Journal of Hydrology

ELSEVIER Journal of Hydrology 168 (1995) 111-125

[3]

Sensitivity o f conceptual rainfal l -runoff algorithms to errors in input data - - case o f the G R 2 M model

J.E. Paturel*, E. Servat, A. Vassiliadis ORSTOM, Antenne Hydrologique, 06 BP 1203 Cidex 1, Abidjan 06, Ivory Coast

Received 7 April 1994; revision accepted 28 September 1994

Abstract

The study concerns the effect of errors in input data on the quality of the results obtained with rainfall-runoff models. A simple conceptual algorithm was selected: two inputs (rain and evapotranspiration), one output (resulting runoff), and two parameters. Systematic and ran- dom errors were considered. The algorithm amplifies the initial systematic error to an extent which depends on the phase of the hydrograph. The model appears less sensitive to errors in evapotranspiration ETP than in rainfall. The distribution curves of random errors in rainfall and in runoff are very similar; there is no distortion of one compared to the other. Tests were carried out on the method of calibration to demonstrate the power of numerical algorithms of this type: they are capable of absorbing the initial error and of determining an optimal solution.

I. Introduction

In hyd ro log i ca l model l ing , and more pa r t i cu la r ly , in mode l l ing the r a i n f a l l - r u n o f f re la t ionship , the p r o b l e m o f e r rors is f requent ly neglected.

However , three m a i n sources o f e r ror can be ident if ied in mode l l ing a na tu ra l sys tem (Ai tken , 1973; Singh, 1988):

(1) the mode l s m a y be on ly a p p r o x i m a t e a n d / o r ideal ized represen ta t ions o f phy- sical systems. I f the obse rved ou tpu t s c anno t be represen ted p rope r ly , there m a y be er rors owing to inadequac ies in the model ;

(2) for ca l ib ra t ion and va l ida t ion o f the models , h is tor ica l or exper imenta l d a t a which themselves include some er rors m a y have been used - - the m o d e l m a y there- fore be affected by d a t a e r rors ( inputs or outputs ) ;

(3) the p a r a m e t e r e s t ima t ion techniques , whe ther a u t o m a t i c or not , p rov ide a set o f

* Corresponding author.

0022-1694/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0022-1694(94)02654-8

112 J.E. Paturel et al. / Journal o f Hydrology 168 (1995) 111-125

parameters which can optimize only one criterion and which is in accordance with previous choices: choice of data, choice of criteria, choice of method of optimization. Furthermore, numerical methods can sometimes select secondary optima. Errors in the estimated parameter values can therefore creep in.

Only errors concerning the inputs for rainfall-runoff models are taken into account in this study. They concern mainly rainfall and evapotranspiration and can influence seriously the quality of results and hence the conclusions stemming from the models.

The causes of these errors may be multiple; measurement errors (owing to the measuring instrument or its close environment) and data sampling errors may be systematic (in this case, the error is always in the same direction and may therefore be detectable) or random (the error follows a distribution law and is practically indetectable).

Mastering the 'noise' introduced in the inputs, on the one hand, and using a simple rainfall-runoff model with few parameters on the other, will give a fairly general view of the sensitivity of conceptual models to errors.

2. Model and data used

This study was carried out in the framework of a broader rainfall-runoff modelling programme led by ORSTOM in the Ivory Coast (Servat, 1993, Servat and Dezetter, 1993). This research was originally directed towards practical use of the rainfall- runoff modelling by developers and managers in developing countries; this influ- enced the choice of tools:

(1) the algorithm must be simple and easy to understand as well as robust, adap- table and accurate;

(2) the model must be compatible with the time and space scales required by the potential users;

(3) the data necessary must be easily accessible to these users. The rainfall-runoff algorithm selected is therefore conceptual, global, working at

monthly increments and with few parameters. The data used come from national hydrometric, raingauging and climatological networks in the Ivory Coast.

2.1. The rainfall-runoff model used

A simple conceptual model was selected: GR2M, developed at CEMAGREF (Edijatno and Michel, 1989; Kabouya, 1990), working at monthly increments. Fig. 1 shows the conceptual diagram of GR2M.

Initially, rainfall and evapotranspiration are balanced. After that, rainfall during one time increment is channelled either towards the ground water reservoir or towards the transfer reservoir.

