Application of a digital filter for modelling river suspended sediment concentrations

Journal of Hydrology, 108 (1989) 267 279

267

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

[4] A P P L I C A T I O N OF A DIGITAL FILTER FOR M O D E L L I N G RIVER S U S P E N D E D S E D I M E N T CONCENTRATIONS

L. ZHANG, D.J. GREGOR and J.-P. VERNET

lnstitut F.-A. Forel, Universitd de Gen~ve, 10, route de Suisse, 1290 Versoix (Switzerland) (Received May 11, 1988; revised and accepted August 29, 1988)

ABSTRACT

Zhang, L., Gregor, D.J. and Vernet, J.-P., 1989. Application of a digital filter for modelling river suspended sediment concentrations. J. Hydrol., 108: 267-279. The relationship between suspended sediment concentration and discharge in a river system is the result of a complex interaction of factors within the basin and of specific storm events. Nevertheless, estimation of suspended sediment concentrations during storm events is essential for prediction of loadings from storm-controlled river basins to the receiving waters. The Venoge River, a tributary of Lake Geneva, Switzerland, with a basin area of 237 km 2, has been monitored during eight storm events between October 1986 and November 1987. A digital filter is used to simulate the suspended sediment response during individual events with the principal factors being antecedent flow, maximum discharge and time during which maximum flow was attained. The measured and calculated sediment concentration curves were assessed using a ~(2-test. It was concluded that the two curves were similar in shape at 95% confidence interval. However, the model is not always capable of simulating the phase lead effect of the sediment distribution curve relative to the hydrograph. This problem is most evident during high-intensity storms at periods of low base flow; and consequently, loadings calculated for individual storms of this type could be overestimated. Fortunately, the model provides good predictions during less-intense, longer-duration events and therefore loadings calculated by summing events for an annual period will tend not to be overestimated. Further work is necessary to evaluate the model for additional events, in particular spring snow melt, and to determine its applicability to other river systems.

INTRODUCTION

Measurements of suspended sediment concentration (SS) and stream discharge (Q) represent the instantaneous response of a river system to a complex interaction of changing basin conditions. These two variables are especially important in storm-dependent watersheds for determining sedimentassociated pollutant transport. Although linear rating curves have been applied to explain this relationship relatively successfully in a general sense, it is clear that the relationship between these two variables during storm events can only be crudely approximated linearly. Wood (1977) identified six types of rating loops for a river system in England, based upon the degree of hysteresis within the relationship. In most cases,

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268 sediment peaked in advance of the hydrograph. Olive and Rieger (1985) considered five small catchments during storm events in New South Wales, Australia. Because of the large spatial and temporal variability in sediment response, the authors concluded that there was no simple, constant relationship between discharge and suspended sediment concentration as suggested by the traditional rating curves. Further discussion was provided by Rieger and Olive (1987). The hysteresis and the lag between peak suspended sediment concentrations and maximum discharge commonly present in these relationships have resulted in the development of other mathematical models to address this question. One example, the exponential and polynomial functions produced by Bfihrer and Wagner (1982), failed to consider entirely the temporal variations of the two variables. Another, frequency domain analysis in the form of spectral analysis (Rieger and Olive, 1986) showed some merit in identifying important frequency components or scale that might be operating in the process. Any method chosen, as was emphasized by Rieger and Olive (1987), must be capable of addressing the dynamic properties that exist between the two variables. The Venoge River, Switzerland, has been studied from October 1986 to November 1987 for the purpose of better understanding the importance of a storm-dominated river basin in determining sediment and sediment-associated pollutant loadings to the receiving water, Lake Geneva. To address this problem, it was necessary to develop a means of simulating the response of suspended sediment concentrations during precipitation events. Here, a digital filter method has been explored for this purpose. SAMPLING AND ANALYSIS The Venoge River is one of principal tributaries of the Lake Geneva drainage basin. It has an area of 237 km 2, consisting of 51% cultivated land, 31% forest, 17% pasture and the remaining (1%) is residential and uncultivated lands (Caloz and Blaser, 1987). The average slope of the river bed, along its 41-km length, is 0.7%. Annual total precipitation within the watershed is about 1000mm (Demierre and Humbert, 1987). In spite of some snow melt in spring, the Venoge is mainly a rain-fed river. It drains rolling land and has its headwaters in the forested J u r a Mountains. The sampling site for the storm events is co-located with a gauging station at Ecublens les Bois, 4.7km upstream of Lake Geneva. Annual mean discharge at this site is 4.76m:~s 1 Suspended sediment samples were collected with two types of apparatus. An ISCO (model 2700) automatic sampler was used to collect time-integrated 1-1. raw water samples at 3 h intervals during five storm events. The intake orifice of the sampler was located adjacent to the water-level weir. The turbulence of the river during the storm events more or less assured the representati,eeness of a single-point sample although this was not confirmed with additional crosssectional samples. A Westfalia continuous flow centrifuge was used to obtain large volumes of suspended sediment, at a rate of 51 min ~, during three storm events. These large samples (the volume of water centrifuged varied in accord-

