ANALYSIS OF THE SCOTT RIVER FLOW IN 1989 (WEST SPITSBERGEN)

Józef Paszczyk, Zdzisław Michalczyk, Institute of Earth Sciences Maria Curie-Sklodowska University Lublin, Poland Stefan Bartoszewski Wyprawy Geograficzne na Spitsbergen UMCS, Lublin 1991 ANALYSIS OF THE SCOTT RIVER FLOW IN 1989 (WEST SPITSBERGEN) In the years of 1986-1990 the field works were carried out in the Bellsund region on Spitsbergen within the Geographical Expeditions of Maria Curie Skłodowska University. They covered, among others, studies on hydrological relations in the Calypsostranda region situated on the forefield of Scott and Renard glaciers (Bartoszewski 1987, 1988, 1989, Michalczyk 1990). The article presents the detailed analysis of the Scott River flow recorded during thefield investigations from July 5 to September 8, 1989. The flow extent was controlled by the water-gauge station situated close to the river estuary into the Fiord Recherche. The results of the continuous observations of water flow intensity are included in the two-hour list in Fig. 1. They were prepared from the constant water level recordings on the limnigraph, read out in 12 terms corresponding to even hours of a day and night. From the calculated consumption curve the hour water levels were changing in to their corresponding flows. Besides the hydrometrie information, the data concerning the air temperature course and rainfall distributions were used in the numerical analysis (Czaban 1990). The Scott River basin covers the area of 10.125 km 2, out of which 5.8 km 2 is occupied by the Scott Glacier filling up the mountain valley situated between Bohlinryggen and Wijkanderberget ridges (Bartoszewski 1988). The lower part of the basin is found within the lifted marine terraces of differentiated hypsometry built of glacial, fluvioglacial and marine silts. Their roof is made of loose Quaternary marine silts. In some places Paleogenic formations from Aspelintoppen shaped as sandstones can be seen (Dallmann et al. 1990). The rocky background is made by Hecla Hoek formations occurring mainly tillites with regroupments of quartzites, fyllites and limestones (Flood et al. 1971, Dallmann et al. 1990). Waters flowing from the Scott Glacier form in its forefield a standstill lakelet, the waters out of which are carried away along a deep ravined river bed cutting a belt of frontal moraines. Two periglacial tributaries get the main river below the ravine. The middle an lower course of the Scott River is of a glen river character with large and frequent profile changes (Bartoszewski 1989, Michalczyk 1990). 97

In the course of Scott flow in 1989 there were periods of water circulation specific conditions. In thefirst spring phasś of the observations, theflows were at 1 m 3 s" 1 and were characterized by a weak ablation rhythm. From July 10 an increase of flow caused by higher air temperature and intense glacier melting could be observed. Next days were characterized by significant day and night flow fluctuations. This rhythm was disturbed by intense rainfalls at the beginning of August. During the sudden rainfall increase the maximum flow value was 8.62 m 3 s" 1 to which a unit flow 850 1 s" 1 km 2 corresponds, (Michalczyk 1990). During the freshet 30 mm water were carried away from the whole basin while the rainfall was only 17.4 mm. In the lowland tributary, the surface flow was 4 mm and in the mountain region it was 50 mm. Another significant rainfall freshet took place in the hydrologie autumn during the last days of the field studies. It occurred in the period of intense cyclone circulation when on 4 and 5 September there were 28,3 mm of rainfall out of which only 20.0 mm were carried away (Michalczyk 1990). During the polar summer, from 10 July to 25 August, proglącial water flow from the Scott basin was 1.9 m 3 s" 1. It corresponds to theflow index of 762 mm and the unit flow 188 1 s - 1 km 2. 1150 mm flow away from the mountain part of the basin and 150 mm from the lowland part (Michalczyk 1990). Throughout the active hydrologie period the mean Scott flow determined in an approximate way from the trend equation course expressed by the second degree polynominal (Fig. 1) was about 1.32 m 3 s"1. The beginning and end of river functioning was determined by the theoretical points of intersection of extreme branches of trend parabolas with the zero level of flow in the second decades of June and September. The estimated total flow from the Scott basin expressed in the form of a water layer was over 900 mm throughout 1989 which corresponds to an ice layer of 1 meter thickness. The characteristic feature of the Scott River system is the evident day and night rhythm of flow changes as a results of day and night temperature fluctuations. This can be seen in the course of the autocorrelation function of the examinedflow series (Fig. 2). The mean day and night rhythm of flow as well as of temperature (Fig. 3) can be presented in the form of sinusoid (Collings 1969): X = X + A [sin (Ph + F)] where: X is the flow in m 3 s~l or air temperature in 0 C, h is the given time of day and night, X is the mean day and night value of flow or temperature, A is the sinusoid amplitude, F is the cycle phase expressed in radians, P is the process angular frequency, also called pulsation. The terms of flow and temperature maxima (y m a x = у + A) and minima (Утт = У A) can be expressed as follows: 98