The ground water reservoir is characterized by its maximal capacity, A, the first parameter of the model. Its lexM, S, depends on the joint action of rainfall and evapotranspiration and the S/A ratio determines the distribution between ground reservoir and transfer reservoir.

The transfer reservoir with its level R is a quadratic type flushing reservoir. It

J.E. Paturel et al. / Journal of Hydrology 168 (1995) 111 125 113

P~otential EvapFotranspiratio~n ~ )

V v l Neutralization of P and E

E' P'

h * E' (1 - k 2) * P'

'~ i~ ~

Soil moisture ~ ~ reservoir of ~ capaci;y ( ~

k 2 * p,

4

I Routing reservoir !of capacity @

h = S/A * (2- S/A) k = S/A

2 R

Discharge - R+B

Fig. 1. Conceptual diagram of GR2M.

depends on only one parameter, B: maximum monthly retention. The discharge obtained f rom the reservoir is calculated f rom the following formula

R 2 Q - - - (1)

R + B

2.2. Data

The observations were made on several catchment areas of the northwest of the Ivory Coast (Fig. 2), located in the savannah zone, characterized by a single rainy season (usually from June to September) and vegetation of the arboraceous savannah type (Avenard et al., 1971).

Table 1 presents the main characteristics of the catchment areas under study, all in the upper basins of the Bandama, Niger and Sassandra rivers. The data available stem from the drought period since the beginning of the 1970s and there are marked interannual variations, both in terms of amounts of rainfall and runoff. As regards land use, the percentage of savannah was high although in some cases sizable tracts of land had been claimed for farming.

3. Methodology

The behaviour of the model was studied as regards relative errors introduced in the

114 J.E. Patureletal./JournalofHy~ology 168 (1995) 111-125

MALl BURKINA . . . .

GUINEA

~HANA

LIBE

Fig. 2. Map of study sites: A, savannah area; B, transition area; C, forest area.

input data: rain and potential evapotranspirat ion (ETP), expressed at monthly incre- ments in both cases.

Two different types of relative errors were studied: systematic errors and random errors.

It was accepted for the purpose of this study that estimates of evapotranspirat ion (Penman, 1948), can be affected only by systematic errors.

The random error generation process used for rainfall was modelled according to the following assumptions:

(1) relative errors are supposed to be independent f rom one month to another; (2) the relative errors follow a Gaussian curve distribution with zero mean and 95%

of values are included between - 3 0 and + 30%. Based on these different assumptions, the methodology selected for the study of

systematic errors as well as random errors was the following. (1) The G R 2 M model was calibrated according to chronological series of several

calendar years (from January to December) .The parameter initialization phase was set at one year so as to include a high water and a low water period. Therefore, the first year was not included in the further processing of results.

(2) During the calibration phase, the depth of runoffwas considered as a zero error

Tab

le 1

C

hara

cter

istic

s o

f th

e st

udy

catc

hmen

t

Cat

chm

ent n

ame

Are

a R

iver

(k

m 2)

ba

sin

ET

P

Stu

dy

Min

. an

nual

M

ax.

annu

al

Min

. an

nual

M

ax.

annu

al

S F

C

H

stat

ion

peri

od

rain

fall

ra

infa

ll

disc

harg

e di

scha

rge

(%)

(%)

(%)

(%)

2 in

the

stu

dy

in t

he s

tudy

in

the

stu

dy

in t

he s

tudy

pe

riod

pe

riod

pe

riod

pe

riod

(m

m)

(mm

) (m

m)

(mm

)

Bag

o6 a

t K

outo

4

70

0

Nig

er

Kor

hogo

81

-85

971

Ban

dam

a at

S~g

u6ki

6le

368

Ban

dam

a K

orho

go

81

85

896

Bou

at

Bor

on

37

10

B

anda

ma

Kor

hogo

81

85

85

3 D

ound

ian

at W

ahir

6 65

0 N

iger

O

dien

n6

80-8

4 10

92

Sien

at

Naf

ana-

Sie

nso

615

Sas

sand

ra

Odi

enn6

77

-84

1038

1377

44

.9

223

60

24

16

0.5

~,~

1541

15

.9

176.

1 68

0.

8 31

0.

6 ,~

14

37

9.63

13

4 N

C

NC

N

C

NC

15

37

55

253

75

13

12

0.3

~.

