O
N THE ROLE OF MODEL STRUCTURE IN
HYDROLOGICAL MODELING
O
N THE ROLE OF MODEL STRUCTURE IN
HYDROLOGICAL MODELING
U
NDERSTANDING MODELS
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 12 januari 2016 om 12:30 uur
door
Shervan G
HARARI
civiel ingenieur, Technische Universiteit Delft geboren te Teheran, Iran.
promotor: Prof. Dr. Ir. H.H.G. Savenije copromotor: Dr. Habil. M. Hrachowitz Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. Dr. Ir. H.H.G. Savenije, Technische Universiteit Delft Dr. Habil. M. Hrachowitz, Technische Universiteit Delft Onafhankelijke leden:
Prof. Dr. rer. nat. Dr. -Ing. A. Bárdossy
Universität Stuttgart, Duitsland Prof. Dr. V. Andréassian IRSTEA, Frankrijk
Prof. Dr. Ir. R. Uijlenhoet Wageningen UR
Prof. Dr. Habil. S. Uhlenbrook Technische Universiteit Delft Dr. Habil. L. Pfister LIST, Luxembourg
Prof. Dr. Ir. N.C. van de Giesen Technische Universiteit Delft, reservelid
Shervan Gharari was fully funded during his PhD by Luxembourg National Research Fund (FNR) under Aides à la Formation-Recherche (AFR) scheme with project number
1383201.
Keywords: rainfall-runoff modeling, model structure, uncertainty, information
Printed by: Ipskamp Drukkers
Front & Back: Designed by Shervan Gharari, adapted from the articles reviewed for this thesis.
Copyright © 2015 by author Shervan Gharari ISBN 978-94-028-0006-7
An electronic version of this dissertation is available at http://repository.tudelft.nl/.
P
REFACE
The aim of writing this thesis is obviously to communicate what I have done during my PhD study at Delft University of Technology. However, my scientific findings are pub-lished or hopefully will be pubpub-lished in peer reviewed journals which will be available for the academic community, students and researchers to read and judge. In this thesis, I aim to present these papers as a consistent story of my research. Besides being a for-mality to obtain my PhD it is a book which later I will be [or hope to be] proud of. So instead of copy-pasting and smoothing the (un)published papers, I decided to take the opportunity to express myself and my opinions in ways which are impossible in peer re-viewed journals. I have tried to make the book as interesting as possible by adding what I have encountered during my PhD life, in different conferences and communicating with other researchers. I apologize beforehand if some parts of the book may seem offensive to the readers. These are my honest ideas at the moment of writing this thesis.
Before going into the scientific discussion, I would like to mention a few persons and organizations who played a significant role during my PhD life.
First of all I should and must be thankful to Prof. Savenije, my supervisor. I ad-mire you for the fact that you know how to deal with every student based on his or her demands, scientific and cultural background. I admire you for promoting your PhD stu-dents to do more and mentioning their names everywhere and every time you talk about our achievements. I admire you for leaving your amazing scientific network open for your students. In a nutshell I owe you a lot.
I would like to thank Prof. Gupta. I learned conceptual thinking from you. You taught me how to break up a complex problem into small pieces and to solve those pieces like a puzzle. It was pleasant to see how you started reformulating the challenge into questions which eventually were much easier to handle. I am thankful for your hospitality during my visit to the University of Arizona; it is a memorable period for me.
I should thank my beloved supervisors. Dr. Hrachowitz, Dr. Fenicia and Dr. Matgen. I am not a statistician but I think having the possibility to work with three nice, capable, passionate and open-minded researchers is fairly rare. Thank you all for tolerating my arrogance and stubbornness!
I must not forget that the opportunity to do my PhD came from Fonds National de la Recherche (FNR) of Luxembourg with Aides à la Formation-Recherche (AFR). I am thankful to NFR for believing in my abilities. I should also thank CRP Lippmann and later LIST and in specific Dr. Pfister.
I keep in mind a short poem from Omar Khayyam, the mathematician, philosopher and poet of 11thcentury who is recognized for Khayam-Pascal triangle and geometric
solution of the cubic equation, translated by Edward FitzGerald: The Revelations of Devout and Learn’d
Who rose before us, and as Prophets burn’d, Are all but Stories, which, awoke from Sleep
They told their fellows, and to Sleep return’d. Shervan Gharari Delft, September 2015
C
ONTENTS
1 Introduction 1
1.1 A brief background . . . 1
1.2 Posing some of the open questions in hydrological modeling. . . 2
1.3 Outline of the thesis. . . 4
2 Rainfall-runoff modeling 5 2.1 What is a model? . . . 5
2.2 How do we build a model? . . . 6
2.3 Conceptual models vs. physically based models . . . 6
2.4 Lumped models vs. distributed models. . . 8
2.5 Complexity, realism and our expert knowledge . . . 8
2.6 Rainfall-runoff models and uncertainty. . . 9
2.7 Scope of this thesis . . . 10
3 reading the landscape, a new vision on an old concept 11 3.1 Introduction . . . 11
3.2 Study catchment . . . 13
3.3 Methods . . . 13
3.4 Results and discussion . . . 22
3.5 Conclusions. . . 33
4 Modeling based on landscapes 35 4.1 Introduction . . . 35
4.2 FLEX-TOPO framework. . . 38
4.3 Landscape classification and data. . . 39
4.4 Model setup. . . 39
4.5 Introducing realism constraints in selecting behavioral parameter sets. . . 41
4.6 Calibration algorithm and objective functions . . . 46
4.7 Model evaluation and parameter evaluation . . . 47
4.8 Results and discussion . . . 48
4.9 Wider implications . . . 54
4.10Conclusions. . . 56
5 Constraining a model 57 5.1 Introduction . . . 57
5.2 Constraints in environmental models. . . 59
5.3 Methodology and algorithm – constraint based search (CBS). . . 61
5.4 Case study . . . 63
5.5 Conclusions. . . 68 vii
6 Towards improved model structural inference 69
Part 1: Theory and methods . . . 69
6.1 Introduction . . . 69
6.2 Information, uncertainty, data, models and learning . . . 71
6.3 How information is coded into dynamical system models . . . 73
6.4 Adding information via the inverse problem of inference. . . 79
6.5 The problem of infereing system architecture. . . 80
Part 2: Case study and illustrative example for conceptual rainfall-runoff mod-els. . . 83
6.6 A recap of part 1: . . . 83
6.7 Establishing the building blocks of the case study. . . 85
6.8 Performance and uncertainty measures . . . 93
6.9 Evaluation . . . 96
6.10Result and discussion. . . 97
6.11General discussion . . . 104
6.12Conclusions: . . . 105
7 Teasing the ritual of hydrology; calibration-validation/evaluation 107 7.1 Introduction . . . 107 7.2 Sub-period calibration . . . 109 7.3 Case study . . . 111 7.4 Results . . . 115 7.5 Discussion . . . 120 7.6 Conclusions. . . 122
8 Triumph of an unfinished battle 123 8.1 The battle itself . . . 123
8.2 Future prospect: Ask what we really do know; information is everything . . 133
8.3 The future prospect: Some social issue!. . . 137
A Appendix 139 B Appendix 143 References 145 Summary 163 Samenvatting 165 Acknowledgment 167
About the author 169
1
I
NTRODUCTION
1.1.
