# Probabilistic Models for the Uncertain Hydrologist

Benjamin Renard

► To cite this version:

Benjamin Renard. Probabilistic Models for the Uncertain Hydrologist. 2024. hal-04745620

HAL Id: hal-04745620

https://hal.inrae.fr/hal-04745620v1

Submitted on 21 Oct 2024

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Distributed under a Creative Commons Attribution 4.0 International License
PROBABILISTIC MODELS FOR THE
UNCERTAIN HYDROLOGIST

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**The Water Cycle**
Water moves around our planet by the processes shown here. The water cycle shapes landscapes, transports minerals, and is essential to most life and ecosystems on the planet.

Benjamin Renard

INRAE, UR RECOVER, Aix-Marseille University
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# Introduction

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crédit: Diego Delso
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# Introduction

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crédit: Diego Delso

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crédit: Aleda12
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# Introduction

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crédit: Diego Delso

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crédit: Irstea

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crédit: Aleda12
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# Introduction

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crédit: Diego Delso

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crédit: Irstea

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crédit: Aleda12

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crédit: Clicgauche
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# Introduction

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crédit: Diego Delso

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crédit: Irstea

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crédit: João Cautela

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crédit: Aleda12

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crédit: Clicgauche
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# Introduction

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crédit: Diego Delso

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crédit: Irstea

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crédit: João Cautela

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crédit: Aleda12

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crédit: Clicgauche

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crédit: RuB
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

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photo: M. Lagouy
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

Réchauffement marqué et augmentation des précipitations

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Changements futurs relativement peu marqués

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Fort réchauffement et fort contraste saisonnier en précipitations

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Fort réchauffement et fort assèchement en été (et en annuel)

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Changement (%)

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crédit: EXPLORE 2
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Diversity of situations

* From flood to drought
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Diversity of situations

* From flood to drought

* Present, past, future
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Diversity of situations

* From flood to drought

* Present, past, future

* From one site to thousands of catchments
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Common Threads

* Water flowing in rivers
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Common Threads

* Water flowing in rivers

* Estimating unknown quantities affected by uncertainty
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Common Threads

* Water flowing in rivers

* Estimating unknown quantities affected by uncertainty

* Uncertainty quantification relies on probabilistic models
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# Introduction

## Three Practical Questions

1. What is the discharge flowing in this river right now?

2. Was the great flood of 1910 in Paris a 100-year event?

3. How much water will flow in French rivers during the summers of the next 50 years, and how warm will it be?

## Common Threads

* Water flowing in rivers

* Estimating unknown quantities affected by uncertainty

* Uncertainty quantification relies on probabilistic models

## Methodological Motto

→ Developing **probabilistic models** for the **uncertain hydrologist**
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# Outline

## ① Uncertainty in Streamflow Data
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# Outline

1. Uncertainty in Streamflow Data

2. Uncertainty in and around Hydrologic Models
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# Outline

1. Uncertainty in Streamflow Data

2. Uncertainty in and around Hydrologic Models

3. Hydrologic Variability
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Uncertainty in streamflow data

La Bar

L'Aa

La Tinée
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# Production of River Discharge Series

* River discharge: a key variable for hydrology... that can't be measured continuously

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# Production of River Discharge Series

* River discharge: a key variable for hydrology... that can't be measured continuously

* But river stage can!

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# Production of River Discharge Series

* River discharge: a key variable for hydrology... that can't be measured continuously

* But river stage can!

* Rating curve: $Q = f(h)$

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# Production of River Discharge Series

* River discharge: a key variable for hydrology... that can't be measured continuously

* But river stage can!

* Rating curve: $Q = f(h)$

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→ How to formulate and estimate rating curves?

→ How to quantify and propagate uncertainties?
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# Rating Curve Formulation

Just fit some flexible curve!

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# Rating Curve Formulation

Just fit some flexible curve!

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# Rating Curve Formulation

Just fit some flexible curve!

1. Use hydraulics: $Q = a(h - b)^c$

with *a*, *b*, *c* related to physical quantities that can be measured (albeit uncertainly)

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# Rating Curve Formulation

Just fit some flexible curve!

1. Use hydraulics: $Q = a(h - b)^c$

with a, b, c related to physical quantities that can be measured (albeit uncertainly)

Rectangular weir

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Rectangular channel

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2. A 'control matrix' to combine multiple controls

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<table><thead><tr><th></th><th colspan="3">Riffle channel floodway</th><th></th></tr></thead><tbody><tr><th>low stage</th><td>1</td><td>0</td><td>0</td><td>a<sub>1</sub>(h - b<sub>1</sub>)<sup>c<sub>1</sub></sup></td></tr><tr><th>medium stage</th><td>0</td><td>1</td><td>0</td><td>a<sub>2</sub>(h - b<sub>2</sub>)<sup>c<sub>2</sub></sup></td></tr><tr><th>high stage</th><td>0</td><td>1</td><td>1</td><td>a<sub>3</sub>(h - b<sub>3</sub>)<sup>c<sub>3</sub></sup></td></tr></tbody></table>
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# Rating Curve Estimation

→ Bayesian Rating Curve estimation (BaRatin)
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# Rating Curve Estimation

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# Rating Curve Estimation

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Prior distribution $p(\kappa, a, c) = p(\theta)$

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Posterior distribution $p(\theta, \gamma | \tilde{h}, \tilde{Q})$

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# Rating Curve Estimation

→ Bayesian Rating Curve estimation (BaRatin)

Prior distribution $p(\kappa, a, c) = p(\theta)$

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Posterior distribution $p(\theta, \gamma | \tilde{h}, \tilde{Q})$

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Probabilistic model linking gaugings ($\tilde{h}_i, \tilde{Q}_i$) and RC $\hat{Q}_i = f_{RC}(h_i|\theta)$:
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# Rating Curve Estimation

→ Bayesian Rating Curve estimation (BaRatin)

Prior distribution $p(\kappa, a, c) = p(\theta)$

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Posterior distribution $p(\theta, \gamma | \tilde{h}, \tilde{Q})$

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Probabilistic model linking gaugings ($\tilde{h}_i$, $\tilde{Q}_i$) and RC $\hat{Q}_i = f_{RC}(h_i|\theta)$:

$$ \tilde{Q}_i = f_{RC}(\tilde{h}_i|\theta) + \delta_i + \varepsilon_i $$
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# Rating Curve Estimation

→ Bayesian Rating Curve estimation (BaRatin)

Prior distribution $p(\kappa, a, c) = p(\theta)$

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Posterior distribution $p(\theta, \gamma | \tilde{h}, \tilde{Q})$

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Probabilistic model linking gaugings ($\tilde{h}_i$, $\tilde{Q}_i$) and RC $\hat{Q}_i = f_{RC}(h_i|\theta)$:

$$ \tilde{Q}_i = f_{RC}(\tilde{h}_i|\theta) + \delta_i + \varepsilon_i $$

Measurement error $\delta_i \sim N(0, u_i)$; $u_i$ assumed known
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# Rating Curve Estimation

