# Presentation: 515316

## 30% Chance of Rain?

- Chaos and Numerical Weather Prediction

**Notes:**
My presentation is about chaos and weather forecasting.  I’m going to focus on one particular aspect of chaos – state dependent sensitive dependence to initial conditions – but chaos and nonlinear dynamical systems theory cuts across many, many aspects of atmospheric, oceanic, and climate science.

There are many ways to forecast the weather . . .

## Outline

- Dynamical system
- State space
- Trajectory
- Attractor
- Lyapunov exponents
- Singular vectors
- State dependent error growth
- Implications of model inadequacy
- What multi-models buys us
- A departure into the real world

- What they are and why
- you should care

- An ensemble-based
- approach

## Dynamical system

- A set of equations giving the time evolution (or ordered evolution) of the state of a system from knowledge of its previous history (in contrast to a stochastic system)

## State space

- A dynamical system’s prognostic variables expressed as a point in a multi-variate space.

## Trajectory

- Given **F** and some **x**(0), one can determine subsequent **x**(*t*), or the trajectory of **x**(*t*), for all *t* by integrating the dynamical system.

- Attractor

- The bounded region of state space towards which initial conditions eventually converge.

## Example: Unforced pendulum

- Dynamical system:

- Change of variable:

- Two first-order ODEs:

- State space

## Trajectory: Unforced pendulum

- y

## Trajectory: Unforced pendulum

- y

- z

## Attractor: Unforced pendulum

- y

- z

## Example: Forced pendulum

- Dynamical system:

- Change of variable:

- Two first-order ODEs:

- State space

## Trajectory: Forced pendulum

- y

## Trajectory: Forced pendulum

- y

- z

## Attractor: Forced pendulum

- y

- z

## Example: Lorenz ‘63

- Dynamical system:

- State space:

## Trajectory: Lorenz ‘63

- x

## Trajectory: Lorenz ‘63

- x

- z

## Attractor: Lorenz ‘63

- x

- z

## Example: NWP

- Dynamical system:

- State space:

## Trajectory: NWP

## Trajectory: NWP

## Trajectory: NWP

## Trajectory: NWP

## A measure of the “amount” of chaos.
Indicate the average rate of error growth for infinitesimal errors over an infinite time.

Can be numerically estimated using linear theory.
Lyapunov exponents are an invariant of the system; they are not state dependent (and need never be realized!)

- Indicate the average rate of error growth for infinitesimal errors over an infinite time.
- Can be numerically estimated using linear theory.
- Lyapunov exponents are an invariant of the system; they are not state dependent (and need never be realized!)

- Lyapunov exponents

## Chaos: SDTIC

**Notes:**
Sensitive dependence to initial conditions

## Slide 25

## Singular vectors

**Notes:**
Here I’ve included a lot of points on the attractor.

Again, the red points have large short term error grown, the blue points have smaller short term error growth, and the black points are regions where all errors shrink.

Again, the game is to figure out when you’re in the red regions (so you know your forecasts are likely to be bad) and when you’re in the blue or black regions (so you know that your forecasts are likely to be good).

## Slide 27

**Notes:**
We can look at this information in a slightly different way.

This is the Lorenz 63 attractor again, and what I’m showing is the evolution of an ensemble-based PDF.  The x-axis is the value of the x-component of the Lorenz 63 system while the y-axis is time.  You can imagine little green lines underneath each of these humps. . .the height of the line represents the density of the green lines.  The higher the hump, the higher the probability of the true system state.

To be clear (because I’ll be coming back to this), this initial red spike corresponds to my initial green histogram.  My forecast green histogram corresponds to these regions out here.

What do we see by considering a longer forecast?  Well, the ensemble spreads out as it moves forward in time, but here we see that it tightens up.  The uncertainty decreases as the ensemble moves through this region of state space.  What is this return of skill (reduction of uncertainty) caused by?  This is when the model trajectory goes through the black region on this plot.  Remember, all errors shrink in this region, and indeed, the uncertainty in our ensemble gets smaller.    

We see that each time the ensemble passes through these regions the PDF tightens up.

We also see that when the ensemble splits between two wings of the butterfly the spread in the ensemble is huge, but there is still a great deal of information in the ensemble (we are here or here).  

What’s the punch line?  Chaotic systems have state-dependent error growth.  We can use ensemble forecasting to tell us whether our forecasts are likely to be good (ensemble/PDF is tight) or not so good (ensemble/PDF is spread out).

