Faculty of Bioscience Engineering Academic year 2014-2015
Assessment of data assimilation on hydraulic simulations of the river Demer
Joline De Smedt Promotor: Prof. Dr. ir. Niko Verhoest Tutor: ir. Katrien Van Eerdenbrugh, ir. Niels De Vleeschouwer
Master’s dissertation submitted in fulfillment of the requirements for the degree of Master of Science in Bioscience Engineering: Land and Water Management
Faculty of Bioscience Engineering Academic year 2014-2015
Assessment of data assimilation on hydraulic simulations of the river Demer
Joline De Smedt Promotor: Prof. Dr. ir. Niko Verhoest Tutor: ir. Katrien Van Eerdenbrugh, ir. Niels De Vleeschouwer
Master’s dissertation submitted in fulfillment of the requirements for the degree of Master of Science in Bioscience Engineering: Land and Water Management De auteur en promotor geven de toelating deze scriptie voor consultatie beschikbaar te stellen en delen ervan te kopi¨erenvoor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting uitdrukkelijk de bron te vermelden bij het aanhalen van resultaten uit deze scriptie.
The author and promoter give the permission to use this thesis for consultation and to copy parts of it for personal use. Every other use is subject to the copyright laws, more specifically the source must be extensively specified when using results from this thesis.
Ghent, June 2015
De promotor, De tutoren,
Prof. dr. ir. Niko Verhoest ir. Katrien Van Eerdenbrugh, ir. Niels De Vleeschouwer
De auteur,
Joline De Smedt Preface
In front of you lies the final piece of the puzzle on how to become a bio-engineer. As with all puzzles, this one also consists of many pieces. Hard work, focus and the necessary complications were a few of those pieces but the majority consisted of making new friends, getting to know my own strengths and weaknesses, enjoying life and gaining more and more interest in the subject of land and water management. The latter would not have been possible without the interesting courses, given by professors and assistants passionate about their field of study. Their devotion and enthusiasm convinced me to attend almost every class, over and over again in the last five years. I would especially like to thank professor Niko Verhoest. Not only did he increase my interest in water related topics, he also provided me with help and guidance on my thesis in this last year. I would also like to thank my tutors Katrien Van Eerdenbrugh and Niels De Vleeschouwer. The patience with which they answered all of my questions and the time they took to help and guide me in the right direction are strongly appreciated. Also the company DHI and in particular Henrik Madsen should not be forgotten. The stay they offered me in their main office in Denmark gave me the opportunity to learn more about the MIKE 11 model as well as provided me with the chance of visiting the beautiful country of Denmark. A special thanks goes to my parents. I understand that it is not self-evident to trust and support a daughter during six years of study, five years of living on her own in Ghent and half a year of living in Sweden on Erasmus. They did this however without ever doubting me or my capacity of completing this adventure. I would also like to thank them and my sister for their patience with me during the completion of my thesis although they did not completely understand what I was doing. Next, I would like to thank Alexandra Van Wesemael. Thank you for sharing the good as well ad the bad moments in our thesis time together, for sheering me up with songs, cookies and advice and for not losing hope in these last months. Finally, I would like to thank all of my friends for supporting me and forcing me to look on the bright side of life, in particular Lotte Moens. Thank you Lotte, for bringing me cake and listening to my endless complaining when I needed it. Also a thank you to Jonas Uyttersprot, who was always there for me even though he had his own thesis to write. I couldn’t have completed this task without the support and guidance of all of them.
Joline De Smedt, June 2015
i ii Abstract
To be able to prepare for the future and protect ourselves from possible disasters, forecasts of future rainfall, temperature, discharge in a river system and many more environmental variables are issued. Data assimilation is in these cases applied to improve the predictions, resulting in a better preparation and protection against upcoming problems. It can thus be said that the impact of data assimilation is significant. In this study, a comparison is made between the two data assimilation algorithms implemented in the DA software of MIKE 11 created by DHI. Moreover both the application of observations of discharge as the application of observations of water level are tested. An analysis is made of the Kalman gain obtained from the EnKF simulation run, while for the WF the amplitude A and the interval of influence are examined. Finally the improvements in the model prediction are validated using a synthetic experiment. The results of this study indicate that, though the time required to complete a model run is considerably larger compared to the WF, a slightly more accurate prediction of the true state is obtained using the EnKF DA algorithm. A substantial improvement is already achieved with the application of the WF DA algorithm despite the poor approximation of the true shape of the Kalman gain by the implemented gain functions. In addition, no significant effect of the shape of the gain function was found. It is however not possible to procure an estimation of the uncertainty of the model run with a WF simulation. Finally, the implementation of DA procedures on the model of the Demer river revealed issues regarding the stability of the DA software. Especially with the application of the EnKF procedure with observations of water level, severe instabilities were encountered using the model of the Demer river.
iii iv Samenvatting
Om de mensheid en zijn omgeving te beschermen tegen mogelijke rampen, worden op regel- matige basis voorspellingen uitgevoerd van verschillende variabelen zoals neerslag, temperatuur, debiet in een rivier en dergelijke meer. Hierbij wordt vaak data assimilatie toegepast om betere voorspellingen te bekomen. Dit leidt tot de mogelijkheid om een betere voorbereiding te treffen en dus op die manier een betere bescherming te garanderen tegen wat in de toekomst zou kunnen voorkomen. In dit onderzoek worden twee data assimilatie procedures bekeken die reeds zijn ge¨ımplementeerd in de MIKE 11 software ontwikkeld door DHI. Daarnaast wordt ook het gebruik van zowel tijdsreeksen van debiet als het gebruik van tijdsreeksen van waterhoogte ge¨evalueerd. Een analyse van de Kalman gain verkregen uit de Ensemble Kalman filter (EnKF) simulaties wordt uitgevoerd. Ook wordt er onderzoek gevoerd naar de amplitude voor de weighting func- tion (WF) simulatie en het interval waarover een invloed merkbaar is. Als laatste worden de verbeteringen in de modelvoorspelling gevalideerd met behulp van een synthetisch experiment. Uit deze studie blijkt dat een iets meer accurate voorspelling wordt bekomen wanneer gebruikt wordt gemaakt van de EnKF procedure dan wanneer de WF procedure wordt aangewend. Hi- ertegenover staat dan wel dat de simulatie met de EnKF procedure beduidend langer duurt. Ook de voorspelling met de WF toont een duidelijke verbetering tegenover de voorspelling zon- der aanwending van data assimilatie, al moet hierbij worden opgemerkt dat de verschillende mogelijke vormen van de gain functies uit de WF geen goede representatie zijn voor het verloop van de Kalman gain. Echter, aangezien wordt aangetoond dat de vorm van de gain functie in dit geval weinig tot geen invloed heeft op de grootte van de aanpassing, vormt dit geen probleem. Daarnaast is het ook zo dat met deze DA procedure het niet mogelijk is om een inschatting te verkrijgen van de onzekerheid op de modelvoorspelling. Een laatste bevinding uit dit onderzoek is dat bij gebruik van de DA procedures mogelijks problemen met de stabiliteit ondervonden worden. Voornamelijk bij de implementatie van de EnKF procedure met gebruik van waterhoogte in het model van de Demer komen ernstige instabiliteiten voor.
v vi Contents
Preface i
Abstract iii
Samenvatting v
List of Figures ix
List of Tables xiii
List of Abbreviations xv
1 Introduction 1
2 Brief introduction to data assimilation 3 2.1 The weighting function ...... 4 2.2 The Kalman filter ...... 6 2.2.1 Discrete Kalman filter ...... 7 2.2.2 Extended Kalman Filter ...... 8 2.2.3 Ensemble Kalman Filter ...... 10
3 Location 13 3.1 The Demer and its catchment ...... 13 3.2 Water management ...... 15 3.3 Calibration and validation data ...... 15
4 The model 17 4.1 Reduced model ...... 18 4.2 Boundaries ...... 20 4.3 Initial conditions ...... 21 4.4 Uncertainty ...... 21 4.5 Model forecast ...... 23
vii Contents viii 5 Methodology 25 5.1 Step 1 and 2: Ensemble Kalman filter ...... 25 5.2 Step 3 to 6: Weighting function ...... 26 5.3 Step 7: Validation ...... 27
6 Results and discussion 29 6.1 EnKF with observations of discharge ...... 29 6.2 EnKF with observations of water level ...... 30 6.3 Analysis of Kalman gain K ...... 30 6.3.1 General observations ...... 32 6.3.2 Evolution of K ...... 36 6.4 Amplitude A ...... 44 6.5 Interval of influence ...... 45 6.6 WF with observations of discharge ...... 48 6.7 WF with observations of water level ...... 57 6.8 Validation ...... 60 6.8.1 WF vs EnKF performance ...... 65 6.8.2 Limitations ...... 68
7 Conclusion 69 7.1 Suggestions for future research ...... 70
Bibliography 73
Additional figures 77 List of Figures
2.1 Illustration of the weighting function for a measurement location (Madsen & Skotner, 2005)...... 6 2.2 Overview of the Kalman filter updating procedure (Welch & Bishop, 1997). . . . 8 2.3 Overview of the Extended Kalman filter updating procedure (Welch & Bishop, 1997)...... 10
3.1 Location of the Demer catchment, its subcatchments and the main tributaries of the Demer in Flanders (Bogman et al., 2013)...... 14 3.2 Discharge on the Demer in the hydraulic stations of Aarschot (number 122), Zichem (number 123) and Diest (number 126) for November 2010...... 16
4.1 Dijle, Demer and its tributaries implemented in the MIKE 11 model of the Demer (Cornu, 2014)...... 19 4.2 Measured discharge and simulated discharge on the Demer in the hydraulic station of Diest (number 126)...... 21 4.3 Schematic representation of the combination of the weighting function data as- similation procedure and the error forecast procedure (DHI, 2014a)...... 23
6.1 Simulation of the EnKF with discharge measurements in both observation points at hydraulic station 122 in Aarschot downstream of the mills, together with the measurement and the simulation without DA at this point...... 31 6.2 Simulation of the EnKF with with discharge measurements in both observation points at hydraulic station 123 in Zichem, together with the measurement and the simulation without DA at this point...... 31 6.3 Maximal and mean absolute Kalman gain of the EnKF simulation with observa- tions of discharge at the two observation points. The mean and standard deviation are taken of the K along the length of the river...... 34 6.4 Maximal and mean absolute Kalman gain of the EnKF simulation with obser- vations of discharge without unstable values at the two observation points. The mean and standard deviation are taken of the K along the length of the river. . . 34 6.5 Simulation without DA and observation together with their standard deviation at both observation points for water level. In the bottom figures the magnitude of the standard deviation is represented...... 35
ix List of Figures x
6.6 Top: Mean evolution of K in space with standard deviation and indication of the important chainages along the river Demer for hydraulic station 122. Bottom: Evolution of K for discharge in time and space at hydraulic station 122. . . . . 37 6.7 Top: Mean evolution of K in space with standard deviation and indication of the important chainages along the river Demer for hydraulic station 123. Bottom: Evolution of K for discharge in time and space at hydraulic station 123. . . . . 37 6.8 Variation of the spread of the 90 % uncertainty interval for the EnKF with mea- surements of discharge relative to the spread of the 90 % uncertainty interval at hydraulic station 122 in time and space...... 41 6.9 Variation of the spread of the 90 % uncertainty interval for the EnKF with mea- surements of discharge relative to the spread of the 90 % uncertainty interval at hydraulic station 123 in time and space...... 41 6.10 Evolution of discharge in time at a chainage of 44 505 m in the river Demer. . . . 42 6.11 Evolution of K for discharge in time and space at hydraulic station 123 with a color scale to bring out the variation in the discharge peak...... 44 6.12 Top: Evolution of K for discharge in space at hydraulic station 122 together with the different shapes of the gain function. Bottom: Evolution of K for discharge in space at hydraulic station 123 together with the different shapes of the gain function...... 47 6.13 Evolution of K in time and space at hydraulic station 122, together with the evolution of discharge in time at station 122...... 49 6.14 Evolution of K in time and space at hydraulic station 123, together with the evolution of discharge in time at station 123...... 49 6.15 Graphs of the discharge in function of time for the three different weighting func- tions with one observation point in hydraulic station 122. The result of the EnKF simulation and the simulation without DA is also presented for each point. The chainages of points 1 to 5 are respectively 37073.5 m, 42004.5 m, 47034m, 51478 m and 56443 m...... 51 6.16 Graphs of the discharge in function of time for the three different weighting func- tions with one observation point in hydraulic station 123. The result of the EnKF simulation and the simulation without DA is also presented for each point. The chainages of points 1 to 5 are respectively 29585.5 m, 31024 m 32553.5 m, 41072 m and 54032 m...... 52 6.17 Indication of the locations of the points in figures 6.15, 6.16, 6.22 and 6.23. The green numbers indicate the locations of the points in figure 6.15, the blue numbers show the locations of the points in figure 6.16, while the locations of figures 6.22 and 6.23 are represented by the green points 1-3 and the blue points 1-5 (numbered as point 4-8 in the graphs). The orange dots represent the locations of both the observation points...... 53 List of Figures xi
6.18 Total assimilation update (value of the gain function at that location multiplied by the innovation) at green points 1-3 in figure 6.17 for the three gain functions of the weighting function with an update at hydraulic station 122...... 54
6.19 Total assimilation update (value of the gain function at that location multiplied by the innovation) at blue points 1-3 in figure 6.17 for the three gain functions of the weighting function with an update at hydraulic station 123...... 54
6.20 Cumulative difference in discharge for the green points 1-3 in figure 6.17 in time. The difference is calculated between the WF with application of the three gain functions (update at hydraulic station 122) and the simulation without DA. . . . 55
6.21 Cumulative difference in discharge for the blue points 1-3 in figure 6.17 in time. The difference is calculated between the WF with application of the three gain functions (update at hydraulic station 123) and the simulation without DA. . . . 55
6.22 (Part 1) Graphs of the discharge in function of time for the weighting function with updating on both observation points. In the observation point at Zichem all three gain functions are implemented while for the observation point at Aarschot a constant gain was implemented. The result of the EnKF simulation and the simulation without DA is also presented for each point. The chainages of points 1 to 8 are respectively 29585.5 m, 31024 m 32553.5 m, 37073.5 m, 42004.5 m, 47034m, 51478 m and 56443 m...... 58
6.23 (Part 2) Graphs of the discharge in function of time for the weighting function with updating on both observation points. In the observation point at Zichem all three gain functions are implemented while for the observation point at Aarschot a constant gain was implemented. The result of the EnKF simulation and the simulation without DA is also presented for each point. The chainages of points 1 to 8 are respectively 29585.5 m, 31024 m 32553.5 m, 37073.5 m, 42004.5 m, 47034m, 51478 m and 56443 m...... 59
6.24 Left: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA and the ’true’ time series of discharge on the one hand and the RSR, NSE and PBIAS of the difference between the WF with updating at all time steps and the ’true’ time series of discharge on the other hand. Right: Evolution in space of the RSR, NSE and PBIAS of the difference between simu- lation without DA, WF with updating at all time steps and WF with updating until the 18th of November, and the ’true’ time series of discharge. Updating with the WF was performed at both observation points...... 61
6.25 Evolution of discharge for three different simulations (without DA, with the WF with updates at all time steps and the WF with updates until the 18th of Novem- ber) and the truth at chainage 44977 m. Updating with the WF was performed at both observation points...... 63 List of Figures xii
6.26 Evolution of discharge for three different simulations (without DA, with the WF with updates at all time steps and the WF with updates until the 18th of Novem- ber) and the truth at chainage 52002 m. Updating with the WF was performed at both observation points...... 63 6.27 Left: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA and the ’true’ time series of discharge on the one hand and the RSR, NSE and PBIAS of the difference between the EnKF with updating at all time steps and the ’true’ time series of discharge on the other hand. Right: Evolution in space of the RSR, NSE and PBIAS of the difference between simula- tion without DA, EnKF with updating at all time steps and EnKF with updating until the 18th of November, and the ’true’ time series of discharge. Updating with the WF and the EnKF was performed at both observation points...... 64 6.28 Left: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA and the ’true’ time series of discharge on the one hand and the RSR, NSE and PBIAS of the difference for both the EnKF and the WF with updating at all time steps with the ’true’ time series of discharge on the other hand. Middle: Same as on the left figure but for the period of 18th of November 12h until the 19th of November 12h. Right: Same as on the middle figure but for the WF and EnKF with updating until the 18th of November. Updating with the WF and the EnKF was performed with both observation points...... 66
A.1 Water level profile and cross-section of the river Demer for the EnKF simulation with the maximum water level (Red line) and the minimal water level (Green line). 78 A.2 Water level profile and cross-section of the river Demer at the mills in Aarschot for the EnKF simulation with the maximum water level (Red line) and the minimum water level (Green line) and indications of the chainage...... 79 A.3 Discharge profile and cross-section of the river Demer for the EnKF simulation with the maximum discharge (Red line) and the minimum discharge (Green line). 80 A.4 Bottom left: Evolution of K for discharge in time and space at hydraulic station 122. Top left: Mean evolution of K in space with standard deviation. Bottom right: Mean evolution of K in time with standard deviation and maximal K. . . 81 A.5 Bottom left: Evolution of K for discharge in time and space at hydraulic station 123. Top left: Mean evolution of K in space with standard deviation. Bottom right: Mean evolution of K in time with standard deviation and maximal K. . . 82 A.6 Change of Kalman gain in space with mean, minimal, maximal Kalman gain and K+std...... 83 List of Tables
4.1 Important hydraulic structures placed on the Demer and included in the MIKE 11 model (Bogman et al., 2011)...... 19
5.1 Description of the hydrologic stations where measurements are available for DA. 26 5.2 Description of the boundary conditions and their characteristics implemented in MIKE 11 for DA...... 27
6.1 Overview of changes in K indicated in figure 6.6 with their cause and chainage. . 38 6.2 Overview of changes in K indicated in figure 6.7 with their cause and chainage. . 38 6.3 Mean values of the goodness-of-fit indicators over a part of the river Demer. For the simulation without DA, the WF and the EnKF the mean was taken over the most upstream part until the hydraulic station 123 at Zichem (US section), in between the two observation points (Middle section) and over the downstream part of the river starting at the hydraulic station 122 at Aarschot (DS section). Updating with the WF and the EnKF was performed with both observation points. 68
xiii xiv xv
List of Abbreviations
A
AR1 First order autoregressive model C
CWF Weighting function with implementation of the constant gain function D
DA Data assimilation
DHI Danish Hydraulic Institute
DS Downstream
DTM Digital terrain model E
EKF Extended Kalman filter
EnKF Ensemble Kalman filter
EWF Weighting function with implementation of the mixed exponential gain function H
HIC Hydrological Information Centre K
KF Kalman filter L
LL Lower limit M
MP Measuring point List of Abbreviations xvi
N
NAM model North American Mesoscale model
NSE Nash-Sutcliffe efficiency P
PBIAS Percent bias R
RSR Root mean squared error-observations standard deviation ratio T
TAW Tweede algemene waterpassing
TWF Weighting function with implementation of the triangular gain function U
UL Upper limit
US Upstream W
WF Weighting function Chapter 1
Introduction
The implementation of models has conquered its place in scientific research by now. Undoubt- edly the use of a model to represent the world around us has a lot of benefits. Among others, it is for example possible to simulate variables that are hard to measure in real life or to run several scenarios to investigate different options. Modelling makes the study of these variables a lot easier, enables us to forecast the future and reveals unknown possibilities. A new theory might be confirmed by the simulations or researchers might get new insights in the remarkable wonders of our surroundings. Although the advantages are numerous, they also possess a num- ber of disadvantages. For instance, all model results are characterised by uncertainty. Processes that are not included or wrongly modelled, an error in the calibration and uncertainty in the parameters can give rise to uncertainty in our model results. However, not only the model predictions contain errors, measurements are also prone to uncertainty because of an inaccu- rate measurement device, climate conditions, human mistakes... Data assimilation (DA) tries to resolve the inaccuracy of the model results as well as the inaccuracy of the observations by combining both model predictions and measurements, based on their uncertainties. This will lead to more accurate results and thus also to better predictions. The improvement of model predictions is especially important in flood forecasting (Vrugt et al., 2006) where the lives of people might depend on this forecast. The same applies to global change (Bengtsson & Shukla, 1988). More accurate predictions will provide a better insight in how we are managing our planet and how planet Earth will respond.
