# Overview ✏️ {.unnumbered}

This part of the book focuses on **hazard assessment** - the process of characterizing climate hazards and their probability distributions.
Understanding climate hazards is fundamental to risk assessment and management across many applications.

## Why Climate Hazards Matter

For virtually every climate risk decision, we need to evaluate the expected outcome:

$$\mathbb{E}(R(a)) = \int_{\mathcal{S}} R(a, \mathbf{s}) p(\mathbf{s}) d\mathbf{s}$$

where $a$ is a decision, $R(a, \mathbf{s})$ is the reward/utility function, and $\mathbf{s}$ represents the state of the climate system.

**The key insight**: We cannot simply use "average" conditions. 
The full probability distribution $p(\mathbf{s})$ matters, especially the tails representing extreme events.

## Motivating Applications

### Engineering Design: Stormwater Infrastructure

**Challenge**: Size a stormwater pipe for adequate drainage

**Process**:
1. Use rainfall-runoff model: $Q = CiA$ (rational method)
2. Design rainfall based on return period $T$: $p(i > i^*) = 1/T$  
3. Size culvert to handle $Q^* = Ci^*A$

**Requirement**: Must know precipitation probability distribution $p(i)$

### Floodplain Mapping

**Challenge**: Delineate areas at risk of riverine flooding

**Process**:
1. Analyze historical streamflow data at gauge
2. Extract 100-year return level (99th percentile) of annual maximum flows
3. Use hydraulic model to map flood extent

**Requirement**: Must characterize extreme streamflow distributions

### Reservoir Operations

**Challenge**: Design reservoir for reliable water supply

**Process**:
1. Consider $N$ years of inflows, releases, and evaporation
2. Count reservoir failures (empty states)
3. Repeat with different inflow sequences (Monte Carlo)
4. Estimate reliability if sampling from true $p(\text{inflow})$

**Requirement**: Must understand inflow variability and extremes

### Index Insurance Pricing

**Challenge**: Price weather-based insurance products

**Process**:
1. Define trigger: payout $X$ if index $I > I^*$
2. Calculate payout probability: $p^* = p(I > I^*)$
3. Set premium: $R = X[\mathbb{E}[p^*] + \lambda \mathbb{V}^{1/2}[p^*]]$

**Requirement**: Must quantify trigger probability and uncertainty

## Case Study: Storm Surge Risk Assessment

Storm surge provides an excellent example of the complexity involved in hazard assessment.

### Physical Drivers

**Tropical cyclone characteristics**:
- Track and intensity
- Size and forward speed  
- Landfall location and timing

**Environmental conditions**:
- Sea level (mean and variability)
- Coastal bathymetry and topography
- Antecedent conditions

### Modeling Chain

1. **Tropical cyclone simulation**: Generate synthetic storm tracks and intensities
2. **Meteorological modeling**: Create wind and pressure fields
3. **Storm surge modeling**: Use hydrodynamic models (ADCIRC, GeoClaw, SFINCS)
4. **Sea level projections**: Account for climate change effects

### Sources of Uncertainty

**Historical data limitations**:
- Short observational records
- Sampling uncertainty for rare events
- Non-stationarity due to climate change

**Model uncertainties**:
- Tropical cyclone behavior in changing climate
- Ice sheet dynamics and sea level rise
- Regional climate patterns

**Recent research** [@sobel_biases:2023] highlights that models may incorrectly simulate tropical responses to greenhouse gas warming, affecting regional projections.

## The Central Challenge

**Fundamental tension** in hazard assessment:

### Historical Data
**Strengths**:
- Measures what we care about directly
- Represents actual physical processes
- Includes complex interactions

**Limitations**:
- Sampling uncertainty for rare events
- Backward-looking (doesn't capture future conditions)
- Limited spatial and temporal coverage

### Model Simulations  
**Strengths**:
- Can explore future climate conditions
- Generate large ensembles for rare event estimation
- Incorporate physical understanding

**Limitations**:
- Model biases and structural uncertainty  
- Simplified representation of complex processes
- Uncertain future forcing scenarios

## Key Technical Challenges

### Sampling Variability
- Observations are noisy, backward-looking, and incompletely measured
- Model estimates can explore unseen conditions but layer in systematic biases
- Rare events are necessarily hard to validate

### Non-Stationarity
- Climate change violates traditional stationarity assumptions
- Relationships between large-scale and local climate may evolve
- Historical records may not represent future conditions

### Multi-Scale Interactions
- Climate hazards emerge from processes across spatial and temporal scales
- Local extremes depend on global patterns and regional processes
- Compound and cascading hazards increase complexity

## Structure of This Part

This part addresses these challenges through several key topics:

- **[Climate Science Fundamentals](climate-science.qmd)**: Physical basis of climate hazards and their characteristics
- **[Extreme Value Theory](extremes.qmd)**: Statistical methods for characterizing rare events  
- **[Downscaling and Bias Correction](downscaling-bias-correction.qmd)**: Connecting global models to local impacts

The overarching goal is integrating information from noisy and/or biased sources to estimate the joint probability distribution of relevant hydroclimatic variables for decision-making under uncertainty.