Homage to Ramon Marga/ef; or Why there is such p/e asure in studyin g nature 21 (J. D. Ros & N. Prat, eds.). 1991 . Oec% gia aquatica. 10: 21-33

USING THE RESCALED RANGE ANALYSIS FOR THE STUDY OF HYDROLOGICAL RECORDS: THE RIVER TER AS AN EXAMPLE

1 1 2 JOAN ARMENGOL , SERGI SABATER , ANTONI VIDAL & FRANCESC 1 SABATER

Departament d'Ecologia. Facultat de Biologia. Universitat de Barcelona. Av. Diagonal, 4 Barcelona. 1 6 5. 08028 Spain

2 Junta d' Aigües. Abastament d' Aigua del Ter. Cardedeu. 08440 Barcelona. Spain

Received: November 1990

SUMMARY Water discharges into the Ter river have a strong influence on the water chemistry and on the dynamics of benthic cornmunities. For this reason, the temporal pattems followed by monthly runoff from 1954 to 1988 have been studied. From a statistical point of view, discharges show a very regular seasonal cycle, but in practice, climatic events introduce a source of variability on runoffs. To determine whether the series of 420 monthly discharges fluctuates with sorne regularity or at random, a rescaled range analilsis was made. The persistence (H) of the series of discharges was measured by the equation R / S = (T / 2) . In fact, H is a measure of the existence of clear trends or periodicities in the records of persistent stocastic processes. An additional measure of the persistence was obtained by means of the relationship D = 2 - H, where D is the local fractal dimensiono For comparative reasons, a random series of the same length, mean and standard deviation was generated and analysed in the same way as the Ter discharges were. The results were H = 0.68 and D = 1 .38 and H = 0.44 and D = 1.52, for the Ter data and for the random series respectively. These results show that the simulated series is a case of ordinary Brownian motion while the Ter discharges series has an intermediate value of persistence. Because of the value of H in the Ter series, there are periods whose values were estimated by the autocorrelation function and the periodogram of the series. The results show the existence of short fluctuations of 3, 6 and 12 months deterrnining the seasonality of the annual cycle, and large fluctuations with periods of 5.5, 8.6, 10.1 and 11.7 months which can be considered an expression of cycles of circa 7 and 1 1 years.

KEYWORD S: River discharge, fractal dimension, Hurst exponent, fractional Brownian motion, persistence .

. . . In the case of the landscape, rain fa lling at random and moving from each point of the surface along the steepest profi le, through eros ion and convergence of runoffs, generates a secondary "hydrologic" relief Will the fractal quality be preserved (MARGALEF, ? 1988)

INTRODUCTION seasonal c1imatic changes. Nevertheless, this pattern is usuall y modified by irregular In Mediterranen ecosystems , annual climatic fluctuations that cause sudden cyc1es are always described in terms of changes in their dynamics. Temporal ponds regular fluctuations associated with and rivers are among the most affected 22 ARMENGOL, SABATER, VI DAL & SABATER

ecosystems because they are mainly taking place (SCHINDLER, 1988; controlled by the periodicity and intensity ARMENGOL, 1988). of the rains. The tumover time of the water In rivers, strong rains increase the acts as a key factor in most of the processes intensity of the advective transport of solid

250

A 200

(") 150 S :::c

a 100

50

3

2

-2

-3

1955 1960 1965 1970 1975 1980 1985

3 I FIGURE 1. River Ter monthly discharges for the 1954 - 1988 periodo A) original series of data in Hm month- .

B) normalized series by Y = . C) the standardized sequence. XO.22 RESCALED RANGE ANALYS IS OF THE TER RIVER 23

and dissolved substances. Because of this, (backward steps) (MARGALEF, 1977). the spates act as huge inputs of external In the Ter river, water flow is the most energy that simplify the structure of the important factor determining the annual river. Most of the organisms are carried variability in the environmental conditions down and the chemical composition of (SABATER & ARMENGOL, 1986) and in water beco mes more similar along the ri ver. the dynamics of phytobenthic (SABATER The annual cyc1e of the cornmunities can et al., 1988) and macroinvertebrate (PUIG be seen as a sequence of populations with et al., 1987) communities. These relation­ short life span, well adapted to seasonal ships are well established for short periods changes but able to reconstruct the of time, such as the annual cyc1e, but for cornmunity after each event (MARGALEF, longer periods it is assumed that c1imatic 1980). If longer periods of time are fluctuations have a similar effect. If we considered, changes in the communities can accept these relationships for the Ter river, be seen as a sequence of periods of the long-term study of a record of organisation (secondary successions) hydrological events as large as possible can followed by quick disruptive episodes provide us with sorne insight into how it

