A 40 mins talk about numerical weather prediction (NWP). The audience are computer science majors in graduate level, taking a course about simulation.

1. Simulating The Weather
13 June 2006
Ting-Shuo Yo

2. Outline
● Properties of Atmospheric Models
● A Brief History
● Numerical Weather Prediction (NWP)
– Parameterization
– Data Assimilation
– Stochastic Weather Models

3. Properties of Atmospheric Models
Properties of a model:
● Dynamic vs Static
● Continuous vs Discrete
● Deterministic vs Stochastic

4. Properties of Atmospheric Models
● Dynamic
Weather changes over time

5. Properties of Atmospheric Models
● Dynamic
● Continuous
Solve differential equations
- Physical systems can be described
by a set of differential equations.
Time-Advance: fixed-increment
(In contrast to discrete-event)

6. Properties of Atmospheric Models
● Dynamic
● Continuous
● Deterministic

∂V  
V⋅∇ V =− ∇ P
∂t
∂h
∇⋅ H V = F
 
∂t

7. Properties of Atmospheric Models
● Dynamic
● Continuous
● Deterministic → Stochastic (after late '90s)

8. Properties of Atmospheric Models
● Dynamic
● Continuous
● Deterministic → Stochastic
Nonliearity in
differential equations
Chaos
Limited predictability

9. Initial condition uncertainty 5-day forecast uncertainty

10. Properties of Atmospheric Models
● The System:

11. Properties of Atmospheric Models
● Dynamic
● Continuous
● Deterministic → Stochastic
● Complex System

12. A Brief History
Early 20th Centary
● Bjerknes, Vilhelm (Norwegian scientist)
– 7 primitive equations
– Weather can be predicted through
computation. (1904)
– Graphic calculus: solve equations
through weather maps.

13. A Brief History (2)
Early 20th Centary (1922)
● Richardson, Lewis Fry
– First numerical weather prediction (NWP)
system
– Calculating techniques: division of space into
grid cells, finite difference solutions of DEs
– Forecast Factory: 64,000 computers (people
who do computations), each one will perform
part of the calculation.

14. A Brief History (3)
Computer Age (1946~)
● von Neumann and Charney
– Applied ENIAC to weather prediction
● Carl-Gustaf Rossby
– The Swedish Institute of Meteorology
– First routine real-time numerical
weather forecasting. (1954)
( US in 1958, Japan in 1959 )

15. Primitive Equations

16. Primitive Equations (2)
Horizontal Equations of Continuity Equation
Motion Conservation of Mass
Newton's 2nd law of motion
Equation of State
The Hydrostatic Equation Property of the ideal gas
Vertical stratification
Water Vapor Equation
Thermodynamic Equation
The 1st law of thermodynamics

17. PE for ECMWF model
East-west wind
North-south wind
Temperature
Humidity
Continuity of mass
Surface pressure

18. ● Simplify systems
– Add extra assumptions, e.g.,
● barotropic model,
● quasi-geostraphic model.
● Increase computing power
– Improved numerical methods
– Parallel computing

19. A Brief History (4)
● Norman Phillips
– First general circulation model (GCM, 1955)
● Primitive equation (PE) models (late '50s ~ '70)
● 1980s : Interaction with other systems (e.g.
ocean, land surface...etc.)

21. Properties of Atmospheric Models
● The System:

22. System of Atmospheric Models
Forcing solar heating,
Adding extra long-wave cooling.....
energy
Atmospheric Model
(dynamic, cloud, precipitation )
Coupling
Ocean model,
plant surface model... Exchanging heat,
momentum and energy

23. Various Atmospheric Models
Resolution
Computing Power Required
Theoretical Models
Operational
Models
Coupled Models
Earth System
Complexity

24. Outline
● Properties of Atmospheric Models
● A Brief History
● Numerical Weather Prediction (NWP)
– Parameterization
– Data Assimilation
– Stochastic Weather Models

25. Parameterization
● What is parameterization?
– Use parameters to represent sub-grid scale
properties.
● Why use parameterization?
– Make computation doable, both on computing
power and numerical stability.

