Climate Change by Numbers

This is the story of climate change.

But told in a way you've never heard before.

Because we're not climate scientists.

We're three mathematicians.

And we're going to use the clarity of numbers

to cut through the complexity and controversy

that surrounds climate change.

Understanding what's happening to the Earth's climate

is perhaps the biggest scientific endeavour

the human race has ever taken on.

From the masses of data, we've chosen just three numbers

that hold the key to understanding climate change.

0.85 degrees.

95%.

And one trillion tonnes.

Just by looking at these crucial numbers,

we're going to try and get to the heart

of the climate change controversy.

They are three numbers that represent what we know

about the past, present and future of Earth's climate.

And it's not just the numbers themselves that are important.

The stories behind them, how they were calculated,

are equally intriguing and revealing.

Ignition sequence starts...

We'll see how the methods,

using everything from the Moon landings...

..to early 20th century cotton mills...

..and motor racing have fed into the numbers we've chosen.

These three numbers

tell an extraordinary story about our climate...

..and take us to the limits of what it is possible for science to know.

Every minute of every day, all over the planet,

scientists are collecting data on the climate.

Around 10,000 weather stations

monitor conditions at the Earth's surface.

Some 1,200 buoys and 4,000 ships

record the temperature of the oceans.

And more than a dozen satellites

continuously observe the Earth's oceans and atmosphere.

All science starts with collecting data.

And when it comes to our climate, we've got masses of it.

But what story about our planet is all that data telling us?

Thousands of scientists are trying to answer that question.

Their results are summarised in a series of huge reports

by the Intergovernmental Panel on Climate Change.

The three numbers we've chosen all come from the IPCC's reports.

Molly!

I'm Doctor Hannah Fry and I use numbers to reveal patterns in data.

I'm looking at one number that answers a critical question...

is climate change really happening?

Our first number is 0.85 degrees.

Now, this number represents what we know about our climate

in the recent past, because it's the number of degrees Celsius

that scientists say our Earth has warmed since the 1880s.

But how can they be so precise?

After all, our climate is complex and extremely varied.

Temperatures change from season to season...

..place to place...

and, even, minute by minute.

As if it wasn't hard enough

to try and find an average temperature of the Earth for now,

we also need to go back in time and compare it to

the average temperature of the Earth in the past,

when we didn't have the luxury of modern measurement techniques.

Working out how the planet's temperature has changed

over more than a century is a huge challenge.

It's a bit like trying to work out

the route I'm taking across this park,

if you only had the route Molly is taking to go on.

You have to identify the trend, my path,

from all those changing temperatures, Molly's path.

And it all starts with the quality of the data.

Now, that's not such a problem for the recent past.

But what about further back in time?

Up until the middle of the 19th century,

the temperature record, as measured by instruments,

is patchy and unreliable and there is some controversy

about how you reconstruct temperatures before this time.

But the record improves from the 1880s...

..due to the efforts of one man.

Now, the key man in this story, the man with a plan,

is a guy called Matthew Fontaine Maury.

Now, Maury was a lieutenant in the US Navy

and from even when he was a small boy,

was obsessed with mathematics and data and analysis.

But, in 1839, Maury had a coaching accident

where he broke his thigh bone and dislocated his kneecap.

And while he was recovering,

he spent his time studying captains' log books.

And the data that he found there

set the path for his next 14 years-worth of work.

So much so, that on 23rd August in 1853 he called together a meeting

of 12 countries surrounding the North Atlantic,

all to talk about one thing.

He wanted to improve the way that data about the oceans was collected.

Captains record all sorts of information in their log books,

things like wind speed and direction,

or the speed and temperature of the sea currents.

Now, this wasn't just interesting to Maury from a scientific perspective,

but also because it was something he could sell

to commercial ship owners.

He found great commercial success

from mapping the position of major sea currents,

like the Gulf stream,

which enabled ships to use the currents to travel faster.

But there was a problem.

Different sailors took the same measurements in different ways.

That was particularly true

for one of the measurements climate scientists are interested in,

sea surface temperature.

Now, the way to measure sea surface temperature

is actually surprisingly simple.

All you do is chuck a bucket over the side of the ship

and get the temperature from it.

But, the problem is that the result that you get

actually depends quite a lot on the type of bucket that you use.

So, let me just take the temperature of this now.

And, in the meantime, I'm going to throw this guy over.

In the early 19th century, some sailors used wooden buckets.

Others used buckets made of canvas.

This meant that the measurements were not consistent.

The wooden bucket is coming out as a surprisingly warm 15.1.

And if we make a comparison,

the canvas bucket, unlike the wooden bucket, isn't insulated,

so things like the air temperature

are going to make a much bigger difference.

The temperature has dropped below 15.1 degrees.

It may not sound like a lot,

but even tiny discrepancies undermine the accuracy of the data.

Now, Maury knew this and so, at his conference in 1853,

he came up with a standardised way for everyone across the world

to measure sea surface temperature.

He wanted everyone to use wooden buckets

and designed special forms for them to fill in with all their data.

Maury also introduced standardisation

to air temperature measurements on land.

That's why our 0.85 degrees Celsius figure is measured from 1880.

It's the date from which the temperature data

is generally well-standardised.

But, despite Maury's efforts, the data was still far from perfect.

Not everyone stuck to the rules.

For example, over time, canvas buckets made a comeback

because they were lighter.

So, there were still errors, some of which were pretty obvious.

