A STATISTICAL STUDY FOR AUTOMATIC CALIBRATION OF A CONCEPTUAL RDNOFF MODEL

STATISTISK STGDIE FÖR AUTOMATISK KALIBRERING AV EN :BEGREPPSMÄSSIG AVRINNINGSMODELL

Sören Svensson

SMIU Rapporter

HYDROLOGI OCR OCEANOGRAFI Nr RRO 10 ( 1977 )

SVERIGES METEOROLOGISKA OCH HYDROLOGISKA INSTITUT

Norrköping 1977

i •

2.

3.

4,

LIST 0F C0NTENTS

Chapter

4. 1

4.2

4-3

4,4

4.5 4.6

5. 1

5.2 5.3

List of contents

SUillI!lary

Introduction

Test catchments

The HBV-model and its free para.meters

Snow routine

Soil moisture routine

Response function

Statistical analysis of the residuals of the HBV-model

Mechanism separation criterion (MSC)

Estimation of the density function

Test of normality and independence

Estimation of the autocorrelation

Significance of the autocorrelation

Analysis of the autocorrelation

A study of response surfaces of criteria of fit

Criteria of fit

Basic assumptions and observations

The response surfaces of criteria of fit and test plottings of Kultsjön

Th f th R2 ·t . e response o e -cri eria to K

2 and C perc

Th f th R2 ·t . e response o e -cri eria to K

1, B and C max route

The response of the R2-criteria to Fe, Sand L /Fe

p of the R2-criteria The response

to csf and S

Th f th R2 ·t . e response o e -cri eria to T and C

0 0

Th f th R2 ·t . e response o e -cri eria to C h and C f w r r

1

2

5 6

8

9 10

13

13

16

17

19

22

23

29

29

31

40

42

42

6.

5.4

5.4.1

The response surfaces of eriteria of and test plottings of Stadarforsen

The response of the R2-eriteria to K

2 and C pere

2 The response of the R -eriteria to K

1, C t and B rou e max

The resporise of the R2-eriteria to K and L

0 UZ

Th f th R2 't . e response o e -eri eria to K

0 and K

1 The response of the to Fe, 8 and L /Fe p The response to csf and 8

of the

R2 't . -eri eria

R2 't . -eri eria

5.4.7 The response of the R2-eriteria to T and C

0 0

5.4.8 The response of the R2-eriteria to C h and C f w rr

5.5 Results of the study Conelusions

Appendix A: Density_funetions

List of symbols

List of referenees

Appendix B:

fit

45

45

45

48

48

52

52

52

55 55

- 1 -

SUMJ'IIARY

A conceptual runoff model is studied in this work. Especially

the residuals of the model (the differences between the compu-

ted hydrograph and the recorded one) are exami-11.ed. The density

function and the autocorrelation function of the residuals are

estimated and tested.

The model must be calibrated for each new catchment, to which

it is applied. Therefore a criterion of the goodness of fit

between the computed and the recorded hydrograph is required.

Some criteria of fit have been examined concerning their sensi-

tivity to changes in the model parameter setting.

Conclusions of the work are: The residuals of the model are

neither stationary nor independent and normally distributed. A

classification based on the different physical processes, which

govern the discharge, makes the residuals of each class more

stationary and in some sense more normally distributed t~an the

residuals of the material without a:ny classification. Further-

more a criterion of good.ness of fit, based on the classifica-

tion above resembles the subjectively judged fit more than the

simple sum of squares criterion, which has become practice in

applications of runoff models.

- 2 -

1 • INTRODUCTION

The purposes of the development of hydrological catchment medels

are mainly (Clarke, 1973):

1 • Forecasting

a) Operational forecast: Estimating streamflow when rain-

falls, losses, streamflows etc.

are given up to the present time.

b) Design forecast: Estimating the flood hydrograph

caused by a hyJ:>othetical heavy

storm (or snowmelt).

2. Extending the discharge redord, where we have got long cli-

matological record but short discharge record.

3. Prediction of the possible effects of proposed pbysical changes to the catchment.

Beföre the model can be taken inta operation, its free parameters

have to be accurately estimated by means of a calibration proce-

dure. This means that the parameters are adjusted until a good

fit is obtained between the computed hydrograph and the observed

one. The procedure can be either a subjective one, relying upon

the hydrologist when deciding which parameters are to be adjusted,

or an automatic one, based on an optimization algorithm.

One major problem concerning the automatic procedure has been the

finding of a matching index, a criterion of fit. This is very im-

portant, because every automatic parameter optimization routine

has to rely upon a nu.merical value of the goodness of fit between

two curves.

The statistical properties of the residuals could be a key toa

better understanding of this problem. The lack of such studies was

pointed out by Clarke (1973).

the prol:iabili ty distribution of the rnodel resj_d_,,1als" BowevE.·r~

one fu::cther possibili ty is to do s. nu.mhe:r: of paE3lliE' r-er sstina-

tions based on independent data and from ths :.r6su::i_ ts of ti1e::;e

estimations get an apprehension of the s i se ,:,f" the ,,onfidenct:c

regions of the model parameters.

The a,im of this wo:r:k is to study the statistical pL·operties sf

the residuals in order to find a more app~opriate criterion oi

fi t between the cornputed and the recorded h;ydJ..0ographo This prob-

lem. was approached in two steps.

1. The calibration period was divid.ed ir1to 11 subperiods'1 in

order to obtain stationary subsets of residuals. Some

statistical properties of these residuals were examined.

2. The response su:rfaces of different criteria of fit were

studied when altering the parameter values.

The EJ3V runoff model was used for the study. This rnodel is deve-=

loped s;c the SJYIHI (Swedish Mete,:;rological and Hydrological Insti-

tute) by Bergström (1976). It is in operational use in same

c1;,,t:;hm.ent~ in Sweden and N"orway.

Tl1is stuc'Qr was financed by the SMHI and it was carried out in

co-o:peratinn with the ])epartment of Mathematics at the Lin-

köping University~

- 4 -

~-100 -= --!~-- 100 km --:1

(j i 30°

. () I

+

i ig. 2.1. Test catchm t th en 8 for th e HBV model~ e applications of the study of

- 5 -

2o TEST CATCBJVJENTS

Two catchments were used in the study: the Kultsjön catchment~

below referred toas Kultsjön, and the Stadarforsen catchment,

below referred toas Stadarforsen (fig. 2.1). Both catchments

are below tirnberline dominated by coniferous forest, and the

soil is mostly moraine ar of a pervious type.

The catchments are representative for large areas in Scandi-

navia.

Table 2.1 Test catchments.

Area Altitude Lakes Catchment River (k:m.2) range (m) %

Stadarforsen Västerdalälven 4 136 835 2,4

Kultsjön 0

Angerm.anälven 1 109 1 040 6

The analysed runoff record of Stadarforsen bega.n 1961-10-01

and ended 1976-03-31. In Kultsjön_ the analysed record bega.n

1961-10-01 and ended 1976-05-18.

- 6 -

3. THE HBV-MODEL AND ITS FREE PARAMETERS

The simulation of discharge by the HBV-model. is made in tbree

steps (fig. 3.1).

1. Snow accumulation and ablation.

2. Soil moisture accounting.

3. Generation of _runoff and transformation~of t~e hydrograph.

PRECIPITA TION

SNOW ROUTINE

RAIN, SNOWMELT, _..__~--i.- EVAPORATION

SOIL MOISTURE ZONE

Fig. 3.1 Schematic representation of the HBV-model.

- 7 -

Table 3 _ •j Parameters of the HBV-model.

Corrections on input va.riables

P correction factor on rainfall corr

P = precipitation-elevation correction lapse

~ = temperature-elevation correction -1apse T general temperature correction

0

Parameters of the snowroutine

snow fall correction factor

= degree-day melt factor water holding capacity

bottom storage under snowpack

= refreezing coefficient

Parameters of the soil moisture routine

Fe = maximum soil moisture storage

L limit for potential evaporation p 6 erupirical coefficient

Parameters of the response function

storage discharge constant of the upper zone

C perc p

w

B max

C route

tr PI 11 n 11

= Il " Il t1 lower limit for slow dxainage of the upper zone

= percolation capacity into the lower zone

part of the lower zone representing lakes and other wet areas

Il

"

maximum base in the transformation function

parameter relating the base in the transfor-mation function to the generated flow

- 8 -

The model para.meters are shown in table 3.1. P , P1

, T1 corr apse apse

and P are set from information outside the calibration procedure w

(maps, experience etc.). The ot:.1ers are calibrated to optimum fit.

3.1 Snow routine

Whenever the air temperature (T) is below a threshold value (T ), all 0

precipitation is regarded as snow and is accu.mulated in the snowpack.

All effects of evaporation and lacking representativenese of the

gauge are put together in one empirical coefficient, the enow fall correc•

tian factor (C8f).

Thus, if T < T then 0

s = C • P, s sf where s = actual snow accumulation (mm),

8

csf = snow fall correction factor, p = precipitation (mm), T = surface air temperature (oc).

Snowmelt is taken care of by the degree-day method:

If T > T, then 0

M = C (T - T ) , 0 0

where M = snowmelt (mm/day),

C = degree-day factor (mm/( 0 c a.ay)). 0

The water retention in the snowpack is described by two parame;.ers.

Owh = waterholding capacity of the anow (% of the snowpack), Sb = bottom storage under the snow (IIIIIl).

Sb has'rarely been used and was therefore omitted in this work.

Refreezing of liquid water in the snowpack:

If T < T and if there is liquid water present in the enowpa.ck, then 0

M = C f • C (T - T ), r r o o

where -M = refreezing rate (mm./a.ay),

C = refreezing coefficient. rfr

- 9 -

The area-elevation distribution of the snowroutine in the HBV-

model is described by Bergström (1976). It contains no fre&

parameters and is therefore not very interesting for the cali::J-

ration of the model.

3.2 Soil moisture routine

The behaviour of the soil moisture zone is illustrated in fig.

i.O

0 0

Fig. 3.2 The contribution.s from rainfall or snowmelt, P, to the soil moisture zone, S , and the upper zone, sm s

UZ

Mathematically, this can be described by the following

equations:

s l UZ 6. p

/:, 8sm 6. p

where P

s UZ

s sm Fe

s s

6. UZ

precipitation or snowmelt (mm), = storage in the upper zone (mm),

computed soil moisture storage (mm), = maximum soil moisture storage in the model (mm), = empirical coefficient,

= amount that passes through the soil moisture zone (mm)'

- 10 -

~ S = amount that is stored in the soil moisture zone (mm), sm

~· p = precipitation or snowmelt fed into the zone mm by mm.