hmax - 24 + [(71/2) F]P h m i n = (3/2) к FP The extreme flow values take place at 6 a.m. and 6 p.m. local time. They are shifted by 4 hours in relation to temperature extremes occurrence. Similar values of shifting can be seen in diagrams of intercorrelation functions of temperature and flow (Fig. 4). Great dependence of water flow from the glacier on air temperature qualifies for seeking correlation and regression relations between these two processes. The data from the meteorological station in the maritime lowlands was used for the analysis of their interrelations. The periods in which glacier melting was caused by heat coming from rainfalls were neglected. The attempt was made to compare mean values of flows and temperatures shifted in relation to each other by 4 hours in the periods of 2, 12, 24 (Fig. 5) and 48 hours. It was stated that independent of averaging values, the correlation coefficients were very high from 0.90 to 0.97 and the calculated parameters of the equations almost constant-included in the range 0.387-0.395. As a result, for all mean periods, the flow - air temperature relation (within their positive values) with an insignificant error can be presented as follows: X = 0.39 t', X is the flow in m 3 s" 1, t is the air temperature in C. The proportionaly coefficient in the equation shows that on the average the air temperature change by ± 1 C increases or decreases the Scott River flow by 0.39 m 3 s" 1. The characteristic night and day and seasonal changes flow from the Scott basin during the hydrologie summer can be analyzed in the category of hazard processes and described by means of time sequence models of ARIMA type (Autoregressive Intergratted Moving Average Process) expressed by the general equation of the form (Box, Jenkins 1983): <pp (B) F p (Bs) V d V? X h = q (B) 0 Q (Bs) a h where: X h is the flow, a h is the hazard process of the zero mean and ended variance (h = 0, ± 1, ± 2). The symbols cpp (B) and^fp (BS) are the corresponding autocorrelation operators of the order p and P, V d and Vfare the difference operators of the order d and D bringing the analyzed flow to stationary. 6 q (B) and 0q(B s ) are the mean mobile operators of the order q and Q. The index s means the fluctuation period expressed as s = 12-multiple of measurements made every two hours. 99

For identification of ARIMA models there were determined the values of flow autocorrelation (Fig. 2) and partial autocorrelation of flows (Fig. 6) functions. In diagrams the broken lines are used to mark the critical area with 95% realiance. The flow autocorrelation function was observed to be decreasing with slowly expiring sinusoidal oscillations, however, the partial autocorrelation function assumes the values exceeding the critical area for h = 1. Such arrangement of both functions indicates the autoregression process of the first orderflows or the mixed process taking into account the cyclic component because of sequencfe repetition of some values of both functions for significant time shifts (delays). Finnaly to describe changeability of flow from the Scott basin, of many alternative models, there was chosen the simplest solution being a combination of the autoregression model of the first order and the temporary component expressed by the mean mobile determined for the measurements distant by h = 12: (1 0. 2 1 1 8 В ) (1 В ) (1 B 1 2 ) X h = (1 0. 7 4 6 2 B 1 2 ) a h, where X h is the natural logarithm of flow amount. Hypothesis of the established model correctness was checked by means of statistics x2, taking into account the estimate of autocorrelation of the model rests and white noise values. The measure of statistics x 2 was 13.854 with probability of assuming higher values p = 73,2%. Deviation of the white noise was 0.10063. All model parameters were significant at the level of a = 0.01. The additional verification of the equation was checking the condition of model rests non-correlation by the autocorrelation function (Fig. 7) and the cumulated periodogram (Fig. 8). The method of a cumulated periodogram is based on the rules of adding the power spectrum quantity of white noise in the frequency range of 0-0.5 cycle and their statistical comparison with corresponding coordinates of addition in relation to the power spectrum quantities established for the rests of ARIMA model and calculated from the real observations of the flow. As there were no significant deviations in the course of both cumulated curves of the periodogram and based on all statistical criteria analyzed it can be said that the established model of the flow time sequence of the Scott River was chosen correctly. In its developed form, the ARIMA model allows for flow calculations in any observation time according to: X h = 1.2118 Xh_, 0.2118 Xh_2 + X h _ 12 1.2118 X h _ 13 + 0.2118 X h _ 14 + + a h т - 0. 7 4 6 2 a h _i2 where X h is the flow natural logarithm, h is the time step of 2 hours, a h is the hazard process of zero and finite variance. 100

