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A principal component regression approach to simulate the bed-evolution of reservoirs
Journal of Hydrology 368 (2009) 30–41
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A principal component regression approach to simulate the bed-evolution of reservoirs T. Kebede Gurmessa a,*, A. Bárdossy b a b
Helmholtz Centre for Environmental Research – UFZ, Department of Aquatic Ecosystem Analysis and Management, Brückstrasse 3a, 39114 Magdeburg, Germany University of Stuttgart, Institute of Hydraulic Engineering, Pfaffenwaldring 61, 70550 Stuttgart, Germany
a r t i c l e
i n f o
Article history: Received 9 April 2008 Received in revised form 13 January 2009 Accepted 21 January 2009
This manuscript was handled by K. Georgakakos, Editor-in-Chief, with the assistance of Ehab A. Meselhe, Associate Editor Keywords: Reservoir sedimentation Spatio-temporal bed-evolution Numerical simulation Multivariate regression Principal components regression
s u m m a r y Long-term simulation of reservoir sedimentation suffers from process complexity, lack of data, as well as high computational cost. This work presents an efficient data-driven modeling approach to simulate the spatio-temporal dynamics of bed-evolution reservoirs using principal components regression. The daily bed-evolution of a validated numerical model was used as an input. The first four principal components contributed to some 90% of the total variance of bed-evolution. Multiple linear regression between the eigenvectors of the first four principal components with the inflow discharge, suspended sediment concentration, and differential discharge was able to reconstruct the spatio-temporal bed-evolution. Predictions with similar initial morphological condition performed reasonably. The work is a step forward to advance the assimilation of numerical and data-driven approaches in modeling long-term sedimentation of reservoirs. Ó 2009 Elsevier B.V. All rights reserved.
Introduction Long-term prediction of the amount and spatial distribution of sedimentation is essential in the planning and management of water storage reservoirs. Physically-based models, data-driven models, or combination of them can be used in order to predict long-term sedimentation of reservoirs. The physically-based modeling approach describes the physical processes involved in flow and transport by solving conservation laws of mass, momentum, and sediment transport. In the long-term simulation of reservoir sedimentation; model, parameter, and data uncertainties are involved. Furthermore, computational cost involved is very high. The process conceptualizations and solution procedures involved in the physically-based long-term simulation of bed-evolution of reservoirs is very challenging. A large number of approximation and simplification are involved in the development and application of numerical models in long-term simulation of reservoir sedimentation. These include those relevant to e.g. the incompressibility, the turbulence closure modeling, the mode of transport, the cohesiveness, the sediment gradation, the erosion and deposition processes, the model dimensions, and the numerical * Corresponding author. Tel.: +49 391 810 9674; fax: +49 391 810 9699. E-mail address: [email protected] (T. Kebede Gurmessa). 0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2009.01.033
techniques. Kebede Gurmessa (2007) presented a review of these aspects and proposed efficient procedures in long-term simulation reservoir sedimentation. Numerical models are widely applied in the state-of the-art modeling of reservoir sedimentation (e.g. Ziegler and Nisbet, 1995; Fang and Rodi, 2003; Wu et al., 2004). On the other hand, the data-driven modeling are based on a limited knowledge of the process and rely on the data describing input and output behaviors. These methods, however, are able to make abstractions and generalizations of the process and often play a complementary role to physically-based models. A simple type of data-driven model is a regression model. Coefficients of regression equation are estimated on the basis of the available existing data. Then, for a given new value of the independent variable, it gives an approximation of an output variable. Data-driven models can be highly non-linear and allow many inputs and outputs. They need a considerable amount of historical data to be trained, and if done properly, they are able not only to approximate practically any given function, but also generalize, providing a correct output of previously unseen inputs (Solomatine, 2002). Statistical methods based on the principal components analysis are used in a wide variety of technical specializations and fields of study. For instance: Hidalgo et al. (2000), Woodhouse et al. (2006) used principal components regression (PCR) procedures for dendrohydrological reconstruction of stream flows; Wotling (2000)
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used principal components (PCs) to regionalize extreme precipitation distribution; Brown (1992) studied the usefulness of principal components analysis (PCA) to laboratory chemical and petrographic data to assess physical and hydraulic properties of the carbonate-rock aquifers in central Pennsylvania, and works of Solow (2003), Pandzˇic´ and Trninic´ (1992), Bauvier (2003) can be mentioned as the application of PCA in atmospheric sciences. More recently, Samani et al. (2007) applied the combination of Levenberg–Marqurdt algorithm and PCA to improve the prediction performance and enhance the training speed of the artificial neural network for determination of aquifer parameters. Al-Alawi et al. (2008) used a combined PCR and artificial neural networks for prediction of ground-level ozone concentration. This work proposes a complementary approach that can be used in reservoir sedimentation prediction based on data-modeling. The long-term simulation of reservoir sedimentation was attempted with the concept of assimilation of physically-based numerical results and regression. The work attempted to address a simplified and computationally efficient approach to predict the spatial distribution of reservoir sedimentation. The use of PCR to simulate the spatio-temporal bed-evolution processes of a reservoir was investigated using a case study on a small daily storage reservoir. Univariate linear regression, and univariate and multivariate PCR were evaluated with respect to their performance to reconstruct the reservoir bed-evolution. Furthermore, the predictive capability of the PCR models were evaluated.
Methodology The study of a data analysis aimed at discovering how independent variables affect other dependent variables is termed as regression. It is commonly used wherever a relationship is to be formulated in any field. The method of least squares is applied for fitting regression models. When highly correlated predictors are used in a multiple linear regression model, multicollinearity (mutual correlation) can become the cause of statistical imprecision and unstable estimation of regression coefficients, incorrect rejection of variables, and numerical inaccuracies in computing the estimates of model coefficients (Jennrich, 1995). When the data matrix is large, methods to reduce the dimensionality of database can be very useful in subsequent statistical analysis. One of the most commonly used method is the projection of the database into new coordinate systems that can express the original database with possibly few dimensions as compared to the original. The central idea of PCA is to reduce the dimensionality of data sets consisting of a large number of interrelated variables, while relating as much as possible of the variation present in the data set. This is achieved by transforming the data into a new set of variables, the PCs, which are uncorrelated, and which are ordered so that the first few retain most of the variation in all of the original variables. The PCs transform an original storm of data to new coordinate systems on the principle of variance maximization. The first PC show the direction of highest variance of the database. The second PC is orthogonal to the first PCs and is in the direction of the second highest variance. The loadings on each PC correspond to its eigenvectors. The eigenvalue represent the variance of the data in the corresponding PCs. The contribution of the components to the total variation reduces from step to step. Wilks (1995), Jolliffe (2002) present the methods of choosing representative PCs without significant loss in variances. The purpose of the PCR is to estimate the values of a variable at the basis of selected PCs of explanatory variables. There are two main reasons for regressing the response variable on the PCs rather than directly on the explanatory variables. Firstly, the explanatory vari-
ables are often highly correlated and cause inaccurate estimations of the least squares regression coefficients. This can be avoided by using the uncorrelated PCs. Secondly, the dimensionality of regression is reduced by taking only a subset of the PCs for prediction. For most practical cases, a sample covariance matrix or correlation matrix are used in the PCA. The choice of correlation or covariance matrices for PCA depends on the nature of data. A major argument for using correlation rather than covariance matrices in the PCA is that the results of analysis for different sets of random variables are more directly comparable than the analysis based on the covariance matrices. The major drawback of PCA based on covariance matrices is the sensitivity of PCs to the units of measurements. Those variables whose variance are the largest will tend to dominate the PCs representing major variance of data. This might however be entirely appropriate if all the elements are measured in the same unit. The scores of PCs for each observation are given by the dot product of the original data with the matrix of eigenvectors. The standard regression model is transformed into PCR model, which replaces the predictor variables by their PCs. Because the PCs are orthogonal, the method overcomes the problem of multicolinearity (see e.g. Wilks, 1995; Jolliffe, 2002). In this work, the spatio-temporal bed-evolution of the numerical results from the numerical simulations were taken for the investigation of PCR modeling approach. For the daily input discharge Q, and suspended sediment concentration SS, the daily depth of bed-evolution at the numerical grid simulation is given by E, having the dimension ðn tÞ; where n is the node number and t is the time in days. The variance–covariance analyses (VCA) were performed on the spatio-temporal bed-evolution matrix E from which the PCA were preformed, as below.
0
E11
E12
BE B 21 B. B. @.
E22 ...
1
0
A
@.
r211 C 12 . . . C B . . . E2t C ) B C 21 r222 . . . B . . .. C .. .. C VCA B .. . . . E1t
.
.
En1 En2 . . . Ent |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} spatio-temporal ev olution
1 C 1t C 2t C C C .. C . A
. . C t1 C t2 . . . r2tt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}
0
v ariance cov ariance matrix
1
a11 a12 . . . a1t C B ) B a21 a22 . . . a2t C C B
. PCA B @ ..
.. .
ð1Þ
..
