Information Content of Lake Tana Using Abstract Introduction Stochastic and Wavelet Analysis Methods Conclusions References Tables Figures Y

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icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Hydrol. Earth Syst. Sci. Discuss., 7, 5525–5546, 2010 Hydrology and www.hydrol-earth-syst-sci-discuss.net/7/5525/2010/ Earth System HESSD doi:10.5194/hessd-7-5525-2010 Sciences 7, 5525–5546, 2010 © Author(s) 2010. CC Attribution 3.0 License. Discussions

Information content This discussion paper is/has been under review for the journal Hydrology and Earth of Lake Tana System Sciences (HESS). Please refer to the corresponding ﬁnal paper in HESS if available. Y. Chebud and A. Melesse

Stage level, volume, and time-frequency Title Page information content of Lake Tana using Abstract Introduction stochastic and wavelet analysis methods Conclusions References Tables Figures Y. Chebud and A. Melesse J I Department of Earth and Environment, Florida International University, FL, USA J I Received: 1 June 2010 – Accepted: 23 June 2010 – Published: 11 August 2010 Back Close Correspondence to: Y. Chebud (ycheb002@ﬁu.edu) Published by Copernicus Publications on behalf of the European Geosciences Union. Full Screen / Esc

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5525 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Abstract HESSD Lake Tana is the largest fresh water body situated in the north western highlands of Ethiopia. It serves for local transport, electric power generation, fishing, ecological 7, 5525–5546, 2010 restoration, recreational purposes, and dry season irrigation supply. Evidence show, 5 the lake has dried at least once at about 15 000–17 000 BP (before present) due to Information content a combination of high evaporation and low precipitation events. Past attempts to ob- of Lake Tana serve historical fluctuation of Lake Tana based on simplistic water balance approach of inflow, out-flow and storage have failed to capture well known events of drawdown and Y. Chebud and rise of the lake that have happened in the last 44 years. This study is aimed at sim- A. Melesse 10 ulating the lake level, specifically extreme events of the lake variation using stochastic approaches. Fourty-four years of daily, monthly and mean annual lake level data has showed a Gaussian variation with goodness of fit at 0.01 significant levels of the Title Page

Konglomorov-Simrnov test. Three stochastic methods were employed, namely pertur- Abstract Introduction bations approach, Monte-Carlo methods and wavelet analysis, and the results were 15 compared with the stage level measurements. The stochastic simulations predicted Conclusions References the lake stage level of the 1972, 1984 and 2002/2003 historical droughts 99% of the Tables Figures time. The information content (frequency) of ﬂuctuation of Lake Tana for various periods was resolved using Wigner’s Time-Frequency Decomposition method. The wavelet analysis agreed with the perturbations and Monte Carlo simulations resolving the time J I

20 (1970s, 1980s and 2000s) in which low frequency and high spectral power ﬂuctuation J I has occurred. In summary, the Monte-Carlo and perturbations methods have shown their superiority for risk analysis over deterministic methods while wavelet analysis has Back Close

met reconstructing stage level historical record at multiple time scales. A further study Full Screen / Esc is recommended on dynamic forecasting of the Lake Tana stage level using a combined 25 approach of the perturbation and wavelet analysis methods. Printer-friendly Version

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5526 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 1 Introduction Lake Tana is the largest fresh water body situated in the north western highlands of HESSD Ethiopia. It serves for local transport, electric power generation, ﬁshing, ecological 7, 5525–5546, 2010 restoration, recreational purposes, and dry season irrigation supply. The lake is fed by 5 four major rivers; Gumara, Megech, Rib and Gilgel Abay. The drainage basin receives an average annual rain fall of 1200 mm which helps to replenish the lake seasonally. Information content The lake has one outlet, where the Blue Nile River begins (Chebud and Melesse, 2009). of Lake Tana Historically, the lake dried at least once at about 15 000–17 000 BP (before present) Y. Chebud and (Hautot et al., 2006). The same report indicated that the reason for desiccation was A. Melesse 10 probably a combination of high evaporation and low precipitation events. Such evi- dences was obtained from sediment core samples of plant remnants that grew and deposited within the present lake area. The recent historical ﬂuctuation of Lake Tana Title Page and shrinkage of wetlands is reported by Kebede et al. (2006b), Chebud and Me- lesse (2009), Wale (2008). All of the studies reported are based on simplistic water Abstract Introduction

