Lecture

Mon., Jan. 22

Given some *hazard*, how do we assess the *damages* or *impacts*?

We can quantify this, for example as \[ \textrm{Damage} = \textrm{Vulnerability} \times \textrm{Exposure} \times \textrm{Hazard} \]

Let’s understand these terms through some examples!

Today

Illustrative examples

How to manage exposure and vulnerability?

Some math

Wrapup

\[ \Pr(\textrm{failure}) = f(\textrm{hazard}, \theta) \]

For a given storm, the total damages can be estimated by summing the damages for each property. \[ \textrm{Total damages} = \sum_{i \in \, \textrm{exposure}} \textrm{Hazard}_i \times \textrm{Vulnerability}_i \]

This is the most straightforward example of the exposure-vulnerability-hazard framework

- Hazard: a real or synthetic storm
- Exposure: all the assets that are insured
- Vulnerability: predicted losses as a function of the hazard

Today

Illustrative examples

How to manage exposure and vulnerability?

Some math

Wrapup

Interventions such as floodproofing can shift the vulnerability curve

Today

Illustrative examples

How to manage exposure and vulnerability?

Some math

Wrapup

The expected value of a function \(f(x)\) of a random variable \(x\) is: \[ \mathbb{E}[f(x)] = \int f(x) p(x) dx \] where \(p(x)\) is the probability density function of \(x\).

In other words, for each possible value of \(x\), calculate \(f(x)\). We then take a weighted average, where the weights are the probability of each value of \(x\).

You cannot get the expected value of a function by plugging the expected value of the random variable into the function. \[ \mathbb{E}[f(x)] \neq f(\mathbb{E}[x]) \]

If we want to calculate the **expected damages** then we can use this formula \[
\mathbb{E}[f(x)] = \int f(x) p(x) dx
\] by defining \(x\) to be the hazard (e.g., flood depth) and \(f(x)\) the damage function.

**Important**

The key assumption we need is that we know the probability density function \(p(x)\)!

In general, \(\int f(x) p(x) dx\) is hard to compute analytically. We can use Monte Carlo methods to approximate this integral.

**Important**

Monte Carlo methods are a wide class of computational algorithms for approximating integrals.

If we draw \(N\) samples independently and identically distributed (i.i.d.) from a probability distribution \(p(x)\), denoted as \(\left\{x_i\right\}_{i=1}^N\) where \(x_i \sim p(x)\), then \[ \mathbb{E}[f(x)] \approx \frac{1}{N} \sum_{i=1}^N f(x_i) \]

**Tip**

Drawing \(N\) samples iid from \(p(x)\) can be *inefficient*. Most of the samples will be in regions where \(f(x)\) is small, so we are wasting computational effort. There are many clever ways around this.

Running regional flood models is computationally expensive. Often, a model may have been run for a few different nominal **return levels.** For example, we might have flood depths at each grid for the nominal 10, 25, 50, 100, 250, and 500 year floods.

Today

Illustrative examples

How to manage exposure and vulnerability?

Some math

Wrapup

- Impacts (damage) depend on hazard, exposure, and vulnerability.
- Exposure and vulnerability are changing rapidly.
- To estimate risk, we need a model for how our system responds to a hazard.
- If we know the probability density function of the hazard, we can use Monte Carlo methods to estimate expected impacts.

- How to think about damages that are not direct damages (e.g., “indirect” damages)?
- How to think about complex systems and risks (e.g., breadbasket failures) that are not easy summarized by a single metric?

- Wednesday: discussion questions for Wing et al. (2020) posted
- Wednesday after class: troubleshoot computing issues
- Friday: lab

Gidaris, I., Padgett, J. E., Barbosa, A. R., Chen, S., Cox, D., Webb, B., & Cerato, A. (2017). Multiple-Hazard Fragility and Restoration Models of Highway Bridges for Regional Risk and Resilience Assessment in the United States: State-of-the-Art Review. *Journal of Structural Engineering*, *143*(3), 04016188. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001672

Jongman, B., Ward, P. J., & Aerts, J. C. J. H. (2012). Global exposure to river and coastal flooding: Long term trends and changes. *Global Environmental Change*, *22*(4), 823–835. https://doi.org/10.1016/j.gloenvcha.2012.07.004

Moel, H. de, Vliet, M. van, & Aerts, J. C. J. H. (2014). Evaluating the effect of flood damage-reducing measures: A case study of the unembanked area of Rotterdam, the Netherlands. *Regional Environmental Change*, *14*(3), 895–908. https://doi.org/10.1007/s10113-013-0420-z

Tedesco, M., McAlpine, S., & Porter, J. R. (2020). Exposure of real estate properties to the 2018 Hurricane Florence flooding. *Natural Hazards and Earth System Sciences*, *20*(3), 907–920. https://doi.org/10.5194/nhess-20-907-2020

Wing, O. E. J., Pinter, N., Bates, P. D., & Kousky, C. (2020). New insights into US flood vulnerability revealed from flood insurance big data. *Nature Communications*, *11*(1, 1), 1444. https://doi.org/10.1038/s41467-020-15264-2