, i.e.
filled with liquid water to its water hol-
ding capacity.Liquid precipitationon dry snow seldom effects runoff, as the water mostly
will be retained in the snowpackand refreeze later
on. So far the method with a thresholdvalue, To, has been used in all the applications of the HBV-model.The same parameter is used as a general correction for representativenessof the temperaturestation in the snowmeltroutine. Examples of parametervalues are given in chapter 7.1.4.
When the precipitationreaches the ground, the temperatureof the soil determines whether it will remain in its original form, freeze or melt. Computations of the soil temperaturecan easily be very complicated,but a simple heat budgetrighthelp to improve the snow accumulationroutine when the ground is snow-free.This has not been attempted in the HEV-model.The effect of frozen ground is discussedfurther in chapter 7.1.8.
7.1.2. Snow fall and eva oration corrections To estimate
the areal totals of snowfall is more difficultthan to esti-
mate rainfall. The catch of the standardgauge is often stronglyreduced due to the aero-dynamiceffect, especiallyat high windspeeds.The variability of snow accumulationover the catchment is also great,
making the repre-
sentativenessof the gauge a critical factor. The effect of the density of the forest cover on snow accurulationwas investigatedin Finland by Seppänen (1961)and in Sweden by Waldenström (1975).The results show an increasingaccumulationwith decreasingdensity of the forest. Similarresults were reported by Dietrich and Meiman (1974)when studyingthe effect of cuttings in a forest. One explanation may be that the interceptionof snow in the trees, which is subjectto strongerexposure to wind and heat exchangewith the atmosphere,will increaseevaporationlosses. The above uncertaintiescause systematicerronin the areal precipitation, as recorded by the precipitationgauges. This is correctedin the HEVmodel by an empiricalcoefficient,Csf' the snow fall correctionfactor. Evaporationand condensationon the snow surface can be calculatedaccording to a formula of the form (Nyberg,1965): E = (C +C •11)(ea el e2
(7.2)
es) • T,
where: Ea
= actual evaporationfrom snow, = wind velocity,
ea
= vapor pressure in the atmosphere,
es
= surfacevapor pressure,
CelandCe2=empiricalcoefficients, =
time period.
The introductionof this formulawill, however,require input variables, representativevalues of which are not generallyat hand in Swedish catchments. Lemmelå and Muusisto (1973)measured the net evaporationduring six melt periods in Finland and reported a total value ranging from - 0.2 to + 9.4 nm/season.Nyberg (1965) arrived at a total amount of evaporationof 25 mm during two months in spring in the extreme north of Sweden. Gray (1973) discussedmany aspects on evaporationfror a snowpackand referred to investigations, which showed
that the amount of evaporationin
winter is negligiblein northern latitudes (wherechinooksdo not prevail). When developingthe snow routine for the HBV-modelit wasfound
important
)4)4
to keep down the demand on input data. Thereforethe variaty evaporationwas neglected in the first aprroach,andth included in the snow fall correctionfactor,
of sncv
average loss as
Sf
The snow fall correctionfactor is thus playing a very imrcrtantrole in the model. As
this parameter is mainly effectingthe total volumes, it is
easy to get a good estimate after just a few test runs. The best way is to study the behaviourof the accumulateddifferencebetween the computedand the recorded springfloodsand to adjust Csf until all systematicerrors are eliminated.Typical values of Csf are given in table Table 7•3• Snow fall correctionvalues, Csf. Gimdalsbyn
0.8C
Kultsjön
1.23
Malgomaj
1.05
Stråms Vattudal
1.12
Filefjell
1.70
As the value of Csf is both dependingon the choice of precipitationstations and their respectiverepresentativeness and accountingfor evaporation from snow, its great variabilityis not surprising.Values lower than one, as in Gimdalsbyn,are possible for the same reason. The Csf-value interactsstronglywith the assumed value of the altitude lapse rate of precipitation,as will be discussed in secticn 7.1.10. The fact that Csf is one of the most criticalparametersin the model was cnnf;r~d when applyingthe HEV-modelto the Filefjellbasin without any previous calibration(BergstrOmand J8nsson, 1976 B). The difficultyto relate this parameterto catchment characteristics is one of the main obstacles on the road towards the applicationof the HBV-modelto ungauged catchments.It is evident that an increaseddensity of the precipitationnetwork would help to stabilizethe Csf-values.
7.1.3. Wind correctionson snow accumulation An attempt was•made to correct the computed snow accumulaticnfor wind velocity when applying the model to the Malgomaj catchment (Eergstr8m and Jönsson, 1976 A). Three daily readings of wind velocity were available
)45
at Stensele,50 km from the divide. The correctionwas made accordingto the followingequation:
ss = P
• (Cul + Cu2 • u),
(7.3)
where: ASs
P
= snow accumulation(mm water equivalent), = recordedprecipitation(mm),
Cul and Cu2 = empiricalcoefficients, u
= wind velocity (m/sec.)
A few combinationsof Cul and C were tested in order to study their u2 effectsonthe variabilityof volume errors over four years. Table 7.4 is showing the volume errors in mm evaluatedfrom the graphicalrepresentation of the accumulateddifference for different springfloodswithout wind correction,with correctionfor average windspeedand with correction for maximum windspeed among the three readings. Table 7.4. The effect of wind correctionson the volume errors in mm during springfloodsin the Malgomaj catchment.
1967
1968
1969
1970
Without correction
- 60
— 20
0
0
Average wind velccity
- 60
- 30
0
- 10
Maximum wind velocity
- 50
- 20
0
5
The results in table 7.4 are based on a Cul-valueof 0.1 correspondingto a 10 % increase in accumulationper m/s windspeed.Cul was adjusted to get comparabletotal volumes over the entire period as the investigationwas made not in order to reduce the total volume error but its variability. The results show that no substantialchange in the variation of the volume errors could be obtained by
equation 7.3.
Accordingto Larson and Peck (197)-) a ten percent correctionper m/s wind velocity is a reasonablefigure for an unshieldedgauge. Their investigation showed that wind corrections
in the accumulationroutine improved
the performanceof the hydrologicmodel of the National Weather Service Forecast System. It is realisticto believe that the lack of improvement when trying to account for wind speed in the HEV-modelis mainly due to poor representativenessof the wind velocity data.
46
7.1.4. Tem erature index methods for snowmeltcon utations The simplestway to c=pute snowmelt from temperaturesis undoubtedlythe degree-daymethod accordingto:
M =C o • (T - To) where: = snowmelt (mm/day), Co = degree-dayfactor (mm/C
day),
T
= surface air temperature (°C),
T
= thresholdvalue of the temperature
o
,o C).
The method has been widely applied in more or less modified form all around the world. Kuztmin (1972) refers to this method as "the simplest and most accurate of all examinedmethods". Other referencesare for example:U.S. Ozaki, Katsuyamaand WataCorps of Engineers (1956) and (1960), Eg:ugawara, nabe (1975) and Popov (1968) (in forests).A modified form was presentedby Martinec (1975). Quick and Pipes (1975) and Anderson (1973)are using the method under certain meteorologicalconditions. This type of index methods has the advantageof being simple and easy to handle. On the other hand it is difficultto generalizeits parametersas they are empiricaland cannot be estimatedthrough physical considerations. Thereforethe degree-dayfactor and the thresholdtemperaturehave to be determinedspecificallyfor each catchment. Bergstr5m (1975) showed examples indicatingthat the degree-dayfactor would increasewith proceedingmelt and introctæedthe
followingequation:
M = Co (1 + Ceff EM) (T - To), where: Ceff = a coefficientaccountingfor the ripenessof the melting snow.
Martinec (1975) obtained a similar effect when relating Co to the density of the snowpack,as the density is correlatedtc the ripeness cf snow. Anderson (1973) introduceda seasonalvariation in the degree-dayfactor3 which results in an increasingCo-value during melt periods in spring. The introductionof Ceff in eq. 7.5 complicatedthe snowroutineconsiderably.
It was necessaryto keep track of the accumulatedmelt, E, of all underlyng,layers when fresh snow was falling on snow that had already started to melt. Snowmeltwas assumed to occur at the top of the topmost layer and two consequtivelayers were consolidatedto one as soon as they reached the same value of EM as shown schematicallyin fig. 7.4.
TEMPERATURE
PRECIP1TATION SNOWPACK T1ME Fig. 7.4. The build-up and consolidationof the layers in the snowroutine accordingto eq. 7.5.
The error function response around the optimum Ceff-valueswere studied in the Gimdalsbyncatchment (BergstrOm,1975) and in the Filefjellcatchment (Bergströmand ,Mnsson, 1976 B). The study in Filefjellis shown in fig. 7.5. It is a good example of how mappings of the error function can revealaninsignificantparameter.TheoptimumCeff-values did not deviate from zero at any To-level.This result confirmedthe investigationsof Ceff in Gimdalsbynand was also supportedby visual inspectionsof the hydrographswhen attemptingnon-zero-valuesof Ceff. The conclusion is thus that although the introductionof Ceff is justifiedby field measurements, its effect cannot be observed in the dischargeat the outlet of the catchment.
The result is important,as it simplifiesthe model
considerablywith positive consequencesfor both programmingefforts and needed computertime.
48
tso
tia
-40
/ 0 5
to 15
2.o 2.5 tw
.so 3o 5s
0
Clool
0.00L
Cloob
0.004
Fig. 7.5. The response of F2 to Co, To and Ceff. Filefjell (1967 - 1971). (The dimension of F2 is (m3/s)2 • 104.)
The values of the snowmelt parameters in the different catchments are shown
in table 7.5. C
was used as an active parameter when applying the model to eff the Gimdalsbyn catchment, but the fit was almost as good when Ceff = 0.
Table 7.5. Snowmelt parameters in the different catchments.
Co (mm/°C.day) Ceff (mm-l)
1 .25
Gimdalsbyn alternative
2.0
1) 1)
(oc)
To
0.01
0
0
0
Kultsjan
3.2
0
0.5
Malgomaj
2.5
0
1.0
Ströms Vattudal
2.5
0
1.0
Filefjell
2.5
0
- 1.0
) Mean in a statistically distributed routine (chapter 7.1.9). As can be seen in fig. 7.5, there is a strong interaction between To and Co, which makes it difficult to draw any firm conclusions about the exact value
49
of these parameters.One further complicationis the areal-elevation distributionwhichwill be discussedin chapter 7.1.10. In spite of this, Co and To seem to be rather stableparameters,a factwhichsimplifies a reasonablefirst estimate in a new catchment. As a comparisonthe U.S. Corps of Engineers (1960) siggestsCo = 2.25 'Im/°C• day)and To = 0 °C in forests and Co = 2.76Tim/0C• day) and To =
-
4.4°C in open fields. The followingfigures were given by the WMO
(1975 A) as a first approximationin practicalcalculations: C
(mm/0C • da )
Dense coniferousforest (crowndensity 0.8 - 1.0):
1.4 - 1.5
Coniferousforest of average density (crown density 0.6 - 0.7) and dense mixed forest:
1.7 - 1.8
Low density coniferousand deciduousforest (averagecrown density):
3-4
The WMO points out that the figures are less accurate for open country than for forests due to increasingrelative effect of radiationand wind velocity. It can also be worth mentioningthat the degree-dayfactor is most likely to be dependenton the latitude and aspect of the catchments and the day of the year.
7.1.5. Alternativemeltfunctions Some alternativemeltfunctionshave also been tested or at least considered in the HBV-model.In the Malgomaj catchmentthe wind velocity was included (Bergströmand Jönssons 1976 A) accordingto:
m = (0m,+ 02 • -11) • (T- To),
(7.6)
where: C =empirical coefficient, m1 C. = empiricalcoefficient, n2
u
= average wind velocity(m/s).
Data on windspeedwere availableat Stenseleabout 50 km from the divide as three readingsper day. The new model was calibrated for the years 1962 - 1966,and 1966 - 1970 were used as an independenttest period. The parametervalues were Cm1 = 1.5' 0m2 = 0.55.The results are comparedto those by the originalmodel in table 7.6.
50
Table 7.6. The effect of wind corrections on the melt funct:ionin the Le.igcmaj catchment.