15 balance approach of inflow, out-flow and storage. The reports invariably indicated the Conclusions References stationarity of the mean lake level in the last 44 years. None of the studies explained how the lake stage level behaved in the 1972, 1984 and 2002/2003 historical droughts Tables Figures nor showed any extreme event. Whereas, the historical flow data showed a decline of the out flow below 500 million cubic meters (MCM) which was only 10% of its annual J I 20 average 5 billion cubic meters (BCM). The challenge of past studies that use deterministic water balance approaches is J I evident from the uncertainty of inputs. First, the major inflow variable (inflow from 4 Back Close major rivers) is gauged at a daily scale (a sample/24 h) interval, therefore most peak flows could have been missed any time of the day. Second, the considerable size of the Full Screen / Esc 25 contributing watershed is un-gauged (Wale, 2008) and is assumed. Third, sub-surface inflow and outflow is often assumed (Kebede et al., 2006b), or estimated using numer- Printer-friendly Version ical models (Chebud and Melesse, 2009) that take into account geological inferences. Interactive Discussion In general, the propagation of error could limit the predictability of Lake Tana using deterministic water balance approaches. 5527 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Considering the scarcity of data and absence of forecasting mechanisms for Lake Tana, simulating the lake level using stochastic methods was relevant. The stochastic HESSD methods hold the statistical variability of the historical lake level and also take into 7, 5525–5546, 2010 account the error term from the probabilistic distribution of the noise. This makes 5 stochastic methods superior, in Lake Tana’s context, over deterministic methods that suffer from poor sampling methods, crude assumptions as well as uncertainties of Information content un-gauged watersheds. The stochastic method uses a historical time series of the of Lake Tana daily lake level which precludes any propagated error from component measurements; rather it uses measurement of the lake level (net balance of storage). Y. Chebud and A. Melesse 10 In fact the stochastic methods could serve to simulate historical events as well as forecast the stage level dynamically. However, a complement using wavelet analysis is also necessary to resolve the return period and the spectral power of the events. So, Title Page the two objectives set for this study were to Abstract Introduction 1. simulate Lake Tana stage level and volumetric balance using stochastic ap- 15 proaches of Monte Carlo and perturbation methods, and Conclusions References

2. observe the time-frequency resolved changes of the spectral power and validate Tables Figures low frequency events of drought or ﬂooding of Lake Tana. The study hypothesis is that a combined approach of stochastic methods and wavelet J I

analysis could improve predictability of Lake Tana water level compared to deterministic J I 20 water balance models. Back Close

2 Study area Full Screen / Esc

◦ 0 ◦ 0 Lake Tana is located at 11 27 N and 37 10 E of Ethiopia at 1790 m a.s.l. The lake Printer-friendly Version has a mean and maximum depth of 8 m and 14 m, respectively. The total drainage area of the Lake Tana basin is about 16 500 km2. It receives about 90% of the inflow Interactive Discussion 25 from the four rivers. The local and regional groundwater inflows contribute only 3% 5528 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion and 7%, respectively (Chebud and Melesse, 2009). The lake is geologically dammed by quaternary and tertiary basalts in the south and western part where the out flowing HESSD Blue Nile River drains out (Kebede et al., 2005b). The damming has cutoff any seepage 7, 5525–5546, 2010 outflow which simplifies modeling of the lake stage (Hautot et al., 2006).

3. Background on Stochastic Approaches Information content 5 3 Background on stochastic approaches 3.1. Random nature of lake stage and the stochastic approach of Lake Tana 3.1 Random natureThe water of lake balance stage approach and theis the stochastic basis for the stochastic approach simulation approach. Inflow Y. Chebud and and outflow determine lake stage (volume) variability which in retrospect is dependent A. Melesse The water balance approach is the basis for the stochastic simulation approach. Inflow on the amount of rainfall, evapotranspiration and, catchment characteristics that and outflow determine lake stage (volume) variability which in retrospect is dependent generates runoff. Rainfall influences the amount of inflow in two ways. First, it is a on the amount of rainfall, evapotranspiration and, catchment characteristics that gen- Title Page 10 erates runoff. Rainfalldirect influencescontributory over the amountthe lake area. of inflow Second in, twoit drives ways. the recharge First, it and is a runo directff and 3 contributory over thehence lake the area.inflow. Considering Second, it river drives inflow the and recharge precipitation and as inputs runoff I(t)and ( L /T) hence , river Abstract Introduction 3 the inflow. Considering river inflow and precipitation as inputs I(t)(L /T ), river outflow 3 outflow at the outlet and evapotranspiration from the lake surface as outputs Q(t) (L ), Conclusions References at the outlet and evapotranspiration from the lake surface as outputs Q(t)(L3), the dif- 3 3 ference in the volumethe difference was formulated in the volume as the was storage formulatedV ( tas)( theL ) storage as shown V(t) (L in) free as shown diagram in free Tables Figures 15 below. diagram below.

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Back Close The equation couldThe be balance formulated could be by formulated balancing balancing the change the change in volume in volume with with time time takingtaking the Full Screen / Esc the difference of thedifference inflow of and the inflow outflow and as outflow indicated as indicated in Eq. in (1),eq. (1), considering considering anan analogous analogous approach byapproach Yevjevich by (1972).Ameniya, Sinceet al. (2005). the variables Since the variables in Eq. (1) in haveeq. (1) theirhave their respective respective Printer-friendly Version 20 mean and noise, themean perturbation and noise, the approach perturbation takes approach the takes form the shown form shown in Eq. in (2)eq. (2) where wherev ,v, i, Interactive Discussion i, and q are noisesand from q are volume,noises from inflow volume, and inflow outflow, and outflow, respectively. respectively. Averaging Averaging eq. Eq. (2) (2) over over several realizations,several realizations, the noise the terms noise will terms be will eliminated be eliminated giving giving only only the the equations equations in terms of the mean values.5529 Detrending mean values from eq. (2) results in a relationship of the noises only, eq. (3). The formulation of the noises of inflow, outflow and lake volume as in eq.(3) is core of the perturbation approach which enables to reformulate for the statistical behavior of the lake fluctuations in terms of the variances and covariance of inflow and outflow.