So, here is the sea surface temperature data

between 1880 and 1980.

And the first thing that you really notice about this graph

is this huge spike that happens,

where it looks like the sea surface temperature's raised

by 0.8 degrees Celsius.

Or, at least, it looks that way

until you realise that this spike happened in 1941

when, during the Second World War,

understandably, sailors didn't much want to go up on deck

with a torch and a bucket to record sea surface temperature levels.

So, instead, during that time,

they used the water that was coming in through the engine room,

which is hence why the data is a lot higher.

Now, after the Second World War,

people gradually started returning to using uninsulated canvas buckets.

But, unfortunately, we don't know who was using them or when.

And so, in all of this big mess of data,

how do we get accurate temperature readings

for land and sea from the past?

The answer is related to a mathematical technique

that was used to help solve one of history's greatest challenges.

At Cape Kennedy, it's a wonderful day for a wonderful event,

the first manned flight to the Moon.

In a mission fraught with difficulties,

one of the biggest was how to navigate

a quarter of a million miles through space to the surface of the Moon.

It's a feat of navigation all the more astonishing

when you consider how difficult finding our way around can be

even down here on the ground.

Working out exactly where you are on the Earth at any point in time

is, actually, a surprisingly difficult problem,

especially if you want really, really precise information.

It's tricky because tracking your position,

just like measuring temperatures over time,

is prone to error.

Not the easiest thing ever.

Take dead reckoning,

timing how long you've travelled in a particular direction

from your last known position.

Three miles an hour. Lovely. Three miles an hour. Hang on one second.

It's easy to drift off course as inaccuracies build up.

Hang on...

Even more hi-tech methods can get it wrong.

Actually, the GPS is putting us over there at the moment,

which is less than ideal.

So, when it comes to navigating,

the problem is which measurement of your position do you trust?

In the 1950s, a young Hungarian-born mathematician, Rudolf Kalman,

devised an elegant algorithm to solve this problem.

Kalman's method uses matrix algebra,

and takes into account all of the errors to give you

the best possible estimate of your position at any point in time.

So, how does Kalman's method work?

In 1969, NASA gave it its ultimate test

in the mission to land men on the Moon.

Ignition sequence start.

Six, five...

Navigating in space poses particular challenges.

We have lift off. Lift off on Apollo 11.

The spacecraft was being tracked by four radar stations on Earth.

Onboard instruments were also estimating its position.

But, each of these measurements could be wrong.

So, how could NASA be sure of Apollo 11's position?

Go. Contol, go.

This is where Kalman's algorithm came in.

Moment by moment,

it compared each position measurement with the others...

..looking for differences that fell outside the expected margin.

We're go. Same time. We're go.

If the algorithm had found significant disagreement,

the mission would have been aborted.

But, it didn't.

And the rest is history.

Tranquillity Base here.

The Eagle has landed.

So, this process is now known as Kalman filtering

and has been used in everything from cleaning up grainy video

to looking for trends in economics.

And a lot of the underlying principles are exactly the same

as you see in the processes used for climate science.

So, knowing when to trust your data

and picking out when the errors are big enough

to flag up a deeper underlying issue.

But, the process in climate science

is, instead, known as homogenisation.

Homogenisation has allowed climate scientists today

to clean up data gathered in the past.

Unreliable measurements can be corrected or discarded.

So, what the homogenisation process is doing, effectively,

is taking all of the data from all of the weather stations

and comparing it on a day-by-day basis.

Now, in doing that,

if a particular data set starts to look a bit unusual,

it will really stand out.

You can see what happens when scientists homogenise a data set

by looking at how they corrected

the unusual jump in sea surface temperature in the early 1940s.

So, once you've applied this homogenisation process,

here is what the sea surface temperature data will look like.

So, we have the original data here in yellow

and the cleaned up version also available in blue.

Now, the first thing that you notice

is that the big jump that we had in 1940 has dramatically reduced.

There is still a bit of a jump

because there was an El Nino that year,

which meant that the sea surface did actually warm.

But the jump that was down to the difference in measurements,

the error in the way that people were measuring,

has been taken away completely from the graph.

All the big scientific groups that work with climate data

use homogenisation methods like this

to try and clean up the records of past temperature.

And it's absolutely vital

that you account for some of these errors in measurement

that have occurred in historical data, otherwise you've got no hope

of finding any kind of underlying patterns in your data.

But, inevitably, as soon as you start applying

these mathematical recipes to clean things up,

other people will start accusing you

of building in biases into your data.

Perhaps the best defence against bias is scientists' own scepticism.

Many different groups work on climate data,

using slightly different homogenisation methods.

And all are subjected to searching scrutiny by their peers.

But, even after homogenising the historical data,

climate scientists face a further problem...

gaps in the temperature record.

Even today, we do not have temperature measurements

for the whole planet.

If you look at where we have temperature data for,

if you split the Earth into a grid,

it becomes very obvious that there are some areas

where we have much more information on than others.

The black squares show where we have hardly any weather data at all.

So ,if you take the Arctic, for example,

it's very obvious there are almost no sample points in the Arctic.

The gaps in places like Africa and the poles can affect how we

calculate the average temperature of the whole planet.

Now, if you take an average across the whole of the Earth

and don't take into account the fact that

you have a lot less data for the Arctic, you're going to end up

with a really biased average and something that doesn't really

represent the Earth properly.

