Like humans, no two floods are alike, so living with them can have its problems. This post considers the unpredictable ‘moods’ of floods and why we need probability simulations to address them. We will look to rainfall as the underlying characteristic of a flood’s personality. But we will soon find that our intuition can be sketchy, our observations sparse, and our methods limited.

## Problems with probability

Humans are notoriously unintuitive when it comes to understanding probability.

### Test your intuition on the birthday problem

A good example of how we understand probability is the birthday problem. That problem asks: What is the probability that a room of 23 people will contain two people with the same birthday? This famous problem has an answer much higher than what most people think. (You will find the answer at the end of this post!)

### How regular is the 1 in 100 flood?

Let’s consider the canonical 1 in 100-year flood. Many non-experts would intuit that this event is somehow periodic, or at least semi-periodic. Real floods are much more variable.

The figure below shows three random examples, where every year in the sequence has a 1% (ie. 1 in 100) chance of a flood. Real flood events are random, they may follow in quick succession or break for prolonged periods.

### How the solitaire card game helps solve flood probability

Whereas the birthday problem has an analytic solution and the flood example is simple, many problems evade analysis. The question ‘what is the probability that …’ may be simple, but the solution is often complicated by the system of which it is asked. For example, finding the theoretical chance of winning at the solitaire card game is not easy (only known to be within the bounds 82% to 91.5%).

One approach that allows us to sidestep complicated analysis is to conduct a simulation. For the example of solitaire, this would need the generation of many random games, along with a strategy for playing them. Providing these two criteria can be met, the approximate answer can be found by tallying the proportion of times a random game was won.

For rainfall, simulation has considerable potential, but implementing it is not without its challenges. The quality of our answer depends on how well we can randomly generate rainfall events and model their catchment response.

Let’s review the complexity of flood events before returning to consider how they might be simulated.

## Factors influencing how we understand floods

Recently, Adelaide experienced wild weather, including lightning (1 fatality), extreme wind (89 km/h), rain (35mm in 30 min), hail (1-2cm) and flash flooding (no fatalities). This type of storm is not uncharacteristic for summer, but does not occur regularly compared to other seasons. The flooding was enough to clog drainage systems in the southern suburbs and other nuisance problems, but did not cause wider impacts due to the dryness of the Adelaide catchments.

There are a few factors that influence how we understand floods. One is that floods are spatially complex. One is that catchments have specific characteristics. And another includes the data limitations of gauge records.

### Rainfall events are spatially complex

Some floods may be local: a small rain-cell of intense rainfall can have significant flow-on effects. Other floods, in larger catchments, may depend on the widespread properties of a storm and how it evolves over the region. Weather radars give us a glimpse into the spatial complexity of storms, but their measurements are indirect and error-laden. This limits confidence in the precision of their rainfall estimates.

Note the absence of a numerical scale in the image, indicating imprecision in the rain rate.

### Why catchments confound our understanding of floods

Even if we had accurate spatial observations of a storm, this would be insufficient for understanding a flood. Catchment characteristics confound flooding (eg. slope, orientation, soil moisture, ground water levels, dam levels). Some of these characteristics depend on the influence of earlier rainfall events. For example, it matters if the catchment was already ‘wet’ due to prior storms. This is because incident rainfall will generate more flow instead of soaking into the soil. The link between catchment wetness and storms requires us to use long continuous sequences of rainfall.

### Measuring wetness using rainfall gauge observations

Whereas radar measurements resolve spatial patterns, gauge records have excellent temporal content. Gauge records show us that rainfall is variable on every conceivable scale: Inter-annual fluctuations, seasonal cycles, storm sequences, rainfall bursts, and raindrop distributions. In urban catchments subdaily gauges are required for analysis due to quick response times. Unfortunately, these records are sparse, with only a handful having length greater than 50 years. Data limitations are a significant constraint on our ability to simulate long rainfall sequences.

## The simulation solution

If I gave you the sequence {6, 4, 1, 1, 2, 5, 3, 6, 3, 1, 4, 4, 2, 4, 3} you might infer that a six-sided dice generated it. If I gave you a 50-year record of 5-minute rainfall at just one location (ie. 5,256,000 observations), what might you infer about the nature of the ‘dice’ ? Simulating rainfall patterns is not a simple task. There can be many different features that need to mimicked (e.g. long-run averages, clustering of events, seasonal fluctuations and correlations to name a few). The problem is even harder for flood events because extremes are infrequent and our observed records are short.

There are many conceptual models of rainfall that rise to the challenge of mimicking long sequences of rainfall patterns. Leonard et al. (2008) give one example, but these models appear simplified when compared to the true generating process of rainfall. Likewise, catchment run-off models used for translating (extreme) rainfall into (extreme) streamflow can be simplistic. These models perform well for a majority of cases, but may struggle to match statistics for every possible site, scale and season.

Despite the relative simplicity of models, the result is that we can generate many hundreds of years of synthetic storm events. These events are then used to calculate flood probabilities. There is considerable effort needed to establish that the models are ‘true’ to reality. Over time, it is likely that simulation models will continue to improve their representation.

## Continuous simulation allows us to realise flood behaviour

Floods have plagued human development over many thousands of years. But it has only been in recent decades that we have had tools that yield solutions to flood probability. This progress has come about due to extraordinary advances in computational power.

In the past we waited (many decades) for the next real event to test our flood defenses. Now computers can continuously simulate and test thousands of synthetic storm sequences, each of which has its own personality. While humans are intuitively poor at understanding probability, computer simulations will increasingly give us the brute force ability to solve the probability problem of floods.

*P.S. The next time you are in the middle of a storm, check out the radar and contemplate its colourful ‘personality’.*

*P.P.S. The next time you are in the middle of a room with 23 people, tell them there is a 50% chance that two of you will share a birthday.*

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