(1)

(2)

(3) Multiplying eq. (3) on both sides by the storage noise ‘v’ and averaging over several 2 realizations, a physically meaningful parameter ( ) or the storage variance (σv ) and icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion in terms of the mean values. Detrending mean values from Eq. (2) results in a relationship of the noises only, Eq. (3). The formulation of the noises of inflow, outflow and HESSD lake volume as in Eq. (3) is the core of the perturbation approach which enables us to 7, 5525–5546, 2010 reformulate for the statistical behavior of the lake fluctuations in terms of the variances 5 and covariance of inflow and outflow. dV (t) Information content = I(t)−Q(t) (1) of Lake Tana dt d(V¯ +v) Y. Chebud and = (I¯+i)−(Q¯ +q) (2) A. Melesse dt d(v) = (i)−(q) (3) dt Title Page

Multiplying Eq. (3) on both sides by the storage noise “v” and averaging over several Abstract Introduction 2 2 10 realizations, a physically meaningful parameter (v ) or the storage variance (σ ) and v Conclusions References the other terms of Eq. (4) namely covariance of the inflow and outflow with the storage changes are derived. The equation relates rate of amplification or decay of storage vari- Tables Figures 2 ance (σv ) with its covariance against inflow and outflow. Similarly, multiplying Eq. (3) on both sides by the inflow noise, averaging it over realizations and rearranging the terms J I 2 2 15 gives (i ) which is the inflow variance (σi ) that shows a relationship with its covariance against the outflow and the rate of change of its covariance with the storage volume. J I Using the same approach, the relationship of the variance of the outflow is found to be Back Close related with its covariance to the storage changes and inflow as shown in Eq. (6) Full Screen / Esc d(v 2) −2(iv)−(qv) (4) dt Printer-friendly Version

d(iv) Interactive Discussion 2 20 = (i )−(iq) (5) dt 5530 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion d(qv) = (iq)−(q2) (6) dt HESSD The relationships in Eqs. (4), (5) and (6) reveal that the variance or covariance from 7, 5525–5546, 2010 any of the inﬂow, outﬂow or storage variables is propagated with time, suggesting the propagation of the error as part of the simulation. Though such error propagation is Information content 5 a limitation of the perturbation approach, it is powerful to inform all possible realiza- of Lake Tana tions. The limitations however could be constrained by prior knowledge of the historical variability and some boundary conditions. An alternative to the classical perturbation Y. Chebud and methods with less error propagation is the Monte-Carlo approach. The Monte Carlo A. Melesse approach (Christopher, 1997) is simplistic that relates probability of future realizations 10 based on past events added to a noise. The relationship could be established discretizing Eq. (1), Title Page

Abstract Introduction Vt+∆t = Vt +∆t[I(t)−Q(t)] (7) From the definitions of the Monte Carlo approach, the probability of realizing the pre- Conclusions References dicted Volume (V ) is the sum of the current probabilities of the volume, inflow and t+∆t Tables Figures 15 outflow. This dependence of current and future probabilistic distributions is reported as a weakness of Monte Carlo approach (Gelhar, 1998). However it is merely inapplica- ble to non-stationary distributions. On the other hand, its simplicity and conformity to J I boundary conditions of realizations makes it appealing. J I

3.2 Wavelet analysis Back Close

Full Screen / Esc 20 Chebud and Melesse (2009), showed the two frequency ranges and the spectral power of Lake Tana but fall short of informing exactly when such changes of frequencies occur. In other words, the information content of the lake variation (the spectral power Printer-friendly Version

and return period) was not resolved in time. This study investigated the potential of Interactive Discussion the time-frequency resolution using wavelet analysis. Wavelet analysis could detect 25 (1) any historical changes in frequency of lake level fluctuation which could help to 5531 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion relate the effect of the frequency with other ecological phenomenon (signatures) since the latter is recorded in the time domain and (2) the wavelet analysis could decompose HESSD the signal as local and global trends which could later be reconstructed and serve for 7, 5525–5546, 2010 lake level forecasting. 5 The wavelet transform can be considered as the convolution of the signal of interest (in this case the lake level fluctuation) with the conjugate of the Fourier transform of a Information content mother wavelet. The mother wavelet has a magnifying as well as discrimination role of Lake Tana over the signal. oreover the type of mother wavelet chosen usually depends on its pattern similarity with the signal type. For hydrological analysis, the Morelet wavelet Y. Chebud and A. Melesse 10 (Fig. 1) is recommended by Tesche et al. (1998), Kang et al. (2007), and Torrence et al. (1997). The normalized Morelet wavelet’s distribution, Eq. (8), has a mean value of zero and is known for its localization in time and frequency information content. The Title Page vertical axis, represented by Ψ(t) is analogous to a density function for deterministic signals (Cohen, 1989) (a measure of amplitude in this case, though unit-less) while the Abstract Introduction 15 horizontal distribution is time whose unit is dependent on the unit of signal with which it is convoluted . Conclusions References

−x2 Tables Figures −1/4 iω0x Ψo(x) = π e e 2 (8)

Ψo is the Morelet wavelet distribution, x constitutes the product of time and is a scale J I term which enables us to expand or shrink the wavelet in order to increase the win- J I 20 dow for analysis. Large scale factors expand mother wavelets (increase windows and suppress high frequencies) which results in higher time but lower frequency resolution. Back Close Whereas low scale values shrink the mother wavelets increasing the frequency content and decreasing time resolution. The wavelet characteristic at diﬀerent scaling factors Full Screen / Esc helps to decompose the lake level signal at localized as well as large scale trends. The 25 practical implementation as described by Torrence et al. (1997) Eq. (9) involves the Printer-friendly Version Fourier transforms of the lake level signal xˆ and the complex conjugate of the Morelet r Interactive Discussion