269 ance with the turbidity but was consistently more than 301.) were required for additional chemical analyses not reported here. The centrifuge, following the procedure outlined by Allan (1979), Ongley and Blachford (1982) and Burrus (1984), provided essentially instantaneous samples over the course of the storm event but at less frequent intervals than the smaller ISCO samples. The submersible pump used for these large volume samples was located in turbulent flow immediately below the weir. The suspended sediment from the ISCO sampler was centrifuged using a bench-top centrifuge in the laboratory. The remaining water was decanted from the sample and the sediment was then freeze-dried and weighed to determine the concentration of suspended sediment. DIGITAL FILTER The storm hydrograph is a realization of a stochastic process, which has two parts, a systematic wave-like function (a drift) and a random irregularity which cannot be predicted except in a probabilistic manner. Suspended sediment response is also a random time series which may or may not relate to the hydrograph. In the case of the existence of a relationship between the hydrograph and the sediment concentration, it has been shown by Wood (1977) and Rieger and Olive (1987), that the relation is dynamic. A digital filter is capable of describing this type of relationship. Operationally, a digital filter is a signal-processing method that removes interference and/or noise from a time series to estimate the expected signal. In other words, it is a mathematical operator that changes a time series into another time series having some desired form (Davis, 1986). Let us consider the time series x(t) which is composed of two components z(t) and c(t), where z(t) is the expected signal and e(t) is the noise process. Also, a predictor y(t) is available for z(t). A digital filter with a pulse response f(t) can be designed to estimate y(t) as a linear function of x(t). We suppose x(t) and y(t) are, respectively, input and output series of a digital filter; y(t) is related to the convolution of x(t) and f(t):

y(t) = ~. f ( i ) * x ( t - i)

(1)

i=O

where f(1), f(2) . . . f(s) are the set of filter coefficients. The difference between the output y(t) and the series expected z(t) should be minimized under the least-squares criterion. The coefficients of f(i) are estimated by minimizing: D =

~, ~ { [ f ( i ) * x ( t -

~ [ y ( t ) - z(t)] 2 : t

1

t

i)]-

z(t)} 2

(2)

1i-0

Equation (2) can be transformed under the least-squares criterion to:

~,f(i)*rxr(j-

i) = rzx(j),

(0 ~< j

~< s)

(3)

270 The terms rxx and r= are referred to as the autocorrelation function of the input, and the cross-correlation between the input and the output series, respectively (McQuillin et al., 1984). To apply a digital filter in modelling sediment~lischarge relationships, it is essential that the time interval between observations of the series be constant within and between events. Additionally, a stochastic sequence which can serve as the input series of the filter [i.e., x(t)] must be found. In hydrology, some empirical formulae of sediment transport exist. If the terms and parameters of these can be expressed reasonably as a function of a hydrological variable, for instance, discharge, they could be regarded as the interfered random time series. In this case, sediment concentration measurements of the ISCO samples will be used as the expected output [i.e.. y(t)]. MODELLING I n p u t sequence

Generally, the relationship between sediment transport and discharge results in increasing concentrations of sediments as the hdyrograph rises. However, as has been shown for a variety of drainage basins (Wood, 1977; Olive and Rieger, 1985), the sediment-carrying capacity of a river depends not only upon discharge, but also upon sediment delivery potential of the basin. The empirical expression for suspended sediment transport is: Sc