There is actually a mathematical solution to this problem

that climate scientists are beginning to use,

but it's one that wasn't even devised by a mathematician.

The attempt to fill in gaps in the temperature data

begins in the gold fields of South Africa in the 1950s,

where a mining engineer was grappling with a problem.

Danie Krige was in charge of the leases

of the country's very valuable gold fields

and was inundated by companies desperate to mine them.

But, until each plot of land had been mined,

he had no way of knowing how valuable each area would be.

What he needed was a systematic way of working out

how much each lease was worth and so turned to spatial statistics.

To understand the challenge Krige faced,

I've come to gold mining country, to Dolaucothi in Wales.

All Krige had to go on were a few scattered core samples

that had been taken across the gold fields

as miners tried to find more gold.

He had to find a way of working out

how much gold there was in each plot of land

with just these few measurements,

just like climate scientists

have to work out the temperature in places

where they don't have measurements.

So, what I'm going to do here

is show you how Danie Krige's method worked

using these as my core samples.

Imagine each of these poles represents a core sample

and the number of lights indicates the amount of gold found in it.

So, our first core sample is giving us a reading of 16 parts per million

all the way up there into the red.

And this core sample is giving us a reading...

..of only six parts per million.

Danie Krige's samples were often around a kilometre apart.

Climate scientists have weather stations

that might be hundreds or even thousands of kilometres apart,

especially in regions like the Arctic.

The problem in each case is the same,

how to fill in the gaps in the data.

So, one more core sample to do

and then I can show you the map.

So, our last reading is only giving us two parts per million.

So, we're still on the gold field,

but we're at a much lower grade of gold than we were before.

But the real question that Danie Krige wanted to ask was

how can you tell what happens in between the core samples?

How can you tell how much gold is in the middle?

His answer was to use maths

to take into account both the amount of gold in each sample

and the distances between them.

So, Krige's method would take the first exciting strike of gold

and look at how far away the neighbouring samples are,

as well as how high the levels of gold found in them are.

This helps estimate how much the gold levels drop off

around each strike.

The process is then repeated over the whole field.

It may not sound like it, but the maths is relatively simple.

Now, it's so powerful,

that this method has been used all across the world

in everything from looking at gold mines to forestry

and even temperature data.

And it's even been named after the great man himself,

now known as Kriging.

Kriging is now being used to throw new light

on the biggest recent climate change controversy -

what's happened to the temperature of the planet

since the turn of the century.

The issue is how you account for gaps

in the record of global temperature.

If you take the UK Met Office's Hadley Centre, for example,

and their data on the changing global temperatures

in the recent past,

they leave blanks in regions where they don't have any information.

But, if you look at the temperature set,

you can see that it demonstrates an effect that's become known

as the pause,

which is that the temperature of the Earth

doesn't appear to have risen since the year 2000.

This pause in the Earth's rising temperature is controversial.

Some climate change sceptics say it shows

that global warming is not real.

But most climate scientists

say they would expect pauses every now and again

within a warming trend.

But whether there even is a pause

depends on how you account for the gaps in the temperature record.

When this data set was Kriged by an independent scientist in 2014,

so that they could take into account

the little data that you have in the Arctic,

he found that the graph changed.

Kriging put more weight

on the few temperature points we have from the Arctic

and there the temperatures are rising fast.

The impact of Kriging on the original incomplete data

is to turn the pause into a small temperature rise.

Now, you might think

that this doesn't necessarily represent reality, either.

But it does demonstrate an important point.

What you do with your data

has an impact on how you make your conclusions.

It's not to say that Kriging the Arctic figures

has really shown that there isn't a pause.

It remains an area of debate.

But, techniques like this offer scientists the only way they have

to overcome the inevitable limitations of incomplete data.

It doesn't matter how much effort scientists go to,

temperature data will never be perfect.

And the trouble is, mathematical manipulation of the raw data

can look like fiddling the figures.

But the techniques that climate scientists have used

are well-understood,

they're open to scrutiny

and they all lead in the same direction.

Three major research groups

have contributed to the IPCC's reconstruction of past temperature.

They've each used slightly different methods

to clean up the historical data

and account for gaps in the temperature record.

And here are their results.

So, in the top left-hand side, you have the results from

the Global Historical Climatology Network.

Top right, you have the results from

the Goddard Institute of Space Studies.

And in the bottom left, you have the results from

the Met Office's Hadley Centre.

Now, just these three graphs show pretty similar results.

They all seem to be showing a very similar shape,

especially when you take into account the fact

that all of the groups were using different techniques.

From there, how did the groups arrive

at an average temperature rise?

This bit is surprisingly simple.

Now, rather than all of the zigging and zagging,

the groups put a line through each of their graphs

and from there it's very easy to just read off

how much the temperature has risen.

These three lines show the trend in the average temperature

since 1880 for each data set.

But the IPCC then took the average of each of these three lines

and came up with the value of 0.85 degrees Celsius,

the most accurate measure that we have

for how much the Earth's temperature has risen by since 1880.

That doesn't mean it's perfect.

The exact figure is always going to be uncertain.

Scientists have done their best to try and compensate

for imperfections in the historical temperature record.

They've applied mathematical methods

to patchy, unreliable and erroneous data.

Now, 0.85 degrees is, itself, just a symbolic figure.

I could have averaged the data in several different ways

and ended up with a slightly different figure every single time.

But that's not really the point.