Potential evaporation is reduced to actual values by the function

in fig. 3. 3.

0

Fig. 3.3 Reduction of potential evaporation, E, to actual, E. p a

Mathematically described by:

E = a

E if S > L p sm - p

s E • sm

p L if S < L, sm p p

where E = potential evaporation, p E = actual evaporation,

a L = limit for potential evaporation.

p

3.3 Response function

Having passed the soil moisture routine the excess water passes

through some reservoirs, where the runoff Q is easily calculated g

in the ma.nner illustrated in fig. 3.4.

- 11 -

0. =K·(S-l I 0 0 ~ Vl'

QI= K(Suz

TRANSF. _ __..-;it FUNCTION

Fig. 3.4 The response function of the HBV-model.

C = percolation capacity, perc E = potential evaporation,

p K = storage discharge para.meter of the upper zone,

0

K1 = slow drainage storage discharge para.meter of the

K2

L UZ

p

p w

Qg

Qo

Q,1

Q,2

Qc 81z s

UZ

6 S UZ

upper zone,

= storage discharge para.meter of the lower zone,

= limit for slow drainage of the upper zone,

= precipitation,

= part of the lower zone, representing wet areas,

= total generated runoff,

= runoff generated from the upper zone,

= slow drainage runoff generated from the upper zone,

= runoff generated from the lower zon8

= total computed runoff,

= storage in the lower zone of the medel,

" Il " upper " " " " inflow in the upper zone.

The transforming function multiplies Q with a weight differing g

intime. Q is transformed inte Q according to fig. 3.5. g C

- 12 -

WEIGh1

TIME TIME

Fig. 3.5 The effect of the transformation function on the generated hydrograph.

This can be expressed as: t -C • Q,g if (B - C • Qg) .::_ 1 B = max route max route q if (B - C • Q ) < 1 max route g

B = base of the triangu.la.r function q

B = maximum base (days), max

(days),

C t = free parameter (days/(m3/s)), DOU e

Qg = generated runoff from the reservoirs (m3/s),

Time = O: the day on which Q is generated (days). g

4. STATISTTCAL Af~ALYSIS OF ~BE RESIDUALS OF THE m:iv-MGD~L

4.1 Mechanism separation criterion (lVISC)"

A study of the hydrographs showed that differe11t pbysical pro-

cesses produced different discha:r·ge patter·ns. Subsequen-tly 8,

study of the residuals showed a similar mixture of different

curve types with different statistical properties.

Chiefly three different processes were assumed.

1. Sno"WIIlelt

2. Rainfall or recession succeding rainfall or sno-wIILelt

(referred toas y-flow)

3. Dry summer- or winter-recession (referred toas low flow).

1'.i:le problem was to find a good cri terion to separa te the days

in order to obtain classes of days with approximately station-

a:.r:y residuals. One method is to divide the calibration period

by visual inspection. The drawbacks of this method are its sub-

jective character and the fact that it does not take the dif-

ferent model mechanisms inta account.

The method was therefore abandoned and the followJ.11g mechaism

separation criterion (MSC) was used.

1. M > 0 snowmelt

> o): y-flow 2. M

3. JVI

where M

s UZ

LIS UZ

< 0

< 0

and

and

(LIS > UZ

( LIS = UZ

0 or S UZ

0 and S UZ

snowmelt (mm),

0) : low flow,

storage in the upper zone of the rnodel (mm),

= inflow in the upper zone of the model (mm).

This MSC has two major drawbacks:

1. It will give different partitions of the calibration

period at different parameter settings. This will

disturb the behavi01.-~1.• of the properties studied.

- 14 -

2. It may be difficult to apply to other models.

The main advantage of the chosen MSC is that it is easy to extract

from the model. An example of the use of the MSC is shown in fig. 4.1.

In Kultsjön the transforming function causes a time lag, which hasa

maximum length of one day (B = 2, C t = 0.00103, chapter 3.3). max rou e This small influence is neglected in the mechanism separation.

In Stadarforsen the time lag is much greater (B = 6, C t = O, max rou e chapter 3.3). Here the main part of the generated runoff is delayed by three days. In consequense of this the MSC was displaced three

days.

Fig. 4.1 (See next page.) Residuals (1/s) and MSC of Stadarforsen 65.10.01 - 69.09.30. 1. Snowmelt

2. Rain or recession succeeding rain or

snowmelt.

3. Dry summer- or winter-recession.

The upper curves show the MSC, the

lower curves show the residuals.

laj M-J -Jvrvu~t f I-'· 250000 ~ ~ . 150000 _.

50000 t, \~~-~

-50000

-150000

0 N D J F M A M J J A s 0 ~~ I I ._j -3

'{ -\_ _ _J ~2 250000 I .l

150000

_.

50000 (\ \Jl

-1 (/\~ t\ ' -50000 \/ 1-v ,,

-150000

F M A M J J A s ('J N D _J F 1'1 I A -7 M _ j\__ __ 1 Jlfl_

vVVl __ J\ __ -250000

150000

50000

~ /I},\ /1 --, r \ I _,.,~--\-,_ .-- -~- '\._ - -._,j - ... /\ -50000 'v \/\

I

I I I ,_ I

,,J

-150000 _J J I

---- - - -- -- I - - - - -,

A s 0 N D I F M A 1'1 l I :_ l ,_,

4.2 Estimation of the density function

Mean and standard deviation of the residu.als were estimated by:

x = 1 i ro (i) - Q (i)l n i=o L'""r c J

s2 ~ ~-1 te [ Cl.,, (i) - Qc (i) - X] 2 where X = mean value of the chosen residu.als (m3/s),

n = number of the chosen residu.als,

~(i)= observed discharge (m3/s),

Q (i)= computed discharge (m3/s), C

S =standarddeviation of the chosen residuals (m3/s).

A_ symmetrical interval, four standard deviations long, was

placed around the mean value. The interval (x - 2s, X+ 2s) was divided inta 40 equ.ally long classes, and the number of residuals

in each class was plotted in a histogram. For comparison a Normal

probability distribution function with X mean and S standard devi-

ation was plotted in the same diagram.

As the interval of four standard deviations is toa short to

contain all the residu.als, the number of exceeding residu.als (NER)

is printed together with each histogram. An exam.ple of a histogram

can be seen in fig. 4.2. (days)

300

200

100

(standard 0

0L...L-'--L.....L.-'--L-L..L-J~

0.L...l.-'-.L...l.-'-'---'-_._~

0......,,_._~_._ ........ __._ ........ o_._ ......... ~ ........ ..._.__._,o deviations)

N - 0 -I I Fig. 4.2 Histogram of the low flow residu.als of Kultsjön 62.10.01 -

- 76.05.1a. Meanvatue = o. 725 (m3 /1,);-i3tanda:rd deviation--=

= 7.07 (m3/s). NER= 145

All histograms are presented in Appendix A.

A preliruinary test was made i.:;e, s1;ud:r it ~h,-, ill::oc',-'-- 'c,_1_ ~te

eae:b clase of __cesiduals (x) dlfI8Ii8U_ ,JJ_g11L L~antl~ :r,-,(,Jli 2,-0 :1 ( The simple t-test w-a,s used fo:c this purpose. SoLe appaie:~-;Jy

signific&nt resul ts were fou11d.

However, the t-test is -based on the assumption of i_ndepeudeuc

and N( µ, cr)-distributed variables. The assumption of .norn19,li ty

is not very critical, but the assumption of independence has

to be fullfilled.

As will be shown later, the residuals are not independent. The

test was carried out with regard paid to the interdependence~

according to Hansen (1971). Thus the quantity .2l_ was inter-2 n

preted as the equivalent number of independent x observa-

tions.

Where n is defined by: X

n J R( T) d T X 0

R(T) = autocorrelation function of the chosen residuals.

Using the t-test in this way showed no significant deviation

from zero on the 5 % level.

However, each class of residuals consists of a number of inde~

pendent continuous time periods. Regarding this, one could

construct a more powerful test.

4.3 Test of normality and independence

Ve,ry often in statistical applications the assumptions of mubu-

ally independent and normally distributed variables are ma,le.

To test these assumptions on the 2 mechanisms the X -test was used.

k I: u = . 1 l=

2 (Y.- np.) l l

residuals of different model

where u k

n

p. l

Y. l

- 18 -

test va.riable,

number of test classes,

= total number of recorded days,

probability fora normally distributed, stochastic variable

to assume a value within class i,

observed number of days in class i.

If the assumptions hold, then U is approximately x2-distributed with the

number of test classes mL"J.us three degrees of freedom.

Whenever np. < 5, the test class ,j,was merged with a neighbouring l

test class, until the probability to fall within test class limits

was greater than 5/n. This validity limit is ta"i

- 19 -

The hypochesis had to be rejected on the 0.7 ja ievel.

Although the material was divided in-i:;o different classesj the

hypothesis H still did not hold. However, the x2-variable (u) 0

was made less by the separation of the different mechanisms.

A visual inspection also shows that the residuals a:r-e more

close to normality in each subclass after the separation.

Therefore H is, in some sense, more valid after the mechanism 0

separation of the material than before.

2 Ji'urthermore, the x -test is very sensitive when used on a mate-rial built upon many observations. The data may be sufficient

in nllJIJ,ber to show their inconsistency with al.mast any hypothe-

sis suggested. (Ha.mon and Hannan, 1963).

The x2-test is unfortunately not able to separate the questions

of distribution and autocorrelation. As will be shown later,

a clearly significant autocorrelation exists in this case.

Moreover, mostly the assumption of normally distributed resi-

duals is not very ciritical. Every result below, obtained by

relying upon Normal distributions may be considered as an

approximation.

4.4 Estimation of the autocorrelation

To estimate the autocorrelation a method described by Jenkins

and Watts (1969) was used.

At first the autocovariance was estimated for each continuous

period j, when one mechanismv.BS working (see chapter 4.1).

where

,.. R. ( T)

J

R. ( T) J

X(t)

N. J

t. J

=

N.-,+t.-1 1 J J

N.-T J

z (x (t+,) t=t.

J

x) ( x ( t) - x) ,

estimated autocovariance of residuals separated

by T days,

'\,(t) - Qc(t) = residual at time t (m3/s), duration of period j (days),

the time at which period j started (days from

the beginning of the discharge record).