= KQ~ST~

(4)

where Sc is the suspended sediment-carrying capacity, Q is the discharge, SD is the sediment delivery potential of the basin, and K, n and m are empirical constants. In this study, all samples have been collected at the Venoge River. The geomorphological structure of the river bed and the drainage basin (e.g., soil texture, sediment composition and land use), although quite variable spatially, generally change very little in a temporal sense. Factors controlling SD are therefore assumed to be constant, except for those determining sediment supply, which are a function of antecedent hydrological conditions of the basin. Thus, setting So as a constant and solving for k and n in such a way as to consider, at least in part, antecedent moisture conditions, allows S O and K to be combined and eqn. (4) to be simplified to: S~, = k Q ~

(5)

Therefore, solving for k and n for each storm event, such t h a t antecedent basin moisture and precipitation intensity and duration are integrally incorporated, allows eqn. (5) to provide the input series for the digital filter. Constant determinations

The response of a river system to precipitation is related to a complex interaction of a variety of factors operating generally within the basin and to

271

t

40

=o

QO

t ......................................................................

:Tmox

To

~"'T

Time

Fig. 1. Definition s k e t c h of t h e p a r a m e t e r s of a s t o r m event.

stochastic storm-dependent variables. In the model, three main factors are considered: antecedent flow (Q0) before a storm event; the time interval between the start of the event and attainment of maximum flow (AT); and the maximum absolute change in discharge for the event (AQ). These terms are defined in Fig. 1. It is evident that Q0 is a function of and therefore indirectly expresses the effect of antecedent basin moisture, particularly for a storm-dependent river system. The ratio of AQ/Qo measures the net relative increase in discharge resulting from the storm event and therefore is largely a function of the total quantity of precipitation received. The duration and intensity of the event, in addition to the total amount of precipitation, are important factors in determining the basin hydrograph response. Both duration and intensity are considered primarily by the AQ/AT ratio. This ratio is high for a high-intensity storm of relatively short duration and low for comparatively light rainfall that continues for a long time; even though both events may produce the same total quantity of precipitation. Additionally, all of these factors are influenced to some extent by seasonal differences within the basin. The two factors, Q0 and AQ/AT are considered as determinant variables in the solutions of k. For any storm event, Q0, Qmaxand AT can be readily determined from the hydrograph. Through a series of experiments, using this information for the Venoge River, k and n have been solved for different types of storm events:

f k =

(0.16 + 5.93AQ/AT) 2 (0.16 + 5.93AQ/AT) 21nl0,

(AQ/Q0) > 2

(6a) (6b)

(0.16 + 5.93AQ/AT)2/lnlO,

(1/Qo)(AQ/AT) < 2E - 2

(6c)

n = 4.46 - 0.79(Qmax/2)-1/2

(7)

272 S t r u c t u r e o f the filter a n d c o m p u t e r p r o g r a m

F i g u r e 2 symbolically displays a typical example of a digital filter. T h e term Z 1 in the figure m e a n s Z-transform with a first delay. To s e a r c h for suitable filter factors, the following steps are followed. First, k and n of the c o r r e s p o n d i n g e v e n t are d e t e r m i n e d as s h o w n in eqns. (6) and (7). Second, the input series f u n c t i o n Sc is o b t a i n e d by using eqn. (5). Third, we c a l c u l a t e the a u t o c o r r e l a t i o n of the i n p u t and c r o s s - c o r r e l a t i o n between the input and o u t p u t respectively. F o u r t h , supposing the filter's length to be (s ÷ 1), the s i m u l t a n e o u s eqns. (3) are set up and solved in t u r n to o b t a i n the values for the filter factors. Finally, the factors and the i n p u t series are t a k e n back to eqn. (1) and the c o n v o l u t i o n gives the c a l c u l a t e d o u t p u t series S~). where: (8)

So = ~ f ( t ) * S c ( t -- i)

