the application of time.series modelling to short.term...

Forest Hydrology and watershed Management - HydrologieForestiere et Amenagement des Bassins Hydrologiques{Proceedings of the Vancouver Symposium, August 1987, Actesdu Co11oque de Vancouver, Aout 1987):IAHS-AISHPub1.no.167,1987.

The application of time.series modelling toshort.term streamwater acidification in uplandScotland

S. J. LANGAN

Dept. Civil Eng. Imperial College, London SW72BU, UKP. G. WHITEHEADInstitute of Hydrology, Wallingford, Oxon OXlOBBB, UK

ABSTRACT Hourly discharge and stream pH have beenmodelled on 3 basins differentiated by land-use. Thebest-fit model for each of the basins has the same simplestructure (first order autoregressive moving averagetype) identified from a calibration run on one month ofhourly observations. The results show that a moorlandbasin and a forested (8 year old Sitka Spruce) basin havebroadly similar water quality. Results from the third,part-forested, basin illustrate how retrogressive manage-ment techniques (e.g. liming) may ameliorate surfacewater acidificati9n, both in terms of magnitude andduration. While the model is not particularly powerfulin terms of predictability, it has been useful in iden-tifying the differences between the 3 basins in terms ofwater quality. The modifying influence of land manage-ment techniques on one of the basins is particularlypertinent to the short-term solution of streamwateracidification.

Hodelisation par series temporelies de I 'acidification acourt terme des cours d'eau dans Ia region montagneused'Ecosse

RESUME On a modelise les debits horaires et le pH del'ecoulement de 3 bassins soumis a differentes utilisa-

tions des terres. Pour chaque bassin, le meilleur modelea une structure simple similaire (type de premier ordre,autoregressifa moyennemobile)qu'ona identifieparetalonnage sur un mois d'observations horaires. Lesresultats demontrent que les eaux d'un bassin de bruyereet d'un bassin boise (sapins Sitka ages de 8 ans) ont desproprietes similaires. Les resultats du troisieme bassinpartiellement boise montrent comment les techniquesd'amenagement retrogressives (par exemple, chaulage)peuvent ameliorer l'importance et la duree del'acidification des eaux de surface. Bien que le modele

presente ne soit pas un outil de prediction efficace, ila ete utile dans l'identification des differences entre

les trois bassins en fonction des proprietes de l'eau.75

76 S.J.Langan & P.G.Whitehead

L'effet modifiant des techniques d'amenagement sur un desbassins offre une solution a court terme al'acidification des cours d'eau.

INTRODUCTION

In the study of water resources the ability to simulate water qual-ity is desirable from two aspects:

(a) To improve our understanding of the processes and dynamicswithin drainage basin systems;

(b) To predict or forecast with the view to managing a basicresource.

Simulation may help to improveidentifying relationships betweenunder study. In identifying suchations of model types.

Where modelling is undertaken with a prior knowledge of theprocesses and their theoretical behaviour a partial system synthesisapproach (Amorocho & Hart, 1964) will provide the appropriate model-ling approach. The use of such models is common in the hydrochem-ical literature (Christophersen et al., 1982; Lam & Bobba, 1984;Galloway et al., 1980). A classic example in the use of thesemodels is provided by the North American ILWAS study (IntegratedLake-Watershed Acidification Study). This model simulates thephysical and chemical transformations occurring in watersheds andlakes, as induced by acid-deposition and internal acid generation.The ILWAS model is highly complex using physical, chemical andbiological theories to derive partial differential equations todescribe the system under investigation. These models can be usefulin explanation and prediction but lose their ability to predictaccurately when system synthesis is incomplete or when some of themodelling assumptions are not met (see Christophersen et al., 1982).This type of model will also be inappropriate if there isinsufficient data to characterise the relevant processes in themodel.

An alternative approach to this system synthesis is the use ofsystem analysis (Amorocho & Hart, 1964). In a system analysisapproach the system processes are replaced by mathematical functionswhich do not attempt to represent the detailed mechanistic behaviourof the system. Where a system is exceedingly complex or where thereis an inadequate knowledge of the processes governing the input-out-put relationship, system analysis can provide a more appropriatemodelling technique. This black-box approach offers relativelysimple but robust models which may provide some initial understand-ing of a system. They are particularly useful where input-outputrelationships are of primary concern. Consider the processes in anupland drainage basin which interrelate to determine the hydrochem-istry of the effluent stream during a storm. When the basinreceives rain it will respond according to the internal structure ofthe basins hydrochemical processes. This internal structure whichgoverns the input-output relationship filters the input signal intoan output response. This typ~ of model has been successfully ap-plied to a variety of environmental studies (e.g. Lai, 1979; White-

our understanding of a system bydifferent components in the systemrelationships there are two gener-

Time-series modelling of stream acidification 77

head, 1979). The development of such models may in turn give aninsight to the processes to which a system synthesis approach may befitted.