A
BRIEF BACKGROUND
Hydrology is the science of water circulation on Earth. The moisture in the atmosphere is condensed and precipitates in the form of rain or snow. The precipitation is partly evaporated via plant transpiration, direct surface evaporation or even snow sublimation. Part of the precipitation is stored in the soil matrix or infiltrates into deeper layers, which eventually feed the aquifers. From the interaction of all these processes come streams and rivers, which act like natural drains of the catchments.
These processes, although simple and clear in mentioning, are very complicated to model. The boundary conditions, governing equations, current states of the system, and – on top of all – the heterogeneity of the mentioned processes, are all difficult to truly measure and therefore uncertain.
Rainfall-runoff processes at a catchment (or basin) scale follow the same complexi-ties. Catchments are open systems, meaning that the level of knowledge on the bound-aries of such systems is poorly understood (Refsgaard and Henriksen,2004). This lack of knowledge stimulates the studies that aim at better understanding catchments charac-teristics in finer detail. Heavily instrumented catchments around the world are examples of such efforts (Uchida et al.,2005).
To find the connection between the gathered data, hydrologists rely on models. Mod-els can be as simple as a regression relation between two time series, or be extremely complex consisting of intricate relations between input and output signals, through in-terconnecting fluxes and states, satisfying mass, energy and momentum balances.
As rainfall-runoff modelers, it is our task to make sense of the available knowledge on the system within a modeling framework, which eventually enables us to fill our gaps of knowledge spatially and temporally about a system (catchment).
The need to know the amount of runoff generated by a rainfall event was the primary drive behind the attempt to develop rainfall-runoff models. The rational method1can be mentioned as a simple example of a hydrological model to predict runoff. Finding
1Which relates discharge to rainfall intensity, area and runoff coefficient.
1
the empirical relation between catchment characteristics, rainfall and runoff was a wayforward. However these empirical models were simplistic; simple enough to generateone output of the system, such as discharge, but not complex enough to keep track of other properties of the system, such as soil moisture or evaporative fluxes.
By increasing computational power it became possible for hydrologists to model the desired system at any desired time scale. The computational capacity boom made it easier (or better said possible) for hydrologist to run models which keep track of the state and fluxes of a system at any given time and spatial scale. It also made it possible for the hydrologists to solve differential equations numerically, which was almost impossible before.
Two main-stream modeling practices emerged from the increased computational ca-pacity: 1- Conceptual bucket models and 2- so-called ’physically based’ models. More-over computational capacity made it possible to implement the models in a distributed fashion, rather than lumped implementation. Lumped models refer to models that look at the catchment as an entity. On the contrary, distributed models account for dis-tributed forcing and catchment characteristics. Disdis-tributed models are essentially lump-ed models applilump-ed at small scale (grid scale). Distribution of a model is an attempt to in-corporate much more details within the modeling framework and have a better under-standing of the spatially varying system behavior of our model. Distribution (or semi-distribution) of a model can be based on desired and different characteristics of a catch-ment or basin (e.g. soil, geology, etc.).
Soon it was realized and acknowledged that the ever-increasing complexity of mod-els introduced significant uncertainty about modeled state variable and fluxes. There are many uncertainties associated with a model and the use of a model. Our knowledge is encapsulated in the package called model, however one should realize that if a model is a combination of expert knowledge, then at the same time it is also a combination of our doubts on what we know. Therefore models themselves are a source of uncertainty.
1.2.
P
OSING SOME OF THE OPEN QUESTIONS IN HYDROLOGI
-CAL MODELING
The battle of modelers against uncertainty and its sources resulted into a significant body of literature. Substantial efforts were deployed to reduce model output uncertainty, using different sources of information, such as multi-observation or signatures, which will be elaborated in the next chapter. But what seems to have received much less at-tention, even today, is the basic question about the uncertainty of rainfall-runoff model concepts themselves. Models are an encapsulated form of our knowledge. However, one should question what knowledge do we have about the system? We learn from the past and make our models, but why should they always be calibrated or adjusted to new circumstances? The honest question remaining to be explored is “what actually do we know of a system if we have to adjust our model to any new case?” As an example: we know that if the saturation level of a catchment is higher, the response of the system to the same amount of rainfall should result in higher runoff production. But can we say “how much exactly?” This remains unanswered or depends on many more factors to be taken into account.
1.2.POSING SOME OF THE OPEN QUESTIONS IN HYDROLOGICAL MODELING
1
3
The other aspect of modeling we should be keen on exploring, is how to formulate all quantitative and qualitative information (extracted from expert knowledge or data) into modeling practice. In the hydrological community, nowadays, there is no doubt that the emphasis is (for scientific purposes) not on fitting the hydrograph but rather on representing hydrological processes. Although studies have been trying to cover this aspect of modeling, in my point of view the aforementioned question remains open to debate.
Models are the brainchild of modelers on how a real system (a catchment) might work. In better words models are basically an expert guess (hypothesis) on how the sys-tem functions. The hypothesis can be put into a neat package called a model and then can be tested against observed data or physical characteristics. If the established hy-pothesis manages to survive the tests designed for its eligibility, then this hyhy-pothesis is more likely to be the explanation of the underlying processes or the system functional-ity. Usually this process is referred to as hypothesis testing. It still remains a matter of disagreement whether models are truly hypothesis testing tools, and furthermore if any hypothesis can be tested by a model, solely.
One of the main reasons behind modeling a natural system is to understand how our set of hypotheses works on a given representation (the model; which is a hypothesis by itself ) of the system. In order to have such a model, the model should and must be as close as possible to reality. The choice of selecting one model above another should be supported by other sources of information and data justifying any additional process in the model. Usually this is referred to as model realism or model consistency in com-parison with belief on how a real system behaves. Although there are many efforts for pushing modeling practices into a direction with more agreement with known underly-ing processes, still the definition of realism and beunderly-ing realistic remains vague.
Following the challenges about model realism, it is also not well understood to which level of complexity or detail a system can be modeled given the available data. In fact there is no metric that can justify a certain level of complexity given our information (any type of information). The notion of complexity itself is also not well understood and debated in hydrological modeling.
What is overlooked, in my point of view, in the development and use of hydrological models [although originally I believed all the hydrological modelers were aware of that] is the nature of hydrological modeling itself. Whether modeling is a scientific, engineer-ing or even artistic practice remains for further debate. Within the scientific community, models are usually known as hypothesis testing tools (although I do not agree about the testing claim). Models can act like a tool whereby a hypothesis can be formulated as a model or part of a model and then can be confronted by observed data. Models are used for process of understanding rather than getting the right answer at any cost. In a sci-entific approach, models are preferred that have a high realism. In engineering this may be different; the result and reliability of the result is the primary concern even if some part of a model is not fully justified. But it seems that in scientific modeling there are many assumptions which are not fully justified, and the only reason for accepting them is to enable ourselves to solve or reduce the dimensions of the problem to a manageable scale. “Do we learn from the model or do we teach the model to do what we want the models to do?” As an example: can we simply rely on a model’s output and postulate
1
that the contribution of groundwater and surface runoff are as given by the model? Theanswers to these questions rely on how we look at the models and the modeling exercise.1.3.