192
Feriod
— 1c,66
1966 — 1970
Without correction
0.7856
0.8316
With correction
0.7879
0.8472
Table 7.6 indicates a very small improvement, which was nct altogether in agreement with the impression from visual inspection of the hydrographs. Due to this fact,together with the complications when introducing a new input variable,the modified form of the melt function was not adopted. Some work on the introduction of short wave radiation into the snowmelt computations was initiated in the Gimdalsbyn catchment, but the results (never published) were questionable. Poor data and lack of information due to the damping in the basin made the interpretation of the results difficult. Studies of the effect of radiation and other meteorological variables on snowmelt would be more fruitful in a catchment with high quality data, a quicker response and less regular snowmelt seasons.
7.1.6. Water retention in the sno
When snowmelt starts. water is
ack
percolating
through the snowpack. This is
actually the most important mechanism for heat transfer from the snow surface (Kuz'min, 1972). Some of this water wi11 be retained by capillary fcrces and thus detain water yield and discharge into the river. Some water may also accumulate under the snow in its lowest layer, further delaying runoff. The water-holding capacity of a snowpack is a function of the aggregate structure of the snowpack itself. The capacity decreases with the melting process along with the metamorphoses of the snow. The figures found in the literature are highly variable. The U.C. Corps of Ingineers (1960) recommend32 - 5 % of the total snowpack during melt. Kuz'min (1972) suggests 35 — 55 % for freshly fallen snow decreasing to 5 — 15 % during active snowmelt. According to the WMO (1975 A) fine, crystalline drifting snow is capatle of containing up to
40 % of liquid water, while large granular snow
at the end of melting has a water-holding caracity of akout 5 —
e %.
In the HBV-model two ways to model water retention in the snowpack have been attempted. First a water holding capacity Cwh must be exceeded before
51
the pack can yield any water. Then, in a few catchments,a bottom storage, Sb, was introducedunder the snowpack.Values of C , and S, from the catchwn ments, where snowmelthas been modelled,are shown in table 7.7.
Table 7.7. Water retentionparametersin the differentcatchments.
Cwh (%)
Sb (mm)
Gimdalsbyn
5
10
Kultsjn
5
0
Malgomaj
10
0
Str5ms Vattudal
5
0
Filefjell
5
10
The values in table 7.7 are rather arbitrary.Frror function studies carried out by Bergstrm (1975) and Bergstri5m and J5nsson (1976 B), fig. 7.6, show
that the interactionis strong between the two parameters.
The sensitivityof the F2-criterionto these parametersis also low, which makes it difficultto draw anything but qualitativeconclusions.
0
30
20
10
40
Cwh 0.9908
0
1.2555 1.0962 1.06 5 1.01
3
1.1115 1
5
1. 497 1.01 7 0.9880 0.996 1.0
8
1.0 46 1.0
10
42
1.
0
1.0249 1.0246 I.
7 0.9903 0.9952
2
. 248
5
.0548
1.0704 1.1187
Fig. 7.6. The response of F2 to S and Ch' Filefjell (1967 - 1971 w b (The dimensionof F"-is (m3/s)2 • 10'-'.)
7.1.7. Refreezin ofli uid water
A refreezingroutine is necessary,if snowmelt is interruptedby the intrusion
52
of cold air. In the HBV-modelthis is simulatedby eq. 7.5, which will automatically yield negativemelt rates at low temperatures.If so, the liquid water content is reduced and added to the snowpack,and in case Ceff > 0, the accumulated melt figure is reduced. The method is very crude, consideringthat melting and freezing are differentprocesses,which occur at differentlevels in the snowpack,but it was adopted in order to avoid more free parameters.The amount of liquid water is also small, so that it will mostly refreeze entirely during a cold period. During a poorly defined melt period, however,with temperature fluctuationsaround freezing,abetter procedure for the computation of refreezingmight be a way of improvingthe performanceof the model.
7.1.8. The effect of frozen round The effect of frozen ground on the runoff process was discussedby the WMO (1975 A). So far no such effect has been consideredin the HBV-model.The areal variablilityof the soil cover and the lack of knowledge of the impact of frozen ground are problems making an attempt to model this process but little tempting. The applicationsof the model so far have led to the rather intuitiveconclusionthat the effect of frozen ground is either accountedfor by the existing free parawters, or it is of minor importancefor the performance of the model.
7.1.9. Statisticaldistributionof the de ree-da factor The snowroutinediscussed so far can be said to simulatethe processes in one single point. It is obvious that the great areal inhomogeneitiesas regards snowpackand meteorologicalvariablesmust be consideredin one way or another. Most critical is the effect of partly snowcoveredand partly bare ground in the basin. One way of modelling this could be to adopt an areal depletioncurve relating snow covered area to the accumulatedmelt, as suggestedby Anderson (1973) and the WMO (1975 A). This method is relatively simple but entails difficultiesif fresh snow is falling on a snowcover whichis already reduced by melting. A more convenientway is to divide the catchment into a number of zones and to distributeone or more of the parametersor variablesover these zones in order to simulatedifferent conditionsin differentparts of the catchment. In fig. 7.7 it is shown how a rectangulardistributionof Co (or T) in the
53
simple degree-daymodel (eq. 7.4) will effect the overall input from snowmeltto the soil, in case five differentzones are used. A constant water-holdingcapacity is assumed togetherwith a constanttemperature above freezing.The part with highest Co-valuewill start to melt quickly and consequentlythe snow will end soon, while the situaticnis the opposite at lcw Co-values.As can be seen, the superpositionof the differentcontributionsgives a completelydifferent input to the underlyingparts of the model than would an entirely lumped procedure.
YIELD LOCAL r 11-- - II
-
-
-
fi — -
-
li
TOTAL -
1- - -
rt-
TIME
Fig. 7.7. The effect of a distributionof Coor T in the exiDression M = Co (T - To) at constant water-holdingcapacity and constanttemperatureabove freezing. A rectangulardistributionof the degree-dayfactor, Co, was assumed when applying the HBV-modelto the Gimdalsbyncatchment (Bergström1975). C was ranging from Co max to C - C in ten different zones. o o int o max Both Co max and Co int were regarded as free parameters.Studies of the error functiontopographyrevealed,however, that this distributionhad very little effect on the model performance (fig. 7.8). A minimum in the error functionwas obtained along a valley definedby the equation 2 C
o max
C = constant,which is identicalto(C o int
o max
+ c)/2 o min
= constant (brokenline in fig. 7.8). The distributionof Co had little effect on the model as long as the average degree-dayfactor remained constant.Once again did error function studieshelp to keep down the complexityof the model as the assumptionsof a variable degree-day factor did not have any positive effect on the model performance.
5)4
co Co
1.1 1.3 max 1.5 1.7 1.9
int
0.50
64\0. 7 8 1.56 \ 0.N60 . 15 1.53 \ 2.12 I. 4 `0. . 5
0.75
2.89 1.79 1.
1.00
3.64 2.44 1.53
0.00 0.25
I. 2
A7\0.67 d66
Fig.7.8.TheresponseofF2tocandc_Gimdalsbyn o int o o max (The dimensionof F` is (m3/s)2 • 105.)
(1961 - 1969).
7.1.10. Area-elevationdistributionof the snowroutine,
When applying the model to a catchment of
considerableelevationrange,
the altitude effect on air temperatureand precipitationcannot be neglected. The usual way of accountingfor the temperaturegradient is to adopt the moist adiabaticlapse rate, Tlapse - 0.6 0C/100 m. This method has been used by the U.S. Corps of Engineers (1960), Quick and Pipes (1975), Sugawaraet.al. (1975) and Kuz'min (1972) among others. The method is a gteat simplificationconside7 ring the complex meteorologicalconditions,such as frequenttemperatureinversions,but it is probably the most practicalway of extrapolationin catchments where all temperaturestationsare situated in the lower parts. A lapse rate of - 0.5 °C/100 m was used in the Kultsjön,Malgomaj and Ströms Vattudal catchmentsand - 0.6 °C/100 m was used when applying the model to Filefjell.The catchmentswere subdividedinto ten elevationzones of equal size, and the temperaturewas adjusted accordingto equation 7.7 before entering the snowroutine. T. ,='T+ Tlapse • All.
(7.7)
where: T. = temperaturein the i:th zone (°C)5 T
= recorded temperature (°C),
AH. = mean altitude differencebetween the temperaturestationsand thF; mean elevation in the i:th zone (100 m).
Tlapse =
elevationlapse rate of temperature (°C/100m).
55
T
has been used as a confined parameter in all catchments. An error lapse function study carried out in Filefjell (Bergström and Jönsson, 1976 B) is shown in fig.
7.9. The interaction with To is not surprising, but it is
interesting to note that both parameters are significantly deviating from zero. Contrary to the statistical distribution discussed in chapter 7.1.9 this routine is improving the performance of the model.
To
0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
T
lapse
Fig.
-0.00
2.4387 3.2802 4.7241 7.0543 9.2334 •
-0.30
1.3079 1.4641 1.9773 2.8460 4.2630 .
-0.60
1.402 1. 011 0.
.2362 1.7852 2.6464 3.9447
-0.90
2.5742 1.7550
.89
-1.20
3.8130 2.9987 2.3162 1.62
1.2149 1.7196
468 0.954
.0139
7.9. The response of F2 to T
and T . Filefjell (1967 - 1971). lapse o (The dimension of F- is (m3/s • 10
It is, however, dangerous to draw anything but qualitative conclusions concerning the correct value of Tlapse from fig.
7.9. The exact position
of the minimum is strongly effected by our basic assumptions, erroneous data and estimates of other parameters.
The correction of precipitation with altitude is even more difficult than are temperature corrections. Observations of the total catch at precipitation stations at different altitudes are normally the only source of information available. Wallen (1951) investigated a number of stations in Sweden in this respect. His results have been the basis of the precipitation lapse rates, Plapse' in the Swedish catchments when applying the HBV-model. Recommendations from the Norwegian Water Resources and Electricity Board were used in the Filefjell basin. Plapse has been used as a confined parameter in all applications of the model.
56
The correctionsof precipitationare made from the average altitudeof the precipitationstationsto the mean altitude in each elevationzone. During the snow free period the snow routine is inactiveand precipitationwas mul, instead of being processedby the snow routine in all tiplied by a factor, P corr was identicalto the average elevaticncorrectionsin elevation zones. P corr the ten elevationzones,, in the Kultsj5n and Filefjell catchments.lapse in Malgomaj and Ströms Vattudal. It would have been It deviated from -P°»lapse better to be more consequentin the asessmentof Pcorr as it is a parameterwhich interacts the
with the parameters in the scil moisture zone and thus complicates
generalizationsof these. In future work Pcorr will thereforebe trea-
ted as a confined parameterwith a value equal to Plapse. Values of Plapse' are presented in table 7.3. lapse and Pcorr
Table 7.8. Precipitationcorrectionparaneters.
lapse
(%/100m)
'lapse
corr
Ku1tsji5n
13
1.33
1.33
Malgonaj
15
0.95
1.0
Str5ms Vattudal
16
1.16
1.0
Filefjell
12
1.4
1.4
Investigationsby Bergströmand J3nsson (1976 B) (fig.7.1C)show that the interactionis strong between the precipitation-elevation correctionfactor, Plapse' and the snow fall correctionfactor, Csf, as these two parameters are effectingthe total snowpack in the catchment.As mentioned earlier it is wise,underthese circumstances,toset a reasonableapproximationon one of the parameters,in this case Plapse, and concentrateon the other, C f, when calibratingthe model. It is realisticto believe that the potential evaporationhas a negative lapse rate with elevationdue to the decreasingvapor pressure deficit. So far, however, no distributionof the values of potentialevaporationaccording to the area-elevationcurve has been attemptedin the applicationsof the HBV-model.
57
Csf
1.3
1.5
1.7
1.9
2.1
lapse 4
2.5785 1.9933 1.5108 .19 3 1.0731
8
1.98321.453
12 16 20
.
4 1.0079 1.0727
1.508 1. 91 .9748 1.0 1 .1 2 .9822 .0 1.02441.0
.3412
.32881.8761
.27091.84192.6675
Fig. 7.10. The responseof F2 to Csf and P ,lapsp. Filefjell (1967 - 1971). (The dimensionof F2 is (m3/s)-•104.) The areal distributionof the snow routine entails difficultiesin the underlyingroutines as they will be subjectsto a distributedinput instead of a lumped one.Attemptsto account for this by
a distributedsoil
moisture zone will be presented in chapter 7.2.4.