Where ωr is the angular frequency below the Niquist frequency ω ≤ 2πr/N∆t, s is the wavelet scaling factor, and r 1,. . . , N − 1 is the frequency index. For normalization Information content 1/2 of Lake Tana 5 purposes, the wavelet should be multiplied by (2πs/∆t) . The wavelet transfer, Wn in Eq. (9) has a physical meaning of similarity index (Torrence et al., 1997) between water Y. Chebud and level signal and pattern of the daughter wavelet (a reproduced signal). The square of A. Melesse 2 the wavelet transform Wn (s, r∆t) at a given localized time and frequency content is known to be an indicator of the lake level ﬂuctuation spectral power, shifted in time by 10 r∆t with a dilation of the mother wavelet by scale s (Torrence et al., 1997). Title Page

Abstract Introduction 4 Methodology Conclusions References For Monte Carlo and Perturbation methods, a mean annual stage level of Lake Tana was derived from 44 years of daily data and used to calculate initial input parameters Tables Figures such as variance, initial stage level and initial volume of the lake. For the wavelet analy- 15 sis, the daily lake stage data collected from the Ethiopian Ministry of Water Resources J I was used. J I The statistical characteristic of the lake level variation was tested fitting different probabilistic distributions on the historical data. Two probabilistic distributions were fitted Back Close using the annual averaged data namely the normal and lognormal. The goodness of fit Full Screen / Esc 20 was tested based on the Konglomorov-Simirnov tests (Chakravart et al., 1967). Appro- priate the stationarity assumption of the annual lake level for Monte Carlo simulation Printer-friendly Version was tested using autocorrelation at different lag times of the annual average lake level data. Interactive Discussion Simulation using the perturbation approach was done discretizing Eqs. (4), (5) and 25 (6) which takes the same format as Eq. (7) except that values discretized at different 5533 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 2 time steps are variance of the volume (σv ), covariance of inflow and volume (iϑ) and covariance of outflow and volume (qϑ). On the other hand, Eq. (7) in its discretized HESSD format is used upon using realizations that were already generated for the perturbation 7, 5525–5546, 2010 approach. Practically, it would mean sampling all realizable values of the inflow I(t), 5 outflow Q(t) and Volume V , based on the historical mean, variance and probabilistic t Information content distributions. The realizations were sampled from the normal probability distribution of Lake Tana which was decided after the fitting in this paper. The wavelet analysis involved two implementations, one intended to reconstruct the Y. Chebud and historical record after resampling and the other for time-frequency resolution of the lake A. Melesse 10 level. Reconstructing the historical signal was done by decomposing existing signals at different sampling rates through a convolution of the mother wavelet at different scaling factors (related to the window width of the mother wavelet). The decomposed Title Page signal was first filtered at high and low frequency filters (to capture the local and global trends respectively), which was later denoised and added back, to reconstruct at the Abstract Introduction

15 respective resolutions and test predictability of lake level. Conclusions References Spectral density was resolved in time and frequency using Wigner methods (Cohen, 1989). The method uses Eq. (8) to create the mother wavelet distribution for a given Tables Figures time period and later calculates the complex conjugate of its Fourier transform. It subsequently convolves it with the Fourier transform of the observed lake level to obtain J I 20 the Wigner intensity distribution in time and frequency Eq. (9). The output is used to derive instantaneous frequency for Wigner distribution as shown in Eq. (10) (Auguer J I et al., 1995–1996). W is the Wigner distribution from Eq. (8), t is time and θ is the Back Close frequency Full Screen / Esc R+∞ θW (t,θ)dθ ω(t) = −∞ (10) R+∞ Printer-friendly Version −∞ W (t,θ)dθ Interactive Discussion