O
UTLINE OF THE THESIS
In this thesis I try to address the open questions about hydrological rainfall-runoff mod-eling posed earlier. Although, due to my limited resources, knowledge and time con-straints, these questions are not fully answered (if any answer exists), I try to pave a path toward a framework or strategy which can: lead to better understanding of our models; better understanding of what we know and what we impose on a model; better inter-action of expert knowledge, model realism, model complexity, and the effect of model structure and our expert knowledge on the outcome of the model.
In chapter2, I focus on the model building steps and on the distinction between conceptual and physically based, lumped and distributed models. A further explanation has been given about realism and complexity and finally uncertainty and its sources in hydrological modeling. Finally, I elaborate how the open questions are addressed in this thesis and which strategy to follow for providing possible explanations.
In chapter3, I elaborate on HAND (height above nearest drainage) as a metric of landscape classification, which then becomes the basis for the delineation of catch-ments based on topographical hydrological response units (HRUs) where distinct hy-drological behavior can be considered for each HRU.
In chapter4, I continue the effort of building a model based on landscape units de-fined in chapter3and tuning it with our expert knowledge, with and without calibration. The effect of our expert knowledge is studied and explored in a semi-distributed model (FLEX-TOPO).
Chapter5is an elaboration of the strategy used in chapter4to select parameter sets which satisfy our perception of how the real system works.
In chapter6, following the result found in chapter4, the importance of model struc-ture in hydrological model is scrutinized. A strategy is developed in order to help evalu-ate the performance of model structure apart from its parametrizations and parameter value.
Chapter7tries to challenge the calibration-validation routine for model evaluation. Instead a strategy is proposed which eventually eliminates the need for a validation (evaluation) period while searching for time consistent parameter sets.
Chapter8is devoted to my critical view about my own work, about the serious and funny discussions I have had during the period of my PhD and what I think would be the future of hydrological modeling in future and ways forward.
2
R
AINFALL
-
RUNOFF MODELING
2.1.
W
HAT IS A MODEL
?
“What is a model?” may seem a trivial question, but to my eyes it looks very important to start with. A model is an abstract simplification and mathematical replica of real system behavior. A model is a tool where we can condense our knowledge about a system and thereby design an artificial experiment. The possible uses of a model can be summarized as:
1. Extrapolation of existing knowledge in space such as application of a model in other catchments (e.g. prediction in ungauged basins, PUB) or interpolation with-in a catchment (e.g. spatial distribution of a model for modelwith-ing soil moisture or evaporation).
2. Extrapolation of existing knowledge in time. Split sample test or calibration-valid-ation is an attempt to measure the strength of a model in reproducing the desired system response outside the calibration period.
3. Prediction of system response to change in system characteristics (e.g. land use change) or input data (e.g. climate change). It should always be from the highest interest for decision makers to understand the effect of their decision on hydro-logical behavior and therefore the future status of a system.
4. Investigation whether or not a set of hypotheses results in a credible result. The latter use of the model is often referred to as hypothesis testing however in my point of view a model cannot serve as a hypothesis testing tool. Hypothesis testing or even evaluation is rather difficult or better said impossible due to the nature of catchments as open systems. For a single hypothesis to be truly tested it should be first isolated and compared with data which are specifically gathered blind from the hypothesis itself. Even then the hypothesis can be said to be more likely rather than to be tested.
2
2.2.
H
OW DO WE BUILD A MODEL
?
The model building practice usually follows a set of routine steps (seeBeven,2001b, the Primer). First, a perceptual model is formed which decides on the possible processes and their interactions; this is the initial level of the modeling process, whereby a modeler describes broadly the way s/he sees a catchment and the most important underlying processes. In the second step, the personal understanding of how a catchment works should be described by some mathematical recipe. This level of model is referred to as a conceptual model, where we conceptualize our perception of system behavior. This conceptualization then is transformed into a computer code. The model then should be tuned or calibrated, which is the unfortunate partner of a hydrological model.
After having a working model the model result should be validated. Validation of a model as a set of superimposed hypotheses, similar to validation of any individual hypothesis, is fairy difficult and almost impossible. However if a model is satisfying its purpose of design it can be accepted as a utility based approach and if the model does not meet its designed objectives, its building steps should be revisited and amended. This recursive procedure can continue until we have a satisfactory model.
As explained in figure2.1, going down the steps of a model building practice, from perceptual model to conceptual model, and from conceptual model to procedural mod-el, and finally parameter identification or model calibration, our approximation of how the system works increases. The ways we decide how to proceed at each step affects the next steps and subsequently the outcome of the model. There are also limitations at each step. A conceptual form of a model would be desired if it can be put into com-puter code or if it needs reasonable computational power. However the increase of the computational capacity over the last few decades has made it possible to come up with much more complicated and elaborated conceptual and procedural models.
In addition to the general model building steps, there are detailed guidelines for de-veloping and amending the model structure problem (Refsgaard and Henriksen,2004). What is usually common between all these detailed guidelines are a kind of recursive check and update of the model concept. The model (our hypothesis) is confronted with observations and if it does not fully satisfy the modeling purpose it should be amended or altered.
The implication of such an approach in scientific hydrological modeling showed it-self as stepwise model building or simply tailoring the model to understand better the interaction of all the hypotheses embedded in the model. Stepwise model building is the crux behind approaches such as FLEX and FUSE, giving the opportunity to the mod-eler to change the model structure and related parameterization (discharge-state rela-tions). The step-wise development is interesting in the sense that it implicitly aims at the balanced compromise between model complexity and available information for a given catchment. Such attempts were done for example by incorporation of groundwa-ter levels in modeling (Fenicia et al.,2008b;Seibert,2003).
2.3.
C
ONCEPTUAL MODELS VS
.
PHYSICALLY BASED MODELS
Following the increase of computational power, two fronts of modeling flourished: one is the top-down approach that looks at the catchment process from an overall point view,
2.3.CONCEPTUAL MODELS VS.PHYSICALLY BASED MODELS
2
7
The perceptual model: Deciding on the processes The conceptual model: Deciding on the equations The procedural model:
Getting the code to run on a computer Model Calibration:
Getting values of parameters Model Validation:
Good idea but difficult in practice Declare Success? Revise perceptions Revise equations Debug code Revise parameter value No Yes
Figure 2.1: The model building steps (Beven,2001b).
whereby first the uncertainty of the processes in a catchment is defined and the model is based on that usually in a lumped fashion. The other is the bottom-up approach as-suming that overall performance is an aggregate of small scale processes, while their mathematical formulations are assumed as physical laws.
Usually in literature, conceptual bucket models are referred to as conceptual rainfall-runoff (CRR) models. These models see the catchment in a schematic fashion. The buckets and parameters in such a model cannot be easily linked with directly observ-able parameters in the field. On the contrary, the so-called ’physically based’ models have parameters which, in principle, can (or are hoped to) be directly observed in situ. The theory is reductionist in that the catchments can be supposed to be split up in the smallest fragments, which can be related to any distinct characteristic of the real system. In fact, it is almost impossible to categorize a model into a bucket model (CRR) or ’physically based’ model, as any model falls somewhere between a truly physically based model and a truly conceptual model (if such absolute models exist!). Moreover there are arguments that any model (except data driven models such as Artificial Neural Network, ANN) is physically-based, as it should satisfy physical principles such as conservation of mass and energy. And on top of that it can be said that all models are conceptual in the sense that they are the brainchild of modelers who designed them.