7.2. The soil moisture zone The conditionsin the soil moisture zone are acting as a controlling agent in the formationof runoff, as exemplifiedin chapter 7, fig. 7.1. Consequentlythe soil moisture accountingpart of the model is where the main contributionto runoff from precipitationand snowmeltis determined. The part of the lower zone, which representslakes and rivers, can also contributedirectly,but this area is mostly small comparedto the total catchment. As mentioned in the introductorydiscussionof chapter 7, interceptionof precipitationin the vegetationwill be accountedfor im131±citly in the soil moisture routine if not modelled separately.interceptionis of course related to the vegetationcover,which is a functionof the time of the year. If we further considerthe very complex laws governingwater
58
retention and transport in a heterogenous soil column, it is evident that the striving for a physically c=rect
representation of the processes in the
soil moisture zone will lead to very complex models. The picture is further complicated by the great areal variability of the vegetation cover and soil characteristics. As the objective is to keep the HBV-model as simple as possible, the soil moisture accounting procedure is developed from greatly simplified assumptions according to the strategy lined out in chapter 3. The routine is based on physical considerations but is adjusted specifically to each catchment by empirical coefficients, i.e. free parameters.
7.2.1. A sim le reservoir a
roach
One of the most simple representations of the processes in the soil is to regard the soil moisture zone as a reservoir, which has to be filled to a certain level, or capacity, Fc, before any water can pass through (fig. 7.11).
RAI N, S NOWMELT
••
••
•
•
•• •
• •••
TO RUNOFF
Fig. 7.11. A simple soil moisture accounting procedure. The parameter Fc can be said to represent the maximum available water in the soil moisture zone, expressed as the difference between field capacity and wilting point. Due to the greatly simplified approach and unavoidable implicit corrections one must, however, be careful not to abuse Fc when relating parameter values to catchment characteristics.
The procedure in fig. 7.11
has some physical relevance but it is incapable
of handling the great areal variability of the soil cover in most catchments Therefore its response will be too
abrupt as soon as the soil moisture
storage is filled to Fc. If the catchment area is divided into a number of
zones in the model, and a distribution of Fc is assumed, a contributing area approach is obtained, where each individual zone with the area Z, will contribute to runoff as soon as its soil moisture storage is filled to its capacity, Fci. The response of this routine will be much less abrupt, as illustrated in fig. 7.12.
f(Fc)
ÆZ (0/0)
Fc(mm) loo
srn Fc(mm)
°/0AREA
)1 Fig. 7.12. Distribution of the simple reservoir soil moisture routine. The distribution of Fc. Schematic representation of contributing areas and soil moisture conditions. Contributing area, EZ., as a function of the soil moisture storage, Ssm.
Fig. 7.12 shows an idealized progression from initially dry conditions. The modelling of evaporation from each zone will complicate the picture so that no unambiguous curve, which relates the contributing area to soil moisture conditions, will exist.
Axelsson (1975) and Becker (1975) showed examples of some modified forms of the above approach. It is also the basis of
the two soil moisture
accounting procedures which have been tested in the HBV-model, as will be discussed in the remaining part of chapter 7.2.
7.2.2. Soil moisture accountin
in the HBV-model
The contributing area concept in fig. 7.12
has been adopted in the soil
moisture accounting procedure of the HBV-model. Due to a lumped evaporation routine,the total soil moisture procedure is of a quasi-distributed
60
rather than distributedcharacter. The soil is assumed to react upon rainfall or snowmelt,AP, in a fashion shown in fig. 7.3, which is in analogy with fig. 7.12.C.
1.0 13,SuzlåP
åSsm / L‘P 0
0
Fc Ssm
Fig. 7.13. The contributionsfrom rainfall or snowmelt,P , to the soil moisture zone, S , and the upper zone, S . sm uz Mathematicallythe procedure can be expressedas: AS uz
AP
S
(TE71)
AS (sm) sm = 1 Fc AP where: P
= precipitationor snowmelt,
S = storage in the upper zone, uz s = actual computed soil moisture storage, sm Fc = maximum soil moisture storage in the rriodel, f3 = empiricalcoefficient. The amount AS will pass through the soil moisture zone and eventually uz contributeto runoff or evaporationfrom the lower zone. AS will contrism bute to the soil moisture storagewith evaporationas the only exit from the system. 13is a free parameterwhichgives the curvesin fig. 7.13 their shapes.At f3-values of plus infinitythe routine will degenerateto the simple reservoirtype in fig. 7.11, while if
equals zero there will be
no active soil moisture storage at all. The routine is thus very economic in terms of free parameters,asa large spectrumof possibilitiescan be
covered when changing 13, and Fc. Potential evaporation is reduced to actual values according to the s:»Lmple function of the total computed soil moisture conditions in fig. 7.14.
Ea/E p 1.0
0 0
Lp Fc
Ssm
Fig. 7.14. Reduction of potential evaporation, Ep, to actual, E. This can be expressed as: E
a
= E p sm
E = E• a p
L
if
S
if
S < L sm
> L sm — p (7.10)
where: E
= potential evaporation,
Ea
= actual evaporation,
S
= actual computed soil moisture storage,
L
sm
= limit for potential evaporation.
Due to its simplicity this method for reduction of potential evaporation has been very popular in hydrological modelling. It was used by Porter and McMahon (1971), Girard, Fortin and Charbonneau (1971) and Dickinson and Douglas (1972) to mention a few authors.
The parameter L
is sometimes said to represent the root zone constant,
above which evaporation will occur at potential rate from a soil profile. If the distributed approach is adopted, it is, however, more realistic to interpret L
as the general moisture conditions above which the entire
catchment has the opportunity to evaporate at potential rate. A general decrease of evaporation with decreasing soil moisture storage is a realistic assumption, anyhow. Kristensen
and Jensen (1975) elaborated this a
62
bit further when accounting for the intensity of the potential evaporation by
different shapes of the curve in fig. 7.14. This modification cannot
be considered in the HBV-model, unless better evaporation data are obtained.
The above soil moisture routine has been used in all test catchments. The parameter values are shown in table 7.9. The procedure,when fitting the model has generally been to start with adjustments of 13,withL
equal to'Fc
and with
Fc at values regarded as reasonable from considerations of the soil and vegetation cover in the catchment. Thereafter L
has been adjusted and finally,
if found necessary, different values of Fc have been tested. This procedure is reflected in the rather regular Fc-and L- values in table 7.9. .P Table 7.9. Parameter values in the soil moisture zone.
Fc
L /Fc
Lilla Tivsjön
100
0.7
3.4
NolsjOn
200
0.8
8.0
Stabby
100
1.0
7.0
Stormyra
50
0.8
4.0
Solmyren
100
1.0
2.25
Gimdalsbyn
200
1.0
1.80
Kultsjön
150
1.0
3.0
Malgomaj
150
1.0
2.0
Ströms Vattudal
150
1.0
1.0
Filefjell
150
1.0
2.0
When analysing the values of the soil moisture parameters, it is important to bear in mind the possible effects of erroneous input data, the inter(see chapter 7.1.10) corr and other implicit corrections. It is also important to remember the role actions with the precipitation correction factor, P
played by
as an exponent in eq. 7.8 and 7.9,making the model little
sensitive to changes in
at high 13-values.When analysing the error func-
tion the latter problem is overcome with a log transformation of the 13-axis.
A general conclusion is that the model can be fitted quite well with a rather crude approximation of Fc and with L /Fc equal to one. The fact that all ft-valueslie between one and eight and thus give a concave shape of the AS /AP -curve (fig. 7.13) is supporting the hypcthesis lined out in fig. uz 7.12.
63
The error function topography of the soil moisture parameters was studied by Bergstr3mand Forsman (1973),BergstrOm (1975)and Bergströmand
Jönsson (1976 B). Two of these studies are shown in fig. 7.15 and 7.16.
Fe
50
100
150
200
250
0.5 3.48 2.952.63 2.34 2.33 1.0 2.45 1.83,1. 8 2.0 2.04 1.5
1120
.06 077 0.96
4.0 1.97 1.72 .34 1.08 1:32 8.0 1.99 1.98 1.76 1.59 1.90
Fig. 7.1 . The response of F2 to Fc and Ø with L /Fc at a constantvalue. Gimdalsbyn(1961 - 1969). (The dimensionof F2 is (m3/s)2 • 105.)
1,zs
1.00
1.02,
0.1.5 0.2.5 1.02.
1 oo
1.00
L /Fc
2. 00
1.o4 0.50 50
100
150
200
4.00 2.50
Fc
Fig. 7.16. The response of F2 to Fc, Ø and L /Fc. Filefjell (1967 - 1971). 2 2 P (The dimensionof F is (m-/s) ' 104 .)
6)4
All studies of the error function have confirmedthat the simple reservoir routine in fig. 7.11 is too rigid a representationof the soil moisture zone, as 3 has a well defined optimum value.
In order further to investigatethe role played by the soil moisture zone, the active and relative active soil moisture storageshave been computed in a few test catchmentsaccordingto: Sact = Fc - S sm . min sact =
(7.11)
act
(7.12)
Fc
where: Sact
= active soil moisture storage (mm),
sact
= relative active soil moisture storage,
sm
= minimum soil moisture storage during the period (mm), min
Sact and sact are thus measures of the maximum reductionof the soil moisture storage,S , during a specificperiod. A small value of s sm sm indicatesthat only a fraction of the storage is playing an active role, which will have consequencesfor the sensitivityof the model to the parameters in the soil moisture zone. Values of Sact and sact are presented in table 7.10. Table 7.10. Active, Sact, and relative active, sact, soil moisture storage in the HBV-model.
Area above timberline (%)
Sact
Gimdalsbyn
0
Malgomaj
sact
Fc (mm)
Number of years
154
0.77
200
12
7
86
0.57
150
12
Str5ms Vattudal
13
90
0.60
150
13
Kultsj6n
51
71
0.47
150
13
Filefjell
86
50
0.33
150
7
(mm)
Consideringthe climatologicalimpact it is not surprisingthat the active soil moisture storage is decreasingwith increasingarea above timberline.In the Filefjellalpine basin only one third of the assumed availablewater is exchanged.This is reflectedby a low sensitivityof
the
model to changes in the parameters L , Fc and
in the soil moisture
zone (fig. 7.16).
From the results in table 7.10 it is obvious that one must be very careful when relating Fc-values to physical characteristics. When designing the model, Fc was meant to represent the maximum available water content of the soil, i.e. the differenobetween just
field capacity and wilting point. If
a part of the soil moisture storage is active, it is, however,
difficult to draw any conclusions concerning the relevance of a chosen Fc-value.
7.2.3. An alternative soil moisture routine
The routine discussed in the previous chapter was based on a contributing area concept with a lumped evaporation procedure. A more logical routine treating evaporation separately in a number of zones with varying values of the maximum soil moisture storage, Fc, was tested in the Stabby catchment. The catchment was divided into ten zones with a distribution of Fc according to fig. 7.17. Each zone was then modelled according to the simple storage concept lined out in chapter 7.2.1, fig. 7.11.
Fci Fc.=Fc I
Fcmax
0
1
2
(( i -0.5)/10)
flICIX
3
4
5
6
7
8
9
10 ZONE,i
Fig. 7.17. Assumed distribution of the Fc-values in the distributed soil moisture routine.
Actual evaporation from each zone, Ea., was computed separately in analogy with fig. 7.14, as shown in fig.17.18. Identical values of the difference Fci - L wereassumed P-
in all zones.
•
•
66
Fc.-L I Pi ,
E /E 1.0
Fc,
Pi
•
Fig. 7.18. Evaporationfrom the different zones in the distributedsoil moisture routine. IfthevalueofFc.-LexceededFc.in any zone, the soil moisture P.; storage was subject to ufireducedevaporationuntil emptied.
The above routine was compared to the one described in chapter 7.2.2. for the snowfreeperiods of 1959, 1960 and 1961. It was calibrated through mappings of the F2-surfacewith differentvalues of a, Fci - L P. = 0 and Fc and Fc. Fitted values were a = 0.4, Fci - L = 150 mm. max max P.; The response surface at this Fcm -value is shoWn in fig. 7.19.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
)
1
P1. 897. 1026. 1185. 1372. 1566.