Over the years, data assimilation (DA) has evolved: from simple direct insertion (Heathman et al., 2003) where model results are directly replaced by observations, and nudging of the model results to the observations in Newtonian nudging (Houser et al., 1998), to more compli- cated forms such as Kalman filtering (Kalman, 1960), where the model error is calculated while simulating the results. This thesis only deals with the updating methods implemented in the software of MIKE 11 designed by the Danish Hydraulic Institute (DHI): the weighting function and the Kalman filtering procedure. The choice for these two DA methods results from the simple, fast calculations obtained with the weighting function on the one hand and the slower but possibly more accurate predictions resulting from the Kalman filter on the other hand. De-
1 Chapter 1. Introduction 2 pending on the application, one updating procedure is more suitable than the other.
This thesis aims to identify what method to use in which situation by comparing both DA procedures in terms of calculation speed, most accurate approximation of the real situation and other factors. The aim of this thesis is also to examine the consequences of applying measurements of water level compared to the utilization of discharge data in the updating procedure. In this study, the MIKE 11 software designed by the Danish Hydraulic Institute (DHI) is applied to a model of the Demer basin in Belgium. The research will be outlined in this thesis in different consecutive steps. After a short introduction to data assimilation in the second chapter, the location of the studied region is described more in detail in the third chapter. A description of the applied model is provided in the fourth chapter while the fifth chapter explains the methodology. The last chapters six and seven respectively deal with the obtained results and provide a discussion and some conclusions on this subject. Chapter 2
Brief introduction to data assimilation
Nowadays, data assimilation can be found in a wide range of research fields. The most impor- tant applications however, are found in the field of numerical weather prediction, climate studies (Kanamitsu, 1989) and flood forecasting as studied in this thesis. If the obtained model forecasts are to be used in decision making, a correct prediction is of great importance. The uncertainty on the forecast should therefore be as small as possible. However, due to uncertainties in the meteorological forcings and model parameters, and errors or oversimplifications in the model physics, a certain level of uncertainty on the model forecasts will be present. This uncertainty can generally be reduced by the use of data assimilation or model state updating. Data assim- ilation (DA) is a technique used to update the modeled observations related to a certain state of a system, using externally obtained data sets. The updating procedure will try to correct for differences between the predicted data and the real situation, represented by observations, when making a new forecast. Based on the uncertainty accompanying these measurements and predictions, the state variables of the system are updated (Plaza Guingla, 2014).
Before data assimilation can be applied, a numerical model must be implemented. A general form of a numerical model without assimilation can be formulated as
xk = M(xk−1, uk, wk) (2.1) where M( · ) is the model operator necessary to bring the model from time step k − 1 to k by solving the mathematical equations. The state vector at time step k − 1 is represented by the variable xk−1, the model forcing at time step k by uk and wk stands for the uncertainty of the modeled system. The latter encloses model uncertainty coming from physical conceptualization and mathematical approximations of the governing processes, the use of non-optimal parameter values and errors in the model forcing. Based on the state at a previous time step, a set of model parameters and forcing data, the model will create an estimate of the state of a system at time k using equation 2.1 (Madsen et al., 2003, 2006; Ridler et al., 2014).
3 Chapter 2. Brief introduction to data assimilation 4
To perform data assimilation, observations of the state variable at different locations have to be available in an observation measurement vector y. The model state can then be related to the observations using the measurement operator H such that
yk = Hk(xk, vk) (2.2) with measurement error vk (Ridler et al., 2014).
Data assimilation procedures appear in different forms and perform an update on different variables, for example model state and model parameters (Madsen & Skotner, 2005; Ridler et al., 2014). A general form for updating of variables based on data assimilation, can be found in equation 2.3.
+ − − xk = xk + Kk(yk − Hkxk ) (2.3) − + Where xk is the a priori state and xk the a posteriori state, i.e. the state value after update. The gain Kk compares the level of reliability that accompanies the forecast and the observations and reflects how much weight should be accounted to each (Ridler et al., 2014). Depending on the data assimilation procedure chosen, this gain will be computed with a specific function.
2.1 The weighting function
As a basic concept to start from, a most simple updating technique called direct insertion is used. Here, the updating consists of directly replacing the model states with the observations as it is assumed that the measurements are 100 % reliable. The gain Kk, defined in equation 2.3 is thus equal to one for this method. A general formula for direct insertion is represented by equation 2.4.
+ − − xk = xk + 1 ∗ (yk − Hkxk ) (2.4) Moreover, because correlation of the variable at one point with nearby points is not taken into account, updating is only executed at points where observations are available (Houser et al., 1998; Walker et al., 2001). This is of course not a correct representation of reality. Since the observations are inserted directly, abrupt changes in the model state can occur. These sudden alterations can lead to instabilities in the simulation. Consequently, this technique has several disadvantages. To cope with these problems, Newtonian nudging provides a solution. In this procedure, the model state is nudged towards the observed state by adding a term to the prognostic equation, that is proportional to the difference between the two states. In its general form, this nudging is represented in equation 2.3. The correction of the model state occurs during a certain period of time surrounding the observations, which can be either gridded observations or randomly spaced observations. Here, the gain in equation 2.3 consists of a nudging factor G, a weighting function W and a quality factor γ. Chapter 2. Brief introduction to data assimilation 5
Stauffer & Seaman (1990) represent the Newtonian nudging scheme as
N P 2 − Wi γi(yk − Hkxk ) x+ = x− + G ∗ i=1 (2.5) k k N P Wi i=1 where subscript i denotes the ith location of a total of N locations with an observation. The first factor G in equation 2.5 represents the magnitude of the nudging term relative to all other model processes, W induces a weighting based on the temporal and spatial variability of the observations while the quality factor accounts for the characteristic error in measurement sys- tems and representativeness (Stauffer & Seaman, 1990; Houser et al., 1998). Consequently, this means that the model results are updated with observations that are corrected for their quality, and at the same time these corrected results are weighted in four dimensions. Furthermore, not only state variables at observed locations but also at non-observed locations can be updated as a function of their distance to the observations. Newtonian nudging thus provides a more accurate representation of reality than direct insertion. Resembling the Newtonian nudging algorithm, the weighting function is implemented in the MIKE 11 software package as a first option to execute data assimilation for real-time flow forecasting. According to Madsen & Skotner (2005), the method is based on an explicit description of the Kalman gain vector. To represent the spatial distribution of model errors, pre-defined functions are implemented in Kk.
In the weighting function, the gain vector Kk, as mentioned in equation 2.3, has a predefined shape and is assumed to be constant. This results into an updating procedure only slightly more expensive than a normal model run. In MIKE 11, the gain vector Kk is called a gain function g.
Different gain functions are possible: a constant (g1), a triangular (g2), and a mixed exponential
(g3) distribution (Madsen & Skotner, 2005). Each gain function g is a function of their chainage, meaning the distance to the upstream end of the river (DHI, 2014a,b).
g1(ξ) = A, ξUS ≤ ξ ≤ ξDS (2.6a) ξ−ξUS A , ξUS ≤ ξ ≤ ξMP ξMP −ξUS g2(ξ) = (2.6b) ξDS −ξ A , ξMP ≤ ξ ≤ ξDS ξDS −ξMP 0 ξMP −ξ ξ = , ξUS ≤ ξ ≤ ξMP −9 02 ξMP −ξUS g3(ξ) = A exp ξ = (2.6c) 2 0 ξ−ξMP ξ = , ξMP ≤ ξ ≤ ξDS ξDS −ξMP
with g(ξ) indicating the gain value at chainage ξ, while ξUS, ξDS and ξMP represent the chainage at the upstream (US) and downstream (DS) bounds of the gain function and the chainage of the measuring point (MP) respectively. A represents the gain amplitude. This amplitude A has a value between zero and one and signifies the reliability of the measurement compared to the model forecast. For an amplitude of one, it is supposed that the measurements are perfect Chapter 2. Brief introduction to data assimilation 6 whereas if the amplitude is near zero, the measurements are believed to be very uncertain and consequently the forecast is attributed a lot more confidence. An illustration of the gain function in case of the weighting function can be found in figure 2.1. The shape and the boundaries of the gain function reflect the correlation between the error of the forecast on the measurement − location with the errors on the nearby locations. The parts of the vector xk corresponding to the measurement variable are explicitly updated by equation 2.3 (e.g. if discharge is measured, the discharge grid points are updated). The other variables in the vector are implicitly updated according to the numerical scheme in the next time step (e.g. if discharge is measured, water level points are updated) (Madsen & Skotner, 2005).
Figure 2.1: Illustration of the weighting function for a measurement location (Madsen & Skotner, 2005).
Applying a constant gain vector, makes the implementation a lot more straightforward. Al- though this simplification is not a realistic assumption, Madsen & Skotner (2005) describe the weighting function as a robust, accurate and efficient updating technique. According to their findings, it provides accurate forecast results when compared to forecasting without updating. Additionally, the updating method is very cost-effective since the computational effort is only slightly more expensive than a normal model run because of the simple structure of the as- similation procedure. However, the weighting function results in a deterministic prediction and thus does not provide any indication of the uncertainty of the model results. Although a short calculation time is desirable, the deterministic approach presents a constraint to the application of this updating procedure. Consequently, DHI also implemented another data assimilation procedure in their MIKE 11 software which is capable of providing a stochastic outcome and hence an estimation of the associated model uncertainty: the Ensemble Kalman filter.
2.2 The Kalman filter
The original Kalman filter (KF) for linear Gaussian processes presented by Kalman (1960) would appear to be the basis for the development of a range of new data assimilation algorithms. In this Chapter 2. Brief introduction to data assimilation 7 section, the original Kalman filter will be discussed as well as the Extended Kalman filter (EKF) and the Ensemble Kalman filter (EnKF). It must be noted however, that only the Ensemble Kalman filter is implemented in the software of MIKE 11 by DHI.
2.2.1 Discrete Kalman filter
In its original form, the Kalman filter was designed by Kalman (1960) for linear processes. Like the weighting function described in the previous section, the Discrete Kalman filter consists of calculating a gain Kk which is then used for updating the model state. As opposed to the constant gain in the weighting function, the gain is in this case a variable function since it will be recalculated every time step. When applying a Kalman filter, this gain Kk is called the Kalman gain. The Kalman filter updating procedure consists of two steps. First, a time update is performed in which the model is run and model results at the next time step are obtained. Second, the model results are updated using the Kalman gain. The model on which the updat- ing procedure will be performed, can again be represented by equation 2.1, in this case a linear equation, and equation 2.2 indicates the relation between the observations and the model state. wk in the first equation and vk in the second respectively represent the process and measure- ment noise. Both variables are assumed to be independent white noise with normal probability − distributions. An update is performed by combining the a priori model state vector xk with − the difference between the vector of predicted observations, Hkxk , and the corresponding actual measurements yk according to equation 2.3. The Kalman gain Kk presented in this equation, is + constructed by least square minimization of the a posteriori estimate error ek , which reflects the + difference between the true state of the model xk and the model forecast after updatingx ˆk . This + corresponds to a minimization of the a posteriori error covariance Pk (Madsen & Skotner, 2005; + Madsen et al., 2003). In the discrete Kalman filter for linear systems, this error covariance Pk is then incorporated in the Kalman gain Kk by formulating the a posteriori covariance matrix in terms of the a priori covariance matrix and then including this a priori covariance matrix − Pk into the Kalman gain:
− T − T −1 Kk = Pk Hk (HkPk Hk + Rk) (2.7) where Rk depicts the covariance matrix of the measurement errors (Madsen et al., 2003). The + updated covariance Pk is then obtained using equation 2.8.
+ − − Pk = Pk − KkHkPk (2.8) When the components of equation 2.7 are analyzed, it can be observed that the error covariance between locations with measurements and nearby points without measurements, is reflected − T in the composition of the Kalman gain Kk function. The first component Pk Hk represents the correlation between the model forecast error at each observation point and the error at all other points (with and without an observation). During the updating step, the surrounding points will also be updated to a higher or lesser degree, according to their correlation with the observation points. The sphere of influence in time and/or space of the observation in one Chapter 2. Brief introduction to data assimilation 8 point will thus be larger if the error correlation with all other points is high. However, if the model forecast errors at different observation points are highly correlated, they will influence one another during updating. This may cancel out precious updates which is not a desired effect and should therefore be avoided. This correlation among the errors at observation points is reflected − T in the second component HkPk Hk . In short, the preferred outcome is to maximize the first − T − T component Pk Hk while at the same time, minimize the second component HkPk Hk (Madsen & Skotner, 2005). An overview of the Discrete Kalman filter updating procedure can be found in figure 2.2. To start with, the initial conditions are imposed on the model. A first correction is applied and these updated variables are again put into the model to obtain the model results at the next time step. Subsequently, the calculated model results are once again updated. This process continues until the last time step is computed.
Figure 2.2: Overview of the Kalman filter updating procedure (Welch & Bishop, 1997).
The Discrete Kalman filter assimilation method offers a big improvement compared to the preceding updating procedures. An important benefit is the explicit integration of the model and data uncertainties into the updating procedure. The fact that an estimation of the uncertainty of the system is provided, offers another advantage (Madsen & Skotner, 2005). However, this method is only suited for linear systems which leads to various modifications and approximations when highly nonlinear models are implemented (Vrugt et al., 2006). This difficulty can be resolved by implementing the Extended Kalman filter (EKF) (Madsen & Skotner, 2005).