Looking at how this number is produced,

you can see that it doesn't matter how you collect your data,

how you measure your data, or how you treat it,

one point still stands overall -

the Earth's temperature has been rising in the last 130 years.

Different groups using different techniques,

each scrutinising the others,

have all arrived at pretty much the same conclusion.

That's why it's now relatively uncontroversial

to say that the Earth's temperature has risen

by just under a degree since the 1880s.

There's far less agreement, though,

on the answer to the big question all this raises -

why did the Earth's temperature rise?

We're going to look at a very different number,

a number that answers one of the most difficult

and controversial questions

in the whole climate change debate.

Just to what extent is the rise in temperature

caused by human activity

and to what extent is it caused by just natural fluctuations?

Scarf! Souvenir! Hat, scarf or the badge!

Get your colours, lads, here!

I'm Professor Norman Fenton...

-Hi, there. How are you?

-Hi.

..a mathematician and lifelong Tottenham Hotspur fan.

From financial services to transport and even football,

I use numbers to work out the most likely causes of different events.

The climate change number I'm looking at

is all about cause and effect.

The scientists have made a big statement.

They say they're 95% sure of the main cause of the Earth's recent warming.

And that cause, they say, is us.

All science involves identifying not just what is happening,

but also why it's happening.

When it comes to the climate,

scientists say they're 95% sure

that over half of the warming in the last 60 years

has been caused by humans.

How can they be so sure?

Well, by using statistics,

we can analyse the most likely cause of something,

whether that's success at football or climate change.

I've been coming to Spurs for over 50 years

and, I have to say, this isn't one of their finest seasons.

But, like most fans, I'm pretty confident

I know which factors are going to be most important

for determining whether they'll play better or worse than expected

in any given season.

Unsurprisingly, there's no shortage of opinions here.

Definitely the manager, you know?

They need to respect the manager.

The manager needs to have, like, respect of the players as well.

If you've got the tactics right

and you've got the players in the right places where they should be.

Well, you need a very good executive board.

Your players need to stay very fit.

Your manager needs to be focused, have a very good philosophy.

Beyond opinion, there is a way to use maths

to work out which factors are the most crucial.

It's called an attribution study

and it's what the IPCC did to arrive at their 95% figure.

All attribution studies start with identifying the factors

that might cause an outcome.

Let's take footballing success.

Here I've got lots of statistics on all the Premiership teams

going back many seasons.

It's interesting, looking at the league tables,

to see how the performance of a team will vary from season to season.

I want to understand which of many possible factors

are the most important cause of this.

Is it the length of time the manager's been with the club?

Is it the injury rate?

Is it how much they spend on players?

I'm going to put all those factors together with many others

and plot my own attribution study.

'It's another bad day for Tottenham at White Hart Lane.

'Full time, Tottenham 1, Stoke 2.'

To work out why some teams win and some lose,

we need the second part of the attribution study.

The different factors we've identified

that could affect the team's performance

are put into a mathematical model.

It's the same process climate scientists use

to try to work out what is driving climate change.

I can now check the accuracy of my model

against teams' past performance.

So, what I've got here, for example,

is I've taken one of the teams, Manchester City,

and I've plotted the actual performance

in terms of points that they achieved in each of the last few seasons.

Now, we look at what the model would have predicted

and you can see it's actually a pretty good prediction

of what actually happened.

And this is true for all the teams in the Premier League.

Now I know I can trust my model,

I can move on to the clever bit -

isolating the factors that make the most difference

to the team's success.

I found that there was one factor

which had far greater impact on performance than any other,

the wage bill.

If I take out the wage bill factor,

it's no longer a good estimate at all.

It's quite a long way off.

And, in fact, we can repeat that for all of the other teams.

Using the same methods as the IPCC,

I can even put an actual figure on how big an effect the wage bill has.

I can say there's a 95% chance that,

if you increase the wage bill by 10%,

there'll be at least one extra point per Premiership season.

I can be so confident because the answer's so clear from my model.

But how can climate scientists be equally sure of their results?

After all, what drives changes in the Earth's climate

is one of the most complex puzzles scientists have ever tried to unlock.

Before trying to work out the impact humans have,

scientists have to account for natural variations

in the Earth's climate.

The key science involves a number of factors,

all of which play a role in changing the climate.

If this was a court case, they'd be our suspects.

Many natural factors are known to cause changes to the climate.

They include the sun.

The energy it emits varies

and this can change the temperature here on Earth.

Volcanic eruptions.

The vast gas clouds they throw up

can cause sharp global cooling

as they affect the chemistry of the upper atmosphere.

And climate cycles, like El Nino,

that can cause global temperature fluctuations lasting many years.

But climate scientists say they're 95% sure that recently,

all these natural factors have been overshadowed by one other.

For most climate scientists, there's one prime suspect in this case - us.

And that's because of a colourless, odourless gas called carbon dioxide

that we're pouring into the atmosphere.

One of the first people to try and unravel the role of carbon dioxide

on changing the Earth's temperature

was a depressed Swedish physicist called Svante Arrhenius.

Arrhenius wasn't interested in the Earth's warming, however,

but cooling.

In 1894, Arrhenius' marriage broke up.

Searching for distraction,

he set his mind to one of the great mysteries of his time,

the origin of the ice ages.

Scientists had long wondered

how the great mountain landscapes of Europe had been formed.

Once, the rugged valleys were thought to be the relics of a biblical flood.

But, in Arrhenius' time,

it was realised that the Earth had been beset by periodic ice ages

over the last 2.5 million years.