5534 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 5 Results and discussion HESSD 5.1 Significance of perturbations, Monte Carlo and deterministic methods 7, 5525–5546, 2010 With the intention to observe the fit of the data to probabilistic models, 44 years of lake elevation (1960–2003) was fitted to normal and lognormal distributions. The data is a Information content 5 monthly averaged from a daily measured value. Among the two probabilistic mod- of Lake Tana els the normal has shown a better goodness of fit at 0.01 significant levels of the 2 Konglomorov-Simrnov test. The coefficient of determination (R ) value was 0.58 and Y. Chebud and 0.47 for the normal and lognormal distributions respectively. Although, the graphical fit A. Melesse (Fig. 2a and b) is indistinguishable, which might be due to the size of annual data, the 10 normal fit was used in this study for Monte Carlo implementation. A test of the periodicity of the data was done using an autocorrelation analysis of Title Page monthly and annual data at all lag times. The analysis (using the integral of Fig. 3) Abstract Introduction showed 1.5 years of memory time and decay of the autocorrelation afterwards with no 2 periodicity of the annual averaged data. As a large water body with an area of 3150 km Conclusions References 15 and mean annual volume of 29 BCM, 1.5 years of memory time for the input and output relationship show randomness of the lake level fluctuation data. So, annual resolution Tables Figures was used for the perturbation as well as Monte Carlo approaches. The perturbation approach of the model used the annually disctetized form of J I Eqs. (3), (4), (5) and (6). Using statistical values of 5% coefficient of variation, 20 000 J I 20 realizations of inflow (river inflow and precipitation added together), out flow (gauged values at outlet added with evapotranspiration), and their respective covariance were Back Close generated. The 1960 mean annual lake volume, derived from the historical lake level Full Screen / Esc and bathymetric relationship of Lake Tana (Wale, 2008), was used as an initial value and updated for each realization of the inflow and outflow. The covariance of the flows Printer-friendly Version 25 with the lake volume was updated in parallel. The Monte Carlo approach of predictions was estimated using the same realizations substituting in Eq. (7). Interactive Discussion ANOVA of the perturbation and Monte Carlo approaches showed nonexistence of significant differences between the two at 1% of significance. Whereas the lake level 5535 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion determined using deterministic water balance approach has shown significant differences with the perturbation as well as the Monte Carlo approach even at a 10% level. HESSD The graphical plot of the lake level as well as volumetric trends showed similar patterns 7, 5525–5546, 2010 (Figs. 4a and b). The stochastic approaches have explained the lowest lake levels 5 around 1972, 1984 and 2002/2003 predicting extreme events of low lake levels. Each of these time periods were well documented drought years in Ethiopian history. Information content The lake level predicted by the perturbation as well as the Monte Carlo approach of Lake Tana reaches the maximum lake level measured at 1987 (Fig. 3a). Adding twice the stan- dard deviation on top of the mean, perturbation approach has shown a predicted level Y. Chebud and A. Melesse 10 of 1789.8 m a.s.l., which was 3.2 m short of flooding the town adjacent to the lake called Bahir Dar. The recurrence period analysis using Monte Carlo and the perturbation approaches showed that a severe flooding could occur every 96 years. The deterministic Title Page approach has extended it to 290 years for flooding. The latter number does not seem to be realistic though historical evidence is needed from other sources. Due to lack of Abstract Introduction 15 information on the position of the datum with respect to the zero out flow, no boundary condition was introduced in this study which would have contributed to the accuracy Conclusions References

of the model. Another presumable source of error in this study could be the accuracy Tables Figures of the interpolation from the bathymetry as it was derived from others studies. The propagation of the error in this respect is obvious though maximum care is taken on J I 20 the reading and resolution. Lastly, it was observed that deterministically modeled lake level and the yearly min- J I imum measured lake level fit at 1% significant test. The graphical plot of Fig. 4a Back Close showed constant lake level depicting a smoothing effect which failed to predict any of the drought years, underestimating the lake level at all times. Full Screen / Esc

25 5.2 Time-frequency decomposition of Lake Tana ﬂuctuation Printer-friendly Version

The choice of scale for the wavelet analysis was made at a range of 1–5 for testing. Interactive Discussion The result showed that Lake Tana has not encountered a change in frequency of its global patterns over the last 44 years as indicated by decomposed and reconstructed 5536 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion signals of Fig. 5a. Second, decomposition at several scales (d1–d5) and the reconstructed signal as shown by the simulated signals in Fig. 5a, (approximated by, a1...a5) HESSD at each scale suggested that Lake Tana’s stage level sampling period interval between 7, 5525–5546, 2010 daily and monthly level (d1...d5 in Fig. 5c) had no differences in simulation and pre- 5 diction at least for the wavelet analysis. Third, the exact fit of the original signal (s) and reconstructed signal (cs) in Fig. 5b) showed that Lake Tana stage level could be Information content forecasted at daily and monthly scale without being affected by the sampling rate. This of Lake Tana is a challenge for deterministic approaches that are dependent on component measurements as mentioned elsewhere. Last, early 1970s, mid 1980s and early 2000 Y. Chebud and A. Melesse 10 are characterized by rapid fluctuations as shown in decomposed signals at all scales (d1...d5) (Fig. 5c). The effects of such perturbation and possible ecological signatures are important research areas that need further analysis. Title Page The spectral power was resolved very well using Wigner time-frequency decomposition. The negative spectral values (Fig. 6, 3-D) that are cross product of the mother Abstract Introduction 15 wavelet (especially Morelet) with the signal need to be interpreted in their absolute value as suggested by Boreghetti et al. (2009). The early 1970s, mid 1980s and early Conclusions References

2000, show low frequency (high return period) events of high spectral power confirm- Tables Figures ing the previous scaled signal decomposition analysis. The intensity of the change in the amplitude could be obtained upon integrating the spectral power over the specific J I 20 time-frequency localization. The spectral power distribution viewed from the 2-D shows the peak occurrence at the 1984 drought year. It is also shown that time and frequency J I (return period) for the case of Lake Tana are related by a polynomial distribution mainly Back Close by the second power 2 (parabolic) as shown in the 2-D plot (Fig. 6) which warns us against the use of direct ranking technique for return period estimation of lake level ex- Full Screen / Esc 25 tremes. The spectral power distribution could help to sharpen questions about whether such distributions of spectral power (in Fig. 6) and noise dominated fluctuation could Printer-friendly Version cause any impact on fisheries and other aquatic life which would be an interest for other studies. Interactive Discussion