2.2.2 Extended Kalman Filter
In case the function M( · ) in equation 2.1 represents a non-linear process and the state vector xk is related to the measurements yk by a non-linear function h( · ) according to equation 2.9, the KF can not be applied because of the need for a linear system. Chapter 2. Brief introduction to data assimilation 9
yk = h(xk, vk) (2.9) with vk again indicating the measurement noise (Welch & Bishop, 1997). Instead the Extended Kalman filter can be implemented. In this updating procedure, the system of equations 2.1 and 2.9 is linearized using Jacobian matrices in the following manner: − xk+1 =x ˆ + Ak(xk − xˆk) + Wkwk k+1 (2.10) − − yk =y ˆk + Hk(xk − xˆk ) + Vkvk where
• xk+1 and yk are the actual state and measurement vectors
− − • xˆk+1 andy ˆk are the a priori estimated state and measurement vectors
• xˆk is the a posteriori estimate of the state at time step k
• wk and vk are random variables representing the process and measurement noise
• Ak[i, j] is the Jacobian matrix of the partial derivatives of M( · ) with respect to x δM (ˆx , u , 0)[i] A [i, j] = k+1,k k k k δx[j]
• Wk[i, j] is the Jacobian matrix of the partial derivatives of M( · ) with respect to w δM (ˆx , u , 0)[i] W [i, j] = k+1,k k k k δw[j]
• Hk[i, j] is the Jacobian matrix of the partial derivatives of h( · ) with respect to x δh (ˆx−, 0)[i] H [i, j] = k k k δx[j]
• Vk[i, j] is the Jacobian matrix of the partial derivatives of h( · ) with respect to v δh (ˆx−, 0)[i] V [i, j] = k k k δv[j] As with the KF, first a time update is performed where the a priori estimate of the state and the a priori covariance estimate are transferred from time step k − 1 to step k: − xˆ = Mk,k−1(ˆxk−1, uk−1, 0) k (2.11) − T T Pk = Ak−1Pk−1Ak−1 + Wk−1Qk−1Wk−1 with Qk−1 representing the process noise covariance at time step k −1. Afterward, the state and covariance estimates are corrected with the measurement update equations. The a posteriori states and covariances are calculated using the system of equations represented in 2.12. − T − T T −1 Kk = Pk Hk [HkPk Hk + VkRkVk ] + − − (2.12) xˆk =x ˆk + Kk(yk − hk(ˆxk , 0)) + − Pk = [I − KkHk]Pk Chapter 2. Brief introduction to data assimilation 10
with the function M( · ) coming from equation 2.1, Hk and Vk representing the measurement
Jacobians at time step k and Rk indicating the measurement noise covariance at step k (Welch & Bishop, 1997). In figure 2.3 the different steps of the updating procedure for the Extended Kalman filter are shown. The basic operations in the EKF are comparable to the operations in the Discrete Kalman filter in figure 2.2.
Figure 2.3: Overview of the Extended Kalman filter updating procedure (Welch & Bishop, 1997).
The problem of linearity in the Discrete Kalman filter can thus be solved by implementing the Extended Kalman filter. This procedure is however limited to mild non-linearity since too strong linearity will induce instabilities in the algorithm (Beven, 2009). Additionally, both the Discrete Kalman updating procedure and the Extended Kalman filter require a huge computational effort when applied to a high-dimensional modelling system. For the EKF, computing the Jacobian matrices becomes expensive in this case and updating the covariance matrix also becomes computationally demanding. This drawback has lead to numerous studies investigating more cost-efficient Kalman filter schemes better applicable to real-time implementations. This research resulted in the development of methods often referred to as sub-optimal Kalman filtering procedures, such as the Ensemble Kalman filter (Madsen & Skotner, 2005; Beven, 2009).
2.2.3 Ensemble Kalman Filter
In the Ensemble Kalman filter (EnKF) the model is again represented by a stochastic formulation of the model equation 2.1. However, to avoid long calculation times as a consequence of evolving the Jacobian matrices, the EnKF uses a Monte Carlo ensemble of state vectors to represent the statistical properties of the state vector. This ensemble is generated by perturbation of the parameters, model forcings, model states... A different realization of the errors on the Chapter 2. Brief introduction to data assimilation 11 perturbed variables results in a new ensemble member. Each of the vectors in the ensemble is then propagated according to the model dynamics and the model error to result in a model forecast, generally taken as the mean value of the ensemble forecast. The error covariance matrix − Pk is estimated from the ensemble as
1 P − = S−(S−)T ,S− = (x− − x¯−) (2.13) k M − 1 k k i,k i,k k − − with Si,k indicating the ith column in Sk and M the ensemble size. To account for measurement errors and thus prevent underestimation of the model error covari- ance, it is necessary that the observations are perturbed (Ridler et al., 2014). This ensemble of possible measurements is shown in equation 2.14.
yi,k = yk + ηi,k (2.14) yk represents the actual measurement vector while ηi,k is the measurement error randomly generated from a distribution with zero mean and covariance matrix Rk. Afterwards, updating of each ensemble state vector is performed according to the updating scheme represented in equation 2.3. The gain vector for measurement i is given by
P + (hi )T Ki = k,i−1 k (2.15) k i + i T 2 hkPk,i−1(hk) + σi In equation 2.15 i represents a running index indicating elements corresponding to the ith up- i date using the ith measurement yk. The ith row of the matrix Hk, shown in equation 2.2, is i presented as hk. The updated state vector and error covariance matrix can then be estimated based on this updated ensemble, using equations 2.13 and 2.14 (Madsen & Ca˜nizares,1999). This method was first introduced by Evensen (1994). In MIKE 11, all calculations comprised in the EnKF are based on Sk. The covariance matrix Pk is never explicitly calculated in the implementation (DHI, 2014a).
Due to the use of an ensemble of state vectors and observations, the dimensions of these matrices for the EnKF implementation differ from the KF implementation. While for the KF the state variable is presented in a vector of as many rows as there are data points, in the EnKF there is an ensemble of state vectors and thus the matrix xk consists of M columns and an equal amount of rows as with the KF implementation. The same is valid for the matrix of observations: yk is a vector with the number of rows equal to the number of observations for the KF method but for the EnKF method the number of columns increases to the amount of ensemble runs M. Consequently, the computational effort of calculating these matrices by running an ensemble of model simulations will be higher for the EnKF. However, since it is not necessary to compute the covariance matrix using the covariance matrix at the previous time step but from the ensemble at that time step, less computational effort is needed (Plaza Guingla, 2014). Additionally, it is not necessary to explicitly calculate the covariance matrix as stated before. This results in an extra reduction of required computational effort. Chapter 2. Brief introduction to data assimilation 12
Although this method shows an improvement compared to the KF and the EKF, problems can arise. First of all, it is not guaranteed that the generated ensemble represents the true uncertainty of the system. Also, ensemble size plays an important role when implementing the EnKF. Using a small ensemble has computational advantages but it may not adequately describe the actual range of variability (Beven, 2009). Moreover, when a considerable amount of observations are applied, as is the case with remote sensing products, the calculation of the
Kalman gain Kk has to be estimated. Otherwise the inversion of very large matrices is required. In the case of remote sensing products, a mismatch of scales can also occur (Sahoo et al., 2013; Draper et al., 2012). Chapter 3
Location
3.1 The Demer and its catchment
As a part of the river basin of the Scheldt, the Demer catchment is located mainly in Flanders (figure 3.1). The catchment area in Flanders amounts to about 1919 km2 of a total catchment area of 2334 km2. The remaining 415 km2 is situated in Wallonia. The soils largely consist of sand, sandy loam or loam. Three main regions with a different topography can be distin- guished in the Demer catchment: the northern area of the Demer catchment belongs to the sandy Campine region in Limburg; the area south of the Demer on the other hand, is part of the sandy-loam and loam region of Haspengouw and Hagenland (Bogman et al., 2011; De Clercq et al., 2006).
The Demer is about 85 km long and is divided over two provinces, the eastern part of Brabant and the southern part of Limburg. The most important tributaries are Herk, Mombeek, the river Gete and Velpe. The spring of the Demer can be found in Ketsingen near Tongeren at a height of about 85 m TAW. From there on, it flows to the North up to Bilzen where it curves to the West. Passing Hasselt, Zichem, Diest and Aarschot, it eventually has its mouth in Werchter, where the river flows into the Dijle. After Bilzen, the valley of the Demer becomes a lot broader and the slope decreases from 0.35 % in the headwaters to 0.097 % near Werchter (Bogman et al., 2011). Downstream the confluence of Dijle and Demer, the river is named Dijle although the catchment of the Demer is more than twice as large as the part of the Dijle before the confluence. This is probably because in the Dijle catchment, approximately 93 % of the water in the river in Leuven originates from springs. Thus, it is clear that the Dijle catchment is more depending on springs than the Demer (Bogman et al., 2011).
The water courses in the sand region are typically low land streams, mainly fed by rain water. This causes a strong reaction of the water level to rainfall and consequently high peak discharges. The brooks in Haspengouw are fed by different springs so the water level of the big tributaries like Gete, Herk and Mombeek maintain a relatively high level even during summer. In the Hageland region an intermediate situation is encountered. The rivers Winge and Velpe still
13 Chapter 3. Location 14 originate from some springs but this is not the case for Motte and Begijnenbeek. These brooks show heavy fluctuations in discharge because of the limited amount of springs. The discharge of the Demer is thus very dependent on the amount of rainfall and the height of the groundwater table (Bogman et al., 2011; Van Passel et al., 2004).
Figure 3.1: Location of the Demer catchment, its subcatchments and the main tributaries of the Demer in Flanders (Bogman et al., 2013). Chapter 3. Location 15
3.2 Water management
The Demer and its environment has always attracted people. Already before the Middle Ages villages could be found on its banks, with small scale agricultural activities surrounding them. After the Middle Ages, agricultural needs led to important deforestation. The inhabitants of the towns and cities on the banks of the Demer were confronted with floods and plans for improvements were constructed. Before the 17th century the Demer was a strongly meandering river. During the 17th and especially the 18th century, the situation changed as the originally meandering Demer was embanked to protect the villages. Since the Demer was an important watercourse for navigation, the river was also straightened out. This process continued until 1980. However, in the 20th century a series of heavy floods still struck the Demer catchment resulting in severe damage. After two serious dike breaks in December of 1965 and 1966, the Belgian government decided to remove the last meandering parts, deepen the river bed and construct dikes with reinforced banks. Due to a rising number of questions on the effect on the environment and the validity of these operations, these works were stopped in 2000 (De Clercq et al., 2006; HIC, 2011). In recent years, more and more scientists are convinced of a new approach. It is now thought that floods cannot be stopped but can be directed to where they cause the least damage. To support this point of view, investments have led to the development of computer models such as the Demer MIKE 11 model. Initially, in the nineties, a combined hydrological-hydraulic model was used offline to investigate the effect of certain measures. From 2000 onwards the models were applied for online real-time forecasts and the first flood predictions were possible. Since then, adjustments have been made to improve these models and their predictions to obtain even more reliable forecasts (HIC, 2011).
3.3 Calibration and validation data
For the calibration and validation of the model, two floods were selected with sufficient mea- surements. The first one took place in November 2010 while the second occurred in January 2011.
From the 11th until the 16th of November 2010 floods occurred on several places in the De- mer catchment. Due to the use of the controlled inundation areas upstream (Schulen and Webbekom), more severe problems were avoided. A smaller discharge peak preceded the major peak with a maximal discharge of about 59 m3/s in both Diest and Zichem and of 64.5 m3/s in Aarschot. The discharges measured in November 2010 in the different hydrologic stations can be seen in figure 3.2. The water level exceeded the alarm level in both Zichem and Aarschot. In the latter the alarm level was even exceeded for six days because of the slow drainage of the controlled inundation areas. The water level remained about 12.9 m TAW. The second flood occurred between the 5th and the 31st of January 2011. During this event, two discharge peaks entered the Demer and caused high water levels. The highest discharge was 64.72 m3/s and occurred in the second wave on the 15th of January (Boeckx et al., 2011). Chapter 3. Location 16
70 Aarschot 122 Zichem 123 60 Diest 126
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0 Nov 01 Nov 06 Nov 11 Nov 16 Nov 21 Nov 26 Dec 01 Date
Figure 3.2: Discharge on the Demer in the hydraulic stations of Aarschot (number 122), Zichem (number 123) and Diest (number 126) for November 2010. Chapter 4
The model
The model that is used in this thesis was originally requested by Flanders Hydraulics Research to cover the Demer basin from Hasselt upstream to Werchter downstream at the confluence with Dijle. In cooperation with Danish Hydraulic Institute (DHI) it was implemented by Inter- national Marine & Dredging Consultants (IMDC) in MIKE 11 software created by DHI. Since its development in 2003, several projects have been issued to improve or update the original hydrodynamic model of the Demer to the changing environment (Bogman et al., 2011).
In MIKE 11, the equations of de Saint-Venant (the continuity equation and the momentum equation) are used to calculate the dynamic flow in a water course. Solving the de Saint-Venant equations is done according to the implicit finite difference scheme developed by Abbott & Ionescu (1967). This scheme consists of a sequence of alternating h and Q-points. The water level in the h-points and the discharge in the Q-points are calculated using the continuity equa- tion and the momentum equation respectively, taking into account the external and internal model boundaries. External boundaries are implemented at the edges of the model in the form of water level or discharge as a function of time or as a rating curve, while internal boundaries consist of a mathematical description of hydraulic structures that have an influence on the hy- drodynamics of the system. Both these boundary conditions are described later. The Abbott & Ionescu scheme implies that there are Q-points which are located between neighboring h-points and at structures, and h-points which can be found at cross-sections or at equidistant intervals if the distance between cross-sections is larger than a maximum distance dx. When the mo- mentum equations are considered together with the internal and external boundary conditions, it forms a system of N algebraical equations, with N the total amount of calculation points, that can be solved to obtain water level and discharge (Van Looveren, 2003; DHI, 2014a). With these points, a river network is constructed.
The cross-sections of a water course, describe the topography of the MIKE 11 model. These cross-sections are defined as a two-dimensional profile of the river, measured perpendicular to the direction of flow in the river. The cross-sections are identified by three variables: the name of the water course, a topoID and a chainage. The topoID is defined as the year in which the
17 Chapter 4. The model 18 topographical measurement has been performed, while the chainage indicates the distance to the upstream end of the water course. For parts of the water course where the geometry was estimated based on a digital terrain model (DTM), the topoID is indicated as ’DTM’. A value for the hydraulic resistance can also be defined for every cross-section. In practice however, a general value for the hydraulic resistance is applied. This resistance can be adjusted for certain cross-sections if necessary (Van Looveren, 2003; DHI, 2014a).
Different hydraulic structures are situated along the Demer and its tributaries. In the model, these structures can be categorized as five types: fixed weirs, adjustable weirs, bridges, divers and lateral spillways. Fixed weirs are implemented as ’weirs’ whereas adjustable weirs are defined as ’control structures’ in the MIKE 11 model. Bridges and divers are combined as ’culverts’ while lateral spillways are represented by ’link channels’. The latter are applied for modelling the flow towards floodplains. In the Demer model, most floodplains are simulated by means of flood branches and reservoirs. The first applied technique implies the use of a parallel river branch to represent floodplains with a strong current. The connection of this flood branch and the main river is implemented as a lateral spillway that simulates the flow over the dike. These spillways are called link channels. Reservoirs on the other hand, are defined as additional storage when the flow over the floodplain is negligible. This is implemented in the model as a flood cell which is connected to the river via a link channel (Van Looveren, 2003).
4.1 Reduced model
The so-called reduced model used in this thesis, does not comprise the whole Demer from its source in Hasselt to the confluence with Dijle in Werchter. It extends from Diest upstream to Werchter downstream, containing also the part of the river Dijle between the hydrologic stations in Wijgmaal and Haacht (Van Looveren, 2003). The different rivers and river branches implemented in the applied MIKE 11 model are represented in figure 4.1. Another significant part of the model are the hydraulic structures. These can have a severe impact on the water level and discharge measured in the river by acting as an internal boundary condition and thus have to be implemented in the model. The most important structures included in the model are shown in table 4.1. Chapter 4. The model 19
Figure 4.1: Dijle, Demer and its tributaries implemented in the MIKE 11 model of the Demer (Cornu, 2014).
Table 4.1: Important hydraulic structures placed on the Demer and included in the MIKE 11 model (Bogman et al., 2011).
Hydraulic structure Chainage [m] Type of hydraulic structure Bridge E. Claesstraat at Zichem 33740 diver Mill in Zichem 33990 2 control structures + fixed weir Bridge in Testelt 36500 diver + fixed weir Bridge Beckerstraat in Aarschot 48250 diver Bridge ’Nieuwbrug’ in Aarschot 48660 diver ’s Hertogen mills in Aarschot 48879 4 divers Sluice complex on ’s Hertogen mills 30/115 2 fixed weirs on Demer-(Molenarm3) in Aarschot Railway bridge in Aarschot 49119 diver Chapter 4. The model 20
4.2 Boundaries
External boundary conditions can be classified either as upstream or downstream boundary conditions. The upstream boundary conditions in the MIKE 11 Demer model are mostly time series of discharge acquired from measurements using rating curves. If no measurements were available for a certain boundary, a hydrogram was calculated using the hydrological model NAM based on the parameters of the subbasin. This is mostly the case for small water courses. More information on the NAM model can be found in DHI (2014a); Madsen (2000). In the MIKE 11 Demer model only one downstream boundary condition is defined. This boundary in the Dijle in Haacht consists of a time series of water level (Van Looveren, 2003). The impact of the boundary condition on the water level and discharge in the Demer is different for every water course. For most water courses, the effects are limited. However, for the upstream boundary in the river Dijle, and to a lesser extent also in the rivers Losting, Motte, Winge and Zwartwater-Hulpe, the response of the Demer is significant. Of course, the part of the Demer upstream of Diest which is not included in the reduced model also has a substantial effect. Since two time series of discharge are available for this boundary, a problem arises. One time series has been measured at the hydrologic station number 126 in Diest in January 2011. The other is internally generated using the upstream part of the full model. Calibration on the full model was performed using the data of November 2010. The two time series are slightly different as can be seen from figure 4.2. Given that in this thesis the reduced model is implemented and that it is more correct to make use of observations, the time series of measured discharge obtained from the hydrologic station number 126 is used. Recalibration of the reduced model is however desired for future use. Chapter 4. The model 21
70 Measurement Diest 126 Simulation Diest 126 60
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Figure 4.2: Measured discharge and simulated discharge on the Demer in the hydraulic station of Diest (number 126).