If So and the m e a s u r e d series c a n n o t be verified from the same p o p u l a t i o n by m e a n s of a ~2-test, we suppose a new l e n g t h for the filter and r e p e a t steps four and five until a set of filter factors are found which c r e a t e a s a t i s f a c t o r y So. The c o m p u t e r p r o g r a m s c h e m e is s h o w n in Fig. 3. After a r e i t e r a t i v e o p e r a t i o n , the solution of eqn. (3) with the length s ÷ 1 = 3, gives the filter factors for the V e n o g e R i v e r of: f(i)

m

=

(0.31, -0.067, - 0.11) ~"~ 1

i n t e g e r (n/k)

<

2

0

i n t e g e r (n/k)

=

2

1 i n t e g e r (n/k)

> 2

~(t)

V

y(t) Fig. 2. Symbolic representation of a digital filter.

(9)

273

begin

7

input: k,n,Si

/

i i

/

[f(1), f(2),

. f(s)]

2+,

f(t)

=

1 So = f(t) * S,

I

not

yes

I

I I Fig. 3. Computer program scheme for the model. A filter of this type is therefore capable of adjusting automatically according to different attributes of the storm event. RESULTS AND DISCUSSION The variables for each storm event are listed in Table 1. The storm events, excluding th at of J u n e 19-22, 1987 (see below) can be roughly divided into two sets. The first set consists of the storms characterized by a gradual rise in the hydrograph: specifically, the events of October 25-27, 1986, March 26-30, 1987, May 15-18, 1987, and J u n e 6-12, 1987. The March and J u n e events are both multi-peaked hydrographs whereas the other two are single peaked. Although both peaks of these two storms are treated separately by the model they are discussed as one complex event. The antecedent flow for these four events ranged between 2.35 and 5.34 m3s 1; whereas the maximum flow ranged from 6.17 to 12.6m3s 1. The AQ/AT values for all of these events are relatively low except for the J u n e event. The second group of storms includes the events of October 23-25, 1986,

ll.23 12.89 17 12.9 7.50 30.43

SD of meas. conc. distribution SD of calc. conc. distribution Degrees of freedom /2 ~2 critical-1 ~2 critical-2

x

AQ[Qo AQ/T

AT

2.42 6.17 3.75 27 1.55 0.14

Q0 Q

/3-2

f3

7.51 6.17 17 25.18 7.50 30.43

1.69 19.2 17.51 35.5 10.36 0.49

/3/2

fU2 f2/2

0 <2

3

>2

/1.2 f2-2

21.92 2.01

Dec. 18-20 1986

0.97 3.07

Oct. 25-27 1986

3.8 5.98 2.18 15 0.57 0.15

[3.2

f1. 2 1"2.2

>2

3

1.04 3.09

Mar. 26-27 1987

results of the storm events

ISCO s a m p l i n g

and/2-test

fl f2

k n Integer (nik)