. C . .. A at1 at2 . . . att |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} eigenv ectors of ev olution
The extraction of eigenvalue–eigenvector pairs demanded 3– 5 h (on the PC AMD 3400+ processor used) depending on the size of the matrix of the spatio-temporal bed-evolution E analyzed. A PCA programmed in FORTRAN was used based on the routines in Press et al. (1992). For multiple regression MATALB was used. Data description The Lautrach reservoir is located on the river Iller of Germany between the towns of Kempten and Memmingen. The reservoir is a daily storage reservoir. Fig. 1 shows the location and aerial view of the Lautrach reservoir. The numerical model used was a depth-integrated advection dispersion equation, in which the suspended sediment module SUBIEF-2D is externally coupled with the hydrodynamic module TELEMAC-2D. The modules are part of the TELMAC-modeling system developed by French Electricity board (EDF-DRD). The data used in the numerical modeling of Lautrach reservoir were obtained from various sources: the daily flow and suspended sediment concentration of the Kempten gauging station managed by Bavarian water management administration, the daily water surface elevation at the weir from the Lech Electricity company, the
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
Fig. 1. Location and aerial view of partly emptied Lautrach reservoir.
reservoir cross-sectional profiles from the study by Al-Zoubi et al. (1997). Kebede Gurmessa (2007) presented the data used for the numerical simulation of the Lautrach reservoir to sufficient detail. The daily bed-evolution data used for the principal component regression modeling were taken from the simulations at the numerical grid points. The bed-evolution model of the Lautrach reservoir after the calibration–validation procedures were used in the prediction. The periods 1988–1992 and 1992–1996 were used as a calibration and validation steps, respectively. Further numerical prediction were also made till 2025. The large spatio-temporal matrix of the daily depth of sediment deposition/erosion at each computational grid points were considered for the statistical analysis. The number of nodes in the numerical meshes were 2614. Fig. 2a shows the daily bed-evolution at all the computational node for the bed-evolution time series of the numerical model results for the simulations 1988–1992–1996–2001. The cumulative bed-evolutions for the periods 1988–1992, 1992–1996, and 1992–2001 are shown in Fig. 2b. In modeling the spatial distribution of sedimentation using regression techniques, the relation
between the averaged daily discharge ðQ Þ and the suspended sediment concentration ðSSÞ inputs were analyzed with regard to the time series of daily depth of deposition/erosion at the computational grid of the physically-based model. The averaged daily discharge and suspended sediment concentration data of the years 1988–2005 taken from the Kempten gauging station were used for the analysis (Kebede Gurmessa, 2007). The analysis conducted using the regression techniques for the spatio-temporal bed-evolution are discussed briefly in the following sections. Analysis and results The Spatio-temporal bed-evolution of Lautrach reservoir analyzed by two approaches were evaluated. Firstly, univariate regression model was conducted between evolutions at each node, and the input daily discharge and suspended sediment concentration. Secondly, PCR, in which the dimensionality of the bed-evolution data were reduced through PCA, and regressions performed between the coefficient of the PCs with the discharge, the sediment concentration, and their differentials.
Fig. 2. Daily and cumulative bed-evolution at the computational grids of the numerical model for the period: 1988–1992, 1992–1996, and 1996–2001.
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
Univariate linear regression Univariate linear regression models were tested between daily bed-evolution at each of computational nodes of the numerical grid and time series of discharge as well as suspended sediment concentrations. For each daily discharge or sediment input, the corresponding daily bed-evolution EðQ ; x; yÞ and/or EðSS; x; yÞ computed at the numerical grid points of the physical models, yielding the regression relation of the form
EðQ ; x; yÞ ¼ b0q þ bq Q;
ð2Þ
EðSS; x; yÞ ¼ b0s þ bs SS:
ð3Þ
where the b0q ; bq ; b0s ; bs are the regression coefficients corresponding to locations (x, y). The spatial distribution of the correlation coefficients (r) between the bed-evolution and discharge is shown in the Fig. 3a and that between bed-evolution and suspended sediment concentration is shown in b. Only those points with non-zero cumulative evolution were integrated in the analysis. From the Fig. 3, there is an indication that the degree of correlation goes well with regions of predominant erosion and deposition. Negative correlations were observed in regions of predominant erosion, whereas positive correlation in regions of predominant deposition. The reconstruction of bed-evolution from the regression prediction equation was tested. Fig. 4 below shows the daily and cumulative depth of sedimentation/erosion reconstructed from the coefficients for the periods 1988–1992, compare with Fig. 2. The
r (Q-E) 0.60 0.47 0.33 0.20 0.07 -0.07 -0.20 -0.33 -0.47 -0.60
33
reconstruction of the bed-evolution from regression equation with discharge was not satisfactory. The prediction with suspended sediment resulted in similar discrepancy, but comparatively better results. PCA and PCR Linear regression between spatio-temporal bed-evolution at each of computational grid points and the input discharge and concentration as well as their fluctuation were performed by first compressing the bed-evolution data using the PCA, which substantially reduced the dimensionality of the bed-evolution data. The principal component analysis performed on the spatio-temporal bedevolution data for various periods indicated that only the first few PCs sufficiently present the database, with an excellent reduction in dimensionality. With only the first four PCs, 89%, 93%, 89.2%, and 86.08% of the variance were explained for the periods 1988–1992, 1992–1996, 1996–2001, and 1988-2001 respectively. This implies the spatio-temporal bed-evolution data of the cases under the study can be expressed well using linear combination of the first four eigenvectors. The regression analysis were therefore conducted making use of this reduction. An example is shown in Fig. 5a the first few PCs represent some 90% of the total variance indicating the possibility of developing a regression model based only on the few PCs. Fig. 5b indicates the time series coefficients (eigenvectors) of the first four PCs for the same period. The coeffi-
r (SS-E) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2
Fig. 3. Spatial distribution of the correlation coefficients for the year 1988–1992.
Fig. 4. Simple regression prediction on the bed-evolution of the year 1988–1992.
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
1
0.5
Coefficients
Variance
0.8
0.6 Variance Cumulative variance 0.4
0
PC1 PC2 PC3 PC4
0.2
0
2
4 6 8 Principal component number
10
−0.5 0
200
400 600 800 1000 1200 1400 PCs time series(d)
Fig. 5. The variances explained by the first 10 PCs and their cumulative and the eigenvectors for the period 1988–1992.
cients represent the weight of the bed-evolution on each day fulfilling the property of orthonormality. The scores of PCs is given by
U ¼ Ea;
ð4Þ
where U is the PCs scores with dimension ½n t; E is the spatiotemporal bed-evolution with dimension ½n t, and a is the eigenvector of the variance–covariance matrix of bed-evolution with the dimension ½t t. The spatial display of the scores of the first two PCs representing 77% of the total variance for the year 1988–1992 for example is shown in Fig. 6. The Figure shows that the spatial display of the scores of the first component is similar to that of the numerical bed-evolution simulation. Various regression models were investigated between the coefficients of the first four PCs and the discharge and sediment inputs. The first model investigated was a simple linear regression based on averaged daily discharge ðQ Þ and the coefficients of the first four PCs; the second was a simple linear regression between the differential discharge ðDQ Þ between consecutive days and the first four PCs; the third was the simple linear regression between daily averaged suspended sediment concentration ðSSÞ and the first four PCs; and the fourth was a multiple regression of the discharge, differential discharge, suspended sediment concentration with the coefficients of the first four PCs. The new PC coefficient matrix, an , for the case of multiple regression with the discharge Q, the differential discharge DQ, and the suspended sediment concentration SS time series is
++++ ++++++++++++++ +++ ++++ ++++++++++ ++++++++ ++ + +++++ + +++++++ ++++ ++++++++ + ++++++ + + +++++++++ ++++++++++++++++++++ + + ++++ +++++ + + +++ + + + + ++++ + +++ + + + + ++ + + ++ + + ++++ + + +++++++ ++++ + + + + + + + + + + + + + + ++ + + + + + +++ + + ++ + + + + + + + + + +++ + + + + + + ++ + + + + + + ++ + + + + + ++ ++++ + + + + + + + + + + + + + + + + + + + ++ + + + + ++ + + + + + +++ +++ + + + + + + +++++ ++ + + + + + + + + + + + + + + + ++ + +++ + + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++++++ +++++++ + + + + + + + + + + +++++++++++++++++++++ ++ + + + + + + + ++ + + ++ +++ + ++ +++++++++++ ++ + + + + + ++ + + + + + + + + + + + + + + ++ +++++++ + +++++++ ++ ++++ ++++ ++ + ++ ++++ +++ + +++ + + + + ++ + + + + + + + ++ +++++ ++++++ ++++ ++++ ++ + ++++++++ + + + + + ++++ + + + + ++ ++++++ +++ ++ ++ +++++++ ++ + ++ +++++++ +++++++++++ ++ ++ + + + + + + + + + + + ++ + + + ++ ++ +++ +++++++++ ++ ++ ++ ++ ++ ++ + ++ + + + + + + + + + + + + + + ++++++++ + ++ +++++++++ ++++++++ +++++++++ + + + + ++ ++ + + + + + + + + + + + + + + + + + +++++++++++++++++++ + +++ + ++ ++++ +++++++ ++++++++ ++ ++ ++++++ +++++ ++++++++ + ++++ + + + + + + + + + + + + + + + + + + ++ + ++++ ++ ++++++++++ +++++ +++ +++++++ ++ ++ + ++ ++++++++++++++ + + + + + + + + + + + + + + + + + + + + + + +++++++++ ++ +++ +++ ++ ++++++ ++ ++++++ ++++++ ++++++ ++++ ++ ++ +++ + + + + ++ ++ ++ +++++ ++ +++++ ++ +++++++ + + + ++ + + + + + + + + + + + + + + +++++++ + ++ ++ ++ +++++ +++++++++++++++++++++++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++++++++++++++++++++++++++++++++ + + + + ++ + ++++ ++++++++++++ ++ ++ + + + + + + + + + + + + ++++++ +++++++++++++ +++ ++++ ++ ++++++++++++++++ +++++++++++++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++ +++ ++++++ ++ ++ ++ ++ +++++ ++++ + +++++ + + ++ + + ++++ ++ + + ++ ++ +++ +++ +++ ++ ++ ++ + + ++ +++ ++ ++ + + ++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++++ ++++++ ++ +++++++++++ + + + + ++ +++++ +++++++ + + + + + + ++++++++ + + ++ + ++ ++ ++ ++ + + + + + + + ++ + + + +++++ + + + + + + + + + + + + + + + + + + ++ + + + ++ ++++++ +++++++ ++++ + + ++++++ ++++++++++ +++ + + + + + + + + + + ++++ + +++ ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + ++++ ++ ++++ + +++ + + + + + + + + + + + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + ++ +++ ++ + + + ++ + + +++ + + + + + + + + + + + + +++ + + + + + + + + ++ + + ++ + + + + + + + + + + + ++ + + + + + + + ++++ ++ + + + + + ++ + + + + + +++ + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + ++ + + + + + + + + + + ++ + +++ + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++ + + + + + +++ + + + + + + + + + + + + + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + ++ + + + + + + + + + + + + + +++ + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + +++ + + + +++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + ++ ++ + + ++ ++ ++ + + + + + + ++ + +++ + + + ++ ++ ++ + + + + + + + + + ++ + ++ + + + + + +++++ +++ + + + + + + + + + + + + + + + + + + ++++++ ++ + + + + + + + + + + ++++++++ + + + + + + + + + + + + + + + + + + +++ ++ +++ + + + + + + + +++++ + + + + + + + + + + + + + + + + + + + +++++++++++ + + + + + ++++++++++ ++ ++ + + + + + + + + + + + + + ++ + + + + + + + + + + ++ ++ + + + ++ + ++++ ++ ++ + + + + ++ + + + + + + + + ++ ++ ++ ++++++ ++ ++ +++ + + + ++ + + + ++ ++ ++ +++++ +++ + + + + + ++ ++ +++++++++++++++ ++++++++ + ++