20.0
736. 756. 80
15.0
691. 6
10.0
689. 664. 664.
5.0
709
0
745.
5.0 10.0
. 720.
87. 1331. 1023. 1183. 1355. 80.
667. 643.
884. 1015. 1160. 78.
648. 641.
64. --) 829. 744. 693. 669. 672.
699
947. 838.
. 7
.
713.
719.
15.0 1128. 983. 889. 828.
3.
778.
878.
852.
78 .
868.
74 •
4.
746.
784.
778.
799.
Fig. 7.19. The response of F2 to a and the differenceFci
at Fcmax Fi
= 150 mm. Stabby (1959 - 1961). (The dimensionof F2 i
(1/s)2 . 10 3)
67
2 The results expressed as R- and F2-values are shown in table 7.11. Due to the low initial variance the R2-value is a misleading criterion of fit for
the period 1959 as shown in fig. 5.1. The total error expressed as
F2 is therefore the most appropriate :)asefor comparison in this invest gation.
Table 7.11. Comparison between the distributed and the original soil moisture routines. (Stabby 1959 - 1961).
R2
Distributed routine Original routine
Total F2 (1/s)2
1959
1960
1961
0.47
0.92
0.54
64•10
- 0.64
0.93
0.80
40.104
Although it was possible to fit both models well to each year individually, the total error was larger
by
the distributed model when making a
simultaneous calibration of all three years. The conclusion is therefore that the distributed model is inferior to the one adopted in the HBV-model.
The sensitivity of the distributed model to the resolution in the subdivision was also tested. The model was run with a subdivision of 2, 4, 6
to 20 zones. There was very little difference from 6 and higher
numbers of zones.
7.2.4. Distribution accordin
to the area-elevation curve
When distributing the snow routine according to the area elevation curve, as discussed in section 7.1.10, not only the snowpack but all underlying components of the model are effected, as snowmelt is simulated differently in the different zones of the catchment. A critical part in this respect is the soil moisture zone. In the HBV-2 and HBV-3 models discussed so far this problem was ignored, and all the snowmelt was fed into one common soil moisture zone. This is,of course,a questionable point,and therefore a distributed soil moisture zone was tested in the sense that separate soil moisture accountings, based on the procedure in chapter 7.2.2, were carried out in each elevation zone (Bergström and Jönsson 1976 A). The investigation was carried out in the KultsjOn catchment (1962 - 1974).
68
The problem with such a differentiationis that the number of free parameters will multiply by the number of elevation zones, making it difficult to obtain stable parameterestimates.In order to avoid this problem, Ø and L /Fc were regarded as invariablewith thealtitudewhile Fc was subjectto a rectangulardistributionaccordingto fig. 7.10.
Fc Fc max •••
Fcm
•n
ALTITUDE
m 1500
1000
500 0
50
100 0/0AREA BELOW I NDICATED ALTITUDE
Fig. 7.20. Distributionof the soil moisture zone accordingto the areaelevation curve in the Kultsj5n catchment. The model was run both with invariableFc-values and with a distribution ranging from 25 mm to 275 mm. The results are summarizedin table 7.12.
69
Table 7.12. Results with a lumped soil moisture routine and one distributed accordingto the area-elevaticncurve. Kultsjön (1962 197)4). R2 1962 - 1966
1966 - 1970
1970 - 1974
Lumped Fc = 150 mm
0.7977
0.8712
0.8402
0.7931
0.8 731
0. 8512
0.7902
0.8753
0.8495
Distributed Fc Fc
max
= 150 = 150
min
mm mm
Distributed Fcmax = 275 mm Fc
min
= 25
mm
It is evident from tabIe 7.12 that the differencesin performancebetween the lumped and the distributedmodels are quite small, and that a lumped model is sufficientin the Kultsjön catchment.This conclusionwas supported by visual inspectionof the hydrographs.
7.3. The res onse function of the HBV-model
The response function is the part of the model which transformsexcess water from the soil moisture zone to runoff. It also includesthe effect of direct precipitationand evaporationon a part pw which representslakes, rivers and other wet areas. The HBV-modelhas a nonlinearresponse function,but some of
its compo-
nents are of a linear, or at least, quasi-linearcharacter.It is sometimes argued that a linear response function should be more convenientbecause of the existenceof analyticalsolutions.The estimationof the parameters in the nonlinear functiondescribedbelow is, however,no great problem. Thereforeit seems
wise not to compromisein the sense that a
physical behaviour,whichweknow is nonlinear,is forced into a linear framework. The most successfulversions of the HBV-modelhave a response functionwhich can be illustrated,in a general form,as in fig. 7.21.
7o
•••
miew
-
- uz -E
Luz
E P
(S-1 tz Q0=5
12)
CII=K1'suz Cperc TRANSF. FUNCTION
Slz
_ _
00.1M
••
••
1•
Qg GfkiSlz
Fig. 7.21. The response function of the HBV-model.
Excess water enters the upper zone. It leaves as runoff through its two outlets or percolates, at a constant rate, Cperc, down to the lower zone, from which it can either evaporate or drain. At last a transformation function is activated to give a proper shape of the hydrograph which is generated by the two reservoirs. Fig. 7.21 is actually showing the HBV-3 version, but with proper parameter settings HBV-1 and HBV-2 can be obtained.
The linkage between the upper zone and the lower zones has been supported by a
work by Waldenström and Andersson (1973). Special high-frequency measure-
ments of soil moisture and ground water in the KassjOån Representative Basin, of which Lilla Tivsjön is a part, showed that recharge of groundwater continues for some time after the termination of rainfall and replenishment of the soil moisture deficit. In the model recharge of the lower zone will continue until the upper zone is empty.
It is tempting to attribute the different runoff components in the response function to the three components often discussed in the handbooks: overland flow, interflow and groundwater flow. (See for example: Gray, 1973.) It must be stressed once agains however, that the fact that the model yields reasonable discharge values, is no confirmation of the physical relevance of the model. It might be possible to achieve similar results with other structures.
7.3.1. The sin le linear reservoir
The corner-stone on the response function is the single linear reservoir
shown in fig. 7.22.
INPUT
STORAGE,S
OUTFLOW Q=K.S
Fig. 7.22. The single linear reservoir.
Its differential equation during a dry period (no input) can be expressed as: d S(t) =
K • S(t),
d t
(7. 13 )
where: S(t) = storage at time t, = storage discharge constant
(recession coefficient).
Eq. 7.13 has the general solution: S(t) = s(t)
• e-K.(t-to)
if we introduce the discharge Q(t) = S(t)-1
(7.14) d t
into eq. 7.13
1 • d S(t) _ - —K Q(t)
and substitute S(t) in eq. 7.14, we obtain Q(t) = Q(t ) e-K.(t-to)
(7.15 )
During a dry period the single linear reservoir will show an exponential decay which is a quality
often detected when analysing hydrographs.
Instead of the storage discharge constant K with the dimension (time-1) the invers, 1/K, with the dimension (time) is sometimes used.
Another possibility is to use the relation: r =
Q (t+1)
Q(t)
'
(7.16)
72
which can be related to K, if one time step is used in eq. 7.15: Q (to + 1) = Q(t)
e-K. 1
or: Q(t) ln
Q(t0+1
1 = K = ln-
(7. 1 7)
In this work K will be used as the characteristicparameterof a reservoir. In some of the investigations of the HBV-model,a K-value, which is directly relating storage in mm to discharge in l/s, is used. This value is, however, not independentof catchment size, which complicatescomparisons and generalizations. The single linear reservoirwill be distortedwhen linked together to form the response function.A linear charactercan, under certain conditions,still be assumed, which will help in the calibrationof the model or at least in finding first estimatesof some parameters.If ln Q is plotted against time during a dry spell, the gradientsof the hydrographat differentmagnitudes of discharge are good first estimatesof the differentrecession coefficients, Ko, Ki and K2 in fig. 7.21. The single linear reservoir is only one of many possible ways of modelling a storage dischargeprocess in nature. Roche (1970) shows examples of other reservoirswith differentphysical interpretations.He points out that the single linear reservoircan be visualizedas a containerwith a porous outlet, thus obtainingeq. 7.13 from Darcy's law.
7.3.2. The u
er zone
A very general interpretationof the configurationof the upper zone in fig. 7.21 could be expressedas follows: If yield from the soil exceeds a certain percolationcapacity,thewater will start to drain through more superficial channels and thus reach the rivers and streams with a higher drainage coefthese channels are ficient, Ki. At a storage in the upper zone exceedingL uz, filled to their capacity,and even more rapid drainage accordingto Ko will start. This explanationis somewhatvague, but the interpretationis not a determiningfactor for the model as long as we are working with a lumped approach,a fact that was discussedby Bergströmand Jönsson (1976 B).
The introductionof Ko in the HBV-3 version of the model is,of course, disturbingthe linearityof the upper zone considerably.Even if Ko is zero, that is if we study the HBV-2 version, the upper zone will be a distortedlinear reservoirdue to the parameter Cperc. Its differential equation during a dry period will be of the form: d S
uz
(t) =-K1 • Suz(t) - Cperc'
d t
(7.18)
if S (t) > 0 uz
where: S (t) = storage in the upper zone at time t, uz Ki = storage dischargeconstant, Cperc = percolationrate to the lower zone. A general solutionto 7.18 is: cPerc + (C Derc S (t) = + S (t )) -K1 • (t-to) uz uz o 1 Ki
(7.19)
The smaller Cpere is,comparedto K, the more the response of the upper zone will approachthat of the single linear reservoirdiscussed in section 7.3.1. The response is shown graphicallyin fig. 7.23.
\
N.
C.pere> 0
td.Z ••
,..
0 TIME Fig. 7.23. The response of Suz during a dry spell, if Ko = 0.
The response of the model to the parameters of the upper zone have been studied by Bergströmand Forsman (1973)and Bergströmand Jönsson (1976 B). The result from a study of Ko, K1 and L in the Filefjellbasin is shown uz in fig. 7.24.
7)4
.oe 10
96
.50
092.
1 2.00 400
K1
600 800
70
1000
oo
K0
12.00
to
2150
300
400
300
1400
and L . Filefjell (1967 - 1971). Fig. 7.24. The response of F2 to K , o uz (The K-values are relating storage in mm to dischargein l/s. The dimensionof F2 is (m3/s)2 ' 104.) It is interestingto note the increasingimportanceof Ki with increasing L -values reflectedby the switch in the main axis of the ellipse formed uz by the isolines.The explanationto this is that at high Luz-valuesthe HBV-3 model will degenerateto HBV-2 with Ko as an inactiveparameter, while at low L -values the effect of Ko is more pronounced.Fig. 7.23 is uz also a good indicationof the significantimprovementof the model when introducinga third runoff coefficientin the Filefjellbasin, as the optimum Ko-value is situatedfar from zero.
7.3.3.The lower zone The lower zone can be said to represent the total groundwater storage of the catchment. Cpere is thus a parameter governing groundwater recharge. The interpretation of this zone is somewhat easier than that of the upper zone. The fact that one slow and at least one quick runoff component have been detected in all investigated catchments,no matter if they are homogenous or heterogenous, is supporting the theory that the slow component is caused by a lower reservoir and not a parallell one. It is true that a model with one runoff component, HBV-1, was used in the Lilla Tivsj3n catchment, but later work (Bergstr3m, 1973) showed that the HBV-2 version is preferable, if higher flows are to be modelled in the catchment.
If we assume that no part of the lower zone is subject to direct precipitation or evaporation, the reservoir will be linear and have a response according to: Slz(t) =
dt
- K2 • Slz (t) + C
perc
if
S (t)>0 uz
Slz(t) dt
= - K2 • Slz (t)
if
(7.20)
S (t) = 0, uz
where: Slz(t) = storage in the lower zone at time t, S (t) = storage in the upper zone.at time t, uz K2 = storage discharge constant for the lower zone, perc
= percolation rate from the upper zone.
The solution to 7.20 is: C C perc ( perc Slz(t) = K2 \ K2 lz
(t) = S
lz
Slz t )
if (t ) • e-K2*(t-to), o
-K .(t-t ) 2 o, if S (t) = 0. uz
The response of Slz is visualized in fig. 7.25.