1667

57

28

4 58

0.

9 41

0.

2

NC

: no

car

togr

aphy

; S:

sav

anna

h; F

: th

in f

ores

t, C

: cu

ltiv

ated

are

a; H

: ha

bita

t.

x~

X

~

t~a

q-n

%tl

116 J.E. Paturel et al. / Journal of Hydrology 168 (1995) 111 125

depth, corresponding to zero error inputs (rainfall and ETP). This will be the refer- ence runoff depth.

(3) The response of the model was studied after applying noise to the inputs, that is after applying a systematic error to one of the inputs (rainfall or ETP), or after applying a random error to monthly rainfall measurements. The gap between the reference runoff depth and the runoff depth generated by the model was then studied.

From the numerical point of view, calibrating the models was carried out following the combined gradient method (Gourdin and Boumahrat, 1993), supplemented with a modified formulation of the Nash criterion (Servat and Dezetter, 1991)

F = ~ 7 ( a c - Qo) 2 ~ 7 (Oo - 0) 2 (2)

This Nash criterion corresponds to the percentage of residual variance after calibra- tion, compared with the total variance of the series considered.

4. Results

Apart from the case of the catchment area of the Doundian at Wahir6, which sometimes behaves slightly differently, all the results obtained were homogeneous, irrespective of area or year.

4.1. Sy s t ema t i c errors

4.1.1. Overes t imat ing rain

In the first place, the monthly evolution of the relative runoff error is considered as a function of the initial relative rainfall error. Fig. 3 shows the results obtained in the case of the Bou at Boron in 1982, which can be generalized for all other catchment areas.

For the months when the water is rising (June and July), a slight drop in the slope is visible, when the amplitude of the rainfall error exceeds 15-20%. For other months, the relative error varies almost linearly as a function of the error in rainfall.

Moreover, in all cases, the slope of the relative runoff error variation is above 1 for each month. The initial error is therefore amplified by the model.

The evolution of relative runoff errors was then studied continuously over a year for a systematic rainfall error of 10%. Fig. 4 shows the average runoff depth for the Doundian at Wahir6 for the years 1981-1984.

Although the values differ from one year to the next, the shape of the relative runoff error curves remains the same in every case. It reveals large relative errors during the water rising phase which is a sensitive period for this type of rainfal l-runoff algorithm (Servat and Dezetter, 1991). After this phase, the error drops gradually.

In the case of a systematic error of 10% in rainfall, the history of resulting relative runoff errors was studied over four consecutive years. No cumulative or amplifying effect was found. No hydrograph distortion was carried over from one year to the

J.E. Paturel et al. I Journal o f Hydrology 168 (1995) 111-125 117

300

2S0

200

i 150

1oo

,~ 5o

0

i

0 i

[ ]

ill I I I

S 10 1S

[ ]

0 [ ]

• i i |

20 25 30

Rainfall relative error ( % )

Fig. 3. Monthly variation of runoff error owing to rainfall overestimation (Bou at Boron, 1982): II, April; IS], June; 0, July; 0, September; a,, December.

next; the hydrographs and their associated error curves are independent. The relative error is low in the dry season and high when the water rises. Fig. 5 shows the results for the Sien at Nafana-Sienso.

There is a slight difference in behaviour concerning the results recorded on the Dioundian at Wahir6 (Fig. 6), where the relative error tends to rise slightly from year to year. However, the observed runoff depth dropped between 1981 and 1984, and this resulted in an increased relative error.

F0

60 .\ 60

SO SO

¢0 \ ,,~ 40 , 40 -

,~ ~o 20 20

,~ I0 " '' I0

,--, ~ ~ e-,

a v e r a g e observed depth of runoff

. . . . . . 1981 - - - 1983

- - 198Z 1904

Fig. 4. Yearly evolution of runoff error owing to a 10% overestimation of rainfall (Doundian at Wahir6).

118 J.E. Paturel et al. / Journal of Hydrology 168 (1995) 111-125

120 ~120

80 ao

4o 4o

0 .......................................... 0

Fig. 5. Evolution over a 4 year period of a runoff error owing to a 10% overestimation of rainfall (Sien at Nafana-Sienso): thick line, observed depth of runoff; thin line, relative runoff error.