4.3 Initial conditions
Determining the initial conditions for the model was done by using a hotstart file. This indicates that for a certain time period prior to the start of the simulation, a false simulation is run. For the MIKE 11 Demer model, the hotstart simulation was executed from the third to the fifth of November 2010 using a time step of one minute. The results obtained from this hotstart simulation are then used as initial conditions for the true simulation (Van Looveren, 2003).
4.4 Uncertainty
When making use of a model, one must realize that there are several sources of uncertainty. This uncertainty occurs as model errors introduced by errors in the boundary conditions but also by errors in the model structure where they are associated with the parameters or modelled processes, structures... Walker et al. (2003) classify the different sources of uncertainty into five possible categories (Loosvelt, 2013):
• Context uncertainty is associated with an identification of the boundaries and the repre- sentation of the system to be modelled.
• Input uncertainty represents the uncertainty on the system data and the external forces that have an influence on the system and its performance. Chapter 4. The model 22
• Model structure uncertainty indicates uncertainty related to the model itself since the model is only a simplified representation of reality and since processes can be incomplete or wrongly modelled.
• Parameter uncertainty arises from the data and the methods used for calibration of the model parameters.
• Model technical uncertainty is associated with the implementation of the model in com- puter software and numerical implementation of the algorithms.
In the MIKE 11 DA module only perturbation of the boundaries is possible so the errors intro- duced by uncertainty in the boundary conditions, i.e the input uncertainty, can be evaluated. Next to context uncertainty, input uncertainty can also be assessed. This is realized by defining a standard deviation for the measurements used for DA. However, a large amount of model uncertainty is neglected since it is not possible to assess parameter uncertainty, model structure and model technical uncertainty.
To determine the input uncertainty, the error on the boundary conditions must be evaluated. This error consists of several factors as the discharge time series are deduced from water level measurements by means of a previously determined rating curve, i.e. a h-Q relationship. Errors both originate from the use of a measuring device as well as from the creation and approximation of the h-Q relation. The water level h was measured using a limnigraph. At the Hydrological Information Centre (HIC), these limnigraphs are mostly radar based measuring devices, pressure probes or ultrasonic probes. To construct the h-Q relation, the flow is frequently measured with a current meter at all possible water levels in different points along the water course. Combining the flow measurements and the cross-sections then leads to h-Q relations (HIC, 2002). According to Le Coz (2012), all acknowledged sources of errors seem to have been examined but no general uncertainty analysis framework has yet been established. It is thus difficult to fully assess the uncertainty on the boundaries. As stated by Le Coz (2012), in many cases an arbitrary relative uncertainty value is fixed. Based on literature and the uncertainty of the measuring devices, the standard deviation of the boundaries was determined on a relative value of 2.5 %. Due to an error 5 % was implemented as a standard deviation, however this had no effects on the results. Due to correlation of the input errors in time and space, this standard deviation can on each time step be divided into a correlation with the error on the previous time step and white noise. In the MIKE 11 Demer model this is implemented as a first-order autoregressive model represented as
k = Ak−1 + ζk (4.1) with A a diagonal correlation matrix containing autoregressive coefficients and ζk representing the white noise. The correlation matrix A is a bxb-matrix with b being the number of boundary conditions. The autoregressive coefficients on the diagonal of A are determined from the corre- lation between the error compared to the mean value of the boundary on a certain time step, Chapter 4. The model 23 and the same error on the previous time step (Madsen et al., 2006; DHI, 2014a).
Errors also occur in the observations of discharge and water level used for data assimilation, i.e. input uncertainty. The discharges were obtained using rating curves as well, so an equal standard deviation of 5 % was assigned to these observations. For the observations of water level applied for data assimilation, a standard deviation of 0.01 m was assessed based on the accuracy of the measuring equipment. The same error as with the standard deviation on discharge caused a value of 0.02 m being implemented as the standard deviation for water level measurements in the model but no effects on the results could be found.
4.5 Model forecast
Data assimilation procedures as described in chapter 2, will perform an update on the state of the river system up to the time of forecast. Updating after time of forecast is executed but is not based on the use of actual state observations at those time steps. Once measurements are no longer available and the time of forecast is reached, the model state at this point in time can be used as initial condition for a model forecast. How the forecast of the model state is made, differs for the weighting function and the Ensemble Kalman filter.
Figure 4.3: Schematic representation of the combination of the weighting function data assimilation procedure and the error forecast procedure (DHI, 2014a).
If the state of the river system at the time of forecast is simply used as initial conditions for a model forecast, the capacity of the system to perform an adequate model prediction will be lim- Chapter 4. The model 24 ited. This is why the weighting function is combined with an error forecast at the measurement points. Figure 4.3 illustrates the principle of this combined approach. In the filtering period, an innovation vector is composed at every update location. An error forecast model defined by the user at each update location gives rise to an update of the state variables from the time of forecast on. This update is executed by propagating the innovation vector in the forecasting period according to equation 2.3. The errors at the measurement locations are predicted by the error forecast models and then distributed to the neighboring points. The error forecast model predicts one step ahead and generally depends on the previous innovations, the predicted model state at the present time step, the previous updated model states and the present and previous model forcings. If no error forecast model is specified, the model run has initial conditions at time of forecast being the updated model state at that time but no other updates after time of forecast will occur. In this case, the influence of the initial conditions will soon fade out (DHI, 2014a; Madsen & Skotner, 2005; DHI, 2014b). In case of the Ensemble Kalman filter, the mem- ory of the model that is propagated into the model forecast is captured in the autoregressive description of the model error. The autocorrelation of the error on the model input thus results in an autocorrelation of the error on the model prediction. This is then applied in the forecast period. After time of forecast the model errors are phased out according to the exponential decay defined by the autoregressive process. Chapter 5
Methodology
As stated in the introduction, the goal of this thesis is firstly to determine the differences between the use of the EnKF and the WF. Secondly, the aim is also to identify how DA using water level data differs from DA using discharge data. The results from this study will for example give an insight in which DA algorithm to apply in which situation and indicate the points of attention in the interpretation of the DA results. Furthermore, it will provide a better assessment of the possible consequences of the use of both DA algorithms and the use of both types of data.
To achieve these goals, a series of consecutive steps are performed:
1. Run EnKF with observations of discharge.
2. Run EnKF with observations of water level.
3. Determine amplitude A for WF.
4. Determine interval of influence for WF.
5. Run WF with observations of discharge.
6. Run WF with observations of water level.
7. Validate performance of DA algorithms
The applied procedure is outlined in this chapter and more in detail information is provided in the next sections.
5.1 Step 1 and 2: Ensemble Kalman filter
In the first and the second step, the EnKF is run with observations of discharge and water level. In November 2010, measurements have been taken at four different hydrologic stations (For more information on this flood see section 3.3). These data are applied as observations to perform DA. The four hydrologic stations are all situated along the river Demer. In table 5.1, the number of the hydrologic station, the location and it’s chainage are represented starting
25 Chapter 5. Methodology 26 from the stations most upstream and going downstream. The two stations most downstream are both located in Aarschot, each on one side of the ’s Hertogen mills.
Water level observations are provided at all four hydrologic stations while discharge observations are only present at two out of four stations (Hydrologic stations number 122 and 123). Next to observations, the boundary conditions are also important for applying DA. Six boundary conditions are implemented in the MIKE 11 DA software. These boundary conditions are all obtained from the flood of November 2010. A summary of the six boundary conditions and their characteristics are shown in table 5.2. More information on the boundary conditions can be found in section 4.2. The uncertainty on these boundary conditions is represented by a stan- dard deviation of five percent and the autocorrelation in the data is implemented as a first order autoregressive model (AR1). The assumption that the length of autocorrelation is the same as for the variable is of course only a crude estimation of the truth.
As a standard deviation on the observations of discharge as well as the boundary conditions, a relative value of five percent has been applied. For the observations of water level two centimetre was taken as a standard deviation. In literature however, a standard deviation of 2.5 percent is mentioned for the application of discharge data and one centimetre for the application of water level data. It was verified that this difference in standard deviation only has a minor effect for the results of the EnKF. Though this assumption is incorrect, the analysis was performed on the results with a standard deviation of five percent and two centimetre since the error was only discovered late in the process. Since no deviation was found, this should not cause any complications. This might actually partially compensate the underestimation of the model spread due to not including other causes of uncertainty such as parameter uncertainty. Of course, this will not cause a better estimation of the covariance of the model errors.
5.2 Step 3 to 6: Weighting function
Since both the WF and the EnKF attempt to improve the model forecast, the correction applied by the WF should be similar to the correction of the EnKF on the model predictions. From equation 2.6a it can be seen that the gain applied in the constant WF is equal to the amplitude A. This thus indicates that the gain of the WF and thus the amplitude can be determined
Table 5.1: Description of the hydrologic stations where measurements are available for DA.
Hydraulic station Location River Chainage [m] 123 Zichem Demer 33 412 128 Testels Demer 36 495 121 Aarschot Demer 47 941 122 Aarschot Demer 49 361 Chapter 5. Methodology 27
Table 5.2: Description of the boundary conditions and their characteristics implemented in MIKE 11 for DA.
Location Chainage [m] Data type Minimum Maximum Demer 28 577.5 Discharge 7.72 m3/s 58.90 m3/s Zwartwater-Hulpe 0 Discharge 1.05 m3/s 3.65 m3/s Motte 0 Discharge 0.43 m3/s 3.81 m3/s Winge 0 Discharge 0.03 m3/s 3.72 m3/s Losting 0 Discharge 0.01 m3/s 1.20 m3/s Dijle 0 Discharge 3.72 m3/s 31.7 m3/s
based on the Kalman gain from the EnKF, assuming that the EnKF DA procedure performs better at predicting the truth. By studying the evolution of the Kalman gain in time and space, several questions can be answered. First of all, from the magnitude of the gain across the system and the variation in magnitude, a value of A for each observation point can be deduced.
Looking at the upstream and downstream distance in which this Kk still has a sufficiently large value, furthermore provides an indication of the width of the interval of influence. Next to the distance, the progress of the gain in space supplies information on the shape of the most suited gain function to apply in the WF. To get an idea of the influence of updating in the tributaries, the evolution of the Kk in a couple of points in these tributaries is also examined.
5.3 Step 7: Validation
The validation in this master thesis was performed by making use of a synthetic experiment. In the first place a perturbation of the boundary conditions is executed. With this perturbed boundary conditions, a single model run is executed to obtain the assumed truth. As with the boundary conditions implemented in the MIKE 11 model, a first order autoregression (AR1) is assumed in the boundaries. The factor φ which represents the autocorrelation in the time series of discharge, is considered to be 0.99. An adjusted value was applied since the true au- tocorrelation value of the boundary discharge is unknown. The conclusions obtained from this experiment will thus be less artificial. At the points where observations of discharge were avail- able in the original experiment, hydraulic stations 122 and 123, the results of the truth run are extracted. These ’true’ results are then transformed into synthetic observations by applying an error with AR1. In the MIKE11 software, no autoregression is assumed in the observations but this is nevertheless applied in the creation of the synthetic observations. The autoregression is necessary since too much oscillation of the data is obtained without the application of AR1. This induces instabilities when applied in DA in the MIKE 11 software. The autoregression coefficient φ was extracted from the autocorrelation of the discharge time series of the obser- vations at each hydraulic station. With the application of the synthetic observations and the original boundary conditions, DA was performed on the model of the Demer river. A DA run was executed using all observations of discharge over the whole time series as well as issuing a Chapter 5. Methodology 28 forecast from the 18th of November at 12h on. The latter time period was chosen since a large difference was observed here between the truth and the weighting function without DA. It was thus considered interesting to observe how well DA could perform in improving the prediction of the truth. In an ideal situation, forecast would be issued at different time periods. Due to time constraints however, no extra simulations were possible. For the WF, the error forecast model applied was an AR1 model where the autocorrelation coefficient φ was estimated using the four hours previous to the time of forecast.
The goodness-of-fit is evaluated based on three indicators. According to Moriasi et al. (2007) the combination of the Nash-Sutcliffe efficiency (NSE), the percent bias (PBIAS) and the ratio of the root mean squared error to the standard deviation of measured data (RSR), in addition to the graphical techniques form a reliable basis for the evaluation of the goodness-of-fit. These true goodness-of-fit measures are calculated as can be observed in formula 5.1, 5.2 and 5.3. Yi sim represents the value of the ith member of the time series of true discharge, whereas Yi is the value of the ith member of the time series of simulated discharge, Y mean indicates the mean of the time series of true discharge and n is the total number of observations.
Pn true sim 2 i=1 (Yi − Yi ) NSE = 1 − Pn true mean 2 (5.1) i=1 (Yi − Y ) The NSE provides an indication of how well the plot of true values versus simulated values data fits the 1:1 line. This measure ranges between -∞ and 1.0, the latter being the optimal value. If the values become negative, the mean true value is a better predictor than the simulated values, which is unacceptable (Moriasi et al., 2007).
Pn true sim i=1 (Yi − Yi ) ∗ 100 P BIAS = Pn true (5.2) i=1 Yi The average tendency of the simulated data to be larger or smaller than their observed counter- parts is represented by the PBIAS. This is expressed as a percentage. The value of this measure should approach 0.0 as close as possible. If the values are positive a model underestimation bias occurs while negative values indicate a model overestimation bias (Moriasi et al., 2007).
q Pn true sim 2 RMSE i=1 (Yi − Yi ) RSR = = (5.3) std pPn true mean 2 true i=1 (Yi − Y ) In the RSR, the root mean squared error (RMSE) is standardized by using the standard deviation of the time series of true discharges. An optimal value for the RSR is 0, which indicates zero residual variation, and can vary until large positive values. A low value of the RSR is aspired since this means a low RMSE value and thus better model performance. Moriasi et al. (2007) determined that model simulations can be judged as satisfactory if NSE > 0.50, the RSR ≤ 0.70 and if PBIAS ≈ 25 % for streamflow measurements. Chapter 6
Results and discussion
After completing the steps described in chapter 5, analysis of the obtained results was per- formed. In this chapter, these results are presented together with additional information and an interpretation.
6.1 EnKF with observations of discharge
To understand the results of the EnKF, both the original simulation as well as the observations on which the EnKF was based, have to be examined. In figures 6.1 and 6.2 the results of the EnKF with discharge observations, together with the simulation without DA and the obser- vations are visualised for both observation points. The first figure represents the evolution in time of the simulations and observations at the hydraulic station number 122 in Aarschot down- stream of the mills. It can be seen that initially the EnKF simulation almost coincides with the observations. For the discharge peak however, the uncertainty on the observations increases due to the use of a relative standard deviation of five percent (see figure 6.5). This makes the EnKF prediction approach the simulation without DA. After the peak, the EnKF simulations again start to follow the observations instead of the simulations without DA since the uncertainty on the observations will again decrease due to decreasing discharges. It can also be noticed that while the observations are initially smaller than the simulation without DA, the opposite is true for the peak and when the discharge starts to decrease the first situation is recovered. At the hydraulic station number 123 at Zichem, the observations and simulations differ much less as can be seen from figure 6.2. This is probably due to the proximity of the upstream border in the Demer. At this border, observations of discharge are applied causing that the influence of these observations is still present at the hydraulic station. Although the deviations are small, the same conclusions as with the first hydraulic station can be drawn. Initially the EnKF prediction approaches the observations closely. When discharge rises, the EnKF simulation leans closer to the simulation without DA and in the end it follows the observations again. A switch of positions is also observed with the observations and the simulation without DA in the rising limb of the hydrograph and again when the discharge decreases after the peak. The simulation without DA thus overestimated the lower values or baseflow and underestimated the higher values or peak
29 Chapter 6. Results and discussion 30
flow for both the observation points.
An important remark has to be made when interpreting the uncertainty coming from the ensem- ble spread on the simulation without DA as shown in figure 6.5. In the model, this uncertainty results entirely from the standard deviation applied at the boundaries of the model. In the MIKE 11 DA software, no uncertainty on the parameters can be defined. This is of course a constraint when assessing the model uncertainty. As a result, the spread of the ensembles on the simulation without DA in reality can be significantly larger.