→ Bayesian Rating Curve estimation (BaRatin)

Prior distribution $p(\kappa, a, c) = p(\theta)$

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Posterior distribution $p(\theta, \gamma | \tilde{h}, \tilde{Q})$

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Probabilistic model linking gaugings ($\tilde{h}_i, \tilde{Q}_i$) and RC $\hat{Q}_i = f_{RC}(h_i|\theta)$:

$$ \tilde{Q}_i = f_{RC}(\tilde{h}_i|\theta) + \delta_i + \varepsilon_i $$

Measurement error $\delta_i \sim N(0, u_i)$; $u_i$ assumed known

Structural error $\varepsilon_i \sim N(0, \sigma_i)$; $\sigma_i = \gamma_1 + \gamma_2 \hat{Q}_i$ to be estimated
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# Operational Tool: BaRatinAGE

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# Operational Tool: BaRatinAGE

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# Operational Tool: BaRatinAGE

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Operational Tool: BaRatinAGE

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* A viable solution to estimate uncertain Q = f(h) rating curves...
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# Operational Tool: BaRatinAGE

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* A viable solution to estimate uncertain $Q = f(h)$ rating curves...

* ... which are the exception rather than the rule!
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Operational Tool: BaRatinAGE

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* A viable solution to estimate uncertain $Q = f(h)$ rating curves...

* ... which are the exception rather than the rule!

① rating shifts

② complex rating curves $Q = f(h, others)$
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# Rating Shift Happens!

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# Rating Shift Happens!

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→ Detecting rating shifts (M. Darienzo, F. Mendez)

From the stage time series

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# Rating Shift Happens!

- Detecting rating shifts (M. Darienzo, F. Mendez)

- Estimating shifting curves (V. Mansanarez)

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Rating Shift Happens!

→ Detecting rating shifts (M. Darienzo, F. Mendez)

→ Estimating shifting curves (V. Mansanarez)

<figure><img src="image_292.png" /></figure>

Hydraulic assumptions on what changes ☰ stable vs. varying RC parameters
<figure><img src="image_293.png" /></figure>

<figure><img src="image_294.png" /></figure>

<figure><img src="image_295.png" /></figure>

<figure><img src="image_296.png" /></figure>

<figure><img src="image_297.png" /></figure>

Complex Rating Curves $Q = f(h, ...)$

→ Vegetation model (E. Perret)

$Q = f(h, V)$ due to seasonal aquatic vegetation growth

<figure><img src="image_298.png" /></figure>
<figure><img src="image_299.png" /></figure>

<figure><img src="image_300.png" /></figure>

<figure><img src="image_301.png" /></figure>

<figure><img src="image_302.png" /></figure>

<figure><img src="image_303.png" /></figure>

# Complex Rating Curves $Q = f(h, ...)$

→ Vegetation model (E. Perret)

$Q = f(h, V)$ due to seasonal aquatic vegetation growth

As $V$ is not measured, $Q = f(h, \chi t)$

<figure><img src="image_304.png" /></figure>

(a) Vegetation cycle

<figure><img src="image_305.png" /></figure>

<figure><img src="image_306.png" /></figure>
<figure><img src="image_307.png" /></figure>

<figure><img src="image_308.png" /></figure>

<figure><img src="image_309.png" /></figure>

<figure><img src="image_310.png" /></figure>

<figure><img src="image_311.png" /></figure>

# Complex Rating Curves $Q = f(h, ...)$

→ Vegetation model (E. Perret)

$Q = f(h, V)$ due to seasonal aquatic vegetation growth

As $V$ is not measured, $Q = f(h, \chi t)$

Possible to incorporate qualitative information on the vegetation state.

<figure><img src="image_312.png" /><figcaption>(a) Vegetation cycle</figcaption></figure>

<figure><img src="image_313.png" /></figure>

<figure><img src="image_314.png" /></figure>

<figure><img src="image_315.png" /></figure>
<figure><img src="image_316.png" /></figure>

<figure><img src="image_317.png" /></figure>

<figure><img src="image_318.png" /></figure>

<figure><img src="image_319.png" /></figure>

<figure><img src="image_320.png" /></figure>

## From BaRatin to BaM

Replacing $Q = f(h)$ by another model does not really change the statistical framework used in BaRatin.

### Development of BaM (Bayesian Modeling)

'Any' model can be plugged in. Examples: sediment transport, optical camera model, chemistry, 1D hydraulics, hydrologic.

<figure><img src="image_321.png" /></figure>
<figure><img src="image_322.png" /></figure>

<figure><img src="image_323.png" /></figure>

<figure><img src="image_324.png" /></figure>

<figure><img src="image_325.png" /></figure>

<figure><img src="image_326.png" /></figure>

# Uncertainty in & around hydrologic models

<figure><img src="image_327.png" /></figure>
<figure><img src="image_328.png" /></figure>

<figure><img src="image_329.png" /></figure>

<figure><img src="image_330.png" /></figure>

<figure><img src="image_331.png" /></figure>

<figure><img src="image_332.png" /></figure>

# Bayesian Total Error Analysis

<figure><img src="image_333.png" /></figure>

$$ \hat{Q}_t = \mathcal{M}(\mathbf{r}_{1 \to t} | \theta) $$
<figure><img src="image_334.png" /></figure>

<figure><img src="image_335.png" /></figure>

<figure><img src="image_336.png" /></figure>

<figure><img src="image_337.png" /></figure>

<figure><img src="image_338.png" /></figure>

Bayesian Total Error Analysis

<figure><img src="image_339.png" /></figure>

$$ \tilde{Q}_t = \mathcal{M}(\mathbf{r}_{1 \to t} | \boldsymbol{\theta}) + \delta_t + \varepsilon_t $$
<figure><img src="image_340.png" /></figure>

<figure><img src="image_341.png" /></figure>

<figure><img src="image_342.png" /></figure>

<figure><img src="image_343.png" /></figure>

<figure><img src="image_344.png" /></figure>

Bayesian Total Error Analysis

<figure><img src="image_345.png" /></figure>

$$ \tilde{Q}_t = \mathcal{M}(\tilde{\mathbf{r}}_{1 \to t} \times \phi_{1 \to t}|\theta) + \delta_t + \varepsilon_t $$

$$ \phi_k \sim \mathcal{LN}(\mu, \sigma) $$
<figure><img src="image_346.png" /></figure>

<figure><img src="image_347.png" /></figure>

<figure><img src="image_348.png" /></figure>

<figure><img src="image_349.png" /></figure>

<figure><img src="image_350.png" /></figure>

Bayesian Total Error Analysis

<figure><img src="image_351.png" /></figure>

$$
\begin{gathered}
\tilde{Q}_t = \mathcal{M}(\tilde{\mathbf{r}}_{1 \to t} \times \phi_{1 \to t} | \theta, \lambda_{1 \to t}) + \delta_t + \varepsilon_t \\
\lambda_k \sim \mathcal{D}(\Lambda)
\end{gathered}
$$
<figure><img src="image_352.png" /></figure>