4.1.2. Underes t imat ing rainfall

U n d e r e s t i m a t i n g rainfal l , na tu ra l ly , leads to unde re s t ima t ing runof f as ca lcu la ted by the models . The re la t ive e r ror is negat ive in this case. The s lope o f the relat ive runof f e r ror is lower than when ra infa l l is overes t imated . The a lgo r i t hm used tends to ampl i fy an ove res t ima t ion o f ra infa l l more than an underes t ima t ion . This is illu- s t ra ted in Fig. 7 for the B a n d a m a at S6gu6ki616 in 1982. The curves are presented for 4 mon ths .

There is a d r o p in the ove res t ima t ion s lope b e y o n d 1 5 - 2 0 % o f relat ive ra infa l l e r ror for the high wa te r months . However , in the case o f unde re s t ima t ion the runof f e r ror curves have a m o r e ' a s y m p t o m a t i c ' shape.

As in the case o f the ove res t ima t ion s tudy, the evo lu t ion o f the flow er ror ca lcu la ted by the mode l over a year was examined , this t ime with a sys temat ic unde re s t ima t ion o f ra infa l l o f 10%. F o r example , Fig. 8 shows the case o f Sien at N a fa na -S i e nso for the years 1978-1982. The m e a n runof f he ight obse rved over this 5 year pe r iod d e m o n s t r a t e s the h i s tory and va r i a t ion o f e r ror t h r o u g h o u t the year.

The obse rva t ions on the Sien at Na fana -S i enso were verified in the o the r c a t chmen t

1oo

80 ~D

g 40

20 ,~.

.IOO

80 _~

60

40

20 ~"

Fig. 6. Evolution over a 4 year period of a runoff error owing to a 10% overestimation of rainfall (Doundian at Wahir~): thick line, observed depth of runoff; thin line, relative runoff error.

w

b,.

¢ 150

i00 k.,

50

, , * ~ :~ -so

-I00

J.E. Paturel et al. / Journal o f Hydrology 168 (1995) 111-125

358

300

250

200

ti, O n

m • I •

[] 15

I ,

Q

I

30

119

r a i n f a l l r e l a t i v e e r r o r (%)

Fig. 7. Monthly variation of runoff error owing to rainfall underestimation (Bandama at S6gu6ki616, 1982): m, May; E3, June; 0, July; <), October.

areas. Although the values differ slightly f rom one year to the next, the shape of the relative flow error curves remains identical in each case. The values are not as high as in the case of systematic positive errors and the general shape of the curves is flatter. The max imum values are reached during the sensitive water rising phase.

A study of the time series of data for four consecutive years showed no cumulative or amplifying effect f rom one year to the next. This confirms the independence of the

8O

6o

w

4O

20 '• 0

~ -20 -40

80

6o

o c

4O 2 o

2o

(l,) 'Im)

average observed depth of runoff

...... 1981. 1983

- - 198Z 1984

Fig. 8. Yearly evolution ofrunofferror owing to a 10% underestimation of rainfall (Sien at Nafana-Sienso).

120 J.E. Paturel et al. /Journal of Hydrology 168 (1995) 111-125

o

,~ -lO

-20 o

g, -30

, ~ - 4 0

-50

-t0

-70 .~, ,= -80

-90

~ ,\ / J " .

. . . . . . DQwith D E = S % - - - - - D Q w i t h D P = - 5 %

- - - - DQ with DE=IS% DQ with DP=-15%

- - DQ with DE:25% ~ DQ with DP:-25%

Fig. 9. Comparative runofferror curves owing to rainfall error and ETP error (Bandama at S6gu6ki61~, 1982).

hydrographs and their associated error curves as was previously observed in the case of rainfall overestimation.

4.1.3. Systematic errors on evapotranspiration

Owing to their structure, the conceptual ra infal l - runoff models behave differently in the case of rainfall overestimation (leading to runoff overestimation) and of eva- potranspirat ion overestimation (leading to runoff underestimation by the model). Conversely, underestimating rainfall leads to underestimating runoff and underesti- mating evapotranspirat ion to overestimating runoff.

Generally speaking, such a model seems less sensitive to an error in evapotranspira- tion than in rainfall. The relative runoff error calculated by the algorithm is in fact lower than that due to a systematic error of the same level on rainfall. Fig. 9 shows the curves obtained for both cases for the Bandama at S6gu6ki+16 with systematic input errors varying f rom 5 to 25%.

