Floods

2nd February 2001    Back to Bridges    back to Home page

Among the problems that confront the designer of structures, floods rank high in importance, in areas where they occur.

In fact, even the normal flow of river water past bridge piers can generate scouring which can bring down a bridge.  The presence of the piers changes the flow, producing acceleration and turbulence.  The lifting and carrying power of a fluid increase as a high power of the speed.  The ancient Romans knew about this, and took precautions.  Foundations need to penetrate to secure ground, and a pavement around piers can help to protect the bed.

In times of spate, the speed of a river can increase greatly, and in times of flood, the width and depth can change dramatically.  In the pages about Severn bridges you can find many examples of bridges which have spans which normally cross fields populated with crops or animals, or even parts of river-side towns.  These spans play their part when the flood-plain lives up to its name, helping to allow the water to pass under the road or railway.

Unfortunately, even these measures often don’t prevent the occasional flooding of people’s homes and workplaces.  One problem is the statistical distribution of magnitudes, whether they be speeds of winds, depths of floods, or magnitudes of earthquakes.  Perhaps it would be possible to build a structure to withstand any imaginable event, but the cost would be completely uneconomic.  The practicable course is to build for a magnitude of event that is expected on average every N years, where N is determined by some kind of authority, using some kind of economic calculation.  For the same reason, it is impossible to purchase and maintain emergency equipment which will cope with any event that will ever happen.

Even this is does not have completely calculable consequences, because random events, however unusual, can happen at any time.  If events are truly random, the time between them follows an exponential formula, which means that if the future is divided into uniform periods, the next event is more likely to occur in the next period than in any other.  This often leads people to conclude that events are connected when they are not.

These lines represent the results of a calculation using pseudo-random numbers.  These numbers are not "truly random", but they pass a number of tests that are considered to indicate a degree of randomness.  The horizontal scale can represent space or time.  If you had been asked to draw some randomly spaced lines, would you have drawn something like this?

In the next diagram, the events have random sizes as well as random times.  Bunching of rarer larger events is apparent, though not as close as the bunching of the commoner smaller events.  Note also that the larger events show up more on the diagram than the more common smaller ones.

There is no "law of averages" that says that when tossing a coin, a run of heads is more likely to be followed by a tail than by another head.  If you are playing a lottery, you might as well choose the numbers 1 2 3 4 5 6 as any other, as all combinations are equally likely.

The next picture shows a typical frequency distribution of 10000 events, with magnitude plotted horizontally.  The largest event, marked with a red arrow, is far bigger than the mean, marked with a blue arrow.

The next pictures show the same type of plot with a logarithmic vertical scale, showing that the more events we observe, the larger the biggest event becomes.  The maxima are shown by arrows in the left-hand plots and by histograms at right.

So even if nothing changes, the deepest recorded flood will on average become bigger over the years, though there may be long periods when no new record is set.  The extreme value of a distribution is in fact a poor guide to the trend.  To see properly what is going on we should look at the mean or the standard deviation, or some other function that uses all the data, rather than one value.  Note how the range of the histogram does not decrease much as we add more events.  The mean and standard deviation are known more and more accurately as we add more data, but the the extremes never become good variables, unless the distribution is strictly bounded.  An example of a bounded distribution would be the timing of the first goal in a football match, which can never happen before the start of the game.

For a normal distribution also, the range is a poor measure of the size of the distribution.  The reason that the maximum is ill defined is found by considering 1000 sets of 1000 events, and the same events in one group of a million.  Clearly, one of the groups of 1000 must include the biggest event in the million.  If you look carefully at the fifth histogram you will see signs of artefacts, always a danger with rare events in statistical calculations.

The pictures below were taken during flooding of the river Severn near Gloucester.  Some older pictures are included for comparison with more normal conditions.  The first two show the road between Westgate and Maisemore.  One picture shows the camber of the road: the second includes the depth indicating posts, showing over two feet of water on the road.  It also shows clear standing waves as the water flows off the road at left.  The third and fourth, taken a few days later, show about eighteen inches less water, and a cyclist making bow waves.  Strangely, the river Severn, the proximate cause of the problem, is on the left, yet the flood water is coming from flooded fields on the right.  They received the water from the river upstream.

Here are some pictures of bridges and their flood spans in the Severn flood plain.  They all have to cope with flooding.

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