There is another way of differentiating physically based models from conceptual models. If a model relies purely on calibration for parameter identification, then it is an absolute conceptual model, and if it relies purely on measured parameters, which
2
can directly be identified, it is an absolute physically based model (keep in mind that such an absolute models might not exist!). Due to the scaling problem of parameters in physically based models and what is measured in the field, some kind of calibration (or upscaling) is always necessary. The need for calibration is rooted in our lack of knowl-edge to fix or justify any parameter or any process value at the modeling scale. This lack of knowledge is the same cause of uncertainty in our models and its results, which will be discussed in the coming paragraphs.
Usually physically based models are blamed for having too many parameters and de-grees of freedom, which is almost impossible to fully grasp by a single (or even multiple) time series such as discharge or soil moisture (irrespective of the fact that soil moisture is almost impossible to measure at the appropriate scale or representative depth). Refs-gaard and Knudsen(1996), were among the first to explore and inter-compare a concep-tual model with physically based models. They concluded that the concepconcep-tual lumped model is capable of reproducing the discharge equally good as a physically based model. They also mentioned that refining a groundwater model to more than a certain size (few kilometers in his case study) would not affect the outcome.
2.4.
L
UMPED MODELS VS
.
DISTRIBUTED MODELS
As mentioned in chapter1, lumped models are models that consider the catchment as an entity. Distributed models are (in principle) lumped models implemented at finer grid scale by which breaking up the system in small compartments. Distributing a model is simply an attempt to interpolate or extrapolate our understanding of how the system works spatially and regenerate the system characteristics and output spatially as well (e.g. soil moisture). This serves the first purpose of model building among all others mentioned in the beginning of this chapter. There are different ways of distributing a model regarding the input forcing and parameters: only the forcing data can be dis-tributed, or only the parameters, or both at the same time. Depending on the availability of data any of these combinations may be preferable.
Semi-distributed models are basically a branch of distributed models in which the model’s cells are deliberately grouped in clusters representing similar process represen-tation. Often this is done merely to avoid unnecessary complexity which cannot be backed by information (data) or available computational power. HRUs or hydrological response units reflect the effort of building a semi-distributed model, which reasonably reproduces the system response at the desired scale while avoiding redundant complex-ity. HRUs can be built based on catchment characteristic such as soil type, geology, land use, topography or combination of all of these. The level of complexity and details of a (semi-) distributed model remains under debate.
2.5.
C
OMPLEXITY
,
REALISM AND OUR EXPERT KNOWLEDGE
In chapter1and earlier in this chapter the words complexity and realism have been men-tioned. Both complexity and realism are hypothetical measures. To my knowledge there is no agreement on a measure that can be defined to indicate the complexity of a model. Although there are many studies trying to make a formal explanation of complexity and its link to uncertainty (Arkesteijn and Pande,2013), the notion of complexity remains
de-2.6.RAINFALL-RUNOFF MODELS AND UNCERTAINTY
2
9
batable under different theories and schools of thoughts. The same is valid for realism of a model and the way it is perceived. The realism of a model is a general hypothesis about that model. Similar to the difficulties of testing or validating a hypothesis, the hypothesis of model realism remains also difficult to proof.
However based on a naïve and intuitive definition, model complexity can be simply considered as the number of parameters or state variables within the model, and realism is the agreement between a model’s behavior and what we assume or perceive to be real system behavior. Usually the modelers are keen to have the most realistic model while avoiding unnecessary complexity that cannot be justified by data.
Many studies have explored different directions to incorporate expert knowledge into modeling practice. A dialogue between experimentalists and modelers is essential for model development. Fulfilling this need, frameworks have been proposed to help the modelers and experimentalists to communicate easier through a kind of flexible mod-eling framework. This strategy, although a step forward, did not fully answer the need for a unified framework for considering the whole range of knowledge, quantitative and qualitative, in the model building practice. Moreover the relation between complexity, realism and our expert knowledge of a system remains unexplored as well.
2.6.
R
AINFALL
-
RUNOFF MODELS AND UNCERTAINTY
Rainfall-runoff modeling deals with various sources of uncertainty. These uncertainties can be summarized generally as lack of knowledge. Uncertainties can be divided into two major categories; uncertainty or lack of knowledge about the real system function-ing, which result in model uncertainties as explained earlier, and uncertainty in the forc-ing data which is fed into the model. As an example of the latter, our understandforc-ing of rainfall as a very heterogeneous process is constrained by the rain gauges, which are just representative of a very small area (the area of a rain gauge is only a few square centime-ters) compared to a catchment area (in the order of square kilomecentime-ters) to be modeled. Therefore we have to assume the rain gauge data to be valid over a wider area in order to be able to run our models. In other words to run our hydrological models we rely on a model (or interpolation techniques) to be able to extrapolate the point measurement rainfall data over the wider area of the catchment. This interpolation/extrapolation of rainfall can be more and more detailed and complicated by bringing in additional data about height, temperature, wind direction, etc. Any other measurement of rainfall data, evaporation or discharge measurement has its own uncertainty and any technique (or model) we use to extrapolate or interpolate those measurements adds to this uncer-tainty.
The uncertainty in the model is usually divided into two categories, the model struc-tural uncertainty and parameter uncertainty. These two sources of uncertainty are inter-related and the relation between them is not well understood either. Parameter uncer-tainty is what has attracted a lot of attention during the last two decade in hydrological models. Generalized Likelihood Uncertainty Estimation (GLUE;Beven and Binley,1992) was one of the first and, most probably, the most famous method for obtaining an idea on parameter uncertainty. Later studies have tried to shed a light on the disintegration of model structural uncertainty and parameter uncertainty. Although valuable, these stud-ies were based on various assumptions to carry out such a disintegration of uncertainty.
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2.7.
S
COPE OF THIS THESIS
The scope of this thesis is spinning around the aforementioned sub-titles of this chap-ter. The chapters of the thesis, although maybe not fully devoted to answering the men-tioned challenges, indirectly touches on these issues. Chapter3,4and5are an attempt to bring the expert knowledge (from any type) to bear and use it in the modeling of a meso-scale catchment in Luxembourg. During this voyage I noticed the importance of both model structure and expert knowledge in the modeling practice. It made me think and explore the whole new research question on the importance of model structure in the hydrological modeling. I try to make use of information theory, although I am lacking basic knowledge in this field. Chapter6is a manifesto for the first steps made in this re-gard. I tried to segregate the information brought by model structure, parameterization and any assumption in the models. Chapter7is devoted to a way of model calibration where more time consistent parameter sets are found using characteristics of defined periods of available time series (sub-periods). It is a nice practice showing how desirable knowledge (information) can be extracted from a set of data in modeling practice.
In chapter8, I elaborate my vision at moment of writing this thesis. It reflects my thoughts about my own work and the future of hydrological modeling and the possible ways forward in my point of view. It also acts like a milestone to which I can look back in future, to see how my thoughts and vision changed and evolved over time.
3
READING THE L ANDSCAPE
,
A NEW
VISION ON AN OLD CONCEPT
3.1.