- 20 -

In order to get an overall estimate of the autocovariance the estimates

from the different periods with the same mechanism working were brought

together as follows,

h N. - '"[ R(,) = Z (-N--

all -r the

periods j

R.(-r)) J

where R(-r) = N

overall estimate of the autocovaxiance,

T ~(N. - ,) = the total number of observations of R(-r) (days). j J

Finally the autocorrelation was estimated by dividing the estimates of

the autocovaxiance by the estimate of the variance.

When there is less than about 400 observations, one should not rely too

much upon the autocorrelation cuxves, because of the extremely skew

probability distribution of the autocovariance.

Kendall and Stuart (1967) actua.lly warned for the use of a correlation

coefficient estimate, based on less than 500 observations. However,

here we have got autocorrelation estimates at surrounding time inter-

vals toa, which gives an indication of the roughness of the estimation

procedu:re. Furthermore the coefficients were estimated from a number of

series differing intime •

.Duxing series of days with rapidly changing MSC the discharge pro-

duced by different physical mechanisms strongly interacts. This may

distuxb the estimation of the autocorrelation function. (See fig. 4.3, where the estimated autocorrelation exceeds 1.) To avoid this phenomenon a

minimum period duration of 5 days was defined. It had to be exceeded without any change in the MSC, if the period was to be used in the

autocorrelation estimation proceduxe. (See fig, 4.4.)

- 2·1 ~ X

1· ,:_

O.::JO

0. 70 \

X. ' 0.50 \ ~ " "

\ I.

X \ I J(

\

0.30 \ I

X

:r.

0.10 X \ \~ _ ,L.__J

0 ~ .. 9 \ 0 -o. 10 .; 00 C\JI ~~ X .... I X I

-0.30 I "I

X -o.so X

X X

Fig. 4.3 Estimate of the autocorrelation of the model residu-als at snowmelt in Stadarforsen 61.10.01 - 76.03.31. Variance of the residuals = ·1. 72 • 1 o3 (rn3 / s) 2 • 95 % confidence limits: x > 400 observations to the left of the line: ,

Minimurn period dUJ'.'a tion = 1 day. X

l(

X 0.90

X

0.70 l(

X

o.so X X

)(

X l(

o. 30 )( lx " I )( )( )( )(

l(

0.10 )(

0 o• d 0 -0.10. ~ CD c-J

. )(

-1 )(

I

-0. 30 "I X

" )( X

-0.50

Fig. 4,4 Estimate of the autocorrelacion of the rnodel residu-als at snowmelt in Stadarforsen 61.10~01 3 76.03.31. Variance of the residuals = 2.35 • 107 (m /s) 2• 95 % confidence limits: x I > 400 observations to the left of the line:

Minimurn period duration = 5 days.

- 22 -

4.5 Significance of the autocorrelation

It was considered an important task to determine confidence intervals

around the autocorrelation function and especially to tel1 when it

significantly differe:i from zero.

One method to approach this problem isa significance test developed

by Anderson (1941).

The hY}lothesis that the series isa so called circular series built

upon N purely random observations of a normally distributed stochas-

tic variable, is tested. This method was not used because of the

difficulty of merging the different significance limits from separate

time series, differing in length inta an overall significance limit.

Assurning that the residuals are normally distributed and that the

estimated mean value is correct, we will obtain:

N.-,-1

= 1 I:J [R~(v) + R.(v+-r) R.(v-,)] (1 Nj V= -N.+-r+1 J J J

-r+lvl) N.

A

where R . ( -r ) J

N. J

J

estimate of the autocovariance function of period j,

duration of period j (days).

A

This method is described by Hjorth (1976). Assurning the R.(-r) of J

different periods j with the same mechanism working as independent,

we get

Var [iicTJ - ,t; -TJ 2 "Vax [R.(T)] all -r J

periods j

Confidence intervals were computed, assurning the R(-r):s to have

normal probability distributions. Then the 95 % confidence interval was given in the form:

R(-r) ~ 1. 96 • v Var [R(-r) ]' When plotted in the autocorrelation graphs all confidence limits

A

had to be divided by R(o) in order to get the correct scale. The

prodedure led toa slight paradox, the upper confidence limit

might exceed one. This is of course impossible, but the autocor-

J

- 23 -

c-o,_r.c:;:ciance g.raph. 'i'he conf'idencc L.1ni t s werG s;('.llU1etricall;:t

_placed. on both s ides of the autocorrela tion est:;_c!!att:.

Analysis of the autocorrelat:'..on

::::h.rring the beginning of the statistfoal study the :.utoco1:T8la-

tion coefficients were believed to contain valuable inforn1a-

tion.

The kind of i ntermi ttent processes descri bed here are not

knmm to be fi tted wi th any standard estirnation methods. .An

aim of this work has been to get unb i ased estima tions. Wi th

increasi,ng number of observations the observed quantity

tums more and more normal , and then an unbiased estimation

is preferable.

The three MSC are:

1 • M > 0 ( snowmelt) ,

2, M 0 and (s or 6S ) > 0 C,·-flow), UZ UZ '

3. M 0 and S = LIS = 0 (low flow). UZ UZ

snowmel t (mm),

s UZ

~ storage in the upper zone of the model

6S UZ

""inflow in the lr lT lt lt Il Il

:uv.ri ng snowrnel t we have got a c l earl y s i gnificant autocorrela-

tion (see fig. 4.5 and 4.6). This shows us that a large resi-dual du.ring one day will gi ve rise to large resi dual s du:ring

the following days. If, for instance, bad representativeness

of the temperature measurements causes false sno-wrne lt one day,

the reservoirs of the model are filled up toan improper level.

This affects the r esiduals du.ring the following days.

X

,c

0.90 X

X

0.70 '\_ " X

0.50: X X

X

0.30 X X

,c

X

0.10 X

0 OX

-0.10 . ..: CD X

-0.30

-0.50

X

X

lx

9 ~

X

XI I

X

- 24 -

lC X X X

X X X X

X

0

ci C\I

lC X

lC

o x x•x x x6 T CD N C\I N (I')

0

ID (I')

X X X X X X X X lC X

X 0 X •

0 .,.

X X X X X X X X lC X X X X X

Fig. 4.5 Estimate of the autocorrelation of the residuals of

Stadarforsen (snowmelt) 61.10.01 - 76.03.31.

Variance of the residuals = 2.35 • 103 (m3/s)2

•

95 % confidence limits: x > 400 observations to the left of the line:

X 0.90

X

0.70 X

X

o.so lC X

0,30 X X X X X

X

0,10 X lC

0 X

-0.10 • . 9 . t0 X X ~X X T

X X --: X ,c 400 observations to the left of the line:

X

T

T

-0. 30

-0.50

- 25 -

cr12 fact.s tha:t~.

"

lund of noise that lies on top of "d_e o.::.scharge r:ecord

in Kultsjön, due to the methofi_ (3ergströrri ·1976; used

when estimating inflow.

Fic-. 4.7 Estimate of the autocorrelation of the residuals of Stadarforsen ( Y-flow·) 6 L 10. 01 - 76. 03. 31

Vari ance of the res i dual s = 33c (m3/s,) 2

95 % confidence limits: x > 400 observations to the left of the lLne:

- 26 -

0.90

0.70

0.50 X

X X

X X 0.30 I X X X

X X X X

X X X

0.10 X (days) X X X

0 0 0 0 0 -0.10. N ~ ~ w

• ci T ~ X - - ~ X X

X

-0.30 X X X X X

X -0.50

X

Fig. 4.8 Estirnate of the autocorrelation of the residuals of Kult-sjön (y-flow) 62.10.01 - 76.05.18.

Variance of the residuals = 258 • (m3/s) 2

95 % confidence lirnits: x > 400 observations to the left of the line:,

I Note that due to the small number of observations the maxi-

mum argument above is just 19 days.

When the MSC shows y-flow (fig. 4.7 and 4.8), we have also got a sig-nificant autocorrelation. However, it is less than in the former case.

A reasonable explanation to this is that the reservoirs may still be-

come filled up toan irnproper level but not up to the same high level

as during snowmelt. This makes the residuals, separated by a shorter

tirne period, independent of each other.

"

0.50 "

0,30

o.w

-0.10

-::i. 30

-::i. sa

0.90

0.70

0.50

0.30

0.10

-o. 101· )( -0.30

-o.so

- 27 -

\

" )'. :~ X X

X X X X

:,(

" " " X " X )< " X )C X

)C

0 0 0

----------................ _____ _ X )C X " X )( X " )( X " -1!. • "6" 0 0 0 0

a:i .

st" N (0 . .

ai c-i 0 ..,. N N C\i (I')

Fig. 4.9 Estimate of the autocorrelation of the residuals of Stadarforsen (low flow) 61.1C:.01 - 76,03.31.

Variance of the res i dual s = 19.3 (m3/s) 2 • 95 o/~ confidence limits: x > 400 observati ons everywhere.

)C

>< X X " ( days)

Fig. 4.10 Est imate of the autocorrelati on of the residuals of

Kultsjön (low flow) 62 .10. 01 - 76. 05.18.

Variance of the residuals = 47 .1 • (m3/s) 2

95 % confidence l imits: x > 400 observations everywhere.

- 28 -

At low flow there are great differences between Stadarforsen and

Kultsjön (fig. 4.9 and 4.10).

The noise of the hydrograph of Kultsjön leads to the assumption

that any autocorrelation in Kultsjön during low flow is masked by

the noise on top of the recorded bydrograph.

In Stadarforsen this is not the case. A nice smooth recession curve

gives us large autocorrelation estimations.

If the reservoirs of the model are filled up toan improper level

during or before the winter recession, this will cause a very

persistant series of residuals.

- 29 -

5. A STUilY OF RESPONSE SURFACES OF CRITERIA OF FIT

5- ·1 Cri teria of fi t

Nash and Sutcliffe (1970) defined the R2-criterion of fit as

the efficiency of the model.

where

l: ( ~ ( t) - ~) 2 - It ( ~ ( t) - Qc ( t) >2 2 t

R =------------------l: (~(t) - ~)2 t

~(t) observed discharge at tirne t (m3/s),

Q (t) computed I! I! ,r It H C

~ ::: arithmetic mean of ~(t).

The R2-criterion of fit is widely spred and it hasa relative

character, which makes it attractive when comparing the fit of

different models, different time periods or different catch-

ments.

These comparisons should not be made between hyclrographs that

differ too much, because of the phenomenon illustrated in fig.

5.1 and 5.2, which could be due to one or both of the follow-

ing explanations.

1. Inevitable errors of roughly constant size cause the

R2-criterion to give high values when applied to hyclro-

graphs with high initial variance and vice versa.

