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UNIVERSITE NICE-SOPHIA ANTIPOLIS

ECOLE DOCTORALE STIC

SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA

COMMUNICATION

T H E S E

pour l’obtention du grade de

Docteur en Sciences

de l’Université Nice-Sophia Antipolis

Mention: Automatique, Traitement du Signal et des Images

présentée et soutenue par

Morgan ABILY

High-resolution modelling with bi-dimensional shallow water equations based

codes – High-resolution topographic data use for flood hazard assessment

over urban and industrial environments –

Thèse dirigée par Philippe GOURBESVILLE,

co-encadrée par Claire-Marie DULUC & Olivier DELESTRE

soutenue le 11 décembre 2015

Jury:

M. Shie-Yui Liong, Professeur - National University of Singapore Rapporteur

M. Nigel Wright, Professeur - De Montfort University Leicester Rapporteur

Mme. Nicole Goutal, Directeur de recherches - Lab. d'Hydraulique de Saint Venant, EDF R&D Examinateur

Mme. Claire-Marie Duluc, Ingénieur/chercheur et chef de Bureau - IRSN Examinateur

M. Olivier Delestre, Maître de conférence - Université Nice Sophia Antipolis Examinateur

Mme. Nathalie Bertrand, Ingénieur/chercheur - IRSN Invitée

M. Philippe Audra, Professeur - Université Nice Sophia Antipolis Président du jury

M. Philippe Gourbesville, Professeur - Université Nice Sophia Antipolis Directeur de thèse

ACKNOWLEDGMENT

Among many reasons I have to express my thank to them, I would like here to thank

Philippe Gourbesville and Claire-Marie Duluc for they pushed me to work on this thesis

project. They gave me the chance to review the work I have accomplished during my first

years of practice as a research engineer at IRSN and Polytech. It is an invaluable luxury.

I would like to express my thanks to the jury of this thesis and particularly to the reviewers for

their commitment and valuable feedbacks.

I am grateful to the two colleagues with whom I work with since our CEMRACS 2013

collaboration: Olivier Delestre and Nathalie Bertrand. This thesis project started after the

CEMRACS summer school and was an extra workload for them as well.

Part of the material used in this thesis has been kindly provided by Nice Côte d’Azur

Metropolis for research purpose.

This work was granted access to (i) the HPC and visualization resources of "Centre de

Calcul Interactif" hosted by "Université Nice Sophia Antipolis" and (ii) the high performance

computation resources of Aix-Marseille Université financed by the project Equip@Meso

(ANR-10-EQPX-29-01) of the program "Investissements d'Avenir" supervised by the Agence

Nationale pour la Recherche. DHI is acknowledge for the sponsored MIKE Powerd by DHI

license file.

Technical support for codes adaptation on high performance computation centers has been

provided by Fabrice Lebas, Yann Richet, Hélène Coullon, Christian Laguerre and Minh-

Hoang Le.

I also take this opportunity to thank the colleagues from BEHRIG and I-CiTy laboratory I had

the pleasure of spending time with these past years. I would like to thanks as well the

numerous scientists (from experienced ones to interns) with whom I had the chance to be in

touch, exchange and collaborate regarding this work. Lastly, I am thankful to my family and

friends. Dany, Didier, Maité, Eugénie, Bernard, Cynthia, Shéba, Tibo, Sébastien, Steph,

Alex(s) and Mag, the love and friendship I receive from you is the best daily support one can

dream of.

NOTE TO THE READER

This PhD work started in November 2013, results from the research framework built from the

collaborative activities initiated in 2011 between the Institute for Radioprotection and Nuclear

Safety (IRSN, Institut de Radioprotection et de Sûreté Nucléaire) and the water department

of the Engineering school Polytech Nice Sophia. Two main topics were defined for the

research work conducted during this collaboration.

Need for runoff modelling with highly detailed information at industrial site scale. The

IRSN wanted to test a specific approach for runoff hazard concern that has been specifically

enhanced in the guide for protection of basic nuclear installation against flooding elaborated

by the IRSN for the French Nuclear Safety Authority (ASN, 2013). In the guide, a specific

runoff Reference Flood Situation (RFS) states that a nuclear installation has to be able to

cope with a one hour long rainfall event of one over hundred years return period. IRSN

wanted to test feasibility of standard 2D Shallow Water Equations (SWEs) based numerical

tools for the runoff RFS. Spreading of High-Resolution (HR) topographic information

gathering techniques goes in this direction of a HR dataset (e.g. Light Detection and Ranging

or imagery based) easily available for a specific study purpose. Consequently, hydraulic

numerical modelling community increasingly uses Digital Elevation Models (DEM) generated

based on airborne HR datasets for urban flooding modelling. For a purpose like local runoff

flood risk modelling over an industrial site which is a complex environment, added value of

High-Resolution (HR) topographic data use that describes in detail the physical properties of

the environment was interesting to test. Moreover, Runoff flow paths influencing above-

ground features are not equally represented in DEM generated based on LiDAR and

photogrammetric data. Lastly, feasibility of HR data in standard 2D numerical modelling tools

might be challenging. Possibilities and challenges of these surface features inclusion in

highly detailed 2D runoff models for runoff flood hazard assessment deserve a specific

consideration and were therefore the key stone which motivated the work of this thesis.

Need to check uncertainties of the High-Resolution overland flow models - Even

though HR classified datasets have high horizontal and vertical accuracy levels (in a range of

few centimeters), this data set is assorted of errors and uncertainties. Moreover, in order to

optimize models creation and numerical computation, hydraulic modellers make choices

regarding procedure for this type of dataset use. These sources of uncertainties might

produce variability in hydraulic flood models outputs. Addressing models output variability

related to model input parameters uncertainty is an active topic that is one of the main

concern for practitioners and decision makers involved in the assessment and development

of flood mitigation strategies. IRSN and Polytech Nice Sophia wish to strengthen the

assessment of confidence level in these deterministic hydraulic models outputs.

SUMMARY

High-resolution (infra-metric) topographic datasets, including LiDAR data and

photogrammetric based classified data, are becoming commonly available at large range of

spatial extent, such as municipality or industrial site scale. This category of dataset is

promising for High-Resolution (HR) Digital Elevation Model (DEM) generation, allowing

inclusion of fine above-ground structures (walls, sidewalks, road gutters, etc.), which might

influence overland flow hydrodynamic in urban environment. DEMs are one key input data in

Hydroinformatics for practitioner willing to perform free surface hydraulic modelling using

standard 2D Shallow Water Equations (SWEs) based numerical codes (e.g. modeller wishing

to assess flood hazard). Nonetheless, several categories of conceptual, technical, and

numerical challenges arise from this type of data use with standard 2D SWEs numerical

codes.

Objective of this thesis is to tackle possibilities, advantages and limits of High-Resolution

(HR) topographic data use within standard categories of 2D hydraulic numerical modelling

tools for flood hazard assessment purpose.

Review of concepts regarding 2D SWEs based numerical modelling and HR topographic

data are presented. Methods to encompass HR surface elevation data in standard modelling

tools are tested and evaluated. Two types of phenomena generating flooding issues are

tested for High-Resolution modelling: (i) intense runoff and (ii) river flood events using in both

cases LiDAR and photo-interpreted datasets. Three scales of spatial extent are tested

ranging from a small industrial site scale to a city district scale (Nice low Var valley, France).

In this thesis, test studies are performed using a wide range of categories of standard

numerical modelling tools based on 2D SWE, from commercial (Mike 21, Mike 21 FM) to

open source (TELEMAC 2D, FullSWOF_2D) codes. Comparison is performed with 2D SWEs

simplified approaches (diffusive wave approximation using Mike SHE code) and with Navier-

Stokes volume of fluid resolution approach (Open FOAM code). Tools and methods for

assessing uncertainties aspects with 2D SWEs based models are developed and tested to

perform a Global Sensitivity Analysis (GSA) related to HR topographic data use.

Chapter 1 of this thesis introduces first, state of the art of HR topographic data gathering

techniques considering their possibility for a HR description of industrial and urban

environment. Then, a second part of this chapter summarizes the background of the

theoretical framework of SWEs, in order to raise questions up regarding validity of the

approach of 2D SWEs based modelling over complex environments. As the framework of

this type of application is different from the one for which SWEs have originally been

designed for, the expected limits that might be encountered for HR topographic data use in

standards codes are enhanced. Indeed, if from a practical point of view, codes relying on

approximation of SWEs are already commonly used for urban environment overland flow

modelling, theoretical questions arise and remain open regarding several conceptual,

mathematical aspects. Mainly, due to high gradient occurrences, boundary conditions and

initial condition that are seldom properly known or to the numerical treatment of

discontinuities (Riemann problem).