<figure><img src="image_353.png" /></figure>

<figure><img src="image_354.png" /></figure>

<figure><img src="image_355.png" /></figure>

<figure><img src="image_356.png" /></figure>

# Recognizing Data Uncertainty

* Horton catchment (1920 km²), Australia

* Calibrate M(θ) using rainfall data from either R1 or R2

* Compare inference schemes ignoring or acknowledging streamflow and rainfall data uncertainties (uQ and uR)

<figure><img src="image_357.png" /></figure>

<figure><img src="image_358.png" /></figure>
<figure><img src="image_359.png" /></figure>

<figure><img src="image_360.png" /></figure>

<figure><img src="image_361.png" /></figure>

<figure><img src="image_362.png" /></figure>

<figure><img src="image_363.png" /></figure>

Recognizing Data Uncertainty

<figure><img src="image_364.png" /></figure>
<figure><img src="image_365.png" /></figure>

<figure><img src="image_366.png" /></figure>

<figure><img src="image_367.png" /></figure>

<figure><img src="image_368.png" /></figure>

<figure><img src="image_369.png" /></figure>

## Recognizing Data Uncertainty

<figure><img src="image_370.png" /></figure>
<figure><img src="image_371.png" /></figure>

<figure><img src="image_372.png" /></figure>

<figure><img src="image_373.png" /></figure>

<figure><img src="image_374.png" /></figure>

<figure><img src="image_375.png" /></figure>

# Recognizing Data Uncertainty

<figure><img src="image_376.png" /></figure>

* Ignoring data uncertainties ⇌ parameters fit to data errors
  reliability of streamflow prediction? parameter regionalization?
<figure><img src="image_377.png" /></figure>

<figure><img src="image_378.png" /></figure>

<figure><img src="image_379.png" /></figure>

<figure><img src="image_380.png" /></figure>

<figure><img src="image_381.png" /></figure>

# Recognizing Data Uncertainty

<figure><img src="image_382.png" /></figure>

* Ignoring data uncertainties ⇌ parameters fit to data errors
  reliability of streamflow prediction? parameter regionalization?
<figure><img src="image_383.png" /></figure>

<figure><img src="image_384.png" /></figure>

<figure><img src="image_385.png" /></figure>

<figure><img src="image_386.png" /></figure>

<figure><img src="image_387.png" /></figure>

# Recognizing Data Uncertainty

<figure><img src="image_388.png" /></figure>

* Ignoring data uncertainties ⇌ parameters fit to data errors
  reliability of streamflow prediction? parameter regionalization?
<figure><img src="image_389.png" /></figure>

<figure><img src="image_390.png" /></figure>

<figure><img src="image_391.png" /></figure>

<figure><img src="image_392.png" /></figure>

<figure><img src="image_393.png" /></figure>

# Recognizing Data Uncertainty

<figure><img src="image_394.png" /></figure>

* Ignoring data uncertainties ⇌ parameters fit to data errors
  reliability of streamflow prediction? parameter regionalization?

* Acknowledging data uncertainties ⇌ parameters more stable
  vaguely correct rather than precisely wrong!
<figure><img src="image_395.png" /></figure>

<figure><img src="image_396.png" /></figure>

<figure><img src="image_397.png" /></figure>

<figure><img src="image_398.png" /></figure>

<figure><img src="image_399.png" /></figure>

# Structural Uncertainty through Stochastic Parameters

→ What do structural errors look like?
<figure><img src="image_400.png" /></figure>

<figure><img src="image_401.png" /></figure>

<figure><img src="image_402.png" /></figure>

<figure><img src="image_403.png" /></figure>

<figure><img src="image_404.png" /></figure>

# Structural Uncertainty through Stochastic Parameters

→ What do structural errors look like?

<figure><img src="image_405.png" /></figure>
<figure><img src="image_406.png" /></figure>

<figure><img src="image_407.png" /></figure>

<figure><img src="image_408.png" /></figure>

<figure><img src="image_409.png" /></figure>

<figure><img src="image_410.png" /></figure>

# Structural Uncertainty through Stochastic Parameters

→ What do structural errors look like?

<figure><img src="image_411.png" /></figure>

<figure><img src="image_412.png" /></figure>

<figure><img src="image_413.png" /></figure>

<figure><img src="image_414.png" /></figure>

* High heteroscedasticity, complex dependence structure

* Structural error model may be as complex as the model itself...
<figure><img src="image_415.png" /></figure>

<figure><img src="image_416.png" /></figure>

<figure><img src="image_417.png" /></figure>

<figure><img src="image_418.png" /></figure>

<figure><img src="image_419.png" /></figure>

Structural Uncertainty through Stochastic Parameters

* Allow $\theta_1$ to vary in time according to : $log(\theta_{1,k}) \stackrel{iid}{\sim} N(\mu, \sigma)$
<figure><img src="image_420.png" /></figure>

<figure><img src="image_421.png" /></figure>

<figure><img src="image_422.png" /></figure>

<figure><img src="image_423.png" /></figure>

<figure><img src="image_424.png" /></figure>

# Structural Uncertainty through Stochastic Parameters

* Allow $\theta_1$ to vary in time according to : $log(\theta_{1,k}) \sim iid N(\mu, \sigma)$

<figure><img src="image_425.png" /></figure>
<figure><img src="image_426.png" /></figure>

<figure><img src="image_427.png" /></figure>

<figure><img src="image_428.png" /></figure>

<figure><img src="image_429.png" /></figure>

<figure><img src="image_430.png" /></figure>

Structural Uncertainty through Stochastic Parameters

* Allow $\theta_1$ to vary in time according to : $log(\theta_{1,k}) \stackrel{iid}{\sim} N(\mu, \sigma)$

* Residuals are smaller and much simpler

<figure><img src="image_431.png" /></figure>
<figure><img src="image_432.png" /></figure>

<figure><img src="image_433.png" /></figure>

<figure><img src="image_434.png" /></figure>

<figure><img src="image_435.png" /></figure>

<figure><img src="image_436.png" /></figure>

Structural Uncertainty through Stochastic Parameters

* Allow $\theta_1$ to vary in time according to : $log(\theta_{1,k}) \stackrel{iid}{\sim} N(\mu, \sigma)$

* Residuals are smaller and much simpler

<figure><img src="image_437.png" /></figure>

→ Stochastic parameters allows using simple statistical models for structural uncertainty, at the cost of a demanding inference
<figure><img src="image_438.png" /></figure>

<figure><img src="image_439.png" /></figure>

<figure><img src="image_440.png" /></figure>

<figure><img src="image_441.png" /></figure>

<figure><img src="image_442.png" /></figure>

# Structural Uncertainty through Stochastic Parameters

* Allow $\theta_1$ to vary in time according to : $log(\theta_{1,k}) \stackrel{iid}{\sim} N(\mu, \sigma)$