U1 (m) 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01
an ¼ b0i þ bqi Q þ bdqi DQ þ bssi SS i ¼ 1; 4;
ð5Þ
where b0i ; bqi ; bdqi , and bssi are the regression coefficients for the component i. The reconstructed bed-evolution En is given by an inverse equation
En ¼ UaTn :
ð6Þ
This holds under the assumption of orthonormality in which the dot product aaT is a unit matrix. This simplifies the matrix inversion problem. The inverse of an orthonormal matrix is equal to its transpose. Loss of orthonormality is to be expected as a result of regression. The approach for the regression in this work is slightly different from the traditional PCR. The eigenvectors in this analysis have a dimension of time as the data matrix considered was a time series. The regressions were made with the eigenvectors rather than the scores. The loss of orthonormality as a result of regression remains unavoidable, in the same manner as when the regression are performed with the scores. Univariate PCR Regression analysis were made for three periods: 1988–1992, 1992–1996 and 1996 onwards, where the outcome of the numerical results were the basis for classification. Univariate regression between the PC scores with the discharge, suspended sediment concentration, and differential discharge were evaluated and found to be unsatisfactory in reconstruction of the spatial dynamics of
++++ ++++++++++++++ +++ ++++ ++++++++++ ++++++++ ++ + +++++ + +++++++ ++++ ++++++++ + ++++++ + + +++++++++ ++++++++++++++++++++ + + +++++++++ + + +++ + + + + ++++ + +++ + + + + ++ + + ++ + + ++++ + + +++++++ ++++ + + + + + + + + + + + + + + ++ + + + + + +++ + + ++ + + + + + + + + + +++ + + + + + + ++ + + + + + + ++ + + + + + ++ ++++ + + + + + + + + + + + + + + + + + + + ++ + + + + ++ + + + + + +++ +++ + + + + + + +++++ ++ + + + + + + + + + + + + + + + ++ + +++ + + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++++++ +++++++ + + + + + + + + + + +++++++++++++++++++++ ++ + + + + + + + ++ + + ++ +++ + ++ +++++++++++ ++ + + + + + ++ + + + + + + + + + + + + + + ++ +++++++ + +++++++ ++ ++++ ++++ ++ + ++ ++++ +++ + +++ + + + + ++ + + + + + + + ++ +++++ ++++++ ++++ ++++ ++ + ++++++++ + + + + + ++++ + + + + ++ ++++++ +++ ++ ++ +++++++ ++ + ++ +++++++ +++++++++++ ++ ++ + + + + + + + + + + + ++ + + + ++ ++ +++ +++++++++ ++ ++ ++ ++ ++ ++ + ++ + + + + + + + + + + + + + + ++++++++ + ++ +++++++++ ++++++++ +++++++++ + + + + ++ ++ + + + + + + + + + + + + + + + + + +++++++++++++++++++ + +++ + ++ ++++ +++++++ ++++++++ ++ ++ ++++++ +++++ ++++++++ + ++++ + + + + + + + + + + + + + + + + + + ++ + ++++ ++ +++++++++++++ +++++ +++ +++++++ ++ ++ + ++ ++++++++++++++ + + + + + + + + + + + + + + + + + + + + + + +++++++++ ++ +++ +++ ++++ + ++ ++++++ ++ ++++++ ++++++ ++++ ++ ++ ++ +++ + + + + ++ ++ ++ +++++ ++ +++++ ++ +++++++ + + + ++ + + + + + + + + + + + + + + +++++++ +++++ + ++ ++ ++ +++++ ++++++++++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++++++++++++++++++++++++++++++++ + + + + ++ + ++++ ++++++++++++ ++ ++ + + + + + + + + + + + + ++++++ +++++++++++++ +++ ++++ ++ ++++++++++++++++ +++++++++++++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++ +++ ++++++ ++ ++ ++ ++ +++++ ++++ + +++++ + + ++ + + ++++ ++ + + ++ ++ +++ +++ +++ ++ ++ ++ + + ++ +++ ++ ++ + + ++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++++ ++++++ ++ +++++++++++ + + + + + ++ +++++ +++++++ + + ++++++++ + + ++ ++ + ++ ++ + + + +++++++ + + + + + + + + + + + + + + + + + + + ++ + + ++++++++++++ + + + + + + ++ ++++++ +++++++ ++++ + ++ ++ + + ++++++ + + + + ++ + ++++ + +++ +++++++++ + ++ + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + ++++ ++ ++++ + +++ + + + + + + + + + + + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + ++ +++ ++ + + + ++ + + +++ + + + + + + + + + + + + +++ + + + + + + + + ++ + + ++ + + + + + + + + + + + ++ + + + + + + + ++++ ++ + + + + + ++ + + + + + +++ + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + ++ + + + + + + + + + + ++ + +++ + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++ + + + + + +++ + + + + + + + + + + + + + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + ++ + + + + + + + + + + + + + +++ + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + +++ + + + +++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + ++ ++ + + ++ ++ ++ + + + + + + ++ + +++ + + + ++ ++ ++ + + + + + + + + + ++ + ++ + + + + + +++++ +++ + + + + + + + + + + + + + + + + + + ++++++ ++ + + + + + + + + + + ++++++++ + + + + + + + + + + + + + + + + + + +++ ++ +++ + + + + + + + +++++ + + + + + + + + + + + + + + + + + + + +++++++++++ + + + + + ++++++++++ ++ ++ + + + + + + + + + + + + + ++ + + + + + + + + + + ++ ++ + + + ++ + ++++ ++ ++ + + + + ++ + + + + + + + + ++ ++ ++ ++++++ ++ ++ +++ + + + ++ + + + ++ ++ ++ +++++ +++ + + + + + ++ ++ +++++++++++++++ ++++++++ + ++
U2 (m) 0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04
Fig. 6. Scores of the first two PCs for the year 1988–1992, representing 77% of the total variance.
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
bed-evolution. The results indicated that the bed-evolution reconstruction by making use of regression of the eigenvectors with the discharge has not sufficiently reproduced the input evolution. The results at the end of the regression models were relatively in agreement with the numerical results. The results with differential discharge DQ showed an improvement as compared to regression performed with the averaged daily discharge Q. The outcome of prediction with suspended sediment and the first few PCs performed better in reconstructing the bed-evolution as compared to that made with the discharge. Multivariate PCR Regression analysis were conducted between the discharge, the differential discharge, and the suspended sediment concentration with the coefficients of the first four PCs representing 90% of the variation. Fig. 7 shows the time series of bed-evolution for all computational nodes, for the period 1988–1992 computed from the regression model, compare with the period 1988–1992 in Fig. 2b. Fig. 8 shows the spatial distribution of sedimentation for the years 1988–1992, in which a shows the numerical and b shows the PCR results. The results obtained using the multiple regression of Q ; DQ and SS, with the PC coefficients, were able to reconstruct the bed-evolution of the numerical simulations in a very good manner. For the period 1988–1992, the correlation coefficient between the spatial bed-evolution of the numerical and the PCR at the end of simulation is 0.9867.
35
Similar analyses were performed for the other periods. Fig. 9 shows the bed-evolution calculated form multiple PCR in reconstructing the bed-evolution from the numerical result for the period 1992–1996, compare with the period 1992–1996 in Fig. 2. Fig. 10 gives the spatial distribution of sedimentation at the end of the simulation period 1992–1996. For the period 1992–1996, the correlation coefficient between the spatial bed-evolution of the numerical and the PCR at the end of simulation is 0.9811. Fig. 11 shows the differential and cumulative evolution for the computational results of the year 1996–2001 obtained on reconstruction using the PCR modeling. Fig. 12 compares the spatial distribution of depth of evolutions from 1996 to 2001. For the period 1996–2001, the correlation coefficient between the spatial bedevolution of the numerical and the PCR at the end of simulation is 0.9905. The multiple linear regression among the first four PCs with the discharge, differential discharge, and suspended sediment concentration were sufficient for a very good reconstruction of spatiotemporal bed-evolution of the numerical results. Simple linear regression of the PCs were unsatisfactory in reconstruction of the bed-evolution. The inclusion of more factors influencing the reservoir sedimentation could result in better performance to reconstruct a data. Table 1 gives the regression coefficients between the eigenvectors of the first four PCs and the discharge, differential discharge and suspended sediment concentration. The coefficients of deter-
Fig. 7. Reconstruction of bed-evolution based on regression between Q, DQ, and SS with the PC coefficients, 1988–1992.
E92-E88(m)
E92-E88(m)
1.20 1.09 0.98 0.88 0.77 0.66 0.55 0.45 0.34 0.23 0.12 0.02 -0.09 -0.20
1.20 1.09 0.98 0.88 0.77 0.66 0.55 0.45 0.34 0.23 0.12 0.02 -0.09 -0.20
Fig. 8. Comparisons of spatial sedimentation between the numerical and the PCR model, 1988–1992.
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
Fig. 9. Reconstruction of bed-evolution based on multiple regression between Q, DQ and SS with the coefficients of PCs, 1992–1996.
E96-E92(m) 1.40 1.26 1.12 0.98 0.85 0.71 0.57 0.43 0.29 0.15 0.02 -0.12 -0.26 -0.40
E96-E92(m) 1.40 1.26 1.12 0.98 0.85 0.71 0.57 0.43 0.29 0.15 0.02 -0.12 -0.26 -0.40
Fig. 10. Comparisons of the spatial distribution of sedimentation using the numerical and the PCR approaches, 1992–1996.
Fig. 11. Reconstruction of bed-evolution based on multiple regression between Q, DQ and SS with the coefficients of PCs, 1996–2001.
mination ðR2 Þ were high for the regressions in the periods 1988– 1992 and 1992–1996. The period 1996–2001 indicated lower coefficient of determination, that can be a result of extreme flood event of the year 1999 showing distinct behavior as compared to the rest of the data series. The results on the use of a multiple linear regression are satisfactory in reconstruction of the bed-evolution of the
Lautrach reservoir for the periods studied. The next section proceeds with the validation, i.e. using the regression parameters, b, in a given period where the model was well reconstructed and applying them in a different period. It was seen that the coefficients were varying for the three periods investigated. This is the influence of the initial morphological conditions, flow, and sus-
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
E01-E96(m) E01-E96(m)
1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00 -1.20
1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00 -1.20
Fig. 12. Comparisons of spatial distribution of sedimentation using the numerical and the PCR approaches, 1996–2001.