I
NTRODUCTION
As mentioned in earlier chapters hydrological behavior, including large scale hydrolog-ical behavior, is still poorly understood, as a result of the lack of realisthydrolog-ically observ-able variobserv-ables on the one hand and the complexity of catchment processes on the other. Catchment topography, readily available as digital elevation models (DEM), has the po-tential to provide important information on catchment processes, particularly due to its inherent co-evolution and diverse feedback processes with hydrology and ecology (Savenije,2010). A number of previous studies investigated the relationships between topography and hydrological behavior in the attempt to identify hydrologically differ-ent functional landscape units and to better characterize model structure, parameter sets as well as metrics of catchment similarity. For example,Winter(2001) classified the catchment into hydrological landscape units (upland, valley side and lowland) exploit-ing the combination of topographic, geological and climatic conditions. Based on this conceptWolock et al.(2004) classified hydrological units for the entire United States of America using GIS data. Topography, land use and geology have also been used to di-rectly infer dominant runoff processes within a catchment (Flügel,1995;Naef et al.,2002; Schmocker-Fackel et al.,2007;Hellebrand and van den Bos,2008;Müller et al.,2009).
Another widely used indicator is the topographical wetness index (Beven and Kirkby, 1979) which is the basis of TOPMODEL and characterizes hydrological behavior based on upslope contributing area and local slope. The topographical wetness index was fur-ther modified byHjerdt et al.(2004), who took into account downstream conditions con-sidering how far a water particle needs to move to lose a specific amount of potential energy. Topography was also used to investigate the relationship of catchment transit times with numerous catchment characteristics such as flow path length, gradient and
This chapter is based onGharari et al.(2011): Gharari, S., Hrachowitz,M., Fenicia, F., Savenije, H. H. G.: Hy-drological landscape classification: investigating the performance of hand based landscape classifications in a central European meso scale catchment. Hydrology and Earth System Sciences 15(11), 3275-3291, 2011.
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connectivity (McGuire et al.,2005;Jencso et al.,2009,2010) or drainage density ( Hra-chowitz et al.,2009,2010) using tracer techniques. Other tracer studies directly linked topography and hydrological behavior (Uhlenbrook et al.,2004;Tetzlaff et al.,2007). A wide range of additional topographical indices have been suggested, describing, among other aspects, the shape, age and stability of a catchment, such as the hypsometric inte-gral (Ritter et al.,2002) and its correlation with catchment processes (Singh et al.,2008). Other studies correlated topographical indices with soil type and hydrological behav-ior (Park and van de Giesen,2004;Lin and Zhou,2008;Pelletier and Rasmussen,2009; Behrens et al.,2010;Detty and McGuire,2010).
In spite of the information content of topography for hydrological process inference, its general usefulness is controversial. It has been argued that climate and geology exert stronger influence on the rainfall runoff behavior of a catchment than topography ( De-vito et al.,2005). Furthermore, it was shown that flow patterns may be dominated by bedrock- rather than surface topography (McDonnell et al.,1996;Tromp-van Meerveld and McDonnell,2006b). According toMcDonnell(2003) the “catchment hydrologist will need to develop hypotheses from non-linear theory that are testable on the basis of ob-servations in nature. This will not come about via model intercomparison studies or DEM analysis”. These comments highlight the perception that DEM analysis alone may be of limited value for gaining deeper understanding of catchment processes and that this needs to be brought into a wider context, accounting for the subtle interplay of to-pography, geology, climate, ecology and hydrology.
In spite of the complexity of catchment processes and due to the frequent lack of data for bottom-up modeling approaches, relatively simple, lumped conceptual models can be sufficient in representing the dominant flow generation processes and model-ing stream flow (cf.Sivapalan et al.,2003;Savenije,2010). However, even for these top-down models additional data, other than precipitation and stream flow, are desirable for enhancing physical significance of model parameters and evaluation (Nalbantis et al., 2011).
Recently, (Rennó et al.,2008) formalized the Height Above the Nearest Drainage (HA-ND and its index is referred as IHANDhereafter) metric and employed it for landscape
classification. This metric may be more adequate to identify hydrologically different landscape units than the traditionally used elevation above mean sea level. HAND calcu-lates the elevation of each point in the catchment above the nearest stream it drains to, following the flow direction. It thus provide us with more informative hydrolgic elevation rather than topographic elevation. Nobre et al.(2011) showed that HAND is a stronger topographical descriptor than height above sea level by analyzing long term piezome-ter data (groundwapiezome-ter behavior). Based on hydrologically meaningful landscape clas-sification introduced by HAND metric,Savenije(2010) suggested that as topographical features are frequently linked to distinct hydrological functioning, they can be used to construct a conceptual catchment model perceived of hydrological units within a catch-ment.
Landscape classification based on HAND is potentially sensitive to different aspects, such as the definition of the threshold for channel initiation when deriving streams from a DEM, the seasonal fluctuations of the channel initiation, and the resolution of the DEM. Furthermore, it is unknown to what extent the observation points and their
den-3.2.STUDY CATCHMENT
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sity (sample size) can affect the landscape classification based on IHAND.
The objectives of this chapter are thus to (1) assess different hydrologically mean-ingful landscape classification tools based on the HAND and further parameters such as slope (Islope) and the distance to the nearest drainage (DIST; and its index is referred
as IDISThereafter), (2) test the sensitivity of HAND-based landscape classification to the
resolutions of the DEM, (3) evaluate the effect of the sample size of the calibration data set on the robustness of HAND-based landscape classification and to (4) analyze the relation of IHANDto the topographical wetness index in a mesoscale catchment in a
tem-perate climate. The landscape classification result of this chapter then will be used as a basis for building the hydrological model based on the FLEX-TOPO approach proposed bySavenije(2010) in following chapters.
3.2.
S
TUDY CATCHMENT
The outlined methodology will be illustrated and tested with a case study using data of the Wark catchment in the Grand Duchy of Luxembourg (figure3.1a). The catchment has an area of 82 km2with the catchment outlet located downstream of the town of Et-telbrück at the confluence with the Alzette River (49.85◦N, 6.10◦E, Fig. 3.1a). With an annual mean precipitation of 850 mm yr−1and an annual mean potential evaporation of 650 mm yr−1the annual mean runoff is approximately 250 mm yr−1. The geology in
the northern part is dominated by schist while the southern part of the catchment is mostly underlain by sandstone and conglomerate. Hillslopes are generally characterized by forest, while plateaus and valley bottoms are mostly used as crop land and pastures, respectively. (Drogue et al.,2002) quantified land use in the catchment as 4.3 % urban areas, 52.7 % agricultural land and 42.9 % forest. In addition they reported that 61 % of catchment is covered by permeable soils while the remainder is characterized by lower permeability substrate. The elevation varies between 195 to 532 m, with a mean value of 380 m (Fig3.1b). The slope of the catchment varies between 0-130 %, with a mean value of 17 % (Fig3.1c).
3.3.
M
ETHODS
3.3.1.