S (t)>0 uz (7.21)
76
S
Slz
LZmax
rc K2
-
I
0 uz >13
uz= 0
TIME
Fig. 7.25. The response of the lower zone to input from the upper zone.
If Suz>O, Slz is raising and is asymptotically approaching the value Cperc/K2. As soon as the upper zone is empty, i.e. S = 0, the lower uz zone will decline exponentially towards zero.
The introduction of a part, pw, representing lakes, rivers and other wet areas, somewhat complicates the picture on the input side. Recharge from the upper zone will occur parallell to direct precipitation and evaporation on the part, pw. The effect of precipitation will be present throughout the year as the pressure of the snowpack on the ice will have the same effect as direct rainfall on a water surface. Evaporation from the part, pw, is assumed to occur at potential rate, as soon as the ice has disappeared
from the lakes. The same values of potential evaporation as from
land are used in combination with a standard date for icefree conditions. (See chapter 7.3.5.) The lower zone has a dead storage which is allowing runoff to cease completely due to evaporation from lakes in dry summers.
One attempt was made to model the lakes as a separate reservoir, through which the response
of both the upper and the lower zon; were routed. The
estimation of the storage discharge curve of this reservoir caused problems, as it interacted with K2 and K1. The work was abandoned but it is strongly felt that the routing through lakes and improvement of the lake evaporation routine is still a point where the model could be improved.
A mapping of the response surface with K2 and Cperc as free parameters was made when applying the model to the Filefjell catchment (Bergstr5m and Jönsson, 1976 B), (fig. 7.26). The optimum parameter values in the
F2-sense of the term did not coincidewith judgementsfrom visual inspection of the two hydrographs.The F2-criterionshowed low sensitivity to changes in K, and Crerc
which is not surprisingas they are
mainly effectingthe low flow recessionswith small but persistantresiduals. The contributionto a sum of squares criterionis therefore rather small compared to timing errors of flood peaks for example.
K2
10
50
90
130
170
210
250
290
perc
1.03691.02911.04301.05951.0769 1.030 .
.9
1.1328 . 436 0.92 3
1.31710. 817 0.923
.98551.0097 . .94 0.9666 2110. 62 0. w760.98111.0449
1.53551.036 0.924 0.90260.90 0.92 0.949 0.9745 5
•
0.90 0.895 0.911 0. 36 0 9590
6
•
0.95z12225_21.,9177 0.93670.9618 (3, 965 0.93350.9323 .94790.9717
Fig. 7.26. The response of F2 to Cperc and K2' Filefjell (1967 1971). (K2 is relating storage in mm to dischargein l/s. The dimensionof F2 is (m2/s)2 ' 104
.)
7.3.4. The transformationfunction The transform.ation function is sometimesnamed the time-areatransformation function,but difficultiesin the physical interpretationof the recessioncoefficients,as will be discussedin the followingchapter, have led to some sceptisismas regards the latter name. The effect of the transformationfunctionon the hydrographis illustra-
78
ted in fig. 7.27, where Q
representsgeneratedrunoff from the upper and the
lower zones in fig. 7.21..
WEIGHT
ag
T1ME
q
TIME
TIME
Fig. 7.27. The effect of the transformationfunction on the generated hydrograph.
Q
is distributedon consequtivedays accordingto a triangularfunction,
with the base B . Bq is a variable, which depends on the magnitude of Q q accordingto: - C B = B q max route • Qg' B = 1,
. Q ) > 1, if (B - C max route g — if (Bmax - Croute • Qg) < 1,
(7.22)
where: = base in the triangularfunction (days), Bmax
= maximum base at low flows (days),
C = free parameter (days/(m3/s)), route = generatedrunoff from the upper and the lower zones (m3/s) b Qg TIME
= 0 is the day on which runoff was generated.
Thus a variable transformationfunction is obtained,which, to some degree, may account for the variationsof time-of-travelwith magnitude of runoff. Eq. 7.22 was introducedwhen the model was applied to the Gimdalsbyncatchment,a stronglydamped forestedbasin with a high percentage of lakes (Bergström,1975). The F2-surfaceat differentvalues of Bmax and Croute is shown in fig. 7.28.
croute
0.32 0.29 0.26 0.23 0.20 0.17 0.14 0.11 0.08 0.05 0.02 0.00
max 10
•
•
•
•
•
•
•
20
•
.
•
•
•
•
•
30
•
•
•
.
o
o.
0.8
.6
1.100.98
•
0.73 0.730. . 4
0.95
•
•
•
40
1.08 . 1 0.71 .
.
1.36 .
•
•
•
50
0.93 1.12 1.36 1.64 1.96 .
.
•
•
•
Fig. 7.28.The response of F2 to Bmax andoCroute*Gimdalsbyn (1961 - 1969). (The dimensionof F2 is (m3/s)- • 105.) A non-variabletransformationfunction is obtained, if Croute = 0. As can be seen from fig. 7.28, the introductionof Croute improvedthe model significantlyin the Gimdalsbyncatchment.In some basins with a quicker response Croute has not been used. From the work on the Filefjellcatchment (BergstrOmand JOnsson, 1976 B) response surfacesof Ko and K1 at differentB -values and with Croute= 0 are shown in fig. 7.29. Although there is some interactionbetween K6 and B
the latter has a very distinct optimum at B
= 2.0.
It is sometimesargued that a more physicallybased time-areatransformation function accordingto Chezy's formula or similar would be preferable. One must, however, bear in mind that for proper use of Chezy's formula not only the basin slope but also the wet perimeter is needed. Otherwise the formula will not account for the relationbetween time of concentration and magnitude of discharge.
80
1.
loo
2.o
1oz
••••
' 2.00
K1
, 400 600 .300
Bq
(000
1.oe 1400
flo 1600 100
2,00
300
400
300
Fig. 7.29. The response of F2 to Ko, K1 and Bqwith Croute = 0• Filefjell (1967 - 1971). (The K-values are relating storage in mm to discharge in l/s. The dimension of F2 is (m3/s)2 • 104.) It can be discussed, whether the same transformation function should be applied to all runoff components. But as long as the physical interpretation of these components is unclear aii te
:-esiatsare
there is little reason to introduce more complexity than necessary for the purpose of the model. There is a possibility, however, that a more complex damping function, based on the laws of fluid dynamics,might simplify the generalization of parameters and thus be a better approach to the problem
of
ungauged catchments.
The technique to damp out the response of one more or less linear reservoir by a triangular weighing function can sometimes be seen used in the opposite way. Dawdy et. al. (1972), for example, obtained the hydrograph by the routing of the time-area histogram through a linear reservoir. By definition the response is identical with the damping of a generated hydrograph, as long as the reservoir is linear. In the stanford IV model (Crawford and Linsley, 1966) a method that resembles the one used in the HBV-model is described. The authors refer to the method as the "channel
time-delay histogram".
7.3.5.Parameter values of the res onse function The recession coefficients in the response function are given
first
estimate from plottings of the logarithms of discharge against time during a dry period (see chapter 7.3.1). This method, referred to as recession analysis, is commonly used when identifying parameters in an exponential response function (see, for example, Burnash et, al., 1973, or Sugawara et. al., 197)4).Applications to the HBV-model were shown by BergstrOm (1972 B), Bergstr5m and JOnsson (1975) and BergstrOm and Jönsson (1976 B). When applying recession analysis to the HBV-model a few points should be stressed. Due to the linkage between the upper zone and the lower zone, the linearity of the upper zone will be distorted and the recession analysis will yield only approximate Ko- and Kl-values.
The coefficient of the lower zone, K2, will be distorted by evaporation. Therefore it is advisable to carry out the analysis during a winter recession. The variable transformation function is another factor which introduces uncertainties into the estimates of the recession coefficients. Due to the above uncertainties recession analysis must be carried out as a first step only and always in combination with visual inspection of the hydrographs. Poor estimates of the coefficients will immediately be revealed when comparing
computed and the observed recession limbs.
Analysis of the hydrograph can also be used to give a first estimate of Cperc as discussed by Bergstr6m and Jönsson (1975), but the value must generally be adjusted after visual comparison between the computed and the observed hydrographs. The same goes for the parameter L in the upper uz zone of the response function. The part of the lower zone, pv, representing lakes, rivers and outflow areas, is determined from the map as the lake percentage plus a correction for swamps and other wet areas. As the work
on
the applications
of the model has been going on for several years, these corrections,
82
being rather arbitrary, have unfortunately not been made according to a consequent rule. The differences are, however, small and of minor importance, as the model is rather insensitive to differences inpas
shown by BergstrOm and
Jönsson (1976 B). Evaporation from the wet parts is assumed to occur as soon as there are icefree conditions in the lakes. Long term observations by Moberg (1967) of ice conditions have been used to determine standard dates for each catchment in Sweden, while surface water temperature recordings were used in the Filefjell catchment.
The remaining parameters in the response function, Croute
and Bmax, are
found by visual inspection of the hydrographs and the accumulated difference-curve.
In table 7.13 the parameters in the response function are shown together with some characteristics of the catchments.
Table 7.13. Response function parameters and catchment characteristics.
Catchment
Lilla
Size km2
Tivsjön
Nolsjön
Area above timberline (%)
pv
L uz
(%)
12.7
0
4.6
18.2
0
2.0
K0
K1
(day-1)
(mm)
-
.._ -
(day
-1
)
K2
Cperc
(day-1)
(mm/day)
0.079
0.194
0.5
Bmax
Croute
day
(days/(m3/s))
_1)
0
4
0 ')
Stabby
6.4
o
0.5
-
-
0 .360
0.131 0
0.9
2
Stormyra
4.o
o
3.0
-
-
0.422
0.126
0.8
2
0
10.0
-
-
0.0192)
0.5
o
15.0
-
Solmyren
27.5
Gimdalsbyn
2178
0
Ku1tsj8n
1109
51
0
'
7.0
Malgomaj
1862
12.0
-
Ströms
3851
13
10.0
-
154
86
10.0
20
Vattudal
Filefjell
A time Poor
lag
estimate
of
7
one day was used due to poor
instead
performance
-
00.:2)
0.014
0 .6
3 ho
-
0.335
0.0234
1.3
2
0.299
o.o399
0.6
2
0
-
0.130
0.0336 0.0281
o.4
5
0.007
0.6
2
0
0.394
of
damping
of the
model.
0.126
in the
HBV-1 model.
0.23 0.0103
When analysing table 7.13 some relation between catchment size and K, can be observed for the small catchments, while for the larger ones it is more difficult to recognize any pattern. K2 shows some relation to the catchment size for the whole sample. When relating
parameters to the
wet area, pw, it is important to note that net inflow, according to eq. 6.1, was modelled in the Kultsjn,
Malgomaj and Strms
Vattudal catchments,
which means that pw must be reduced to a rather small figure, as the reservoir is representing the major part of the lakes. The only conclusion as concerns lakes is therefore their drastic effect on the recession coefficients, Bmax and Croute in the Gimdalsbyn catchment.
Cperc is a parameter which is surprisingly stable in all catchments. It is also the experience that Cperc causes a minimum of trouble when fitting the model. Due to its small variability it is hard to relate the parameter to any of the catchment characteristics presented in table 7.13 or 4.1.
The distinction in fig. 7.21 between generation of runoff and transformation of the hydrograph with a time-area concept was given a critical discussion by BergstrOm and JOnsson (1976 B). The main point was that if the concept was true, the recession coefficients would be independent of the catchment size, as all areal effects would be handled by the time-area function. The pattern in table 7.13 is rather irregular, but there seems to be some increase of the coefficients with very small basins, a result supported by a work by Persson (1976). Instead of firmly sticking to the timearea concept, it might therefore be less hazardous to regard the response function as a wholeness which accounts for the whole conglomerate of runoff processes on the ground, in the ground, and down through the system of streams, rivers and lakes without any specification of each individual process.
7.4. Com utational details
All the computations in the model are carried out on a daily basis. Daily totals of precipitation or snowmelt are fed into the soil moisture zone, and daily totals of discharge are leaving the transformation function.
