Pete's Potpourri
Numerical Weather Prediction inFAQ

Source: www.scn.org/~bm733/inFAQ.htm

Weather forecast computer modeling is known in the trade as numerical weather prediction (NWP). As an amateur, I find it interesting because it's at the intersection of three areas that are themselves individually interesting -- first, weather forecasting and the prediction of events that appear unpredictable; second, computer modeling and parallel computing; and third, non-linear systems and chaos theory.

The first two parts of this inFAQ are loosely adapted from The Handy Weather Answer Book, by Walter A. Lyons, Ph.D. (Visible Ink Press, 1997); the part on deterministic chaos is cobbled together from multiple sources; and the fourth part is largely cribbed from USA Today at www.usatoday.com/weather/wmodlist.htm.



Who formulated the ideas of fronts and air masses?

During World War I, Norwegian meteorologists were largely cut off from weather information outside of their neutral country due to restrictions imposed by the warring nations of Europe. In response Norway established a dense network of weather stations within their own country. Led by the father and son team of Vilhem and Jacob Bjerknes, a dedicated group of scientists now known as the the Bergen School went to work analyzing the resulting data.

From this work they developed the concept of air masses and the weather fronts between air masses. They studied instabilities on the polar front (the demarcation between polar and tropical air) and from this developed the basic theory of mid-latitude storms. This theory has become a cornerstone of modern meteorology.


So what is numerical weather prediction?

Because of their academic backgrounds in the study of fluid dynamics, the Bergen School scientists understood that air as a fluid obeys the fundamental physical laws for fluids, called the hydrodynamic equations. These include the equations of motion, the thermodynamic energy equation, the hydrostatic equation, the equation of mass conservation, and the Boyle-Charles equation of state.

In theory, then, one can use these equations to create a mathematical model of the atmosphere, plug in data on the past and current state of the atmosphere, and solve the equations to predict a future state -- numerical weather prediction. But of course it's not that easy.

Like any other mathematical modeling of a complex dynamic system (and the earth's atmosphere is a very complex system) NWP requires the solving of the above mentioned nonlinear partial differential equations -- not possible by precise analytical methods, but rather done by numerical approximation -- classically by iterating to an acceptably close approximation, but now by matrix methods. To do this for any practical purpose requires a huge amount of computation. Further, it might be noted here that because of the nonlinear nature of these equations, tiny differences in the data that are "plugged in" to the equations -- that is, defining the initial state -- will yield huge differences in the results. This "sensitive dependence on initial conditions" is the hallmark of a chaotic system.


Some real-world problems NWP must overcome

This latter problem is significant. In spite of all the world-wide weather stations, weather buoys, observations from ships and aircraft, the use of weather balloons and radiosondes, doppler radar, and satellite information, some of the actual data needed or available for initializing any given computer run of a model either is missing or does not fit the model grid either in time or space. The methods developed to deal with this are referred to as data assimilation.

Further, atmospheric processes that happen on scales smaller than that of the model's grid scale but that signicantly affect the atmosphere (such as the large amount of convection that can occur in thunderstorms, cloud formation and the release of latent heat, etc.) must be accounted for. The procedure to do this that is incorporated into the models is called parameterization. As the Canadian Meteorlogical office states, "Parameterizations can be (and usually are) complex models in their own right."


How does resolution affect the model?

Now obviously the finer the resolution of the model grid the more accurately the model reflects the actual atmosphere, so all else being equal, the more accurate the prediction is that uses that model. But equally obviously the finer the resolution, the more numbers that have to be crunched on the same computers.

So, in practice, models that cover large areas (like the whole Northern hemisphere) tend to have coarser resolution than those that cover smaller areas (like just the USA) and so are not going to be as accurate in the small scale. Further, it's worth noting that models that work with smaller areas can predict only for shorter time periods, since as time passes it's inevitable that weather from outside the model area (and therefore not accounted for in the model) will have influenced the weather inside the model area.

One way to overcome this limitation is by nesting a finer grid for a limited area of interest inside a larger, coarser grid. This method is widely used, but adds is own complications which must be accounted for.


Okay. So how's NWP work?

The process goes approximately like this (assuming you've already developed a mathematical model):

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.

All that produces a numerical prediction. An actual forecast takes much more work -- more about that later.


Who first tried NWP?

Although the fundamental notions of numerical weather prediction were first stated by Vilhelm Bjerknes in 1904, it was in 1922 that Lewis F. Richardson -- "a great scientist whose originality mixed with eccentricity," to quote Mandelbrot -- formally proposed that weather could be predicted by solving the "equations of atmospheric motion." He soon realized that the amount of calculation would be formidable, so he proposed a weather prediction center in which a giant circular amphitheater would contain some 26,000 accountants equipped with calculators who would make their additions and subtractions as commanded by a sort of conductor.

Richardson's first attempts failed because the method predicted pressure changes far larger than any that had ever been observed. This was later found to result from the way he approximated the solutions to the equations. His idea -- which, it should be emphasized, was basically sound -- was thus dismissed and forgotten for over 20 years.

NWP would have to wait for the proper tool.



When were the first practical attempts at NWP?

The electronic computer was conceived in the 1940's, when mathematician John von Neumann developed the prototype of the stored program electronic machine, the forerunner of today's modern computers. He turned his interests to NWP and formed a meteorology project in 1946 at Princeton's Institute for Advanced Study. There meteorologist Jule Charney began working on the problem of numerical weather prediction.

