Trillions of sums, but we can't predict a white Christmas

The 'big freeze' is set to return today, but forecasts that look even 10 days ahead always need an element of luck

When it comes to dreaming of a white Christmas, don't believe all you read in the papers – such as the assertion, prominent yesterday, that it's certain to happen. It might, say the latest forecasts. They also predict the return of the "big freeze" today, which may or may not last as long as a month. But the truth is that modern weather forecasting is all down to sums – and to forecast accurately a weather event 10 days away is simply too big a sum to carry out. Even for the brightest mathematical prodigy.

To calculate current forecasts, which stretch six days or 144 hours ahead, half a million pieces of data have to be put into a mathematical model of the atmosphere run on one of the world's fastest supercomputers at the Met Office's HQ in Exeter.

The computer then performs 7,500 trillion calculations every minute for 90 minutes to carry out the forecast, for a total of 675,000 trillion calculations (or 675 times 1015). The further out you go, the more the number of sums increases, and the less reliable the forecast becomes. Ten days away is simply too far, the Met Office says, for any forecast to be accurate.

Gone are the days when forecasting was based on checking the barometer, poring over old records, looking at present weather conditions and glancing quizzically at the sky. For the past half-century, since the advent of modern computers, foretelling the sunshine, the wind and the rain or snow has been based on a system known as numerical weather prediction (NWP), which tries to make what were once educated guesses into a more exact science.

NWP takes observations of conditions throughout the atmosphere at a given moment, such as the wind speed, temperature, humidity and barometric pressure, and calculates how they will influence each other, using the laws of physics, to develop a weather pattern at a subsequent moment. It was first conceived by the British mathematician Lewis Fry Richardson in 1922, but it was not until modern computers became available in the 1950s that the huge amount of data necessary to model atmospheric conditions could be handled in real time.

Since then, the science has steadily improved with increasing computer power, and the Met Office says it can now give a five-day forecast as good as its two-day forecast was 30 years ago. But it can never be perfect – because to be perfect, you would need an infinite amount of data.

It's all down to the fact that a tiny difference at the beginning of an enormous calculation can make at enormous difference to the calculation's end, a phenomenon known as "sensitive dependence on initial conditions".

This is sometimes illustrated with the much-misunderstood metaphor of the butterfly's wing – the idea that the flap of a butterfly's wing in China can eventually be responsible for a hurricane in the Atlantic.

It does not mean that the butterfly has directly caused the hurricane, rather that the flapping of its wings one way or another – a tiny difference at the beginning of a process – might have a huge effect at the end, such as the presence or absence of a hurricane. The atmosphere is continuously full of tiny movements and forces, and, to model their reaction upon each other and predict the outcome, you would have to know them all. "The atmosphere is very complicated," said Dave Britton, a Met Office spokesman. "If you wanted accurately to predict the weather every time, you would have to know where every single air molecule was and where every single water droplet was and how the wind was blowing in every part of the atmosphere. The amount of data would in effect be infinite, and then you would have to calculate how it would all work out."

Meteorologists are surprised at just how accurate they can be, at short time scales of up to six days, using an amount of weather data a long way short of infinity.

It is collected by constructing a grid across the world, one "box" of which is shown in the graphic, which goes up into the atmosphere for up to 70 levels (and also down into the ocean for up to 30 levels.)

The illustration is of a grid of 100km squares, but actual forecasts are done at a finer resolution: 40km squares for the world weather, and 12km squares for Britain.

In terms of accuracy, the Met Office says the next-day forecast is "good advice 85 to 86 per cent of the time". But the farther out you get, the more the accuracy diminishes, and yesterday's widely reported assertion by an independent weather forecaster based in Wales that a snowy Yuletide is a racing certainty is regarded as scientific nonsense.

"It's simply too early to tell," Mr Britton said. "The six- to 15-day forecast looks like it will remain cold, but which will be the less cold days, and whether it will rain or snow on Christmas Day, is very difficult to say."

Forecasting: A history

A red sky at night, they used to say, is a sailor's delight (or a shepherd's) - that is, a guarantee of fine weather on the morrow.

That sort of folklore, the observation of repeated patterns and the hope that they would repeat again, was the basis of attempted weather forecasting for thousands of years. It was not until the mid-19th century that scientific weather forecasting was developed, led by Admiral Robert FitzRoy, the senior naval officer who 25 years earlier had commanded the survey ship HMS Beagle on which Charles Darwin sailed to South America as resident naturalist.

In 1854 Fitzroy was appointed as head of what eventually became the Meteorolgoical Office and began the systematic and scientific collection of weather data; the invention of the electric telegraph was also found to be a great help in forecasting as it could quickly send news of distant weather conditions.

But it was to be another century before the advent of the computer was to make possible numerical weather prediction, the method which turned forecasting (at least in the short term) into something approaching an exact science.