2. rr·he calibration of the model and the model i tself

favour fit during periods with high initial variance,

and so the relative fit is bound to be worse during

periods with low initial variance.

0..(1/s) I. 0 0

200

- 30 -

OBSEAVE:D HYDROGflAFti

-4f-- COMPUTEO HYOROGRAPH

R2 =- 0.64

MAY JU 1-1 JUL AUG ' SEP OCT

F:c. 5.1. Lcw P.2-value as a result cf lo,-, initial variance (Stab'c:::r, 19::?;. (From Bergström, 1976,)

Q(L/~) 600

,oo

200

0 BSERVED HYDFWGRAP+-1

COMPUTE O HYDROCiRAPH

a~--..-----~--=~~-~~_..,~-f"-..!t,:,.,,!=.~---.-----1 A:,..,Pc--:R'-'---'---'--'~'---'--....:....;=....:..c..._ ...__....:J:....U;...L"--_._--'-'=--">--'--.-J

- 31 -

Let u2 E,ssume that the R2-c:citeri& of ·ilfferent, pnysical p1:0--

cesses do give a more accurate measure of the goodness oi fit

between the computed and the :recorded l--:i.yc,l',.,g-:::. Then the

problem of merging the three cri teris, im;o one arises. One

possibili ty of doing this i8 simply t0 add u;i "'.;l1s three cri teria,

thus achieving the R -criterion. sum

5.2 Basic assumptions and observations

The MSC gives different classifications of the material at

different parameter settings. This sometimes made the R2-

criteria vary unexpectedly (especially the R~-criterion). If

a couple of days are moved to the low flow class from the

other classes, this will probably make the initial variance

(F2 ) of the low flow class greater, but it might possibly not 0

influence the sum ~ (~(t) - Qc(t)) to the same extent. Thus, the R~-cri terion will grow perhaps wi thout a..n.y visible change

in the hydrograph, a phenomenon similar to the one illustrated

in fig. 5.1 and 5.2.

The HBV-model has always been calibrated by visual inspection

of the computed and recorded hydrographs. This means that a

considerable skill in parameter setting has been obtained

during the years. For example, recently on the first try when

calibrating the HBV-model fora new catchment the R2-value of w

the four year ca.l_ibration period was greater than 0.8.

This implies that a fairly good parä.!Jleter setting could be

obtained by a QUalified guess based on experience from other

applications. It is likely that an automatic parameter optimi-

zation routine would accomplish this too.

This assurnption made the work easier, while only the region

around the optimum parameter setting had to be examined. The

subjectively found optimum (tab. 5.1) was used as the actual one, and the parameters were varied around this central point.

- 32 -

Table 5, i Criginal settings of the free parameters of the HBV-model.

Parameter Kultsjön Stadarforsen

p 1.330 1. 136 corr

T ( 0 c) 0.5 o.o 0

C f' s__,_ 1. 23 0.90

C (mm/(0 c

0 day)) 3.2 2.0

C ·wh 0.05 0.05

sb (mm) o.o 0.0

C rfr 1 • 0 1 • 0

Fe (mm) 150 150

L (mm) 150 150 p

s 3.0 1. 5

K (1/(s • mm)) 0 12500 0

K1 (1/(s • mm)) 4000 3500

K2 (1/(s • mm)) 300 350

Luz (mm) co 15

C (mm/day) 1.3 1 .o perc 1l (days) 2.0 6.o max

C (day . s/m3) 0.00103 0.00000 route

- 33 -

!fote that K in Kultsjön is zero. That version of the :i"TIV-o

model is older than the one used in Stadarforsen. :t is occa-

sionally used when the catchment behaviour is deterrnined to be

sufficiently simple. Because there were no initial values of

K and~ , a variation of these parameters in Kul tsjön was 0 UZ

avoided.

During the original subjective calibration 8-year periods were

used, but due to the limited capacity of tte computer only

4-year periods were used in the study of the response of the

R2-criteria. This limits the value of the comparisons between

the model behaviour of the original parameter setting and the

test settings of this study. The periods studied are in

Stadarforsen 1965-10.01--1969-09-30 and in Kultsjön 1962-10-01

--1966-09-30.

Two, sometimes three, parameters were varied simultaneously,

and the R2-criteria were computed at the different parameter

settings. This resul ted in tables and "three-dimensional 1'

d . f . R2 2 "'- d iagrams o iso- graphs. The R -curves were only u..i.awn aroun

the optimurn point.

If there was a substantial difference between the R2-values at

the optimum and at the original parameter setting,

the cooputed hydrograph corresponding to the

optimum values of the ~arameters was plotted. These hydxo-

graphs were later judged by the model calibraters in order to

estimate the goodness of fit at the new parameter settings.

-·· 34 -

K2 100 200 300 400 500 600 700 800 900 1000

C perc

')

12

15

18

21

24

27

33

\ . \

. 727 ~J_JO , 729 . 727 . 725 , 723',----_ 721 ~ .

,727 732 .73~30 .728 ,726 .724

\ . 725 \. 731 . 733 . 733 . 10,~o . 728 • 726 \. 722 \ 730 . 734 . 734 . 733 . 73~"'--_:z30 . 728

, \ \ , 7 3 5 · 7 3 4 . 7 3 3 · 7 ~ 7 30

K2 100 200 300 400 500 600 700 800 900 1000

C perc

9

12

15

18

21

24

27

33

.524 .524 .524 .520 . 519

Fig. 5,3 The response of the R2-criteria to K2 (1/(s • mm)) and

C (10-1 mm/day) at Kultsj ön. (See chapter 5,3; 5.3.1) perc

- )) -

H2 3

K2 100 200 300 ~00 500 600 '700 800 900 1000

C perc

9 .145 .233 .254 .248 .232 . 12 .161 .274 . 306 .306 ,293 . 15 .151 . 286 .328 .332 .323 . 18 .141 .292 .344 . 35_1 . 21 .113 .285 • 47 ~.360-:=.=;-. -;:,•

--- --- --- / .316 24 .079 .265 .33 .354 .3Sö .354 ,347 .337 .327 27 .067 .260 350 .341

30 .024 .229 .307 .353 .341 .334

33 .009 .220 .300 .334 . . 358 · . 347 36 .336 . 341 • 341 .339 .335

R2 w

K2 100 200 300 400 500 600 700 800 900 1000

C perc

9 • 777 .783 .783 .782 .700 . 12 .780 .788 .789 .788 . 786 . 15 .781 .792 ,792 790 . 18 .780 .791 .795 .795 . 793 ">

.796 .796 ~

21 • 777 .791 .795 . . ~ . 24 • 772 .']89 .795 .797 .796 .795 .793

-~~787 27 .766 .7 .794 .796 .797 .796 .794 .793 .792 790

30 .759 .781 791 .795 .796 .796 .795 .793 .792 .790

33 .751 • 776 .78 .792 ,794 .795 .795 ,794 .792 .791

36 ~ ,794 ,794 ,793 ,792 .791

Fig. 5.3

::-:,f K1_,1_j_ ts 7ör.!.

the j_r_j_ tig_l valucs :)f the ,1,u•;ir_,_ei:,,~::-•s spe tab. 5. 1.

The response of tica ii'~ -cri teria to K0

and C - - - - - - - - - - - = - - _p~r~

')

C::hc va:r·ia";ic:i of e,',ch p_-'- -c.~:-i tericn was c.omputed when al tering K2

and

C , Since t~'lese •ia.1:·ameters kLve thcir grcatest influence on ::..ow per8 ~ -

flov, the R':-criterion was considercd most important. ln fig. 5.3 the . -s ? /

n;-criterion lndicates )

tha-t C and K2

should be ir.creased. The r, perc

i:rrei;ular i,h2.:;ie of ~}'e R~-rc-sponse surfacc is duc to thc variable ') -

cl;,,ssificai;ion ·'Jy the MSC (chapter 5.2).

An increase in C will have ~he effect of increasing the overall perc

flow during low· flow. It will 2.lso cause a shorter duration of the

pea.1.::: flows. l:m increase : . .L i::? ,.,ill accelerate the low flow recession.

A test :;ilr::nting was made at;

K - ''·rJ·n 1;1s • 2 - ,_,, ,..,,

C perc.

w.:--1:'..ch is the o:;itir:rwn point du.ring low flow.

Tne ne,., hydrograph was considerecl to be somewha t ·c,etter than the

old one.

? The response of the R--criteria to K

1, :B and C t

max rou e

The R;-criterion in fig. 5-4 ob-rjously points towards a decrease in K. a...nd C t • A decrease in K

1 ·,1ill ::na.ke the hydrograph morc d.amped. , rou e

and will also cause a consiuerable number oi' days to movc from the

low flow ciass to the Y-flow class. The latter is the reason ?

why we get such odd. informati on :..':::-om the B.3-cri terion here.

C . 1 too, affects the variancc cf t°'J-? hyrirograpL A l ow C t roui;e rou e value will damp the discharge peaks, ~rh:~:::..e a :'ligi value .-,ill make

the hydrograph v~ i n a 111orc rap:, c. way.

7

2

·)

5.Li. 'l'he response of tl1e RL-criteria to K 1 (l;(s · :mn)),

r\ (days) and C (clav · s;m3 ) at Kultsjön. max · route C

(See chapter 5,3; 5,3.2)

--'---, I -

,-,

'-'route 0 • 00 1 day / ( ,n J / s ) ,

f: na.x 2 days.

Eere the rrwde L demonstrates an unability to move rapi dly between

k"ign a..Dd medium flows by consistently underesti.mating high flows.

This underestimation was ccmsidered serious and ca.used the ealibra-

02:cs to ,1uö.ge the plotti.:.'l.g to be not as good as the original one.

This Ja,ck ·:::i-f a K parameter is likely to be the cause of this 0

d.arrrped b e2haviour of the 11'.odel, since one further storage di se har ge

para.meter w(mld increase t;Je s l ope of the recessi on.

The response of the R2-cri teria to Fe, L /Fe and B - - - - - - - - - - - - - - - - - E - - - - -

The parameters varied i.r. fig. 5.5 affect the evaporation. They also affect the lcvel of the flow peaks succeedi21g dry periods.

·')

TLe c:ptirrur:, of the RL -cirterion seems to be: w

Fe 200 mm,

L /Fe p

O. 6 - ·-L p

2 or 4.

