Analyse de sensibilité de modèles spatialisés - application à l'analyse coût-bénéfice de projets de prévention des inondations

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Nathalie Saint-Geours

PhD thesis defended on November 29, 2012

Supervised by: Christian Lavergne Jean-Stéphane Bailly – Frédéric Grelot

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Introduction

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Introduction The Orb case-study Spatial inputs Scale issues General conclusion

Spatially distributed models

Numerical models

= blackbox

to describe, understand, support decision making, forecast

Nov. 29th 2012 Sensitivity analysis of spatially distributed models 2 / 55

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Numerical models

= blackbox

spatially distributed outputs

numerical model inputs

outputs

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Numerical models

= blackbox

numerical model inputs

outputs

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Numerical models

= blackbox

Spatially distributed models spatially distributed inputs

numerical model

outputs

other inputs

spatial inputs +

u1=8

u3=5 u4=2.3 u2=0.1

u5=-3

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Numerical models

= blackbox

Spatially distributed models spatially distributed inputs spatially distributed outputs

numerical model

other inputs

spatial inputs +

u1=8

u3=5 u4=2.3 u2=0.1

u5=-3

spatial output

(8)

Numerical models

= blackbox

numerical model

other inputs

spatial inputs +

u1=8

u3=5 u4=2.3 u2=0.1

u5=-3

spatial output

spatial scale issues !

(9)

Numerical models

= blackbox

numerical model

other inputs

spatial inputs +

u1=8

u3=5 u4=2.3 u2=0.1

u5=-3

spatial output

spatial scale issues ! point output

(10)

Numerical models

= blackbox

numerical model

other inputs spatial inputs +

u1=8

u3=5 u4=2.3 u2=0.1

u5=-3

spatial output

spatial scale issues ! point output

aggregated output

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Uncertainty in spatial modelling

All models are wrong, some are useful (G. Box)

Many sources of uncertainty

lack of knowledge natural variability measurement errors model assumptions. . .

Spatial uncertainty

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Spatial uncertainty

spatial dependence spatial scale issues

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Spatial uncertainty

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Spatial uncertainty

spatial structure of uncertainty

spatial scale issues

z p(z)

spatial structure of uncertainty

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Spatial uncertainty

spatial structure of uncertainty spatial dependence

z p(z)

spatial dependence

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Spatial uncertainty

z p(z)

scale issues

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Spatial uncertainty

what impact on the uncertainty of model outputs?

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Sensitivity analysis

How the uncertainty of a model output can be apportioned to different sources of uncertainty in the model inputs

Model

Inputs Output

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Model

Inputs Output

Modelling uncertainty on model inputs

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Model

Inputs Output

Propagating uncertainty

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Model

Inputs Output

Resulting uncertainty on model output

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Model

Inputs Output

Ranking sources of uncertainty (sensitivity indices)

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Ranking sources of uncertainty (sensitivity indices) Sensitivity analysis is not model validation.

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Ranking sources of scalar uncertainty

(sensitivity indices)

Resulting scalar uncertainty

on model output Propagating

scalar uncertainty

Modelling uncertainty on scalar inputs

for scalar models only

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Ranking sources of spatial uncertainty

Resulting spatial uncertainty

spatial uncertainty

Modelling uncertainty on spatial inputs

what about spatial models?

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Research questions

investigate the use of sensitivity analysis

for spatial models

VB-GSA: Variance-Based Global Sensitivity Analysis (Sobol’ 1991)

low CPU models

2 research questions

1 how to compute sensitivity indices forspatial inputs?

2 how to account forscale issueswith VB-GSA?

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Research questions

Research goal

for spatial models

(Sobol’ 1991) low CPU models

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Research questions

Research goal

for spatial models

Scope

low CPU models

questions 2 how to account forscale issueswith VB-GSA?