This ensemble approach to forecasting is widely used in weather prediction.

## t=0

- 7/22

- t=3days

## t=0

- 7/22

- t=3days

## t=0

- 7/22

- t=3days

## t=0

- 7/22

- t=3days

## t=0

- 7/22

- t=3days

## t=0

- 7/22

- t=3days

**Notes:**
World’s greatest ensemble slide.

Perfect model.

Can’t use standard linear techniques for propagating uncertainty forward because of the importance of nonlinearities over the timescale of interest.

Can’t propagate the full initial PDF forward because of computational limitations.

This *is* the forecast.  Even with a perfect model, imperfect observations will render a correct deterministic forecast unobtainable.  The forecast PDF *is* the relevant forecast.

## Now you have to make a choice. 

- ...or you take the red pill, and try to use ensemble guidance knowing all models are wrong.

- “Remember that all I am offering is the truth. Nothing more” - Morpheus

- You take the blue pill and the lecture ends, you wake up in your bed and happily do mathematics...

**Notes:**
The implications of this are not pretty, especially if you like clean theory.  The mathematicians in the room should probably take the blue pill.

Just as initial condition error kills our ability to produce correct deterministic forecasts, model error kills our ability to produce correct probabilistic forecasts.  

This is for two reasons:
When our model is wrong, we can never obtain the correct initial uncertainty distribution.  That uncertainty distribution is dictated by the system (truth) dynamics.  Since we don’t know those dynamics we can never know the correct initial distribution.
 Even if we had a correct initial uncertainty distribution, propagating ensembles drawn from that distribution with an inadequate model will result in an incorrect forecast distribution.

Examples. . .

## We can only aspire to the limitations

- imposed by chaos!

- 7/22

- t=0

- t=3days

**Notes:**
We know how to solve the chaos problem...if only we were so lucky!

## Goal is to move the blue PDF

- 7/22

- Goal is to move the **blue PDF**

- Towards the **green PDF**

- t=0

- t=3days

**Notes:**
This is the real world.  The blue curves are the ensemble forecast from an imperfect model.  

Probabilistic forecast is wrong, but we’d never know for any single realization of the forecast.

Traditionally, the aim is to make the model better by driving the black curve towards the red curve.

In the proposed probabilistic approach, the aim is to drive the blue PDF towards the green PDF.  

In addition to it being better posed than the deterministic problem, there is clearly much more information to exploit in an ensemble than in a deterministic forecast.

- 3d attractor pictures with 1 model
- Put an ensemble on there
- Multi-models
- Multi-models with ensembles

## A brief foray into the real world

## 43

- 23

- 46
- 29

- 29
- 5

- 32
- 6

- Minneapolis,
- Mar 9 – Mar 15
- 2005

**Notes:**
I want to show you some examples of forecasts gone wrong where the ensemble uncertainty information would have been useful.

This is the WCCO forecast from March 9 through March 15.  Notice that you have temperatures in the 40s, then a quick dip into the low 30s as some patch of cold air moves through, and then back up into the 40s.

I’m going to show you the actual temperatures one day at a time, and also show you how the forecasts changed each day.

This Tuesday verified reasonably well, and you don’t see a lot change in the new forecast.

Wednesday was much warmer than expected, and you can see that the new forecast thinks things are going to get much colder on Thursday and the cold is going to stick around for two days instead of just one.

Thursday wasn’t as cold as they thought on Wednesday, and the  new forecast still thinks Friday is going to be cold.

And it turns out Friday is cold (it warmed right back up over the weekend).

What’s the point?  Focus on the forecast from Tuesday.  Their Friday forecast ended up being off by almost 10 degrees.  Could they have known that at the time?

Let’s look at the ensemble.

## Slide 40

**Notes:**
Another spaghetti diagram.  Remember, this is showing the same contour pulled from each of the ensemble members.  

This is a contour of temperature at low levels (it turns out there was a temperature front that was well represented by this contour). To the north of the contours is cold air, to the south is warmer air.  

It is the model forecast for that Friday that was available when WCCO made their Tuesday forecast (that top forecast).

If you took just the high-resolution forecast (the black line), Minneapolis is on the warm side of the front.  However, note that in almost half of the ensemble members, Minneapolis is on the cold side of the front! 

Back when they were making the forecast on Tuesday, they knew that there was a great deal of uncertainty in the Friday forecast, but that uncertainty was not communicated to the user.