* Residuals are smaller and much simpler

* In prediction, randomly sample $log(\theta_1)$ values from $N(\hat{\mu}, \hat{\sigma})$

<figure><img src="image_443.png" /></figure>

→ Stochastic parameters allows using simple statistical models for structural uncertainty, at the cost of a demanding inference
<figure><img src="image_444.png" /></figure>

<figure><img src="image_445.png" /></figure>

<figure><img src="image_446.png" /></figure>

<figure><img src="image_447.png" /></figure>

<figure><img src="image_448.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \boldsymbol{\lambda}_{1 \to t}) + \varepsilon_t$
<figure><img src="image_449.png" /></figure>

<figure><img src="image_450.png" /></figure>

<figure><img src="image_451.png" /></figure>

<figure><img src="image_452.png" /></figure>

<figure><img src="image_453.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \boldsymbol{\lambda}_{1 \to t}) + \varepsilon_t$

<figure><img src="image_454.png" /></figure>
<figure><img src="image_455.png" /></figure>

<figure><img src="image_456.png" /></figure>

<figure><img src="image_457.png" /></figure>

<figure><img src="image_458.png" /></figure>

<figure><img src="image_459.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \lambda_{1 \to t}) + \varepsilon_t$

<figure><img src="image_460.png" /></figure>

* main sources of predictive uncertainty ⇌ ways to reduce it
<figure><img src="image_461.png" /></figure>

<figure><img src="image_462.png" /></figure>

<figure><img src="image_463.png" /></figure>

<figure><img src="image_464.png" /></figure>

<figure><img src="image_465.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \lambda_{1 \to t}) + \varepsilon_t$

<figure><img src="image_466.png" /></figure>

* main sources of predictive uncertainty ⇌ ways to reduce it

* specific sources may be turned off or replaced
<figure><img src="image_467.png" /></figure>

<figure><img src="image_468.png" /></figure>

<figure><img src="image_469.png" /></figure>

<figure><img src="image_470.png" /></figure>

<figure><img src="image_471.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \boldsymbol{\lambda}_{1 \to t}) + \varepsilon_t$

## → Requirements:

* data uncertainties are known beforehand

If you don't know the quality of data, you won't know the quality of the model
<figure><img src="image_472.png" /></figure>

<figure><img src="image_473.png" /></figure>

<figure><img src="image_474.png" /></figure>

<figure><img src="image_475.png" /></figure>

<figure><img src="image_476.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \boldsymbol{\lambda}_{1 \to t}) + \varepsilon_t$

## → Requirements:

1. data uncertainties are known beforehand

   If you don't know the quality of data, you won't know the quality of the model

2. data uncertainties are not too large

   A minimal data precision may be necessary to estimate structural uncertainty
<figure><img src="image_477.png" /></figure>

<figure><img src="image_478.png" /></figure>

<figure><img src="image_479.png" /></figure>

<figure><img src="image_480.png" /></figure>

<figure><img src="image_481.png" /></figure>

# Decomposing Predictive Uncertainty

Putting everything together... $\hat{Q}_t = \mathcal{M}(\tilde{r}_{1 \to t} \times \phi_{1 \to t} | \theta, \lambda_{1 \to t}) + \varepsilon_t$

→ Requirements:

① data uncertainties are known beforehand

If you don't know the quality of data, you won't know the quality of the model

② data uncertainties are not too large

A minimal data precision may be necessary to estimate structural uncertainty

Otherwise, the problem is **ill-posed** due to **non-identifiable parameters**
<figure><img src="image_482.png" /></figure>

<figure><img src="image_483.png" /></figure>

<figure><img src="image_484.png" /></figure>

<figure><img src="image_485.png" /></figure>

<figure><img src="image_486.png" /></figure>

# Hydrologic Variability

<figure><img src="image_487.png" /></figure>

crédit: HydroSHEDS
<figure><img src="image_488.png" /></figure>

<figure><img src="image_489.png" /></figure>

<figure><img src="image_490.png" /></figure>

<figure><img src="image_491.png" /></figure>

<figure><img src="image_492.png" /></figure>

# Estimating Distributions

Managing water resources and risks often requires estimating distributions:

* Dam design: T-year quantile from the **marginal** distribution of streamflow

<figure><img src="image_493.png" /></figure>
<figure><img src="image_494.png" /></figure>

<figure><img src="image_495.png" /></figure>

<figure><img src="image_496.png" /></figure>

<figure><img src="image_497.png" /></figure>

<figure><img src="image_498.png" /></figure>

# Estimating Distributions

Managing water resources and risks often requires estimating distributions:

* Dam design: T-year quantile from the **marginal** distribution of streamflow

* Does climate change compromise dam design? **time-varying distribution**

<figure><img src="image_499.png" /></figure>
<figure><img src="image_500.png" /></figure>

<figure><img src="image_501.png" /></figure>

<figure><img src="image_502.png" /></figure>

<figure><img src="image_503.png" /></figure>

<figure><img src="image_504.png" /></figure>

# Estimating Distributions

Managing water resources and risks often requires estimating distributions:

* Dam design: T-year quantile from the **marginal** distribution of streamflow

* Does climate change compromise dam design? **time-varying distribution**

* Adapt dam operation to climate state: **conditional distribution** of streamflow given some predictor.

<figure><img src="image_505.png" /></figure>
<figure><img src="image_506.png" /></figure>

<figure><img src="image_507.png" /></figure>

<figure><img src="image_508.png" /></figure>

<figure><img src="image_509.png" /></figure>

<figure><img src="image_510.png" /></figure>

# Estimating Distributions

Managing water resources and risks often requires estimating distributions:

* Dam design: T-year quantile from the **marginal** distribution of streamflow

* Does climate change compromise dam design? **time-varying distribution**

* Adapt dam operation to climate state: **conditional distribution** of streamflow given some predictor.

* Large hydroelectricity companies ask this at the national scale: **space-and-time varying distribution**

<figure><img src="image_511.png" /></figure>
<figure><img src="image_512.png" /></figure>

<figure><img src="image_513.png" /></figure>

<figure><img src="image_514.png" /></figure>

<figure><img src="image_515.png" /></figure>

<figure><img src="image_516.png" /></figure>

# Estimating Distributions

Managing water resources and risks often requires estimating distributions:

* Dam design: T-year quantile from the **marginal** distribution of streamflow

* Does climate change compromise dam design? **time-varying distribution**

* Adapt dam operation to climate state: **conditional distribution** of streamflow given some predictor.

* Large hydroelectricity companies ask this at the national scale: **space-and-time varying distribution**

* Renewable energy production depends on the **joint distribution** of wind, solar radiation and streamflow.

<figure><img src="image_517.png" /></figure>
<figure><img src="image_518.png" /></figure>

<figure><img src="image_519.png" /></figure>

<figure><img src="image_520.png" /></figure>

<figure><img src="image_521.png" /></figure>

<figure><img src="image_522.png" /></figure>

# Estimating a Marginal Distribution

A simple statistical problem, plagued with sampling uncertainty.