The object of this report is to model streamwater acidity overstorm periods on 3 drainage basins with different land-use.

THE TIME-SERIES TECHNIQUE

In this investigation both the series input and output of the systemis known, what remains is to identify and estimate the transferfunction; where the system (drainage basin) receives a stimulus(discharge) and responds to it according to the internal structure(hydrochemical processes) of the system which determines the systemoutput (acidity, expressed as hydrogen ion concentration, H+). Thisflow-acidity relation, represented by a transfer function has nophysical realization, it is a mathematical approximation which willbehave in the same way as the system behaves over the same timeincrements as that of the data. In the single input-single outputmodel applied here there are two components, a process model and anoise model. The observed input Ut causes most of the output varia-tion (Yt), however superimposed on this process model is astochastic input (at) which arises from random disturbances to thesystem. Such uncertainties in the measurement of pH. The generalform of the model is:

Yt = Xt + at (1)

Where the output at time t (Yt) is given by the sum of the deter-ministic (Xt) and stochastic (at) outputs and where Xt is assumed tobe generated by an autoregressive moving average (ARMA) discretetime model (Box & Jenkins, 1970). Further details of the model andtheory are given in Young (1974) and Whitehead et al. (1986). Inthis investigation the model was applied through a computer aidedpackage, CAPTAIN (Venn & Day, 1977).

STUDY AREA

Loch Dee has a remote setting in the Galloway Hills in S.W. Scot-land. The drainage basin is typical of much of upland Scotlandcomprising part of a glaciated valley which has steep and ruggedsides with a broad valley floor. The climate is best described ascool and wet. Annual rainfall is 2200 mm. The drainage basinconsists of three major sub basins, the control, Dargall lane (2.1km2) with a vegetation community dominated by grass, heather andbracken. The White Laggan (5.6 km2) with 27% afforestation, plantedin 1976 with Sitka Spruce (Picea sitchensis). This basin has hadsome management techniques employed to improve water quality; theseinclude the backfilling of forest drains, liming and the creation ofa 'buffer zone' between the trees and major watercourses. The third

basin, Green Burn (2.5 km2) has 70% of the area plantedwith sitkaspruce of similar age to those of White Laggan.


The data used in this investigation consist of hourly streamdischarge measured by a float operated chart recorder on calibratedstretches of each stream and continuous pH measurements from probesimmersed in the stream coupled to continuous recorders on the streambank, both measured over a period of 5 months.

APPLICATIONS AND RESULTS

A CAPTAIN analysis was run on hourly data from Dargall Lane andGreen Burn for the month of November 1983 and for White Laggan overthe month of March 1984. The general form of the model was the samefor all 3 basins, a first order ARMA model. ie. one autoregressiveand one moving average parameter. The general form of the model maybe written as:

Yt = a1Yt-1 + boUt (2)

where Yt is the Hydrogen ion concentration (~eq H+l-1) and Ut is thestream discharge (m3s-1)at time t. Whilst there was no time lagfor Dargall Lane it was necessary to incorporate a time lag of 1hour for both the White Laggan and Green Burn models. The specifi-cation of the three models are given in Table 1. From the modelparameters it is also possible to calculate the systems mean re-sponse time (T) and the gain of the system (G). Mean response time,equation (3) and system gain, equation (4) have been defined byWhitehead et al. (1986):

T = I/ln(-a1) (3)

G = boll + a1 (4)

A system gain of 2.0 would imply (for this study) that for every1 m3s-1 increase in discharge there would be an increase of 2.0 ~eqH+ 1-1. The value of these two parameters for each model is givenin Table 1. Also included in Table 1 are % variance values; %

variance in this case referring to the amount of variance in theobserved H+ concentration which can be accounted for by the model.

Figures 1 to 3 show graphically how the models match the observeddata. In both the graphical output and in the variance figures(Table 2) some degree of dissimilarity is evident. This is due inpart to the stepped nature of the input data compared to thesmoothed model output data. The stepped appearance of the observeddata is due to the conversion of discrete pH values (to one decimalplace) to H+ concentration. However, in calculating model outputvalues the CAPTAIN package uses fractions of the input and output,this has the effect of smoothing the model output and producing acontinuum of H+ values.