Principal parameters

TABLE t

10.39 10.39 35 25 18.43 47.26

4.77 10.6 5.83 21 1.22 0.28

/3/2

/1/2 f2/2

<2

1

3.26 2.64

Mar. 27 30 1987

7.44 7.31 23 23.82 12.79 38.34

5.34 8.01 2.67 21 0.50 0.13

/3.2

f1. 2 f2.2

~2

3

0.84 2.88

M a y 15 18 1987

2.35 5.49 3.14 9 1.34 0.35

/3/2

/1/2 f2/2

<2

i

4.97 3.15

June 7 1987

9.64 9.39 30 31.62 18.43 47.26

4.64 12.6 7.96 21 1.72 0.38

/3/2

/1]2 f2/2

<2

0

5.80 2.48

J u n e 7 10 1987

2 6.37 4.37 9 2A9 0.49

/1/2 f2/2 f3/2

<2

0

21.28 3.05

Oct. 23-24 1986

12.6 16.4 3.8 21 0.30 0.18

2.25 18.4 t6.15 35.5 7.18 0.45

/3/2

18.81 2.06 0 <2 /1/2 f2/2

0.66 2.20 3 .~2 fl.2

f2-2 f3.2

Nov. 12-15 1987

J u n e 19--22 1987

Centrifuge sampling

b~

275 December 19-20, 1986, and November 12-15, 1987. These are characterized by somewhat steeper hydrographs with single peaks. The antecedent flows ranged from 1.69 to 2.25mas -1 whereas the maximum flows varied between 6.37 and 19.2 m 3s-1. It is not ew or t hy t hat the October 23-25, 1986, event which had a maximum discharge of only 6.37mas t produced suspended sediment concentrations of nearly 700 mg 1- 1, nearly as high as the other two events which have much higher maximum flows. This shows the potential effect of rainfall intensity which is mirrored by the higher AQ/AT values. As noted above, two of the events, March 26-30 and June 6-12, 1987, are distinct in that the hydrographs are characterized by two peaks. These are treated as two separate events for the purpose of the model so that the variables can be reset for the second rise of the storm. However, the z2-test is processed for the entire period of each of these events, thereby demonstrating the ability of the model to deal with a multi-peaked hydrograph as long as the variables are reset at the new antecedent flow. The three events sampled with the centrifuge, October 23-25, 1986, June 19-22, 1987, and November 12-15, 1987, have a limited number of sampling points. Consequently, a measured suspended sediment concentration curve c a nn o t be produced and the goodness-of-fit test is not possible for these three cases. Two of these events, October 23 25, 1986, and November 12 15, 1987, are relatively simple hydrographs. The J u n e event, in contrast, characterizes an event of long duration and irregular flow, which resulted in several peaks in the hydrograph. This event is also unique in t hat it had the highest Q0 (12.6mas 1) of all of the storms. This was a result of a number of events preceding, in rapid succession, this storm. Figure 4a h shows the hydrographs and the measured and simulated suspended sediment concentrations. Intuitively, it appears that the two suspended sediment c o n c e n t r a t i o n curves for Fig. 4a e are similar. A hypothesis was established, using the z2-test (significant level a = 10%, lower limit Z~ = Z~l ~/2) and upper limit Z~ = Z(~/2~, 2 to determine whether the two sample variances are from the same population. The results of the tests, for the storms sampled with the ISCO sampler, are shown in Table 1. As noted above, this has not been feasible with the limited number of samples available from the centrifuge. The ratios of the two variances (as Z2) are tested against the critical values for the given degrees of freedom with a confidence probability of 95%. It is noted th at the test suggests t hat the two curves are similar in shape in all cases. Despite the similarity between the curves, there is obviously a phase lead factor in the actual suspended sediment concentrations relative to the hydrograph, which is not being reproduced by the model. This is most notable in Fig. 4b, d, e and f. There is no evident common factor determining this phase lead in these events and therefore it will be necessary to conduct furt her experimentation with the model to see whether the filter can be designed to take this factor into account. In these events, the sediment curves lead the hydrograph by 3-9 h. Although the shapes of the two curves are comparable

Q

~

Q

~

Q

Su$-sed (mgl

')

o~ o~ ~o ~o

o~ o~

o~

o

o

o

o

S~s-sed (mgl -~) o

o

o

o

o

o

o

o

o 2 i

~r

o (~s-')

o (~';*-')

Sus-s~d (mgl -*)

,

;

,

,

i

Sus-se~ (mgl -~)

t

,

~-T----

i

i

1

~

I

i

i

, Z

9Lg

0 14:00

100

2OO

~0

500-

,

,

,

O:(X)

. . . .

,

,

,

Oct 2 3 - 2 ¢

1987 ( 3 h )

0:00

i

1987 ( 3 h )

JunO7-10

"' " " " ' " ' ,

°°l";:""

....

o:oo

600 "

700

0

40,

80-

}20 -

160

,

14~0

..........

calculated

-.. ."'."-,.,.,

....

OOO

2

25

3

3.5

4

4.5

5

5.5

0

.

,

,

,

.

,

.

/y/'

(

(h)

12:00

o

100

20O

300

40O

500

6O0

700

900

so -~

70

8OO

C

i

8O

90

6

o

-

C

]00

110

6.5

2

3

4

5

6

8

9

10

11

12

13

1 ,

.

.

.

i

.

12:00

,

*

i

. . . . . . (3h)

12:(30 Nov 1 2 - 1 5 . 1 9 8 7

. . . . . . .

.

.

................ ,

22!00 . . . . . Jun. 1 9 - 2 2 1987 { 3 h )

measured

,

,

12:00

I

'

'

Q calcu~oted

22!00

-,.