26. Parameterization
Much of the weather
occurs at scales smaller
than those resolved by
the weather forecast
model. Model must
treat, or “parameterize”
the effects of the
sub-grid scale on the
resolved scale.
Source: MODIS

27. A lot happens inside a grid box
Rocky Mountains
Approximate
size of one
grid box in
NCEP
ensemble
system
Denver
Source: accessmaps.com

28. Systems to be Parameterized
● Land surface
● Cloud micro-physics
● Turbulent diffusion and interactions with
surface
● Orographic drag
● Radiative transfer

29. Parameterization Example
Cloud Micro-physics
1.Build a detailed cloud model from observation.
2.Build a classifier for its output as a function
with limited complexity.
3.Use this function in a mature regional/global
model.
4.Repeat step 1 ~ 3 till satisfied.

30. NWP Flowchart

31. Data Assimilation
Data assimilation is the process through which
real world observations:
● Enter the model's forecast cycles
● Provide a safeguard against model error growth
● Contribute to the initial conditions for the next
model run

32. Data Assimilation (2)
A model analysis is not made from observations
alone. Rather, observations are used to make
SMALL corrections to a short-range
forecast, which is assumed to be good.
Example:
● Obs. at 09:00 -> initial condition for a 3-hr
forecast.
● At 12:00, new obs. is corrected by the forecast
done at 09:00, and then used for correcting the
forecast for next 3-hr.

33. Data Assimilation (3)

34. Data Assimilation Procedure
● Ingesting the data
● Decoding coded observations
● Weeding out bad data
● Comparing the data to the model's short-range
"first-guess" fields
● Interpolating the data (in the form of model
corrections) onto the model grid for making the
forecast

35. Data Assimilation Procedure

36. Two Fundamental Difficulties in
Data Assimilation
● Transferring information from the scattered
locations and times of the observations to the
model grid, while at the same time.
● Preserving the interrelated physical, dynamical,
and numerical consistency in the short-range
forecast, since these are essential to making
consistently good NWP forecasts

37. Current state-of-the-art data assimilation:
4-dimensional variational assimilation (4D-Var)
Courtesy
ECMWF

38. 4D-VAR
● Introduced in November 1997
● The influence of an observation in space and
time is controlled by the model dynamics which
increases its realism of the spreading out of the
information.
● The algorithm is designed to find a compromise
between the previous forecast at the beginning
of the time window, the observations, and the
model evolution inside the time window.

39. Stochastic Weather Models
● Stochastic-dynamic approach (climate prediction)
- Ensemble of multiple models
- Stochastic differential equations
● Pure probabilistic approach
For events hardly sketch by dynamics (e.g.
precipitation)

40. Initial condition uncertainty 5-day forecast uncertainty

41. Ensemble forecasts
● Generating initial conditions: Each center has
adopted their own approximate way of sampling
from initial condition pdf.
● Breeding (NCEP)
● Singular vector (ECMWF)
● Perturbed observation (Canada)
● Stochastic-dynamic ensemble work just beginning
(e.g., Buizza et al. 1999)
● Many attempts to post-process ensemble
forecasts to provide reliable probability forecasts.

42. Pure Stochastic Weather Models
● Bayesian Network
Antonio et.al. (2002)
The performance is fair for
the preliminary study.

43. Conclusion
● Dynamic, continuous, and deterministic models
● Early development:
– Dynamic equations
– Numerical methods
● Recent development:
– Parameterization
– Data assimilation
– Probabilistic (stochastic) approach

44. Questions?

45. Reference
● P. N. Edwin, A Brief History of Atmospheric General Ciculation Modeling. General Circulation
Model Development: Past, Present and Future, D. Randall, ed. (Academic Press, San Diego,
2000), pp. 67-90.
● A. Arakawa, Future Development of General Circulation Models. General Circulation Model
Development: Past, Present and Future, D. Randall, ed. (Academic Press, San Diego, 2000),
pp. 721-780.
● T. N. Krishnamurti and L. Bounoua, An Introduction to Numerical Weather Prediction
Techniques. CRC Press, Boca Raton, 1996, pp. 121-150.
● J. R. Holton, An introduction to dynamic meteorology. (3rd Edition). International Geophysics
Series, Academic Press, San Diego, 1992.
● Wilks, D. S., and R. L. Wilby 1999 The weather generation game: a review of stochastic weather
models. Progr. Phys. Geogr., 23, 329—357.
● Antonio S. Cofiño, R. Cano, C. Sordo, José Manuel Gutiérrez: Bayesian Networks for
Probabilistic Weather Prediction. ECAI 2002: 695-699

46. Okay. So how's NWP work?
1. First settle on the area to be looked at and define a grid with
an appropriate resolution.
2. Then gather weather readings for each grid point
(temperature, humidity, barometric pressure, wind speed and
direction, precipitation, etc.) at a number of different altitudes;
3. run your assimilation scheme to initialize the data so it fits
your model;
4. now run your model by stepping it forward in time -- but not
too far;
5. and go back to Step 2 again.
6. When you've finally stepped forward as far as the forecast
outlook, publish your prediction to the world.
7. And finally, analyze and verify how accurately your model
predicted the actual weather and revise it accordingly.

