Modeling water flow of the Rhine River using seasonal long memory

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1 WATER RESOURCES RESEARCH, VOL. 39, NO. 5, 1132, doi: /2002wr001697, 2003 Modeling water flow of the Rhine River using seasonal long memory Michael Lohre, Philipp Sibbertsen, and Tamara Könning Fachbereich Statistik, Universität Dortmund, Dortmund, Germany Received 3 September 2002; revised 27 December 2002; accepted 6 March 2003; published 17 May [1] The discharge of the Rhine River is modeled by using flexible seasonal long-memory models. The memory parameters are estimated by log periodogram regression for every seasonal frequency separately. It turns out that these models fit well the long-term behavior of the river. Significant long-range dependence was estimated at annual and semiannual frequencies. These results are robust against elimination of possible deterministic seasonal structures. INDEX TERMS: 1860 Hydrology: Runoff and streamflow; 1869 Hydrology: Stochastic processes; 1655 Global Change: Water cycles (1836); KEYWORDS: Rhine River, seasonal models, long memory, log periodogram regression Citation: Lohre, M., P. Sibbertsen, and T. Könning, Modeling water flow of the Rhine River using seasonal long memory, Water Resour. Res., 39(5), 1132, doi: /2002wr001697, Introduction [2] Considering the minimal water flow of the Nile River, the hydrologist Hurst [1951] observed strong dependencies between observations far away from each other. The minimal water flow of the Nile River shows a persistent behavior and can be described by long-memory models. A first approach for modeling long-range dependencies was given by Mandelbrot and van Ness [1968]. They introduced the fractional Brownian motion, a generalization of the standard Brownian motion using self-similar processes. Following this approach, Mandelbrot and Wallis [1969] modeled several geophysical records including the river flow of the St. Lawrence River, Loire, Mississippi and the Rhine River. Using rescaled range based methods for estimating the dependence structure of the data they found that long-memory models fit the data well for all rivers except the Rhine. Thus long-memory time series seem to be appropriate for modeling water flow data. [3] In general, a stationary process X t is said to exhibit long memory or long-range dependence, if CovðX t ; X tþk Þ :¼ R k k!1 L 1 ðkþjkj 2d ; d 2 ð0; 1=2Þ; where L 1 (k) is slowly varying as jkj!1. This means that the correlations of the process decay slowly with a hyperbolical rate. Equivalently long-range dependence can be defined by using the spectral density of the process. Let f(l) be the spectral density of the stationary process X t. Then X t is said to exhibit long-memory, if f ðlþ jlj!0 L 2 ðlþjlj 2d ; d 2 ð0; 1=2Þ; ð1þ where L 2 (l) is slowly varying for jlj!0. [4] A first model for long-memory processes was the fractional Brownian motion introduced by Mandelbrot and van Ness [1968] as mentioned above. Independently, Granger and Joyeux [1980] and Hosking [1981] introduced Copyright 2003 by the American Geophysical Union /03/2002WR SWC 3-1 ARFIMA models, a generalization of the classical Box and Jenkins [1976] ARIMA models that allows also for a fractional degree of differencing. These processes are defined by ðbþð1 BÞ d X t ¼ CðBÞe t ; where B is the backshift operator, (x) and C(x) define the AR and MA polynomials with roots outside the unit circle, respectively, and e t is white noise. The term (1 B) d is defined by ð1 BÞ d ¼ Xd k¼0 d k ð 1Þ k B k ; where the binomial coefficient is expressed in terms of the gamma function c k :¼ d ðd þ 1Þ ¼ k ðk þ 1Þ ðd k þ 1Þ : ð3þ [5] For a more detailed discussion of long-memory models and their applications, see, for example, Beran [1994], Robinson [1994], and Sibbertsen [1999]. The problem of modeling river flow is extensively investigated by Lawrance and Kottegoda [1977]. They summarize statistical properties of hydrological time series and show advantages of short- and long-memory models. An overview about statistical modeling in hydrology is given by Hipel and McLeod [1994], where the emphasis lies on modelbased techniques including long-memory or seasonal models among others. [6] A surprising finding of Mandelbrot and Wallis [1969] is that all considered rivers exhibit long-range dependence, but the Rhine River does not. From a hydrological point of view one would expect that the Rhine River shows similar long-run characteristics. Mandelbrot and Wallis [1969] used monthly data from the gauge Basel at Switzerland whereas in our paper the gauges Kaub directly before the mouth of the Mosel River and the gauge Cologne behind the orifice are considered. So different long-run characteristics might occur because of the geographical distance of these gauges. ð2þ