In the particular case of GR2M, this lower sensitivity to errors in evapotranspira- tion may be due to the architecture of the model. Evapotranspirat ion affects the runoff calculation only indirectly, through the S reservoir level, which then com- mands the propor t ion of rain feeding the transfer reservoir R. However, rain is taken into account twice in the calculation of runoff: indirectly, in the S ground reservoir level calculation, and directly, as the quantity of rain which feeds the R transfer reservoir.

4.2. Random errors

Only the random 'noise' in rainfall was taken into account here. The characteristics of the r andom error generation model are the following:

J.E. Paturel et al. / Journal o f Hydrology 168 (1995) 111 125 121

80

60

,~ 4O

2O

0

~ -20

~ -4o

-60

-80

Fig. 10. Evolution over a 4 year period of a runoff error owing to random rainfall error (Bandama at S6gu6ki616, 1982): thick line, rainfall relative error (%); thin line, discharge relative error (%).

(1) independent relative errors between months; (2) generated relative errors are distributed according to a Gaussian law of zero

mean such that 95% of values are between - 3 0 and + 30%. For each catchment area studied, a period of four consecutive years was selected.

For each of these 'catchment area-period' pairs, over 300 simulations were applied with the GR2M after random noise on rainfall. Fig. 10 provides an example of the comparative evolution of relative random error on rainfall and relative error on runoff depths calculated by GR2M for the Bandama at S6gu6ki616.

A thorough observation of the results show the following. (1) There is no linear or exponential correlation between the random rainfall error

and the resulting relative runoff error. (2) The season or time of the year is important as regards reduction or amplification

of the effect of random error in rainfall on calculated runoff. For whatever the rainfall error may be during the dry season (from November to April), the corresponding relative runoff error is always very slight. This probably reflects the low rainfall and runoff measurements at that time of the year. A marked amplification of rainfall errors occurs only during the periods of rising water (May-July), flooding (August-September) and sometimes flood recession (October).

(3) As observed so far, amplification of relative errors in runoff occurs in the same way as random error in the corresponding rainfall; this means that an overestimation (or underestimation) of rainfall leads to an overestimation (or underestimation) of the calculated runoff.

Using these numerous simulations, the distribution curves of random relative rainfall error and of the resulting relative runoff error for the GR2M model were traced for each catchment area. Fig. 11 shows the case of the Sien at Nafana-Sienso. The corresponding statistics of.the distribution are presented in Table 2. The obser- vations for the Sien at Nafana-Sienso are valid for the other catchment areas.

The distribution curve of runoff errors shows a slight positive asymmetry and a

122 J.E. Paturel et al. / Journal of Hydrology 168 (1995) 111-125

4000

¢ N

'G 2000

-50 0

relative error (%)

- B - rainl

, --o-discl

0 -100 +50 +100

Fig. 11. Distribution curves of random rainfall error and the resulting runoff error (Sien at Nafana-Sienso): m, rainfall; O, discharge.

sharper peak than the rainfall error curve. The statistical characteristics summed up in Table 2 confirm this observation. The algorithm therefore reacts correctly to the random error model selected: the output error distribution curve follows closely the input distribution curve and shows no distortion.

The asymmetry of these curves may be explained by the fact that the model input overestimation leads to a stronger amplification of error than an underestimation does. Therefore, higher values for relative runoff errors result from positive rainfall errors.

4.3. Error balancing by calibration

The objective of calibrating a rainfall-runoff model is twofold (Sorooshian and Dracup, 1980):

(1) to obtain a 'conceptually realistic and unique' set of parameters which best reflect an understanding of the physical system;

(2) to obtain a series of parameters providing the closest correlation between simulations using the algorithm, and hydrographs observed during the calibration period.