In chapter 2, three case study are used to give a proof of concept of HR topographic data

use feasibility, (i) to produce a HR DEM for intense flood simulations in complex

environment, and (ii) to integrate this HR DEM information in standard 2D SWEs based

codes. Feasibility, performances and relevance of the HR modelling are evaluated with a

selection of different codes approximating the 2D SWEs based on various spatial

discretization strategies (structured and non-structured) and with different numerical

approaches (finite differences, finite elements, finite volumes). A comparison is conducted

over computed maximal water depth and water deep evolution. The results confirmed the

feasibility of these tools use for the studied specific purpose of HR modelling. Tested

categories of 2D SWEs based codes, show in a large extent similar results in water depth

calculation under important optimization procedure. Actually, major requirements were

involved to get comparable results with a reasonable balance/ratio between mesh generation

procedure - computational time - numerical parameters optimization (e.g. for wet/dry

treatment).

The inclusion of detailed/thin features in DEM and in hydraulic models lead to considerable

differences in local overland flow depth calculations compared to HR models that do not

describe the industrial or urban environment with such level of detail. Moreover, added value

of fine features inclusion in DEM is clearly observed disregarding resolution used for their

inclusion (either 0.3 m or 1 m). Indeed, tests to include fine features (extruding their elevation

information on DEM), through an over sizing their horizontal extent to 1 m, lead to good

results with respect to their inclusion at a finer resolution.

LiDAR and photo-interpreted HR datasets are tested to compare their ease of use for HR

DEM devoted to hydraulic purpose elaboration. Results point out differences, notably

regarding ways and possibilities to integrate HR topographic dataset in 2D SWEs based

codes. Moreover, due to meshing algorithm properties, the over constraints created by the

density of vectors (in case of a HR urban environment) lead to errors and difficulties for non-

structured mesh generation. For these reasons, the use of a structured mesh representing

the HR DEM is found to be a more efficient compromise.

Chapter 3 focuses on uncertainties related to model inputs, and more specifically on

uncertainties related to one type of inputs: HR topographic data use and inclusion in 2D

SWEs based codes. The aim is:

(i) to be a proof of concept of spatial Global Sensitivity Analysis (GSA) applicability to 2D

flood modelling studies using developed method and tools and implementing them on High

Performance Computing (HPC) structures;

(ii) to quantify uncertainties related to HR topographic data use, spatially discriminating

relative weight of uncertainties related to HR dataset internal errors with respect to modeller

choices for HR dataset integration in models.

The Uncertainty Analysis (UA) leads to conclusive results on: output variability quantification,

nonlinear behaviour of the model, and on spatial heterogeneity. The considered uncertain

parameters related to the HR topographic data accuracy and to the inclusion in hydraulic

models, influence the variability of calculated overland flow maximal water depth is found to

be considerable. This stresses out the point that even though hydraulic parameters are

assumed to be fully known in our simulations, the uncertainty related to HR topographic data

use cannot be omitted and needs to be assessed and understood.

Sensitivity indices (Sobol) are calculated at given points of interest, enhancing the relative

weight of each uncertain parameter on variability of calculated overland flow. Sobol index

maps production is achieved. The spatial distribution of Si illustrates the spatial variability

and the major influence of the modeller choices, when using the HR topographic data in 2D

hydraulic models with respect to the influence of HR dataset accuracy.

TABLE OF CONTENTS

Introduction ............................................................................................................................. 15

Chapter I - Theoretical background ....................................................................................... 21

Part. 1. High-Resolution topographic data in urban environment .............................................. 24

1.1 LiDAR ..................................................................................................................................................... 24

1.2 Photogrammetry and photo-interpreted datasets ...................................................................................... 26

1.1.2 Photogrammetry ................................................................................................................................ 26

1.1.3 Object based classification: photo-interpretation .............................................................................. 27

1.3 Focus on spatial discretization ................................................................................................................. 31

1.4 Feeback for HR topographic data use in 2D urban flood modelling ........................................................ 34

Part. 2. Numerical modelling of free surface flow: approximating solution of SWEs ............... 36

2.1 From Hypothesis in the physical description of the phenomena to mathematical formulation................ 36

2.1.1 From flow observation to de Saint-Venant hypothesis and mathematical formulation ..................... 36

2.1.2 Validity and limits of these hypothesis .............................................................................................. 40

2.2 Numerical methods to approach solution of the SWEs system ................................................................ 41

2.2.1 Introduction to concepts for a numerical method .............................................................................. 42

2.2.2 Standard numerical methods ............................................................................................................. 44

2.3 Tested commercial numerical codes for HR topographic data use .......................................................... 50

Chapter I conclusions: foreseen challenges related to HR topographic data use with 2D SWEs

based numerical modelling codes ................................................................................................... 53

Chapter II – Case study of High-Resolution topographic data use with 2D SWEs based

numerical modelling tools ....................................................................................................... 55

Part. 3. High-Resolution runoff simulation at an industrial site scale ........................................ 58

3.1 Test case setup ......................................................................................................................................... 61

3.1.1 Presentation of mathematical and numerical approaches .................................................................. 61

3.1.2 Site configuration and spatial discretization ...................................................................................... 62

3.1.3 Runoff scenarios and models parameterization ................................................................................. 66

3.1.4 Performance assessment methodology .............................................................................................. 68

3.2 Results ...................................................................................................................................................... 72

3.2.1 Rainfall events scenarios (S1 and S2) ............................................................................................... 72

3.2.2 Initial 0.1 m water depth scenario (S3) .............................................................................................. 77

3.2.3 Indicators of computation reliability ................................................................................................. 81

3.3 Discussion and perspectives ..................................................................................................................... 83

3.3.1 Discretization and high topographic gradients .................................................................................. 84

3.3.2 Flow regime changes treatment ......................................................................................................... 85

3.3.3 Threshold for complete 2D SWEs resolution .................................................................................... 85

3.3.4 Computation reliability ...................................................................................................................... 86

3.4 Complementary tests and concluding remarks ......................................................................................... 86

3.4.1 Complementary tests ......................................................................................................................... 86

3.4.2 Concluding remarks .......................................................................................................................... 89

Part. 4. High-resolution topographic data use over larger urban areas ..................................... 91

4.1 HR runoff simulation over an urban area ................................................................................................. 93

4.1.1 Site configuration and runoff scenarios specificities ......................................................................... 93

4.1.2 Presentation of HR topographic datasets ........................................................................................... 94

4.1.3 High-Resolution DSMs for overland flow modelling purpose .......................................................... 96

4.1.4 Impact of fine above-ground features inclusion in DSMs for runoff simulations ........................... 100

4.1.5 Outcomes ......................................................................................................................................... 104

4.2 HR flood river event simulation over the Low Var valley ..................................................................... 106

4.2.1 Site, river flood event scenario and code ......................................................................................... 106

4.2.2 Method for High-Resolution photo-interpreted data use for High-Resolution hydraulic modelling 108

4.2.3 Feedback from High-Resolution river flood modelling .................................................................. 110

Chapter II conclusions ................................................................................................................... 113

Chapter III - Uncertainties related to High-Resolution topographic data use ................... 117

Part. 5. Methodology for Uncertainty Analysis and spatialized Global Sensitivity Analysis .. 124

5.1 HR classified topographic data and case study .................................................................................. 124

5.2 Concepts of UA, SA and implemented spatial GSA approach .............................................................. 126

5.2.1 Uncertainty and Sensitivity Analysis .............................................................................................. 126

5.2.2 Implemented spatial GSA ................................................................................................................ 129

5.3 Parametric environment and 2D SWEs based code ............................................................................... 134

Part. 6. Results of UA and GSA applied to HR topographic data use with 2D Flood models 136

6.1 UA results .............................................................................................................................................. 137

6.1.1 Analysis at points of interest ........................................................................................................... 137

6.1.2 Spatial analysis ................................................................................................................................ 138

6.2 Spatial Global Sensitivity Analysis results ............................................................................................ 140

6.2.1 Analysis at points of interest ........................................................................................................... 140

6.2.2 Spatial analysis ................................................................................................................................ 141

Part. 7. Discussion .......................................................................................................................... 145

7.1 Outcomes ............................................................................................................................................... 145

7.2 Limits of the implemented spatial GSA approach ................................................................................. 146

Chapter III Summary and conclusions ........................................................................................ 148

General conclusion and prospects ........................................................................................ 151

Conclusions and recommendations .............................................................................................. 152

Method and good practices for HR topographic data use (T1) .............................................................. 152

Uncertainties related to HR topographic data use: method and tools for uncertainty analysis in HR

2D hydraulic modelling (T2) ..................................................................................................................... 156

Holistic view and prospects ........................................................................................................... 159

High-resolution modelling ......................................................................................................................... 159

Uncertainty and Sensitivity Analysis with 2D SWEs based models and prospects .............................. 160

Perspectives for spatial discretization and communication on HR flood modelling results ................ 162

Bibliography .......................................................................................................................... 165

List of abbreviations .............................................................................................................. 177

Table of figures ...................................................................................................................... 178

Table of tables ........................................................................................................................ 181

Annexes .................................................................................................................................. 183

15

INTRODUCTION

Over urban and industrialized areas, flood events might result in severe human, economic

and environmental consequences (Dawson et al., 2008). For flood hazard assessment,

numerical models help decision makers to mitigate the risk. Numerical modelling tools are

based on conceptualization of complex natural phenomena, using physical and mathematical

hypothesis. In hydraulics, for flood hazard assessment, numerical models aim at describing

free surface behaviour (mainly elevation and discharge) according to an engineering

description, to provide decision makers information regarding flood hazard estimations.