## 91

- 49

- 77
- 44

- 61
- 43

- 56
- 37

- Minneapolis,
- Apr 28 – May 4
- 2005

**Notes:**
Here’s a similar example just a week or so ago. 

The Wednesday forecast says there will be some warm air around that will quickly move out and leave temps in the mid-50s.

They blew that Wednesday forecast by 10 degrees, but they were still confident in their new forecast that the warm air would move out of the area: there is almost no difference between the Wednesday and Thursday forecast.

But they blew the Thursday forecast by 22 degrees!  Note that the new forecast is still relatively unchanged.

The forecast for Friday was quite good, as was the forecast for the weekend.

Turn your attention back to the Wednesday forecast.  The busts on Wednesday and Thursday are again consistent with the information in the ensemble (I’m not going to show you this time). . .there was a significant amount of spread in the ensemble forecast, so one would expect large errors.

Notice that the Friday and Saturday forecasts issued on the Wednesday are quite good.  Again, this information was contained in the ensemble.  There was a lot of uncertainty in the movement of the warm air, but all the ensemble members agreed on what would happen once that warm air left the region. . .this is an example of “the return of skill” that we saw in the Lorenz 63 system (see picture). 

## Slide 42

**Notes:**
Use this plot to emphasize the return of skill you can obtain due to state dependent error growth.

## 44

- Minneapolis,
- Mar 22 – Mar 28
- 2005

- 17

- 51
- 66

- 66
- 36

- 63

**Notes:**
Back to our WCCO forecasts.

This is over the period March 22 through March 28.

The one day forecast isn’t too bad, and we’ve got a pretty good sized change in the new Wednesday forecast.

The two day forecast is so-so, and the new Wednesday forecast has changed a great deal.

The verification for the new one day forecast is still off by 6 degrees, and the original 3 day forecast is off by 20 degrees (the original 4 day is off by 10 degrees).

Is this another example of not conveying the information contained in the ensemble?  Unfortunately not.

## Slide 44

**Notes:**
This is the forecast of that Wednesday that was available when the first forecast was launched (the one with the 20 degree error).

I’ve plotted low level temperature contours again.  Remember, these are the same contour pulled from each ensemble member.  

There was a temperature front more or less associated with this contour, with cold air to the north and warmer air to the south. 

The forecast put the front two states away in this forecast, when in reality Minneapolis ended up on the south side of the front with a 66 degree day instead of a 46 degree day.  

The spread in the ensemble is not consistent with an error this larger, making this the most insidious types of model error: the ensemble members are tightly packed, but in the wrong place. 

## Scale Dependent Error Growth

**Avg. Deterministic Error**
**(% of Climo. Mean’s Error)**

**200**
**150**
**100**
**50**
**0**

**FL350 Winds**

**CFLOS ****(synoptic)**

**Supercell T-Storm**

**CFLOS ****(convective)**

**Notes:**
Forecast errors grow at small scales first, then cascade up to larger scale motions.  The average error in a deterministic forecast increases during the forecast period and eventually saturates at twice the average error of a climatological forecast (Tribbia and Baumhefner, 2004).  The rate at which the error grows increases with the scale of the element being forecast (Lorenz, 1969; Sivillo et al., 1997).

The error for elements tied to the large scale, such as flight level winds at 35,000ft, grows very slowly and doesn’t reach saturation for many days.
The error for elements tied to the small scale grows extremely rapidly and can saturate within a few hours.  An example is prediction of cloud free line of sight (CFLOS) in a convective regime where clouds are kilometers in size or less.

Deterministic Realm:  The uncertainty is minimal (or potential error is very small), so providing a purely deterministic forecast to a customer is acceptable.  This realm is shorter for smaller scale forecast elements
Stochastic Realm:  Once the error start to grow rapidly, uncertainty (potential error) has increased to the point where it is critical to include a description of the forecast uncertainty in order to optimize warfighter decision.

BOTTOM LINE:  The forecast lead time for transition from the deterministic to stochastic realm depends upon what forecast element may impact the customer.  And since some weather situations are more predictable than others, it is critical to model the error growth in an ensemble to capture the flow-dependent uncertainty.

## Scale Dependent Error Growth

**NCEP Global, 15-Member Ensemble   00Z, 19 Nov 2006**
**500Hpa GPH, 522 and 564 Contour Spaghetti **

**6 Hours                                24 Hours                                48 Hours**

**1 Day                                      4 Days                                    7 Days**

**NCEP CONUS, 21-Member Ensemble   09Z, 19 Nov 2006**
**Surface Temp, 0********C Isotherm Spaghetti **