On trips through northern Europe,

he studied the vast glacial landscapes that surrounded him

and wanted to know how the Earth could possibly have undergone

such monumental change.

What had caused the planet to cool down so dramatically?

Scientific understanding advances by developing theories

and then testing them.

It was already widely accepted that so-called greenhouse gases

worked like a huge blanket around the Earth, keeping it warm.

Arrhenius developed a theory

that changes in the concentrations of these gases,

in particular carbon dioxide,

might also have caused the planet to cool.

The only way he could test his theory was to use maths

to work out the relationship

between changing levels of carbon dioxide in the air

and the Earth's temperature.

It was painstaking work.

Every calculation had to be written out by hand.

Arrhenius himself described it as tedious.

But, eventually, he had his answer.

He predicted that a halving of carbon dioxide in the atmosphere

could lower the temperature by over four degrees

and, perhaps, trigger an ice age.

Almost as an afterthought, he also calculated

that a doubling of carbon dioxide

could increase the temperature by the same amount.

Eventually, it would turn out that changing carbon dioxide levels

weren't the main cause of the ice ages.

But, using maths, Arrhenius had established

the crucial underlying relationship

between carbon dioxide in the atmosphere

and the temperature of the planet.

Much of Arrhenius' efforts and the related work that follows

can be summarised in one simple equation.

This enables you to calculate the heating effect

that comes from raising carbon dioxide above its base level.

It's one of the fundamental building blocks of climate science.

The equation shows that the heating effect, represented by Delta F,

rises in proportion to the amount of carbon dioxide in the atmosphere.

Put simply, you can't raise carbon dioxide levels

without heating the atmosphere.

But there are many factors that influence the climate,

each with their own equations.

The rate of energy coming from the sun.

The cooling effect of volcanic eruptions.

Human pollution from things such as industry and agriculture.

Ocean currents. Cloud cover. Wind speeds.

All of which influence each other in a web of complex interactions.

Unlike my football study,

modelling the climate is unbelievably complicated.

So, how do climate scientists create a model

that accurately represents the complex interactions

of all these different factors?

The answer comes from the very earliest days of weather forecasting.

It's going to be a dull and wet start to the day.

Well, after a few quite exciting days of weather,

today's been a bit nondescript.

And, as the day goes on, I think you're going to find

these showers will become heavier and more frequent

and many of them could well turn out later on in the day

to be fairly thundery with some...

-Oh, these

-BLEEP.

-Let's do it again.

One of the earliest pioneers of weather forecasting

was a man called Lewis Fry Richardson.

At the start of the 20th century,

he set out to revolutionise weather forecasting using maths.

Our climate is governed by the circulation of the atmosphere

and Richardson recognised just how complex this system was,

declaring that, "the atmosphere is like London.

"There's more going on than anyone can properly attend to."

Yet, despite this complexity,

he wanted to find a way to unravel its secrets.

Richardson had an idea of how to do this that was revolutionary.

Using the rows of the theatre as his template,

he thought of dividing the world into grid squares.

This would break the problem down

into a series of discreet and achievable tasks.

He imagined positioning people within each square

would only have to solve the calculations

relevant to the weather in their area.

A director, standing at the centre,

would take in the results of all the calculations to form a forecast.

Richardson made just one attempt to put his ideas into practice,

retrospectively trying to calculate

the weather over Europe for a particular day.

But his calculations took him six weeks to complete

and they were far from accurate.

Despite this failure, Richardson was ahead of his time.

By dividing the world into grid squares,

he had made the crucial theoretical advance

that would not only revolutionise weather forecasting,

but also allow scientists to model the climate.

All that was needed was enough computing power

to put it into action.

Fry Richardson had calculated

that he'd need over 60,000 people using slide rules

in order to predict the next day's weather before it arrived.

I'm sure he wished he'd had access to this,

the world's most powerful meteorological super computer,

part of the European Weather Centre here in Reading.

It may be noisy,

but it can perform over one thousand trillion calculations every second.

The world's biggest super computers are now used to model the climate.

Just like Richardson, they divide the world into a grid

and solve the complex equations

governing the climate for each square.

As computers get more powerful,

the squares get smaller

and the models get better at representing reality.

No computer is ever powerful enough to simulate it

in as much detail as scientists would like.

But this method has allowed scientists

to build a model for factors that affect the climate,

the crucial second step of an attribution study.

However impressive our super computers are,

however much the climate models exploit

the very limits of our technology,

climate modelling remains a simplification.

Which raises the question -

how can scientists be confident

that their simplified models accurately capture reality?

When I made a model for football success,

I was able to check it against the past results of dozens of teams.

But climate scientists have only one Earth

and one set of past data to check their models against,

so they're always looking out for new opportunities to test their models.

In June 1991, they found a big one.

On the Philippine island of Luzon,

a volcano called Mount Pinatubo erupted.

It spewed 20 million tonnes of sulphur dioxide and ash

more than 12 miles up into the atmosphere.

It was one of the most devastating eruptions of the 20th century.

But climate scientists at NASA

realised it also offered a chance to test their climate model.

Could their model predict

the effects of the gases given off on the climate?

After adding the eruption into their model,

it predicted that, over the next 19 months,

there would be an average global cooling of around half a degree.

As the real data came in month by month,

it matched the model's predictions.

It was good evidence that climate modelling could be reliable.