A

3

2

3 B

2

o

-1

-2

-3

1955 1960 1965 1970 1975 1980 1985

FIGURE 2. Simulated sequence of 420 values (35 years x 12 months) nonnally distributed with the same mean and standard deviation as the data of figure lB . B) the standardized series. 24 ARMENGOL, SABATER, VIDAL & SABATER

functions from an ecological point of view. (MANDELB ROT & WALLIS, 1969); for For longer periods it is well established that this reason the existence of temporal the water discharges affect the chemical "cycles" has been studied here by mean s of pollution and the eutrophication levels in autocorrelation and spectral analysis. the reservoirs situated in the middle stretch of the river (VIDAL, 1977, 1987, 1989; ARMENGOL et al., 1986). MATERIAL AND METHODS Long-term records of hydrological events show erratic behaviour at different The record of hydrological data used in time scales (FEDER, 1988). These records this study is composed by the monthly can be studied by means of the rescaled discharges at Roda de Ter station from range analysis (HURST et al., 1965) and 1954 to 1988 (Fig. l). The data were characterized by an exponent H, called the obtained from the Dirección General de Hurst exponent, which is a measure of the Obras Hidráulicas gauging yearbooks persistence or "memory" of one series (D.G.O.H., 1972, 1973, 1974 and 1975; (MANDELBROT & VAN NESS, 1968). D.G.O.H., 1976) for the period 1965- 1977, MANDELBROT (1985) has shown that and were provided by the Servei Hidrologic this record is a self-affine fractal curve with de la Generalitat de Cata]unya for the a dimension D = 2 - H. For a practical periods 1954- 1 965 and 1978-1 988. perspective, the H exponent measures how Roda de Ter is situated in the middle far a temporal series of data departs from a stretch of the Ter river, 88.6 Km from the random series. source (river length 200,3 Km), where it 2 We have used a record of 35 years of has a drainage area of 1,476.8 Km (48. 11 monthly average discharges to analyse the % of the total). At this point the Ter is a persistence in the data sequence. The aim fourth order river. This station was selected of this study is to determine the temporal because it is situated aboye the reservoirs, trend followed by the series and how this thus excluding their regulatory effect, and key factor carried weight on the river at the same time the drainage area is large considered as an ecosystem. It is well enough to avoid the effect of sudden known that wet and dry years appear in discharge due to a local rainfall. clusters, the "Joseph effect" The hydrological characteristics of the

100

B A

75 75

..c::..., s:: o 8 50 "'...... 8 :r:

25 25

JF MAMJ JASo ND

FIG URE Average annual cycle showing the mean monthly discharges with confidence interval: A) rol' 3. 95 % Ter discharges (Fig. lB) and B) for random series (Fig. 2 A). RESCALED RANGE ANALYSIS OF THE TER RIVER 25

3 Ter are typical of the Mediterranean region, for the period studied was 4l.3 3.74 Hm ±3 but the proximity of the Pyrenees also has with ranges between 23 l.4 Hm and 0.70 3 an influence. The annual cyc1e is Hm , while, if the data are grouped by 3 characterized by two periods of maximum years, a mean of 554.6 35.7 Hm is ± 3 water flow; the first, in May (28.6 6.0 obtained with a maximum of 1180 Hm in 3 1 ± 3 Hm month- , p <0.05 for all the data of 1972 and a rninimum of 258 Hm in the this paper), is the highest due to the effect year 1957. of the spring rains plus the thaw of the The Ter hydrological record does not snow accumulated in the Pyrenees, while follow a Gaussian distribution (the kurtosis the second, in November is less important is 4.9 12 and the skewness is 2.001). For ¡ 3 - (17.6 ± 5.7 Hm month ) and exc1usively this reason, it is not possible to use many of due to the autumn rains. the parametrical statistical methods. In Two minima are situated in between, the order to transform the original data into a 3 I lower in August (1 1.4 2.6 Hm month- new series with a near normality ± d and the other in February (14.2 3.9 Hm distribution, a power transformation has 1 ± month- ). The average monthly water flow been sought by means of the Box & Cox