Figures 4 to 7 show how well these models were able to predictstreamwater acidity over additional data sets, under differingconditions. In general the models show a very good agreement be-tween the observed and modelled series. The main areas of deviation

TABLE 1 Summary details of the models

Model parameters and theirstandard deviation

al S. Dev. bO S. Dev.

....f-<.

E3(J)I(/)(J)I-jf-<o(J)(/)

E30Q,(J).....

.....f-<o

::J

\Q

0

t-t,

(/)

c-t-

I-j

(J)

~E3

~C)f-<.

Q,f-<.

t-t,f-<o

C)~

c-t-f-<o

0::J

]~

Dargall Lane -0.7652 0.0496 8.7489 1.822

White Laggan -0.7864 0.0767 0.8697 0.0256

Whitehead et al., -0.680 0.012 0.659 0.022(1986)

Green Burn -0.8208 0.0271 7.7390 1.0614

Time delay in hours

::, Mean respone tim: in hoursPercent varlance ln H+ accounted for by Q

TD' Model MR" PV'"h h Gain %

0 Yt = -0.7652Yt-l + 8.7489Ut 3.7 37.3 81

1 Yt =-0.7864Yt-l + 0.8697Ut-l 5.2 4.1 94

0 Yt =-0.680Yt-l + 0.6599Ut 2.6 2.1 93

1 Yt =-0.8208Yt-l+ 7.7390Ut-l 6.0 43.2 86


are at the extremes of concentration. From Table 2 and from thefigures the influence of varying the prediction time is evidentlyseen. Also apparent from Table 2 is the consistently better fit ofthe data given by the Green Burn model in comparison to the DargallLane model, these differences are further discussed in section 5.

Input Series (Discharge)

I ... a

..,E..e>e.<:" 0.. Nis ~

°,

~1

i~i

Output Series (Acidity)

J

7tT..2>-

+:r

600

Model1ed and - Ob..rY8d

Ob.ervod Acidity ... Modelladr.i. ..fl.

.',

j'1 r i'.'\ '

V I.I \1" I:' "'" .: l

R )1 '-, , ."" '.! ~ . ~ ,

Iv '", ~r. ! ~"

.!!

'I~I" J~ T

'"T

'50 200

7- ~tT..2>-

+:r "

',-

Hourly Values

FIG.1 Observed input, output and modelled series forDarga11 Lane, November 1983.


Examination of the Dargall Lane simulation for November 1983(Fig.l) shows that the model adequately explains the rising andfalling levels of H+ during storms. It is unable however to repli-cate the first storm peak. Note that the subsequent storms duringNovember all reach the same 'acidity-plateau' concentration, theseobservations are in direct contrast to the Green Burn simulation for


.:- I'.. II

"~ 'II 1\

.c

I

I I

~ g \ i\\ I .\j

a JOO

Output Series (Acidity)

r:r="- 0-. ~

+:I:

III

tI

~

\1\~

T'00

T<0°

r600

r'00

~-J

'00

Modelled and Observed Acidity

Observed

Modelled

'-r:r 0" 0

~ ~+:I:

~ \if f~ir l:i~ i'j: f

r\J

\

~~.~

c 30° '00

Hourly Valu85

FIG.2 Observed input, output and modelled series forWhite Laggan, March 1984.


November 1983 (Fig.3). On Green Burn the same 4 storm peaks gener-ate concentrations of acidity proportional to the amount of flow.Again the model adequately describes the general pattern of observedacidities but fails to predict the first acidity peak.

The model fit during January 1984 for both Dargall Lane and GreenBurn (Figs 4 & 5) are very good with the models accountinq for a

~

~i

j'C - i0" ~I

2" ~l

+~ ,~

L\, A

, ''V'" ;" '00"

'. "....

E~

2'..~" -. "is 0

'i0" C~ 0~'+~

~l


'" '"

Dutput Series (Acidity)

000

II~I III

rL- '"

I

J~J

-J

.

\

' Modelled and Observed Acidity

I.ir . - Observed

I \ Q J\.. ...;. Modelled

I L; rt

\n! \ FU: . . .

I...W ~.. : \ r!: .I .. '.J..."""" .

I """'" ~............

) -~"""","""-'..J

00

Hourly Values

~O

FIG.3 Observed input, output and modelled series forGreen Burn, November 1983.

TABLE 2


Percent variance explained during different model runs

t

'~ ~l A~~ ~ r<.""

i! . ~,I l \ ,I'

\~l

1\.\~jl[~ (' I I~1,1[""'" \J I,

I5~

I1DO

I150

I200

I250

b

T300

FIG.4 Modelled streamwater acidity for Darga11 Lane,January 1984.