120 l!llll,

s:'wo test plottings are made, one at S = 2 and one at r3 = 4. They do not differ :nuch fron, each other, but they differ from the original

plotting. The decrease in the L /Fe ratio causes an increase in the p

Eivaporation. This l e ads to an 1mder e stimation of the high flow

peaks of sUlillller. Thus, the test plott ings were i nferi or to the

origi.nal one. The medium and. low summer flows are perhaps a bit

better. on the test plotti.r..gs than on the original one. Agai n the

lack of a third storage diseharge parameter (K) i s obvious. 0

:Pig. ").') (See next page). ,·,

The reSp:)nse of the R"-criteria to Fe, L /Fe and 13 at Kultsjön. p

1.0

0.6 .,/L---7"-'----,'1,---,.,C--'-~ s ..._ ..._s ,,_, ,,_,s

R~ R. 2 _L., -) __ 1,r:::::.------;.-----,.,.:::----,oi

Fig. 5.5 (See chapter 5.3; 5.3.3)

.. 40 -

Tl. ~ th -, 2 . t . ' C , ne response 01 e h -cri eria i,;o · f ano. 13 s_

p and C sf l)oth affect the los ses between precipi tation and runo::'.'f.

The differences ·oetween the c,ptima of these response surfaces and

the R2-values of the original parameter setting was small and no

test plotting was made.

n V

si'

1.0

: • 1

1 • 2

1 • 3

1 • 4

1.0

1 • 1

1 • 2

1.3

1.4

2 P.1

, I • 2. 4, 8.

.699 .690 .671 . 651

~ ~m:-~ri.s .700 . ~ ...... ' •, "- .. L 'I3]__./[ ~?___:>])1 3 =- ·-·-·-----~----,686

.c-01

.396

.334

.680

.603

2.

•)

n"' 2

.678 .675

.608 .610

4.

,466

·16.

.637

.689

,706

,672

.609

-359

. 377

,375

,·,

C si'

'1 • 0

1.1

1 • 2

1.3

1.4

1. 0

1.1

1. 2

1.3

? n3 1 • 2 .. 4, 8, 16.

.317\ .323 .319 . 315,, .306 \,_:/ //

.313 .319 .315 (:309 ,305

~8 .315 \ ··, . 312/ 310 .305 ·-., --- - ____ ,

.303 .301 .3os .304 .302 .. -----"-..

:·299 .20 .305 .301,.,.--:-299

,., c:.. 4. 8. 16.

,1§5-- .774-.. __ !766 .748 ,728 .,,.. ----.... ,,,.-;: .. ----~---------- "'·• .. __ • 783 /0',, Li-·'" 790., ':1:J6 • 758

I • ..._ ""-.. l ·, '-, '-

:Jtl '\.____!]})_ ~ __}23 ) • 768 ·-, . --~--~----~ ------ _ . ..,....,., .-·

• 753 -··:'i-C~---++.:+---:763 . 750

,708 .722 ,730 .724 ,712

5,6 The response of the R~-criteria to r and S at Kultsjön, ....,sf

- 41 -R2

1

C 0 1.4 1. 7 2.0 2.3 2.6 2.9 3.2 3.5 3.6

T 0

-1.0 ,670 ,714 ,672 .59e

-0.5 ,612 .709~ .612 ,539 .466 .399

o.o ,468 .610 • 70 , • 716 ,668 ,618 ,550

0.5 • 221 .425 ,532 ,633 .689 ,706 , 717 .697 .665

1.0 ,473 .546 ,582 ,608 .620

1,5 .211 .251 ,307 ,357 ,395

R2 2

" "o 1. 4 1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8

To

-1.0 .513 ,482 ,55~67 :_. / . -0.5 .449 ,478 -492 .551 • 556 .555 .515 -492 .446

o.o .292 .419 .446 .503 .549 .517

0,5 -570 -318 .450 ,447 -492 .540 ~585

1.0 .433 ,452 ,499 .511 .553

1.5 .344 ,398 ,408 .467 ,500

R2 3

C 1.4 1.7 2.0 2.3 2.6 2.9 3.2 3.5 3,8 0 T

0

-1.0 ,277 ,299

-0,5 .273 .285 .336 .337 ~314

o.o .263 .278 ,337 .338

0.5 .243 .265 .274 .281 .323

1.0 ,286 .292 22

1.5 .274 .282 .285 .288 -295

R2 W

C 0 1.4 1. 7 2.0 2.3 2.6 2.9 3.2 3,5 3,8

T 0

-1.0 , 745 .765 ,746 .704 . -0.5 • 706 ,571

0.0 .610 .697

0.5 .471 . 766

1 .o .656 ,697 ,724 • 741 • 753

1,5 -498 ,540 , 575 .609 .636

Fig. 5.7 The response of the R2-criteria to T and C 0 (mm/( 0 c • day)) at Kultsjön, (See chaiter 5.3; 5.3.5)

- 42 -

mh f th R2 't . t T ~ e response o e -cri eria -o 0

and C 0

T affects the starts of the mel t periods, "11hile C affects the 0 0

overall melt ratio (fig. 5.7;.

A test plotting was made at:

T 0.0 oc 0 '

C ') , - I (0 . day)., ~.o mm/ C 0

The test plotting showed out to be inferior to the original plot-

ting. There seems to be nc way to :nake the model fit the recorded

hydrograph on both peak flow and medium flow. One further degree

of freedorn is needeä..

The response of the R2-criteria to C ~ and C f

W11 r r

The paramete:rs Cwh and Crfr influence the behaviour of the rnodel,

when a rnelt period is restarted after a short interruption by toa

low ternperatures. The use of the Crfr parameter started during

this study of the HBV-model, and the value of having such a para-

meter was questioned.

As the response surfaces (fi6', 5 ., f'.) show, C f does not affect the r r R

2-criteria very rnuch. so the C f. parameter seems to be of no use

· r r here.

The optirnum value af C ~ is obviously 0.05. No test plotting was Wll

rnade, since the optimum values of Cwh and_ C f do not deviate from r_r the original ones.

C,00

0.10

o. 15

0.20

crfr

Cwh

0,00

0,05

0.10

0.15

0.20

,701

,709

,714

.709

,700

o.oo

.566

.566

.564

.562

-556

,701 ,70~ • 70·: . 701

,704 . 7ce("-=~.l;r-~ 7 ~ J .7~7 -----:tCTt---;-t"tOr--:fi~~~- 677

.665 ,GGo .GG1 .644

.635 .6:;o .62.e .604

R2 2

0.05 0.10 0.20 0,40 1, 00

.566 .566 • 5(;(; . 66

.562 59 ,556 .556 .551

)61 ,557 750 ~-54/43

,547 .543 .54•J ,534

0rfr o.oo 0,05 0,10 0,20 0.40 1.00

Cwh

0.00 .326 ,;,26 .326 .326 .326 •

0.05 .328 .;, ,6 .31 / .~314 .313 ----0.10 .323 ~~-295 0.15 .320 ~286

0.20 .313 .287 .284 .283 .281

0rrr o.oo 0.05 0.10 0.20 0.40 1.00

r ~wh

0.00

0.05

0.10

0.15

0.20

,770

• T(8

,784

,785

• 783

.770 .770 .770 . 770

:-r~. 7 ·'3 5 .. 7P.7 ,785 .790

~ 7~__JSC-. -~ 778

• T,'3 . 77'1_>~. 765

.761 .760 .759 .74e

1 r .. \ ... , 50'.)

1. 0 . 6'./ ) .895 , 8;•6 .8?5 .89'.J

1.5 ' . STJ

- 45 -

5.4 The response surfaces of criteria 01 fit and test plot-tings of Stadarforsen

In Stadarforsen runoff data of better q_uality tha.n in Kultsjön

were available. For the interpretation of the model parameters

see chapter 3 and for the initial values of the para.meters see

tab, 5. 1 •

5. 4. 1 The response of the R2-criteria to K

2 and C

- - - - - - - - - - - _p~r~

There is no difference between the optimum parameter setting

here (fig. 5,9) and the original one, and so no test plotting

was made.

There has been difficulties in finding the optimum setting of

K2 and C • Eut note that there isa clearly observab~e opti-perc mum in the dry sum.mer and winter recession class here.

5.4.2 The response of the R2-criteria to K1

, C and B route max

The original parameter setting is within the limits of accept-

a.nce in fig. 5.10. The R2-criterion does not vary much when w

B max

is varied in a region around the original parameter set-

ting. To study this apparent independence a test plotting was

made at:

K. 3 500 1/s • mm, I

day/(m3/s), C route 0

B 5 days, max

where there isa small improvernent of the R2-criterion but the w

new plotting had approximately the same fit as the original one,

as judged b;y the calibrators.

6

6

5

Fig. 5,10

8 max

i J >c ro ute

~,-,J.~~~~-~---+ TSOO

fa-,,,L,L....,,4.:;ff""'-,.L'----"?'------,~ 2500

,~,

7

6

5

;rhe response

(day • s/m3)

5.4; 5.4.2)

of the Hc::-criteria to K1

(1/(s · mm)), C t rou e

and J (days) at Stadarforsen. (See chapter max

.i:>'ig. 5. 11

K0

7~00 10000 12'./JO 1

I. ll.Z

5 ,796 ,850 .870 .0'/5 ,670

10

15

20

25

30

,794

,790

,783

• 774

,851

,850

,846

,838

R2 2

K0

7500 10000 12500 15000 17500 20000 22500 25000

L UZ

5 ,794 ,752 .692 .628 .567

10 ,819 ,809 • 782 • 753 , 721

15 ,8

20

25

30

R2 3

K0

7500 10000 12500 15000 17500 20000 22500 25000

L UZ

5

10

15

20

25

30

~6 ,5 7

5 1\ .543 ~ ,496 .504

,483 .492

.508 ,487

r:,-

550 ,549

.505 .505

,496 .496

,453

.524

,549

. 512 .51.1 ,516 .519

.496 .504 .504 ,504

,491 ,492 ,492

K0

7500 10000 12500 15000 17500 20000 22500 25000

L UZ

5 .sG7 ,895 .902 ,900 .894

10 • 867

15 .865

20 • 861

25 • 855

30

')

(1/ ( s u.m)) 'I'he response of the -::;~ -cri teria to K . 0 , \

and T (mmJ at Stadarforse11. (See chapter 5. 4; 5.4. J_j 77'7

- 48 -

5.4.3 Th f th R2 ·t . t K e response o e -cri eria o 0

andL UZ

K affects the top flow recession rate and L the level, at which 0 UZ

this recession rate is activated. A great improvement of the R2-

criteria can be seen here. It is only the R~-criterion that is not

improved when increasing K and L • (Fig. 5.11). 0 UZ

A test plotting was made at:

K 22 500 1/s • IllIIl, 0

L 25 IllIIl. UZ

The plotting corresponds to the information available through the

R2-criteria. The result isa better overall fit, eHpecially at

high flows but perhaps there is same deterioration at low flow.