A

..c:: s:= 100 O E ...... en 50 8 ::c:

o 1955 1960 1965 1970 1975 1980 1985

100

B

..c:: 75 � t:: O 8 ...... en 50 E ::c:

O 1955 1960 1965 1970 1975 1980 1985

FIGURE 4. Mean annual water flow with 95 % confidence interval: A) for Ter series (Fig. lB) and B) simulated sequence (Fig. 2A). 26 ARMENGOL, SABATER, VIDAL & SABATER

method (BOX & COX, 1964). The c10sest RESULTS transformation to near normality of the data 0.22 is y = x . The new series (Fig. 1) has a Not all the rainfall runs immediately into mean of 2.112 and an standard deviation of the rivers; a certain amount of water 0.449, while the kurtosis and the skewness remains stored in the watershed and measure 0.058 and 0.028, respectively. contributes to the discharge in the driest Finally, the series used in the analysis has months. In the same way, during dry been standardized (Fig. 1). periods, the waterflow is higher than To illustrate how far the hydrological expected due to rainfall. As a first series departs from a random record, a new approach, there is a buffer effect on Gaussian series of the same length, mean discharge due to the storage capacity of the and standard deviation has been drainage basin. The rescaled range analysis constructed, but following a random walk measures the importance of this effect by process (Fig. 2). means of the Hurst exponent. For comparison of the two series the If X l , X2, ... XT is a series of monthly seasonal and annual distribution of the data discharges for a period of time T, it may be are shown in figures 3 and 4. characterized by the mean ex), the standard

15 B

10

-15 1955 1960 1965 1970 1975 1980 1985

FIGURE 5. Accumulated departures from (zero) mean for: A) the Ter hydrological data, and B) the random series. RESCALED RANGE ANALYSIS OF THE TER RIVER 27

deviation (S) and the range (R), which is generating this kind of curve for a given the difference between the maximum and value of H, although there are many other the minimum departures of the water flux ways of doing so (e.g. KIRKBY, 1987). (Xi) from the mean (x). Formally, . From the practical point of view, the case H = 0.5 is ordinary Brownian motion, and when 0.5 H :::; 1 the curves show a _ T < = X ) persistence x L ( i / T (1) as high as the value of H. i=l Finally, it has been demonstrated (MANDELBROT, 1985) that these curves are statistically self-affine scaled sets and i=l their fractal dimensions are not uniquely defined. In other words, there is no global T _ = (X -X) (3) fractal dimension, but there are local ones x(t) L i i=l and this is because the fractional Brownian motions are scaled in two variables; in this T T R(T) = Max X (i) - Min X (i) (4) case river discharge and time. For each case i=l i=l the fractal dimension is:

The rescaled range, RlS, IS well D = 2 - H (6) described by the empirical relation (HURST et al., 1965): At this point it is interesting to describe how figures 5a and 5b were obtained. We H R / S = (a T) (5) began with two series of data and after their transformation into fractional Brownian where H is the Hurst exponent. motions they had the appearance of profiles In figure 5 the accumulated departures or chains of mountains. In fact, these are from the mean, X (t), are plotted against generated irrespective of whether the data time for the standardized discharges of the series are real or simulated. Can we Ter river (Fig. 5a) and for the random consider that these profiles reflect series (Fig. 5 b). The figures obtained are something intrinsic or characteristic of the two cases of fra ctional Brownian motion, basin? and are generalized random functions, X (t), Obviously they cannot be considered as each one characterized by the exponent H, an idealized reconstruction because we which ranges between O and 1 have time series and the basin has a spatial (MANDELBROT & VAN NESS, 1968; plan. Nevertheless, our tentative answer is MANDELBROT, 1987). Fractional affirmative. Can the path of the runoff to Brownian motions are very useful to the rivers be imagined from figures 5a and describe time series followed by various 5b? The answer is yes, we only have to natural phenomena such as river discharge, imagine how the water runs through the rainfall, tree rings, distribution of varves in artificial landscapes of figure 5. The lake sediments or the frequency of sun random series (Fig. 5 b) shows a more spots (FEDER, 1988). This kind of motion abrupt relief, high peaks and narrow can also be used to generate new surfaces valleys, suggesting a quick response to like watersheds or continents, mountain rainfalls and low storage capacity. In other profiles, the shape of lakes or the form, words, there is no persistence or memory. number and distribution of islands in On the other hand, the real series (Fig. 5 a) archipelagos (MANDELBROT, 1975). has a more even profile, with large plateaus VOS S (1985) has introduced an algorithm and wide valleys, where water can be called successive random addition for retained or stored for more time, meaning 28 ARMENGOL, SABATER, VIDAL & SABATER

6 persistence or memory. 10 5 A In practice the Hurst exponent is 10

ca1culated fitting the power equation � . H 10 RJS=(aT) to a set of values of RJS and T .o 3 obtained from different time lags. In figure Z 10 2 6 the results for the Ter hydrological data, 10 ' H = 0.68, and random series, H = 0.44, are lO ° 2 3 10 lO' 10 10 shown. As was expected, the random series b • T has a value of H near 0.5 which 6 corresponds to ordinary random motion. On 10 the other hand, the Ter data show an B 10 • intermediate value of H which is very ' 10 ---., similar to the value found by HURST � Z . � (1965), H = 0.72 using 94 series of 10 discharges ranging between 10 and 100 10 ' years of data. For comparison, the Loire, lO ° 2 3 using the mean monthly flow between 1863 10 101 10 10 b • T and 1966, has H = 0.69, the Rhine at Basle for the period 1808- 1 966 has H = 0.5, FIGURE 7. Number of "boxes" N (b) as a function of while for the Nile the value of 0.9 is the the time scale obtained for the accumulative highest measured to date (MANDELBROT normalized departures of the Ter series (Fig, 5A), and of the simulated seguence (Fig. 5B , both cases a & W ALLIS, 1969). ) In 3 minimum "box" of T = 1 month and a = 0.001 Hm It is possible to achieve a complemen­ have been used and its size was increased by 13 tary measure of the Hurst exponents for the different multiplicative values of b. The curves are D hydrological and random series by means fitted to the eguation N (b) = b- . In the Ter series (A) 1.38 of the relationship D = 2 - H, in which D is the box dimension is with a correlation coefficient of 0.997. In contrast, in the random series the local fractal dimension of each record (B) these values are 1,52 and 0.998 respectively.

2 10 ,------.------,------� of accumulated departures, x(t) (Fig. 5), • considered as a Brownian function. As these functions are self-affine record s scaled in two variables, x(t) and t, only local fractal dimensions depending on time lags can be ca1culated. The box counting method to determine the fractal dimension '1' , monlhs considers that if we cover the x (t) record s - with a grid of boxes of width bt in time, ----r the box :::.------and of length ba in ordinates, � B  dimension (D) is defined by the equation, D • N (b; a', t) = b- (7) • e �

where N (b; a' , t) is the number of boxes we need for covering the record, a and t are T, monlhs the minimum values of the parameters x (t) / S as a function T fIGURE 6. R 01' lag (filled eireles) for the series of figure 1 B (A) and 2 A (B). The lines and t respectively, while b is a multipli­ H represent fits of the Hurst law R / S = (aT) to the cative factor which determines the (A), 0,68, observed R / S. For the Ter series H is with increasing size of the boxes. In practice, a correlation coefficient of 0.955, while for the random series H is 0.44, with a value of r = 0.915. the box dimension (D) is obtained by fi tting RESCALED RANGE ANALYSIS OF THE TER RIVER 29