T
ERMSThe HAND-based hydrologic landscape classification distinguishes three hydrologically, ecologically and morphologically different landscape units, which, in the following, will be referred to as wetland, hillslope and plateau (cf.Rennó et al.,2008;Nobre et al.,2011). The use of these terms might seem inconsistent as they originate from different dis-ciplines – ecology (wetland), hydrology (hillslope) and morphology (plateau) – where they do have clear definitions. These terms were nevertheless deliberately chosen as they highlight distinct hydrological landscapes with different rainfall-runoff behavior (cf. Savenije,2010). Note that in other physio-climatic regions more or different landscape categories may be necessary to adequately describe the landscape. The classification with proposed dominant runoff process is limited to the Wark catchment and may not be valid for other catchments. The terminology used is defined as follows:
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DIST [m] 1730 0 (e) Slope (%) [-] 130 0 (c) HAND [m] 202 0 (d) Elevation [m] 532 195 Ettelbrück (b)N
2 4 0 1 Kilometers (a)Figure 3.1: (a) Location of the Wark Catchment in the Grand Duchy of Luxembourg, (b) digital elevation model (DEM) (c) Islope(%) of the Wark catchment [–], (d) IHANDof the Wark catchment [m] and (e) the distance to the nearest drainage (IDIST) for the Wark catchment derived from a DEM with resolution of 5 m × 5 m.
to the other two landscapes entities. In classical ecological terms they refer to the land where saturation with water is a dominant factor influencing the animal and plant species of that area (Cowardin et al.,1979). From a hydrological point of view, wetlands comprise a broader type of landscape units than the commonly used terms: riparian zones or valley bottom areas. They can be seen as areas which, due to the shallow depth of the water table, have limited residual storage capac-ity and therefore demonstrate a fast response to precipitation, independent from their location in the catchment. The term shallow in this regard means that in a normal wet season the groundwater table reaches the surface during heavy rain-fall events. The predominant locations of wetlands, however, require a subdivision of this class into flat wetlands, which are characterized by modest slopes, such as stream source areas and valley bottoms and sloped wetlands in hollows close to streams where hillslopes end in valley bottoms or steep headwater regions, but
3.3.METHODS
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15
which can nevertheless be characterized by considerably sloped terrain along the flow direction of the stream. Thus, while both wetland types exhibit relatively low IHAND, they are distinguished by different slope angles. The dominant flow
gener-ation process for wetlands is saturgener-ation overland flow.
2. Hillslopes are areas which connect concave and convex landscapes (Chorley et al., 1984). The widespread perception that floods are mainly generated on hillslopes (cfBeven,2010) makes them a crucial element in landscape analysis. The co-evolution of ecology and hydrology, and thus the presence of preferential flow paths (Weiler and McDonnell,2004), such as root canals, animal burrows, fissures and cracks, makes rapid subsurface flow the most effective and dominant runoff process of hillslopes as it fulfills the two functions essential for developing and maintaining their topographical appearance, i.e. drainage and moisture retention (Savenije,2010).
3. Plateaus are flat or undulating landscape units relatively high above streams. Due to the low gradients and comparably deep groundwater levels, plateaus mainly ful-fill storage (both soil and surface) and evaporation functions, with mainly vertical flow processes, in particular deep percolation (Savenije,2010).
3.3.2.
T
OPOGRAPHICALD
ATALandscape classification in the Wark catchment is based on a 5 m × 5 m DEM with a vertical resolution of 0.01 m (figure3.1b). The slope of each grid cell (Islope) was
cal-culated using the average maximum technique (figure3.1cBurrough and McDonnell, 1998). The flow direction network has been derived from the DEM using a D8 algorithm (O’Callaghan and Mark,1984;Jenson and Domingue,1988). Although IHANDis
sensi-tive to the stream initiation threshold, the threshold upslope contributing area has been fixed at a value of 10 ha. This value has been selected to maintain a close correspondence between the derived stream network and the mapped stream network. The value is also in the range of stream initiation thresholds reported by others (e.g.Montgomery and Di-etrich,1988). The relative height, i.e. IHAND, was then calculated from the elevation of
each raster cell above nearest grid cell flagged as stream cell following the flow direction (figure3.1d). Similarly, distance to the nearest drainage (IDIST) was also computed along
the flow path to the nearest stream cell (figure3.1e).
During a field campaign (16-20 November 2010), 5611 points in the catchment, here-after referred to as sampling points, were mapped using GPS way points along various paths throughout the catchment and in-situ visually classified into the three landscape units – wetland, hillslope and plateau – in order to establish a “ground truth” according to expert knowledge of hydrological dominant behavior (figure3.2). The resolution of the observed points is 5 m along the walking paths. The points were collected walking along downhill to uphill transects. The reason for this strategy was to have continuous IHANDvalues from the lowest, near river, to the highest on the plateau. Transects were
selected in different parts of the catchment: in the headwaters, in steep valley bottoms and subtle sloped areas in the southern part. Vegetation was also helpful to indicate each class, for example in the Wark catchment the valley bottoms are covered with grass, hillslopes are covered with forest and plateaus are mostly used for agriculture.
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Elevation [m] 532 195 Kilometers 0 1 2 4 Plateau Wetland HillslopeN
Figure 3.2: The observation points in the Wark Catchment and their corresponding landscape calsses.
Table 3.1: Criteria for land classification using IHANDand Islope.
Low IHAND High IHAND
Low Islope Wetland (flat) Plateau
High Islope Wetland (sloped) hillslope
3.3.3.
L
ANDSCAPE CLASSIFICATION BASED ONHAND,
SLOPE ANDDIST
The landscape classification of each point in the catchment has been identified accord-ing to IHAND, Islope and IDIST. A cell with a steep slope (high Islope) was classified ashillslope or sloped wetland and a cell with a low slope (low Islope) was classified either as
flat wetland or plateau, depending on IHANDor IDIST(table3.1). To separate IHAND, Islope
and IDISTinto high or low categories, thresholds had to be introduced. The thresholds
were adjusted in a way that the modeled landscape classes corresponded sufficiently well with the landscape classes of the observed sampling points.
In reality, the boundary between the different landscape units may not be sharply defined. The transition from one landscape category to another may have to be deter-mined by fuzzy thresholds. The fact that transition is not sudden, reflects, similar to fuzzy set theory, the modeler’s and observer’s “degree of belief” (cf.Bárdossy et al.,1990) that a point belongs to a certain landscape unit, as shown in the illustrative example in figure3.3. In the present case this results in three different percentages for one cell indicating to what extent a cell belongs to a landscape unit. Here, the fuzzy nature of
3.3.METHODS
3
17
the parameters is considered by using a two parameter cumulative Normal distribution function: Fnormal(x|µ,σ) = 1 2 · 1 + erf(px − µ 2σ2) ¸ (3.1) whereµ is the mean (i.e. IHAND, Islope, IDIST) andσ is standard deviation. Both
parame-tersµ and σ are introduced as free calibration parameters in the landscape classification model. Note, that for very small standard deviations the model can be considered as crisp or deterministic as the transition between high and low IHANDor Islopeor IDIST
be-comes immediate and sharp.
I
slope/I
slope,max 0 0.2 0.4 0.6 0.8 1Probability [-]
0 0.2 0.4 0.6 0.8 1 High Islope Low IslopeFigure 3.3: An example of fuzzy classification for high and low Islope. In the central of the graph the classification is uncertain while at the extremes the uncertainty is the lowest.
Three landscape classification models have been designed using IHAND, Islope, IDIST
and a combination of them:
1- The first classification model, based on IHANDand Islope(Model ID: MSH(Rennó
et al.,2008)) uses the four fuzzy threshold parametersµHAND,σHAND,µslopeandσslope.