Table 1 Parameter estimation for multiple regression analysis between coefficient of the PCs with discharge, change in discharge, and suspended sediment concentration. Year
PC No.
b0
bq
bdq
bss
R2
1988–1992
PC1 PC2 PC3 PC4
8.649e004 2.057e002 1.505e003 5.908e003
9.778e006 4.638e004 4.042e005 9.895e005
1.062e004 4.890e004 8.339e005 2.730e005
1.778e001 6.725e002 2.245e002 1.999e002
9.344e001 4.733e001 1.012e002 1.782e002
1992–1996
PC1 PC2 PC3 PC4
2.795e003 1.066e002 1.603e002 3.164e004
7.435e005 2.515e004 3.111e004 4.241e005
1.085e004 1.769e004 2.133e004 1.155e004
8.408e002 1.028e002 1.399e002 9.562e003
9.202e001 2.799e001 3.303e001 3.626e002
1996–2001
PC1 PC2 PC3 PC4
2.412e003 1.581e003 4.602e003 3.954e003
1.0367e004 6.302e005 4.632e006 1.051e005
1.956e004 1.258e004 7.291e005 1.193e005
1.028e002 1.961e002 8.127e003 1.317e002
9.333e002 5.177e002 5.878e003 2.823e003
pended sediment conditions that enforced the system to response differently. PCR model validation The capability of the PCR model coefficient to be applied in a period other than that of the reconstruction were investigated. The use of the coefficient of the periods 1988–1992 and 1992– 1996, (Table 1), for the modeling of the bed-evolution of the period 2001–2005 was not satisfactory. The use of the coefficients of the
period 1996–2001, for the period of 2001–2005, was however satisfactory in validating the bed-evolution using the PCR approach. This is an indication of model sensitivity to the regression coefficients which can very much be influenced by initial morphological conditions. Fig. 13 shows the comparison of the bed-evolution between numerical simulation and the PCR based on the regression coefficient of the period 1996–2001. The spatial distribution of sedimentation calculated based on the numerical and statistical method for the simulation period 2001–2005 is shown in Fig. 14.
Fig. 13. Validation of the cumulative bed-evolution of PCR as compared to the numerical simulation, 2001–2005.
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
E05-E01(m) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 -0.20 -0.30
E05-E01(m) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 -0.20 -0.30
Fig. 14. The spatial distribution of sedimentation for the numerical and the PCR approaches, validation step 2001–2005.
The validation step is quite satisfactory. The regression coefficients were determined from the sample data that is not representing the whole behavior of the population data and discrepancies are unavoidable in using the regression coefficients for predictive purposes. PCR model predictive capability Long-term prediction for the period 2005–2025 were performed by generating the new eigenvectors an using the regression coefficients estimated for the period 1996–2001 and the predicted discharge and suspended sediment concentration till 2025 (Kebede Gurmessa, 2007). Fig. 15 indicates the bed-evolution prediction. The starting period for both numerical simulation and PCR was 1996. The spatial distribution of sedimentation prediction at the end of the 2025 for numerical as well as PCR modeling is shown in the Fig. 16. The correlation coefficient of the spatial display of bed-evolution between the numerical prediction and the PC prediction at the end of the simulation is 0.84. The prediction at the end of 2025 shows the spatial distribution of sedimentation of the numerical model and the statistical model are quite similar. Typical spatial displays of the absolute differences of the bedevolution between the numerical and PCR approaches is shown in Fig. 17. For the steps of the reconstruction, the performance was very good with most differences under 10 cm (Fig. 17a and b). For the validation step, the display of the differences mostly ranged
within 20 cm, showing reasonable performance (Fig. 17c). For long-term prediction, the absolute difference in bed-evolution mostly ranged within 75 cm. This is reasonable considering the 30 years of simulation for which the daily bed-evolution accumulated. The rate of the differences was lower than 0:07 mm=d for most of the points. The cumulative overestimations/underestimations at some points were high. Assuming linear PC coefficients for the time steps of validation/prediction for which the erosion– deposition dynamics had changed increased the discrepancies. Further investigation can be made on using time dependent regression coefficients which may improve the performance of the PCR approach for prediction purposes. The highest differences were at the locations of the highest bed-evolution dynamics. The difference in bed-evolution between the numerical and PCR changed sign ðÞ without any definite trend of the erosion and deposition patterns. Discussion The bed morphological process of reservoirs is a complex dynamic process that normally approaches an equilibrium stage through time. The PCR showed an ever increasing/decreasing trend, where as, the numerical model showed an approach towards equilibrium transport through time. Generally, the results are satisfactory. Further research need to be conducted in order to improve the method. The predictive capability of the PCR may be improved by integrating the whole range
Fig. 15. Predication of the bed-evolution using the numerical and the PCR approach, 1996–2025.
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
E25-E96(m)
E25-E96(m)
2.50 2.12 1.73 1.35 0.96 0.58 0.19 -0.19 -0.58 -0.96 -1.35 -1.73 -2.12 -2.50
2.50 2.12 1.73 1.35 0.96 0.58 0.19 -0.19 -0.58 -0.96 -1.35 -1.73 -2.12 -2.50
39
Fig. 16. Prediction of bed-evolution using the numerical and the PCR approach, 1996–2025.
|Num-PCR| 92-96 (m) 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
|Num-PCR| 96-01 (m) 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
|Num-PCR| 01-05 (m)
|Num-PCR| 96-25 (m) 1.80 1.65 1.50 1.35 1.20 1.05 0.90 0.75 0.60 0.45 0.30 0.15
0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
Fig. 17. Absolute differences in the bed-evolution between the numerical and the PCR approaches.
of sedimentation behavior (population data) rather than using the sample data on which the regressions were performed. Secondly, the consideration of dynamic nature of bed-evolution, by using a
dynamic regression coefficient may improve the result satisfactorily. The use of an autoregressive moving average accounting for the trend and periodicity of the date may improve the predictive
40
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
capacity of the PCR. The regression model can also be investigated at various time scales other than the daily basis used in this work. The assimilation of data and dynamics, in this sense, reusing the numerical model and the statistical model in series can help to overcome the challenges of morphological modeling with a reasonable accuracy and is an aspect that needs further investigation. A methodology in which the PCR is based on measurement data could be formulated by conducting direct measurement of depth of sedimentation/erosion at sensitive location representing most of the bed-evolution dynamics as a response to flow and sediment inputs to reservoirs. It was found that the reservoir bed-evolution data is characterized by high reduction of dimensionality on PCA. The analysis on the time series vectors of bed-evolution ½n t and spatial bedevolution series vectors ½t n indicated that the first few PCs are sufficient to describe the large data matrix. The effect of time series of input discharge, sediment concentration, and their factors as an explanatory variables were regressed with eigenvectors having the dimension of time. For the reconstruction as well as prediction eigenvectors were recalculated using multiple regression model. The complex and non-linear bed-evolution patterns were handled with the linear PCR modeling. There exist an overestimation and/or underestimation in the calculated bed-evolution by the PCR model as compared to the numerical for each time steps. The aggregate evolution better simulated the bed-evolution of the reservoir studied. The method is therefore superior in simulating the bed-evolution in a long-term as compared to evolution of each time step. Fig. 18 shows the scatter plot of the bed-evolution between the numerical and PCR model for selected days. The scatter plot indicates that the trend at these specific time steps are reasonable. There were often scatters that does not ideally lie on the 45 line through the origin. In addition to the above mentioned reasons, the causes of discrepancy can result from the use of linear regression models in deriving new eigenvectors. The orthogonality of the estimated eigenvectors used in the reconstruction and prediction steps were not ensured. A test made on the dot product of the new eigenvectors of different PCs were in the order of 1E-2, indicating the loss of orthogonality as a result of regression model can be neglected. Furthermore, the normality of the eigenvectors of the first PC was close to unity for all the analysis. It is an open question whether the use of non-linear regression approaches to predict the new eigenvectors will improve the validity of the PCA constraints and subsequent improvements in the bed-evolution prediction. The method of assimilating the numerical method with the data-driven model can well be applied in various aspects of flow,
sediment transport, and water quality modeling. Hence, the challenges in data deficiencies and computational effort can better be managed. The concepts can further be enriched by considering the non-linear regression or training methods like ANN, with the PCR (Al-Alawi et al., 2008; Samani et al., 2007). Conclusion The spatial and temporal dynamics of reservoir sedimentation was investigated by using simple linear regression, multiple linear regression, and PCR. The PCR was found to be more effective in dealing with the spatio-temporal bed-evolution processes. The regression approach was able to reconstruct the results of the physically-based numerical simulations. The regression models performed much better for the multivariate regressions as compared to the univariate regressions. The multivariate PCR, based on the regression of the coefficients of the first four PCs with the discharge, differential discharge, and suspended sediment concentration were able to reconstruct the spatio-temporal bed-evolution efficiently. It was found that the initial morphological conditions have significant influence on the effectiveness of the PCR. Using the coefficients of the PCR model from a certain period with a different initial morphological condition to other period with a different initial morphological condition performed poorly in the prediction. The prediction for times other than the reconstruction under the same initial geometric conditions yielded acceptable results. Further research can be undertaken to enrich the research in consequent steps. The methods investigated can be further developed towards the integrated modeling of the data and the dynamics in which prediction using the regression models can be reused for a geometric updating of the numerical modeling. Non-linear regression techniques can be investigated for accounting the non-linearities existing in sediment transport processes. Acknowledgements This work was accomplished as part of the doctoral study of Tesfaye Kebede Gurmessa at institute of hydraulic engineering at the university of Stuttgart, Germany. Tesfaye Kebede Gurmessa would like to thank Prof. Bernhard Westrich for his advices during the time. Tesfaye Kebede Gurmessa also extends his thank to the German Federal Ministry of Education and Research for the study grant. The authors also heartily thank associate editor and three other anonymous reviewers for their comments.
−4
7
−4
x 10
10
6
x 10
8 6
4
PCA(m)
PCA(m)
5
3 2
4 2
1 0
0 −1 −2
0
2 Numerical(m)
4
6 −4
x 10
−2 −5
0
5 Numerical(m)
10
Fig. 18. Sample of scatter plot of the bed-evolution between the numerical and PCR model for selected times.