Considering that the modeller knows in detail what is the chain of concepts, leading from

hypothesis to results, good practice is to provide numerical model results with description of

performance and limits of these simulations. The aim is to provide to the stakeholders, what

are the deviations between what has been modelled and the reality (Cunge, 2003; Cunge,

2012). In the context of flood events modelling over an urban environment, bi-dimensional

Shallow Water Equations (2D SWEs) based modelling tools are commonly used in studies

even though the framework for such application goes straight from some of the 2D SWEs

system underlying assumptions (see chapter 1).

Indeed, for practical flood modelling applications over urban and industrialized areas,

standard deterministic free surface hydraulic modelling approaches most commonly rely

either on (i) 2D SWEs system, (ii) simplified version of 2D SWEs system (e.g. diffusive wave

approximation (Moussa and Bocquillon, 2000; Fewtreel, 2011)) or (iii) multiple porosity

shallow water approaches (Sanders et al., 2008; Guinot, 2012). Compare to (i), approach (ii)

is a simplification of the mathematical description of the flow whereas approach (iii) is based

on a simplification of the geometry description that includes a term in the calculation to

represent sub-grid topographic variations. These approaches are different in terms of

conceptual description of flow behaviour and of computational cost. They require dissimilar

quantity and type of input data. At cities or at large suburbs scales, these approaches give

overall comparable results (Guinot, 2012; Dottori et al., 2013; Guinot and Delenne, 2014).

However, limitation appears at fine scale as diffusive wave approach does not include inertial

effects and porosity shallow water approaches average local effects (Kim et al., 2015).

Therefore, if one aim is to provide High-Resolution (HR) description of overland flow

properties reached during a flood event at fine scales (street, compound or building scales),

codes based on 2D SWEs system using detailed description of the environment are required.

Indeed, above-ground surface features (buildings, walls, sidewalks, etc.) that influence

overland flow path are densely present. Furthermore, these structures have a high level of

diversity, ranging from a few meters (buildings, sidewalks, roundabouts, crossroads, etc.) to

16

a few centimeters width (walls, road gutters, etc.). It creates a complex environment highly

influencing the overland flow properties.

Detailed information about flooding hazard are required in mega-cities flood resilience

context (Djordjevic et al., 2011), and for nuclear plant flood risk safety assessment (ASN,

2013). The use of High-Resolution (HR) numerical modelling should provide valuable insight

for flood hazard assessment (Gourbesville, 2004, 2009). Obviously, to perform HR models of

complex environments, an accurate description of the topography is compulsory. To describe

in detail overland flow, the level of detail of the Digital Elevation Model (DEM) should include

above-ground features influencing flow paths.

Urban reconstruction relying on airborne topographic data gathering technologies, such as

imagery and Light Detection and Ranging (LiDAR) scans, are intensively used by geomatics

communities (Musialski et al., 2013). Indeed, modern aerial transportation vectors, such as

Unmanned Aerial Vehicles (UVA), make HR LiDAR or imagery based datasets affordable in

terms of acquisition time and financial cost (Remondino et al., 2011; Nex and Remondino,

2014; Leitã et al., 2015). During the last decade, topographic datasets created based on

LiDAR and photogrammetry technologies have become widely used by other communities

such as urban planners (for 3D reconstruction approach) and consulting companies for

various applied study purposes including flood risk studies. These technologies allow to

produce DEMs with a high accuracy level (Mastin et al., 2009; Lafarge et al., 2010; Lafarge

and Mallet, 2011). Among HR topographic data, photogrammetry technology allows to

process to an object based classification to produce a 3D classified topographic dataset

(Andres, 2012).

Therefore, to understand or to predict surface flow properties during an extreme flood event,

models based on 2D Shallow Water Equations (SWEs) using HR description of the urban

environment are often used in practical engineering applications (Mandlburger et al., 2008;

Aktaruzzaman and Schmidt, 2009; Erpicum et al., 2010; Tsubaki and Fujita, 2010; Fewtrell et

al., 2011). In that case, the main role of hydraulic models is to accurately describe overland

flow's maximal water depth reached at some specific points or area of interest. If most of

modern 2D SWEs codes integrate strategies to perform computation using parallelization

strategies of codes to take advantage of High Performance-Computing (HPC) power for

computational swiftness (Sørensen et al., 2010; Moulinec et al., 2011; Cordier et al., 2013),

several aspects requires to be addressed for a pertinent and optimized HR modelling. HR

topographic information is considered as “Big Data”, requiring development of method for

their efficient implementation in hydraulic free surface numerical modelling tools. The

17

operational possibilities and issues to integrate the HR topographic data in the numerical

hydraulic models have to be assessed.

The objectives of the research presented in this thesis are to address feasibility, added value

and limits of HR topographic data use with standard hydraulic numerical codes. The main

concerns are to assess the validity of such an approach and the requirements related to

specificities of the HR topographic data for HR hydraulic modelling. Moreover, information

will be provided for practical aspects and ease of use for standard applications.

Through the following objectives:

the first target (T1) is to develop method and to provide a list of good practices for HR

urban flooding event modelling;

the second target (T2) focuses on quantifying and ranking uncertainties related to HR

topographic data use in 2D SWEs based models developing operational tools and

method to carry out an uncertainty analysis.

Framework for T1

From an operational point of view, SWEs based codes are broadly used over urban

environment, even though theoretical questions regarding several conceptual and

mathematical aspects remain open. In fact, such framework is far from the one for what

SWEs were originally been designed for, and it stresses out the fact that limits might be

expected and encountered. Therefore, relevance, feasibility, added values and challenges of

HR flood modelling in complex environment should be tested.

This target, tackles the problematic of high density topographic information inclusion in

standard 2D modelling tools and a second subtask is the assessment of possibilities and

impacts of fine features inclusion.

Different sets of HR topographic data gathered from (i) a LiDAR and (ii) a photogrammetric

campaign are tested. The standard 2D numerical modelling tools used in our studies are

based on 2D SWEs resolution. This category of modelling tools has various numerical

strategies to solve 2D SWEs and discretize the spatial information in different ways. The aim

is not to benchmark performance of different codes, this has already done by Hunter (2008).

Here, the main interest has been keen on assessing possibilities and limits of strategies for

spatial discretization used by modelling tools, investigating on HR DSM use with regular grid

meshing and non-structured meshing approaches.

Framework for T2

18

Dealing with uncertainties in hydraulic models is an advancing concern for both practitioners

(Iooss, 2011) and new guidance (ASN, 2013). Identification, classification and quantification

of the impact of sources of uncertainties on a given model output are a set of analysis steps

which will enable to analyze uncertainties behaviour in a given modelling problem, to

elaborate methods to reduce uncertainties on a model output and to communicate on

relevant uncertainties. Sources of uncertainties in hydraulic models come from (i) hypothesis

in mathematical description of the natural phenomena, (ii) from input parameters of the

model, (iii) from numerical aspects when solving the model. Input parameters are of prime

interest for applied practitioners willing to decrease uncertainties in the results (Iooss, 2011;

Iooss and Lemaître, 2015). Input parameters of hydraulic models have hydrological,

hydraulic, topographic and numerical nature.

Although HR classified datasets are of high horizontal and vertical accuracy (in a range of

few centimeters), produced HR DEMs are assorted with the same types of errors as coarser

DEMs. Errors are due to limitations in measurement techniques and to operational

restrictions. These errors can be categorized (Fisher and Tate, 2006; Wechsler, 2007) as

follow:

(i) systematic, due to bias in measurement and processing;

(ii) nuggets (or blunder), which are local abnormal values resulting from equipment or

user failure, or to occurrence of abnormal phenomena in the gathering process

(e.g. birds passing between the ground and the measurement device);

(iii) random variations, due to measurement/operation inherent limits.

Moreover, amount of data that compose a HR classified topographic dataset is massive.