In the computor program amounts of water are processed through the different procedures in the same order as they are shown in fig. 7.1. In the soil moisture zone the separation between contributions to runoff and soil moisture storage shown in fig. 7.13caused problems due to the non-
84
linear
character of the function. Therefore precipitation and snowmelt are
fed into this routine mm by mm with subsequent adjustmenbsof the soil moisture state. In the evaporation routine actual evaporation is estimated from the arithmetic mean of the computed soil moisture conditions before and after the processing of rain or snowmelt. Actual evaporation is further reduced in proportion to the number of elevation zones with snowcover, so that evaporation will cease completely, if the entire catchment is modelled as snowcovered. In the upper zone Cperc is satisfied before any runoff is computed,and in the lower zone the contribution from the upper zone, precipitation and evaporation are accounted for before the outflow is computed. Of course it can be arguedthat recharge and outflow of the lower zone should be more integrated, but,due to the slow response of this zone,the long term effects of recharge and evaporation are more important than are th€ir time of occurrance during one day. When distributing the model according to the area-elevation curve of the catchment,the position of the lakes is important for the computed volumes due to the precipitation lapse parameter, Plapse. So far the lakes have been treated as if they were situated in the lowest parts of all the catchments.
During the work
the variable transformation function it was found
on
to integer values, as, particularly at low B q values, the switch from one B -value to another caused discontinuities in
necessary not to restrict B the computed hydrograph. B
is therefore allowed to vary continuously
according to eq. 7.22, and the histogram in the damping function is computed as the area below
a
triangle for each specific day, as seen in fig. 7.30.
WEIGHT /
0.5
0.0
0
1
2
DAY
3
Fig. 7.30. The transformation function at B
= 3•5•
When applying the variable transformation function , great care must be taken not to violate the principle of conservation of matter. The best way was found to be to spread out the generated runoff , Qg
,
on the subsequent
days according to the weights in fig. 7.30 and sum up all contributions for each day separately. When starting a simulation some assuptions concerning the initial conditions of the different storages of the model must be made. In those catchments, where only snowfree conditions were modelled, the soil moisture storage was assumed to be filled to its capacity, Fc, as the simulations started just after the termination of the snowmelt period. In the other catchments simulations started in the autumn and the soil moisture storage was generally reduced to two thirds of Fc. This assumption may have some effect on the snowmelt volumes the following spring, a fact that is worth considering when analysing the results. Initial values of the upper and lower zones can be found quite easily from the recession coefficients and some realistic assumptions about the contributions from each zone. Erroneous initial values in these zones will be detected after the first run and can thus be corrected. If the calibration and test aremade in chronological order, and throughout the year, the above problems are limited to th first year of the calibration period, as for all the consecutive periods the conditions can be transferred from the preceeding period. The work on the HBV-model was carried out on a SAAB D 22 computer until June
1975,when
this was replaced by a SAAB D 23. If organized in the
above way with computations on a daily basis, the simulation of one year (365 days) requires approximately 16 seconds in the SAAB D 23. The model, however, is not larger than that it can be programmed on a modern desk calculator, which simplifies the access but increases the time for each computation.
86
8. APPLICATIONS
The model has so far been applied to a variety of catchments in Sweden and Norway as described in chapter
4. The main purpose has been to test the
capability of the model to reconstruct a given hydrograph after calibration and to verify its performance as a forecasting tool. The simulation of discharge without any calibration, i.e. the application of the model to ungauged catchments was touched upon when testing the model in the Filefjell basin (Bergström and Jansson, 1976 B). The results looked promising but more experience will be needed before this can be made on a routine basis.
8.1. Reconstruction of the h dro ra h In appendix 1 a sample of simulated hydrographs is
shown together with
recorded discharge.The time periods used in the applications to Gimdalsbyn, Ku1tsj5n, Malgomaj and Strms ber and ending on
Vattudal are beginning on the first of Octo-
the 30th of September in order to avoid carry over
effects due to snow storage. For the same reason the simulations in Filefjell start
on , the first of September. Among all test catchments Solmyren
stands out as difficult to model (fig. A 13) having poor representation of the recession limbs. The response function of the HBV-2 model is evidently too simple for thin complex hydrograph. A study of the hydrological conditions of the different subcatchments in Solmyren carried out by Häggström, Jansson, Runesson and Simonides (1972) showed that the runoff characteristics are highly variable, which results in a hydrograph with several runoff components. In some of the small catchments, the model tends to overestimate the runoff in autumn (for example fig. A
4 , A 5 and A 9 ), a problem which
might be caused by the crude soil moisture accounting procedure,but evaporation data as a source of error cannot be neglected.
The independent test period in the Gimdalsbyn catchment (fig. A 16) shows a flood in summer, which is poorly modelled. No explanation for this has been found but poor parameter estimates
cannot be excluded, as no counter-
part to this flood occurred in any of the summer periods used for calibration.
87
The plottings from Kultsjön,Malgomaj, StrOms Vattudal and Filefjellare showingthe conditionsin the differentparts of the model during the simulationas graphs of snowpack,snowcoveredarea, yield from the snow routine, soil moisture storage and evaporation(fig. Al7 - A 21). In the three catchments
of
Kultsjön,Malgomaj and Ströms Vattudal the model was
applied for operationalpurposes. Thereforea rather low density of meteorological stations and poor runoff data
had to be accepted.Particularly
the Malgomaj catchmentis poorly covered (Bergströmand JOnsspn 1976 A), which is reflected in the performanceof the model (fig. A 18). The question of how large catchmentswe can model with one model structureis of vital importanceif our aim is to develop a modelwhicb is easy to handle and simple to calibrate.The applicationof the HBV-2 version to the Kultsjön and Malgomaj catchmentscreated the opportunityto study this problem (Bergströmand Jönsson, 1976 A). The model was therefore recalibratedfor the entire catchmentarea as
one
model with identical
meteorologicalstationsas when calibratingthe two catchmentsseparately. It was found that the HBV-3 version gave the best results, as a third runoff componentwas detected.The results, comparedto those obtained after the calibrationand the test of each catchmentseparately,are shown in table 8.1. Table 8.1. Comparisonsbetween the applicationof one model and two to the Kultsjön and Malgomaj catchments.Independent test periods underlined. R2
1962 - 1966
1966 - 1970
1970 - 1974
One model
0.8184
0.8701
0.8648
Two models
0.8436
0.8796
0.8834
The results in terms of R2-values in table 8.1 are in agreementwith the impressionsfromvisual inspectionof the hydrographs.They are interesting as they show that we can extract more informationand thus obtain a better model if we have the possibilityto split the catchmentinto subcatchments.The independenttest period for the separatedapplicationis shown in appendix 1, fig. A 19.
38
8.1.1. R2-values
A summary of the results in all test catchments expressed as R2-values according to eq. 5.3 is shown in table d.2.As discussed in chapter 5.1.1, this can be a misleading criterion of fit, especially if the period is short and the climatic variability is high. This is the situation in some of the small catchments, resulting in highly variable and sometimes discouraging
R2-values (for example Stabby 1959 and 1963, Stormyra 196)4).
Table 8.2. Results from the test catchments, expressed as R2-values.
Catchment
Period
Lilla TivsjOn 1969
Noisjön
Stabby
Stormyra
Calibration
Test
x
0.98
1968
x
0.84
1967
x
0.97
1971
x
0.81
1970
x
0.91
1969
x
0.97
1968
x
0.90
1967
x
0.88
1959
x
- 0.64
1960
x
0.93
1961
x
0.80
1962
x
0.76
1963
x
- 0.90
1964
x
0.84
1965
x
0.93
1966
x
0.64
1967
x
0.62
1963
x
0.87 0.24
1964
Solmyren
R2
1965
x
0.77
1966
x
0.63
1967
x
0.61
1968
x
0.10
1969
x
0.42
1971
x
0.91
1970
x
0.80
1969
x
0.73
Catchment Gimdalsbyn
Period
Malgomaj
o.86
x.
0.86
1962-66
0.80
1966-70
0.87
1970-74
o.84
1974-75
0.89
1962-66
x
0.79
1966-70
x
0.83
1962-66
x x
0.88 x
Filefjell
0.88
1962-66
x
0.84
1966-70
x
0.88
1970-74 StrOms Vattudal
0.79 0.84
1966-70 1970-74
Ströms Vattudal
0.91
1969-73
1970-74 Kultsjön + Malgomaj
Test
1961-65 1965-69
Kultsjan
Calibration
x
0.88
1962-66
x
0.84
1966-70
x
0.90
1970-74
x
0.83
1974-75
x
0.91
1967-71
x
1971-74
0.88 x
o.86
8.1.2. Scatter dia rams of eak flows A scatter diagram of peak flows is a plotting of computed peak flows against observedpeak flows as describedin chapter 5.1.3. Such diagrams have been constructedfor independenttest periods in the Kultsjön,Malgomaj, Str6ms Vattudal and Filefjell catchments.In Gimdalsbynthe total period for calibrationand test of the model was used in order to increasethe sample size. Only fairly distinctpeaks have been analysed,andminor timing errorsbetweenthe computed and observed dischargæhave been ignored. The results are presented in fig. 8.1.
90
oc(rn3/s
GIMDALSBYN(1961-1973)
)
160 120 80 40
•.
0
etc(m3/s) 400
••
80
40
120
160
200
0.r(m3/s)
KULTSJÖN(1970 1974) -
320 240 160 80
0
80
160
240
320
400
Qr(m3/s)
MALGOMAJ(1970-1974)
(m3/s) 400 320 240 160 80 0
0
80
160
240
320
400
Qr(m3/s)
(1970-1974) MALGOMAJ+KULTSJÖN
Q (m3/s) c 800 640 480 320
„4 • .. .
160
•
, •
s
••• 7: •
320
160
480
640
800
Qr(m3/s)
(1970-1974) STRÖMSVATTUDAL
ac(m3/s) 480 360 240
•
120
:*••••:,
0 0
C1c(m3/s)
120
240
cir(m3/s)
480
360
FILEFJELL (1971-1974)
50 40 30 20
10
0
10
20
30
40
50
Qr(m3/5)
Fig. 8.1. Scatter diagrams of peak flows. (Qc = computed discharge, Qr = observed discharge.)
92
Two catchments, Malgomaj and Filefjell, are standing out as difficult to model as far as flood peaks are concerned. In the Filefjell catchment the model underestimates
high
peaks while in Malgomaj the bias is less pro-
nounced.
8.1.3. Flow duration curves A flow duration curve is a graphical representation of the cumulative relative frequency of runoff. Such curves have been established and compared for the computed and observed hydrographs in the Kultsjön, Malgomaj, Str8ms Vattudal and Filefjell catchments. Only independent test periods have been analysed. They are shown in fig. 8.2. It is interesting to analyse the flow duration curves, the scatter diagrams and the plotted hydrographs simultaneously. The agreement between the flow duration curves look very good for Malgomaj and Ströms Vattudal, which, particularly for Malgomaj, is in conflict with the conclusion when analysing the scatter diagrams. It is obvious that the analysis of flow duration curves requires careful attention, if conclusions concerning high flows are to be drawn. A
combination of flow duration curves and scatter diagrams of peak
flows is preferable. A further observation is that all graphs show a slight bias on the low side at low flows, which is worth attention in future applications. Neither scatter diagrams nor flow duration curves have been used when calibrating the model. In future work they will, however, be incorporated in the calibration process, as they have proved to give valuable additional information about the performance of the model and as they are very easy to analyse.
93
KULTSJMI 70.10.01-74.09.30 400
RECORDED
CMS IN
x COMPUTED
320
RUNOFF 240
160
80
PERCENT OF TIMEEQUALLED OREXCEEDED MALGOMAJ 70.10.01-74.09.30 400
BECORDED
CMS IN
COMPUTED
320
RUNOFF 240
160
80
0 C11
CO
CO
CS 7
PERCENT OF TIMEEQUALLED OR EXCEEDED Fig.
8. . Flow duration
curves.
(CMS = m3/s.)
.(Continued)
94
STROMSVD 70.10.01-74.09.30. 400
RECORDED
CMS
x COMPUTED
IN 320
RUNOFF
240
160
80
PERCENTOF TIMEEQUALLED OR EXCEEDED FILEFJELL 71.09.01-74.09.20 =
50
RECORDED x COMPUTED
1-4
Li
IED
40
= cc
30
20
10
0
PERCENTOF TIMEEQUALLED OR EXCEEDED Fig.
8.2.
(Continued)
Flow duration
curves.