2 SWC 3-2 LOHRE ET AL.: MODELING WATER FLOW WITH SEASONAL LONG MEMORY Figure 1. Series of minimum and maximum flow at (left) Kaub and (right) Cologne. A reason for this discrepancy might be at first that the gauge Basel is only influenced by the Alps whereas at Kaub and Cologne the middle German highlands with their affluents to the Rhine influence strongly the behavior of the river. This is due to strong rainfalls in the winter. For a detailed discussion of the discharge regime of the Rhine river, see Gerhard [1976]. The influence of the Alps by snowmelt in spring as well as the influence of the highlands and winter rainfalls result in a complicate seasonal structure of the discharge at the gauges Kaub and Cologne. [7] In this paper we do not eliminate any seasonal structure in advance. We consider again the long-term behavior of the Rhine River by using monthly data and estimating the dependence structure as well as the seasonal pattern by periodogram-based methods at the same time. It turns out that a seasonal model allowing for long-range dependence fits the data well. Although this model gives good results there is still some evidence of deterministic seasonality in the residuals. Therefore we also eliminate deterministic seasonality by subtracting the seasonal means out of the original data later on. If the modelled long memory is only an artifact of any deterministic influences it should vanish afterwards. On the other hand if we have real long- range dependence in the data the structure should still be visible. [8] The paper is organized as follows: In the next section we describe the data which show evidence of long memory by considering the sample autocorrelation function and the periodogram. In section 3 a seasonal long-memory model is fitted to the data and the dependence structure of the water flow is estimated. The possibility of deterministic seasonality is also taken under account. Results are summarized in the final section. 2. Description of the Data [9] The Bundesanstalt für Gewässerkunde records daily river flow of the Rhine at several gauges. We analyze the data of two gauges because of their geographical position: the Kaub gauge near Koblenz is the last gauge before the mouth of the Mosel river, the Cologne gauge is behind it. For both gauges we have daily data from January 1900 to December For every data set we calculated the monthly minima and maxima, leading to a series length of 1188 observations for each gauge. Figure 1 shows the series. [10] The main reason for considering both data sets rather than one single is the importance of the Mosel River concerning the water flow of the Rhine River. Because of the hard subsoil and thus the low storage effect of the Mosel area, rainwater immediately flows off in the river channel of the Rhine. Whereas one maximum of the flow data is generated by thaw another maximum is caused by strong rainfalls in winter. The maximum generated by the rainfalls should be observable even more at gauges behind the Mosel orifice. Taking these arguments into account, it seems appropriate to fit a seasonal model to the data. Considering the sample autocorrelations of the series, we see that they are typical for a seasonal structure with long-range dependence: the autocorrelations display a sinusoidal pattern and, furthermore, the absolute values of the autocorrelations decay slowly (see Figure 2). This points to a long-memory structure which is analyzed in the next section. From hereafter we use the logarithm of each series in our analysis due to the fact that the variance is reduced and the assumption of weak stationarity can be established. 3. Seasonal Filtering [11] Following the standard procedure oscillations in monthly data are removed by taking the twelfth differences, i.e., passing the series through the linear filter (1 B 12 ). Such a treatment is not appropriate here: the troughs that occur at each seasonal frequency in the periodogram suggest a possible overdifferencing. Thus a filter with a fractional degree of differencing seems to be more appropriate.