Consequently, to handle HR dataset and to avoid prohibitive computational time, hydraulic

modeller has to make choices to integrate this type of data in the hydraulic model. However,

this may decrease HR DEM quality and can introduce uncertainty (Tsubaki and Kawahara,

2013; Abily et al., 2015c, 2016a, 2016b). As summarized in Dottori (2013), Tsubaki and

Kawahara (2013), and Sanders (2007), HR flood models effects of uncertainties related to

HR topographic data use on simulated flow is not yet quantitatively understood.

Consequently, our objective here is to define, quantify and rank the uncertainties related to

the use of HR topographic data in HR flood modelling over densely urbanized areas. The

aims are (i) to apply an Uncertainty Analysis (UA) and spatial Global Sensitivity Analysis

(GSA) approaches in a 2D HR flood model having spatial inputs and outputs, and (ii) to

producing sensitivity maps.

19

The first chapter of the thesis presents the state of the art of HR topographic data gathering

techniques considered as relevant with applications aiming at a HR description of industrial

and urban environment (part 1). The focus is on technique suitable to balance spatial extent

and resolution requirement to include fine overland flow influencing structures (walls,

sidewalks, road gutters, etc.) in HR datasets. The concepts and the state of the art of

standard spatial discretization procedure for inclusion of topographic data in 2D numerical

flood models are presented. The second part of this chapter (part 2) reviews the background

of the theoretical framework of SWEs, in order to raise questions up regarding validity of the

approach of 2D SWEs based modelling over complex environments. The limits and

challenges regarding conceptual, mathematical and numerical aspects that should be

expected with HR topographic data use in standards codes are presented.

The second chapter tackles the target 1 (T1). A methodology and the good practices for HR

urban flooding event modelling are presented in parts 3 and 4. Three case study are

considered for our purpose at different scales: (i) over a small (60,000 m²) fictitious industrial

site using a created 0.1 m resolution DSM, (ii) over a real site in Nice city (France) using a

LiDAR and a 3D photo-interpreted dataset at a larger scale (600,000 m²) and (iii) over the

low Var valley in Nice using 3D photo-interpreted dataset at a large scale (17.8 km²).

Moreover, different types of flood scenarios were tested: (i) and (ii) are simulations of intense

rainfall events and (iii) is a river flood event. The first part of this chapter (part 3) focuses on

the validity, the relevance and the limits of HR flood modelling in complex environment. The

idea is to check numerical solving of 2D SWEs over complex topographies having high

topographic gradient and leading to overland flow with challenging properties for numerical

codes. Therefore, the challenging case study (i) of HR flood risk modelling due to local

intense runoff is over an accurately described industrial site is used. Several standard

numerical 2D SWEs based codes relying on different numerical methods and spatial

discretization strategies are tested. In the second part of this chapter (part 4), the

problematic of high density topographic information inclusion in standard 2D modelling tools

and the assessment of possibilities and impact of fine features inclusion at different scales

for different types of 2D SWEs based codes are presented. Study relies on case study (ii)

and (iii).

The third chapter is devoted to target two (T2): a spatial Global Sensitivity Analysis (GSA)

method for 2D HR flood modelling focusing on the spatial ranking of uncertain parameters

related to the use of HR topographic data is introduced. The case study (iii) is used here. The

objective is to study uncertainties related to two categories of uncertain parameters

(measurement errors and uncertainties related to operator choices) with regards to the use of

HR classified topographic data in a 2D urban flood model. The fact that spatial inputs and

20

outputs are involved in our uncertainty analysis study is an important concern for the

methodology application. A spatial GSA is implemented to produce sensitivity maps based

on Sobol index computation. The first part of the chapter (part 5) introduces the test case

context for the uncertainty analysis, enhances description of used HR topographic dataset,

and gives general overview of uncertainty analysis methods and concepts. Lastly

implemented methodology for the spatial GSA and developed tools are described. The

second part of the chapter (part 6) presents results, first at points of interest, then at spatial

levels. Eventually, outcomes and limits of our approach are then discussed (part 7).

21

CHAPTER I - THEORETICAL BACKGROUND

The specificity of densely urbanized or industrialized environments relies in the fact that size

of above-ground features influencing overland flow path, ranges from macro elements (e.g.

buildings) to fine ones (e.g. walls, sidewalks, road curbs, roundabouts, etc.). If one aim is to

use free surface hydraulic numerical model to assess in detail the flood risks in these

environments (e.g. due to intense rainfall events or to river overbanking), influence of these

features has to be considered.

A Digital Elevation Model (DEM) can be the spatial discretization of the continuous variation

of the elevation of the ground; DEM is then called a Digital Terrain Model (DTM). A DEM can

also represent the elevation of the ground plus the elevation of the above-ground features on

it; DEM is then called a Digital Surface Model (DSM). High-Resolution Digital Elevation

Models (HR DEMs) allow the detailed representation of surface features.

In a DEM, resolution of a topographic dataset gives a single elevation value (z) for a given

cell area, whatever are in the reality the changes of z properties within this area. The z value

can be either: an averaged value of the several elevations information gathered within a cell

area; or a given point value applied for the whole area of the cell. Consequently, in a DEM,

the physical properties of z are reduced to the resolution of the cell size.

This resolution aspect has to be kept in mind as being a limiting factor in terms of accuracy of

the topographic representation. Indeed, even if a topographic data gathering technology is

able to provide at a given point an accurate measurement of z, once the z value is averaged

to the resolution of the DEM cells, the use of the original level of accuracy of the technique to

characterize the accuracy of the DEM does not make sense anymore. This is particularly the

case if in the reality the characteristic of z varies significantly within the cell area. Similarly,

when an inaccurate set of z measurement is used, interpolated and then discretized to a

higher resolution than the accuracy level of the technique to create a DEM, it would not make

any sense either.

Accordingly, this work considers that the concept of High-Resolution (HR) of the topographic

dataset depends on the scale and abruptness of change in physical properties of the

elevation with respect to the spatial resolution of the cell. Indeed, if the topography of the

system that is intended to be represented has an important spatial extent, and if the spatial

variations of the topography are not important with respect to the resolution, a few meters

discretization can be considered as HR. For instance, Bates (2003) modelled a flood plain

22

overland flow, over a river reach of 12 km, using a DEM having a 4 m resolution. Bates

(2003) work was considered as based on HR topographic data. Erpicum (2010) uses

topographic data over urban area and considered that data information with a resolution

between 4 m and 1 m is of HR for such type of environment. In the case of an urban or

industrial environment, a topographic dataset is considered to be of HR when it allows to

include in the topographic information elevation of infra-metric elements (Le Bris et al., 2013).

To achieve a horizontal topographical resolution fine enough to represent overland flow

influencing structures in urban type environment, the interval of gathered points z data

should be in the range of 0.1 m to 0.4 m for a HR DEM generation including fine features

(Ole, 2004; Tsubaki, 2013).

For our concern, we will present in this chapter, topographic data gathering technologies - (i)

Light Detection And Ranging (LiDAR) and (ii) photogrammetry – these techniques allow to

represent infra-metric information. A focus is given on features classification carried out by

photo-interpretation process, that allows to have high accuracy and highly detailed

topographic information (Mastin et al., 2009; Andres, 2012; Larfarge et al., 2010; Lafarge and

Mallet, 2011). Photo-interpreted HR datasets allow to generate HR DEMs including classes

of impervious above-ground features (see chapter 2 and Abily et al., 2014a, 2015b).

Produced HR DEMs can have a vertical and horizontal accuracy up to 0.1 m (Fewtrell et al.,

2011). HR DEMs generated on photo-interpreted datasets based can include above-ground

features elevation information depending on modeller selection among classes.

As presented in the introduction of this thesis, hydraulic numerical modelling community

increasingly starts to use HR DSM information from airborne technologies to model urban

flood scenarios (Tsubaki and Fujita, 2010) to understand or to predict surface flow properties

during an extreme flood event. Objective of numerical approaches used in the SWEs codes

is to approximate the solution (when existing) of equations as faithfully as possible by a

method where the unknowns are the values of hydraulic variables (water depth and velocities

or discharges) in a finite number of points (nodes) of the studied domain, and in a finite

number of instances during the considered period of time (spatial and temporal

discretization). In the 2D SWEs based codes, the topographic information is discretized

(either using the DEM or performing a second discretization based on the DEM) to be

included in the computation through the use of computational grid (or mesh).

The first part (part 1) of this theoretical chapter introduces in a first section the specificities of

topographic data gathering techniques estimated to be suitable with the HR DEM production

for our urban flood modelling purpose. The second section emphasizes the principles and

23

the common practices to include DEM information in a 2D hydraulic code, the computational

grid generation.

The second part (part 2) of this chapter recalls and summarizes in its first section the basics

behind 2D free surface modelling using numerical codes approximating the 2D SWEs

solutions and then gives an overview of standard numerical methods to approximate the

solution of the SWEs system.