Ex: estimate GEV($\mu, \sigma, \xi$) with 40 years of data.

<figure><img src="image_523.png" /></figure>
<figure><img src="image_524.png" /></figure>

<figure><img src="image_525.png" /></figure>

<figure><img src="image_526.png" /></figure>

<figure><img src="image_527.png" /></figure>

<figure><img src="image_528.png" /></figure>

# Estimating a Marginal Distribution

A simple statistical problem, plagued with sampling uncertainty.

Ex: estimate $GEV(\mu, \sigma, \xi)$ with 40 years of data.

A possible solution: use ancient data such as historical information, flood marks, old stage series, etc.

<figure><img src="image_529.png" /></figure>

① Hydraulics to translate this information in terms of streamflow.

② Adapted statistical methods (censoring, data uncertainties, etc.)
<figure><img src="image_530.png" /></figure>

<figure><img src="image_531.png" /></figure>

<figure><img src="image_532.png" /></figure>

<figure><img src="image_533.png" /></figure>

<figure><img src="image_534.png" /></figure>

# Estimating a Marginal Distribution

A simple statistical problem, plagued with sampling uncertainty.

Ex: estimate $GEV(\mu, \sigma, \xi)$ with 40 years of data.

A possible solution: use ancient data such as historical information, flood marks, old stage series, etc.

<figure><img src="image_535.png" /></figure>

1. Hydraulics to translate this information in terms of streamflow.

2. Adapted statistical methods (censoring, data uncertainties, etc.)

<figure><img src="image_536.png" /></figure>

(M. Lucas PhD)
<figure><img src="image_537.png" /></figure>

<figure><img src="image_538.png" /></figure>

<figure><img src="image_539.png" /></figure>

<figure><img src="image_540.png" /></figure>

<figure><img src="image_541.png" /></figure>

# Estimating a Time-Varying Distribution

→ Time-varying models

$$M_0: Y(t) \sim \text{GEV}(\mu, \sigma, \xi)$$

$$M_1: Y(t) \sim \text{GEV}(\mu \times (1 + \lambda x(t)), \sigma, \xi)$$
<figure><img src="image_542.png" /></figure>

<figure><img src="image_543.png" /></figure>

<figure><img src="image_544.png" /></figure>

<figure><img src="image_545.png" /></figure>

<figure><img src="image_546.png" /></figure>

# Estimating a Time-Varying Distribution

→ Time-varying models

$$M_0: Y(t) \sim \text{GEV}(\mu, \sigma, \xi)$$

$$M_1: Y(t) \sim \text{GEV}(\mu \times (1 + \lambda x(t)), \sigma, \xi)$$

<figure><img src="image_547.png" /></figure>

(M. Badjana PhD)
<figure><img src="image_548.png" /></figure>

<figure><img src="image_549.png" /></figure>

<figure><img src="image_550.png" /></figure>

<figure><img src="image_551.png" /></figure>

<figure><img src="image_552.png" /></figure>

# Estimating a Time-Varying Distribution

→ Time-varying models

$$M_0: Y(t) \sim \text{GEV}(\mu, \sigma, \xi)$$

$$M_1: Y(t) \sim \text{GEV}(\mu \times (1 + \lambda x(t)), \sigma, \xi)$$

<figure><img src="image_553.png" /></figure>

(M. Badjana PhD)

A simple statistical problem, plagued with sampling uncertainty!
<figure><img src="image_554.png" /></figure>

<figure><img src="image_555.png" /></figure>

<figure><img src="image_556.png" /></figure>

<figure><img src="image_557.png" /></figure>

<figure><img src="image_558.png" /></figure>

# Estimating Space-and-Time-Varying Distributions

→ Space-and-time-varying models (X. Sun PhD)

$$ \mathcal{M}_{\text{loc}} : Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda(s)x(t)), \sigma(s), \xi(s)) $$
<figure><img src="image_559.png" /></figure>

<figure><img src="image_560.png" /></figure>

<figure><img src="image_561.png" /></figure>

<figure><img src="image_562.png" /></figure>

<figure><img src="image_563.png" /></figure>

# Estimating Space-and-Time-Varying Distributions

→ Space-and-time-varying models (X. Sun PhD)

$$ \mathcal{M}_{\text{loc}} : Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda(s)x(t)), \sigma(s), \xi(s)) $$

$$ \mathcal{M}_{\text{reg}} : Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda x(t)), \sigma(s), \xi) $$
<figure><img src="image_564.png" /></figure>

<figure><img src="image_565.png" /></figure>

<figure><img src="image_566.png" /></figure>

<figure><img src="image_567.png" /></figure>

<figure><img src="image_568.png" /></figure>

# Estimating Space-and-Time-Varying Distributions

→ Space-and-time-varying models (X. Sun PhD)

$$ \mathcal{M}_{\text{loc}} : Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda(s)x(t)), \sigma(s), \xi(s)) $$

$$ \mathcal{M}_{\text{reg}} : Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda x(t)), \sigma(s), \xi) $$

* All sites contribute to the estimation of **regional parameters**

* Less stringent hypothesis possible (hierarchical models)

* Need to describe **spatial dependence** (elliptical copulas)
<figure><img src="image_569.png" /></figure>

<figure><img src="image_570.png" /></figure>

<figure><img src="image_571.png" /></figure>

<figure><img src="image_572.png" /></figure>

<figure><img src="image_573.png" /></figure>

# Estimating Space-and-Time-Varying Distributions

→ Space-and-time-varying models (X. Sun PhD)

$M_{loc}: Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda(s)x(t)), \sigma(s), \xi(s))$

$M_{reg}: Y(s, t) \sim \text{GEV}(\mu(s) \times (1 + \lambda x(t)), \sigma(s), \xi)$

* All sites contribute to the estimation of **regional parameters**

* Less stringent hypothesis possible (hierarchical models)

* Need to describe **spatial dependence** (elliptical copulas)

<figure><img src="image_574.png" /><figcaption>(b) DJF, 0.99-quantile, Strong La Niña (SOI=20) vs. Neutral phase(SOI=0)</figcaption></figure>
<figure><img src="image_575.png" /></figure>

<figure><img src="image_576.png" /></figure>

<figure><img src="image_577.png" /></figure>

<figure><img src="image_578.png" /></figure>

<figure><img src="image_579.png" /></figure>

# Hidden Climate Indices Models

→ Standard vs. Hidden Climate Index model

$$M_{\text{stand.}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda x(t)))$$
<figure><img src="image_580.png" /></figure>

<figure><img src="image_581.png" /></figure>

<figure><img src="image_582.png" /></figure>

<figure><img src="image_583.png" /></figure>

<figure><img src="image_584.png" /></figure>

# Hidden Climate Indices Models

→ Standard vs. Hidden Climate Index model

$$M_{\text{stand.}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda x(t)))$$

$$M_{\text{hidden}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda \tau(t)))$$
<figure><img src="image_585.png" /></figure>