15 −4
x 10
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
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Contents lists available at ScienceDirect
Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol
A principal component regression approach to simulate the bed-evolution of reservoirs T. Kebede Gurmessa a,*, A. Bárdossy b a b
Helmholtz Centre for Environmental Research – UFZ, Department of Aquatic Ecosystem Analysis and Management, Brückstrasse 3a, 39114 Magdeburg, Germany University of Stuttgart, Institute of Hydraulic Engineering, Pfaffenwaldring 61, 70550 Stuttgart, Germany
a r t i c l e
i n f o
Article history: Received 9 April 2008 Received in revised form 13 January 2009 Accepted 21 January 2009
This manuscript was handled by K. Georgakakos, Editor-in-Chief, with the assistance of Ehab A. Meselhe, Associate Editor Keywords: Reservoir sedimentation Spatio-temporal bed-evolution Numerical simulation Multivariate regression Principal components regression
s u m m a r y Long-term simulation of reservoir sedimentation suffers from process complexity, lack of data, as well as high computational cost. This work presents an efficient data-driven modeling approach to simulate the spatio-temporal dynamics of bed-evolution reservoirs using principal components regression. The daily bed-evolution of a validated numerical model was used as an input. The first four principal components contributed to some 90% of the total variance of bed-evolution. Multiple linear regression between the eigenvectors of the first four principal components with the inflow discharge, suspended sediment concentration, and differential discharge was able to reconstruct the spatio-temporal bed-evolution. Predictions with similar initial morphological condition performed reasonably. The work is a step forward to advance the assimilation of numerical and data-driven approaches in modeling long-term sedimentation of reservoirs. Ó 2009 Elsevier B.V. All rights reserved.
Introduction Long-term prediction of the amount and spatial distribution of sedimentation is essential in the planning and management of water storage reservoirs. Physically-based models, data-driven models, or combination of them can be used in order to predict long-term sedimentation of reservoirs. The physically-based modeling approach describes the physical processes involved in flow and transport by solving conservation laws of mass, momentum, and sediment transport. In the long-term simulation of reservoir sedimentation; model, parameter, and data uncertainties are involved. Furthermore, computational cost involved is very high. The process conceptualizations and solution procedures involved in the physically-based long-term simulation of bed-evolution of reservoirs is very challenging. A large number of approximation and simplification are involved in the development and application of numerical models in long-term simulation of reservoir sedimentation. These include those relevant to e.g. the incompressibility, the turbulence closure modeling, the mode of transport, the cohesiveness, the sediment gradation, the erosion and deposition processes, the model dimensions, and the numerical * Corresponding author. Tel.: +49 391 810 9674; fax: +49 391 810 9699. E-mail address: [email protected] (T. Kebede Gurmessa). 0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2009.01.033
techniques. Kebede Gurmessa (2007) presented a review of these aspects and proposed efficient procedures in long-term simulation reservoir sedimentation. Numerical models are widely applied in the state-of the-art modeling of reservoir sedimentation (e.g. Ziegler and Nisbet, 1995; Fang and Rodi, 2003; Wu et al., 2004). On the other hand, the data-driven modeling are based on a limited knowledge of the process and rely on the data describing input and output behaviors. These methods, however, are able to make abstractions and generalizations of the process and often play a complementary role to physically-based models. A simple type of data-driven model is a regression model. Coefficients of regression equation are estimated on the basis of the available existing data. Then, for a given new value of the independent variable, it gives an approximation of an output variable. Data-driven models can be highly non-linear and allow many inputs and outputs. They need a considerable amount of historical data to be trained, and if done properly, they are able not only to approximate practically any given function, but also generalize, providing a correct output of previously unseen inputs (Solomatine, 2002). Statistical methods based on the principal components analysis are used in a wide variety of technical specializations and fields of study. For instance: Hidalgo et al. (2000), Woodhouse et al. (2006) used principal components regression (PCR) procedures for dendrohydrological reconstruction of stream flows; Wotling (2000)
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used principal components (PCs) to regionalize extreme precipitation distribution; Brown (1992) studied the usefulness of principal components analysis (PCA) to laboratory chemical and petrographic data to assess physical and hydraulic properties of the carbonate-rock aquifers in central Pennsylvania, and works of Solow (2003), Pandzˇic´ and Trninic´ (1992), Bauvier (2003) can be mentioned as the application of PCA in atmospheric sciences. More recently, Samani et al. (2007) applied the combination of Levenberg–Marqurdt algorithm and PCA to improve the prediction performance and enhance the training speed of the artificial neural network for determination of aquifer parameters. Al-Alawi et al. (2008) used a combined PCR and artificial neural networks for prediction of ground-level ozone concentration. This work proposes a complementary approach that can be used in reservoir sedimentation prediction based on data-modeling. The long-term simulation of reservoir sedimentation was attempted with the concept of assimilation of physically-based numerical results and regression. The work attempted to address a simplified and computationally efficient approach to predict the spatial distribution of reservoir sedimentation. The use of PCR to simulate the spatio-temporal bed-evolution processes of a reservoir was investigated using a case study on a small daily storage reservoir. Univariate linear regression, and univariate and multivariate PCR were evaluated with respect to their performance to reconstruct the reservoir bed-evolution. Furthermore, the predictive capability of the PCR models were evaluated.
Methodology The study of a data analysis aimed at discovering how independent variables affect other dependent variables is termed as regression. It is commonly used wherever a relationship is to be formulated in any field. The method of least squares is applied for fitting regression models. When highly correlated predictors are used in a multiple linear regression model, multicollinearity (mutual correlation) can become the cause of statistical imprecision and unstable estimation of regression coefficients, incorrect rejection of variables, and numerical inaccuracies in computing the estimates of model coefficients (Jennrich, 1995). When the data matrix is large, methods to reduce the dimensionality of database can be very useful in subsequent statistical analysis. One of the most commonly used method is the projection of the database into new coordinate systems that can express the original database with possibly few dimensions as compared to the original. The central idea of PCA is to reduce the dimensionality of data sets consisting of a large number of interrelated variables, while relating as much as possible of the variation present in the data set. This is achieved by transforming the data into a new set of variables, the PCs, which are uncorrelated, and which are ordered so that the first few retain most of the variation in all of the original variables. The PCs transform an original storm of data to new coordinate systems on the principle of variance maximization. The first PC show the direction of highest variance of the database. The second PC is orthogonal to the first PCs and is in the direction of the second highest variance. The loadings on each PC correspond to its eigenvectors. The eigenvalue represent the variance of the data in the corresponding PCs. The contribution of the components to the total variation reduces from step to step. Wilks (1995), Jolliffe (2002) present the methods of choosing representative PCs without significant loss in variances. The purpose of the PCR is to estimate the values of a variable at the basis of selected PCs of explanatory variables. There are two main reasons for regressing the response variable on the PCs rather than directly on the explanatory variables. Firstly, the explanatory vari-
ables are often highly correlated and cause inaccurate estimations of the least squares regression coefficients. This can be avoided by using the uncorrelated PCs. Secondly, the dimensionality of regression is reduced by taking only a subset of the PCs for prediction. For most practical cases, a sample covariance matrix or correlation matrix are used in the PCA. The choice of correlation or covariance matrices for PCA depends on the nature of data. A major argument for using correlation rather than covariance matrices in the PCA is that the results of analysis for different sets of random variables are more directly comparable than the analysis based on the covariance matrices. The major drawback of PCA based on covariance matrices is the sensitivity of PCs to the units of measurements. Those variables whose variance are the largest will tend to dominate the PCs representing major variance of data. This might however be entirely appropriate if all the elements are measured in the same unit. The scores of PCs for each observation are given by the dot product of the original data with the matrix of eigenvectors. The standard regression model is transformed into PCR model, which replaces the predictor variables by their PCs. Because the PCs are orthogonal, the method overcomes the problem of multicolinearity (see e.g. Wilks, 1995; Jolliffe, 2002). In this work, the spatio-temporal bed-evolution of the numerical results from the numerical simulations were taken for the investigation of PCR modeling approach. For the daily input discharge Q, and suspended sediment concentration SS, the daily depth of bed-evolution at the numerical grid simulation is given by E, having the dimension ðn tÞ; where n is the node number and t is the time in days. The variance–covariance analyses (VCA) were performed on the spatio-temporal bed-evolution matrix E from which the PCA were preformed, as below.
0
E11
E12
BE B 21 B. B. @.
E22 ...
1
0
A
@.
r211 C 12 . . . C B . . . E2t C ) B C 21 r222 . . . B . . .. C .. .. C VCA B .. . . . E1t
.
.
En1 En2 . . . Ent |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} spatio-temporal ev olution
1 C 1t C 2t C C C .. C . A
. . C t1 C t2 . . . r2tt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}
0
v ariance cov ariance matrix
1
a11 a12 . . . a1t C B ) B a21 a22 . . . a2t C C B
. PCA B @ ..
.. .
ð1Þ
..
. C . .. A at1 at2 . . . att |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} eigenv ectors of ev olution
The extraction of eigenvalue–eigenvector pairs demanded 3– 5 h (on the PC AMD 3400+ processor used) depending on the size of the matrix of the spatio-temporal bed-evolution E analyzed. A PCA programmed in FORTRAN was used based on the routines in Press et al. (1992). For multiple regression MATALB was used. Data description The Lautrach reservoir is located on the river Iller of Germany between the towns of Kempten and Memmingen. The reservoir is a daily storage reservoir. Fig. 1 shows the location and aerial view of the Lautrach reservoir. The numerical model used was a depth-integrated advection dispersion equation, in which the suspended sediment module SUBIEF-2D is externally coupled with the hydrodynamic module TELEMAC-2D. The modules are part of the TELMAC-modeling system developed by French Electricity board (EDF-DRD). The data used in the numerical modeling of Lautrach reservoir were obtained from various sources: the daily flow and suspended sediment concentration of the Kempten gauging station managed by Bavarian water management administration, the daily water surface elevation at the weir from the Lech Electricity company, the
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Fig. 1. Location and aerial view of partly emptied Lautrach reservoir.
reservoir cross-sectional profiles from the study by Al-Zoubi et al. (1997). Kebede Gurmessa (2007) presented the data used for the numerical simulation of the Lautrach reservoir to sufficient detail. The daily bed-evolution data used for the principal component regression modeling were taken from the simulations at the numerical grid points. The bed-evolution model of the Lautrach reservoir after the calibration–validation procedures were used in the prediction. The periods 1988–1992 and 1992–1996 were used as a calibration and validation steps, respectively. Further numerical prediction were also made till 2025. The large spatio-temporal matrix of the daily depth of sediment deposition/erosion at each computational grid points were considered for the statistical analysis. The number of nodes in the numerical meshes were 2614. Fig. 2a shows the daily bed-evolution at all the computational node for the bed-evolution time series of the numerical model results for the simulations 1988–1992–1996–2001. The cumulative bed-evolutions for the periods 1988–1992, 1992–1996, and 1992–2001 are shown in Fig. 2b. In modeling the spatial distribution of sedimentation using regression techniques, the relation
between the averaged daily discharge ðQ Þ and the suspended sediment concentration ðSSÞ inputs were analyzed with regard to the time series of daily depth of deposition/erosion at the computational grid of the physically-based model. The averaged daily discharge and suspended sediment concentration data of the years 1988–2005 taken from the Kempten gauging station were used for the analysis (Kebede Gurmessa, 2007). The analysis conducted using the regression techniques for the spatio-temporal bed-evolution are discussed briefly in the following sections. Analysis and results The Spatio-temporal bed-evolution of Lautrach reservoir analyzed by two approaches were evaluated. Firstly, univariate regression model was conducted between evolutions at each node, and the input daily discharge and suspended sediment concentration. Secondly, PCR, in which the dimensionality of the bed-evolution data were reduced through PCA, and regressions performed between the coefficient of the PCs with the discharge, the sediment concentration, and their differentials.