47. Is the weather even predictable or is
the atmosphere chaotic?
That's a loaded question. We all know that weather forecasters are right only
part of the time, and that they often give their predictions as percentages of
possibilities. So can forecasters actually predict the weather or are they not
doing much more than just playing the odds?
Part of the answer appears trivially easy -- if the sun is shining and the only
clouds in the sky are nice little puffy ones, then even we can predict that the
weather for the afternoon will stay nice -- probably. So of course the
weathermen are actually doing their jobs (tho' they do play the odds).
But in spite of the predictability of the weather -- at least in the short-term -- the
atmosphere is in fact chaotic, not in the usual sense of "random, disordered,
and unpredictable," but rather, with the technical meaning of a deterministic
chaotic system, that is, a system that is ordered and predictable, but in such
a complex way that its patterns of order are revealed only with new
mathematical tools.

48. Who first studied deterministic chaos?
Well, not so new. The French mathematical genius Poincaré studied the
problem of determined but apparently unsolvable dynamic systems a
hundred years ago working with the three-body problem. And the
American Birkhoff and many others also studied chaotic systems in
various contexts.
But its principles were serendipitously rediscovered in the early 1960s by
the meteorologist Edward Lorenz of MIT. While working with a simplified
model in fluid dynamics, he solved the same equations twice with
seemingly identical data, but the second run through, trying to save a
little computer time, he truncated his data from six to three decimal
places, thinking it would make no difference to the outcome. He was
surprised to get totally different solutions. He had rediscovered "sensitive
dependence on initial conditions."
A 2-D image of a Lorenz attractor. Lorenz went on to elaborate the
principles of chaotic systems, and is now considered to be the father of
this area of study. He is usually credited with having coined the term
"butterfly effect" -- can the flap of a butterfly's wings in Brazil spawn a
tornado in Texas?

49. Characteristics of a chaotic system
Deterministic chaotic behavior is found
throughout the natural world -- from the way
faucets drip to how bodies in space orbit each
other; from how chemicals react to the way the
heart beats; from the spread of epidemics of
disease to the ecology of predator-prey
relationships; and, of course, in the dynamics of
the earth's atmosphere.

50. Characteristics of a chaotic system (2)
Sensitive dependence on initial conditions -- starting from extremely similar but
slightly different initial conditions they will rapidly move to different states.
From this principle follow these two:
* exponential amplification of errors -- any mistakes in describing the initial
state of a system will therefore guarantee completely erroneous results; and
* unpredictability of long-term behavior -- even extremely accurate starting
data will not allow you to get long-term results: instead, you have to stop
after a bit, measure your resulting data, plug them back into your model, and
continue on.
Local instability, but global stability -- in the smallest scale the behavior is
completely unpredictable, while in the large scale the behavior of the system
"falls back into itself," that is, restabilizes.
Aperiodic -- the phenomenon never repeats itself exactly (tho' it may come
close).
Non-random -- although the phenomenon may at some level contain random
elements, it is not essentially random, just chaotic.

51. How many models are there?
Today, worldwide, there are at least a couple of
dozen computer forecast models in use. They
can be categorized by their:
● resolution;
● outlook or time-frame -- short-range, meaning
one to two days out, and medium-range going
out from three to seven days; and
● forecast area or scale -- global (which usually
means the Northern hemisphere), national, and
relocatable.

52. How good are these models and the
predictions based on them?
The short answer is, Not too bad, and a lot better than
forecasting without them. The longer answer is in three parts:
* Some of the models are much better at particular things than
others; for example, as the USA Today article points out, the
AVN "tends to perform better than the others in certain
situations, such as strong low pressure near the East Coast,"
and "the ETA has outperformed all the others in forecasting
amounts of precipitation." For more on this subject, here's a
slide show from NCEP.
* The models are getting better and better as they are
validated, updated, and replaced -- the "new" MRF has
replaced the old (1995), and the ETA is replacing the NGM.
* That's why they'll always need the weather man -- to interpret
and collate the various computer predictions, add local
knowledge, look out the window, and come up with a real
forecast.