3 LOHRE ET AL.: MODELING WATER FLOW WITH SEASONAL LONG MEMORY SWC 3-3 Figure 2. Sample autocorrelations of maximum flow at (left) Kaub and (right) Cologne. [12] Therefore we apply a fractional integrated seasonal ARMA model to the data. This class of models is a natural extension to the class of ARFIMA models as described in the introduction. In detail we apply an approach to seasonal filtering given by Hassler [1994]: for s even (in our case it is s = 12 because we have monthly data) consider the following factorization of the seasonal filter in terms of its unit roots, ð1 B s s=2 1 Þ ¼ ð1 BÞð1 þ BÞ Y 1 e i2pj=s B 1 e i2pj=s B : j¼1 Then a flexible seasonal filter can be defined by convolution: d s : ¼ ð1 BÞ d0 ð1 þ BÞ d s=2 Y n o dj: 1 e i2pj=s B 1 e i2pj=s B s=2 1 j¼1 The expansion of each factor is real valued following Andel [1986]: 1 e iw d B 1 e iw d X 1 B ¼ b j B j j¼0 b j ¼ Xj k¼0 c j k c k cosfðj 2kÞwg ð4þ with c k being defined in (3). Note that c k depends on the memory parameter d. Now the flexible ARFISMA(0,d s,0) process can be defined in terms of the seasonal filter, ds X t ¼ u t ; where u t is a process with positive and bounded spectral density. So the impact of each seasonal frequency can be ð5þ modeled by different memory parameters. The main properties of flexible ARFISMA models are as follows. (1) The process is stationary if and only if d j < 0.5, j =0,1,..., s/2. (2) The process has long-memory if d j > 0 for at least one j, j =0,1,..., s/2. (3) If d j > 0 the spectrum is unbounded at the corresponding seasonal frequency 2pj/s. If d j < 0 the spectrum has a zero, j =0,1,..., s/2. [13] For further details, see Hassler [1994]. The spectrum of the seasonal process in (5) is given by f X ðlþ ¼ 4sin 2 l d0 4sin 2 l 2 2 p ds=2 2 s=2 1 Y 4sin 2 l 2 þ pj 4sin 2 l pj s 2 s j¼1 dj f u ðlþ: [14] Because of its complicated structure autocorrelations cannot be calculated explicitly. Giraits and Leipus [1995] show that the behavior is governed by the largest d j and autocovariances g k can be approximated by g k C Xs=2 ½ Š k 2dj 1 cos k 2pj s j¼0 ðk!1þ: with a suitable constant C. In the neighborhood of a seasonal frequency equation (6) can be approximated by f X ð6þ ð2pj=s þ lþ l 2dj K j f u ð2pj=sþ ðl! 0Þ; ð7þ where K j is a constant that depends on the remaining frequencies and f u is the bounded spectral density of the process u t. For positive values of d the right-hand side of (7) becomes unbounded, for negative values we obtain a zero at 2pj/s. [15] In order to determine the magnitude of d j several parametric and semiparametric procedures have been pro-

4 SWC 3-4 LOHRE ET AL.: MODELING WATER FLOW WITH SEASONAL LONG MEMORY Figure 3. Periodogram of maximum flow at (left) Kaub and (right) Cologne. posed in the literature. Hassler [1994] extended Geweke and Porter-Hudak s [1983] idea of log periodogram based estimation of the memory parameter to the class of seasonal models. This is a semiparametric estimator using the special shape (1) of the spectral density of long-memory processes. Estimating the spectral density by the periodogram and taking logarithms on both sides of the equation gives a linear regression model in the memory parameter d. This estimator has widely been considered in the literature [see, e.g., Robinson, 1995; Hurvich and Beltrao, 1993, 1994] so that we omit the theoretical details here and just adopt it for our model. Using monthly data one obtains the origin and six seasonal frequencies 2pj/12, j = 1,..., 6, with possibly seven different memory parameters. This leads to a modification of the original estimation procedure: the regression equation has to be evaluated around all of these frequencies. We get equations of the type ln I X l j;i ¼ bj þ d j R i þ e j;i ; j ¼ 0; 1...; 6; i ¼ 1;...; m; ð8þ where I X (l j;i ) denotes the periodogram of the series evaluated at Fourier frequency l j;i where index j denotes the seasonal frequency and the following i, i =1,..., m, frequencies are used. b j is a constant depending on the spectral density f u at frequency 2pj/s, R i = ln (4 sin 2 (pi/n)) are the regressors and e j;i is the error term. By means of (8) we can estimate the memory parameter for each seasonal frequency separately. [16] Figure 3 shows the periodogram of the series of maximum flow at Kaub and Cologne. The different magnitudes of the peaks at the origin and at seasonal frequencies possibly indicate the use of different memory parameters. Therefore we apply a flexible ARFISMA(0, d s, 0)-model for every series and estimate each memory parameter separately using the method