5537 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 6 Conclusion and recommendation At the outset, this study aimed simulate the stage level and volume of Lake Tana using HESSD stochastic approaches. It also intended to simulate a time-frequency break down of the 7, 5525–5546, 2010 spectral power using wavelet analysis. Probabilistically, the result showed that the two 5 stochastic approaches (Monte Carlo and perturbations approach) were significantly Information content different from deterministic water balance simulations 99% of the time. Whereas the of Lake Tana perturbation and Monte Carlo approaches have shown no significant differences at 99% of the time. Y. Chebud and For the perturbation and Monte Carlo approaches the mean annual lake level pre- A. Melesse 10 diction is relevant (for its stationrity) and hence the method serves to predict extreme events inclusive of the timing. The wavelet analysis added key knowledge on three issues. First, daily and monthly stage level of Lake Tana could be simulated easily Title Page using Morelet mother wavelet as it was shown on the fit between the observed and constructed signal. It has reproduced the observed data as exact fit and hence it is Abstract Introduction

15 recommended for further employment on forecasting of the lake level at monthly or Conclusions References daily level where neither perturbations nor Monte Carlo method could work. Second, the wavelet analysis predicts at ﬂexible sampling rate which outweighs both determin- Tables Figures istic as well as the other stochastic methods for prediction purposes which reduces error propagation. So, complementarity of the wavelet analysis with the perturbation J I 20 or Monte Carlo approach makes it appealing for a combined forecast of Lake Tana. The daily and monthly forecast is best tackled by wavelet analysis while annual lake J I level, especially extreme events, are well captured by perturbations or Monte Carlo Back Close methods. In summary, the combined use of stochastic wavelet analysis has shown the potential for employment in operational forecasting. And a further study is recom- Full Screen / Esc 25 mended to develop a dynamic approach combining perturbations and wavelet methods for operational forecasting of mean annual and daily stage level of Lake Tana. Last, the Printer-friendly Version Wigner time-frequency resolution had an added value on the spectral power content Interactive Discussion (integrated locally in time and return period) which helps for interdisciplinary analysis cross correlating it with other ecological events around the lake. 5538 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Acknowledgements. The authors would like to acknowledge the Ethiopian Minstry of Water Resources for sharing the data. HESSD 7, 5525–5546, 2010 References Auguer, F., Flandrin, P., Goncalves,` P., and Lemoine, O.: Time-Frequency Tool Box for use with Information content 5 Matlab, Reference Guide; Rice University, USA, 1995–1996. of Lake Tana Chakravarti, L. and Roy, J.: Handbook of Methods of Applied Statistics, Volume I, John Wiley and Sons, pp. 392–394, 1967. Y. Chebud and Cohen, L.: Time-Frequency Distributions-A Review, Proceedings of the IEEE, 77(7), July 1989. A. Melesse Chebud, Y. A. and Melesse, A. M.: Modeling lake stage and water balance of Lake Tana, 10 Ethiopia, Hydrol. Process., 23, 3534–3544, 2009. Christopher, Z. M.: Monte Carlo simulation, Sage Publications, Thousand Oaks, California, USA viii, 103 pp., 1997. Title Page Gelhar, L. W.: Stochastic subsurface hydrology, Printice hall, USA, 390 pp., 1993. Abstract Introduction Hautot, S., Whaler, K., Gebru, W., and Desissa, M.: The Structure of a Mesozoic basin Be- 15 neath the Lake Tana area, Ethiopia, revealed by mangetotelluric imaging, The African Earth Conclusions References Sciences, 44(3), 331–338, 2006. Kang, S. and Lin, H.: Wavelet Analysis of Hydrological and Water Quality Signals in Agricultural Tables Figures Watershed, J. Hydrol., 338, 1–14, 2007. Kebede, S. and Travi, Y.: Water balance of Lake Tana and its sensitivity to ﬂuctuations in rainfall, J I 20 Blue Nile basin, Ethiopia, J. Hydrol., 316, 233–247, 2005a. Kebede, S., Trav, Y., Alemayehu, T., and Ayenew, T.: Groundwater recharge, circulation and J I geochemical evolution in the source region of the Blue Nile River, Ethiopia, Appl. Geochem., 20(9), 1658–1676, 2005b. Back Close Tesche, T. and Gnauck, T.: Wavelet Analysis of Ecological Time Series for Riverine Lakes, Sta- 25 tistical Methods for Hydrologic Systems, Numerics: Stochastic Methods, 302, 1–21, 1998. Full Screen / Esc Torrence, C. and Compo, P. G.: A practical Gide to Wavelet Analysis, B. Am. Meteorol. Soc., 79(1), 61–78, 1997. Printer-friendly Version Yevjevich, V.: Stochastic Processes in Hydrology, Water Resources Publications, Fort Collins, pp. 276, Colorado, USA, 1972. Interactive Discussion 30 Wale, A.: Hydrological balance of Lake Tana, M.Sc Thesis, International Institute for Geo- Information Science and Earth Observation (ITC), The Netherlands, 2008. 5539 3.2. Wavelet Analysis Chebud and Melesse (2009), showed the two frequency ranges and the spectral power of Lake Tana but fall short of informing exactly when such changes of frequencies occur. In other words, the information content of the lake variation (the spectral power and return period) was not resolved in time. In this study, investigation was made on the potential of the time-frequency resolution using wavelet analysis. Wavelet analysis could detect (1) any historical changes in frequency of lake level fluctuation which could help to relate the effect of the frequency with other ecological phenomenon (signatures) since the latter is recorded in the time domain and (2) The wavelet analysis could support to decompose the signal as local and global trend which could later be reconstructed and serve for lake level forecasting.