6.2 EnKF with observations of water level
From this run of the EnKF, no results were obtained due to instabilities in the model run. It was attempted to simulate the EnKF with observations of water level with two models. The first was the reduced model used for the EnKF with observations of discharge and the sec- ond was the online model received from Flanders Hydraulics Research. This last model was designed to run online simulations and is therefore adapted to be more stable. Unfortunately also this model did not succeed in completing the simulation successfully. The instability in the model run appears to be originating from a problem near the mills at Aarschot. At this point in the model of the river Demer, the water level decreases to the bottom of the river bed and thus runs dry. Although the MIKE 11 software is provided with a method to cope with this kind of problem, the dry river in this case causes severe instabilities and terminates the model run. In the simulation without DA and the EnKF simulation with observations of discharge, this problem also occurred though less extreme. In figure 6.1 strong and unrealis- tic fluctuations in the simulation without DA appear around a discharge of 13 to 15 m3/s as a consequence of these instabilities. In this discharge range, the water level reaches a height resulting in a dry river at the mills at Aarschot. For this model run however the effects were limited and a normal simulation could be executed. The same effect was observed in the model run of the EnKF simulation with observations of discharge. This is not visible in figures 6.1 or 6.2 as a result of the averaging of different ensemble runs in the EnKF data assimilation method.
Since the cause of the complication is known, adaptations to stabilize model structure can probably be found. Due to time constraints, no further investigation of this problem was included in this thesis.
6.3 Analysis of Kalman gain K
For the application of the weighting function DA procedure, it is necessary to define a gain function, an amplitude A and an interval of influence per observation point. To obtain a better insight in the shape of the gain function, the magnitude of A and the width of the interval of influence, the Kalman gain derived from the EnKF with discharge measurements is studied. Chapter 6. Results and discussion 31
Hydraulic station 122 70
EnKF simulation Measurement 60 Simulation without DA
50
40
30 Discharge [m³/s] Discharge
20
10
0 Nov 01 Nov 06 Nov 11 Nov 16 Nov 21 Nov 26 Dec 01 Date
Figure 6.1: Simulation of the EnKF with discharge measurements in both observation points at hy- draulic station 122 in Aarschot downstream of the mills, together with the measurement and the simulation without DA at this point.
Hydraulic station 123 60
EnKF simulation Measurement 50 Simulation without DA
40
30
Discharge [m³/s] Discharge 20
10
0 Nov 01 Nov 06 Nov 11 Nov 16 Nov 21 Nov 26 Dec 01 Date
Figure 6.2: Simulation of the EnKF with with discharge measurements in both observation points at hydraulic station 123 in Zichem, together with the measurement and the simulation without DA at this point. Chapter 6. Results and discussion 32
To better comprehend the influence of different factors on the value of K, an overview of the formula for the calculation of the Kalman gain is given in equation 6.1.
− T − T −1 Kk = Pk Hk (HkPk Hk + Rk) (6.1)
The magnitude of K is influence by a change in the components of the formula. The term − T Pk Hk in the numerator depicts the covariance of an observation point with another location in the river system. It is thus a matrix consisting of as many rows as there are points in the river system and as many colums as there are observation points. The first term in the denominator − T HkPk Hk indicates a matrix containing the variance and covariance of all observation points. The second term in the denominator Rk refers to the variance of the measurements at the ob- servation point.
Four variables can induce a change in Kalman gain: the model uncertainty on the observation point, the measurement uncertainty on the observation point, the model uncertainty at the considered location and the correlation between the ensemble errors at the observation point and the ensemble errors at the considered location. The effect of a rise in one of these four factors is described below. The opposite effect is of course true when a decrease in the variable is observed.
• The model uncertainty on the observation point is included in both the covariance term in the nominator as in the first term in the denominator. A higher model uncertainty at the observation point will lead to a higher covariance term in the nominator as well as a higher covariance term in the denominator. Depending on the relative rise of both terms, the K will increase or decrease.
• A change in the measurement uncertainty on the observation point affects the second term in the denominator. A higher measurement uncertainty will thus lead to a lower K.
• The model uncertainty at the considered location determines the value of the covariance term in the nominator. A higher model uncertainty at a certain location will induce a higher covariance at this location and thus the K will increase.
• And finally, an increase in the correlation between the ensemble errors at the observation point and the ensemble errors at the considered location will cause the covariance term in the nominator to increase. This will result in a higher value for K at the considered location.
6.3.1 General observations
A visualisation of the change in time of the mean absolute value over all chainages of the Kalman gain is presented in figure 6.3. It is clear from this figure that the instabilities that occur in the simulation without DA are also present in the Kalman gain of the EnKF with discharge measurements. As a results of the presence of instabilities in the different ensemble members of the EnKF simulation, these fluctuations also appear in the Kalman gain. Especially in hydraulic Chapter 6. Results and discussion 33 station 122, the values of the K are severely influenced by the instabilities in the model. This is probably because of the proximity of the mills at Aarschot. The values of K at hydraulic station 123 also fluctuate but to a lesser extent. To obtain correct results without the influence of the instabilities, the values at the unstable points in time are removed from the data set at both hydraulic stations. These unstable time periods were determined using a visual assessment of the unstable zones in the upper and lower limit of the 90 % uncertainty interval around the EnKF simulation at the observation points.
It can be noticed immediately that the values displayed in figures 6.3 and 6.4 are very small. As explained before, these figures show the mean absolute evolution of the Kalman gain in time. The standard deviation on this mean absolute value and the maximal absolute K values are also displayed. The two upper figures indicate the K for water level in the two observation points at the hydraulic stations. The two lower figures display the K values for discharge in the two hydraulic station where observations were available. The small values can be explained by the extensive availability of observation data in time. Discharge measurements are provided every 15 minutes at both measurement points. Since updates are therefore performed at a time interval of 15 minutes, the spread of the ensemble remains small. The influence of the small ensemble spread apparently even compensates for a possible high correlation at certain points, resulting in a small covariance. Also the variance in the denominator of equation 6.1 is influenced by the small ensemble spread. Since the Rk term remains the same however, 6.1 shows that a small ensemble spread will also keep the Kalman gain small.
For the analysis of the Kalman gain, only the K values for discharge will be examined. The relationship between discharge and water level is a non-linear one and a correction for the update of water level with measurements of discharge is comprised in the Kalman gain. This makes the interpretation of the influence of updating and the relation between the gain and the evolution of water level very complex. No study of the Kalman gain for water level will thus be performed in this thesis.
In general, it can be noticed that the Kalman gain for the measurement at hydraulic station 123 has a larger value than K for the measurement at hydraulic station 122. An explanation for this observation can be found in figure 6.5. The measurement values of discharge in the hydraulic station 122 at Aarschot are clearly larger than in the hydraulic station 123 at Zichem. The uncertainty on these measurements is defined as a standard deviation of five percent. Con- sequently, the uncertainty on the observations in station 122 is larger than on the observations in station 123. In equation 6.1 this results in an Rk matrix for hydraulic station 122 containing larger values than the Rk matrix for hydraulic station 123. The uncertainty on the simulations on the other hand remains roughly the same in both observation points as is shown in the bottom graphs of figure 6.5. This is presumably combined with a smaller correlation of the ensemble error at the observation point with the ensemble errors at other locations for hydraulic station 122 than for hydraulic station 123. This smaller correlation results in a lower covariance Chapter 6. Results and discussion 34
K for h (station 122) #10-3 K for h (station 123) 0.06 2.5 Mean abs K Mean abs K 0.05 K-std 2 K-std K+std K+std 0.04 Maximal abs K Maximal abs K 1.5 0.03 1
0.02 Kalman gain Kalman Kalman gain Kalman 0.5 0.01
0 0
-0.01 -0.5 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Date Date
K for Q (station 122) K for Q (station 123) 2.5 0.025 Mean abs K Mean abs K 2 K-std 0.02 K-std K+std K+std Maximal abs K Maximal abs K 1.5 0.015
1 0.01 Kalman gain Kalman Kalman gain Kalman 0.5 0.005
0 0
-0.5 -0.005 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Date Date
Figure 6.3: Maximal and mean absolute Kalman gain of the EnKF simulation with observations of discharge at the two observation points. The mean and standard deviation are taken of the K along the length of the river.
#10-4 K for h (station 122) #10-3 K for h (station 123) 10 2.5 Mean abs K Mean abs K 8 K-std 2 K-std K+std K+std Maximal abs K Maximal abs K 6 1.5
4 1 Kalman gain Kalman Kalman gain Kalman 2 0.5
0 0
-2 -0.5 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Date Date
#10-3 K for Q (station 122) #10-3 K for Q (station 123) 7 14 Mean abs K Mean abs K 6 12 K-std K-std K+std K+std 5 10 Maximal abs K Maximal abs K 4 8
3 6
2 4
Kalman gain Kalman gain Kalman 1 2
0 0
-1 -2 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Nov 05 Nov 10 Nov 15 Nov 20 Nov 25 Date Date
Figure 6.4: Maximal and mean absolute Kalman gain of the EnKF simulation with observations of discharge without unstable values at the two observation points. The mean and standard deviation are taken of the K along the length of the river. Chapter 6. Results and discussion 35
Figure 6.5: Simulation without DA and observation together with their standard deviation at both observation points for water level. In the bottom figures the magnitude of the standard deviation is represented. Chapter 6. Results and discussion 36 at station 122 than at station 123. Consequently, the Kalman gain for the hydraulic station 122 at Aarschot remains smaller than for the hydraulic station 123 at Zichem. The difference in discharge between the simulation without DA and the observation is however significantly greater for all time steps in the case of the hydraulic station number 122. This is illustrated in figures 6.1 and 6.2. As a result, the update on the simulation without DA for the case of Aarschot will be considerably larger although the Kalman gain for this hydraulic station has a lower value.
6.3.2 Evolution of K
When examining the general course of the K for both measurement points in space from figure A.4 and A.5, a different general pattern is observed. The elapse for both of the hydraulic stations is not what would be expected. A maximal Kalman gain at the observation point and gradu- ally decreasing values when moving to the upstream and downstream boundaries is the general anticipated pattern if no interference of the updates due to both observation points is present. This shape is therefore also implemented in the predefined weighting functions in the MIKE 11 DA editor. Although part of it can be recognized in both stations, deviations are observed. For the hydraulic station 122 at Aarschot, the K value downstream of the observation point shows a gradual decline. The upstream arm however, depicts a quite constant pattern with a sudden drop at chainage 35600 as explained before. This drop is of important magnitude and thus masques the gradual decline of K values in the upstream direction. The K in Zichem also appears to follow an expected course on the downstream side. The upstream part on the other hand shows increasing values of K instead of decreasing. An explanation for this phenomenon is found in the proximity of the upstream boundary. During updating, the boundary has a strong influence on the points upstream of the observation point. This leads to a large spread of the ensemble members and consequently on the uncertainty around the EnKF simulation at the up- stream part. This results in a higher covariance and a higher K at the upstream end is observed.
For every abrupt change in the Kalman gain along the river Demer an underlying reason is present. In figures A.4 and A.5 the change of K with the chainage is visualized. An indication of the most important variations in Kalman gain can be found in figure 6.6 for the hydraulic station 122 most downstream at Aarschot while for the hydraulic station 123 at Zichem the variations in K are indicated in figure 6.7. In addition, figures 6.8 and 6.9 show the ratio of the ensemble spread at a certain point and a certain time step, to the ensemble spread at the observation point at the same time step. This indicates if the ensemble spread at a location is smaller or larger than the ensemble spread at the observation point, which helps to interpret the change in covariance. A summary of the different changes in K, their causes and chainage can be found in table 6.1 for hydraulic station 122 and table 6.2 for hydraulic station 123. A more complete explanation can be found in the next paragraphs.
At the hydraulic station number 122 in Aarschot downstream of the mills, several regions can be distinguished in the EnKF pattern along the chainage axis: Chapter 6. Results and discussion 37
Figure 6.6: Top: Mean evolution of K in space with standard deviation and indication of the important chainages along the river Demer for hydraulic station 122. Bottom: Evolution of K for discharge in time and space at hydraulic station 122.
Figure 6.7: Top: Mean evolution of K in space with standard deviation and indication of the important chainages along the river Demer for hydraulic station 123. Bottom: Evolution of K for discharge in time and space at hydraulic station 123. Chapter 6. Results and discussion 38
Table 6.1: Overview of changes in K indicated in figure 6.6 with their cause and chainage.
Number of point Cause of change in K Chainage [m] in figure 6.6 1 Mill at Zichem 33990 2 River Hulpe and Leigracht 35372 - 35660 3 Floods 44344, 44445 - 44637 and 44768 4 Inflow flood control area and river Motte 47100 - 52840 5 Mills at Aarschot 48879 6 Incorrect drainage of flood control area 52848 7 River Winge 58041
Table 6.2: Overview of changes in K indicated in figure 6.7 with their cause and chainage.
Number of point Cause of change in K Chainage [m] in figure 6.7 1 Incorrect drainage of flood control area 31240 2 Mill at Zichem 33990 3 River Hulpe and Leigracht 35372 - 35660 4 Floods 44344, 44445 - 44637 and 44768 5 Mills at Aarschot 48879 6 Incorrect drainage of flood control area 52848 7 River Winge 58041 Chapter 6. Results and discussion 39
• The first region most upstream extends to a chainage of about 35600 m. The K values in this region all have a similar magnitude of about 0.0005. This zone is interrupted at a chainage of 33990 m by a dip in Kalman gain (Number 1 in figure 6.6). At this point the water flows into the side branch of the mill at Zichem. This removal of water lowers the discharge over a small distance which results in a sudden drop of K. A quite big drop in relative uncertainty can be observed in figure 6.8. This leads to a decrease in covariance in the nominator of K. In this covariance however, the model uncertainty on the observation point is also included. This factor does not show a decrease, leading to a more restricted decrease in covariance than in the variance of the EnKF at those locations. Consequently, the drop in K is not as intense as the drop in the variance of the EnKF simulation. A steep increase of the K is observed once the side branch connects to the river Demer again after about 600 m.
• A new zone of more or less constant Kalman gain starts at a chainage of 35600 m. This starting point is indicated as number 2 in figure 6.6. At this location a combined influence of the river Hulpe and Leigracht leads to this pattern in K. The Leigracht drains a part of the water of the Demer to the flood control areas while the Hulpe river flows into the Demer river and in this way causes an uplift in discharge. A slight increase in relative uncertainty when passing this chainage from upstream to downstream can be observed for most time steps in figure 6.8. Around the time of peak discharge however, a fall in relative uncertainty shows at this point. Next to an alteration in uncertainty, the change in Kalman gain is probably amplified by a strong correlation between the ensemble errors at the points in the Demer river close to the mouth with the Hulpe river and the ensemble errors at the observation point. This causes an intensification of the alteration in K.
• Number 3 in figure 6.6 indicates a gradual decrease in K during the time of the peak in the hydrograph. At chainages 44344, 44445 until 44637 and 44768 m the water level exceeds the height of the dikes occasionally. This leads to an overflow of the water over the dikes into the flood control areas. Figure 6.10 illustrates an example of discharge in the river Demer in this area, at a chainage of 44505 m. The course of discharge at this point shows a distinct cut-off at the peak. Logically the overflow of water over the dikes happens at high water levels and discharges. As a consequence, the change in Kalman gain is only visible during the time window of peak discharge in the bottom graph of figure 6.6. In a very small time zone around the maximum of the peak discharge, the effects of the floods are clearly visible. The blue square during the time of maximal discharge at the chainages where flooding occurs, indicates a severe drop in K due to a small drop in discharge because of the floods. Although in figure 6.8 the graph indicates a peak of uncertainty, the K at this point in time and space has a low value. As an explanation a low correlation of the ensemble errors between this location and the location of the hydraulic station is presumed since flooding only occurs at these chainages. This compensates for the large model variance and therefore leads to a lower covariance and thus a lower K. Chapter 6. Results and discussion 40
• Another region of similar K values extends from around chainage 47100 until about 52840 m. A first jump in Kalman gain around chainage 47100 m is caused by a inflow of water from the flood control area at the time of the discharge peak at a chainage of 47165 m. At this point, the variance of the EnKF simulation within the time window of peak discharge increases as can be observed in figure 6.8. This leads to a slight increase in the mean K but is clearly observed in the minimal and maximal K in figure A.6 since this phenomenon mostly occurs at extreme discharges. The second rise at point 4 in figure 6.6 more downstream in K (chainage 47919) can be attributed to the outflow of the river Motte. This river contributes about 2.5 m3/s to the discharge of the Demer. Another small rise of K is observed and the combination of both results in a small rise in K which is most visible during receding of the discharge peak. Downstream from this location, a constriction of the water occurs. For a distance of about 1200 m no flood control areas are present so the water is forced to stay within the river banks. For about 3700 more meters no floods take place, leading to a steady K up to point 6 in figure 6.6. The slightly decreasing trend of K in this area is probably due to a receding correlation with the observation point while moving further away from it.
• The location of the change in Kalman gain at point 6 coincides with an outflow of water from the Demer into the flood control area. In the simulation, the outflow occurs via a drainage construction to the Demer. Since the goal of such a construction consists of emptying the flood control areas, this is probably an incorrect setting in the model. In the relative uncertainty on the EnKF presented in figure 6.8 this leads to a limited decrease and a very small decrease in K for most time steps. Around the time window of peak discharge however, a substantial increase in variation is observed. This is translated in only a slight rise in K, indicating a low correlation of the ensemble errors at these points with the ensemble errors at hydraulic station 122.
• Point 5 in the upper graph in figure 6.6 at a chainage of 48879 m shows the effects of the mills at Aarschot. The side branch diverts a part of the water flowing in the river Demer and causes the discharge to decrease quickly. After approximately 300 m the side branch of the mill reconnects to the river and a sudden increase of discharge and hence also K occurs, pointing out a high correlation.