the equation (7) to a series of values of (MA6) and 1 year (MA 12). In each case, a N(b) and b by linear regression. certain amount of information is lost but Figures 7 a and 7b show the results we the most important cycles persist (Fig. 9 obtained using the box-counting method b-d). The results show the existence of high from data in figure 5. Because time is an correlations for periods of 1, 2 and 3 years, integer, the minimum box width selected then there is a gap until around 5.5, 8.6, was t = 1 month, while a value of 0.00 1 10.1 and 11,7 years. These longer term 3 Hm was chosen for height. The obtained fluctuations can be considered as an results for these values of a and t are expression of the cycles of cirea 7 and 11 indicated in figures 7 a and 7 b. The fractal years that are characteristic in many dimension for the Ter hydrological record climatic events in all the Mediterranean is 1.38 and for the random series 1.52. Both region (MARGALEF, 1977). Because of results comply with the expected fr actal the high correlations obtained for short time values according to equation (6) showing periods, these cycles do not appear in the slight differences due to the statistical ACF. For this reason the periodogram and approach. the spectral density of the Ter discharges, Since both H and D change when logarithmically transformed, have been another scale of boxes is taken, the ca1culated by Fourier analysis. In the evolution of the box dimension for other spectral density a Tukey window sizes of boxes is considered. The value of (DENMAN, 1975) with a span of 12 was D remains almost constant when the height used. The results (Fig. 10) show little of the box is between 0.0 1 and 0.000 1 difference from the values obtained by 3 Hm . However, D grows very quickly ACF for long-term cycles, that is 5.8, 8.7 when the time scale is changed and values and 11.7 years. Nevertheless, this method higher than two months are considered as gives a more precise measure of the the minimum width of the boxes. In figure importance of seasonal cycles. The cycles 8 the values of D are always higher for of 3 and 6 months are the most persistent, random series than for the Ter discharge series, but both curves grow asymptotically towards a theoretical maximum of D = 2

(MANDELBROT, 1987). In conclusion, 8 1.B o there is a loss of the sensitivity of method when the selected time lag is longer than the minimum time intervals of the data. High values of persistence can be considered as a measure of the strength of the periodical cycles throughout time - the "Joseph effect"- but they do not give information about the frequency of such cycles. To determine these frequencies, we have ca1culated the autocorrelation fu nction

(ACF) and the periodogram of the original 1.2 series of hydrological data. The results of 10 20 30 10 50 60 70 80 90 100 1 10 the ACF are shown in fi gure 9. Because of b * t the noise of the series, the ACF (Fig. 9 a) FIGURE 8. Evolution of the box-dímension as a does not show any characteristic periodo To function of the time scale of the "boxes": A) for the avoid this effect, the series was filtered by Ter data (filled circles) and B) for the random moving average (MA) process using three sequence (hollow circles). In both cases a mínimum value of 0.001 for x (t) have been taken. The data are different spans: 3 months (MA3), 6 months fitted to a logaríthmic equation to show their trend .. 30 ARMENGOL, SABATER, VIDAL & SABATER

but there are also other short cycles of produced by small differences In the around these periods which can be seasonality of each year.

1,0

0,8 A

0,6

0,4

0,2

0,0

-0,2

0,8 MA(3)

fIl fIl fIl fIl fIl ..c ..c ..c fIl fIl s.... s.... 0,6 ...., ...., ...... s.... fIl s.... ro ro r:: r:: r:: ro s.... ro Q) Q) � o o o Q) ro Q) >. >. • o.-< Q) 0,4 S >. >. >. S S l'-_ +' C') co_ o u 0,2 � ;j <+-< 0,0 � • o.-< -0,2 +'

...... ro Q) 0,8 H H o 0,6 u o +' 0,4 ;j

0,0

-0,2

0,8

0,6

0,4

0,2

0,0

-0,2

O 12 24 36 48 60 72 84 96 108 120 132 Lag number

FIGURE 9. Autocorrelation functions of the River Ter monthly water flow (A) and the same series but filtered by a moving average process with a 3 month span MA (3), 6 month span MA (6) and a 1 year span MA (12). The periods which represent the most conspicuous peaks are indicated. RESCALED RANGE ANALYSIS OF THE TER RIVER 31