Fig. 2. Daily and cumulative bed-evolution at the computational grids of the numerical model for the period: 1988–1992, 1992–1996, and 1996–2001.
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
Univariate linear regression Univariate linear regression models were tested between daily bed-evolution at each of computational nodes of the numerical grid and time series of discharge as well as suspended sediment concentrations. For each daily discharge or sediment input, the corresponding daily bed-evolution EðQ ; x; yÞ and/or EðSS; x; yÞ computed at the numerical grid points of the physical models, yielding the regression relation of the form
EðQ ; x; yÞ ¼ b0q þ bq Q;
ð2Þ
EðSS; x; yÞ ¼ b0s þ bs SS:
ð3Þ
where the b0q ; bq ; b0s ; bs are the regression coefficients corresponding to locations (x, y). The spatial distribution of the correlation coefficients (r) between the bed-evolution and discharge is shown in the Fig. 3a and that between bed-evolution and suspended sediment concentration is shown in b. Only those points with non-zero cumulative evolution were integrated in the analysis. From the Fig. 3, there is an indication that the degree of correlation goes well with regions of predominant erosion and deposition. Negative correlations were observed in regions of predominant erosion, whereas positive correlation in regions of predominant deposition. The reconstruction of bed-evolution from the regression prediction equation was tested. Fig. 4 below shows the daily and cumulative depth of sedimentation/erosion reconstructed from the coefficients for the periods 1988–1992, compare with Fig. 2. The
r (Q-E) 0.60 0.47 0.33 0.20 0.07 -0.07 -0.20 -0.33 -0.47 -0.60
33
reconstruction of the bed-evolution from regression equation with discharge was not satisfactory. The prediction with suspended sediment resulted in similar discrepancy, but comparatively better results. PCA and PCR Linear regression between spatio-temporal bed-evolution at each of computational grid points and the input discharge and concentration as well as their fluctuation were performed by first compressing the bed-evolution data using the PCA, which substantially reduced the dimensionality of the bed-evolution data. The principal component analysis performed on the spatio-temporal bedevolution data for various periods indicated that only the first few PCs sufficiently present the database, with an excellent reduction in dimensionality. With only the first four PCs, 89%, 93%, 89.2%, and 86.08% of the variance were explained for the periods 1988–1992, 1992–1996, 1996–2001, and 1988-2001 respectively. This implies the spatio-temporal bed-evolution data of the cases under the study can be expressed well using linear combination of the first four eigenvectors. The regression analysis were therefore conducted making use of this reduction. An example is shown in Fig. 5a the first few PCs represent some 90% of the total variance indicating the possibility of developing a regression model based only on the few PCs. Fig. 5b indicates the time series coefficients (eigenvectors) of the first four PCs for the same period. The coeffi-
r (SS-E) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2
Fig. 3. Spatial distribution of the correlation coefficients for the year 1988–1992.
Fig. 4. Simple regression prediction on the bed-evolution of the year 1988–1992.
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1
0.5
Coefficients
Variance
0.8
0.6 Variance Cumulative variance 0.4
0
PC1 PC2 PC3 PC4
0.2
0
2
4 6 8 Principal component number
10
−0.5 0
200
400 600 800 1000 1200 1400 PCs time series(d)
Fig. 5. The variances explained by the first 10 PCs and their cumulative and the eigenvectors for the period 1988–1992.
cients represent the weight of the bed-evolution on each day fulfilling the property of orthonormality. The scores of PCs is given by
U ¼ Ea;
ð4Þ
where U is the PCs scores with dimension ½n t; E is the spatiotemporal bed-evolution with dimension ½n t, and a is the eigenvector of the variance–covariance matrix of bed-evolution with the dimension ½t t. The spatial display of the scores of the first two PCs representing 77% of the total variance for the year 1988–1992 for example is shown in Fig. 6. The Figure shows that the spatial display of the scores of the first component is similar to that of the numerical bed-evolution simulation. Various regression models were investigated between the coefficients of the first four PCs and the discharge and sediment inputs. The first model investigated was a simple linear regression based on averaged daily discharge ðQ Þ and the coefficients of the first four PCs; the second was a simple linear regression between the differential discharge ðDQ Þ between consecutive days and the first four PCs; the third was the simple linear regression between daily averaged suspended sediment concentration ðSSÞ and the first four PCs; and the fourth was a multiple regression of the discharge, differential discharge, suspended sediment concentration with the coefficients of the first four PCs. The new PC coefficient matrix, an , for the case of multiple regression with the discharge Q, the differential discharge DQ, and the suspended sediment concentration SS time series is
++++ ++++++++++++++ +++ ++++ ++++++++++ ++++++++ ++ + +++++ + +++++++ ++++ ++++++++ + ++++++ + + +++++++++ ++++++++++++++++++++ + + ++++ +++++ + + +++ + + + + ++++ + +++ + + + + ++ + + ++ + + ++++ + + +++++++ ++++ + + + + + + + + + + + + + + ++ + + + + + +++ + + ++ + + + + + + + + + +++ + + + + + + ++ + + + + + + ++ + + + + + ++ ++++ + + + + + + + + + + + + + + + + + + + ++ + + + + ++ + + + + + +++ +++ + + + + + + +++++ ++ + + + + + + + + + + + + + + + ++ + +++ + + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++++++ +++++++ + + + + + + + + + + +++++++++++++++++++++ ++ + + + + + + + ++ + + ++ +++ + ++ +++++++++++ ++ + + + + + ++ + + + + + + + + + + + + + + ++ +++++++ + +++++++ ++ ++++ ++++ ++ + ++ ++++ +++ + +++ + + + + ++ + + + + + + + ++ +++++ ++++++ ++++ ++++ ++ + ++++++++ + + + + + ++++ + + + + ++ ++++++ +++ ++ ++ +++++++ ++ + ++ +++++++ +++++++++++ ++ ++ + + + + + + + + + + + ++ + + + ++ ++ +++ +++++++++ ++ ++ ++ ++ ++ ++ + ++ + + + + + + + + + + + + + + ++++++++ + ++ +++++++++ ++++++++ +++++++++ + + + + ++ ++ + + + + + + + + + + + + + + + + + +++++++++++++++++++ + +++ + ++ ++++ +++++++ ++++++++ ++ ++ ++++++ +++++ ++++++++ + ++++ + + + + + + + + + + + + + + + + + + ++ + ++++ ++ ++++++++++ +++++ +++ +++++++ ++ ++ + ++ ++++++++++++++ + + + + + + + + + + + + + + + + + + + + + + +++++++++ ++ +++ +++ ++ ++++++ ++ ++++++ ++++++ ++++++ ++++ ++ ++ +++ + + + + ++ ++ ++ +++++ ++ +++++ ++ +++++++ + + + ++ + + + + + + + + + + + + + + +++++++ + ++ ++ ++ +++++ +++++++++++++++++++++++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++++++++++++++++++++++++++++++++ + + + + ++ + ++++ ++++++++++++ ++ ++ + + + + + + + + + + + + ++++++ +++++++++++++ +++ ++++ ++ ++++++++++++++++ +++++++++++++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++ +++ ++++++ ++ ++ ++ ++ +++++ ++++ + +++++ + + ++ + + ++++ ++ + + ++ ++ +++ +++ +++ ++ ++ ++ + + ++ +++ ++ ++ + + ++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++++ ++++++ ++ +++++++++++ + + + + ++ +++++ +++++++ + + + + + + ++++++++ + + ++ + ++ ++ ++ ++ + + + + + + + ++ + + + +++++ + + + + + + + + + + + + + + + + + + ++ + + + ++ ++++++ +++++++ ++++ + + ++++++ ++++++++++ +++ + + + + + + + + + + ++++ + +++ ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + ++++ ++ ++++ + +++ + + + + + + + + + + + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + ++ +++ ++ + + + ++ + + +++ + + + + + + + + + + + + +++ + + + + + + + + ++ + + ++ + + + + + + + + + + + ++ + + + + + + + ++++ ++ + + + + + ++ + + + + + +++ + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + ++ + + + + + + + + + + ++ + +++ + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++ + + + + + +++ + + + + + + + + + + + + + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + ++ + + + + + + + + + + + + + +++ + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + +++ + + + +++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + ++ ++ + + ++ ++ ++ + + + + + + ++ + +++ + + + ++ ++ ++ + + + + + + + + + ++ + ++ + + + + + +++++ +++ + + + + + + + + + + + + + + + + + + ++++++ ++ + + + + + + + + + + ++++++++ + + + + + + + + + + + + + + + + + + +++ ++ +++ + + + + + + + +++++ + + + + + + + + + + + + + + + + + + + +++++++++++ + + + + + ++++++++++ ++ ++ + + + + + + + + + + + + + ++ + + + + + + + + + + ++ ++ + + + ++ + ++++ ++ ++ + + + + ++ + + + + + + + + ++ ++ ++ ++++++ ++ ++ +++ + + + ++ + + + ++ ++ ++ +++++ +++ + + + + + ++ ++ +++++++++++++++ ++++++++ + ++
U1 (m) 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01
an ¼ b0i þ bqi Q þ bdqi DQ þ bssi SS i ¼ 1; 4;
ð5Þ
where b0i ; bqi ; bdqi , and bssi are the regression coefficients for the component i. The reconstructed bed-evolution En is given by an inverse equation
En ¼ UaTn :
ð6Þ