5.4.4 The response of the R2-criteria to K0

and K1

In fig. 5.12 we have got an example of

practically governs the R2-criterion. w

the optimum of the R2-criterion. w

K 16 500 1/s • mm, 0

K1 3 500 1/s • mm.

2 how the snowmelt (R1

) class

A test plotting was made at

It showed out to be somewhat inferior to the original plotting. The

difference, however, was considered to be of minor importance.

Fig. 5.12 The response of the R2-criteria to K0

(1/(s • mm)) and K1 (1/(s • mm)) at Stadarforsen. (See next page)

~ ig. 5. 12

, -..,_,

K 0

8500 10500

Kl

1500 .820 .360

2500 .819 .858

3500 .819 .857

4500 .820 .B5T

5500 .821 .857

2 R1

12;,00

8Sl

879

79

78

78

R2 2

,i,1, 5C" 2-6: ·~ ~ J C 1..- .. ,! ~ _',.)

1· = '.il . C94 I I

\::i:,J \

.69:+ I ,.

I I

·~ . .393 ") . C,/ .. · ,:7:_ .893 .S9:; 89c, • c, c',

.SS8\ .892 . .

K0

8500 10500 12500 14500 16500 18500 20500

1500

2500

. 765 . 744

3500 .835 .836

4500 .824 .827

5500 .810 .816

K 8500 10500 0

Kl

1500 .380 .414

2500 .452 .466

3500 .528 .543 -----

R2 3

12500 14500

.440 .458

.475 .479

,550 .549

4500 ~560

5500 5 , I • 542

.794

16500

.472

.478

.549

-557

.539

.800 . 789

.803 . 794

18500 20500

,550 551

.555 .552

K 8500 10500 12500 14500 16500 18500 20500 0

K 1

1500

2500

3500

l:500

5500

.8tl2

.823

.6iL

. f.6~

.882

901 .90'

.90,1 I

. 90~ \

.903

.916 920 .921

.'?l'T 92':' r'•)":I ,::,'.___)

-917 ,q22 .921, .923 .921

,9lb . 02~~ • 9:__:~ .923 , 92]

.914 ()2,J .922

Fig. 5.13

- 50 -

The response of tne R2 -eri teria to Fe (mn ), L /Fe p

and 8 at Stadarforsen. (See enapter 5-4; 5.4.5)

5. 14

o.e5

0,90

0,95

1.00

C

csf

0.80

0.85

0.90

0,95

1.00

~

csf

0,80

0,85

0.90

0.95

1.00

.600 ,796 ,799 ,818 ,833

rl 2

0.5 1.0 2.0 4,0 5.0

.284 .11, 1i~~ \ .no .62,1 .262 .707 5 \ ,791 .696

·'"I/ '\; \\ace .235 . 713· .205 .684 d8~6 \a20 .. 729 , 151 .657,1 .s50\ I Je20 . 732:

2 R3

o.s 1.0 2.0 4.0 8.0

.518 .519 .525 ,440 .254

-5~ ~}~1 .473 .310

.s, \ ·rr 1 ·~t .503 ,355 ~o ll:,Pd4 I ·:p .. 522 ,397 • ., • •◄ 1 · ) . "

~I I ' I I ,498 \l,574, .::,5å ,534 ,430

-~"-,e respomw of the ~,/ -cri teria t o C &id s~ at Stadar-

- 52 -

5. 4. 5 The ~e~p~n~e _ o: ~h~ ~2 =c~i ~e~i~ ~o _ F~, _ \/~c _ ~ci _1~ _ 'I'he danger of letting the R2-criterion define the parameLer optimum is

1,,r

accentuated in fig. 5.13. A test plotting was made at: \ Fe= 150 mm,

~p a :5° ,m J This is the n.2-o::_JtiI;i.wi-1. The model now underestimates y-flow. From the

w response surface it is clear that the R~-criterion was deteriorated by

this modification of the parameters. This gives an unacceptable error

in the accwnulated flow after the four calibration years.

The R -criterion SUlll

(chapter 5.1), however, has i ts optimUlll at:

Fe 150 mm,

L = 150 mm, p s "' 2,

which is very close to the original setting.

5 4 6 Th f th R2 ·t . t C d • • e response o e -cri eri a o sf an 6

The D -criterion (chapter 5.1) shows optimu.m (comp.are fig. 5.14) at: sum

S 2.0,

csf 0.9.

No test plotting was made, because the original parameter val ues

are close to this optimum~ and further more the test plotting of

chapter 5.4.5 showed no improvement of the nydrograph.

The response of the R2 -cri teria to '=' and C 0 0

According to fig. 5,15 C seemed to be a bit too smal l , and a 0

test plotting was made at:

T 0

C 0

0 °c '

2,3 mm/( 0 c · day).

It shows a small improveruent.

5. 15

o.c

0.5

T 0

-1.0

-0.5

C

o.o

0.5

1,0

" T

0

-1.0

0.0

- 1

,724 .645 ,590 ,533 .~HO

,733 .733 .697 .534 .6C2

,657 . 721 ~~~:oo .597 ,639 ~'2:,541

,502 .55n ,6C'i ,656 .G35 ,720

1/ l3

1.4 1. 7 2 .0 2.3 ?.6 2.9

.6;:4 ,590 •. 1,:36 .403 . 357

;tp~? .553 ,491

~i.5:S 640 ,6;7 ,560 ~

.569 . 520 ,)05 .G15 .616 .605

.665 .Gcc

4725 . 72~,

~')-~ • j L.J .7'2t

3.2 3.5

- 54 -

Crfr 0.00 0,C5 0.10 O.?O 0,40 C.80

Cwh

o.oo

0.05

0.10

0, 15.

,9C:,

.922

0.20 .912 .874 .8~7 ,826 ,778

R2 2

crfr 0,00 0.05 0.10 0.20 0,40 0.80

cwh

o.oo ,740 ,740 ,740 .740 .740 . 0.05 • 756 -756 .755

0.10 , 773 .746 • 726

0.15 .780 ,733

0.20 • 777 .710 .

crfr o,oo 0.05 0.10 0.20 0.40 o.so Cwh

o.oo

0.05

0.10

0.15 .644

.663 .663 .663 .663 • ~ .652 .650 .656 .657 .659 • .651 1 • .625

, 581

0.20 .643 .622 .612 ,585 .532

crfr o.oo 0.05 0.10 0.20 0.40 o.eo

Cwh

0. 00 • 919 t-9. ........ ~'-"-"---!~

0.05

o. 10

0.15

0.20

,929

,935

.935

,929

:tig. 5.1s ~ 2 ·t . t C l'Le response ur tl1e ..:"!'. -cri eria o and C ~ w.: rtr

at Stadarforsen. (See c:'!1apter 5,4; 5,4,8\_

'_'Le s-"ttings cf C::rfr and

,~ettinc then seeEted to be (:'ig. 5- -1 C ·_

C ·rLc C wh

().05,

0.05.

,?J .. _ •:,-,- ...... .:. -· ..L

.A plotting was made a.t th:;_s -::i:: int. ::::'he tes-:: -)lotting manageci

to center the spring floods better t:C-1a.r:,, the c,rigir1"":'._ ::_:ilo:ting

and was t!lerefore considered to be i)ette:,r than the original

one,

5.5 Resu'ts of the study ')

It is obvious that the RL-critcrion is nota good criterion of w

fit for our purposes. The sno'WI!lelt peri od g0Yern2 tie behavi-

our of the R2-cri terion too much, }'or cxample, the ;::-parameter

w of Stadarforsen (and to some extent the one oi:' Kultsjöri too) ., is badly optimi·zec1 by the R"--cri ter-; on.

1,:l

-~

J\Ilother disadvantage of the studied c:ri teria is -:;:1e H...::: -- 3 cri teriot:'.. It ~-s cisturbed by t!lc fact that ch.fferc:--it parame-

ter set tings give r1; ,,. _;rent c l ass i fications of data. J3oth the

initial variance and the sum of the squareo_ residuals may l;e

greatl y changed by this phenoEienon. This -:ou:1 lead to a

ch.ange in R~ • not from a change : : :::. -: , but- "'· __ J.J a rearrange-,, m-2nt of the d2,ta 2.vaila'::Jl e.

In order to develci, aI1 acceptabJc cri tcrion of fi t t r_e s1un o:f

,de H~, :.he R; E,nd the R~-cri teria was also studied. The weak-ness of R; de;cri-Ded above, .:-1owever. c.oes in::::_uence tl:is

/

criterion too.

To overcome tl:is :irawback i t is suggested tL2.t tl,c c2-assifica-

tl.on of d.ata shoul ci not -be chan_ged ,iuring cocparisor_1_ of diffe-

rent parameter settings.

Parameter

~

Kc2 (1/(s • mm)) (rnm/day)

perc

{

K1

(1/(s • mm))

C t ( days/m3) rou e 3 (days)

max

t1·0 (1/(s • mm)) (mm) UZ ~~

(1/(s • mm))

(1/(s • mm))

te (mm)

L /Fe p s

tf f 'l.' (oc) ~: (m.m/( 0 c • day))

f~wh trfr

Original setting

350

1. 0

3500

0.0000

6

12500

15

12500

3500

150

1 • 0

1.5

1.5

C).90

0.0

2.0

0.05

1.0

ST.AJ)ARFORSEN

2 t· R1-op lJilum R~-optimum

300

1.0

1500

0.0000

5

25000

30

16500

3500

150

1.0

8.0

8.0

0.85

2.3

0.05

0.05

500

1.5

3500

0.0005

7

25000

25

8500

2500

150

1 • 0

2.0

2.0

0.95

0.5

2.6

0.10

0.05

R~-optimum

350

1 • 0

3500

0.0010

5

7500

5

8500

4500

150

1 .o 1 .o

1.0

0.85

- 0.5

1. 7

0.05

1 .o

R -optimum SUill

350

1.0

3500

0.0000

5

22500

25

14500

4500

150

1. 0

2.0

2.0

0.90

o.o

0. 1 ll

n.05

R2-optimwn

w

350

1 .o

3500

0.0000

5

22500

25

16500

3500

150

1.0

4,0

0,/J5

0. Cl

Table 5. 2 'l'he optimum parameter settings of Stadarforsen as judged by the di fforent cri kria of fi t.