Unfortunately, opportunities to test the models against data

are few and far between.

And, as a mathematician, I find that frustrating.

What's reassuring is that the underlying physics

on which the models are based is robust.

So, despite their limitations,

the models offer a powerful tool to identify the main causes of warming.

It's a process of elimination.

To show you what I mean, let's take the example of the sun.

If the cycles of the sun were a major cause

of the rise in temperature we've measured, then what we should see

would be all the layers of the Earth's atmosphere

warming together like this.

This is called a fingerprint,

a characteristic pattern

that would point to the sun's influence as the cause.

What we actually have from the measurements of the past 60 years

is that only the lower levels of the atmosphere have warmed,

while the upper levels have cooled.

So, what we're actually seeing in the atmosphere

is an entirely different fingerprint.

What the models show is that's a pattern which only fits well

with the main cause of the warming being human activity.

That's human activity primarily

in the form of burning fossil fuels

that release carbon dioxide into the atmosphere.

And since the 1970s, the human fingerprint has become more obvious.

From the loss of sea ice in the Arctic,

increasing frequency of heat waves,

to the warming and acidification of the oceans,

the models predict all of these patterns

only as a result of increasing greenhouse gases like carbon dioxide.

The evidence that human activity

is the major cause of recent warming is compelling.

But the models can go one step further

and help us put a figure on the level of certainty behind this statement.

The yellow line on this graph is the real world data.

This is how much warming we've measured across the world since 1951,

0.6 degrees.

Firstly, in red,

let's look at how the climate models

expected the global temperatures to change

when taking into account all known factors.

The shading shows the amount of fluctuation around the average

that they would expect to happen.

The most obvious features are a general rise,

the result of the increasing carbon dioxide levels,

with some sharp dips caused by big volcanic eruptions.

But look what happens if we run our models

without any human influences like greenhouse gases.

So now, only natural forces are included in our model data.

The line doesn't match the real data well at all.

This is what the model suggests our climate would be like

if there was no human impact on it at all.

The models say that, without any human influence,

global temperature would not have risen significantly

over the past 60 years.

It's as clear as taking the wage bill out of my football prediction.

The models also help scientists put a figure

on how certain they are of human impact on the climate.

From the models, they found there was a greater than 99% probability

that more than half of the warming was due to human activity.

Given that high level of certainty,

how did the IPCC arrive at its slightly lower 95% certainty figure?

All of us who work with mathematical models

know that they're simplifications,

so we have to take into account their limitations.

That's why the IPCC downgraded its final conclusion

from 99% to greater than 95% sure

that humans have caused more than half the recent warming.

All science proceeds by producing theories and then testing them.

But, when it comes to our climate,

it's impossible to test the influence of different factors on the planet.

That's why scientists have turned to maths to help model the climate.

It's not perfect,

but it is the only way to put a figure

on how sure we are the Earth's warming is down to human activity.

Come on, you Spurs!

Years of scientific research and statistical analysis

have brought us as far as 95%

and that's close enough for most people to believe it.

That just leaves the question -

what's going to happen with our climate in the future?

I'm Professor David Spiegelhalter

and I use numbers to try to help organisations

like the Health Service predict the future.

I'm looking at one number

that aims to give us a clear guide

to how our actions now might affect the climate.

The number I'm looking at is one trillion.

This rather unimaginably big number

may be crucial to the future of our planet.

It's the best estimate that climate scientists have made

of the number of tonnes of carbon that we could burn

before we run the risk of causing what's been called

dangerous climate change.

That's defined as an average warming across the globe

of more than two degrees Celsius.

All fossil fuels contain carbon.

When we burn them, it converts this carbon

into the carbon dioxide that warms the atmosphere.

So, the trillion tonnes figure

puts a limit on the amount of fossil fuels we can burn.

In effect, this gives the world a budget.

It says that, if we want to avoid a two-degrees rise,

then we can't afford to spend, or burn,

more than a trillion tonnes of carbon.

And that's a total going right back

to the beginning of industrialisation.

A trillion tonnes sounds like a lot.

But, the trouble is,

we've already burnt around half a trillion tonnes

and that's given us almost a degree of warming.

And if we carry on the way we're going,

we'll burn the other half a trillion tonnes

in about 30 years.

The implications are profound.

We've already identified several trillion tonnes

of fossil fuel reserves buried inside the Earth.

So, to keep warming below two degrees

will probably mean leaving most of those reserves in the ground.

Before we take such drastic action,

I'd like to know a bit more about the trillion tonnes figure.

Where does this number come from?

And how much confidence should we have in it?

The one trillion tonne limit

is based on being able to predict the future.

That may make it sound unscientific

but, for centuries, people have been working on ways

to make predictions using statistics.

The history of statistics and prediction

has been driven by incentives.

In fact, the first people who worked on probability and statistics

were either advising gamblers or pricing up pensions.

So, I think, if you really want to know who's making good predictions,

look at people who are putting their money where their mouth is.

And there's a lot of money in motor racing.

And a lot of effort to try to predict the future,

because winning isn't just about driving fast.

It's also about making the right decisions,

what to do as the race unfolds,

the weather changes

and the unexpected happens.

And this is where prediction and statistics comes in.

There are far too many variables

for the decision to be left to the driver

or, sometimes, even to the people at the race track.

It needs a dedicated race strategist.

In the 2005 Monaco Grand Prix,

Kimi Raikkonen was in the lead after 25 laps,

when there was a six-car pile up.