The classification rules for the models are as below: The probability (Pslope) of having
high values of Islope:
Pslope= Fnormal(Islope|µslope,σslope). (3.2)
The probability (PHAND) of having high values of IHAND:
PHAND= Fnormal(IHAND|µHAND,σHAND). (3.3)
Thus, the probability of being hillslope (PHillslope) is the same as the probability of high
Islopevalues and high IHANDvalues:
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Likewise, the probability of being a plateau (PPlateau) is defined as the probability of high
IHANDvalues and low values for Islope:
PPlateau= (1 − Pslope)PHAND. (3.5)
Similarly the probability of being wetland (PWetland) is defined as the probability of low
IHANDvalues and high (sloped wetland) or low (flat wetland) Islope:
PWetland= Pslope(1 − PHAND) + (1 − Pslope)(1 − PHAND) = (1 − PHAND) (3.6)
where the first term reflects the probability of being sloped wetland Pwetland,sloped and
the second the probability of flat wetland Pwetland,flat.
2- The second model uses Islope (µslope,σslope) and distance (µDIST,σDIST) to the
nearest drainage (Model ID: MSD) to classify the landscape: The probability (Pslope) of
having high values of Islopeis similar to equation3.2. The probability (PDIST) of having
high IDIST:
PDIST= Fnormal(IDIST|µDIST,σDIST). (3.7)
The probability of wetland, hillslope and plateau can be calculated identical to equation 3.4to3.6with the difference that probability of high IHAND(PHAND) should be replace
with probability of high IDIST(PDIST).
3- For the third landscape classification model a combination of IHANDand IDISTis
used. IHANDis normalized (IHAND∗ ) to range from 0 to 1 by dividing the IHANDvalue of
each grid cell by the maximum IHANDvalue (IHAND,max). The same is done for distance
to the nearest drainage (IDIST∗ ). The multiplication of these two matrices results in a third matrix, the IHDindex (Model ID: MSHD). The values for this new matrix are low and the
distribution is highly skewed with more than 86 % of the raster cells showing a value below 0.1. The procedure is as below briefly:
IHAND∗ = IHAND IHAND,max (3.8) IDIST∗ = IDIST IDIST,max (3.9) IHD= (IHAND∗ IDIST∗ ) t. (3.10)
The power of the matrix has been chosen at 0.1 (t = 0.1). IHDis a new matrix and its value
is multiplication of normalized IHANDvalues (IHAND∗ ) and IDIST∗ values (IDIST∗ ) which was
power transformed. The probability (Pslope) of having high values of Islopeis identical to
equation3.2. The probability (PHD) of having high IHDcan be written as:
PHD= Fnormal(IHD|µHD,σHD). (3.11)
Similar to the previous two models the probability of wetland, hillslope and plateau can be calculated identical to equation3.4to3.6with the difference that probability of high IHAND(PHAND) should be replaced with probability of high IHD(PHD).
3.3.METHODS
3
19
3.3.4.
M
ODEL CALIBRATIONThe model calibration procedure has been designed to minimize an objective function (O). The objective function allows to evaluate the goodness of fit based on the probability that a modeled point belongs to the same class as the respective observed point and is defined as: O = " 1 − PNH i =1PHillslope,i NH # + " 1 − PNP i =1PPlateau,i NP # + " 1 − PNW i =1PWetland,i NW # . (3.12) where PHillslope,i, PPlateau,iand PWetland,iare the probabilities of observed hillslope,
plate-au and wetland grid cells i , to be classified by the model as hillslope, plateplate-au or wetland respectively. NH, NP and NWare the numbers of observed grid cells for the hillslope,
plateau and wetland classes.
For crisp models the probability of a certain point is 1 for one class and 0 for the two other classes. For fuzzy models the probability of a modeled point is divided into three classes summing up to unity, while leaving the observed sample points crisp, i.e. the observed points are clearly defined as wetland, hillslope or plateau. The idea behind this approach is to let the model decide about the hydrological behavior of a cell which may not be unique given the large number of crisp observed points. The fact that the objective function is made up of three parts helps to calibrate the model based on a normalized value where all the classes participate equally; even when the proportion of one landscape class is large compared to others, and the in situ observed points are not the same number for all classes. Note, that although the maximum value of the objective function can be 3, in practice it will not exceed 2 because with an extremely unrealistic set of parameters the entire basin will be classified as one unit. As a result, the objective function for that unit will be zero and for the remaining classes will each sum up to unity. Calibration of the models has been done using Monte-Carlo sampling, i.e. the pa-rameters were sampled, in absence of further prior information, from uniform distri-butions, within predefined threshold ranges (Islope[-] ∈ [0, 0.2], IHAND[m] ∈ [0, 20], IDIST
[m] ∈ [0, 100], IHD[-] ∈ [0, 1]) in 20000 Monte-Carlo realizations. Similar to the idea
be-hind the GLUE (Beven and Binley,1992), it has been assumed that there is, due to equifi-nality, no single best model parameter set. A range of acceptable (i.e. behavioral) sample rates (ASR;Li et al.,2010) is tested in sensitivity analysis. The parameters are reported based on the best performance and their likelihood weighted 95 % uncertainty interval (2.5-97.5 %) for an ASR of 5 %, whereby the value of the objective function is used as like-lihood measure. For sensitivity analysis ASR between 1-10 % was used.
3.3.5.
O
PTIMAL EFFECTIVEDEM
WINDOW SIZE AND RESOLUTIONThe 5 m × 5 m resolution of the DEM allowed a relatively accurate representation of the catchment topography in detail. However, high resolution DEMs can introduce a bias in the results as hydrologically negligible local landscape features, such as steep, small scale rock outcrops, can cause certain grid cells to be inappropriately classified. To re-duce this problem the DEM has been smoothed using a weighed mean value for each cell using a 2-D Normal distribution (truncated at a radius of 3 times the standard devi-ation) with varying standard deviation1σSMof 0.5, 1, 1.5, 2, 5, 10 grid cells (equivalent to
3
2.5, 5, 7.5, 10, 25, 50 m). This allows the removal of “noise” in the landscape while keep-ing the high DEM resolution (cf.Hrachowitz and Weiler,2011). Furthermore the effect of lower DEM resolutions (10, 20, 50 or 100 m) on the model parameters and performances has been investigated, to test which DEM resolution is necessary to provide acceptable model results. For model runs with lower resolution no filter was used as it was assumed that local landscape features would automatically average out in the process of resam-pling the DEM at lower resolutions.
3.3.6.
S
ENSITIVITY TO CALIBRATION POINT SAMPLE SIZEThe effect of different calibration point sample sizes on the robustness and predictive power of the models was assessed by cross-validation/evaluation. More specifically, re-peated random sub-sampling evaluation was used to investigate how best fit parameter sets change for calibration point sample sizes of 2806, 1122, 561, 281, 112, 56 and 28 points (i.e. 50, 20, 10, 5, 2, 1 and 0.5 % of the complete of 5611 calibration points which consisted of 1501 (26.8 %), 1385 (24.6 %), 2725 (48.6 %) points for wetland, hillslopes and plateau respectively). 100 random sub-samples for each of the sample sizes (50 to 0.5 %) were drawn from the complete set of 5611 calibration points. The best parameter set for each of the 100 sub-samples was then estimated by 500 Monte-Carlo realizations. Thus, a central parameter estimate together with a spread around that central value was ob-tained from the 100 sub-samples for each of the samples sizes. The objective function for the remaining points not used for calibration was then predicted using the 100 in-dividual parameter sets as an evaluation set. The mean and spread of the deviation of the evaluation point objective function (O) from the calibration point objective function was used as an indicator for the predictive power of models with different calibration point sample sizes, i.e. the closer the evaluation objective function is to the calibration objective function the higher is the predictive power of the models at a given calibration sample size. Likewise, the robustness of the models was further assessed by relating the 100 central parameter estimates and their spreads to the respective sample sizes, i.e. the higher the spread in the parameter estimates, the less robust or the more sensitive the model is to the chosen calibration points, indicating a too small calibration point sample size.