24

PART. 1. HIGH-RESOLUTION TOPOGRAPHIC DATA IN URBAN

ENVIRONMENT

Generally, for topographic information gathering campaign, the technologies -LiDAR or

photogrammetry- are settled on a vector of transportation that can be terrestrial (e.g. cars) or

aerial: Unmanned Aerial Vehicle (UAV, e.g. drones), specific flight (plane or helicopter) or

satellite. For our range of applications vectors compatible with required balance between

resolution and spatial extent are UAV and specific flight campaign. Terrestrial vectors allow

to produce HR and high accuracy datasets (Hervieu and Soheilian, 2013), but are discarded

here due to the prohibitive size of the spatial extent to cover for an application over an urban

area. Nevertheless, it has to be noticed that over a smaller extent (industrial site or district

scale) such type of vector can be suitable. For instance, Fewtrell (2011) uses a HR

topographic dataset gathered using a terrestrial vector for a modelling of an urban flooding

over a district. Remote sensing from satellite can provide information for application at large

scale (Weng, 2012), but is here discarded as the resolution and the vertical accuracy for an

urban environment will not be sufficient for the scope of our study compare to resolution and

accuracy that can be reached when using specific flights (Sanders, 2007). Indeed, specific

flight for topographic data gathering campaign over urban environment using planes or

drones allow to offer the best balance in terms of possibility to get a resolution/accuracy and

spatial extent compatible for HR modelling (Küng et al., 2011).

1.1 LIDAR

LiDAR is a scanner system that uses a laser to pulse a beam that will be reflected by objects

on its way and that will be received by a sensor embedded on the scanner system. This

procedure enables to provide information of distance between the reached targets and the

sensor by multiplying the speed of light by the time it takes for the light to transmit from and

return back to the sensor (Priestnall et al., 2000; Weitkamp, 2005). The nature of LiDAR data

offers the potential for extracting surface information for many ranges of applications

(Priestnall et al., 2000).

When a LiDAR is mounted on board of a flying vector, the gathering system is composed of

combined technologies (Figure 1.1) that allow to accurately georeference the LiDAR system

(Habib et al., 2005; Gervaix, 2010). The material making up the system is generally:

an accurate GPS system, allowing to locate the aircraft with a centimetric precision;

an Inertial Measurement Unit (IMU), to take consider the aeronef movements during

the flight;

25

the LiDAR scanner, that emits and receives back the beam, measuring the distance,

the time and the angle of the scanning;

a computer to store the data.

This equipment can be mounted on board an aeronef such as a plane or a helicopter but,

mainly due to IMU important weight, LiDAR system can be mounted only on an UAV (such

as drones) powerful enough to carry the weight of the whole system, discarding the ultra-light

UAV (see Leitão et al., 2015).

The LiDAR system provides row information under the form of a geo-referenced point cloud.

The LiDAR pulses can lead to single, multiple or waveform returns. In all these cases, the

level of energy returned to the captor is different and can possibly be analyzed in case of

multi-return or waveform return signals. Then, the first return will describe the first objects

encountered by the beam (e.g. vegetation), whereas the last one will represent reflection

from the ground surface. A classification of the points is then necessary and can be carried

out through the use of specific software to discriminate elevation information of above-ground

structures from the ground elevation information. LiDAR ground filtering algorithms within

these software make different assumptions about ground characteristics to discriminate

ground, non-ground features and above-ground objects (e.g. bridges, short walls, mixed

areas, etc.). Abdullah (2012) gives illustration and application of a LiDAR filtering procedure

for bridges and elevated roads removal in urban areas. Nevertheless, Meng (2010)

underlined that complex conditions such as dense, various size and shape of above-ground

features environments, lead to errors in the differentiation. These complex conditions are

likely to occur in case of an urban environment.

In an aerial campaign, the point cloud density will depend on the following parameters in the

methodology for the aerial LiDAR topographic data gathering campaign: (i) Laser pulse rate

(Hz), (ii) flight height/speed ratio, and (iii) scan angle. With recently developed LiDAR

sensors, precision range can reach 2 to 3 cm (Lemmens, 2007).

The accuracy of LiDAR points highly depends on the accuracy of GPS and IMU systems.

Airborne GPS is able to yield results having an accuracy up to 5 cm horizontally and 10 cm

vertically, while IMU can generate altitude with accuracy within a couple of centimeters

(Fisher and Tate, 2006; Liu, 2008).

26

Figure 1.1. Schematization of a LiDAR system mounted on a plane (from Gervaix 2010).

1.2 PHOTOGRAMMETRY AND PHOTO-INTERPRETED DATASETS

1.1.2 Photogrammetry

Aerial photogrammetry technology allows to measure 3D coordinates of a space and its

objects (features) using 2D pictures taken from different positions. The overlapping between

pictures allows to calculate 3D properties of space and features based on stereoscopy

principle (Baltsavias, 1999; Eagles, 2004; Liu, 2008) as conceptualized in figure 1.2. To

measure accurately ground and features elevation, a step of aerotriangulation calculation is

compulsory, requiring information on picture properties regarding their position, orientation

and bonding (or tie) points. The orientation on how the camera lens was pointed varies in

time depending on the lens rotation due to plane or UAV roll, pitch and yaw that lead to lens

rotation angles (respectively called omega, phi and kappa). The use of ground control points

allows to geo-reference the dataset.

A low flight elevation, a high number of aerial pictures with different points of view and high

levels of overlapping, allow to increase the accuracy and the reliability of the 3D coordinates

measurement (Küng et al., 2011). Indeed, sensitivity tests on parameters photogrammetric

influencing dataset quality: (i) flight altitude, (ii) image overlapping, (iii) camera pitch and (iv)

weather conditions, confirmed the major influence of flight altitude on dataset quality (Leitão

et al., 2015).

27

Figure 1.2. Stereoscopy principle in photogrammetry to get ground or object x, y, and z properties (from Linder,

2006).

In photogrammetry, the spatial resolution is the size of a pixel at the ground level. It has to be

distinguished to the spectral resolution which is related to the number of spectral bands and

gathered simultaneously (see Egels and Kasser, 2004). At a given spatial resolution, an

object having a size three times bigger than the pixel size can be identified and interpreted.

1.1.3 Object based classification: photo-interpretation

For 3D classified dataset creation, a photo-interpretation step is necessary. Photo-

interpretation allows creation of vectorial information based on photogrammetric dataset

(Egels and Kasser, 2004; Linder, 2006). A photo-interpreted dataset is composed of classes

of points, polylines and polygons digitalized based on photogrammetric data. Figure 1.3

illustrates the visualization of a sub-part of a photo-interpreted dataset composed of 50

classes of polylines and polygones. Important aspects in the photo-interpretation process are

the classes’ definition, the photo-interpretation techniques and the dataset quality used for

the photo-interpretation. These aspects will impact the design of the output classified dataset

(Lu and Weng, 2007).

28

Figure 1.3. Visualization of elevation information of a photo-interpreted dataset gathered over an urban area

(Nice, France). Details of this specific dataset are given in chapter 2.

The step of classes’ definition has to be elaborated prior to the photo-interpretation step. The

number, the nature and criteria for the definition of classes will depend on the objectives of

the photo-interpretation campaign.

Photo-interpretation techniques can be made (i) automatically by algorithm use, (ii) manually

by a human operator on a Digital Photogrammetric Workstation (DPW) or (iii) by a

combination of the two methods. The level of accuracy is higher when the photo-

interpretation is done by a human operator on a DPW, but more resources are needed as the

process becomes highly time consuming (Zhou et al., 2004; Lafarge, 2010). Eventually, the

3D classification of features based on photo-interpretation allows to get 3D High-Resolution

topographic data over territory that offers large and adaptable perspectives for its exploitation

for different purposes (Andres, 2012).

Usually, when a photo-interpreted classified dataset is provided to a user, the data is

assorted with a global mean error value and with a percentage of photo-interpretation

accuracy. The mean error value encompasses errors, due to material accuracy limits, to

biases and to nuggets (or blunder) that compose error within the row photogrammetric data.

Furthermore, a percentage of accuracy representing errors in photo-interpretation is

generally provided. This percentage of accuracy represents errors in photo-interpretation

29

which results from feature misinterpretation addition or omission. This percentage of

accuracy results from the photo-interpreted dataset comparison with field ground

measurements of elevation over sub-domains of the photo-interpretation campaign (Figure

1.4). This process of control is time consuming as often based on manual operation and

control, resource requiring (field measurement campaign) and subject to operator

interpretation (Andres, 2012).

Figure 1.4. Illustration of field and photo-interpreted measurement comparison that are performed to control

the level of accuracy of the photo-interpretation process.

A typical workflow to illustrate the process to achieve the photo-interpretation is given in

figure 1.5. With this figure, idea is not to go into details into the description of this workflow,

that can vary depending on the campaigns. Nevertheless, it is interesting for a non-specialist

in geomatics to understand that three loops are interconnected in this process. First, the data

gathering/measurement. Impact of camera properties is the main issue here. Second, loop is

the treatment of geo-referencing/calibration part. Last part of the process is the photo-

interpretation part itself.