The safety car came out and the team had to decide very quickly

should Raikkonen come into the pits or should they leave him out there

until the race restarted?

And this would decide whether he won or not.

They didn't know what to do

and then a two-word e-mail came in from the chief strategist

who was in England.

And the e-mail said, "Stay out."

So, Raikkonen stayed out

and the people who came into the pits got all jammed up,

so Raikkonen went on to win the race.

And all because of the power of prediction.

So, how did Raikkonen's strategist

predict the outcome of different strategies?

They used to just use gut feelings, just their instincts.

But now, with the huge amount of data available,

they can do something much more sophisticated.

Throughout the race,

each car streams performance data back to the team...

..from tyre fatigue to fuel consumption.

The team then plugs this data into a mathematical model of the race.

They can constantly make changing predictions as the race proceeds,

as the positions change,

as the lap times change,

they can predict the possible outcomes

if they do a particular action.

You know, for example, just come in for a pit stop.

And then they can choose the strategy

that maximises the chance of the best possible result.

As Raikkonen's victory shows,

the predictions made by the models can be extremely powerful.

And it's only possible thanks to a mathematical technique

that we now use for all sorts of future predictions.

This technique that motor racing teams use

to decide what strategy

will maximise the chances of winning a race,

it's exactly the same

as the technique I use for medical predictions

and climate scientists use

to predict what might happen to the planet.

And it's all due to a stroke of misfortune

that befell one particular mathematician

just after the Second World War.

Coffee, please.

In 1946, brilliant mathematician and physicist Stanislaw Ulam

was struck down by a severe bout of ill health.

He was hospitalised for weeks

and only had the card game solitaire for entertainment.

The aim of the game

is to sort a randomly shuffled pack of cards into four piles,

according to a set of rules.

Whether Ulam could successfully finish the game

depended on the order of the cards he was dealt.

As he played, his instinct was to begin to pick apart the game

and analyse it mathematically.

He became obsessed with trying to predict

whether a game would be successful.

Ulam hoped he could calculate

the probabilities of different outcomes from the very first deal.

But he quickly realised this approach would get him nowhere.

The problem was there were just too many possible combinations,

leading to ever increasingly complex calculations and equations

that became impossible to solve,

no matter how brilliant a mathematician you were.

But Ulam didn't give up.

He came up with an entirely different kind of method

to solve the problem.

In fact, it was one that hardly involved maths at all.

Let me demonstrate with an analogy.

Ahead of me, between me and the wall,

I can just about make out a sheet of Perspex.

There's a hole cut out of the centre in a certain shape.

But, from here, there's no way I can tell what that shape is.

But I can find out with a little help.

Imagine that working out the shape of the hole

is equivalent to Ulam trying to predict outcomes in solitaire.

There's no way I can work out the answer with maths.

I need to play the game.

In this case, the equivalent of a round of solitaire

is a shot with a paintball gun.

After a couple of shots, I've got a few through against the wall.

But, although I know there is a hole in the Perspex,

I've still no idea what the shape is.

It's like after I've just played a couple of rounds of the game.

I'm still none the wiser about what outcomes to expect.

But if I do this...

Now, with enough shots,

I'm beginning to get a picture of what the shape might be.

Looks like a rough sort of diamond to me.

It's great fun!

This is the same principle as playing the game thousands of times.

With enough sample runs,

we can begin to build an idea of what outcomes to expect.

At first sight, it sounds like no solution at all.

Who could sit around actually playing solitaire millions of times

to find an answer?

What turned Ulam's ingenious idea into a useful tool was his timing.

The computer had just been invented.

That meant you didn't have to play the game for real.

Instead, it could be played hundreds of times inside a computer

and the computer could then say

which starting hands were most likely to lead to a successful game.

He called his technique the Monte Carlo method,

after the casino where his uncle made

so many repeated and random attempts to predict the future.

And it turned out to have uses

well beyond predicting the outcome of card games.

The beauty of Monte Carlo is that, even in complex systems,

it tells us not only what is likely to happen,

but how likely it is to happen.

In climate science,

the equivalent of shooting the paintballs

is running climate models hundreds of times.

Each time a model is run,

it comes up with a different prediction

about the future of the Earth's climate.

The results of many different climate models can then be combined.

As the models are run over and over again,

we can see where the results cluster.

This is the Monte Carlo process in action.

The pattern of lines shows us a range of possibilities.

But not only that.

Where the lines are densest,

this is what the models are saying is the most likely thing to happen.

Of course, it doesn't tell us exactly what the future holds.

That would be impossible. The future is inherently unpredictable.

But, the Monte Carlo method gives us an idea,

not only of what the outcomes might be,

but how likely they are.

And this is where our crucial number comes from.

The graph shows the model's predictions

of how much the climate will warm

as a result of us burning one trillion tonnes of carbon.

The most likely outcome

is just below two degrees Celsius of warming.

Now we understand where the one trillion tonnes figure comes from,

we need to consider the second part of the prediction.

Why should we worry about a rise of two degrees Celsius?

When we think about what a small average rise

in global temperature might mean to us humans,

perhaps the first thing to think of is weather,

because we don't experience climate on a day to day basis,

we experience weather.

And, sometimes, it can hit us really hard.

Just a small average temperature rise

can hide very noticeable changes in weather,

especially dangerous extremes.

As the average rises,

the tropics will experience more devastating rain storms,

whilst areas, including the Mediterranean,

will have more droughts.