~

TGOO

FIG,S Modelled streamwater acidity for Green Burn,January 1984.

T700

November Januar y February March

Darga11 Lane 81 92 89 96

Green Burn 86 95 95 97

0"'D-q

""cc

I-CJ' 0

2-- N

+:I:

00

n

0"

e'


00

~"

'O

il- 0

... :;;, .! I\!',. :

~ ~~'~ r:'~ ~.-.: ~I :-'1..

.

'.< ,

.IJ... N~ ~-\:'I; I: 'I li1

'[ 1_\.~~~{ R! -

r I~Lc:~rl"l~ \ rV ~,J~~ If-nr ~ r .~~

0

,0

00

~

\ 00 200 300 400 500 500

FIG.6 Modelled streamwater acidity for Dargall Lane,February 1984.

"D ,

\.JLi I

I

{

0 I,~: \

,:- ~- ~ r~ ' ': :1.

.

~

.

~

.

' \rtv

\t\, ~i Y" f

.~ :11 .

~ -\!~~~ i

. ~ 500i ~ 000I I I 400' I I 300" I I 200;; < , ! 000

0

FIG.? Modelled streamwater acidity for Green Burn,February 1984.

wide range of acidity with different flow conditions. The majordifferences arise from the stepped nature of the original data.Predictions for the February storms after the first storm are lessreliable, the models overpredicting the observed acidity (see sec-tion 5). This gives an indication of the models' simplicity andsuggests that a 'factor' other than that described by the model isaffecting streamwater acidity during the latter part of February.Figure 2 shows the White Laggan model generated on data from March1984. The liming of White Laggan during 1983 had the effect ofeliminating a flow-acidity response in the basin until most of thelime had been flushed from the basin, a point which is discussed


further in section 5. However, by March 1984 a flow-acidity rela-tionship (albeit small) was restored, enabling the application ofthe model given in Table 1. From Fig.2 the model can be seen togive a good fit over a range of flow conditions in the White Laggan.This run again showed the versatility and power of the model underdiffering conditions. Comparison of these results with those from1982 (Whitehead et al., 1986) reinforce the nature of the relation-ship between discharged acidity in White Laggan.

DISCUSSION

Both the graphical output and Table 2 illustrate the efficiency withwhich the simple models fit the observed data. Such a good associa-tion suggests there is a close relationship between H+ concentrationand flow and the models appear particularly effective at forecastingacidity over storm periods.

During both November (used as the calibration period) and overthe predicted data series the percentage explained variance of theH+ flow relationship is consistently higher for Green Burn thanDargall Lane. This probably reflects a real difference in thehydrochemical processes of the drainage basins. The drains of GreenBurn promote a rapid flushing and evacuation of water from all partsof the basin. Dargall Lane however with its armchair like shape andpools of surface water will take longer to flush the acidity fromall parts of the basin. For the Green Burn November forecast themodel failed to predict the first Autumnal flush following a longdry spell (Fig.3). Whitehead et al. (1986), using the same model-ling strategy for data from Birkenes, Norway, noticed a similarfailure of the model in these circumstances.

Model failure also occurs during the February runs. From thecontinuous data it is not readily apparent why this is so.Reference to weekly chemical data also collected at Loch Dee, sug-gest that a sea-salt episode occurred at this time. During thisweek a high amount of H+ was exchanged and leached out of the basin(Langan, 1986). With the present simple model structure there is noattempt made to incorporate the role of sea-salt as this will have amarkedly different flow-acidity regime.

Comparison of the basins' mean response times and delay factors(Table 1) shows Dargall Lane to respond the fastest. This impliesthat for an increase in flow Dargall Lane will respond the fastestin terms of an increase in acidity. This reflects the poor abilityof the basin to buffer any acidity. The longer response times forGreen Burn and White Laggan indicate that these drainage basins areable to buffer some acidity. In the case of Green Burn this buffer-ing capacity is eventually exhausted giving rise to a high gain,whilst White Laggan is able to maintain an effective buffer againstacidity.

The results from the White Laggan model are interesting in thecontrast they provide to the other two models. The White Lagganmodel suggests the basin to be much better buffered than the other

two basins. The similarity between the two sets of results (forWhite Laggan) confirms the magnitude of the gain for the basin whichis markedly smaller than for either of the other basins. The dif-


ference in mean response times between the two studies is an indica-tion of the different antecedent conditions during the calibrationof the models.