<figure><img src="image_586.png" /></figure>

<figure><img src="image_587.png" /></figure>

<figure><img src="image_588.png" /></figure>

<figure><img src="image_589.png" /></figure>

# Hidden Climate Indices Models

→ Standard vs. Hidden Climate Index model

$$M_{\text{stand.}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda x(t)))$$

$$M_{\text{hidden}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda \tau(t)))$$

<figure><img src="image_590.png" /></figure>

<figure><img src="image_591.png" /></figure>
<figure><img src="image_592.png" /></figure>

<figure><img src="image_593.png" /></figure>

<figure><img src="image_594.png" /></figure>

<figure><img src="image_595.png" /></figure>

<figure><img src="image_596.png" /></figure>

# Hidden Climate Indices Models

→ Standard vs. Hidden Climate Index model

$$M_{\text{stand.}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda x(t)))$$

$$M_{\text{hidden}} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda \tau(t)))$$

<figure><img src="image_597.png" /></figure>

<figure><img src="image_598.png" /></figure>

<figure><img src="image_599.png" /></figure>
<figure><img src="image_600.png" /></figure>

<figure><img src="image_601.png" /></figure>

<figure><img src="image_602.png" /></figure>

<figure><img src="image_603.png" /></figure>

<figure><img src="image_604.png" /></figure>

# Hidden Climate Indices Models

→ Generalization to non-homogeneous regions

$$M_0 : Y(s, t) \sim D(\theta(s) \times (1 + \lambda\tau(t)))$$
<figure><img src="image_605.png" /></figure>

<figure><img src="image_606.png" /></figure>

<figure><img src="image_607.png" /></figure>

<figure><img src="image_608.png" /></figure>

<figure><img src="image_609.png" /></figure>

# Hidden Climate Indices Models

→ Generalization to non-homogeneous regions

$$ \mathcal{M}_0 : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda\tau(t))) $$

$$ \mathcal{M} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda_1(s)\tau_1(t) + \dots + \lambda_K(s)\tau_K(t))) $$
<figure><img src="image_610.png" /></figure>

<figure><img src="image_611.png" /></figure>

<figure><img src="image_612.png" /></figure>

<figure><img src="image_613.png" /></figure>

<figure><img src="image_614.png" /></figure>

# Hidden Climate Indices Models

→ Generalization to non-homogeneous regions

$$ \mathcal{M}_0 : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda\tau(t))) $$

$$ \mathcal{M} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda_1(s)\tau_1(t) + \dots + \lambda_K(s)\tau_K(t))) $$

→ Multi-variate extension + large-scale scalability
<figure><img src="image_615.png" /></figure>

<figure><img src="image_616.png" /></figure>

<figure><img src="image_617.png" /></figure>

<figure><img src="image_618.png" /></figure>

<figure><img src="image_619.png" /></figure>

# Hidden Climate Indices Models

→ A global-scale analysis of floods and heavy precipitation

<figure><img src="image_620.png" /></figure>

P

Q
<figure><img src="image_621.png" /></figure>

<figure><img src="image_622.png" /></figure>

<figure><img src="image_623.png" /></figure>

<figure><img src="image_624.png" /></figure>

<figure><img src="image_625.png" /></figure>

# Hidden Climate Indices Models

→ A global-scale analysis of floods and heavy precipitation

<figure><img src="image_626.png" /></figure>
<figure><img src="image_627.png" /></figure>

<figure><img src="image_628.png" /></figure>

<figure><img src="image_629.png" /></figure>

<figure><img src="image_630.png" /></figure>

<figure><img src="image_631.png" /></figure>

# Hidden Climate Indices Models

→ A global-scale analysis of floods and heavy precipitation

Mean sea-level pressure

Zonal wind (850hPa)

<figure><img src="image_632.png" /></figure>

<figure><img src="image_633.png" /></figure>

<figure><img src="image_634.png" /></figure>

<figure><img src="image_635.png" /></figure>
<figure><img src="image_636.png" /></figure>

<figure><img src="image_637.png" /></figure>

<figure><img src="image_638.png" /></figure>

<figure><img src="image_639.png" /></figure>

<figure><img src="image_640.png" /></figure>

# Hidden Climate Indices Models

→ A global-scale analysis of floods and heavy precipitation

Mean sea-level pressure

<figure><img src="image_641.png" /></figure>

Zonal wind (850hPa)

<figure><img src="image_642.png" /></figure>

<figure><img src="image_643.png" /></figure>

<figure><img src="image_644.png" /></figure>
<figure><img src="image_645.png" /></figure>

<figure><img src="image_646.png" /></figure>

<figure><img src="image_647.png" /></figure>

<figure><img src="image_648.png" /></figure>

<figure><img src="image_649.png" /></figure>

# Hidden Climate Indices Models

→ A global-scale analysis of floods and heavy precipitation

<figure><img src="image_650.png" /></figure>

SON 1840

<figure><img src="image_651.png" /></figure>

<figure><img src="image_652.png" /></figure>

<figure><img src="image_653.png" /></figure>

<figure><img src="image_654.png" /></figure>

<figure><img src="image_655.png" /></figure>

<figure><img src="image_656.png" /></figure>
<figure><img src="image_657.png" /></figure>

<figure><img src="image_658.png" /></figure>

<figure><img src="image_659.png" /></figure>

<figure><img src="image_660.png" /></figure>

<figure><img src="image_661.png" /></figure>

Conclusion

<figure><img src="image_662.png" /></figure>
<figure><img src="image_663.png" /></figure>

<figure><img src="image_664.png" /></figure>

<figure><img src="image_665.png" /></figure>

<figure><img src="image_666.png" /></figure>

<figure><img src="image_667.png" /></figure>

# Summary

* Development of probabilistic models enabling the production of uncertain hydrologic predictions
<figure><img src="image_668.png" /></figure>

<figure><img src="image_669.png" /></figure>

<figure><img src="image_670.png" /></figure>

<figure><img src="image_671.png" /></figure>

<figure><img src="image_672.png" /></figure>

# Summary

* Development of probabilistic models enabling the production of uncertain hydrologic predictions

* Key contributions:

① Uncertainties affecting streamflow time series

② Uncertainties affecting hydrologic models

③ Modeling hydrologic variability in space, time and between variables
<figure><img src="image_673.png" /></figure>

<figure><img src="image_674.png" /></figure>

<figure><img src="image_675.png" /></figure>

<figure><img src="image_676.png" /></figure>

<figure><img src="image_677.png" /></figure>

## Perspectives

→ Uncertainties in Hydrologic Data and Models
<figure><img src="image_678.png" /></figure>

<figure><img src="image_679.png" /></figure>

<figure><img src="image_680.png" /></figure>

<figure><img src="image_681.png" /></figure>

<figure><img src="image_682.png" /></figure>

# Perspectives

## Uncertainties in Hydrologic Data and Models

### Treatment of structural errors

* Further assess the interest of stochastic parameters

* Evaluate non-zero-mean residual models (conditional biases)
<figure><img src="image_683.png" /></figure>