• The last change in Kalman gain of the hydraulic station 122 at Aarschot can be attributed to the flowing of the Winge into the Demer river. At chainage 58041 m the discharge increases when going downstream due to the water of the Winge that is added to the river Demer. Since the influence of the upstream boundary causes a high variation in the region downstream of the river Winge (see figure 6.8) the K downstream of the confluence is for most time steps higher than upstream. As with the variation in K at point 5, the magnitude of this change is limited, presumably due to a low correlation. Point 7 in figure 6.6 shows this variation in Kalman gain. Chapter 6. Results and discussion 41
Figure 6.8: Variation of the spread of the 90 % uncertainty interval for the EnKF with measurements of discharge relative to the spread of the 90 % uncertainty interval at hydraulic station 122 in time and space.
Figure 6.9: Variation of the spread of the 90 % uncertainty interval for the EnKF with measurements of discharge relative to the spread of the 90 % uncertainty interval at hydraulic station 123 in time and space. Chapter 6. Results and discussion 42
60
50
40
30
Discharge [m³/s] Discharge 20
10
0 Nov 05 Nov 09 Nov 13 Nov 17 Nov 21 Nov 25 Date
Figure 6.10: Evolution of discharge in time at a chainage of 44 505 m in the river Demer.
The division of the K in different chainage zones disappears around the time window of the peak discharge for hydraulic station 122 in figure 6.6. Although it looks like no zones can be defined, the actual cause of the disappearing of the intersections are the low values of K in the vicinity of the discharge peak. The substantial reduction of the K values as shown in figure 6.13 results from an increasing uncertainty of the simulations at the time interval of the peak. This leads − T to an increase in covariance and variance which corresponds to an increase in both the Pk Hk − T and HkPk Hk term of equation 6.1 respectively. Since the uncertainty on the observations is defined as a relative standard deviation of five percent, uncertainty on the observations will also augment as the discharge rises. This is reflected in an increase in the Rk term of equation 6.1. The increase of the denominator of this fraction will finally be more significant, resulting in a lower Kalman gain. Figure 6.5 shows the uncertainty on the observations together with the uncertainty on the simulations without DA. As a result of higher observation uncertainty, logical reasoning also shows that the Kalman gain values will decrease around the time interval of the peak discharge. A graph of the Kalman gain with a different scaling to obtain a better representation of the values in the time period of the peak is displayed in figure 6.11. The regions noticed in figure 6.6 can all be identified in figure 6.11 after rescaling.
In the hydraulic station 123 at Zichem, the evolution of the Kalman gain is influenced by similar hydraulic constructions and river characteristics. However, since the point at which the observation is taken deviates from the first, the extent of this influence is different.
• The first point in figure 6.7 marks a slight decrease in Kalman gain. The cause for this alteration in K is a drainage construction where water flows in and out of the Demer at the time of peak discharge. At the maximal outflow almost 1.9 m3/s is removed. Since the purpose of this channel is to drain the flood control areas, the water should only be able to pass in the direction of the Demer. This variation in K thus probably indicates an Chapter 6. Results and discussion 43
incorrect implementation of the drainage construction. From figure 6.9 it shows that the reduction in K at this point is associated with a decrease in relative uncertainty and that it is most apparent and abrupt at the time of peak discharge. The change in K at this point is not as visible for the hydraulic station number 122. This can be explained by a lower correlation with the hydraulic station 122 at Aarschot since it is further away than the hydraulic station 123 at Zichem.
• More downstream, at chainage 33990 m the presence of the mill at Zichem causes the discharge and hence the Kalman gain to drop and rise again. In figure 6.7 this is indicated as point number 2. The same reasons for the change in K as provided for the hydraulic station number 122 can also be applied here.
• The third point shows a gradual decrease of K starting at chainage 35350 m. Here, two influences are combined again. The first influence comes from the flow of the river Hulpe into the Demer river. The second indicates the drainage performed by the Leigracht. Both are discussed for the hydraulic station 122. A combination of the two previously discussed items, forces the discharge to alter. In figure 6.9 it shows that, especially around the time of peak discharge, the uncertainty on the EnKF simulations decreases with increasing chainage. For the time steps out of this time window of peak discharge, a slight increase in uncertainty can be noticed. In K however, a drop of values can be observed for all time steps. This indicates that the increase in covariance is compensated by a larger increase in variance in equation 2.7, presumably because of a higher correlation around the time of peak discharge of the ensemble errors at this location with the ensemble errors at the hydraulic station 123 at Zichem.
• At chainages 44344, 44445 until 44637 and 44768, the floods as described for hydraulic station 122 are less pronounced in the K of the hydraulic station 123. In figure 6.7 the impact of the floods on K is indicated as number 4. Since the K of the observations at Zichem is larger than the K of the observations at Aarschot, the effect of the floods is not as clear in the first.
• For the mills at Aarschot the same observation can be made. It is however still clear that the side branch of the mills induces a drop in discharge and K as shown in figure 6.7, number 5.
• The sixth variation in K indicated, can again be explained by the outflow of water from the Demer into the flood control areas through the drainage construction.
• The effect of the Winge flowing into the Demer is still present as well in the K of the observation in Zichem. Number 7 in figure 6.7 again points out chainage 58041 at which the Winge flows into the Demer.
For both figure 6.8 and 6.9 a higher spread of the ensemble members can be observed at both upstream and downstream ends. Since different ensembles are composed at the boundaries, the variation here is large. When approaching the middle section of the river, the different Chapter 6. Results and discussion 44
Figure 6.11: Evolution of K for discharge in time and space at hydraulic station 123 with a color scale to bring out the variation in the discharge peak.
ensembles lie closer to each other than at the boundaries resulting in a smaller variance. At the time interval of peak discharge the uncertainty remains high when moving from the up- stream or downstream part to the middle of the examined length of the river Demer. This can be explained by the relative uncertainty defined at the boundary conditions. Higher dis- charges as encountered around the time of peak discharge, result in a high relative uncertainty at the boundaries during this time frame and thus lead to a larger spread of the ensemble members.
In short: depending on the presence of a hydraulic structure, the variation in cross-section and other factors, the hydraulic behaviour in a certain point in the Demer can change. These changes in discharge are reflected in the magnitude of the Kalman gain along the Demer river, as was indicated in the previous analysis. Also the variation in discharge due to the migration of the flood peak is included in the change of the Kalman gain at different points in time. This results in an irregular shape of the Kalman gain. Moreover, a different pattern of K along the river is observed at different time steps since a variation in discharge is encountered in time. This for example leads to a somewhat different pattern of K along the river for a situation with low discharge than for a situation with high discharge.
6.4 Amplitude A
For the implementation of the weighting function in the MIKE 11 data assimilation editor, an amplitude A must be defined for every observation point. The value for this amplitude, is deter- mined based on Ensemble Kalman filter. A first attempt was made to estimate a value for the amplitude by examining the values of the K of the EnKF with discharge measurements. The result files obtained for the EnKF simulation with the MIKE 11 DA editor contain two matrices Chapter 6. Results and discussion 45 of K values for every time step and every chainage. One comprises the updates for the first observation point, hydraulic station 122 at Aarschot, and the second contains the updates for the second observation point, hydraulic station 123 at Zichem. To learn the total value of K for the update at a point in time and space, both matrices must be summed. The resulting update however still contains values in the order of magnitude of 10−3. The cause of these small values can be found in the frequent update as stated before. This is of course too small to result in a correct update of the simulated states since only a very small correction towards the observa- tions would be implemented. Unless a small update is of course performed in every single time step. As can be seen from the simulation with the EnKF in figures 6.1 and 6.2 the updating of the simulation without DA probably requires a higher value of A. However, as will be discussed later in this thesis, small updating values were also obtained with the implementation of the WF.
A second and more successful method consists of applying the definition of the amplitude. This definition states that the amplitude is based on the magnitude of the uncertainty of both the simulation and the observation. Depending on the reliability of the observations compared to the reliability of the simulations, A adopts a value between 0 and 1. If the observations are to be trusted completely A becomes 1. If the simulations are fully reliable, A becomes 0. This results in a formula for the calculation of an estimation of A based on the uncertainty of the observations and the simulations without DA as outlined in equation 6.2.
s A∗ = sim (6.2) ssim + sobs ∗ where A denotes an estimation of the amplitude, ssim the standard deviation of the simulation without DA at a certain point in time for a specific observation point and sobs the standard deviation of the observation at a certain point in time for a specific observation point.
Application of this formula for every observation point results in a series of estimations of the amplitude at the different time steps. From this sequence the average value over time is taken per observation point. The mean estimation of the amplitude can then be taken as an approximation of the amplitude for implementation in the weighting function. In the weighting function with discharge measurements two amplitudes must be determined, one for each observation point. For the observation at the hydraulic station number 122 at Aarschot downstream of the mills, the amplitude is estimated as 0.3465. A similar value of 0.3892 is obtained for the observation at the hydraulic station number 123 at Zichem. It can thus be concluded (since A is smaller than 0.5) that on average, the simulations without DA are trusted more that the observations for both observation points in this case.
6.5 Interval of influence
An estimate of the amplitude alone is not sufficient to run the WF in the MIKE 11 DA editor. A gain function must also be chosen to represent the course of the correlation of the ensemble errors in space. In figure 6.12 the three different options for the gain function are fitted to the Chapter 6. Results and discussion 46 shape of the Kalman gain obtained from the EnKF simulation with observations of discharge. The amplitude which is depicted in both graphs, does not correspond to the values described in the previous section. These figures are merely composed to demonstrate the shape of the three different functions. They provide a better insight in the fit of the gain functions to the Kalman gain of the EnKF with observations of discharge. However, they do not show an accurate repre- sentation of the magnitude of the gain applied in the weighting function. Moreover, the shape of the Kalman gain in figure 6.12 is obtained using the mean and the standard deviation of the K value during the time period of the rising limb. As it is most important to obtain an adequate prediction of the rise in discharge to produce an accurate forecast of the peak discharge, the shape of the gain function was fit to the mean of this period. From figures 6.13 and 6.14 it can be observed that this period is situated from time steps 2250 to 2500, which corresponds to approximately the 12th of November at 20h until the 13th of November at 16h.
In the upper graph of figure 6.12, it can be observed that the K at the hydraulic station number 122 at Aarschot displays a more or less constant gain between the upstream boundary and the jump in K induced by the rivers Hulpe and Motte flowing into the river Demer. From this chainage on, another approximately constant level of K is encountered until the drop caused by the mills at Aarschot. After the renewed connection of the side branch at the mills, a gradually decreasing K can be observed until the confluence with the river Dijle downstream. The interval of influence obviously must include the observation point. At this point updating is assumed to be at a maximum value because of the evident high correlation with itself, being 1. From the observation point at station 122 towards upstream and downstream, the theoretical correlation pattern follows a constant, triangular or mixed exponential course as presented by equations 2.6. In the case of the K for the hydraulic station 122 a constant gain is determined for the upstream side of the river while at the downstream side a triangular gain would be more appropriate. A remark in this context is that if an update is performed upstream, the points more downstream will automatically be updated as well. During a few time steps, the influence of the corrected states upstream proceeds downstream. As a result, adjusting the state more downstream hap- pens only after a short time period. This should however not impose any problems. Hence, a constant gain function is applied for the WF at the observation point at Aarschot with an interval that stretches from the chainage of the observation point itself until chainage 35660 where K changes. Another reason for the choice of the upstream interval is that errors occurred when implementing a broader interval stretching both upstream and downstream for the WF with updating in both observation points. This is probably the consequence of a conflicting update from the upstream observation point with the update from the downstream observation point in the interval in between the two update locations.
The Kalman gain for the hydraulic station number 123 at Zichem shows a different pattern. For most of the studied length of the Demer river, a triangular form of K is observed. Once the mills at Aarschot are reached, a more constant course is followed downstream. By continuing the same reasoning as for the hydraulic station 122 at Aarschot, the upstream section of the river is Chapter 6. Results and discussion 47
Figure 6.12: Top: Evolution of K for discharge in space at hydraulic station 122 together with the different shapes of the gain function. Bottom: Evolution of K for discharge in space at hydraulic station 123 together with the different shapes of the gain function. Chapter 6. Results and discussion 48 chosen as the interval for updating. No gain function can however be selected that corresponds to the pattern of K upstream of the observation point in Zichem. All gain function have thus been implemented to investigate the accuracy of correspondence of the three functions to the true situation.
6.6 WF with observations of discharge
The results of the WF with observations of discharge and an amplitude, gain function and inter- val of influence as described in the previous sections, are discussed in the next paragraph. First, the effects of an update in a single observation point are studied for both observation points of discharge. Afterwards, the results of the WF applied in both observation points simultaneously is examined. For the observation point in Zichem all three different gain functions are imple- mented since no conclusion could be drawn from the shape of the Kalman gain.
First of all, as a general remark it can be observed in all points illustrated in figures 6.15, 6.16, 6.22 and 6.23 that the results of the three WF with different gain functions are superimposed on one another. This observation results from a rapid decrease in the innovation term in equation 2.3. Initially, a substantial update is performed in the first time steps. In the following time steps, the influence of the previous corrections is still present due to a strong correlation of the discharge values in time and space. Consequently, the update will diminish and the innovations become small. This reduces the impact of the amplitude drastically which leads to the appear- ance of the three weighting functions coinciding. A very small distinction between the three gain functions can however still be made. Figures 6.18 and 6.19 show the update performed at the green points 1, 2 and 3 and the blue points 1, 2 and 3 respectively in figure 6.17. This shows that the variation in amplitude between the three functions increases when moving further away from the observation point. For points 1 and 2 in both simulations with one update point, the constant gain function has a larger value than the triangular function, which in turn has a larger value than the mixed exponential function. The mixed exponential function in point 3 however has a larger value than the triangular function but they are both still smaller than the value of the constant gain. From the shape of these functions and the distance of the point to the observation point in relation to the reach of the interval of influence, this rank in gain functions at each point can be explained. Despite the small variations in gain function, the effect of the difference in amplitude is cancelled out by the insignificant magnitude of the innovation. In addition, this small innovation causes the WF simulations to approach the observations very closely on the location of the observation point.
For the WF with one observation of discharge at the hydraulic station 122 at Aarschot, the results of the different gain functions can be observed in figure 6.15. The first graph shows the outcome of the three gain functions together with the EnKF simulation, the simulation without DA and the observations at hydraulic station 123. In the three following graphs, the results at three points within the interval of influence of the gain function are presented from Chapter 6. Results and discussion 49
Figure 6.13: Evolution of K in time and space at hydraulic station 122, together with the evolution of discharge in time at station 122.
Figure 6.14: Evolution of K in time and space at hydraulic station 123, together with the evolution of discharge in time at station 123. Chapter 6. Results and discussion 50 upstream to downstream. The last two graphs show the course of the WF simulations at two points downstream of the interval of influence. One is situated close to hydraulic station number 122 while the other is located near the downstream end of the river. The location of the points in the river Demer can be observed in figure 6.17. For most graphs, it is clear that the estimation of the peak discharge is higher than with the EnKF simulation. In the rising limb of the dis- charge peak however, both the WF simulations and the EnKF simulation follow approximately the same course. Past the time interval of peak discharge, the WF simulations approach the simulation without DA closer than the EnKF simulation. In the graph representing the point at hydraulic station number 123 the simulations without DA follow the same evolution as the outcome of the three weighting functions. This is due to the location of the upstream end of the interval of influence at a point more downstream than the hydraulic station 123 at Zichem. Most of the time no update is performed at this point since the water flows downstream and thus in most cases will not affect the evolution of discharge at the points more upstream of the interval. Points 1 to 3 are situated from upstream to downstream with the third point closest to the lo- cation of the observation point. Figure 6.15 shows that the WF simulations are located close to the simulation without DA in point 1. While moving to points 2 and 3, they seem to divert from the simulation without DA. This observation can be attributed to the peak of discharge which is propagated through the model when proceeding to the next time step. In the points situated more upstream, the propagated discharge originates from a location more upstream which was not included in the interval of influence and thus encountered no update. Consequently the points situated more downstream experience a cumulative effect as can be observed from figure 6.20 and 6.21. These figures also demonstrate the flattening and widening of the peak discharge when moving more downstream due to the roughness in the river and possible floods along its path. The cumulative updates of discharge performed by the WF are also influenced by this flatter and broader peak discharge and as a result, shows the same pattern. Especially in figure 6.20 this effect is clearly visible. Downstream of the observation point in hydraulic station 122, the influence of the DA procedure is restricted to the propagation of the updated discharges upstream in the interval of influence. This effect is illustrated in points 4 and 5 in figure 6.15. Before as well as after the peak, the WF simulations approach the EnKF simulation closely. As can be deduced from the closer estimation of the peak discharge to the simulation without DA in point 5 than in point 4, the influence of updating reduces while moving more downstream.