8.2. H drolo ical forecastin
Hydrological forecasting means the utilization of hydrological and meteorological information for the prediction of future discharge of a river. One way of doing this is by means of a conceptual runoff model, historical climate series and a weather forecast as indicated in fig.
8.3.
CONCEPTUAL MODEL SNOW CONDIT IONS
CLIMATE RECORDS
SOIL
HYDROLOG1CAL
CONDIT1ONS
FORECAST
RESPONSE FUNCTION
METEOROLOG1CAL FORECAST
Fig. 8.3. The main factors in a hydrological forecast.
The conceptual model is accounting for the memories and dynamics of the hydrological system and is thus restricting the possible effects of future meteorological conditions. The historical meteorological series can be used for long or shortrange forecasts by the simulation of possible alternative outcomes. Weather forecasts can be used for short range hydrological forecasts, i.e. in Sweden five days or less, or for the simulation of the first days of a long range forecast.
The relative importance of each factor in fig. 8.3 is highly variable from catchment to cathment and from season to season. If a catchment has a damped response, such as in the Gimdalsbyn catchment, the potential for a successful forecast is good, as visualized in fig.
8.4.
which gives the results by the HBV-model run with four different climate series starting from identical initial conditions.
(m3/s) 80 v. .1
1962
.\ .\.
60
—
\‘ \\
—1963 1965
40
\\
OBSERVED 1970
• ..
....
......
••-••
20
may
june
july
aug.
Fig. 8.4. Hydrograph response to different meteorological conditions in a damped catchment. Gimdalsbyn (1970).
The dynamics of the response function was the dominating factor, and therefore the different meteorological conditions had little effect on the hydrograph until the springflood had passed.
The opposite situation, in the Ströms Vattudal catchment, is illustrated in fig. 8.5. The same procedure as above was repeated, but the meteorological conditions strongly effected the shape of the springflood. In the latter case the importance of the meteorological forecast is obvious, but the accumulated snowpack is still a limiting factor.
Fig. 8.4 is an indication of the importance of correct initial conditions in
the different components of the model as they may have a dominating
influence on the future computed hydrograph. Therefore, before going any further into forecasting procedures, the problem of adjustments of these conditions will be discussed.
97
Q(m3/s)
360
240 -----
1963 1964 1965
t.
1966 OBSERVED 1970
•
120
may
june
july
aug.
Fig. 8.5. Hydrograph response to different meteorological conditions in a weakly damped catchment (StrOms Vattudals 1970).
8.2.1. Updating
If the forecast is initiated with erroneous discharge values, that is with improper state in the response function, the persistance will cause systematic errors in the forecast for a period of timewlach is depending on the characteristics of the hydrograph. If the snowmelt routine has overestimated the melt for a
part
of the snowmelt season, the remaining period is like-
ly to be underestimated as the snow budget is biased. Furthermore, if the
model -as overetimated
cor prec
a ral='
,
routie
moisture
r
are cor-2cteu ix nr
.2
'
0 avcd
mtan,-by wnich erronecus model cchthe above type. -Dnecan say
e=r,
i that knowl sIgeof the observ d l'i_scharge
used to adju:_ tne =riories
g. 8.3) in order to improve its future performancr,.
the
When updatin
åata, _he nex
improper 2oni1ticns in the
lue
,e
flood
ation
the model th
the model are $Ioall,a
possiblities
to adjust the free parameters in
these have been found through calibration over a
long period and shall be rarded
as time—invariant for each specific
catchment. The adjustment of the conditions in the response function is a mcre tempting possibility, as we can figure out, fairly well,,the corrections needed by looking at the recession coefficients. The method has its drawback, however, in the fact that the
deviation might be caused by errors.in preci-
pitation or in the snowmelt routine thus effecting the snowpack and the soil moisture storage as well.
The simplest and safeSt way is to assume that all the errors originate from the input, preipitation,
temperature or evaporation. Automatic methods t,o
correct these have been sought when working
on
the HBV-model, but again a
quasi-automatic meLhod with visual inspection as an important source of
ri--
formation has been considered the best way to solve the problem. Simultaneous inspections of the hydrographs, the records of temperature and precipitation together with computed snowpack are used in the search for the cause of the deviation and the period during which tO 'correct the data. Once this is determined, the model is repeatedly restarted from a given date with an automatic routine adjusting the data in small steps until an acceptable agreement is obtained. If the initial discharge value is the main problem, the deviation on this single day can be used as criterion of fit. If, for example, during a snowmelt period, we want to delete an erroneous peak in order to correct the snowpack, the R2-value can be used in the search for the best corrections.
An example of updating in the Gimdalsbyn catchment is shown in fig. 8.6. The springflood was overestimated, which was corrected by reduction of the precipitation for some period during snow accumulation. The timing of the flood was finally corrected slightly by adjustment6of the temperature values.
Q(m3/s) 80
RECORDED HYDROGRAPH ———COMPUTED _u_ UPDATED
60
40
20
Fig. 8.6.Example of updating (Gimdalsbyn,1970).
8.2.2. Short ran e forecastin Short range forecasts
were
carried out with the HBV-modelas a case
study in the Kultsjön catchment (Bergströmand Jilsson, 1976 A). Input to the model were forecastedtemperatureand precipitationvalues obtained from a barotropicmeteorologicalmodel at the SMHI. This model delivers forecastsof five days' duration for a number of meteorologicalstations in Sweden. Unfortunatelythe network is not very dense, and the coverage of Kultsjön is rather poor. The case study wasbased on meteorological forecastsin Storlien about 250 km from the Kultsjön catchnent.The temperatur values were correctedby a long term relationbetween Storlien and Klimpfjäll,the temperaturestation in the Kultsjön catchment.No such correctionwas carried out for the precipitationvalues, but this would have been more appropriate,as the average catch in Storlien is higher than that of the stationsused in the applicationto the Kultsjön catchment. The springfloodin 1975and an extreme flood in Septemberthe same year were subjectsto the investigation.The results from sequencesof forecasts
ioc
Q (m3/s) 250 RECORDED COMPUTED UPDAT ED FORECAST 150
50
Q (m3/s)
30 APR.
10 MAY
20 MAY
DATE
10 MAY
20 MAY
DAT E
250
150
50
400 SNOWPACK (mm) 20 APR.
30 APR.
1±-1.
a(m3/s) 250
/ 1 t
1
150
/
\ \
t i /
1
1
I
i
I
i 1 1 l
/"
/
‘
•N
// \
\
--
1
• \/
1
/
\
/
i
/
,
/
1
/
\
\
\ \
50
400 SNOWPACK (rnm)
20 APR.
30APR.
10MAY
20 MAY
DATE
3 Q(m/s) I 1
/ 1
l
/
/ I 1
I 1 I 1
I i
150
/1
- -
1
/
I , 'I ‘
1
t I
I I I
"
1
i
1
50
I
/
/ #
/
,,
/ \
/
I I i
\ 1
11
-
I
/ //
I
I ,
/
)
,
.....,
SS -
400
30 MAY
9 JUN.
19JUN.
1.JUL.
DATE
Fig. 8.7. Short range forecastsof inflow to the Kultsjönreservoirin the spring of 1975. (Computedsnowpack in water equivalents.)
RECO.RDED COMPUTED , UPDATED FORECAST 150
50
10 SEP
20 SEP
30 SEP
DATE
10 SEP
20 SEP
30 SEP
DAT E
3 (m is) 250
150
50
Fig.
8.8. Short
range
September
forecasts
1975.
of
inflow
to
the
Kultsj&I
reservoir
in
are presented in fig. 8.7 and
8.8. The springflood was predicted fairly
well as long as snowmelt was dominating, while uncertain precipitation forecasts caused proble=
for both periods. The long distance to the mete-
orological station, on which the forecasts were based, makes the precipitation values very uncertain. In these applications they would hardly be any betterwith a correction factor, accounting for the systematic deviations in catch. The temperature has a much more favourable spatial correlation pattern and is thus easier to forecast. When analysing the results it must be born in mind that the Kultsjön catchment is an extreme one because of its quick response. Therefore the forecasts are very susceptible to the meteorological input and the support by the dynamics of the response function is small. More experience with short range forecasting will be gained in the future, as the model will be used operationally for this purpose, beginning in the spring of 1976. Particularly the calibration of the meteorological forecasting procedures to appropriate meteorological stations have to be considered.
8.2.3. Lon
ran e forecastin
Long range forecasting can be carried out along the lines touched upon in fig.
8.4 and 8.5. If the model is updated and run with alternative climate
series, an array of outcomes can be obtained for further analysis. This method was applied by Danielsson and Wretborn (1975) and was used in the first tentative long range forecasts and Jönsson,
by
the HBV-model in
1975 (BergstrOm
1976 A). In the winter of 1976 the method was taken into ope-
ration for the prediction of springflood volumes in the Ströms Vattudal and Kults,»n catchments. An example of a long range forecast, issued on the lst of March and stretching to the 31st of July, is shown in fig. 8.9. Meteorological data from
1975 were used. The extreme low curve is a simulation with temperatures according to the year 1975 and the corresponding period of the years 1962 -
with zero precipitation in order to arrive at a lower boundry.
lo4
3 Q(rn/s) 24000
12000
MAR.
APR.
MAY
JUN.
JUL.
Fig. 8.9. Long range forecast of inflow to the StrOms Vattudal reservoir in 1976.
A forecast of the above type can be of good help when operating a reservoir, but the outer limits must be treated with care. If, for a moment, we neglect the effect of poor model performance and erroneous data for the forecasted
period, two principle sources of uncertainty have to be
considered in the forecast. One is the climatic variability, which is well reflected in the distributions of outcomes from the simulations. Another effect is the one of biased initial conditions in the model, which cannot be detected by analysis of the outcomes. If, for example, the snowpack is overestimated in the model on the date when the forecast is issued, all the simulationswill suffer from this. The establishment of confidence limits on the outcome is not an easy task and has not been attempted except for the determination of upper and lower extremes, based on the sample of simulated volumes. An investigation of the reliability of the above long range forecasting zhe Ströms Vattudal catchment is shown in table 8.3. Data
from 1962 to 1974, excluding the year subject to forecasting, were used for the establishment of the average and extreme outcomes over different periods. From the table can be seen that the observed outcomes are exceeding the high extremes for all periods in 1972 and are falling below the low extreme for one period in 1974. The poor results in 1972 are easier understood, if fig. A 20 in appendix 1 is studied. The wet period in July is somewhat extreme and had no counterpart during the same periods of the other years. This wet period has been included in the forecasts for the other years and also in the forecast in 1976 (fig. 8.9).
1D5
Table 8.3. Results from forecastswith the HBV-modelin Stri5msVattudal, based on eleven years of meteorologicaldata. HBV-2
Observedoutcome
EQ(m3/s) Period
EQ(m3/s)
high
mean
1.2. - 31.7.
15 800
12 960
10 250
1.3. - 31.7.
17 620
14 210
11 580
1.4. - 31.7.
16 560
13 64o
lo 770
1.5. - 31.7. 16.5.- 31.7.
16 280 14 290
13 410 lo 650
il 320
1.6. - 31.7.
9.690
7.020
l ow
1971
8 330 5 64c
15 570 14 860 14 54o 14 030 11 130 6 680
1972
1.2. - 31.7.
17 450
14 470
11 740
1 8 5 30
1.3. - 31.7.
15 370
13 110
10 630
18 45o
1.4. - 31.7.
15 560
13 240
10 490
18 070
1.5. - 31.7.
13 870
12 46o
lo 550
17 110
16.5.- 31.7. 1.6. 31.7.
11 030
9 490
7 390
17 810
7 800
6 110
4 880
10 420
1.2. - 31.7.
15 470
12 48o
9 780
15 400
1.3. - 31.7.
16 340
12 960
10 380
1.4. - 31.7.
14 600
11 540
1.5.- 31.7. 16.5.- 31.7.
14 420 14 900
12 060
11 270
8 710 9 45o 9 000
14 650 13 680
1.6.- 31.7.
lo 360
7 780
6 280
7 150
1.2. - 31.7.
20 630
17 920
14 630
14 980
1.3. - 31.7.
21 200
17 990
15 120
1.4. - 31.7.
18 610
15 810
12 690
14 490 14 260
1.5. - 31.7.
14 740
12 080
9 940
11 750
16.5.- 31.7. 1.6. - 31.7.