30

Figure 1.5. Illustration of workflow process to produce a photo-interpreted dataset as described in Linder

(2006).

It has to be mentioned that both type LiDAR and Photogrammetric topographic data

gathering techniques, when mounted on aerial vectors, are not well suited to gathered

underwater bathymetry. New possibilities of post-treatment of these techniques to gather

river bathymetry are developing (see Feurer et al., 2008). Nevertheless, beside for really low

flow condition, issues are still likely to occur due to LiDAR inability to penetrate water masses

(Podhoroanyi and Fedorcak, 2015) and due to visibility through water that will make

photogrammetry use not relevant. Moreover, limits arises related to combine effects of the

transportation vector (here aerial) and of the physical properties of the ground and its objects

(e.g. slope, dense vegetation or narrow streets). It decreases the accuracy of the datasets.

31

1.3 FOCUS ON SPATIAL DISCRETIZATION

In computing codes, the physical domain (Ω) can have 1, 2 or 3 dimensions in space. The

discretization of Ω in 1D, 2D or 3D is respectively associated to variables x, y and z and

called a mesh or a computational grid. The computational grid then represents the continuum

where the governing partial differential equations are replaced by constructed discretized

forms/solved by numerical methods (see part 2). For numerical resolution of the 2D SWEs

system, the continuous variable topography information/elevation (z) is necessary for the

computation, and therefore spatially discretized according to a 1D, 2D or 3D meshing

process.

A mesh that arises from the discretization of z within Ω is composed of referenced points

(computational points or nodes) and of cells (or elements) that link the points together. A

mesh is characterized by its dimension - 1D to 3D -, and the geometry of its cells that can be

flat elements (triangles, rectangles or polygons) or elements in volume (pyramids,

tetrahedron, cubes, etc.), respectively for 2D or for 3D. As recalled in Weatherill (1992), the

mesh has to represent accurately the geometrical boundaries, and “gap” in the computational

domain cannot occur.

Main classification criteria of types of meshes are following:

If elements have identical/regular size to discretize the Ω, the mesh is said to be

structured, whereas if the mesh is composed of elements having different sizes (but

always with the same geometry) they are qualified as non-structured (Figure 1.6). In a

structured mesh, all interior nodes -not located on a boundary of Ω- have an equal

number of adjacent elements. Hybrid meshing exists, Ω being then discretized

mixing structured and non-structured sub-domains.

If properties (size/shape) of elements constituting a mesh evolve with time, the mesh

is referenced as Adaptive Mesh Refinment (AMR) while a mesh that has constant

properties in time is referred as non-adaptive.

Parameters of a computational grid such as area or volume of the cells (resolution) and

number of elements are inherent properties of the mesh. The smaller are the areas or

volumes of the elements, the more the discretization gets close to the continuous variable

(z), but the more the total number of elements increases. By increasing the number of cells in

the mesh, the computational coast increases, not only because of the increased number of

computational points in space, but also due to the temporal discretization that decreases – if

32

dt is adaptive in the numerical scheme - to fit with a numerical stability criterion (CFL

criterion, see part 2, section 2.2.1).

Figure 1.6. Illustration of structured (left) and non-structured (right) meshing.

In 2D free surface modelling, the different types of mesh -structured, non-structured and

adaptive- are used in industrial codes.

Structured computational grids, such as the commonly used Cartesian structured mesh,

have the main advantages that they are often easy to use. Indeed, the DEM representing the

domain can be almost straight forwardly used as a computational grid (assuming that the

33

DEM representing the domain is considered as already suited for the hydraulic modelling

application). For practical application degradation or resampling of a HR DEM can occur due

to limitations in computation resources or in data handling.

Two main disadvantages arise from the use of structured mesh. First the regular size of

elements implies that the highest mesh resolution one can expect from the discretization

procedure is the same over Ω. Hence, in areas where the physical properties of the

phenomena wished to be modelled, or where the variable (elevation) does not vary, there will

be an unnecessary over-discretization of Ω. Consequently it will involve a high

computational cost along with the storage of potentially unnecessary information. Second,

disadvantage of a structured Cartesian mesh is that if the flow or any singularity is orientated,

in the worst case, plus or minus 45° compared to the x or y direction, the computation will

artificially go through a stepwise zig-zag processing (Ma et al., 2015). As a result, the

number of cells, and therefore the length over which the water will flow, is artificially

increased by this process.

Non-structured computational grids rely on a set of computational points that constitute

the set of cells that all have the same shape (most commonly triangles in 2D), but that have

variable sizes (and therefore variable areas). Most common practice is to generate first a

plane mesh according to x and y directions. This process requires to provide vectorial spatial

information such as polygons, lines or points over the domain where the modeller wants the

mesh to be refined. Then z values from the DEM are applied to the mesh nodes. Another

approach, offers the possibility to direcly give criterions such as z gradient from the DEM for

mesh generation and refinement.

Compared to structured meshes, the flexibility regarding mesh cell size allows to decrease

the number of computational points where there is no need for an accurate discretization of

the variable (e.g. areas where elevation is almost constant) or where averaging assumptions

are estimated to be fair. Automatic methods for non-structured mesh generation are

reviewed in Löhner (1997) and Owens (1998). As generalized in Löhner (1997), an automatic

non-structured grid generator requires:

description of the bounding surface and of the domain to be gridded;

description of elements to be generated (nature, size, shape, orientation, growing

ratio criterion);

grid generation techniques.

Most commonly used grid generation techniques in 2D hydraulic non-structured mesh

generation rely on, advancing front method (George and Seveno 1994), Delaunay and

34

constrained Delaunay triangulation methods (Weatherill, 1992), or hybrid techniques. These

techniques will not be reviewed here but it is interesting to mention that a limit of these

meshing algorithms is that they are not well suited for over-constrained domain mesh

generation. These meshing techniques are implemented in commercial and commonly used

codes mesh generator (Mike 21 mesh generator DHI (2007b)) and BlueKenue for TELEMAC

CHC (2010), see conclusion of this chapter).

Adaptive Mesh Refinement (AMR) is a discretization that evolves with space and time. The

aim of this type of approach is (i) to reduce computational time by optimizing number of

computational points to numerical constraints related to flow properties (e.g. CFL) and (ii) to

improve accuracy of the solution. Two main types of approaches are used in 2D SWEs

based flood modelling.

Block-structured adaptive mesh refinement. This type of approach is a nesting of

multiple levels of evolving patches of structured sub-grids that are pre-generated.

Coarse grid or finer nested sub-grids are used depending on flow dynamic properties

as these properties can impact numerical aspects in the solution computation. The

patchwork of grids is user-chosen pre-specified refinement ratios. A modern and well-

described method of AMR applied over the well documented Malpasset dam break

case can be found in Georges (2011).

The other approach is a sequence of grid operations that re-generate the non-

structured mesh during the computation, again depending on flow dynamic

properties. Main steps in mesh regeneration are: (i) node movement, (ii) edge

splitting, (iii) edge collapsing, and (iv) node movement (Tam, 2000).

1.4 FEEBACK FOR HR TOPOGRAPHIC DATA USE IN 2D URBAN FLOOD

MODELLING

This first part of chapter 1 introduced the concept of HR topographic datasets and the spatial

discretization processes that will influence both, possibilities and accuracy of HR topographic

data inclusion within flood models. As a reminder, goals of the research presented in this

thesis is to develop a method and good practices for High-Resolution (HR) topographic data

use (T1) and to focus on uncertainties related to HR topographic use and inclusion in 2D

flood models (T2).

Within T1 framework it is set that for the spatial extent of our applications of interest, namely

urban and industrial sites HR 2D overland flow modelling, LiDAR or photogrammetry

technologies settled on an aerial vector are the best suited to gathered HR topographic

datasets. As enhanced in this part, qualitative difference between LiDAR and

35

Photogrammetric based HR datasets rely in the interpretation/classification possibilities that

are more important in photogrammetry. Photo-interpreted dataset offers a broader range of

possibilities for HR DEM design, in accordance with descriptions of the above-ground

structure that will influence overland flow. Indeed classification of above-ground features

being more extensive in photo-interpreted datasets, it will allow hydraulic modeller to design

its HR DEM having a control on which elevation information should be included in it. This is

especially relevant for complex environment such as urban and industrial sites, where an

important diversity of above-ground elements exists. These techniques are sometimes used

in a combinatory way to gather HR datasets in urban areas (Zhou et al., 2004; Abdullah et

al., 2012). LiDAR and photo-interpreted datasets will be tested in our study in chapter 2.