The second aspect is usually privileged during calibration exercises, which means that the set of parameters obtained may be considered optimal, subject to a whole array

Table 2 Statistical characteristics of rainfall and resulting rainfall runoff error distribution curves

Characteristic Rainfall curve Runoff curve

Mean 0.00049 0.01639 Median 0.00108 0.00028 Standard deviation 0.15245 0.20272

J.E. Paturel et al. / Journal o f Hydrology 168 (1995) 111 125 123

120

°" /i J"I

- - calibrated depth of runoff

_ _ _ " r e f e r e n c e " depth _ . without calibration of ronoff depth of runoff

Fig. 12. Compar ison of reference runoff curves, and curves obtained with the calibrated and non-calibrated G R 2 M resulting from a 10% overestimation of rainfall (Sien at Nafana-Sienso).

of constraints determined by the calibration method (manual or automatic, type of numerical algorithm), the numerical criterion selected and the nature of the data.

The period selected to carry out the calibration is also a determining factor for the set of parameters obtained to be significant. This period should include sufficient 'hydrological variability', i.e. varied rainfall and runoff conditions.

In the particular case of data used in the f ramework of this study, pertaining to the Sudanese savannah area, and in view of the numerical criterion defined above, it seemed worthwhile to examine how well the calibration method (the combined gradient method) could compensate by itself for some input data errors.

A first calibration of the model bounded a reference depth. After that a systematic positive noise of 10% was applied to rainfall. The depth obtained with this 'noisy' data is called 'non-calibrated depth' . An automatic calibration was applied to the 'noisy data - - ra infa l l - runoff model ' system, which determined a 'calibrated depth' .

Fig. 12 shows the case of Sien at Nafana-Sienso from 1978 to 1991 which is typical of the general behaviour of tests carried out in different catchment areas. This curve shows to what extent the calibration method can compensate input data errors. The calibrated depth is, in fact, very close to the reference depth, which is not the case of the non-calibrated depth.

In the case of random errors, the numerical criterion values obtained without calibration are very close to those obtained with calibration. The calibrated depths are themselves very close to the reference depths. Fig. 13 shows the case of the Bagoe at Kouto in 1984 and 1985. The non-calibrated numerical criterion is of 0.0363, i.e. 3.6% of residual variance, against 0.019 for the calibrated numerical criterion, i.e. 1.9% of residual variance.

These minimal gaps compared with the reference depths were also observed on longer series. They may be explained by the fact that r andom errors follow a normal zero mean distribution.

120

I00

80

20

! 60

o

b,- 40

, ~ ; , , i I I I I I ~ I I ; ; ; ; I I I I I I

without calibration depth of runoff

- "relerence" depth of runoff

- - - calibrated depth of runoff

124 J.E. Paturel et al. / Journal o f Hydrology 168 (1995) 111-125

Fig. 13. Comparison of reference runoff curves, and curves obtained with the calibrated and non-calibrated GR2M resulting from a 10% random rainfall error (Bago6 at Kouto).

5. Conclusion

The present study concerns only a particular case, since it applies only to the G R 2 M model. Any generalization would require the use of many other rainfal l - runoff models, as well as the multiplication of data and therefore of catchment areas.

However, this study is instructive and of general applicability, because of the simple and standard architecture of the global conceptual model used.

In the case of systematic errors on rainfall, the behaviour of an algorithm of this type is given below.

(1) Amplification of the initial error by the model. It is noticed both for over- estimation and underestimation.

(2) The behaviour varies according to the phases of the hydrograph. Error ampli- fication peaks occur during rising floods and drop during flood recession and in the low water period.

(3) There is no error amplification cumulatively over years, apar t f rom the case of the Doundian catchment area at Wahir6, where a slight trend of increasing error was observed, probably due to a drop in observed runoff heights and the associated loss of accuracy of the model.

These observations also apply in the case of systematic errors in evapotranspiration. However, an error on rainfall has the opposite effect as an error in evapotranspirat ion as regards the depth calculated by algorithm; the model is also less sensitive to an error in evapotranspirat ion than to an error in rainfall because of its very structure.

When a random error process is applied to rainfall, the results confirm the above observations both as regards amplification of the initial error and the evolution according to the phase of the hydrograph as well as to the absence of error amplifica- tion between years.

J.E. Paturel et al. / Journal of Hydrology 168 (1995) 111-125 125

Multiplication of simulations related to the random error process showed that even when the algorithm amplifies errors, it does not induce distortions, as the error distribution curves on rainfall and on runoff had much the same shape.

Lastly, a final test of the automatic calibration method revealed the power of such a numerical algorithm, which is capable of 'absorbing ' systematic input errors, so that an 'opt imal ' solution is obtained.

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