A further contrast is provided by using the model to calculatethe discharge needed to produce a concentration of 10 ~eg H+ 1-1 (pH<5.0). Table 3 shows the results and the percentage of time theflows are greater than this value. This gives an indication of thetime each stream has an acidity> 10 ~eg H+ 1-1 (flow derived frommean daily flow duration curves).

TABLE 3 Flow-acidity duration figures

The implications of these results are interesting and primarilyreflect differences in the hydrology and geology of the basins. TheDargall Lane basin as already discussed has longer retention times,the water being transmitted through nutrient deficient peats whichmaintain streamwater acidity rather than neutralise any incomingacidity. During very high flows the presence of a scum on the watersuggests the acidity is of an organic form (fluvic acid). Thedischarge value to attain an acidity value of 10 ~eq H+ 1-1 isapproximately the same for Green Burn as Dargall Lane. In GreenBurn the percentage time of lower flows is greater than that forDargall Lane this being a result of the generally steeper basin andpreferential movement of water down the forest drains. This ex-plains the lower percentage of time in which higher acidities pre-vail.

The large discrepancy between White Laggan and the other twostreams may be partly explained by different flow regimes. However,there is little evidence to support this from flow duration curves.More plausible explanations are differences in geology or landmanagement. Weekly stream water chemical background informationintimates that the White Laggan has a greater ability to bufferstream acidity as given by the lower weekly mean and median H+concentrations. However, the differences between streams is notsufficient to account for the differences described here. The

background chemical data suggests that acidity related ions arecontrolled to a large extent by the flow conditions. Futhermore thebackground data are from above most of the lime in White Laggan andthe data presented here are from below all of the liming applica-tions. The most significant difference is thus likely to be ex-plained by the liming programme. Section 4 has already describedthe difficulties in establishing a flow-acidity relationship for

Station Threshold Discharge Time Exceeded

H+>lO !leq 1-1m3s-1 %

Dargall Lane 0.24 18

White Laggan 3.50 0.5Green Burn 0.30 15


earlier periods due to the applications of limestone powder. It islikely that the diminished effect of the limestone lasted over asubstantial period providing an effective buffer against streamwateracidity.

CONCLUSION

This report has shown the application of simple time-series tech-niques to be a useful tool in the initial analysis of field data.Secondly the analysis has provided important information on thedrainage basin dynamics in terms of response times and gain factorsand also indicated differences between the dynamics of sea-salt andacidic events. Preliminary evaluation of the effect of liming ofthe watercourses has also been provided. In combining the modelresults with drainage basin hydrology it has been possible to pro-duce H+ duration curves for the different streams.

ACKNOWLEDGEMENTS This work was carried out whilst S.J.Langan wasin receipt of a NERC grant with the University of St. Andrews (Dept.of Geography) and the Solway River Purification Board.

REFERENCES

Amorocho, J. & Hart, W.S. (1964) A critique ofhydrological systems investigation. Trans.

Box, G.E.P. & Jenkins, A.M. (1970) Time-Seriesing and Control. Holden-Day, San Francisco.

Christophersen, N., Seip, H.M. & Wright R.F. (1982) A model forstreamwater chemistry. Wat. Resour. Res. 18, 977-996.

Lai, W.P. (1979) Transfer function modelling: Relationship betweentime-series variables. Catalog 22. published by Geo-Abstracts,Norwich, England.

Lam, D.C.L. & Bobba, A.G. (1985) Modelling watershed runoff andbasin acidification. In Hydrological and hydrogeochemical mecha-nisms and model approaches to acidification of ecologicalsystems. Nordic Hydrological Programme Report No. 10, 205-216

Langan, S.J. (1986) Atmospheric deposition and water quality at LochDee. Unpublished PhD Thesis, Univ. St. Andrews.

Venn, M.W. & Day, B. (1977) Computer aided procedure for time-seriesanalysis and identification of noisy processes. Inst. ofHydrology Report No. 39.

Whitehead, P.G. (1979) Applications of recursive estimation tech-niques to time variable hydrological systems. J. Hydrol. 40,1-16.

Whitehead, P.G., Neal, C., Seden-Perriton S., Christophersen N. &Langan S.J. (1986) A time-series approach to modelling streamacidity. J. Hydrol. 85, 281-303.

Young, P.C. (1974) A recursive approach to time-series analysis.Bull. Inst. Math. Appl. 10, 209-224.

current methods in

AGU 45, 307-321.

Analysis. Forecast-

the application of time.series modelling to short.term...

Documents