Figure 6.16 shows the three WF simulations with a different gain function for the up- date with observations of discharge in Zichem, together with the simulation without DA and the EnKF simulation at six different points in the river Demer. The location of these points is indicated as the orange dot most downstream for the hydraulic station 122 and the blue dots with numbers 1 to 5 in figure 6.17. In comparison with figure 6.15, the distinction between the different curves in figure 6.16 is less clear due to the small zone between the observation and the simulation without DA. Because of this, only a limited innovation is created. This observa- tion contributes to the restricted assimilation update due to the correlation of discharge values in time and space and results in an even smaller innovation. At the hydraulic station 122 at Chapter 6. Results and discussion 51 Graphs of the discharge inThe function result of of time the forrespectively the EnKF 37073.5 three simulation m, different and 42004.5 weighting m, the functions 47034m, simulation with 51478 one without m observation DA and point is 56443 in m. also hydraulic presented station 122. for each point. The chainages of points 1 to 5 are Figure 6.15: Chapter 6. Results and discussion 52 iue6.16: Figure h euto h nFsmlto n h iuainwtotD sas rsne o ahpit h hiae fpit o5are 5 to 1 points of chainages The point. each for 123. station presented m. hydraulic also 54032 in and is point m DA observation 41072 without one m, with simulation 32553.5 functions the m weighting 31024 and different m, simulation three 29585.5 EnKF the respectively for the time of of result function The in discharge the of Graphs Chapter 6. Results and discussion 53 Indication of the locationsfigure of 6.15, the the blue points numbersthe in show green the figures points locations 6.15, 1-3 of 6.16,observation and the points. 6.22 points the in and blue figure points 6.23. 6.16, 1-5 while The (numbered the as green locations point numbers of 4-8 figures indicate 6.22 in the and the locations 6.23 graphs). are of represented The the by orange points dots in represent the locations of both the Figure 6.17: Chapter 6. Results and discussion 54
Figure 6.18: Total assimilation update (value of the gain function at that location multiplied by the innovation) at green points 1-3 in figure 6.17 for the three gain functions of the weighting function with an update at hydraulic station 122.
Figure 6.19: Total assimilation update (value of the gain function at that location multiplied by the innovation) at blue points 1-3 in figure 6.17 for the three gain functions of the weighting function with an update at hydraulic station 123. Chapter 6. Results and discussion 55
Figure 6.20: Cumulative difference in discharge for the green points 1-3 in figure 6.17 in time. The dif- ference is calculated between the WF with application of the three gain functions (update at hydraulic station 122) and the simulation without DA.
Figure 6.21: Cumulative difference in discharge for the blue points 1-3 in figure 6.17 in time. The dif- ference is calculated between the WF with application of the three gain functions (update at hydraulic station 123) and the simulation without DA. Chapter 6. Results and discussion 56
Aarschot, a limited deviation from the simulation without DA is observed. An explanation for this can be found in the considerable distance between both observation points. This restricts the influence of the hydraulic station 123 at Zichem on the one downstream at Aarschot. For points 1, 2 and 3 similar conclusions can be drawn as with the first three points in figure 6.15, for the update with the WF at the observation point in Aarschot. Although it is not as evident as in figure 6.15, the difference between the WF simulations and the simulation without DA at point 3 is again larger than at point 2, which is in turn larger than at point 1. Again the points more downstream encounter a stronger influence by the propagation of the previous upstream updates through the model. For points 4 and 5 downstream of the observation point, the di- version of the WF simulations from the simulation without DA is small. This is also the case on the location of the observation point in the first graph. Once again this is probably due to the considerable distance between the location of these points and the location of the hydraulic station 123, combined with the influence of the occurrence of floods and side rivers encountered along the way. At point 4 the distance between the WF simulations and the simulation without DA before and at the time of peak discharge is slightly bigger than at point 5 because of a the closer location of point 4 to the hydraulic station 123. After the time of peak discharge, the distance at both points remains approximately the same.
The results of the weighting function simulation with an update performed at both observation points, are represented in figures 6.22 and 6.23. In each graph, the weighting function with a constant gain function for the update at Aarschot and the three different gain functions for the update at Zichem together with the simulation without DA and the EnKF simulation are shown. In points 1 to 3 as well as in the hydraulic station 123 at Zichem, the difference between the WF simulations and the simulation without DA is very small. This is also the case for the WF with only one update at Zichem as discussed previously and can be attributed to the same causes. It can also be stated for this data assimilation procedure that the difference between the WF simulation and the simulation without DA in the points more downstream is more significant than for the points more upstream. This is the case for points 1 to 3 as well as for points 4 to 6. When compared to the WF with an update at one point in Aarschot, the difference between the WF simulations and the simulation without DA in points 4, 5 and 6 is larger than the same difference for points 1 to 3 in the WF with only one update point. This can be explained by the use of the second updating point. The updates at one time step in the points upstream of the hydraulic station 123 at Zichem are propagated downstream with the movement of the discharge peak and influence the points more downstream. Although this effect was limited in the WF with one observation at Zichem, the cumulative effect with the update from the observation point at Aarschot is clearly visible. At the hydraulic station 122 at Aarschot, the WF simulations are all piled on top of the observations. This phenomenon is also observed for the WF with one observation at Aarschot. Due to the strong correlation of discharge in time and space, the updates become small and the WF simulations are kept close to the observations. Downstream of the hydraulic station 122 at Aarschot no more assimila- tion updates are performed. The only possible update in this region is the correction due to Chapter 6. Results and discussion 57 the propagation of the updated discharges originating from more upstream. At the location of points 7 and 8 the influence of the update at station 123, but mostly the influence of the update at station 122 is still strongly felt as can be seen from figure 6.23. Though at most points the WF simulations are located closer to the simulation without DA than the EnKF simulation, in these two points the WF simulations almost coincide with the EnKF simulation. In figure 6.15 at points 4 and 5, the WF simulations exhibit a similar evolution in time. The influence of the update at the hydraulic station 123 at Zichem has probably completely disappeared at this distance. Since the WF simulations at peak discharge at point 8 are closer to the simulation without DA than at point 7, it shows that the influence of the assimilation update at hydraulic station 122 is also decreasing more downstream.
From the previous analysis of the different simulations of the WF, a few concepts can be observed for all simulation. First of all it was pointed out that the WF simulations with application of a different gain function all coincided. Moreover, the resulting innovations obtained were extremely small. These observations indicate that the magnitude of the amplitude and consequently also the shape of the gain for the WF do not have a substantial effect on the update of the simulation without DA. The width of the interval of updating thus seems to be the parameter with the largest influence on the obtained results. As a comment on this it should be noted that only the upstream distance to the observation point seems to have an effect since the discharge values downstream are influenced by the updated values resulting from the propagation of the upstream updates. Secondly it was observed that the influence of the WF increases when moving more downstream in the interval of updating. This can be explained by the accumulated effect of the propagation of the discharge peak and the update due to the WF in the interval. The last general remark implies that the downstream influence of the WF DA algorithm is limited to the propagation of the discharge peak. When moving downstream, this update is combined with the effects of floods and tributaries, leading to a decreasing update.
6.7 WF with observations of water level
Although the Ensemble Kalman filter with observations of water level could not be executed as explained in section 6.2, it would still be possible to implement the WF with observations of water level. In this case it would be necessary to define four amplitudes since four observations of water level are available. These amplitudes could be calculated using the same formula as described for the WF with observations of discharge in section 6.4. Since no Kalman gain is available for this run it would not be possible to determine an interval of influence for the weighting function with water level based on the shape of the Kalman gain. Nevertheless, the intervals of influence for each observation point could be defined as starting at the observation point and moving upstream until the next observation point or the upstream boundary was encountered. This would already provide a first idea of the update of the WF with observations of water level. Due to time constraints however no such experiments were performed. Chapter 6. Results and discussion 58 iue6.22: Figure h euto h nFsmlto n h iuainwtotD sas rsne o ahpit h hiae fpit o8are 8 to 1 points of chainages The m. 56443 implemented. point. and was each m gain for 51478 constant presented 47034m, a also m, Aarschot 42004.5 is m, at DA 37073.5 point without observation m, observation simulation the 32553.5 the In the m for points. 31024 and observation while m, simulation both 29585.5 implemented EnKF on respectively are updating the with functions of function gain weighting result the three The for all time Zichem of function at in point discharge the of Graphs 1) (Part Chapter 6. Results and discussion 59 (Part 2) Graphs of the dischargepoint in at function of Zichem time all forThe three the result weighting gain function of functions with the updating arerespectively on EnKF implemented 29585.5 both simulation m, while observation and 31024 points. for m the In the 32553.5 the simulation observation m, observation without point 37073.5 DA at m, is 42004.5 Aarschot m, also a 47034m, presented constant 51478 for gain m each was and point. implemented. 56443 m. The chainages of points 1 to 8 are Figure 6.23: Chapter 6. Results and discussion 60
6.8 Validation
The RMSE-observations standard deviation ratio (RSR), the Nash-Sutcliffe efficiency (NSE) and the percent bias (PBIAS) of the difference between the simulation without DA and the WF with discharge observations over all time steps, and the ’true’ time series of discharge is represented on the left graphs in figure 6.24. The evolution of RSR, NSE and PBIAS in space for the difference between the simulation without DA, WF with updating at all time steps and WF with updating until the 18th of November, and the ’true’ time series of discharge is represented on the right side in figure 6.24. For most points, the changes in RSR, NSE and PBIAS can be related to changes or hydraulic structures in the river system as was done for the K. Only the changes important for the evaluation of the performance of the simulations will be discussed here.
From the graphs illustrating the evolution of RSR and NSE along the river Demer for the simu- lation without DA and the WF with updating at all points, it is apparent that the WF performs better in estimating the truth than can be said from the simulation without DA. Everywhere the RSR is lower while the NSE is closer to one for the WF compared to the simulation without DA. At the upstream chainages however, an important influence of the boundary condition is observed. Since this boundary condition is not updated, it will lead to a diversion of the WF simulations from the truth and thus cause a higher value of RSR and a lower value of NSE at the upstream side of the river. The graph in the lower left corner of figure 6.24 shows the PBIAS for both the simulation without DA and the WF simulation. Upstream a decreasing positive bias can be observed for the WF simulation which then over a short distance becomes negative, followed by a rise in PBIAS value. From a chainage of about 47000 m an approximately constant value of 1 is observed until the downstream end of the river. This indicates an underestimation of the truth upstream, shortly changing into an overestimation and finally underestimating the truth in the downstream end of the river again. The PBIAS value for the simulation without DA on the other hand shows a quite constant course with two small decreases and one large dip. This considerable decrease in PBIAS is caused by floods occurring around these chainages. Due to the floods, a broader and lower peak discharge is obtained in the true discharge which causes an overestimation by the simulations without DA instead of the underestimation at the other chainages. A lower bias (closer to zero) is obtained for the WF simulation compared to the bias at different places in the river. For most points upstream of a chainage of about 44000 m the bias in the WF is smaller than the bias in the simulation without DA, as would be expected. The upstream effect in PBIAS of the WF is again caused by the influence of the upstream boundary. Downstream of the floods, both simulations show a more or less equal model underestimation bias.
On the left side of figure 6.24), the evolution of the RSR, NSE and PBIAS in space is calculated using the values of discharge between the 18th of November 12h until the 19th of November 12h. Almost everywhere a lower RSR and a NSE closer to one is found for the WF simulations as for the simulations without DA. This indicates that the WF simulations will deviate less from the Chapter 6. Results and discussion 61
Sim no DA -3 -3 #10 #10 WF 6 1.2 WF no updates Sim no DA 5 WF 1
4 0.8
3 0.6
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NSE NSE 0.9994 0.992 Sim no DA Sim no DA 0.9992 WF 0.99 WF WF no updates
3 4 5 6 2.5 3 3.5 4 4.5 5 5.5 6 Chainage [m] #104 Chainage [m] #104
1.5 0.4 Sim no DA WF 0.3 WF no updates 1 0.2
0.5 0.1 PBIAS [%] PBIAS PBIAS [%] PBIAS 0 0 Sim no DA -0.1 WF -0.5 -0.2 2 3 4 5 6 2 3 4 5 6 Chainage [m] #104 Chainage [m] #104
Figure 6.24: Left: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA and the ’true’ time series of discharge on the one hand and the RSR, NSE and PBIAS of the difference between the WF with updating at all time steps and the ’true’ time series of discharge on the other hand. Right: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA, WF with updating at all time steps and WF with updating until the 18th of November, and the ’true’ time series of discharge. Updating with the WF was performed at both observation points. Chapter 6. Results and discussion 62 truth than the simulation without DA. Also, the WF simulation with updating in all time steps performs better than the WF with updating until the 18th of November. Of course, these results were to be expected. The indicators at the upstream river side are in this case also influenced by the upstream boundary condition. At this upstream side of the river, it can be noticed that the WF with updates until the 18th of November performs worse than the simulation without DA. The reason for this illogical observation is not that the WF performs a lot worse than at other places but that the simulation without DA performs a lot better than at other places. In between the chainages of about 44000 and 47000 the RSR shows low values and the NSE approaches one for all curves. In this interval in space, floods occur as already mentioned in section 6.3.2. This leads to a flattening of the peak and thus creates a broad, flat peak discharge. Since the observed time interval and the four preceding hours are situated in this broad and flat peak discharge, all three methods will be able to predict the discharges in this time interval exceptionally well. This is illustrated in figure 6.25. Approximately the same reasoning leads to the explanation of the coinciding of both RSR and NSE for the WF simulations from around chainage 47000 to downstream. Since the discharge peak becomes broader and flatter due to friction in the river and possible floods, both the WF simulations will be equally capable of predicting the true discharge. For the WF with updating at all time steps, only limited updating will be necessary once the simulation is on the peak. With the WF with updating until the 18th of November the four preceding hours are used to construct the updating after time of forecast and thus a more or less constant discharge will be pursued as was the case in the four preceding hours. An illustration of the evolution of discharge for the different simulations around this time period is show in figure 6.26. The evolution of the PBIAS in the graph in the lower right corner of figure 6.24 corresponds to the findings in the two upper graphs. The lowest bias is found for the WF with updates at every point in time, with the exception of the upstream boundary. The second lowest bias occurs for the WF with updating until the 18th of November. Both WF simulations also show the same bias from a chainage of 47000 m downstream onwards. From this graph the extra observation can be made that only the WF with updates until the 18th of November show an overestimation bias until a chainage of about 47000 m. The other two simulations exhibit an underestimation of the truth over the whole length of the river system, except in the interval where floods occur.
Figure 6.27 illustrates the evolution of the same indicators as in figure 6.24 but for the EnKF simulation. The left side of the figure is again calculated using the whole time series while the graphs on the right side only include 24 hours (18th of November 12h until 19th of November 12h). All three indicators are calculated for the difference between the simulation without DA and the truth as well as for the difference between the EnKF simulation and the truth. In the right graphs also the difference between the EnKF with updates until the 18th of November and the truth is included. In general it can be observed that the EnKF shows a better performance in predicting the true discharge than the simulation without DA for both time periods. In the upstream part of the graphs on the right side however, he RSR and the NSE indicate a worse prediction with the EnKF with updates until the 18th of November than with the simulation Chapter 6. Results and discussion 63
Sim no DA 55 WF WF no updates True discharge 50
45
40 Discharge [m³/s] Discharge
35
30
Nov 13 Nov 15 Nov 17 Nov 19 Nov 21 Nov 23 Nov 25 Date
Figure 6.25: Evolution of discharge for three different simulations (without DA, with the WF with updates at all time steps and the WF with updates until the 18th of November) and the truth at chainage 44977 m. Updating with the WF was performed at both observation points.
65 Sim no DA WF 60 WF no updates True discharge 55
50
45
Discharge [m³/s] Discharge 40
35
30
Nov 13 Nov 15 Nov 17 Nov 19 Nov 21 Nov 23 Nov 25 Date
Figure 6.26: Evolution of discharge for three different simulations (without DA, with the WF with updates at all time steps and the WF with updates until the 18th of November) and the truth at chainage 52002 m. Updating with the WF was performed at both observation points. Chapter 6. Results and discussion 64
Sim no DA -3 -3 #10 #10 EnKF 6 1.2 EnKF no updates Sim no DA 5 EnKF 1
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NSE NSE 0.9994 0.992 Sim no DA Sim no DA 0.9992 EnKF 0.99 EnKF EnKF no updates
2.5 3 3.5 4 4.5 5 5.5 6 2.5 3 3.5 4 4.5 5 5.5 6 Chainage [m] #104 Chainage [m] #104
1.5 0.4 Sim no DA EnKF 0.3 EnKF no updates 1 0.2
0.5 0.1 PBIAS [%] PBIAS PBIAS [%] PBIAS 0 0 Sim no DA -0.1 EnKF -0.5 -0.2 2 3 4 5 6 2 3 4 5 6 Chainage [m] #104 Chainage [m] #104
Figure 6.27: Left: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA and the ’true’ time series of discharge on the one hand and the RSR, NSE and PBIAS of the difference between the EnKF with updating at all time steps and the ’true’ time series of discharge on the other hand. Right: Evolution in space of the RSR, NSE and PBIAS of the difference between simulation without DA, EnKF with updating at all time steps and EnKF with updating until the 18th of November, and the ’true’ time series of discharge. Updating with the WF and the EnKF was performed at both observation points. Chapter 6. Results and discussion 65 without DA. The PBIAS also shows a strong negative bias upstream compared to the other two simulations. This observation can be explained by a smaller width of the hydrograph peak upstream than downstream. The updating is stopped at the moment the peak discharge starts to decrease. Since no further updating is performed, the EnKF will predict a broader peak and thus perform badly. Further downstream the peak will be broader resulting in a better prediction by the EnKF. In the downstream area, again a coinciding of the two EnKF simulations can be noticed, indicating that updating after the 18th of November did not improve the predictions downstream.