<figure><img src="image_684.png" /></figure>

<figure><img src="image_685.png" /></figure>

<figure><img src="image_686.png" /></figure>

<figure><img src="image_687.png" /></figure>

# Perspectives

## Uncertainties in Hydrologic Data and Models

### ① Treatment of structural errors

* Further assess the interest of stochastic parameters

* Evaluate non-zero-mean residual models (conditional biases)

### ② Distributed models

* Hydrologic (SMASH) or Hydraulics

* Define spatialized error models
<figure><img src="image_688.png" /></figure>

<figure><img src="image_689.png" /></figure>

<figure><img src="image_690.png" /></figure>

<figure><img src="image_691.png" /></figure>

<figure><img src="image_692.png" /></figure>

# Perspectives

## Uncertainties in Hydrologic Data and Models

### ① Treatment of structural errors

* Further assess the interest of stochastic parameters

* Evaluate non-zero-mean residual models (conditional biases)

### ② Distributed models

* Hydrologic (SMASH) or Hydraulics

* Define spatialized error models

### ③ Beyond streamflow series

* What other data types are valuable to constrain model calibration?

* Sporadic gaugings, flood videos, flood marks, satellite (SWOT), etc.
<figure><img src="image_693.png" /></figure>

<figure><img src="image_694.png" /></figure>

<figure><img src="image_695.png" /></figure>

<figure><img src="image_696.png" /></figure>

<figure><img src="image_697.png" /></figure>

# Perspectives

→ Hydrologic Variability
<figure><img src="image_698.png" /></figure>

<figure><img src="image_699.png" /></figure>

<figure><img src="image_700.png" /></figure>

<figure><img src="image_701.png" /></figure>

<figure><img src="image_702.png" /></figure>

# Perspectives

## Hydrologic Variability

### Methodological questions

* Using Hidden Climate Indices to derive spatial extremes models?

* Clarify the link between HCI models and Machine Learning approaches such as probabilistic PCA
<figure><img src="image_703.png" /></figure>

<figure><img src="image_704.png" /></figure>

<figure><img src="image_705.png" /></figure>

<figure><img src="image_706.png" /></figure>

<figure><img src="image_707.png" /></figure>

# Perspectives

<figure><img src="image_708.png" /></figure>

## ① Methodological questions

* Using Hidden Climate Indices to derive spatial extremes models?

* Clarify the link between HCI models and Machine Learning approaches such as probabilistic PCA

## ② Applications

* Global droughts (low flows and precipitation deficit)

* Relation with the wildfire hazard: a testbed for multi-risk assessment?
<figure><img src="image_709.png" /></figure>

<figure><img src="image_710.png" /></figure>

<figure><img src="image_711.png" /></figure>

<figure><img src="image_712.png" /></figure>

<figure><img src="image_713.png" /></figure>

## Key aspects

<figure><img src="image_714.png" /></figure>

* Probabilistic modeling is central
<figure><img src="image_715.png" /></figure>

<figure><img src="image_716.png" /></figure>

<figure><img src="image_717.png" /></figure>

<figure><img src="image_718.png" /></figure>

<figure><img src="image_719.png" /></figure>

Key aspects

<figure><img src="image_720.png" /></figure>

* Probabilistic modeling is central

* Data are invaluable, and models should adapt to data
<figure><img src="image_721.png" /></figure>

<figure><img src="image_722.png" /></figure>

<figure><img src="image_723.png" /></figure>

<figure><img src="image_724.png" /></figure>

<figure><img src="image_725.png" /></figure>

Key aspects

<figure><img src="image_726.png" /></figure>

* Probabilistic modeling is central

* Data are invaluable, and models should adapt to data

* Coding is an important component
<figure><img src="image_727.png" /></figure>

<figure><img src="image_728.png" /></figure>

<figure><img src="image_729.png" /></figure>

<figure><img src="image_730.png" /></figure>

<figure><img src="image_731.png" /></figure>

## Key aspects

<figure><img src="image_732.png" /></figure>

<figure><img src="image_733.png" /></figure>

<figure><img src="image_734.png" /></figure>

<figure><img src="image_735.png" /></figure>

<figure><img src="image_736.png" /></figure>

* Probabilistic modeling is central

* Data are invaluable, and models should adapt to data

* Coding is an important component

* Operational transfer as both an output and an input of research work
<figure><img src="image_737.png" /></figure>

<figure><img src="image_738.png" /></figure>

<figure><img src="image_739.png" /></figure>

<figure><img src="image_740.png" /></figure>

<figure><img src="image_741.png" /></figure>

And most importantly... Thank You!

<figure><img src="image_742.png" /></figure>
<figure><img src="image_743.png" /></figure>

# Xtra
<figure><img src="image_744.png" /></figure>

# Complex Rating Curves $Q = f(h, ...)$

→ Stage-Fall-Discharge model (V. Mansanarez)

$Q = f(h_1, h_2)$ due to variable backwater (dam, tide)

Used operationally for dam-influenced stations

Work still in progress for tide-influenced stations

<figure><img src="image_745.png" /></figure>
<figure><img src="image_746.png" /></figure>

# Estimating a Time-Varying Distribution

→ Trend detection studies

Drought duration, 1968-2008

<figure><img src="image_747.png" /></figure>

Snowmelt flows start, 1961-2006

<figure><img src="image_748.png" /></figure>

(I. Giuntoli, A. Bard)
<figure><img src="image_749.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_750.png" /></figure>
<figure><img src="image_751.png" /></figure>

# Error Sources in Hydrologic Modeling

Reality

<figure><img src="image_752.png" /></figure>

True Input(s)

<figure><img src="image_753.png" /></figure>

True processes

<figure><img src="image_754.png" /></figure>

True response(s)

<figure><img src="image_755.png" /></figure>

Model

Conceptualized
processes

<figure><img src="image_756.png" /></figure>
<figure><img src="image_757.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_758.png" /></figure>

Structural errors
due to the imperfect nature of the model

<figure><img src="image_759.png" /></figure>

<figure><img src="image_760.png" /></figure>
<figure><img src="image_761.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_762.png" /></figure>

Structural errors

due to the imperfect nature of the model

<figure><img src="image_763.png" /></figure>

<figure><img src="image_764.png" /></figure>

<figure><img src="image_765.png" /></figure>
<figure><img src="image_766.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_767.png" /></figure>

Input errors

e.g. rainfall measurement + sampling errors

Structural errors

due to the imperfect nature of the model

<figure><img src="image_768.png" /></figure>

<figure><img src="image_769.png" /></figure>

<figure><img src="image_770.png" /></figure>

<figure><img src="image_771.png" /></figure>

<figure><img src="image_772.png" /></figure>
<figure><img src="image_773.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_774.png" /></figure>
<figure><img src="image_775.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_776.png" /></figure>
<figure><img src="image_777.png" /></figure>

