We all know the water cycle. We’ve seen this ubiquitous image with the green earth, and the blue water, and the arrows and the smiley face on the sun. So you probably know that the majority of the rain that falls from the sky ends up washing along the ground, following gravity until it eventually finds a stream or channel. The more it rains, the higher the creeks and rivers rise. And if they rise too much, you’ve got a flood. Now, almost everyone agrees, flooding is bad. Most years it’s the number one natural disaster in the US by dollars of damage. And flooding is particularly capricious: difficult to characterize without detailed engineering studies, very localized, and depends on many unrelated factors.

In the 1960’s, we started to recognize the fact that development was exacerbating flood risks by putting more and more expensive homes and businesses near rivers and creeks. Everyone wants a waterfront view, right? So the federal government came up with the National Flood Insurance Program to create a larger pool for flood risk and incentivize management of the floodplain. But what is the floodplain? Well, you can’t let people go uninsured because the risks are too high, but you also can’t force every single person who owns real estate to pay flood insurance premiums. Ultimately you have to draw a line in the sand, or on a map in this case, that says: As a society how much flooding risk are we willing to tolerate? And the answer ended up being the 100-Year Flood: an often misunderstood combination of statistical hydrology and societal risk tolerance (two of my favorite topics). But it’s actually a pretty simple concept. 100 years is a return period, and the inverse of that is the annual exceedance probability. A 100-year flood has an annual exceedance probability of 1%. Let’s say our 100-year storm is a flow of 10,000 cubic feet per second in the river. That means in a given year, there’s a 1% probability that a flow of 10,000 cfs or greater will occur.

Using a return period is misleading, because it makes storms seem cyclical. It also hints at the possibility that we could predict the future, that somehow if a storm of a particular magnitude was to occur, that we might have a period of security before it happened again, or even that if a particular magnitude of storm had not occurred in quite some time, that we might be more "due" for it to occur than if it had occurred more recently. No one calls snake eyes the 36-roll dice throw, and you don’t call a full house the 700-deal poker hand. If a 100-year flood occurred last year, there’s still a 1% chance that it could happen this year. In fact, many hydrologists have stopped using return period altogether in favor of just calling flood magnitudes by their annual exceedance probability, just to avoid this confusing notion that storms are on some kind of schedule. So, to answer our question from before, the floodplain is the area of land that can reasonably be expected to have a 1% chance of flooding for a given year. This has been mapped for the entire U.S. and you can go on FEMA’s website to look at the floodplain maps for any area. But, how were these maps developed, and how do we know where the floodplain is? This is where the engineering comes in.

Let’s try a simple example first. Take the height of all the men or women in your group of friends or office. We know that height is normally distributed, the variation fits this particular shape of distribution that is ubiquitous in the natural world. To fit your data to a normal distribution, the only thing you need to know is the average height, and the standard deviation, and these are both simple formulas. Once you fit your data, you can make a reasonable guess at, for example, the percentage of the population that is over 7 feet tall, even if you don’t have anyone in your sample this tall. This works the same way with hydrology.

The U.S. Geological Survey maintains thousands of stream gages across the U.S. There is a gage on Williamson Creek in south Austin near my house. It uses air pressure to measure the depth of the stream, and then it converts that to flow. You can log on to the USGS website and take a look at the data from any stream gage in the U.S. Once you have this data, it’s pretty simple to figure out the 100-year flood, even if you don’t have a 100 years of data. You can take the peak flow from each year and fit that data to a probability distribution. This way, even if we don’t have a very long period of record, we can make a reasonable guess at the magnitude for lower probability floods.

But what if you don’t have a stream gage in your watershed? Stream gages aren’t that common, and they’re expensive to maintain, but rainfall gages are everywhere. You may even have one at your house that you check obsessively and brag to your colleagues about. Well now we’re really getting into the engineering part of it, because somehow we have to use rainfall data to come up with flows in a stream.

We usually only care about peak annual flow, or what’s the highest point the stream reached in a given year. There is a really simple way called the Rational Method of converting a rainfall intensity into a peak flow. It’s based on the idea that volumetric inflow (rainfall depth times area) is equal to outflow, minus any water that soaks into the ground or evaporates or gets trapped in a puddle, etc. And this is where you first start getting a sense that maybe engineering hydrology isn’t the most exact of sciences. Because not only is this single C value supposed to encompass all of the factors that can affect the peak flow coming from the watershed (e.g. vegetation, slope, soil type, amount of impervious cover, etc.) there’s also this thing about the units that I love. None of the imperial units for these values agree. But, if you do the math, the conversion factor is just about one, so we ignore it. We just leave it off and say close enough, and what’s hilarious to me is, that is the perfect acknowledgement of how imprecise engineering hydrology really is.

If you have a more complicated area or maybe you need to understand how flow varies over time, you have to continue taking steps away from reality until you end up with an entire hydrologic model. This can include developing a design storm, calculating how much runoff results from the rain, routing the runoff over the ground and through any streams and reservoirs, and we do this for each time step, sometimes minute by minute, using computer software.

Hydrology is something many engineers devote their entire careers to. Water is incredibly important to us, and it shapes almost every facet of our lives, but it’s almost never in the right place at the right time. Sometimes there’s not enough, like in a drought or just an arid region, but we also need to be prepared for the times when there’s too much water, a flood. Rainfall and streamflow have tremendous variability and it’s the engineer’s job to characterize that so that we can make rational and intelligent decisions about how we develop the world around us.