Application of ARIMA models makes the analysis of the observation material easy, allows for completing the measurements data and helps to give the short term forecast of water flow from the glaciated basins of Spitsbergen in the period of active hydrologie season. Translated by Maria Charmas REFERENCES Bartoszewski S., 1987: Dynamika odpływu powierzchniowego w zlewniach rzek lodowcowych Scotta, Biomli i Tjórn podczas lata 1986 r. (Spitsbergen). XIV Sympozjum Polarne, Lublin Bartoszewski S., 1988: Warunki kształtowania się odpływu w zlewni rzeki Scotta (Zachodni Spitsbergen). Wyprawy Geograficzne na Spitsbergen, UMCS Lublin Bartoszewski S., 1989: Charakterystyka odpływu ze zlewni lodowca Scotta (Zachodni Spitsbergen). Wyprawy Geograficzne na Spitsbergen, U M C S Lublin Box G.E.P., Jenkins G.M., 1983: Analiza szeregów czasowych. PWN, Warszawa Collings M. R., 1969: Temperature analysis of a stream. U.S. Geol. Survey Prof. Paper 650 В Dallmann W. K Hjelle A., Ohta Y Salvigsen O., Bjornerud M.B., Hauser E.C., Maher H. D., Craddock C., 1990: Geological map of Svalbard 1:100 000, Sheet B 1 1 G Van Keulenfjorden. Norsk Polarinstitutt, Oslo Flood В., Nagy J., Winsnes T. S., 1971: Geological map of Svalbard, Sheet 1G, Spitsbergen southern part. Norsk Polarinstitutt, Oslo Michalczyk Z., 1990: Hydrographical characteristic of Calypsostranda. Wyprawy Geograficzne na Spitsbergen. UMCS Lublin. STRESZCZENIE W opracowaniu poddano szczegółowej analizie przepływy rzeki Scotta zarejestrowane w czasie badań terenowych trwających od 5 lipca do 8 września 1989 r. W całym czynnym okresie hydrologicznym odpływ rzeki Scotta, określony z przebiegu równania trendu, wyrażonego przez wielomian drugiego stopnia, wynosił 1,32 m 3 s" 1. Początek i koniec funkcjonowania rzeki został wyznaczony teoretycznie przez punkty przecięcia skrajnych gałęzi paraboli trendu z zerowym poziomem przepływu. Momenty te przypadają na drugie dekady czerwca i września. Silne uzależnienia odpływu wody z lodowca od temperatury powietrza upoważnia do poszukiwania związków korelacyjnych i regresyjnych zachodzących między tymi wartościami. Formułę zależności przepływów (x) temperatura (t) powietrza, w zakresie dodatnich wartości temperatury, można przedstawić wzorem: x = 0,39 t. Współczynnik proporcjonalności wskazuje, że zmiana temperatury powietrza + 1 C zwiększa lub zmniejsza przepływ rzeki Scotta o 0,39 m 3 s" 1. Charakterystyczna dobowa i sezonowa zmienność odpływu ze zlewni Scotta może być analizowana w kategoriach procesów losowych i opisana przy pomocy modeli ciągów czasowych typu ARIMA. Stosowanie modeli A R I M A ułatwia analizę materiałów obserwacyjnych, pozwala uzupełniać brakujące dane pomiarowe oraz może stanowić podstawę do formułowania krótkookresowej prognozy odpływu wód ze zlewni glacjalnych Spitsbergenu w okresie czynnego sezonu hydrologicznego. 101

X h = 1 A 8 0 7 2 + 5. 3 0 6 5 8E - 3h - 9.50815E - 6 h 2 1600 8 SEPTEMBER 1989 Fig. 1. The Scott River flows (Q - the chosen time) and trend equation (X h ). -Q5H о 40 80 120 160 200 Fig. 2. The autocorrelation function of Scott River flows: r - the correlation coefficient, h - hours 102

T h = V219 + 0.701 (0.261 В h+v3527) h 1 1 1 1 1 1 1 1 1 1 1 1 0 2 4 6 8 10 12 14 16 18 20 22 24 В Fig. 3. The day and night courses of air temperature (A) and flow (B).

1-1 г. 054 - j i IluIZT^ ~ ТТ п ~ п ~ 0.5 Ч 10 20 "зо~ h г 50 Fig. 6. The function of partial autocorrelation of Scott River flow. Fig. 8. The cumulated periodogram of ARIMA model deviation (B). Ю5

п - и - - U П - "О" i i ^' 11 i i ТТ",,,, -Г 10 Г" 20 j 30 М) lb и г. 12 18 24 30 I 36 Fig. 7. The autocorrelation function (A) and partial autocorrelation of rests of ARIMA model deviation (B). 106