The wavelet transform can be considered as the convolution of the signal of interest (in this case the lake level fluctuation) with the conjugate of the Fourier transform of a mother wavelet. The mother wavelet has a magnifying as well as discrimination role over the signal. And the type of mother wavelet chosen usually depends on its pattern similarity with the signal type. For hydrological analysis, the Morelet wavelet (Figure 1) was recommended by Tesche et al. (1998), Kang et al. (2007), and Torrence et al. (1997). The normalized Morelet wavelet’s distribution, eq. (8), has mean value of zero and known for its localization in time and frequency information | content. Paper Discussion | Paper Discussion | The Paper Discussion | vertical Paper Discussion axis, represented by Ψ(t) is analogous to a density function for deterministic signals HESSD 7, 5525–5546, 2010 (Cohen,1989) (a measure of amplitude in this case though unit-less) while the horizontal distribution is time whose unit is dependent on the unit of signal to beInformation content of Lake Tana convoluted with. Y. Chebud and MORELET WAVELET A. Melesse

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FigureFig. 1. 1.A MoreletA Morelet mother mother wavelet w densityavelet function density distribution function used distrib forution decomposing used for the decomposing water J I level signal.the water level signal Back Close Full Screen / Esc

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5540 5. Results and Discussion 5.1. Significance of Perturbations, Monte Carlo and deterministic methods With the intention to observe the fit of the data to probabilistic models, 44 years of lake icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion elevation (1960 – 2003) was fitted to a normal and log normal distributions. The data is a monthly averaged from a daily measured value. Among the two probabilistic models HESSD the normal has shown a better goodness of fit at 0.01 significant levels of the 7, 5525–5546, 2010 2 Konglomorov-Simrnov test. The coefficient of determination (R ) value was 0.58 and Information content 0.47 for the normal and lognormal distributions respectively. Though, the graphical fit of Lake Tana (Figures 2a and 2b) is indistinguishable which might be due to the size of annual data, Y. Chebud and the normal fit was used in this study for Monte Carlo implementation. A. Melesse

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J I Figure 2 a) Normal fit to lake level probabilistic distribution b) log normal distribution fit Fig. 2. (a) Normal ﬁt to lake level probabilistic distribution, (b) lognormal distribution ﬁt. J I A test on the periodicity of the data was done using an autocorrelation analysis of monthly and annual data at all lag times. The analysis (using integral of figure 3) Back Close showed 1.5 years of memory time and decay of the autocorrelation afterwards with no Full Screen / Esc 2 periodicity of the annual averaged data. As large water body with an area of 3150 Km Printer-friendly Version and mean annual volume of 29BCM, 1.5 years of memory time for the input and output relationship show randomness of the lake level fluctuation data. So, annual resolution Interactive Discussion was used for the perturbation as well as Monte Carlo approaches. 5541

Figure 3 Autocorrelation of annual averaged lake level for a lag time of 1 to 12 years 5. Results and Discussion 5.1. Significance of Perturbations, Monte Carlo and deterministic methods With the intention to observe the fit of the data to probabilistic models, 44 years of lake elevation (1960 – 2003) was fitted to a normal and log normal distributions. The data is a monthly averaged from a daily measured value. Among the two probabilistic models the normal has shown a better goodness of fit at 0.01 significant levels of the 2 Konglomorov-Simrnov test. The coefficient of determination (R ) value was 0.58 and 0.47 for the normal and lognormal distributions respectively. Though, the graphical fit (Figures 2a and 2b) is indistinguishable which might be due to the size of annual data, the normal fit was used in this study for Monte Carlo implementation.

Figure 2 a) Normal fit to lake level probabilistic distribution b) log normal distribution fit

A test on the periodicity of the data was done using an autocorrelation analysis of monthly and annual data at all lag times. The analysis (using integral of figure 3)

showed 1.5 years of memory time and decay of the | autocorrelation Paper Discussion | Paper Discussion | Paper Discussion | afterwards Paper Discussion with no 2 periodicity of the annual averaged data. As large water body with an area of 3150HESSD Km and mean annual volume of 29BCM, 1.5 years of memory time for the input 7,and 5525–5546, output 2010

relationship show randomness of the lake level fluctuation data. So, annualInformation resolution content was used for the perturbation as well as Monte Carlo approaches. of Lake Tana Y. Chebud and A. Melesse

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J I Fig. 3. Autocorrelation of annual averaged lake level for a lag time of 1 to 12 years. Figure 3 Autocorrelation of annual averaged lake level for a lag time of 1 to Back12 years Close

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5542 The perturbation approach of the model used the annually disctetized form of equations (3), (4), (5) and (6). Using statistical values of 5% coefficient of variation, 20, 0000 realizations of inflow (river inflow and precipitation added together), out flow (gauged values at outlet added with evapotranspiration), and their respective covariance were generated. The 1960 mean annual lake volume, derived from the historical lake level and bathymetric relationship of Lake Tana (Wale, 2008), was used as an initial value and updated for each realization of the inflow and outflow. The covariance of the flows with the lake volume was updated in parallel. The Monte Carlo approach of predictions was estimated using same realizations substituting in equation (7).

icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion ANOVA of the perturbation and Monte Carlo approaches showed nonexistence of significant differences between the two at 1% of significance. Whereas the lake level HESSD determined using deterministic water balance approach has shown significant 7, 5525–5546, 2010 differences with the perturbation as well as the Monte Carlo approach even at 10% level. The graphical plot of the lake level as well as volumetric trends showed similar Information content of Lake Tana patterns (Figures 4a and 4b). The stochastic approaches have explained lowest lake levels around 1972, 1984 and 2002/2003 predicting extreme events of low lake levels. Y. Chebud and A. Melesse Each of these time periods were well documented drought years in Ethiopian history.