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Research goal

for spatial models

Scope

low CPU models

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An inductive approach

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One case-study

flood risk model on the Orb river floodplain

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One case-study

first tests of sensitivity

analysis

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One case-study

analysis

methodological problems &

observations

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One case-study

analysis

observations

Theoretical framework

to explain observations to elaborate methods

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One case-study

analysis

observations

to explain observations to elaborate methods new properties

of sensitivity indices

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One case-study

analysis

observations

to explain observations to elaborate methods new properties

of sensitivity indices

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One case-study

analysis

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One case-study

analysis Inputs ^ModelOutput Modelling

uncertainty on model inputs

Estimating sensitivity indices

preliminary step:

delimit the model under study

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One case-study

preliminary step:

but: initial model not fully suitable to our needs

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One case-study

preliminary step:

but: initial model not fully suitable to our needs better describe the model

(not presented)

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investigate the use of sensitivity analysis for spatial models

VB-GSA: Variance-Based Global Sensitivity Analysis low CPU models

1 how to deal withspatial inputsin VB-GSA?

Applied goals

design a flood risk model on the Orb floodplain

perform sensitivity analysis

Focus on

maps of flood damages at different scales

uncertainty in spatial data

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Research goal

Scope

Applied goals

Focus on

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Applied goals

Focus on

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Outline of the presentation

2 The Orb case-study

3 Variance-based sensitivity indices for spatial inputs

4 Spatial scale issues

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The Orb case-study

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The Orb floodplain (Hérault)

(required by project funders - EU Floods Directive)

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−→ assessment of flood damage reduction

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Terrain elevation

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Terrain elevation Hazard map (water depths)

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Terrain elevation Elements at risk

(landuse map) Hazard map (water depths)

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Depth-damage functions

+

water depth

€

corn vineyard house

damage cm

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+

water depth

€

damage cm

numerical model

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+

water depth

€

damage cm

numerical model

Expected damages (for one flood event)

€

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(landuse map)

+

water depth

€

damage cm

numerical model

€

Hazard maps

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(landuse map)

+

water depth

€

damage cm

numerical model

€

Hazard maps

Frequencies of flood events

30 years 50 years 100 years

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(landuse map)

+

water depth

€

damage cm

numerical model Hazard maps

Frequencies of flood events

30 years 50 years 100 years

Expected annual damages Total : 12.2 M€ /year

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numerical model

Expected annual damages Total : 12.2 M€ /year Terrain

elevation Elements at risk

(landuse map)

+

numerical model

Hazard maps Frequencies of

flood events

water depth

€

damage cm

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(landuse map)

+

numerical model

flood events

Before After

water depth

€

damage cm

12.2M€ /y 5.7M€ /y

After Before

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(landuse map)

+

numerical model

flood events

water depth

€

damage cm

12.2M€ /y 5.7M€ /y

After Before

Annual avoided damages (ΔAAD)

6.5 M€/y

- =

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Orb case-study: model output

Map ofexpected annual avoidedflood damages

What spatialsupport (=aggregated area) for model output?

Orb river

Mediterranean sea ΔAAD [k€/year]

4 km N

1 0

< -6 -6 to -2 -2 to -0.1

-0.1 t o 0.1

4 to 6 > 8 0.5 to 2

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Orb river

4 km N

1 0

< -6 -6 to -2 -2 to -0.1

-0.1 t o 0.1

4 to 6 > 8 0.5 to 2

house of Mr Poujol 1.73 k€ /year

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Orb case-study: model output

Map ofexpected annual avoidedflood damages

Orb river

4 km N

1 0

< -6 -6 to -2 -2 to -0.1

-0.1 t o 0.1

4 to 6 > 8 0.5 to 2

221 k€ /year Sérignan district

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Orb river

4 km N

1 0

< -6 -6 to -2 -2 to -0.1

-0.1 t o 0.1

4 to 6 > 8 0.5 to 2

6523 k€ /year total floodplain

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Main characteristics of the Orb case-study

A spatially distributed model

∀ν ⊂R², Yν = Z

x∈ν

Y(x)dx

apoint-basedmodel:

∀x∈R², Y(x) =Floc(U,Z(x))

adeterministicmodel

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A spatially distributed model

a point-basedmodel:

a deterministicmodel

numerical model

other inputs Z1(x)

Z2(x)

Z3(x) U1 ... Uk

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A spatially distributed model a spatially additivemodel:

∀ν ⊂R², Yν = Z

x∈ν

Y(x)dx

numerical model

other inputs Z1(x)

Z2(x)

Z3(x) U1 ... Uk

aggregated output

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∀ν ⊂R², Yν = Z

x∈ν

Y(x)dx

numerical model

other inputs Z1(x)

Z2(x)

Z3(x) U1 ... Uk

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∀ν ⊂R², Yν = Z

x∈ν

Y(x)dx

numerical model

other inputs Z1(x)

Z2(x)

Z3(x) U1 ... Uk

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1^st research question

Variance-based sensitivity indices for spatial inputs

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Problem

Ranking sources of scalar uncertainty

Resulting scalar uncertainty

scalar uncertainty

Modelling uncertainty on scalar inputs

for scalar models only

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Ranking sources of spatial uncertainty

Resulting spatial uncertainty

spatial uncertainty

what about spatial models?