Moreover, HR topographic datasets errors have been briefly introduced in this chapter and

within T2 framework, will be detailed in chapter 3 in order to compare impact of errors in HR

topographic dataset and modeller choices in HR topographic data integration effects on flood

modelling results.

Structured and non-structured approaches are selected as other discretization strategies

(AMR) are not commonly used in practical applications (reader can refer to Georges (2011)

or Unterweger (2015) for applied research cases using AMR). Structured and non-structured

meshing processes will be tested to assess if they offer the same possibilities for HR

topographic data integration within the 2D hydraulic codes (chapter 2). Idea is to compare

performance of these two discretization strategies in terms of accuracy of HR urban flood

models building. Moreover, ease of use and computational efficiency will be regarded as

well.

Preconceived idea is that photo-interpreted dataset might be efficient for non-structured

mesh generation as the data is vectorialized and should offer interesting possibilities for non-

structured mesh generation. Another idea arising from the theoretical background would be

the assumed advantage of non-structured grid compared to structured ones. Case study

studies in chapter 2 will illustrate that these preconceived assumptions are not confirmed.

36

PART. 2. NUMERICAL MODELLING OF FREE SURFACE FLOW:

APPROXIMATING SOLUTION OF SWES

2.1 FROM HYPOTHESIS IN THE PHYSICAL DESCRIPTION OF THE PHENOMENA

TO MATHEMATICAL FORMULATION

2.1.1 From flow observation to de Saint-Venant hypothesis and

mathematical formulation

Observation of channel flow to de Saint-Venant hypothesis

In nature, examples of free surface water flow complexity are observable and numerous (e.g.

flood event, runoff over urban area, etc.). In parallel there are needs for humans to use water

resources and to protect themselves from flood hazard resulting from natural intense events.

Engineering interest in knowing water stage and discharge along a given canal reach has

conducted Barré de Saint-Venant to formulate a simplification framework from observation of

flow behaviour which lead to an idealistic situation or concept where flow behaviour can be

described and understood for practical perspectives (de Saint-Venant, 1871).

As reminded in Cunge (2012), basics behind the simplified idealistic situation is to switch

from local detailed scale to a more macroscopic (several hundred meters) one. Then, at such

a scale the only forces which are considered are gravity, inertial and resistance forces.

Therefore, simplification introduced by de Saint-Venant are that (i) the water surface is the

same over one cross section, (ii) it can be considered that flow has one privileged direction

and that the flow velocity is the same over one vertical, (iii) hydrostatic pressure hypothesis

and (iv) energy losses can be represented using empirical formula (Chézy like formulas).

Originally, validity of this simplified framework is for a flow along an inclined channel of

constant slope and cross sections.

Shallow Water Equations

Laws of mechanics can be summed up as three principles: (i) mass conservation, (ii)

variation of momentum and (iii) total energy conservation. Applying above mentioned

hypothesis to mechanics laws, lead to the Shallow Water Equations system (SWEs) eq. (1).

Writing equations system in one dimension over a control volume included between two

rectangular cross sections separated by the distance dx for a given time interval dt; we have:

37

(1)

where: g is the acceleration of gravity constant, h(t, x) the water depth and u(t, x) the mean

flow velocity. The system of equations of Partial Differential Equations (PDEs) expressed in

eq. (1) does not consider any source terms (no friction included here and no variation of

topography) and is therefore called a homogeneous writing of the system. Adding source

terms eq. (2), the SWEs is called non-homogeneous system and writes as follow in 2D:

(2)

where, the unknowns are the velocities vector components u(x, y, t), v(x, y, t) [m/s] and the

water height h(x, y, t). The subscript x (respectively y) stands for the x-direction (respectively

y-direction).

and

are the ground slopes and S

fx

and S

fy

are

the friction terms. Component I of the momentum equations is the time evolution, II is the

convection term, III is the hydrostatic pressure, IV is the transversal component (in 2D only)

and the source term V includes the slope and the energy loses related to resistance (friction)

against channel boundaries.

Analytical solutions of this system of equations exist only for a few theoretical cases where

initial and boundary conditions are known (e.g. SWASHES library compiling a couple of 1D

and 2D theoretical cases, see (Delestre et al., 2013)) or in case of backwater curve

occurrences. Nevertheless for cases of flood event which are of prime interest for

practitioners, no general analytical solution exists. Indeed, the perfect knowledge of

information of initial and boundary conditions can only be assumed or approached in applied

natural cases. Therefore, from a mathematical point of view, the exact solution of this system

cannot be obtained in such a context. Consequently, the exact solution can only be

approximated with a numerical method.

As it will be explained in section 1.2, the numerical resolution of SWEs system can be

computationally resource demanding. Simplified versions of the SWEs system exist and are

made from simplifying hypothesis regarding terms in the momentum equation of the SWEs

system. Most commonly used approximate models are the kinematic waves, where the

momentum equation (from eq. (2)) is reduced to the expression of the term V, and the

38

diffusive wave approximation, where the momentum equation then is reduced to termes

III+V. Underlying simplification assumptions are restricting the applicability domain of the

arising models, but these simplified models are suited for specific problems (Moussa and

Bocquillon, 2000):

o kinematic wave, is suited to represent flow transfer in condition where

changes in steep slope condition, but fails in case of flat or inverse slope

conditions. The Kinematic wave approach is often used in hydrology for water

transfer conceptualization modelling e.g. in HEC-HMS code (USACE-HEC,

2000).

o Diffusive wave approximation is commonly used for computation when inertial

terms effects are negligible with respect to gravitational, friction and pressure

terms in case of supercritical flow for instance. Hydrological applications at a

catchment scale can be based on this approach that can be coupled with a full

resolution of the SWEs in the river bed e.g. in Mike SHE code (Abbott et al.,

1986; DHI, 2007).

Properties of the SWEs system and concept of friction for energy dissipation

Conservation and hyperbolicity are two of the main properties of the SWEs system as

explained below. The SWEs system can be written in one dimension under vectorial

conservative form in an Euclidian space as follow:

with (3)

where U is the vector of conservative variables, F(U) the flux, S(U, t, x) the vector of source

terms, R is rainfall source term and I is infiltration source term. This system writes also under

the form:

(4)

with F’(U) being the Jacobian matrix of F(U):

39

.

Showing that when the water depth is not null, F’(U), is diagonizable according to

eigenvalues:

and

. These eigenvalues are the propagation

velocities of surface waves. In a surface flow, the surface waves will propagate differently

depending on the ratio between gravitational and inertial force (Froude number). The Froude

number is defined as follow:

(5)

where u is the velocity (m.s

-1

), g the acceleration of the gravity constant and h the water

depth. Figure 2.1.1 illustrates

o If Fr>1, gravitational force drives the flow properties. Flow regime is qualified

as supercritical (or torrential). In this type of flow, surface waves (λ

1

and λ

2

)

follow the flow direction. We have an upstream control.

o If Fr<1, the mass of the flow drives the flow properties. Flow regime is

qualified as subcritical (or fluvial). In this situation, λ

1

and λ

2

move in both

upstream and downstream directions. We have a downstream control.

o If Fr=1, flow regime is qualified as critical. A flow cannot be critical over a

domain. It can be sub-critical and then it becomes supercritical through a

critical point. In that case the flow is said to be transcritical. It often happens

over a weir or through a Venturi. After a transcritical flow, we might also have

energy dissipation through a hydraulic jump (Figure 2.1). Through a hydraulic

jump, flow becomes subcritical.

Figure 2.1.1. Illustrates the flow regime changes where, as previously mentioned,

eigenvalues sign changes depending on the type of flow: subcritical, critical or supercritical.

Key points regarding the physical meaning of the celerity, or wave propagation speed, is

summarized by Guinot (2012) as follow: ”The celerity is the speed at which the variations in

U propagate. A perturbation appearing in the profile of U at a given time propagates at speed

of λ. The celerity can be viewed as the speed at which “information”, or “signals” created by

variations in U, propagates in space”. For someone moving at speed λ, U is invariant

(Riemann invariant), herby reducing the PDEs to ordinary differential equations that are

curves along which the perturbation propagates (characteristic curves).

40

Figure 2.1. Representation of possible eigenvalues/surface waves direction of propagation depending on the

flow regime.

2.1.2 Validity and limits of these hypothesis

Initial and Boundary conditions

From a mathematical point of view the solution of the SWEs can be approximated over a

calculation domain of finite length only if the problem is well-posed. Well-posed problem

requires that the solution exists, is unique and that the initial condition that is a "function of

the solution" over the domain at time t=0 is known. Moreover one boundary condition has to

be specified for each characteristic that enters the domain at the boundaries during the

whole time of calculation (Cunge, 2012; Guinot, 2012). The number of characteristics

entering the domain is function of the sign of the eigenvalues that depends on the flow

regime. If the flow is supercritical (we have an upstream control), both eigenvalues are

imposed upstream. If the flow is subcritical one is imposed upstream and the other one

downstream.