described above. [17] The extension of log periodogram regression to seasonal models results in further difficulties: due to the symmetry and periodicity of the spectral density around zero and p it is sufficient to use frequencies on only one side of the pole. At the other frequencies it is possible to use frequencies on both sides of the pole. Note that the spectral density of such a process is not necessarily symmetric around a pole in (0, p). Arteche and Robinson [2000] provide a test on spectral symmetry in order to decide if the underlying spectral density is symmetric around a pole. Although in our case the null hypothesis of spectral symmetry cannot be rejected at the 5% level at any seasonal frequency we provide estimates for the memory parameter from the left and right hand side of the pole not to loose any information. [18] The choice of a suitable bandwidth is a delicate problem, since regression equation (8) is only valid around a small neighborhood of each seasonal frequency. The number of frequencies used in the regression should be of order n 0.5 to n 0.8. Since theoretical aspects are not the main concern of this paper we decided to use a bandwidth with maximum length. We can compute periodogram ordinates at n/2 = 594 Fourier frequencies. Between two seasonal frequencies we have n/(2 * 6) = 99 Fourier frequencies, so that for each estimation procedure a bandwidth of m = 49 Fourier frequencies can be used in order to avoid influences of adjacent seasonal frequencies. Note that in our case m n 0.55, so the bandwidth is rather small compared to the series length and a bias due to inaccuracy of (8) is avoided. Arteche and Robinson [2000] show that the estimator ^d j is asymptotically normal distributed with variance p2 24m ; so that the chosen bandwidth leads to an asymptotic standard error of for each series. Table 1 shows the regression estimates for the different series. We estimated from the right side as well as the left side of the seasonal frequency. Because we cannot reject the hypothesis of spectral symmetry the given value is the mean of the right and left estimates. It gives us a clear long-memory structure for all series. [19] We obtain two main results: First, it can be seen that every series contains a long-memory structure because at

5 LOHRE ET AL.: MODELING WATER FLOW WITH SEASONAL LONG MEMORY SWC 3-5 Table 1. Regression Estimates for Different Series and Frequencies Series Seasonal Frequency 0 p/6 p/3 p/2 2p/3 5p/6 p kaub.min kaub.max cologne.min cologne.max least one estimated parameter is significantly greater than zero. Second, considering the minimum discharge, there are not any substantial differences between the gauges at Kaub and Cologne. We get evidence that the river flow is governed by its yearly oscillations due to the high estimates at the first seasonal frequency. However, the size of the estimates for the maxima is slightly different: at Kaub large values appear at seasonal frequency p/3, i.e., half-year oscillations determine the magnitude of flow at Kaub gauge. Regression estimates propose that the maximum flow might display two persistent local maxima per year. This is consistent with our observation that a large discharge is caused by rainfalls in winter as well as by thaw in early summer. At Cologne gauge the highest estimates can be found at the first two seasonal frequencies. This gets along with our conjecture that the influence caused by thaw should be superimposed by winter rainfalls and inflow of the Mosel river as mentioned above. Note that long-range analysis by Mandelbrot and Wallis [1969] was done at gauge Basel in Switzerland so different results might occur because of the varying seasonal structure. Nevertheless seasonal adjustment via differencing is not appropriate for this data and therefore has to be taken into account. [20] Because of the size of the memory parameters it seems reasonable to test whether these parameters can be seen as zero or not. If we can assume those parameters to be zero which have estimates quite near of it this would simplify the structure of our filter because we do not have to take them into account any more. So this would be helpful in the sense of parsimonious modelling. If it turns out that the hypothesis of d = 0 can be rejected for the relevant seasonal frequencies our conclusions would be strengthened. With the relevant seasonal frequencies we mean here and in what follows those frequencies which have a clearly positive memory parameter as described in the previous paragraph. These is through all series the first and/or second