Numerical Weather Prediction

Because weather is so hard to predict, meteorology was far distanced from “noble science” until very recently.  Whereas astronomers could predict the exact date and location of the next solar eclipse, it was considered foolish to even try to predict the weather.  The climate of even one or two days ahead resisted any serious attempts at forecasts and was generally attributed to acts of God.  Yet I would guess that this stubborn unpredictability secretly nagged at many scientists who were trained to believe in a “clockwork universe.”  Although Benjamin Franklin was curious about the weather and studied it on and off throughout his life[2], the subject was generally considered taboo for serious scientists until the late 1800’s.

Lewis Fry Richardson can be considered the father of what is now known as numerical weather prediction (NWP).  In his 1922 book Weather Prediction by Numerical Process, Richardson developed a technique for weather prediction that was astonishingly ahead of its time.  The method he pioneered is the foundation of modern weather forecasting, so it is worth studying it in some detail.  What Richardson proposed was to divide the earth’s surface into a grid, with each grid cell the base of a vertical column of atmosphere (figure 1).  Each column was then divided into several layers, making a three-dimensional grid of atmospheric boxes.  Differential equations were known that govern various aspects of atmospheric physics such as pressure, thermodynamics, and fluid dynamics.[3]  The basic idea was that if you knew the values of certain environmental variables at the center of each grid box, you could use the physics equations to calculate their values at a short time later. 

Figure 1: Richardson’s numerical weather prediction grid over part of Europe.[4]

The main practical issue was, and continues to be, actually making those calculations.  For one thing, calculations must be done for every single box in the grid – a huge amount of work.  For another, most of the equations are non-linear differential equations which can only be solved numerically through a process that is fairly complex, differs for each equation, and is computationally intensive.  With his extensive knowledge of both meteorology and mathematics, Richardson managed to develop a remarkable technique that simplified the equations as much as possible while maintaining their most important properties.  This balancing act between ease of computation and physical reality remains a central theme of numerical weather prediction.

At the time, the only conceivable way of making this type of weather forecasting a reality was to employ hordes of human “computers” to do the necessary calculations before the future they were predicting had already passed.  Richardson outlined a vision[5] of thousands of such workers armed with slide rules, performing calculations around the clock, passing results to each other and telegraphing forecasts around the world.  There was a “large pulpit” with “the man in charge of the whole theatre... one of his duties is to maintain a uniform speed of progress in all parts of the globe.”[6]  He even included a “research department, where they invent improvements.”[7]  His description is remarkably similar to descriptions of modern multiple-processor supercomputers used in weather forecasting today.  The fundamental notion that each process in such an interconnected system must progress at a uniform speed, and therefore the slowest process is the limiting factor for the overall speed, is now known as Amdahl’s Law.  The importance of “research departments” using up some of the computing time is now universally acknowledged.  And relaying forecasts around the world is now of course done routinely via the Internet.

To test his technique and provide an example of how to use it, Richardson performed the calculations himself for just two adjacent columns of air with 5 vertical levels each.[8]  The data he used was a set of observations of pressure (P), wind velocity (M), and air density at several heights taken throughout one day in 1910 by hot air balloonists.[9]  Richardson used one set of data as the initial conditions for calculating the state of each variable in each grid box 6 hours later.  He could then compare these results with the actual data taken later that day in 1910.  This technique is now routinely used to measure the ability of forecasting models (among other things) and is known as a “reforecast”.

Richardson concluded by outlining the three major areas for improvement which continue to form the basis for progress in weather forecasting: better scientific knowledge, more observational data, and faster computing ability.[10]  In 1922, there were major problems with all three.  It took Richardson 6 weeks of on and off work to perform his calculations and double-check them,[11] and when he finished he found that his results did not even agree with the measurements!  Richardson recognized the “glaring error” but published his book anyway because he felt that the ideas were significant and he understood that the hot air balloon data and/or atmospheric equations used might be faulty.[12]  (In fact, both the data[13] and one of the physical processes[14] were probably erroneous.)  He also published despite his estimate that he would need 64,000 human calculators[15] to make his vision a reality (a later estimate put the figure around 200,000)[16].  He knew that this was not very practical and that few would take the idea seriously, despite the significant economic value of good weather forecasts.  After all, the current methods of weather forecasting could predict with some accuracy about 3 days ahead, and Richardson’s meager 6-hour forecast attempt had failed.  Soon after publishing Weather Prediction by Numerical Process, he left the field.[17]

Hardly anyone read, much less understood, Richardson’s work because it was so mathematically intense and because the methods he defined were useless without sufficient computing power (figure 2).  But meteorologists continued to further their knowledge of the behavior of atmospheric systems and more instruments for observing weather conditions were deployed.  The standard technique for forecasting – based on reading “weather maps” displaying atmospheric variables such as temperature and pressure – evolved into two schools of thought.  One method was to look at the current conditions and use what was known about the large-scale physics of the situation to make a prediction.  The other method, called the “analogue” method, involved directly comparing the current meteorological situation to past ones (analogues).  Over time, meteorologists had built up enough of a collection of weather maps to make these comparisons feasible.  By the time of WWII, numerical weather prediction was still deemed a fantasy and most weather forecasters employed some mixture of the two weather map techniques.

Figure 2: Weather Prediction by Numerical Process. [18]

It is extremely interesting to note that the decision of when to proceed with D-Day in June of 1944 depended on the weather forecast, and more specifically which weather forecast.  The invasion depended on calm enough weather and clear enough skies for boats to launch, airplanes to bomb, and soldiers to disembark.  And two opposing schools of forecasting were giving opposing weather forecasts for the day in question.  Sverre Petterssen, who had been a professor at MIT and the author of the first textbook on weather forecasting, knew the merits of forecasting by physics.  Irving Krick, who had taught at Caltech but was not a very respected scientist, thought that he could predict based on past analogues.  The two had completely different forecasts, but both defended their approaches staunchly – leaving it up to the judgment of higher officers to decide who to believe.  Fortunately, they followed Petterssen’s approach, delaying the invasion by one day and managing to attack in the time span between two storms.  (As it happened, Krick managed to get all the media attention for the successful weather forecast.)[19]

The D-day invasion was a significant point in the history of weather forecasting for several reasons.  For one thing, it showed the world how crucial accurate weather forecasts could be – making a clear case for a continuing stream of funding.  Secondly, it was one of the first situations where large operations had substantially changed plans solely on the predictions of weather forecasters.  Right before the originally scheduled D-day, the invasion force held off preparations despite clear, calm weather.  And right before the newly chosen day, they prepared to attack despite the storm then going on.[20]  Only fairly recently have most institutions decided to put this much trust in weather forecasts.[21]  Finally, the success of Pettersen’s method of prediction strengthened the case for its worth.  Atmospheric scientists would continue to refine it, while increasing their disregard for the analogue method.



[1] Weather map graphics adapted from images available at
[2] Cox, 5.
[3] Kimura, 1405.
[4] Richardson, 184.
[5] Ibid, 219.
[6] Ibid.
[7] Ibid.
[8] Richardson, 17.
[9] Cox 159.
[10] Kalnay, E., Lord, S. J., and McPherson, R. D., 2766.
[11] Richardson, 219.
[12] Cox, 160.
[13] Introduction to Richardson, vii.
[14] Kimura, 1407.
[15] Richardson, 219.
[16] Cox, 161.
[17] Cox, 162.
[18] Photo taken by the author.
[19] Cox, 189-97.
[20] Cox, 195.
[21] Kalnay, E., Lord, S. J., and McPherson, R. D., 2754.



Numerical Weather Prediction Computers Meet Weather Forecasting
Futures Implications References