'For comparison the original parameter setting ( the optimu.m of an 8 year period as junged 1,y tr1 e

calibraters) is also printed in this table.

c--l.f'\

KULTSJÖN

Original R2 opti- R2 opti- R2 opti- R opti- R2 opti-1 2 3 sum w F,:,earn,,ter setting mwn mwn rnum ffilUil mum " ----··-~··-~-· "'"--~-- --~~ ~~--

I ( ; .· (" 111ni)) 300 rioo 2110 600 450 450 2

>)

' 1 wr,_ (n1m/day) 1. 3 3,G 2 •. 2.7 2.4 2.6

k_ I (1/(s . mm)) 4000 3000 3000 6000 3000 3000

(; (,bJ s/m3) u.00105 o.onos O.ll01 0 0,001 IJ,0005 i r,Ju te ', B (days) ,; 2 2 3 2 2 c'.. \_ msx

.. jFc ( rnm) 150 200 200 100 200 200 ! .

~/Fe 1 .o o.6 0_6 o.6 o.6 o.6 3.0 1 • 0 4.u 2.0 4.0 2.0

,· (B 3 .o 1.0 4.0 2.0 4.0 2.0 \c tsr 1. 23 1.2 1 . 1 1.0 1 • 1 1 • 1 r

(oc) ~:o

0.5 ·- 0.5 0.5 o.o o.o o.o

(rnm/( 0 c . day)) 3.2 2.0 3.5 3.8 2.9 2.6 \ 0

i C wh 0.05 0,05 o.u5

I_I, ()7 0.05 0.05 •;

\C rfr 1. Cl 1.U 0. 0:i 1.-.,.05 (1. 4 1 .o

Tat,12 5, 3 The optimum parameter settings of Kultsjön as judged liy the different cri teria of fi t.

For comparison the original parameter setting (the optimum of an 8 year period as judged

by the calibrators) is also printed in this table.

- 58 -

The optimu.m parameter settings, as described by the dif:erent R2-

cri teria and the R -cri terion, .ruay be studied ir ta0. 5. 2 a:1d sum

5.3, where they are compared with results fro□ calibrations '-'Y

visual inspection. Note that the RBV-model is usuaEy c8.librs:tecl

during 8-year periods, but :iue to the computer capaci ty ,:,nly

4-year periods were used in the study of the response surfaces c-f

the R2-criteria. We see that the R -criterion is ;rrcstlv closer sum '

to the original setting than -the R2-criterion (tab. 5.4). w

In Kultsjön, however, any studied parameter optimu.r:1 sugges tec. ::i.y

any of the criteria did not give better hydrographs than the original-

ly pJ_.:)tted one. Possible explanations to this are the bad qual.:.. t:,,-

of both the runoff data and the climate data and the inco□pleteness

of the model type used (the lack of a K -parameter). 0

In Stadarforsen the R -criterion agrees with the visual inspection sum

surprisingly well. If same sort c: fixed classification manages to

make the R~-criterion more reliatle, it is obvicus that the HBV-

rnodel could be automatically ca.:.ibratec in Stadarforsen,

Whether the model can be autornatica:::_ ly calibrated for any catchment,

isa much harder question. This study covers only two catchments and

furtherrnore only four years of each one.

The impression of the author is that the R -criterion or same .-, sum

other linear combination of the R'--cri teria might be the basis oi a

useful crit"rion of fi t. Other q_uantities, such as the SWD c-E" the

residuals and the ratio of over- or underestimations of the hydxo-

graph, may perhaps take part in a criterion oi fit too, since

these two quantities are often used during the subjecti ve caliLra-

tion.

K u

L UZ

K 0

i, /Fe -.:,'

C . ., s.r

T 0

(; C

,~ . , .. ,h

C rf~

·:.1J.L..:

X X

X

· . .x.

X:

X

X y

X X

6. CONCLUSIONS

The residuals of the H:BV-model are nei ther independent nor s·uationa-

ry distributed dUTing the :Jear. Yet this has been assumed by 1;1any

calibrators of hydrological models. A way to get closer to these

assu.mptions is to base a classification of the calibration data on

the different processes governing the c.ischarge and to consider

each class to be a separate set of residuals. By doing this we do

not get rid of the autocorrelation, but the residuals become more

stationary distributed. In fact it is impossible to get rid of the

autocorrelation of the model residuals, since one single climate

measurement er.ror affects the level of the model storage and thereby

the discharge during a series of days.

Knowing that a classification helps irl making the residuals rnore

stationary, the R~-criterion of fit (chapter 5.1) was computed for l

each class. The sum of these R~-criteria (the R -criterion) showed i sum 2

out to be a better criterion of fit than the formerly used R -w

criterion. Concerning the possibility of automatic calibration of

the .d13V-model at any catchment, data of good Qual ity must be demanded.

There must also be a sufficient number of degrees of freedom in the

model. It might otherwise compensate an unability to fo::i_J_ow the

observed hydrograph by producing a too da.mped hydrograph, overesti-

mating low flows and underestimating high ones.

If these demands are full : j lled, the R -cri teri;'):'.1 gives a better SUID

representation of the goouness of fit tha.rc the R2-criterio~. The w

response surfaces had mostly a regular elliptic shape. Therefore it

seems plausible that an opti.rnization algorit:bm such as Rosebrock 1 s

(1960) method or Powell's (1964) method with slight :nodifications may perform acceptably.

The demand above of many degrees of freedom contains a dil emma. l f a

model is equipped with a sufficient number of degrees of freedom, it

will be able to reconstruct almost any di scharge record from any

climate record. ]ut the headpoint in making a model is not to make

it comrlex in ord.er to f:_t tJ: .. e calitratior. pericö c::1ly, (si:1ce

this is ~10 guarantee for fi t dur.i.ng :;,ther perioc.s), but to make

it si:1:1;Je and yet ~it 80th the ,::::al i.bra t: ion r>eriod a.ni tl:'.e inde-

pendent ueriod.s.

APP:s'lmIX A

The distributions of the residuals (differences between the

computed and the recorded hydrogTaphs) are shown below. Histo-

gTams showing the residuals both separated and not separated

by the MSC (chapter 4.1) are plotted.

A description of the method used when constructing these

histograms can be föund in chapter 4.2. The prescribed minimum

period duration is explained in chapter 4.4.

(days)

600

Fig. A. 1 Histogram s:,owing the residuals of Stadarforsen

61.10.01 - 76.03.31. Prescribed minimum period dl.lration: 1 day.

I'-'Iean = -589 1/ s.

Standard deviation= 22 503 1/s. Number of residu.als = 5 273. Number of exceeding residuals (NER) 337.

A 2

120

80

C\I I

(days)

0 0

0 ...: 0 . C\I

(standard deviations)

Fig. A.2 Histogram showing the snowmelt residuals of Stadarforsen,

61.10.01 - 76.03.31.

60

40

C'I I

Prescribed minimum period duxation: 1 day.

Mean = 1 385 1/s. Standard deviation= 41 431 1/s. Number of residuals = 1 156.

NER= 76.

(days)

0 - ci I 0 0 N (standard deviations)

Fig. A.3 Histogram showing the snowmelt residuals of Stadarforsen,

61.10.01 - 76.03.31. Prescribed minimum period duxation: 5 days.

Mean = -4 645 1/s. Standard deviation= 48 485 1/s.

Number of residuals ~ 794.

NER= 49.

300

200

120

80

A 3

( days)

(standard 0

0 o deviations) N

Fig. A.4 Histogram showing the y-flow residuals of Stadar-

forsen 61.10.01 - 76.03.31.

(days)

Prescribed minimum period duration: 1 d.ay

Mea.n = 16 1/s.

Standard deviation= 17 553 1/s. Number of residua.ls = 2 008.

NER= 116.

0 . C

(standard deviations)

Fig. A. 5 Histogram showing the y -flow residua.ls of Stadar-forsen 61.10.01 - 76.03.31. Prescribed minimum period duration: 5 days.

Mean = -590 1/s. Standard deviation= 18 185 1/s.

Number of residua.ls = 1 650.

NER= 94.

A 4

(days)

120

BO

40

0 0 0 0 _. 0

. .... c-.i I

Fig. A,6 Histogram showing the low flow residuals of Stadar-

forsen 61.10.01 - 76.03.31.

120

80

(Il I

Prescribed minimum period duration: 1 day,

Mean = -743 1/s. Standard deviation= 5 629 1/s.

Number of residuals = 2 097,

NER= 102.

(days)

..... I

0

c:i C)

.,...; C)

(Il

Fig, A.7 Histogram showing the low flow residuals of Stadar-

m forsen 61.10.01 - 76.03.31, Prescribed minimum period dUJ'..'ation: 5 days. Mean = 1 188 1/s.

Standard deviation= 4 393 1/s.

Nu.mber of residuals = 1 780.

NER = 114.

(standard deviations)

(standard deviations)

A 5

( days)

120

80

+o

~:::J......Jl...l......L..1-1.....L...J.....L__LL...L.J-Ll...l...LL..L..LJLL.Ll-Ll..l..LL.l....LJ_J_..t::1....Cn:::J (standard ~ deviations)

C\l I

600

+oo

200

C\I I

0 -I 0 ci 0 N Fig. A.8 Histogram showing the low flow residuals of Stadar-

forsen 61.10.01 - 76.03.31.

(days)

0

.... I

Prescribed minimum period duration: 20 days

Mean = -1 578 1/s. Standard deviation= 3 890 1/s.

Number of residuals = 1 260.

NER= 69.

0 0

ci

(standard deviations)

0

csi

Fig. A.9 Histogram showing the residuals of Kultsjön

62.10.01 - 76.05.18. P1:'23cribed minimum period duration: 1 day.

Mea.n = 374 1/s. Standard deviation= 16 905 1/s.

Number of residuals = 4 977,

NER= 274,

A b

( days)

120

80

Fig, A.10 Histogram showing the sno-wrnelt residuals of Kultsjön

62.10.01 - 76.05.18.

60

+o

20

c-., I

Prescribed minimum period duration: 1 day

Mean = -545 1/s. Standard deviation= 30 198 1/s,

Number of residuals = 1 185,

NER =74,

(days)

-I 0 ö 0 0 N (standard deviations)

Fig. A.11 Histogram showing the snowmelt residuals of Kultsjön

62.10.01 - 76,05.1 8. Prescribed mmimum period duration: 5 days.

Mean = -2 400 1/s.

Standard deviation = 33 605 1/s.

Number of residuals = 884,

NER = 50.

..

60

40

20

iO

20

C\I I

(days)

.... 1

0

ci 0 . ....