12 600
9 040 4 240
6 740
9 42o 6 14o
-
1973
12 46o
11 050
19 74
6 770
2 990
A very tempting approach to long range hydrological forecasting is the use of stochastic models for the generation of meteorological data to the model. Stochastic generation of temperature and precipitation is, however, a complex problem due to the involved correlation pattern between the two variables, and no such method is, according to the author's knowledge, at present in operational use.
8.2.4. 0 erational s stems In the spring of 1976 the HBV-model was entering an operational phase. A system for direct on-line operation of the model for the reservoirs in Kultsj5n, Malgomaj and Str5ms Vattudal was being developed by the Kraftdata AB in cooperation with the river regulation company (Ångermanälvens Vattenregleringsf5retag) and the SMHI. Forecasts were also issued directly by the SMHI. The rapid transmission and processing of field data is of vital importance in hydrological forecasting. This can actually be an argument for restrictions in the model on the demands of input data. So far the model is operated with manned meteorological stations, reporting the most recent data by telephone. In the future more emphasis has to be put on this problem, at least if automatic stations in remote areas are incorporated in the work, and if the number of forecasts per day is increasing. Operational systems also require good access to computer and practical routines for updating and forecasting. Particularly the updating procedure requires a system with some built-in safety functions, as the observed field data will be manipulated. When running the model operationally it may be convenient to save data on the internal variables, i.e. the conditions in the model, at some intervals in order to be prepared for restarts in the updating procedure.
107
9. CONCLUSIONS
A very general but important conclusion from the work
on
conceptual run-
off models at the SMHI is that the simulation of river flow from meteorological data can be made with surprisingly simple models. Detailed subroutines, which may be justified by field measurements, can often be greatly simplified,when incorporated into a runoff model,without any degeneration of the model performance.
The HBV-2 and HBV-3 versions of the HBV-model have proved to be capable of reconstruction of a hydrograph from data of precipitation, temperature and potential evaporation in several test catchments, if the parameters of the model are adjusted during a calibration period. Generalization of these parameters is a difficult problem due to interactions and implicit corrections.More experience is therefore needed before the HBV-model can be applied to ungauged catchments.
The model can also be used for hydrological forecasting. The performance of the model for this purpose is increased,if a reliable updating routine is incorporated for adjustments of the initial conditions of the model before performing the forecast. Meteorological forecasts and recorded climatic series can be used in shortrange
and longrange
hydrological forecasts.
The model has not been used for predictions of the effects of future physical changes in a catchment. As long as the physical interpretation of some of the components of the structure is unclear, the only approach to this problem are empirical relations between parameters and catchment characteristics which requires a lot more experience
of
applications to catch-
ments of different types.
Proper assessment of the parameters of the model is of outmost importance for its performance. A good model with a potential for close reconstruction of discharge will be of little value, if it is poorly calibrated. The many aspects on a good agreement between a computed hydrograph and an observed one make
the use of automatic calibration and a single numerical verifi-
cation criteriquestionable. a rather subjective process.
The calibration procedure is therefore still
o8
The model has been used for operational hydrological forecasting since the spring of 1976. Future work will probably be concentrated both on the development of better forecasting procedures and more efficient systems for data collection, updating and forecasting. The model will also be applied to some other three
catchments for operational purposes. At present work is in progress on catchments of the river Västerdalälven, the river Stora Luleälv and
the river Ljusnan.
109
APPENDIX1
PLOTTINGSOF COMPUTEDAND RECORDED HYDROGRAPHS
lic
LIST OF SYMBOLS
ACC. DIFF. EVP MELT
accumulated difference between the computed and the observed hydrographs. computed actual evaporation. yield from the snow routine (including rainfall when the ground is partly snowcovered). recorded precipitation, areal means.
SM
computed soil moisture storage.
SNOWCOV
computed snow covered area.
SP
computed average snowpack.
TEMP
recorded temperature, areal means. computed and recorded discharge.
111
L.TIVSJdN
ACC.D1FF. (l/s) 3000
NEDRE 1969
2000 1000 0 1000 -2n0 3000
OBSERVED HYDROGRAPH COMPUTEDNYDROGRAPH
"/INI0 40.0
200
0
(mm) 30 20 10 MAY
APR
Fig.
A 1.
Fitted
JUN
period
ACC.D1FF. (1/s)
JUL
in
the
AUG
Lilla
L.TIVSJÖN
SEP
Tivsjffil
OCT
NOV
catchment.
NEDRE 1968
3000 2000 1000 0 -1000 2000
a
OBSERVED HYDROGRAPH
3000 ( t /s ) 600
COMPUTED HYDROGRAPH
400
200
0
(mm) 30 20 10 0
Fig.
APR
A 2.
MAY
Test
period
, I JUL
JUN
in
the
Lilla
AUG
Tivsj5n
EP
catchment.
OCT
NOV
LTIVSJÖN NEDRE 1967 ACC.D1FF. (t/s) 3000 2000 1000 0 —1000 —2000 —3000 Gl(L/5) 1200
0 BSERVED HYDROGRAPH —X--
COMPUTED HYDROGRAPH
1000
800
600
4 00
200
0
p (mm) 30 20 10 0
Fig.
APR
A 3. Test
MAY
period
JUL
JUN
in the
Lilla
Tivsjön
AUG
SEP
catchment.
OCT
NOV
113
ACC.DIFF. (1/s) 3000
NOLSJÖN
NEDRE
1971
2000 1000 0 1000 2000 3000 a ((/s) —x—
OBSERVED HYDROGRAPH COMPUTED HYDROGRAPH
400
200
0 P (men) 40 -
20
PR
Fig.
A
MAY
JUN
JUL
AUG
SEP
OCT
NOV
Fitted period in the Nolsjön catchment.
)4•
ACC.DI FF. ((/s)
NOLSJÖN NEDRE 1970
3000 2000 1000 0 1000 2 000 - 3000
a
OBSERVED HYDROGRAPH
((/s) --
X--
COMPUTED HYDROGRAPH
4 00
200
0 P(mm) 60
40 20
0
APR
MAY
JUN
JUL
AUG
S P
Fig. A 5. Fitted period in the Nolsjön catchment.
OCT
NOV
ACC.DI FF. (L/5 )
NOLSJÖN
NEDRE 1969
3000 2000 1000 0 —1000 —2000 —3000 0. (lfs) 600
OBSERVED HYDROGRAPH COMPUTED HYDROGRAPH
400
200
(rnrn 40
20
APR Fig.
A
6. Test
period
JUN
MAY
in the
ACC.DIFF.(1./s) 30 00
Nolsjön
JUL
AUG
SEP
OCT
NOV
OCT
NOV
catchment.
NOLSJÖN NEDRE 1968
20 00 100 0 0 1000 2000 3000 BSERVED HYDROGRAPH
x
(l/s)
COMPUTED HYDROGRAPH
4 00
200
0 P (mm)_ 40
20
0
Fig.
APR
A 7. Test
period
MAY
in the
JUN
Nolsjön
JUL
AUG
catchment.
SEP
115
NOLSJÖN NEDRE ACC.DIFF.
1967
(l/s)
6000
4000
2000
—2000
Q (115)
OBSERvED HYDROGRAPH
1200
COMPUTED HYDROGRAPH
1000
800
600
400
200
P(mm) 40
20
APR
Fig.
MAY
A 8. Test period
JUN
in the
JUL
Nolsjn
AUG
catchment.
SEP
OCT
NOV
n6
STABBY 1959 ACC.DIFF.(l/s) 2000
0
—2000
OBSERVED HYDROGRAPH COMPUTED HYDROGRAPH
Oh(L/s) 400
200
0 P(mm) 40
20
0
Fig.
A
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
9. Fitted period in the Stabby catchment.
STABBY 1960 ACC. DIFF. (t/s)
2000 0
—2000
0 BSERVED HYDROGRAPH
et (1.4) —X--
400
COMPUTED HYDROGRAPH
200
0 P(mm) 40 20
APR
MAY
JUN
JUL
AUG
SEP
Fig. A 10. Fitted period in the Stabby catchment.
OCT
NOV
117 STABBY
1965
ACC.DIFF.(l/s) 2000
0
—2000
a (l/s)
0 BSERVED HYDROGRAPH
400
COMPUTED HYDROGRAPH
200
0 P(mm) 40
20
0
Fig.
APR
MAY
JUL
JUN
A 11. Test period
in the
Stabby
STORMYRA ACC. DIFF. (Ijs
SEP
AUG
OCT
NOV
catchment.
1963
)
2000
0
—2000
OBSERVED HYDROGRAPH
a (L/s)
COMPUTED HYDROGRAPH
400
200
0 P(mm) 40—
20
0
Fig.
APR
MAY
A 12. Fitted
JUN
period
JUL
in the
AUG
Stormyra
SEP
catchment.
OCT
NOV
118
ACC. DIFF.
(1/
SOLMYREN
s)
1970
30 00 2000 1000 0 -100 2000 3000
(l /s OBSERVED COM PUT ED
1000
HYDROGRAPH HYDROGRAPH
800
600
400
200
P (mm) 30 r 20 I10 APR
MAY
JUN
JUL
AU G
Fig. A 13. Test period in the Solmyrencatchment.
S EP
OCT
NOV
GIMDALSBYN 1961 -1965
TEMP (°C +20
r_. I-
-20
L
f
Q (m3/s) r100
,Askt
PrW,-
-
NY. "
----
# •
• •
ACC DIFF(mm)
COMPUTED
- 7+150
RECORDED
tO 5C J-150
ONDJ
FMAMJ
J
AS
0 NIDIJIFIMIAIMIJ
1.1 lAiSIOIN
DJFM
A;MiJ
;.1 1A S1OS
D J F1M AIMIJ
J
A S;
P(mm) 30
SP(mm) 300 '
Fig. A 14. Fitted period in the Gimdalsbyncatchment.
GIMDALSBYN 1965 -1969
TEMP (°C ) +20 ±0 -20 Q (m31s)
ACC.DIFF(mm, +150
COMPUTED
100
RECORDED tO 50 -150
ONDJ
FMAMJJ
ASONDJFMAMJJ
ASONDJ
FMAMJ
J
ASONDJ
FMAMJ
J
AS
P(mm) 30
SP(mm) 300
Fig. A 15. Fitted period in the Gimdalsbyncatchment.
GIMDALSBYN 1969 -1973
TEMP (°C ) +20
-20 Q m 100
s
ACC.DIFF(mm) -
COMPUTED
-j+150
RECORDED _O 50 --150
ONDJ
FMAMJJ
ASONDJ
FMAMJJ
ASONDJ
FMAMJ
J
P(mm) 30
ASONDJ
FMAMJ
JIAS SP(mm) 300
Fig. A 16. Test period in the Gimdalsbyn catchment.
120
Fi ures in the followin
a es:
Fig. A 17. Test period in the Kultsjön catchment. Fig. A 18. Test period in the Malgomaj catchment. Fig. A 19. Test period in the Malgomaj + Kultsjön catchment. Fig. A 20. Test period in the Stråms Vattudal catchment. Fig. A 21. Test period in the Filefjellcatchment.
SMHI HBV-3
KULTSJUN70.10.01-74.09.30
TEMP(C) 20 10 0 -10 -20
ACC.DIFF (MM)
(L/S)
isa loo so
COMPUTED HYDROGRAPH REC1RDEDHYDROGRAPH
20000Q 160000 120000 DL
7,1 C33 03 •
80000
111
-.., en= i—co
40000
o
.cp.
V ONDJF
P (MM) 30 20 10
rT
MJJASOND
MJJASONDJFMAM'JJASONDJF
AMJJAS SP (MM) 300
cp= i; SNOWCOV co 100 HJ
75
IC3
1C) JJ
co=
cj
50 0 10 20 30
MELT (MM)
SM (MM) 200
100
EVP (MM 8.0
4.0
SMHI HBV-3
ALGOMAJ 70.10.01-74.09.30
TEMP (C) 20 10
-4`
0 -10 -20
ACC.DIFF
(L/S)-
(MM) 150
200000
RECORDED HYDROGFIRP
100
160000 -50
120000
-100 —1 rrl
•
90r
-150
80000
-17L.40000 Cfl
ONDJFM
3A91-1
P (MM) =
=
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