6.8.1 WF vs EnKF performance
Contrary to what was expected, figure 6.28 shows that a similar performance is achieved by the WF as with the EnKF. An explanation for this equally well performance can be found in the small innovations, leading to small updating. Due to an extensive availability of observations in time, the updates remain small and are thus comparable for both updating procedures. More- over, updating in the upstream section propagates more downstream due to the migration of the discharge peak, which might decrease the importance of the shape of the gain function. A difference between the two updating procedures in the goodness-of-fit indicators can nevertheless be observed in the upstream part of the river. Especially when the indicators are calculated using the whole time period (graphs on the left) and for the 24 hours after the time of forecast, with updating until the 18th of November (graphs on the right). In the three graphs to the left, the WF performs worse than the EnKF upstream. Since the indicator value for the WF simulation at the upstream side corresponds to the indicator value of the simulation without DA, this inadequate performance is probably due to the influence of the upstream boundary. For the EnKF an ensemble is run. Due to an averaging of the different ensembles runs, this effect is not as strong as with the WF simulation. The opposite is observed upstream in the graphs on the right side of figure 6.28. Here, the EnKF displays a lower value for the NSE, a higher value for the RSR and a higher positive value for the PBIAS, indicating a worse performance than the WF. This can be explained by the shape of the discharge hydrograph upstream. At the time of forecast (the 18th of November at 12h) the true discharge starts to decrease, approaching the simulation without DA. The EnKF simulation continues to predict higher values, moving further away from the truth since no further updates are performed. The WF on the other hand was already situated closer to the simulation without DA and thus is located closer to the true discharge. If updating is done over the whole time period and only the 24 hour time period is observed, no clear difference is noticeable upstream.
Downstream the indicators for both DA procedures almost coincide, leading to the conclusion that the upstream differences in updating originate from the exclusive assimilation updating upstream with the observations of hydraulic station 123. Since further upstream no updated discharges are available which can proceed downstream, no other adjustments are made. The downstream equality on the other hand can be explained by the cumulative effect of assimila- tion updating by the hydraulic station 122 downstream, together with the adjustments of the Chapter 6. Results and discussion 66 iue6.28: Figure h 9ho oebr1h ih:Sm so h idefiuebtfrteW n nFwt paigutlte1t fNovember. of 18th the until until updating with 12h with November steps EnKF of time 18th and all of WF at period the updating the for points. with for observation but but WF both figure with figure the middle left performed and the was the EnKF on EnKF on the the as as and both Same Same WF for Middle: Right: the difference discharge with hand. of the 12h. Updating series other November of time the ’true’ of PBIAS on the 19th and and discharge DA the of NSE without series simulation RSR, between time the difference ’true’ the and the of hand PBIAS and one NSE the RSR, the on of space in Evolution Left: Chapter 6. Results and discussion 67 discharge due to the updated values originating from upstream that proceed downstream in the river. The effect of downstream propagating of updated discharges can be examined by per- forming DA with a single observation point upstream and comparing this to DA results from an update using two observation points. For the WF a slight difference downstream was noticeable but with the EnKF this was not yet tested. Furthermore it can be seen that if the indicators are calculated over the whole time period, the RSR and bias show a broader range of values than if only the 24 hour time period is used. This indicates that during the considered time period the DA simulations deviated less from the true discharge values than during the whole time period.
The indicators represented in figure 6.28 are summarized in table 6.3. A mean value is taken over three different sections: The first extending from upstream until the hydraulic station 123 at Zichem, the second one in between the two hydraulic stations and the third one extending from the hydraulic station 122 at Aarschot until the downstream boundary. From this table it can be deduced that both DA procedures as well as the simulation without DA perform quite well. According to the targets postulated by Moriasi et al. (2007) the performance of all three simulations is extremely well: all RSR values are much closer to zero than 0.70, all NSE values approach one much better than 0.5 and the PBIAS values are all much smaller than 25 % (even smaller than 1 %). The EnKF simulation however performs slightly better than both other sim- ulations in all sections but the difference with the WF simulation is very small. Comparing the different sections shows that in the middle and downstream section a more accurate prediction of the truth is provided. In the downstream section a strong model underestimation bias is observed, even more so in the WF simulation than in the EnKF simulation. An explanation can be found in the flattening and broadening of the hydrograph peak when moving downstream. As stated before, the upstream section renders the worst performance due to the exclusive assimi- lation upstream with the observations at hydraulic station 123. Furthermore it can be noticed that only the EnKF simulation in the upstream section presents a mean model overestimation bias.
When looking at the overall picture of the performance of both the WF and the EnKF DA algorithm, some observations are generally valid. According to the values of the goodness-of-fit indicators, the simulation without DA as well as the simulations with DA showed an adequate performance of the prediction of the truth. However, both DA algorithms indicate that a considerable improvement in prediction capacity can be attained when DA is implemented. The EnKF simulation seemed to predict the truth better than the WF simulation, though apparently the updating by the WF can perform almost equally well as the updating with the EnKF procedure. The slight difference was most apparent in the upstream section. More downstream, almost no distinction could be made between the performance of the WF and the EnKF. Even for both DA procedures with updating at all time steps and the DA procedures with updating until the 18th of November, almost no difference in the 24 hour prediction was observed downstream. Chapter 6. Results and discussion 68
Table 6.3: Mean values of the goodness-of-fit indicators over a part of the river Demer. For the sim- ulation without DA, the WF and the EnKF the mean was taken over the most upstream part until the hydraulic station 123 at Zichem (US section), in between the two observation points (Middle section) and over the downstream part of the river starting at the hydraulic station 122 at Aarschot (DS section). Updating with the WF and the EnKF was performed with both observation points.
RSR Upstream section Middle section Downstream section Sim without DA 0.0046 0.0023 0.0018 WF 0.0027 0.0006 0.0007 EnKF 0.0016 0.0005 0.0007 NSE Upstream section Middle section Downstream section Sim without DA 0.9900 0.9951 0.9958 WF 0.9965 0.9997 0.9993 EnKF 0.9991 0.9998 0.9994 PBIAS Upstream section Middle section Downstream section Sim without DA 0.9548 0.7173 0.8575 WF 0.5903 0.2464 0.9568 EnKF -0.2749 0.1049 0.8047
6.8.2 Limitations
It can be noted that the validation with the synthetic experiment has its limitations. For example: the autocorrelation of the errors of the synthetic observations are based on the provided observations in the two hydraulic stations, while the boundary conditions used in the original DA procedures are also applied in the synthetic experiment. The true situation might actually be more different. Moreover, this synthetic experiment only represents and validates one possible truth. Other ’true’ time series of discharge can be composed and a forecast of other time periods can be implemented, possibly leading to more complete conclusions. A validation with a real validation data set would thus be preferred. Chapter 7
Conclusion
For the EnKF DA procedure with the application of water level data severe instabilities oc- curred. When discharge data was applied, instabilities were also encountered but they were considerably smaller. These instabilities terminated the model runs repeatedly when using wa- ter level observations even though measures were taken to increase the stability. This indicates that the EnKF DA method implemented in the MIKE 11 software is highly sensitive to insta- bilities. Adjustments have been made to improve the stability of the system but for complex river systems, stability problems are still encountered. In the WF DA procedure these stability issues were experienced significantly less frequent. As a partial answer to the question posed in the introduction it can be concluded that the application of observations of water level leads to a more unstable simulation than the application of observations of discharge. More research could not be performed since no simulations with DA procedures using observations of water level were implemented.
From the simulation with the EnKF DA procedure it could be observed that the changes in the river structure and the hydraulic constructions present induced variations in K. Since it is important to obtain an accurate representation of the different influences in the actual river system, these changes in K can not be avoided. The gain functions implemented in the MIKE 11 DA editor for the WF however did not provide a correct approximation of the shape of K. On the other hand it was shown that the shape of the gain function (and thus also the amplitude) did not have a substantial influence on the results of the WF DA algorithm. Only the width of the upstream interval of updating was concluded to have a considerable influence on the results. As a result, the amplitude as described in DHI (2014a) appears to lose its importance. This conclusion seems to somewhat contradict the description of the weighting function as presented in DHI (2014a).
The second goal of this thesis was to compare both DA procedures in terms of calculation speed, most accurate approximation of the real situation and other factors. In general it can be said that the EnKF simulations provide a slightly better estimation of the truth than the WF simulations as could be observed in the validation experiment. More research should however
69 Chapter 7. Conclusion 70 be performed to be certain of this conclusion. The long calculation time necessary for the EnKF DA procedure can be seen as a disadvantage. For the runs executed in this thesis for 25 days with a time step of one minute, it took the EnKF about 36 hours while the WF could perform a simulation in less than half an hour. Considerable improvements can nevertheless be obtained with application of the WF compared to the simulation without DA. Even when an update is only performed for part of the time period, a substantial improvement was observed for the simulations of the next 24 hours. Since the application of the EnKF only provided a small improvement of the estimation of the truth, in this case the WF simulation would be preferred. Of course, no estimation of the uncertainty is obtained when executing the WF DA procedure. Moreover, when forecasting a certain time period no difference could be seen in the simulations with or without updating after the time of forecast for the downstream part of the river.
7.1 Suggestions for future research
First of all it is necessary to study the instabilities in the DA procedure. What is their origin and how is it possible to counteract these problems even more than is applied now? Possibly some points of attention for the construction of a model can be pointed out to construct a more stable model while maintaining the necessary complexity of the model. This would already solve a lot of problems encountered in the course of this thesis. Also on the model uncertainty more research is desirable. The discharge data applied in this experiment was obtained using water level data. These data were then transformed using a h − Q relationship. Moreover, no uncertainty on the model parameters was taken into account.
From this experiment, the amplitude and shape of the gain for the WF DA procedure appear to have no influence on the results. If however more in depth research would show that in different circumstances they do have a substantial influence, a combination of the EnKF and the WF procedures would be a possible option. This procedure could lead to shorter calculation periods and more accurate predictions. Though the differences between both DA procedures appeared to be small in this experiment, it can be otherwise under different circumstances (for example when observations are available less frequently). As the different hydraulic constructions and the structure of the river system have a substantial influence on the shape of the gain function, it might be that this shape remains more or less the same for a certain time period. If it would then be possible to implement a gain function of the modellers choice in the WF DA procedure, simulations with a shorter run time than for the EnKF simulation and more accurate predic- tions than with the original WF simulation could be performed. Of course this would not allow an estimation of the uncertainty on the model results. The suggested procedure also requires an estimation of the magnitude of the gain. An estimation can be obtained by applying the suggested formula in section 6.4. A thorough investigation of the accurateness of this formula is however still required. If implementation of the original gain functions is maintained, more research should be executed on the circumstances where the amplitude has a substantial effect on the assimilation update and the extent of this effect. Moreover, the influence of the width Chapter 7. Conclusion 71 and the location of the interval of updating should be investigated. This would clarify whether an update in the downstream area is necessary. If not, instabilities resulting from the updates from two observation points hindering each other can be avoided.
Another research possibility consists of investigating other options of DA procedures since se- vere limitations are observed for both implemented DA procedures: the EnKF DA procedure does not include parameter uncertainty, requires a long calculation time and shows instabilities, while the WF DA procedure is only influenced by the choice of the upstream interval of updating.
From the suggestions described in the previous paragraph, the need for more research in the field of data assimilation in hydraulic models is clearly stated. Since the implementation of a relatively simple DA procedure as the WF can already lead to a substantial improvement, application of these procedures is definitely crucial. An even better forecast of discharges and water levels in a river system can probably be obtained with more research and would evidently lead to improved predictions of the time and extent of for example a flood. Logically a lot of damage and costs can be avoided in this way, resulting in a beneficial effect for all. Chapter 7. Conclusion 72 Bibliography
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G. Welch & G. Bishop (1997). An introduction to the Kalman filter. Department of computer science, University of North Carolina at Chapel Hill. Bibliography 76 Additional figures
77 Appendix . Additional figures 78 iueA.1: Figure
ae ee rfieadcosscino h ie ee o h nFsmlto ihtemxmmwtrlvl(e ie n h minimal the and line) (Red level water maximum the with simulation EnKF the for Demer line). river (Green the level of water cross-section and profile level Water Water level [meter] 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 4.0 6.0 8.0 0.0 5000.0 10000.0 Upstream distance to confluence with X Dijle, 1:0, to confluence distance Upstream Water level in the river Demer - 10-11-2010 14: 10-11-2010 - Demer river the in level Water 15000.0 20000.0 Y 1:2156 00:00 00:00 25000.0 30000.0 [m]
Appendix . Additional figures 79
44565 44565 44565
44637 44637 44637
[m]
44768 44768 44768
15000.0 44854 44854 44854
44940 44940 44940 44940 44940 44940
45014 45014 45014
45131 45131 45131
45219 45219 45219 45219 45219 45219
45364 45364 45364 45364 45364 45364
45465 45465 45465
45564 45564 45564
45645 45645 45645 45645 45645 45645 45667 45667 45667 45670 45670 45670
45751 45751 45751 45751 45751 45751
14000.0
46049 46049 46049 46049 46049 46049
46140 46140 46140
46330 46330 46330
46440 46440 46440 46440 46440 46440
13000.0 46879 46879 46879 46879 46879 46879
46984 46984 46984
47084 47084 47084 47084 47084 47084
00:00 47165 47165 47165 47165 47165 47165 1:2156 Y
47351 47351 47351 47351 47351 47351
47563 47563 47563
47653 47653 47653 47653 47653 47653
47752 47752 47752
12000.0
47863 47863 47863
47917 47917 47917 47917 47917 47917
47941 47941 47941 47963 47963 47963
48075 48075 48075
48142 48142 48142
48216 48216 48216
48305 48305 48305
48392 48392 48392
48504 48504 48504
48621 48621 48621
48701 48701 48701
48794 48794 48794 48794 48794
48794 11000.0 48850 48850 48850
48874 48874 48874 48893 48893 48893
48959 48959 48959 48993 48993 48993
49060 49060 49060 49060 49060 49060
49104 49104 49104
49141 49141 49141
49220 49220 49220
49267 49267 49267
49361 49361 49361
49374 49374 49374
49404 49404 49404 49404 49404
Water level intheriver Demer -10-11-2010 14: 49404
Upstreamdistance confluenceto 1:0, Dijle, X with
49504 49504 49504
49597 49597 49597
49646 49646 49646
49695 49695 49695
49743 49743 49743
49791 49791 49791
49839 49839 49839 10000.0
49937 49937 49937
50040 50040 50040 50040 50040 50040
50089 50089 50089
50142 50142 50142
50193 50193 50193
50249 50249 50249
50280 50280 50280
50321 50321 50321 50321 50321 50321
50368 50368 50368
50415 50415 50415
50461 50461 50461
50509 50509 50509 50509 50509 50509
50608 50608 50608
50649 50649 50649
50696 50696 50696
50744 50744 50744
50837 50837 50837 9000.0
50885 50885 50885
50932 50932 50932
50981 50981 50981 50981 50981 50981
51065 51065 51065
51150 51150 51150 51150 51150 51150
51198 51198 51198
51247 51247 51247
51293 51293 51293
51346 51346 51346 51346 51346 51346
51450 51450 51450
51506 51506 51506 9.0 8.0 7.0 6.0 5.0 8000.0
19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.0 [meter] Water level Water Water level profile and(Red cross-section line) of and the the river minimum Demer water level at (Green the line) mills and in indications Aarschot of for the chainage. the EnKF simulation with the maximum water level Figure A.2: Appendix . Additional figures 80 iueA.3: Figure
icag rfieadcosscino h ie ee o h nFsmlto ihtemxmmdshre(e ie n h minimum the and line) (Red discharge maximum the with simulation EnKF the for Demer river line). the (Green discharge of cross-section and profile Discharge Heigth of the cross-section [meter] 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 4.0 6.0 8.0 0.0 5000.0 10000.0 Discharge in the river Demer - 10-11-2010 14:00 10-11-2010 - Demer river the in Discharge Upstream distance to confluence with 1:0Dijle, to confluence distance Upstream 15000.0 20000.0 :00 :00 25000.0 30000.0 [m] [m^3/s] 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 5.0
Discharge Appendix . Additional figures 81 in space with standard K . K in time with standard deviation and maximal K for discharge in time and space at hydraulic station 122. Top left: Mean evolution of K Bottom left: Evolution of deviation. Bottom right: Mean evolution of Figure A.4: Appendix . Additional figures 82 iueA.5: Figure eito.Bto ih:Ma vlto of evolution Mean right: Bottom deviation. of Evolution left: Bottom K o icag ntm n pc thdalcsain13 o et eneouinof evolution Mean left: Top 123. station hydraulic at space and time in discharge for K ntm ihsadr eito n maximal and deviation standard with time in K . K nsaewt standard with space in Appendix . Additional figures 83
Figure A.6: Change of Kalman gain in space with mean, minimal, maximal Kalman gain and K+std.