(standard deviations)

0 c-,i

Fig. A.12 Histogram showing the y-flow residuals of Kultsjön

62.10.01 - 76.05.18.

Prescribed minimum period duration: 1 day.

Mean = 411 1/s.

Standard deviation= 14 652 1/s.

Number of residuals = 914,

NER = 47. f ( days) :

(standard deviations)

0

0 L...L....c;;;::L...L--=l.......1.-'--'-'---'-.1...J~..L....IL....L...J.......aL..L..J_L..J......L....I......J......L...I......J..-J....J_J....J.....L..J-l::::1....J~ 0

c-.i I

0 . ..... I

0

0 0

N

Fig. A.13 Histogram showing the y-flow residuals of Kultsjön

62.10.01 - 76.05.18.

Prescribed minimum period duration: 5 days.

Mean = 1 178 1/s,

Standard deviation= 16 067 1/s.

Number of residuals = 607.

NER= 36.

A 8

I ( day s)

300 l I

~ 200

100

_._, - ---· I , 1 r--.,._R ... i : ,---_ • . . -H-- .· I 1-titt ! . 7

! I ' lii'·,. lliwnn , i 1· ~i I I , -I !, I ' ...._ __

l..-1.-.I.....I..-L...J..J......L..1-.1......L-L.L....L _ _.__._...1..-1......1..,...;.._.L....L_LI I I 1 :k, N I

0 . -I -,

C

0 0

N

(standard ,::.evid. t:'..ons)

Fig. A.14 r:istogram showing the i.ow flow residuals of Kultsjön

62.10.01 - 76.05.18.

C\I I

(days)

Prescri bed mir1imum period duration: 1 day.

Hean = 725 1/s.

Standard deviation = 7 068 1/s.

}Jumber of reaiduals = 2 876.

0 0

c-.i

(standard deviation)

Fig. A.15 Histogram showing thc l ow flow resici.uals of Kul tsjöE

62.10.01 - 76.05.18.

Prescribed minirnUl!l. perioci. duration: 5 d.ays.

Mean = 530 J./s.

Standard deviation= 6 863 1/s.

Number of residuals = 2 545.

NER= 123.

120

80

C'II I

(days)

0 - ci I 0 0 C'II

A 9

(standard deviations)

Fig. A.16 Histogram showing the low flow residuals of Kultsjön

62.10.01 - 76.05.18.

Prescribed minimum period duration: 20 days.

Mean = 370 1/s.

Standard deviation= 6 637 1/s.

Number of residuals = 2 117.

NER.= 100.

- 1

AppendL: B

LIST OF SYJVIBOLS

B max

B q

C 0

C perc

C rfr

C route

E a

E p

Fe

F 2 0

K 0

L UZ

M

MSC

n

N

NER

N T

N. J

n X

maximum base in the transformation function

actual base in the transformation function

degree-day melt factor

percolation capacity

refreezing coefficient

parameter in the transformation function

snowfall correction factor

water holding capacity of snow

actual evaporation

potential evaporation

maximum soil moisture capacity in the model

initial variance

stora.ge discharge parameter of the upper zone

slow drainage stora.ge discharge parameter of the upper zone

stora.ge discharge J"''.L'am.eter of the lower zone

limit for potential evaporation

limit for slow drainage of the upper zone

snowrnelt

hlechanism separation criterion

tr,tal number of observations in a class

number of observations in a continous period in a class

number of residuals not contained in the histo-gram. of the estimated density function

total number of observations of R(,)

duration of period j

sum of autocorrelation coefficients

B 2

p

p COIT

p lapse

p. l

p w

R sum

R2 w

R2 1

R2 2

R2 3

R( T)

R( T)

R.(T) J

s

s s

s sm s

UZ

T

T 0

precipi tatic,n

rainfall corrsc __ ur, J.c,c t,:.:c.·

area elevation corr

T lapse

t

t. J

u

Var(X)

X

X

s

y

]J

a

T

area elevation correction of ternperatlll'e

time

starting time for the j:th period

test variable

variance of the stochastic variable X

a residual regarded as a stochastic variable

mean value of the residuals

parameter of the soil moistlll'e zone

indicates rain or recession succeeding rain or snowmelt

mean value

standard deviation

time step

:B 3

B 4

LIST 0F REFERENCES

Andersson, R L (1941)

Bergström, S ( 1976)

Clarke, R T ( 1973)

Hamon & Harman (1963)

Hansen, E (1971)

Hjorth, u ( 1976)

Jenkins & Watts (1961)

Kendall & Stuart (1967)

Nash & Sutcliffe (1970)

Powell, M J Il (1964)

Rosenbrock, H H (1960)

Rudemo & Råde (1970)

Distributioi,. _ 1 "'""'--' ial co:e:c·r.=.,.a ;:;ion c:oefi'icient. Arm. Math. Statis-G, S No. 1

Development and application of a conceptual runoff model for Scandinavian catchments. SMHI Rapporter Hydrologi och 0ceanografi Nr RRC 7

A review of same mathematical models used in bydrology, with observation on their calibration and use. Journal of Hydrology" No. 19

Estimating relations between time series, J. Geophys. Res., 81 (21). 6033-6041

Analyse af hydrologiske tidsserier. Laboratoriet for Hydraulik, Ilanmarks Tekniske H~jskole

Stokastiska processer. Kompendium i matematisk statistik III. Matematiska institutionen, Universitetet i Linköping.

Spectral analysis and its applications

The advanced theory of statistics. Val. 2, 2nd Edn

River flow forecasting through conceptual models. Part I. A discussion of principles • .:: au.mal of Hydrology No. 10

An efficient method of finding the minimum of a function of several variables without calculating derivates. The Computer Journal, Val. 7, p. 155

An automatic method for finding the greatest or least value of a function. The Computer Journal, Val. 3, p. 175

Sannolikhetslära och statistik med tekniska tillämpningar. Ilel 1

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Nr 2

Nr 3

Nr 4

Nr 5

Nr 6

Nr 7

Nr 8

Nr 9

Nr 10

Nr 11

Nr 1~

Nr 13

Notiser och pre~iminlira rapporter

Serie HYDi.JL0GI

Sundberg-Falkemnark, M Om isbärighet. Stockholm 1963

Forsman, A Snösmältning och avrinning. Stockholm 1963

Karström, U Infrarödteknik i hydrologisk tillämpning: Värmebilder som hjälpmedel i recipientundersökningar. Stockholm 1966

Moberg, A Svenska sJoars isläggnings- och islossningstidpunkter 1911/12 - 1960/61. Del 1. Redovisning av observationsmaterial. Stockholm 1967.

Ehlin, U & Nyberg, L Hydrografiska undersökningar i Nordmalingsfjärden. Stockholm 1968

Milanov, T Avkylningsproblem i recipienter vid utsläpp av kylvatten. Stockholm 1969

Ehlin, U & Zachrisson, G Spridningen i Vänerns nordvästra del av suspenderat materiel från skredet i Norsälven i april 1969. Stockholm 1969

Ehlert, K Målarens hydrologi och inverkan på denna av alternativa vattenavledningar från Mälaren. Stockholm 1970

Ehlin, U & Carlsson, B Hydrologiska observationer i Vänern 1959 - 1968 jämte sammanfattande synpunkter. Stockholm 1970

Ehlin, U & Carlsson, B Hydrologiska observationer i Vänern 17 - 21 mars 1969. Stockholm 1970

Milanov, T Termisk spridning av kylvat\t=nutsläpp från Karlsharrmsverket. Stockholm 1971

Persson, M Ryd~'~ 2..ogi ska undersökningar i Lappträskets represen ta ti va område. Rapport I. Stockholm 1971.

Persson, M Hydroloe:;iska undersökningar i Lappträskets representativa område. Rapport II: Snömiitningar med snörör och snökuddar. Stockholm 1971.

Nr 14

Nr 15

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Nr 17

Nr 18

Nr 19

Nr 20

Nr 21

Nr 22

Nr 23

Nr 24

Nr 25

Heu.in, L Hydrologi:_;ka w1(:E:Y';.-~Ök:-1ine;SLr i ·✓ cle~lS rc1

Nr 2G

Nr 27

Nr 28

Nr 29

Nr 30

Ber ge; 1.,röm, S The application of a simple r2,inf'ull-nrnc,f':~ mcde~- ·"o " 2:1:cri-ment w~ th incor:1_piete data coverage. Stockholm lC:172

W~ndahl, T i bergstrand, E Oceano~rafiska förhållanden 1 svenska kustvatten. Stockholm 1973

Ehlin, U Kylvattenutsläpp 1 SJoar och ha,,. Stockholm 1973

Andersson, U-M & Waldenström, A Mark- och grundvattenstudier i Kassjöåns representativa område. Stockholm 1973

Milanov, T Hydrologiska undersökningar i Kassjöåns representativ~ område. Markvattenstudier i Kassjöåns område. Rapport II. Stockholm 1973

Nr RllG 1

Nr RHO 2

Nr RHO 3

Nr RHO 4

Nr Rl!O 5

Nr EU-JO 6

Nr RHO 7

Nr RHO 8

Nr R.c"9:0 9

Nr RHO 10

Weil, J G Verification cf heated wo.ter jet nurnericaJ model. Stock:-1clrr. 197 1~

Svensson, J CalculaLion of poison concentrations from a hypothet::.ce.l accident off tbe Swedish coast. Stockholm 1974

Vasseur, B Temperaturförhållanden i ~venska kustvatten. Stockholm 1975

Svenssor., ~ Beräkning av effekt ::. v vattentransport genoo Sunninge sund till Byfjord.er .. Stockholm 1975

Bergströo, S & Jönsson, S The application of the HBV-rwdel ':, O the F::.leCell researcL bo.sin. Norrköping l976

Wilmot, . ..,, A numerical nocel of the effects of reactor cooling water on fjordcirculation, I och !I (bilaga till I). Norrköping 19'(6

Bergström, S Development and applic2.tion of a conccpt,1al rur.off medel for Scandinavian ca-cchmcr.ts. Norrköpir.e 1 976

Svensson, ,J Se1uinars at SHJ-a 1976-03-29--:J4-U1 on ~ifumerical Y:odels of the Spreading of Cooling Water Norrköping 1976

Simons, .J & :2unkquist, L ~:: Svensson, J Application of a rn; ,':e r i c2.l I:J.odel to Lake V:'~r1.err:: Norrköping 197'7

Svensson, S A Statistical Study for Automatic Calil)ration of a Conceptual Runoff J'llodel Norrköping 1977

Tom sida