Britain will suffer more flooding.

But it's not simply the fact

that these events might be more frequent that is a concern.

It's critical that we understand extreme weather events.

How often and how hard they might hit us.

And this is the worry with a climate that might warm

by as much as two degrees, that it would disrupt that ability,

because the method we use to predict extreme weather

uses a particular type of statistics

that's very sensitive to this type of change.

A method, in fact,

that was developed for something quite different.

In the early 1920s, the cotton mills of England were a vital industry.

But the looms often sat idle for as much as a third of the time.

The problem was that cotton threads kept snapping.

Every snap could stop production for hours,

so they needed to work out why the threads broke

and what they could do to stop it happening.

Fortunately, they had the good sense to call in a statistician.

The newly-formed British Cotton Industry Research Association,

charged with improving all aspects of the industry,

dispatched one Leonard Tippett to investigate.

Tippett admitted he was woefully inexperienced.

But, in the best traditions of British statistics,

he managed to combine careful collection of data

with elegant statistical analysis.

He toured the mills of Lancashire,

carefully recording breakage rates

and studying the strength of individual fibres.

Common sense said that,

if the average strength of the fibres in one thread

was higher than another,

you'd think there'd be fewer breakages.

But Tippett discovered

that it wasn't the average strength that was important,

it was the weakest thread that really mattered.

It's like the old saying, a chain is as strong as its weakest link.

It's the extremes that make all the difference.

Tippett's breakthrough came when he realised

he could use the data he'd gathered

about the strength of the most common threads

to predict how often the very weakest threads would be found.

In other words, he'd invented a method of using numbers

to predict extreme events from the spread of less extreme events.

Tippett's insights from the cotton industry

led to what is called extreme value theory

and it turned out to be amazingly powerful.

What used to be considered just unpredictable

could be analysed mathematically.

And his breakthrough turned out to be vital

in the understanding of extreme weather.

In 1953, a huge storm surge

was driven down the North Sea towards London,

devastating coastal areas.

Over 300 people died in Britain alone.

After the floods,

it was decided something had to be done to protect London,

in case it ever happened again.

It was time to put extreme value theory to the test.

This extraordinary piece of engineering

was conceived in the 1960s

after catastrophic and fatal floods in 1953.

These really were extreme events, literally, a perfect storm,

when rare conditions combined to create a terrible night.

In order to ensure the barrier would do its job,

the planners had to predict the most extreme storm surge

that could be expected in the future,

more extreme and unusual than anything that had been seen before.

Extreme value theory, together with classic British record keeping,

was the answer.

They had a century's-worth of data on extreme high tides.

And using Tippett's models,

this allowed them to gauge the chance of events occurring

that were so extreme they'd never occurred before.

The Thames Barrier was built to stop a once-in-a-thousand years event

and, so far, we've not seen a storm

come anywhere near testing its limits.

But Tippett's method has an Achilles heel.

Its predictions are based on the assumption

that the future will be similar to the past.

Extreme value theory uses the frequency of fairly extreme events

to give us a good idea of the chances

of really extreme events,

things we haven't observed, even.

But the problem with climate change is that the patterns alter.

When the planners were designing the Barrier,

they had 100 years of data about storms

to base their prediction of the 1,000-year storm.

But, if the climate changes,

the pattern of storms may well change, too.

And that will mean the data on past storms will no longer be relevant.

Without it, the predictions made by extreme value theory

will be unreliable.

The average shift might not actually seem that impressive.

But it's what happens in the extremes

that is so important to us

and these become a lot less predictable.

We can't just tweak the extreme value theory.

So, as our climate changes,

not only are we likely to suffer more frequent extreme weather,

we'll also lose the tool that has allowed us

to prepare for such eventualities.

Climate scientists have used maths and statistics

to give us their most likely prediction of the future.

Sticking to a trillion tonnes of carbon

should cause less than two degrees of warming.

But, given the inherent unpredictability of the future,

and the imperfections of our climate models,

how sure can we be that that prediction is right?

With the help of techniques like the Monte Carlo method,

the climate scientists have put a number on their certainty.

They are at least 66% sure.

That means there's a sting in the tail of the trillion tonnes figure.

Climate scientists tell us

that, if you burn a trillion tonnes of carbon,

they can be 66% certain that warming should stay below two degrees.

But there's another way of looking at this.

We could say that they think there's a one-in-three chance

that warming will be more than two degrees.

So, the rather sobering conclusion is that,

even if we burn less than a trillion tonnes,

we're not guaranteed to keep warming below this level.

Yet we also know that it would take huge changes to our lives

to keep to the trillion tonnes limit.

So, how should we react?

It's always really difficult to know what to do

when we're uncertain about the future.

Usually, we might try to work out the chances

of something bad happening

and do what we can to avoid it

or to protect ourselves against it.

So, there's a very good chance we'll get old and so we buy a pension.

There's a small chance we'll have a road accident,

but we wear a seatbelt.

In each case, we weigh up the risk and the reward.

That calculation relies on the quality of information available.

Scientists have collected and analysed data,

come up with plausible theories

and used mathematical models to make predictions.

But, with the climate,

we can't do experiments to test those predictions.

Only time will tell how accurate they are.

And, if we want to influence our future,

we can't wait to find out.

We have to choose on the basis of what we know now.

When it comes to the climate,

the scientists have done the calculations for us.

But now, it's up to us to decide what action to take.