This holds under the assumption of orthonormality in which the dot product aaT is a unit matrix. This simplifies the matrix inversion problem. The inverse of an orthonormal matrix is equal to its transpose. Loss of orthonormality is to be expected as a result of regression. The approach for the regression in this work is slightly different from the traditional PCR. The eigenvectors in this analysis have a dimension of time as the data matrix considered was a time series. The regressions were made with the eigenvectors rather than the scores. The loss of orthonormality as a result of regression remains unavoidable, in the same manner as when the regression are performed with the scores. Univariate PCR Regression analysis were made for three periods: 1988–1992, 1992–1996 and 1996 onwards, where the outcome of the numerical results were the basis for classification. Univariate regression between the PC scores with the discharge, suspended sediment concentration, and differential discharge were evaluated and found to be unsatisfactory in reconstruction of the spatial dynamics of
++++ ++++++++++++++ +++ ++++ ++++++++++ ++++++++ ++ + +++++ + +++++++ ++++ ++++++++ + ++++++ + + +++++++++ ++++++++++++++++++++ + + +++++++++ + + +++ + + + + ++++ + +++ + + + + ++ + + ++ + + ++++ + + +++++++ ++++ + + + + + + + + + + + + + + ++ + + + + + +++ + + ++ + + + + + + + + + +++ + + + + + + ++ + + + + + + ++ + + + + + ++ ++++ + + + + + + + + + + + + + + + + + + + ++ + + + + ++ + + + + + +++ +++ + + + + + + +++++ ++ + + + + + + + + + + + + + + + ++ + +++ + + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++++++ +++++++ + + + + + + + + + + +++++++++++++++++++++ ++ + + + + + + + ++ + + ++ +++ + ++ +++++++++++ ++ + + + + + ++ + + + + + + + + + + + + + + ++ +++++++ + +++++++ ++ ++++ ++++ ++ + ++ ++++ +++ + +++ + + + + ++ + + + + + + + ++ +++++ ++++++ ++++ ++++ ++ + ++++++++ + + + + + ++++ + + + + ++ ++++++ +++ ++ ++ +++++++ ++ + ++ +++++++ +++++++++++ ++ ++ + + + + + + + + + + + ++ + + + ++ ++ +++ +++++++++ ++ ++ ++ ++ ++ ++ + ++ + + + + + + + + + + + + + + ++++++++ + ++ +++++++++ ++++++++ +++++++++ + + + + ++ ++ + + + + + + + + + + + + + + + + + +++++++++++++++++++ + +++ + ++ ++++ +++++++ ++++++++ ++ ++ ++++++ +++++ ++++++++ + ++++ + + + + + + + + + + + + + + + + + + ++ + ++++ ++ +++++++++++++ +++++ +++ +++++++ ++ ++ + ++ ++++++++++++++ + + + + + + + + + + + + + + + + + + + + + + +++++++++ ++ +++ +++ ++++ + ++ ++++++ ++ ++++++ ++++++ ++++ ++ ++ ++ +++ + + + + ++ ++ ++ +++++ ++ +++++ ++ +++++++ + + + ++ + + + + + + + + + + + + + + +++++++ +++++ + ++ ++ ++ +++++ ++++++++++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++++++++++++++++++++++++++++++++ + + + + ++ + ++++ ++++++++++++ ++ ++ + + + + + + + + + + + + ++++++ +++++++++++++ +++ ++++ ++ ++++++++++++++++ +++++++++++++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++ +++ ++++++ ++ ++ ++ ++ +++++ ++++ + +++++ + + ++ + + ++++ ++ + + ++ ++ +++ +++ +++ ++ ++ ++ + + ++ +++ ++ ++ + + ++ + +++ + + + + + + + + + + + + + + + + + + + + + + ++++ ++++++ ++ +++++++++++ + + + + + ++ +++++ +++++++ + + ++++++++ + + ++ ++ + ++ ++ + + + +++++++ + + + + + + + + + + + + + + + + + + + ++ + + ++++++++++++ + + + + + + ++ ++++++ +++++++ ++++ + ++ ++ + + ++++++ + + + + ++ + ++++ + +++ +++++++++ + ++ + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + ++++ ++ ++++ + +++ + + + + + + + + + + + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + ++ +++ ++ + + + ++ + + +++ + + + + + + + + + + + + +++ + + + + + + + + ++ + + ++ + + + + + + + + + + + ++ + + + + + + + ++++ ++ + + + + + ++ + + + + + +++ + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + ++ + + + + + + + + + + ++ + +++ + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + ++ + + + + + +++ + + + + + + + + + + + + + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++ + + ++ + + + + + + + + + + + + + +++ + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + +++ + + + +++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + ++ ++ + + ++ ++ ++ + + + + + + ++ + +++ + + + ++ ++ ++ + + + + + + + + + ++ + ++ + + + + + +++++ +++ + + + + + + + + + + + + + + + + + + ++++++ ++ + + + + + + + + + + ++++++++ + + + + + + + + + + + + + + + + + + +++ ++ +++ + + + + + + + +++++ + + + + + + + + + + + + + + + + + + + +++++++++++ + + + + + ++++++++++ ++ ++ + + + + + + + + + + + + + ++ + + + + + + + + + + ++ ++ + + + ++ + ++++ ++ ++ + + + + ++ + + + + + + + + ++ ++ ++ ++++++ ++ ++ +++ + + + ++ + + + ++ ++ ++ +++++ +++ + + + + + ++ ++ +++++++++++++++ ++++++++ + ++
U2 (m) 0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04
Fig. 6. Scores of the first two PCs for the year 1988–1992, representing 77% of the total variance.
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
bed-evolution. The results indicated that the bed-evolution reconstruction by making use of regression of the eigenvectors with the discharge has not sufficiently reproduced the input evolution. The results at the end of the regression models were relatively in agreement with the numerical results. The results with differential discharge DQ showed an improvement as compared to regression performed with the averaged daily discharge Q. The outcome of prediction with suspended sediment and the first few PCs performed better in reconstructing the bed-evolution as compared to that made with the discharge. Multivariate PCR Regression analysis were conducted between the discharge, the differential discharge, and the suspended sediment concentration with the coefficients of the first four PCs representing 90% of the variation. Fig. 7 shows the time series of bed-evolution for all computational nodes, for the period 1988–1992 computed from the regression model, compare with the period 1988–1992 in Fig. 2b. Fig. 8 shows the spatial distribution of sedimentation for the years 1988–1992, in which a shows the numerical and b shows the PCR results. The results obtained using the multiple regression of Q ; DQ and SS, with the PC coefficients, were able to reconstruct the bed-evolution of the numerical simulations in a very good manner. For the period 1988–1992, the correlation coefficient between the spatial bed-evolution of the numerical and the PCR at the end of simulation is 0.9867.
35
Similar analyses were performed for the other periods. Fig. 9 shows the bed-evolution calculated form multiple PCR in reconstructing the bed-evolution from the numerical result for the period 1992–1996, compare with the period 1992–1996 in Fig. 2. Fig. 10 gives the spatial distribution of sedimentation at the end of the simulation period 1992–1996. For the period 1992–1996, the correlation coefficient between the spatial bed-evolution of the numerical and the PCR at the end of simulation is 0.9811. Fig. 11 shows the differential and cumulative evolution for the computational results of the year 1996–2001 obtained on reconstruction using the PCR modeling. Fig. 12 compares the spatial distribution of depth of evolutions from 1996 to 2001. For the period 1996–2001, the correlation coefficient between the spatial bedevolution of the numerical and the PCR at the end of simulation is 0.9905. The multiple linear regression among the first four PCs with the discharge, differential discharge, and suspended sediment concentration were sufficient for a very good reconstruction of spatiotemporal bed-evolution of the numerical results. Simple linear regression of the PCs were unsatisfactory in reconstruction of the bed-evolution. The inclusion of more factors influencing the reservoir sedimentation could result in better performance to reconstruct a data. Table 1 gives the regression coefficients between the eigenvectors of the first four PCs and the discharge, differential discharge and suspended sediment concentration. The coefficients of deter-
Fig. 7. Reconstruction of bed-evolution based on regression between Q, DQ, and SS with the PC coefficients, 1988–1992.
E92-E88(m)
E92-E88(m)
1.20 1.09 0.98 0.88 0.77 0.66 0.55 0.45 0.34 0.23 0.12 0.02 -0.09 -0.20
1.20 1.09 0.98 0.88 0.77 0.66 0.55 0.45 0.34 0.23 0.12 0.02 -0.09 -0.20
Fig. 8. Comparisons of spatial sedimentation between the numerical and the PCR model, 1988–1992.
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
Fig. 9. Reconstruction of bed-evolution based on multiple regression between Q, DQ and SS with the coefficients of PCs, 1992–1996.
E96-E92(m) 1.40 1.26 1.12 0.98 0.85 0.71 0.57 0.43 0.29 0.15 0.02 -0.12 -0.26 -0.40
E96-E92(m) 1.40 1.26 1.12 0.98 0.85 0.71 0.57 0.43 0.29 0.15 0.02 -0.12 -0.26 -0.40
Fig. 10. Comparisons of the spatial distribution of sedimentation using the numerical and the PCR approaches, 1992–1996.
Fig. 11. Reconstruction of bed-evolution based on multiple regression between Q, DQ and SS with the coefficients of PCs, 1996–2001.
mination ðR2 Þ were high for the regressions in the periods 1988– 1992 and 1992–1996. The period 1996–2001 indicated lower coefficient of determination, that can be a result of extreme flood event of the year 1999 showing distinct behavior as compared to the rest of the data series. The results on the use of a multiple linear regression are satisfactory in reconstruction of the bed-evolution of the
Lautrach reservoir for the periods studied. The next section proceeds with the validation, i.e. using the regression parameters, b, in a given period where the model was well reconstructed and applying them in a different period. It was seen that the coefficients were varying for the three periods investigated. This is the influence of the initial morphological conditions, flow, and sus-
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
E01-E96(m) E01-E96(m)
1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00 -1.20
1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00 -1.20
Fig. 12. Comparisons of spatial distribution of sedimentation using the numerical and the PCR approaches, 1996–2001.