Beside for simple cases (e.g. canal or backwater curve influence), in real practical cases with

the objective to assess flood event extent in 2D, these conditions are seldom fully achieved,

due to incomplete knowledge of these initial and boundary conditions.

Transcritical flow occurrences lead to a division of the solution domain in two subdomains

separated by a stationary discontinuity. Indeed, as summarized in Sart (2010), transcritical

condition leads to sign change in the slowest eigenvalue leading to a so called shock speed.

As mentioned in previous section, hyperbolic properties of 2D SWEs allow discontinuous

41

solutions such as hydraulic jump (Hervouet, 2007), but then a Riemann problem occurs

(Guinot, 2012).

Parametrisation of energy losses

In the SWEs system, S

f

represents energy losses which are assumed to represent energy

dissipation (turbulence). Originally S

f

is considered using Chézy empiric formula (eq. 6) or its

derivates such as Manning formula (eq. 7). Therefore what has been conceptualized in the

SWEs system is energy losses related to resistance (friction) against channel boundary.

, (6)

, (7)

, (8)

where u is the flow mean velocity, k is the Chézy coefficient, n is the Manning coefficient, R

is the hydraulic radius, A is the wetted area, P is the wetted perimeter and S the slope. It has

to be emphasized here that this empirical formulation of energy losses introducing one

parameter in the SWEs system has been found to be empirically valid for steady-state flow

over experimental channel.

As a partial conclusion, it is impossible to exactly solve the SWEs but only in the best case to

approximate solution of the system if, the system is well posed, to guaranty from a

mathematical point of view condition of existence of the solution. In fact, in practical cases

the boundary and initial conditions are not well known, furthermore important topographic

gradient occurs, and wet/dry of cells in 2D overland flow simulations are frequent. These

aspects might lead to issues at least in the conservation aspects.

2.2 NUMERICAL METHODS TO APPROACH SOLUTION OF THE SWES SYSTEM

Numerical approaches to solve the set of PDE constituting the SWEs system are numerous

(see reviews in Toro et al.,1994; Bouchut, 2004; Hervouet, 2007; YU-E, 2007; Novak et al.,

2010; Guinot, 2012). Aim of this section is modestly to introduce concepts of most commonly

used numerical approaches in standard codes. In the context of this thesis this will help

reader not familiar with these concepts to understand them and their limitations.

42

2.2.1 Introduction to concepts for a numerical method

To approach the SWEs (eq. (1) and (2)) system solution, it is required to use numerical

methods which allow to reach an approximate numerical solution (Cunge, 2012). Objectives

of numerical method step is to approximate the set of PDE as faithfully as possible by a

system of equations, where the unknowns are the values of hydraulic variables in a finite

number of points (nodes) of the studied domain, and in a finite number of instants during the

considered period of time (spatial and temporal discretization). It has to be reminded that

numerical models, whichever would be the numerical approach that will be used, can only, in

the best case, approximate solution of the original equation. This is related to the

discretization and to the incomplete knowledge of the spatio-temporal variation of boundary

conditions as mentioned in previous section (2.1).

Then, available methods are: central/semi-implicit, forward/explicit and backward/implicit in

space and/or in time.

With central methods (res. semi-implicit methods), solution at a point x

i

(resp. at a

time t

n

) is calculated from points x

i+1

and x

i-1

, to find solution in x

i

, (resp. from times t

n+1

et t

n-1

to find solution at t

n

).

With forward methods (resp. explicit method), solution at x

i

(resp. at t

n

) is calculated

from solution at points x

i-1

(resp. t

n-1

). The numerical solution is then calculated going

"forward" in space and/or time.

With backward methods (resp. implicit methods), solution at x

i

(resp. at t

n

) is

calculated from solution at points x

i+1

(resp. at t

n+1

) that are still not known. The

numerical solution is then calculated going "backward" in space and/or time.

Properties of a numerical scheme

A numerical scheme is defined as a combination between a choice in the equations, a choice

in the discretization strategy and a choice of a numerical method. The application of a

numerical scheme should lead to the treatment of the Partial Differential system of Equations

(PDEs). In order to ensure the efficiency of a numerical scheme as illustrated in figure 2.2,

following properties have to be verified (Lax and Richtmyer, 1956):

Conservation: a numerical scheme has to conserve physical quantities such as

mass and momentum.

Consistency: a finite differences scheme or operation is consistent if the scheme

reduces to the original differential or partial differential equations as the increment in

the independent variables vanish (dx

0 and dt 0). The difference between the

discretized equation and the original equation is called truncation error.

43

Stability: a stable numerical scheme prevents unlimited growth of numerical errors

during computation. This property commonly implies important restrictions on the CFL

condition, as the stability of the scheme often depends on it (CFL restriction is then a

necessary condition but not sufficient to insure existence of stability). There are

different ways to study the stability of a numerical method: e.g. a Von

Neumann/Fourier stability analysis which is based on a Fourier decomposition of the

the numerical error. The stability can also be studied considering the positivity

preservation of some variables (such as water height, a pollutant concentration, etc.).

A checking of the TVB (Total Variation Bounded) can also be performed to control

that the overall amount of oscillation remains bounded. Most of the stability criteria

are equivalent and conduct to the CFL condition.

Convergence: the discrete solution U approaches the exact solution U (x, t) of the

differential equation at every point and time of the space when dx

0 and dt 0.

The equivalence theorem of Lax (Lax and Richtmyer, 1956) states that for a correctly

posed initial value problem and a consistent discretization, stability is a necessary

and sufficient condition for convergence.

Figure 2.2. Schematic view of necessary properties of a numerical scheme.

The numerical method can introduce mathematical terms not originally present in the

equations, which are terms introducing numerical diffusion and/or dispersion. Numerical

diffusion and dispersion phenomena will respectively smooth and create spurious oscillations

in the numerical solution (as shown in figure 2.3).

As explained for the convergence, a numerical result can be improved by increasing the

number of cells and thus by decreasing the space step along with the time step. However, it

is not always feasible due to lack of data, to important CPU cost, etc. Thus, the convergence

can be increased and the truncation error decreased by increasing the order of the numerical

method.

, (8)

44

where is the truncation error and then the scheme is said to be of order in time and of

order in time. There are different ways to increase the order of a scheme which will depend

on the type of scheme used (finite differences, finite elements or finite volumes).

As previously mentioned, without the source term (homogeneous system), the SWEs

system, can be written as a system of two transport equations where the transport velocities

are the eigenvalues

and

(where u is the velocity of the fluid, and

the wave velocity). Depending on the flow regime, the surface waves can go either

upstream and downstream (subcritical or fluvial flow) or both can go downstream

(supercritical or torrential flow). Therefore, all the information in case of a torrential flow go

downstream whereas, in a fluvial flow, information goes both upstream and downstream.

Numerical schemes can be built in order to detect the direction (sign) of the characteristics to

be purposely decentered (backward or forward) depending on where the information is

coming from. This category of numerical scheme is called upwinded.

Figure 2.3. Effects of numerical diffusion (up) and numerical dispersion (down) over profile under convection

(from Guinot, 2005).

2.2.2 Standard numerical methods

Various numerical methods exist to approximate the solutions of the SWEs. The most

commonly used methods, namely finite differences, finite volumes and finite elements

methods, are introduced in this section.

Finite differences

Finite differences method is a numerical technique built to approximate solutions of PDE. A

finite space of grid functions is defined and equations of the continuous function are

45

converted to algebraic equations (using Taylor series development or the definition of the

derivative). It results in a discretized form of the SWEs system. A relationship equation or

system (numerical scheme) linking the values of the unknowns at the considered discrete

points (close enough) is solved using computers algorithm. This numerical method has been

historically the first numerical method to be used in hydraulics due to its stability (under

conditions, see below), its robustness and simplicity. Moreover in terms of practical

implementation, finite differences schemes allow swift computation due to the simplicity of

matrix manipulations that are often diagonalizable (e.g. Abbott-Ionescu or Preissmann

schemes). Drawback of finite differences method is, as it will be explained later, that

linearized approaches have difficulties to treat discontinuities in the solution domain that

occur in case of transcritical flow followed by hydraulic jump, due to the inherent hyperbolic

nature of the SWEs system.

Designed finite differences schemes have to produce a well-posed problem. Depending on

the scheme, it would require different number of points for computation and different

numbers of discretized equations have to be provided along with the correct number of

boundary conditions to close the problem. Illustration can be given with the space centered,

time implicit Preissman scheme developed in the 60’s for hydraulic applications (Cunge and

Wegner, 1964; Cunge, 1966; Cunge et al., 1980). Under fully explicit writing, the values of

the two unknowns flow variables of the SWEs (e.g. h and u under non-conservative form) at