After figuring out why Richardson's first attempts 25 years earlier had failed, Charney was able to formulate equations that could be solved on a modern digital computer. The first successful numerical prediction of weather was made in April 1950, using the ENIAC computer at Maryland's Aberdeen Proving Ground. Within several years research groups worldwide were experimenting with "weather by the numbers." The first operational weather predictions, using an IBM 701 computer, were begun in May 1955 in a joint Air Force, Navy, and Weather Bureau project.

What role do computers play now?

Because of the extent of the computation required, meteorologists have invariably required the biggest and fastest computers to do their numerical modeling. NWP has advanced greatly in six decades, in large part due to the spectacular growth in speed and capacity of digital computers. To give you a feel for it, one of the first commercial computers used for atmospheric research was the IBM 1620 -- it could perform about 1,000 (10^3) additions per second -- while today's massively parallel supercomputers can clip along in the low teraflops -- trillions (10^12) of floating point operations per second -- a billion (10^9) times faster.

With the high speed number crunching of NWP, atmospheric scientists use and generate huge amounts of data. The National Center for Atmospheric Research estimated that in 1997 they maintained computer files totaling 30 terabytes -- 30 trillion (10^12) bytes -- of data. In late 2000 that number had grown to over 200 terabytes. And by early 2003, total data at their Mass Storage Section had continued growing exponentially to over a petabyte -- that's 1024 terabytes, or a mega-gigabyte, if you will.

But there are inherent limits that even the fastest computers can't overcome.



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.


Who first studied deterministic chaos?

Well, not so new. The French mathematical genius Poincare 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."

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? (But see the note.)

(James Yorke of the University of Maryland is credited with having spawned this -- somewhat misleading -- new use of the word "chaos.")


What are the 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.

But all these seemingly unrelated phenomena share certain characteristics in common:

• 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. (Sorry if that seems circular.)

Reading through this list you can see some of the inherent difficulties that NWP has to deal with, and why there are necessary limits on what numerical modeling can do.



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.

What models are in common use?

Models that are in common use in the United States, with their resolution, outlook, and scale:

• The NGM (Nested Grid Model with 80Km resolution), ETA (29Km with the ETA-22 on the way), and AVN (Aviation Model, 100Km resolution) are short-range national models:
  
  
  • The NGM uses the common approach of a smaller grid "nested" inside a larger one for the area of interest, in this case the U.S. It is a short range model that predicts for 2 days ahead, producing forecasts every 6 hours, predicting variables such as temperature at various altitudes, amount of precipitation, position of upper level toughs and ridges, and the position of surface high and low pressure areas.
     
  • The ETA is named after the Eta coordinate system, a mathematical system that takes into account topographical features such as mountains. The ETA is similar to the NGM in that it forecasts the same atmospheric variables; however, because of the ETA's better resolution (29 Km to the NGM's 80 Km), and its coordinate system, the ETA has a much more accurate picture of the terrain across the USA. It's still too soon to tell if the ETA gives a more accurate forecast than the NGM for all forecast variables; but according to the National Center for Environmental Prediction (NCEP), the ETA has outperformed all the others in forecasting amounts of precipitation. The ETA is forecast to completely replace the NGM by the end of 1999.
     
  • One of the oldest operational models used today, the AVN gives short range forecasts like the NGM and ETA, but it also forecasts into the medium range -- up to 72 hours ahead. Although its 100Km resolution isn't as good as the NGM or ETA, it still provides valuable insight -- the AVN tends to perform better than the others in certain situations, such as strong low pressure near the East Coast.
  
  
  
• The MRF (Medium Range Forecast) is a 150Km resolution medium-range global model. Here's a non-mathematical but jargon-filled technical introduction, [but link broken 6-2004] which, if studied, will give some idea of the complexity of the modeling process.
   
• The ECMWF (European Centre for Medium-Range Weather Forecasting) and UKMET (United Kingdom Meteorological Office) refer to European 75Km resolution medium-range global models whose output is widely used here in the States.
   
• The Global Ocean Model forecasts seasonal changes in oceanic variables, such as sea surface temperature and ocean currents. The ocean model is coupled with an atmospheric model to help determine how forecasted changes in oceanic variables will affect the atmosphere. This model tandem is used to forecast long range seasonal or yearly variations of the ocean and the atmosphere -- including events such as an El Nino warming event in the Pacific Ocean. For more information, see this short paper from the Advanced Computing Lab at Los Alamos National Laboratory.
   
• The MM5 (Mesoscale Model version 5) is an experimental model developed at Penn State and being used at various universities. See http://www.atmos.washington.edu/mm5rt/info.html or this overview, for example.

To turn a numerical prediction into a forecast more work is involved. Not strictly speaking a numerical model, Model Output Statistics (MOS) are generated by forecast model postprocessors. They are used to help overcome the weaknesses of numerical models by developing statistical relations between model predictions and observed weather. These relations are then used to translate the model predictions directly to specific surface weather forecasts. For example, two types of plots are forecast surface conditions from the NGM model, and 7-day maximum temperature predictions from the MRF.


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.

The Web makes some fine resources available. Here are a few mostly about numerical weather prediction:

• There's a great little online tutorial at Texas A&M that takes you through a simplified example of the process.
   
• UCAR's Meteorology Education and Training website makes quite a few somewhat technical training resources available either online or for download, including those for NWP.
   
• The ECMWF provides a short primer on Forecasting by computer.
   
• The British Meteorological Office provides a comprehensive discussion of NWP. (I've linked to a few of their good pages above.)
   
• As for the models themselves,
  • try two sites from the University of Maryland:--
       the World Wide Web Weather Interface,
       and a collection of various weather products used by forecasters;
  • or, Unisys Weather which provides plots and maps from the models.
  
  
• And one decent chaos link.