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Fig. 4a.FigureLake level4a. Lake based level on perturbations,based on perturbations, Monte-Carlo Monte and- deterministicCarlo and deterministic methods. methods J I

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Interactive Discussion Figure4b. Simulated volume based on Monte-Carlo and Perturbations methods

5543 The perturbation approach of the model used the annually disctetized form of equations (3), (4), (5) and (6). Using statistical values of 5% coefficient of variation, 20, 0000 realizations of inflow (river inflow and precipitation added together), out flow (gauged values at outlet added with evapotranspiration), and their respective covariance were generated. The 1960 mean annual lake volume, derived from the historical lake level and bathymetric relationship of Lake Tana (Wale, 2008), was used as an initial value and updated for each realization of the inflow and outflow. The covariance of the flows with the lake volume was updated in parallel. The Monte Carlo approach of predictions was estimated using same realizations substituting in equation (7).

ANOVA of the perturbation and Monte Carlo approaches showed nonexistence of significant differences between the two at 1% of significance. Whereas the lake level determined using deterministic water balance approach has shown significant differences with the perturbation as well as the Monte Carlo approach even at 10% level. The graphical plot of the lake level as well as volumetric trends showed similar patterns (Figures 4a and 4b). The stochastic approaches have explained lowest lake levels around 1972, 1984 and 2002/2003 predicting extreme events of low lake levels.

Each of these time periods were well documented drought years in | Ethiopian Paper Discussion | Paper Discussion history | Paper Discussion . | Paper Discussion HESSD 7, 5525–5546, 2010

Information content of Lake Tana

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Fig. 4b. SimulatedFigure4b volume. Simulated based volume on Monte-Carlo based on and Monte perturbations-Carlo and methods.Perturbations methods J I

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and possible ecological signatures are important research areas for further analysis. Information content of Lake Tana

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Fig. 5. (a) Reconstructed signals of lake level a1... a5 based on decompositions at scales Back Close of one upFigure to 5. (b)5a.Decomposed Reconstructed signals ﬂuctuation signals for the of scales lake of 1 level to 5. (c) a1..a5Fit of the based on reconstructed and original lake level signal (Cs/s). Full Screen / Esc decompositions at scales of one up to 5. Figure 5b) Decomposed signals fluctuation for the scales of 1 to 5. Figure 5c) Fit of the reconstructed and Printer-friendly Version original lake level signal (Cs / s) Interactive Discussion

The spectral power was resolved5545 very well using Wigner time-frequency decomposition. The negative spectral values (figure 6, 3D) that are cross product of the mother wavelet (especially Morelet) with the signal need to be interpreted in their absolute value as suggested by Boreghetti et al. (2009). The early 1970’s, mid 1980’s and early 2000, show low frequency (high return period) events of high spectral power confirming the previous scaled signal decomposition analysis. The intensity of the change in the amplitude could be obtained upon integrating the spectral power over the specific time-frequency localization. The spectral power distribution viewed from the 2D shows the peak occurrence at the 1984 drought year. It was also shown that time and frequency (return period) for the case of Lake Tana are related by a polynomial distribution mainly by the second power 2 (parabolic) as shown in the 2D plot (Figure 6) which is warns the use of direct ranking technique for return period estimation of lake level extremes. The spectral power distribution could help to sharpen questions whether such distributions of spectral power (in Figure 6a or 6b) and noise dominated fluctuation could cause any impact on fisheries and other aquatic life which would be an interest for other studies.

HESSD 7, 5525–5546, 2010

Information content of Lake Tana

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J I Figure 6 spectral power 3D under frequency-time resolution of Annual Lake Level Fig. 6. Spectral power under frequency-time resolution of annual lake level spectra, 3-D and J I 2-D. spectra, 3D and 2D Back Close

6. Conclusion and Recommendation Full Screen / Esc At the outset, this study aimed at simulating the stage level and volume of Lake Tana Printer-friendly Version using stochastic approaches. It has also intended to simulate a time-frequency break Interactive Discussion down of the spectral power using wavelet analysis. Probabilistically, the result showed that the two stochastic approaches (Monte Carlo and Perturbations approach) were significantly different from deterministic5546 water balance simulations 99% of the times. Whereas the perturbation and Monte Carlo approaches have shown no significant differences at 99% of the times.

For the perturbation and Monte Carlo approaches the mean annual lake level prediction is relevant (for its stationrity) and hence the method serves predicting extreme events inclusive of the timing. The wavelet analysis added key knowledge on three issues. First, daily and monthly stage level of Lake Tana could be simulated easily using Morelet mother wavelet as it was shown on the fit between the observed and constructed signal. It has reproduced the observed data as exact fit and hence it is recommended for further employment on forecasting of the lake level at monthly or daily level where neither perturbations nor Monte Carlo method could work. Second, the wavelet analysis predicts at flexible sampling rate which outweighs both deterministic as well as the other stochastic methods for prediction purposes reducing error propagation. So, complementarity of the wavelet analysis with the perturbation, or Monte Carlo approach makes it appealing for a combined forecasting of Lake Tana. The daily and monthly forecast is best tackled by wavelet analysis while annual lake level especially extreme events are well captured by perturbations or Monte Carlo