2

4

6

8

12

14

16

18

12.5

13

135

14

14.5

t5

155

16

~65

17

o

~

o

Fig. 4. Hydrograph and measured and calculated suspended sediment concentration curves (ISCO samples) for storm events of (a) October 25 27, 1986, (b) December 18-20, 1986, (c) M a r c h 26-30, 1987, (d) May 15-18, 1987, (e) J u n e 7 10, 1987; and (centrifuge samples) for storm events of(f) October 23-24, 1987, (g) J u n e 1~22, 1987, (h) November 12 14, 1987.

~

i

T

2OO

240

28O

278 a n d t h e model is c a p a b l e of p r e d i c t i n g the m a x i m u m s e d i m e n t c o n c e n t r a t i o n w i t h i n a n event, t h e i n a b i l i t y to a d j u s t for t h e lag c o n s i s t e n t l y is u n f o r t u n a t e . T h i s lag f a c t o r is i m p o r t a n t in the q u a n t i f i c a t i o n of s e d i m e n t a n d s e d i m e n t a s s o c i a t e d p o l l u t a n t loads in t h a t e s t i m a t e s u s i n g the model o u t p u t would frequently pair the maximum sediment concentration with a much higher d i s c h a r g e t h a n h a d a c t u a l l y occurred. C o n s e q u e n t l y , the loads c a l c u l a t e d for the i n d i v i d u a l s t o r m s w o u l d be o v e r e s t i m a t e d . T h i s s i t u a t i o n is typified by the M a y 15-18, 1987, event. In this case, a fairly steep rise in t h e h y d r o g r a p h coincides w i t h a n a l r e a d y h i g h b a s e flow (5.34m~s 1) in the s p r i n g s e a s o n to r e s u l t in the s e d i m e n t c u r v e p e a k leading the h y d r o g r a p h p e a k by 7~8h. A l t h o u g h this l o a d - e s t i m a t i n g e r r o r could be c r i t i c a l for i n d i v i d u a l storms, the g e n e r a l c a p a b i l i t y of the model to a p p r o x i m a t e e v e n t s s u g g e s t s that: a n n u a l loadings c a l c u l a t e d u s i n g t h e model o u t p u t would n o t be s e r i o u s l y in error. T h u s for o u r purpose, t h a t of e v a l u a t i n g the r e l a t i v e i m p o r t a n c e of stormd e p e n d e n t r i v e r b a s i n s to t o t a l a n n u a l loads to a r e c e i v i n g w a t e r b o d y , the filter p r o v i d e s an a c c e p t a b l e a p p r o a c h for e s t i m a t i n g s e d i m e n t c o n c e n t r a t i o n s d u r i n g u n m e a s u r e d s t o r m events. T h e definition of t h e p a r a m e t e r s a n d filter f a c t o r s r e m a i n confined to the d a t a n o w a v a i l a b l e for t h e V e n o g e River. E v a l u a t i o n u n d e r a d d i t i o n a l e v e n t s is w a r r a n t e d , e s p e c i a l l y for the high-flow s n o w m e l t e v e n t s w h i c h o c c u r u s u a l l y in c o n j u n c t i o n w i t h a p r e c i p i t a t i o n event. F u r t h e r , it is n e c e s s a r y to d e t e r m i n e t h e a p p l i c a b i l i t y of this model to o t h e r rivers; b o t h t h o s e w h i c h are s i m i l a r to the V e n o g e R i v e r a n d t h o s e w h i c h a r e not, to assess its g e n e r a l applicability. ACKNOWLEDGEMENTS We t h a n k W. S t r y j e n s k a a n d o t h e r s of t h e I n s t i t u t e F.-A. Forel w h o assisted w i t h the s a m p l i n g w o r k . F u n d i n g for this w o r k w a s p r o v i d e d by C I P E L ( C o m m i s s i o n I n t e r n a t i o n a l e p o u r la P r o t e c t i o n des E a u x du L ~ m a n c o n t r e le Pollution). We also t h a n k Dr. A. E1 S h a r r a w i for helpful r e v i e w c o m m e n t s .

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