3.3.7.
S
ENSITIVITY TO THE LOCATION OF CALIBRATION POINTSAs the topography of the Wark catchment sharply changes from undulating hills in the western part to plateaus above steep, incised valleys in the eastern part (figure 3.1), this allowed assessing the robustness of the landscape classification models to chang-ing landscape structures. That is, the ability of the model to correctly predict landscape classes when it was calibrated in a structurally different landscape. Here this was done by splitting the Wark Catchment into four zones; North, East, West and South, by using mean latitude and longitude (the mean of maximum and minimum of latitude within the catchment and the same procedure for longitude). While the eastern part of the catchment has very pronounced landscape features with sharp hillslopes and narrow valleys, the western part is characterized by a comparably subdued profile with wider valleys. The models were subsequently calibrated using observed points from one zone, while the observed points in the remaining zones were predicted. The changes in
objec-3.3.METHODS
3
21
tive functions and parameter sets were then used as indicators of the model sensitivity to changing landscapes.
3.3.8.
C
OMPARISON BETWEEN THE TOPOGRAPHICAL WETNESS INDEX ANDDIFFERENT LANDSCAPE CLASSES
As mentioned above the land classification aims at categorizing the catchment into hy-drologically similar zones. For this study the land classification was based on visual ob-servation. In reality it is expected that the position of the groundwater table can provide a more objective selection criterion as the groundwater for wetlands can be assumed to be shallower than the groundwater for plateaus and hillslopes. To see how well the model predicts the likely position of the groundwater table, hereafter referred to as indi-cator of wetness of each landscape, the models and their result were compared to the To-pographical Wetness Index (ITW), which is the base for TOPMODEL (Beven and Kirkby,
1979). The ITWis defined as follows:
ITW= ln(
A
tanβ) (3.13)
where A is the upstream contributing area andβ is local slope. The principle behind TOPMODEL is that locations with similar wetness indices are considered to have similar hydrological behavior.
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3.4.
R
ESULTS AND DISCUSSION
3.4.1.
C
OMPARING THE PERFORMANCE OF DIFFERENT MODELS FOR THEORIGINAL
DEM
In order to identify the most adequate landscape classification model, the three mod-els (MSH, MSD, MSHD) were run with the original 5 m × 5 m DEM. Model MSH, which
is equivalent to the original HAND-based model (Rennó et al.,2008), is found to be the most adequate model with an objective function (equation3.12value of O = 0.527, while the objective function values for the MSDand MSHDmodels are moderately higher
with values of 0.702 and 0.584, respectively. For the model MSH the best fit
parame-ter values for slope and HAND are found to be µslope= 0.129 (95 % uncertainty
inter-val: 0.096–0.166) withσslope= 0.002 (95 % uncertainty interval: 0.001–0.039) andµHAND
= 5.9 m (95 % uncertainty interval: 3.2–8.9 m) withσHAND= 0.23 m (95 % uncertainty
in-terval: 0.05–2.9 m). Correspondingly, for model MSDthe parameter values for slope and
DIST areµslope= 0.127 (95 % uncertainty interval: 0.102–0.150) withσslope= 0.001 (95 %
uncertainty interval: 0–0.026) andµDIST= 62.6 m (95 % uncertainty interval: 42.6–84.5 m)
withσDIST= 2.80 (95 % uncertainty interval: 0.3–22.5 m). While for MSHD, the parameter
values for slope and and the normalized metric of HAND and DIST are, µslope= 0.135
(95 % uncertainty interval: 0.092–0.183) withσslope= 0.004 (95 % uncertainty interval:
0–0.044) andµHD= 0.512 (95 % uncertainty interval: 0.454–0.585) withσHD= 0 (95 %
un-certainty interval: 0.001–0.075). Since in MSHtheσ values for the Normal distribution
are very low, these results suggest that all grid cells with Islope< 0.129 and IHAND< 5.9 m
are to be classified as flat wetlands, while grid cells with Islope> 0.129 and IHAND< 5.9
m are classified as sloped wetlands. Grid cells with Islope> 0.129 and IHAND> 5.9 m are
defined as hillslopes while those with Islope< 0.129 and IHAND> 5.9 m represent plateaus.
The classified landscapes are illustrated in figure3.4. The worst performance was obtained with model MSD. This model cannot mimic flat wetland and especially
head-water, narrow valley bottoms and wide valleys simultaneously. For headwaters and wide valleys the model needs to use a high distance from the stream to correctly model the observed point, however for narrow valley bottoms the distance should be as little as possible not to overlap with neighboring hillslopes. This causes a poor performance of MSD. The model which used HAND performs the best; it can predict the headwater as
well as wide and narrow valley bottoms better than MSD.
One problem which is obvious in Figure3.4is the noise within a specific landscape. Some raster cells with very high resolution have completely different characteristic from their neighboring cells. For example a cell (which may be a road or other human inter-ference) may have a zero slope and be classified as plateau while its neighboring cells having steep slopes are classified as hillslope.
The relatively low spread for both parameters, HAND and slope, in the MSH high-lights that the landscape units can be classified with a surprisingly low fuzziness, i.e. there is only limited uncertainty if a landscape element belongs to one class or to an-other and it shows that a crisp model withσslope=σHAND= 0 (Model ID: MSHcrispwould
produce results very close, in terms of model performance and parameter estimates, to those from the fuzzy approach.
3.4.RESULTS AND DISCUSSION
3
23N
(b)
(d)
(c)
Elevation [m] 532 195 0 1 2 4 Kilometers Wetland-sloped Hillslope Plateau Wetland-flat 0 250 500 meters(a)
Figure 3.4: Comparison between different models for land classification in a headwater of one of the tributaries of the Wark. (a) The location of the headwater in the Wark catchment; (b) model using HAND and slope (MSH); (c) model using DIST and slope (MSD); (d) model using HAND, DIST and slope (MSHD). White areas represent areas which are classified in more than one class (uncertain areas).
classification than DIST or combination of HAND and DIST. It shows that additional or similar parameters do not necessarily lead to equally good representations of landscape units as shown in figure3.4, where several areas of obvious landscape misclassification can be seen, especially for MSD. This underlines the potential of HAND to characterize
the landscapes as it is believed to be originated, other than elevation, from the feedback processes within the catchment (Savenije,2010).
3.4.2.
E
FFECT OF SMOOTHING ON MODEL PERFORMANCE AND PARAMETERSRelatively prominent, though small scale, landscape features, such as rock outcrops or hollows, can be present in landscapes of any type. However, up to a certain size they do not significantly change the appearance of the overall landscape or the associated dominant runoff process. Thus they should be smoothed out in order to reduce noise in the resulting landscape classification. Here it is found that, with increasing characteris-tic smoothing scale fromµSM= 0 (original 5 m × 5 m DEM) to σSM= 10–25 m, equivalent