Table 1 Parameter estimation for multiple regression analysis between coefficient of the PCs with discharge, change in discharge, and suspended sediment concentration. Year
PC No.
b0
bq
bdq
bss
R2
1988–1992
PC1 PC2 PC3 PC4
8.649e004 2.057e002 1.505e003 5.908e003
9.778e006 4.638e004 4.042e005 9.895e005
1.062e004 4.890e004 8.339e005 2.730e005
1.778e001 6.725e002 2.245e002 1.999e002
9.344e001 4.733e001 1.012e002 1.782e002
1992–1996
PC1 PC2 PC3 PC4
2.795e003 1.066e002 1.603e002 3.164e004
7.435e005 2.515e004 3.111e004 4.241e005
1.085e004 1.769e004 2.133e004 1.155e004
8.408e002 1.028e002 1.399e002 9.562e003
9.202e001 2.799e001 3.303e001 3.626e002
1996–2001
PC1 PC2 PC3 PC4
2.412e003 1.581e003 4.602e003 3.954e003
1.0367e004 6.302e005 4.632e006 1.051e005
1.956e004 1.258e004 7.291e005 1.193e005
1.028e002 1.961e002 8.127e003 1.317e002
9.333e002 5.177e002 5.878e003 2.823e003
pended sediment conditions that enforced the system to response differently. PCR model validation The capability of the PCR model coefficient to be applied in a period other than that of the reconstruction were investigated. The use of the coefficient of the periods 1988–1992 and 1992– 1996, (Table 1), for the modeling of the bed-evolution of the period 2001–2005 was not satisfactory. The use of the coefficients of the
period 1996–2001, for the period of 2001–2005, was however satisfactory in validating the bed-evolution using the PCR approach. This is an indication of model sensitivity to the regression coefficients which can very much be influenced by initial morphological conditions. Fig. 13 shows the comparison of the bed-evolution between numerical simulation and the PCR based on the regression coefficient of the period 1996–2001. The spatial distribution of sedimentation calculated based on the numerical and statistical method for the simulation period 2001–2005 is shown in Fig. 14.
Fig. 13. Validation of the cumulative bed-evolution of PCR as compared to the numerical simulation, 2001–2005.
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
E05-E01(m) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 -0.20 -0.30
E05-E01(m) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 -0.20 -0.30
Fig. 14. The spatial distribution of sedimentation for the numerical and the PCR approaches, validation step 2001–2005.
The validation step is quite satisfactory. The regression coefficients were determined from the sample data that is not representing the whole behavior of the population data and discrepancies are unavoidable in using the regression coefficients for predictive purposes. PCR model predictive capability Long-term prediction for the period 2005–2025 were performed by generating the new eigenvectors an using the regression coefficients estimated for the period 1996–2001 and the predicted discharge and suspended sediment concentration till 2025 (Kebede Gurmessa, 2007). Fig. 15 indicates the bed-evolution prediction. The starting period for both numerical simulation and PCR was 1996. The spatial distribution of sedimentation prediction at the end of the 2025 for numerical as well as PCR modeling is shown in the Fig. 16. The correlation coefficient of the spatial display of bed-evolution between the numerical prediction and the PC prediction at the end of the simulation is 0.84. The prediction at the end of 2025 shows the spatial distribution of sedimentation of the numerical model and the statistical model are quite similar. Typical spatial displays of the absolute differences of the bedevolution between the numerical and PCR approaches is shown in Fig. 17. For the steps of the reconstruction, the performance was very good with most differences under 10 cm (Fig. 17a and b). For the validation step, the display of the differences mostly ranged
within 20 cm, showing reasonable performance (Fig. 17c). For long-term prediction, the absolute difference in bed-evolution mostly ranged within 75 cm. This is reasonable considering the 30 years of simulation for which the daily bed-evolution accumulated. The rate of the differences was lower than 0:07 mm=d for most of the points. The cumulative overestimations/underestimations at some points were high. Assuming linear PC coefficients for the time steps of validation/prediction for which the erosion– deposition dynamics had changed increased the discrepancies. Further investigation can be made on using time dependent regression coefficients which may improve the performance of the PCR approach for prediction purposes. The highest differences were at the locations of the highest bed-evolution dynamics. The difference in bed-evolution between the numerical and PCR changed sign ðÞ without any definite trend of the erosion and deposition patterns. Discussion The bed morphological process of reservoirs is a complex dynamic process that normally approaches an equilibrium stage through time. The PCR showed an ever increasing/decreasing trend, where as, the numerical model showed an approach towards equilibrium transport through time. Generally, the results are satisfactory. Further research need to be conducted in order to improve the method. The predictive capability of the PCR may be improved by integrating the whole range
Fig. 15. Predication of the bed-evolution using the numerical and the PCR approach, 1996–2025.
T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
E25-E96(m)
E25-E96(m)
2.50 2.12 1.73 1.35 0.96 0.58 0.19 -0.19 -0.58 -0.96 -1.35 -1.73 -2.12 -2.50
2.50 2.12 1.73 1.35 0.96 0.58 0.19 -0.19 -0.58 -0.96 -1.35 -1.73 -2.12 -2.50
39
Fig. 16. Prediction of bed-evolution using the numerical and the PCR approach, 1996–2025.
|Num-PCR| 92-96 (m) 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
|Num-PCR| 96-01 (m) 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
|Num-PCR| 01-05 (m)
|Num-PCR| 96-25 (m) 1.80 1.65 1.50 1.35 1.20 1.05 0.90 0.75 0.60 0.45 0.30 0.15
0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
Fig. 17. Absolute differences in the bed-evolution between the numerical and the PCR approaches.
of sedimentation behavior (population data) rather than using the sample data on which the regressions were performed. Secondly, the consideration of dynamic nature of bed-evolution, by using a
dynamic regression coefficient may improve the result satisfactorily. The use of an autoregressive moving average accounting for the trend and periodicity of the date may improve the predictive
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T. Kebede Gurmessa, A. Bárdossy / Journal of Hydrology 368 (2009) 30–41
capacity of the PCR. The regression model can also be investigated at various time scales other than the daily basis used in this work. The assimilation of data and dynamics, in this sense, reusing the numerical model and the statistical model in series can help to overcome the challenges of morphological modeling with a reasonable accuracy and is an aspect that needs further investigation. A methodology in which the PCR is based on measurement data could be formulated by conducting direct measurement of depth of sedimentation/erosion at sensitive location representing most of the bed-evolution dynamics as a response to flow and sediment inputs to reservoirs. It was found that the reservoir bed-evolution data is characterized by high reduction of dimensionality on PCA. The analysis on the time series vectors of bed-evolution ½n t and spatial bedevolution series vectors ½t n indicated that the first few PCs are sufficient to describe the large data matrix. The effect of time series of input discharge, sediment concentration, and their factors as an explanatory variables were regressed with eigenvectors having the dimension of time. For the reconstruction as well as prediction eigenvectors were recalculated using multiple regression model. The complex and non-linear bed-evolution patterns were handled with the linear PCR modeling. There exist an overestimation and/or underestimation in the calculated bed-evolution by the PCR model as compared to the numerical for each time steps. The aggregate evolution better simulated the bed-evolution of the reservoir studied. The method is therefore superior in simulating the bed-evolution in a long-term as compared to evolution of each time step. Fig. 18 shows the scatter plot of the bed-evolution between the numerical and PCR model for selected days. The scatter plot indicates that the trend at these specific time steps are reasonable. There were often scatters that does not ideally lie on the 45 line through the origin. In addition to the above mentioned reasons, the causes of discrepancy can result from the use of linear regression models in deriving new eigenvectors. The orthogonality of the estimated eigenvectors used in the reconstruction and prediction steps were not ensured. A test made on the dot product of the new eigenvectors of different PCs were in the order of 1E-2, indicating the loss of orthogonality as a result of regression model can be neglected. Furthermore, the normality of the eigenvectors of the first PC was close to unity for all the analysis. It is an open question whether the use of non-linear regression approaches to predict the new eigenvectors will improve the validity of the PCA constraints and subsequent improvements in the bed-evolution prediction. The method of assimilating the numerical method with the data-driven model can well be applied in various aspects of flow,
sediment transport, and water quality modeling. Hence, the challenges in data deficiencies and computational effort can better be managed. The concepts can further be enriched by considering the non-linear regression or training methods like ANN, with the PCR (Al-Alawi et al., 2008; Samani et al., 2007). Conclusion The spatial and temporal dynamics of reservoir sedimentation was investigated by using simple linear regression, multiple linear regression, and PCR. The PCR was found to be more effective in dealing with the spatio-temporal bed-evolution processes. The regression approach was able to reconstruct the results of the physically-based numerical simulations. The regression models performed much better for the multivariate regressions as compared to the univariate regressions. The multivariate PCR, based on the regression of the coefficients of the first four PCs with the discharge, differential discharge, and suspended sediment concentration were able to reconstruct the spatio-temporal bed-evolution efficiently. It was found that the initial morphological conditions have significant influence on the effectiveness of the PCR. Using the coefficients of the PCR model from a certain period with a different initial morphological condition to other period with a different initial morphological condition performed poorly in the prediction. The prediction for times other than the reconstruction under the same initial geometric conditions yielded acceptable results. Further research can be undertaken to enrich the research in consequent steps. The methods investigated can be further developed towards the integrated modeling of the data and the dynamics in which prediction using the regression models can be reused for a geometric updating of the numerical modeling. Non-linear regression techniques can be investigated for accounting the non-linearities existing in sediment transport processes. Acknowledgements This work was accomplished as part of the doctoral study of Tesfaye Kebede Gurmessa at institute of hydraulic engineering at the university of Stuttgart, Germany. Tesfaye Kebede Gurmessa would like to thank Prof. Bernhard Westrich for his advices during the time. Tesfaye Kebede Gurmessa also extends his thank to the German Federal Ministry of Education and Research for the study grant. The authors also heartily thank associate editor and three other anonymous reviewers for their comments.
−4
7
−4
x 10
10
6
x 10
8 6
4
PCA(m)
PCA(m)
5
3 2
4 2
1 0
0 −1 −2
0
2 Numerical(m)
4
6 −4
x 10
−2 −5
0
5 Numerical(m)
10
Fig. 18. Sample of scatter plot of the bed-evolution between the numerical and PCR model for selected times.
15 −4
x 10
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