seasonal frequency. For doing this we are using the approach of Arteche [2002]. Here a test for this null hypothesis is provided. We omit the technical details of the test here and just concentrate on the results. As expected it comes out that all frequencies but the relevant frequencies can be neglected because the null hypothesis of d = 0 can not be rejected to the 5% level. Including the relevant frequencies the null is now being rejected to the 5%- level. This clearly indicates a long-memory structure at these seasonal frequencies. [21] Our conclusions depend on the quality of the estimates. So we finally have to verify whether the chosen model fits the data. Therefore we computed the residual process û t and check its characteristics via the periodogram. Remember that we estimated the memory parameters d j of the process d12 X t ¼ u t : [22] If we pass X t through a filter d 12 with estimated coefficients then the remaining process û t should neither exhibit long memory nor a seasonal structure. Because some of the estimates are close to zero those d j can be neglected when computing the residuals. This reduces the filter. Thus in fact we pass X t through a filter ~ d12 containing only those d j which are significantly positive. Figure 4 shows the sample autocorrelation function and the periodo- ð9þ Figure 4. Sample autocorrelation and periodogram of residual series for maximum flow at Kaub.

6 SWC 3-6 LOHRE ET AL.: MODELING WATER FLOW WITH SEASONAL LONG MEMORY Table 2. Estimated ARMA Coefficients for Different Residual Series Table 3. Regression Estimates for Different Deseasonalized Series and Frequencies AR MA Seasonal Frequency kaub.min kaub.max , cologne.min , cologne.max , Series 0 p/6 p/3 p/2 2p/3 5p/6 p kaub.max cologne.max gram of the estimated residual series for the maximum discharge at Kaub. The sinusoidal structure is removed and taking absolute values, autocorrelations decay to zero more rapidly. For the other series we obtain similar results that are not presented here to save space. The periodogram of the residual series indicates that the estimates are meaningful since the peaks at seasonal frequencies are reduced and neither a seasonal nor a long-memory structure can be identified any longer. [23] We fit ARMA(p, q) models to the residual series by using the Bayes-Schwarz information criterium. Plausible fits are obtained for orders up to p = 3 and q = 3. The exact ARMA coefficients are given in Table 2. It can be seen that all estimated coefficients are significantly smaller than one. Thus nonstationarities as for example unit roots do not occur. For the minimum discharges Figure 5 shows the theoretical spectral density using our estimates for the memory parameters and for the ARMA coefficients compared with the periodogram of the series. It can be seen that both seem to be quite similar. [24] Because not all peaks in the periodogram are removed by our filter some deterministic seasonality might be present in the data. This could also cause some spurious long memory. To exclude this we eliminate the seasonal means out of the data. This means that we subtract from every data point the monthly average of the corresponding month. This is a well known method to eliminate deterministic seasonality. If there is the proposed long-memory structure present in the data this structure will remain. Therefore the whole analysis is done again for the deseasonalized series. Because the results are quite similar to the first study we give only the results for the maximal series to save space. The estimates are given in Table 3. [25] As we see the memory structure is still present. The memory parameter of the half-year frequency are still clearly greater than zero and thus indicating long memory. Again fitting ARMA processes to the residuals gives plausible fits for orders up to p = 3 and q = 3. The coefficients for the series of the maxima are in Table 4. Again they are clearly lower than one. [26] Thus we see that our findings of long memory at the first and second seasonal frequency is robust against deterministic seasonality. 4. Summary [27] From a hydrological point of view the water flow of the Rhine river should exhibit long-range dependence because of the storing influence of Lake Constance and the lakes of the Swiss Alps. Furthermore differences should occur between gauges because of their differing distances to the Lake Constance and because of different sensitivity to Figure 5. Periodogram (dotted line) and estimated theoretical spectral density (solid line) of minimum flow at (left) Kaub and (right) Cologne.