DISCUSSION morphology (the slopes); the third, climatic factors (snow accumulation); the fourth, To date, the morphology of the microc1imatic factors (soil moisture), and so catchment has been considered the main on, until all possible factors have been cause of the persistence in hydrological taken into account. The fractal nature of the fluctuations. This is the simplest version of basin may also contribute to the fractal the storage model: the water runs off more behavior of the river discharge (FEDER, slowly in the floodplains and even reliefs 1988). The result is that the discharge of a than in mountainous areas with high slopes. river depends not only on recent In a more realistic model the discharges are precipitation but also on earlier rainfalls the sum of a series of independent and, at a theoretical limit, it will be processes in decreasing order of independent of recent rainfall. importance. For instance, the first factor In lakes, the intensity of the fluctuations may be the existence of natural stores of produced by c1imatic oscillations is geological origin (karsts); the second, regulated by their size (GOLDMAN et al.,

8 ro A � 15 � O '"O .-O r... 10 Q,) o.. s::: ....J 5

10 � ,5 .83 years .-+oJ III B e 8 Q,) '"O - 6 ro months r... +oJ o Q,) 4 months o- 3 ID

s:: 2 ....J

O 0.0 0.1 0.2 0.3 0.4 0.5 Frequency

FIGURE 10. Periodogram (A) and spectral density (B) plots of the results obtained by a Fourier analysis from the original Ter data, logarithmically transformed. The periods which correspond to the Iargest cycles are indicated. 32 ARMENGOL, SABATER, VIDAL & SABATER

1988; POWELL, 1989). In a similar way, The species try to couple their biological the size of the catchment is an important cycles to temporal scales of variabi lity of factor determining the persistence of the physical ( = climatic) events. First of aH, hydrological series due to the possibility of they need to know if prediction is possible. 0.5 1 integrating events taking place in different If it is, that means < H < or O < H 0.5, subbasins. < and then natural selection can acto It is interesting to consider the effect of This coupling is as strong as the value of various human activities on the basin to H. determine what effects may be expected on In conclusion, it is easy to be a prophet persistence. The channeling of rivers leads in the Nile with H = 0.9, and it rnay be to destruction of fIoodplains; the pumping possible to have sorne chance in the Ter (H of groundwater destroys various natural = 0.73), but it is better not to bet on rivers stores, while the construction of reservoirs with a value of H close to 0.5. increase them. In a similar way, the areas without vegetation and with eroded soils do not have water retention capacity and the persistence is close to values of 0.5. ACKNOWLEDGEMENTS As for the communities, hydrological events are a key factor in determining their We express our gratitude to Professor R. persistence in the river. According to Margalef for giving us an idea of the value MARGALEF (1980), the organisms make of the study of environmental fIuctuations their own Fourier analysis to foresee the in understanding the structure and dynarnics environmental fluctuations. In fact, first of cornrnunities. His ideas (MARGALEF, they make a factorial analysis to identify 1977, 1980, 1983, 1988) were a stimulus the key factor and the correl ated ones. for the study of the Ter river hydrological Accordingly, communities in a river are an series. It is a pleasure to dedicate this paper example of physical - biological coupling. to hirn.