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Problem

Model

Inputs Output

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scalar input U

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scalar input U

probability density function

u

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scalar input U

u p(u)

random sample u⁽¹⁾ = 12.3 u⁽²⁾ = 4.5 u⁽³⁾ = 8.9 u⁽⁴⁾ = 7.4 u⁽⁵⁾ = 48.5 u⁽⁶⁾= 23 u⁽⁷⁾ = 8.5 u⁽⁸⁾ = 10.5

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scalar input U

u

random sample u⁽²⁾ = 4.5 u⁽³⁾ = 8.9 u⁽⁴⁾ = 7.4 u⁽⁵⁾ = 48.5 u⁽⁶⁾= 23 u⁽⁷⁾ = 8.5 u⁽⁸⁾ = 10.5

spatial input Z(x)

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scalar input U

u p(u)

random sample u⁽¹⁾ = 12.3 u⁽²⁾ = 4.5 u⁽³⁾ = 8.9 u⁽⁴⁾ = 7.4 u⁽⁵⁾ = 48.5 u⁽⁶⁾= 23 u⁽⁷⁾ = 8.5 u⁽⁸⁾ = 10.5

spatial input Z(x)

stochastic process

2D

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scalar input U

u

random sample u⁽²⁾ = 4.5 u⁽³⁾ = 8.9 u⁽⁴⁾ = 7.4 u⁽⁵⁾ = 48.5 u⁽⁶⁾= 23 u⁽⁷⁾ = 8.5 u⁽⁸⁾ = 10.5

spatial input Z(x)

stochastic process

2D

n random realisations

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grid of 5m resolution from aerial photography

measure & interpolation errors 2D Gaussian error field variogram model (estimated)

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Terrain elevation (DEM)

2D Gaussian error field variogram model (estimated)

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Modelling uncertainty

range a variogram model

h η

nugget σ²

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range a variogram model

h η

nugget σ²

n=100 random realisations of DEM geostatistical

algorithm

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Vector data

from remote sensing + field surveys

classification errors confusion matrix

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Landuse map

Vector data

confusion matrix

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Landuse map

Vector data

confusion matrix vineyard corn corn

vineyard 0.82 0.18 0.18 0.82

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Landuse map

Vector data

confusion matrix vineyard corn corn

vineyard 0.82 0.18 0.18 0.82

n=1000 random realisations of landuse map Monte Carlo

simulation

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Model

Inputs Output

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Model

Inputs Output

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What uncertainty should be propagated?

uncertainty on the set of random realisations (fixedP2D) additional uncertainty on stochastic processP2D (second-level)

stochastic process

2D

n random realisations of spatial input Z(x)

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uncertainty on some map attributes

stochastic process

2D

map attributes (scalar descriptors)

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uncertainty on the set of random realisations (fixedP_2D)

stochastic process

2D

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uncertainty on the set of random realisations (fixedP_2D) additional uncertainty on stochastic processP2D (second-level)

stochastic process

2D

vineyard corn corn vineyard 0.82 0.18

0.18 0.82

a h

η σ²

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Benchmark of methods

7 methods reviewed

CPU cost? / many spatial inputs? / anyP_2D? / meta-modelling? main idea: spatial input Z(x)→somescalar parameters

stochastic process

2D

vineyard corn corn vineyard 0.82 0.18

0.18 0.82

a h

η σ²

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Benchmark of methods 7 methods reviewed

Macro Parameter Dimension Reduction Map Labelling Joint Meta-model Trigger Input

Map Attributes Second

Level

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7 methods reviewed +comparisonon analytical test cases:

CPU cost? / many spatial inputs? / anyP_2D? / meta-modelling?

Level

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main idea: spatial input Z(x)→somescalar parameters

Level

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main idea: spatial input Z(x)→somescalar parameters

Level

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