7 LOHRE ET AL.: MODELING WATER FLOW WITH SEASONAL LONG MEMORY SWC 3-7 Table 4. Estimated AR Coefficients for Different Deseasonalized Residual Series rainfall and inflow impacts. However the analysis of Mandelbrot and Wallis [1969] did not support the thesis of longrange dependence in the Rhine data. We reanalyzed the river flow using long series of minimum and maximum flow at two different gauges. We applied a seasonal long-memory model to the data and showed that such models result in adequate fits. In contrast to the findings of Mandelbrot and Wallis [1969] we got evidence for long-range dependence in the following sense: minimum flow depends on yearly oscillations. At Kaub gauge the maximum flow is governed by half-year cycles whereas at Cologne gauge a one-year cycle is predominant. [28] Acknowledgments. First we would like to thank Hans Jürgen Liebscher from the Bundesanstalt für Gewässerkunde for providing the prepared data and for his hydrological advice. The authors are grateful for the remarks of two anonymous referees which considerable improved the paper. They also thank Christoph Helwig for computational assistance. The support of Volkswagenstiftung and Deutsche Forschungsgemeinschaft is gratefully acknowledged. References AR Coefficients kaub.max , , cologne.max , , Andel, J., Long memory time series models, Kybernetika, 22, , Arteche, J., Semiparametric robust tests on seasonal or cyclical long memory time series, J. Time Ser. Anal., 23, , Arteche, J., and P. M. Robinson, Semiparametric inference in seasonal and cyclical long memory processes, J. Time Ser. Anal., 21, 1 25, Beran, J., Statistics for Long-memory Processes, Chapman and Hall, New York, Box, G. E. P., and G. M. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, Calif., Gerhard, H., Abflussregime, in Das Rheingebiet, pp , Int. Kommission für die Hydrol. des Rheingebietes, Herausgeber, Germany, Geweke, J., and S. Porter-Hudak, The estimation and application of long memory time series models, J. Time Ser. Anal., 4, , Giraits, L., and R. Leipus, A generalized fractionally differencing approach in long-memory modelling, Lithuanian Math. J., 35, 53 65, Granger, C. W. J., and R. Joyeux, An introduction to long-range time series models and fractional differencing, J. Time Ser. Anal., 1, 15 30, Hassler, U., (Mis)specification of long memory in seasonal time series, J. Time Ser. Anal., 15, 19 30, Hipel, K. W., and A. I. McLeod, Time Series Modelling of Water Resources and Environmental Systems, Elsevier Sci., New York, Hosking, J. R. M., Fractional differencing, Biometrika, 68, , Hurst, H. E., Long-term storage of capacity of reservoirs, Trans. Am. Soc. Civ. Eng., 116, , Hurvich, C. M., and K. I. Beltrao, Asymptotics for the low- frequency ordinates of the periodogram of a long-memory time series, J. Time Ser. Anal., 14, , Hurvich, C. M., and K. I. Beltrao, Automatic semiparametric estimation of the memory parameter of a long-memory time series, J. Time Ser. Anal., 15, , Lawrance, A. J., and N. T. Kottegoda, Stochastic modelling of riverflow time series, J.R. Stat. Soc., Ser. A, 140, 1 31, Mandelbrot, B. B., and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, , Mandelbrot, B. B., and J. R. Wallis, Some long-run properties of geophysical records, Water Resour. Res., 5, , Robinson, P. M., Time series with strong dependence, in Advances in Econometrics, 6th World Congress of the Econometric Society, vol. 1, pp , Cambridge Univ. Press, New York, Robinson, P. M., Log-periodogram regression of time series with long range dependence, Ann. Stat., 23, , Sibbertsen, P., Robuste Parameterschätzung im linearen Regressionsmodell, VWF, Berlin, Germany, T. Könning, M. Lohre, and P. Sibbertsen, Fachbereich Statistik, Universität Dortmund, D Dortmund, Germany. (sibberts@statistik. uni-dortmund.de)