REFERENCES

ARMENGOL, J. 1988. Manipulació fi sica i orglmica HURST, H.E, BLACK, R. & SIMAIKA, Y.M. 1965. de masses d'aigua continentals. Tretze Congres de Long-Term Storage: An experimental Study. Metges i Biolegs de Llengua Catalana. Llibre de Constable. London. Ponencies: 194-207. KIRKBY, M.J. 1987. The Hurst effect and its ARMENGOL, J., CRESPO, M., MORGUÍ, J.A. & implications for extrapolating process rates. Earth VIDAL, A. 1986. Phosphorus budgets and forms Surface Processes and Landforms, 12:57-67 of phosphorus in the Sau reservoir sediment: An MANDELBROT, B. B. 1975. Stochastic models of interpretation of the limnological record. Earth relief, the shape and the fractal dimension of Hydrobiologia, 143: 33 1 -336. coastlines, and the number-area rule for islands. BOX, G.E.P. & COX, D.R. 1964. An analysis of Proc. Nat. Acad. Sci. USA., 72:3825-3828 transformations. J. Royal Statistical Soco (B), 26: MANDELBROT, B. B. 1985. Self-affine fractals and 211-252. fractal dimensiono Phys. Scr., 32:257-260 DENMAN, K.L. 1975. Spectral Analysis: A Summary MANDELBROT, B. B. 1987. Los objetos fractales. of the Theory and Techniques. Fish. Mar. Servo Tusquets. Barcelona Res. Dev. Tech. Rep. , 539 MANDELBROT, B. B. & VAN NESS, J. W. 1968. D.G.O.H. 1972, 1973, 1974, 1975. Aforos. Cuenca Fractional Brownian motions, fractional noises and JO del Pirineo Oriental. Anuarios 66-67 a 69-70. applications. SIAM Review, 10:422-437 D.G .O.H. 1976. Aforos y aguas subterraneas. 10 MANDELBROT, B. B. & WALLIS, R.1968. Noah, 1. Cuenca del Pirineo Oriental. Período 1970-7 1 a Joseph and operational hydrology. Water 1973-74. Resources Research, 4:909-9 18 FEDER, J. 1988. Fractals. Plenum Press. NewYork MARGALEF, R. 1977. Ecología. Omega. Barcelona GOLDMAN c.R., JASSBY, A. & POWELL, T.M. MARGALEF, R. 1980. La biosfe ra, entre la 1989. Interannual fluctuations in primary termodinámica y el juego. Omega. Barcelona. production: Meteorological forcing at two MARGALEF, R. 1983. Limnología. Omega. subalpine lakes. Limnol. Oceanogr., 34:308-32 1. Barcelona RESCALED RANGE ANALYSIS OF THE TER RIVER 33

MARGALEF, R 1988. An Image of the Biosphere. SABATER, S., SABATER, F. & ARMENGOL, J. Unpublished manuscript. 1988. Relationships between Diatom Assemblages POWELL, T.M. 1989. Physical and biological scales and Physico- chemical Variables in the River Ter of variability in Lakes, Estuaries and the Coastal (N.E. Spain). Int. Revue ges. Hydrobiol., Ocean. In: Perspectives in Ecological Theory. (J. 73: 171-179 Roughgarden, M. May & S.A. Levin, eds.): VIDAL, A. 1977. Eutrofización del embalse de Sau en 157- 176 Princeton University Press. Princeton el transcurso de sus primeros años (1963- 1 972). PUIG, M. A; GONZÁLEZ, G. & RECASENS, L. Trab. II Asamblea Nac. Geodesia y Geofísica: 1987. Modelos de distribución de plecópteros, 1955-1997. efemerópteros, tricópteros y simúlidos en el río VIDAL, A. 1987. Impacto de distintas fuentes de Ter. Limnetica, 3: 125-132 fósforo en la eutrofización del embalse de Sau. SAUPE, D. 1988. AIgorithms for random fractals. In: XVIII Jornadas Comité Español de la Detergencia, The science of fractal images (H.O. Peitgen & D. Tensoactivos y Afines. Barcelona. Saupe, eds.): 71-136. Springer. New York. VIDAL, A. 1989. El cas de l'embassament de Sau. In: SCHINDLER, J. E. 1988. Freshwater Ecosystem: A Sistemes Naturals. Historia Natural deIs Paisos Perspective. In: Concepts of Ecosystem Ecology. Catalans. XIV. (J. Terradas, N. Prat, A. Escarré & (L.R Pomeroy & J.1. Alberts, eds.): 57-74. R. Margalef, eds.): 434-441. Enciclopedia Springer. New York Catalana. Barcelona. SABATER, F. & ARMENGOL, J. 1986. Chernical VOSS, R F. 1985. Random fractal s forgeries. In: characterization of the Ter river. Limnetica, Fundamental Algorithms in Computer Graphics. 2:75-84 (RA. Eamshaw, ed.): 805-835. Springer. Berlin.