# Error Sources in Hydrologic Modeling

<figure><img src="image_778.png" /></figure>

+ Parametric uncertainty
<figure><img src="image_779.png" /></figure>

Systematic Rating Curve Errors

<figure><img src="image_780.png" /></figure>

<figure><img src="image_781.png" /></figure>
<figure><img src="image_782.png" /></figure>

Rainfall Uncertainty

<figure><img src="image_783.png" /></figure>

<figure><img src="image_784.png" /></figure>

<figure><img src="image_785.png" /></figure>
<figure><img src="image_786.png" /></figure>

Non-identifiability

Model: $\hat{y} = \theta_1\theta_2x$

<figure><img src="image_787.png" /></figure>

Identifying $\theta_1$ and $\theta_2$ based on this information is an ill-posed problem.
<figure><img src="image_788.png" /></figure>

Non-identifiability

Model: $\hat{y} = \theta_1\theta_2x$

<figure><img src="image_789.png" /></figure>

Identifying $\theta_1$ and $\theta_2$ based on this information is an ill-posed problem.
Non-identifiability and ill-posedness are frequently encountered in the
calibration of hydrologic models
<figure><img src="image_790.png" /></figure>

Non-identifiability

Solution 1: 'under-parameterize', $\hat{y} = \gamma x$ with $\gamma = \theta_1\theta_2$

<figure><img src="image_791.png" /></figure>
<figure><img src="image_792.png" /></figure>

Non-identifiability

Solution 2: use prior, $\theta_1 \sim N(1, 0.1^2)$

<figure><img src="image_793.png" /></figure>
<figure><img src="image_794.png" /></figure>

# Non-identifiability

## Solution 2: use prior, $\theta_1 \sim N(1, 0.1^2)$

<figure><img src="image_795.png" /></figure>

## Solution 3: use other data (not more of the same!)

$\hat{y} = \theta_1\theta_2 x$ and $\hat{y}' = \exp(\theta_1 x')$

<figure><img src="image_796.png" /></figure>
<figure><img src="image_797.png" /></figure>

# HCI framework

<figure><img src="image_798.png" /><figcaption>4. Temporal processes for HCI</figcaption></figure>

<figure><img src="image_799.png" /><figcaption>3. Spatial processes for HCI effects</figcaption></figure>

<figure><img src="image_800.png" /><figcaption>2. Parameter model</figcaption></figure>

Observed data
Inferred variable
Derived variable

**Stochastic link:** B is a realization from a distribution D with parameters A

**Deterministic link:** C is computed from A and B

Varies in time

Varies in space

<caption>1. Data model</caption>
<figure><img src="image_801.png" /></figure>

# Hidden Climate Indices Models

→ Generalization to non-homogeneous regions

$$ \mathcal{M}_0 : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda\tau(t))) $$
<figure><img src="image_802.png" /></figure>

Hidden Climate Indices Models

→ Generalization to non-homogeneous regions

$$
\mathcal{M}_0 : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda\tau(t)))
$$

$$
\mathcal{M} : Y(s, t) \sim \mathcal{D} \left( \theta(s) \times \left( 1 + \lambda_1(s) \tau_1(t) + \dots + \lambda_K(s) \tau_K(t) \right) \right)
$$
<figure><img src="image_803.png" /></figure>

# Hidden Climate Indices Models

→ Generalization to non-homogeneous regions

$$ \mathcal{M}_0 : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda\tau(t))) $$

$$ \mathcal{M} : Y(s, t) \sim \mathcal{D}(\theta(s) \times (1 + \lambda_1(s)\tau_1(t) + \dots + \lambda_K(s)\tau_K(t))) $$

<figure><img src="image_804.png" /></figure>
<figure><img src="image_805.png" /></figure>

# Hot-and-Dry Australian Summers

<figure><img src="image_806.png" /></figure>

<figure><img src="image_807.png" /></figure>

<figure><img src="image_808.png" /></figure>

<figure><img src="image_809.png" /></figure>

<figure><img src="image_810.png" /></figure>

Effect on...

<figure><img src="image_811.png" /></figure>

<figure><img src="image_812.png" /></figure>

<figure><img src="image_813.png" /></figure>

<figure><img src="image_814.png" /></figure>

<figure><img src="image_815.png" /></figure>

Effect on...

<figure><img src="image_816.png" /></figure>

<figure><img src="image_817.png" /></figure>

<figure><img src="image_818.png" /></figure>

<figure><img src="image_819.png" /></figure>
<figure><img src="image_820.png" /></figure>

# Flood Marks Case Study

<figure><img src="image_821.png" /></figure>

<figure><img src="image_822.png" /></figure>

$$ \begin{cases} Q(s, t) \sim \text{GEV} \left( e^{\mu_0(s)} \times \left( 1 + \sum_{k=1}^{K} \mu_k(s) \tau_k(t) \right) \right), e^{\mu_0(s)} \times e^{\gamma(s)}, \xi(s) \\ O(r, t) \sim \text{B} \left( \text{logit}^{-1} \left( \lambda_0(r) + \sum_{k=1}^{K} \lambda_k(r) \tau_k(t) \right) \right) \end{cases} $$

Site status

<figure><img src="image_823.png" /></figure>

<figure><img src="image_824.png" /></figure>

<figure><img src="image_825.png" /></figure>

<figure><img src="image_826.png" /></figure>

<figure><img src="image_827.png" /></figure>

<figure><img src="image_828.png" /></figure>

<figure><img src="image_829.png" /></figure>

<figure><img src="image_830.png" /></figure>

<figure><img src="image_831.png" /></figure>

<figure><img src="image_832.png" /></figure>

Flood mark

Probability of occurrence

<figure><img src="image_833.png" /></figure>

1710

<figure><img src="image_834.png" /></figure>

1855

<figure><img src="image_835.png" /></figure>

1866

<figure><img src="image_836.png" /></figure>
<figure><img src="image_837.png" /></figure>

# Floods and Heavy Precipitation at the Global Scale

<figure><img src="image_838.png" /><figcaption>Available series length at 1884 sites</figcaption></figure>

<figure><img src="image_839.png" /></figure>

$$ egin{cases} P(s,t) \sim \text{Beta}(\mu_P(s,t), \nu_P(s,t)); Q(s,t) \sim \text{Beta}(\mu_Q(s,t), \nu_Q(s,t)) \\ \operatorname{logit}(\mu_P(s,t)) = \zeta_{\mu_P}(s) + \sum_{k=1}^{K} \lambda_{k,P}(s) \tau_k(t) + \sum_{k=1}^{K} \theta_{k,P}(s) \delta_k(t) \\ \operatorname{logit}(\mu_Q(s,t)) = \zeta_{\mu_Q}(s) + \sum_{k=1}^{K} \lambda_{k,Q}(s) \tau_k(t) + \sum_{k=1}^{K} \theta_{k,Q}(s) \omega_k(t) \end{cases} $$