Mathematics of weather forecasting

• SUMMARY: Accurate weather forecasting requires the use of advanced mathematical models and powerful supercomputers to handle the vast number of calculations.

Weather prediction, or forecasting, is the application of science and technology to predict the future state of the atmosphere at a given location using available past and present data from the surrounding area. The word “weather” describes the state of the atmosphere at a particular time, or short time period, while the word “climate” is an average of these conditions over long time periods—often months or years. The weather is typically described in terms of temperature, wind speed, wind direction, air pressure, density, and atmospheric composition (for example, water vapor, liquid water, or carbon dioxide content). The intensity of solar and terrestrially emitted radiation is also a fundamental determining factor. A forecast typically includes the prediction of these meteorological variables and helps people make more informed daily decisions that may be affected by the weather. Moreover, it helps predict dangerous weather phenomena, such as hurricanes, which might endanger human life.

History

People have tried to forecast the weather for thousands of years and throughout history, farmers, hunters, warriors, shepherds, and sailors understood the importance of accurate weather predictions for planning daily activities. Ancient civilizations appealed to the gods of the skyhe Egyptians looked to the sun god Ra, the Greeks sought out Zeus, and in the ancient Nordic culture, Thor was believed to govern the air with its thunder, lightning, wind, rain, and fair weather. The Aztecs used human sacrifice to satisfy the rain god Tlaloc, while Native American and Aboriginal Australians performed rain dances.

The Babylonians were predicting the weather from cloud patterns as well as astrology by 650 B.C.E., but the earliest scientific approach to weather prediction occurred circa 340 B.C.E. when Aristotle described his theories about the earth sciences and weather patterns in Meteorologica. The ancient Greeks invented the term “meteorology,” which derives from the Greek word meteoron which refers to any phenomenon in the sky. The Greek philosopher Theophrastus, one of Aristotle’s successors, compiled the ultimate weather text The Book of Signs, which contained a collection of weather lore and forecast signs and served as the definitive weather book for over 2000 years.

Weather forecasting advanced little from these ancient times to the Renaissance. Beginning in the fifteenth century, Leonardo da Vinci designed an instrument for measuring humidity, Galileo Galilei invented the thermometer, and his student Evangelista Torricelli came up with the barometer. With these tools, people could objectively monitor the atmosphere. In 1687, Sir Isaac Newton published the physics and mathematics that govern the motion of all bodies and can be used to accurately describe the atmosphere. To this day, his principles are the foundation for modern mathematical analysis and computer prediction of weather.

However, scientifically accurate weather forecasting was not feasible until the early twentieth century, when meteorologists were able to collect and organize data about current weather conditions from observation stations in a timely fashion. Vilhelm and Jacob Bjerknes developed a weather station network in the 1920s that allowed for the collection of regional weather data. The data collected by the network could be transmitted nearly instantaneously by use of the telegraph, invented in the 1830s by Samuel F. B. Morse. This system allowed knowledge of the weather conditions upwind to be incorporated into downwind forecasts, improving their quality.

Great progress was made in the science of meteorology during the twentieth century. The possibility of numerical weather prediction was proposed by Lewis Fry Richardson in 1922, although computers did not yet exist. It was consequently impossible to perform the vast number of calculations required to produce a forecast before the predicted events actually occurred. Practical use of numerical weather prediction began in 1955, spurred by the development of programmable electronic computers. 

The ability to accurately forecast the weather played a crucial role in one of the most momentous decisions in military history. In early June 1944, Allied military commanders anxiously studied weather patterns over the English Channel as the massive invasion of France, years in the planning, would shortly commence. Rough weather had the potential to wreak havoc with the already-risky invasion. Originally set for June 5, Allied weather personnel advised a postponement until the next day. The forecasters accurately predicted that a small break in weather would occur on June 6. Supreme Allied Commander General Dwight D. Eisenhower had to decide whether to accept this forecast or to wait until the next period of favorable conditions several weeks later. Eisenhower’s enemy counterparts, the German Army High Command, decided that the weather was too foul for Allies to launch their forces and relaxed their defensive posture. Instead, Eisenhower decided to launch the invasion and achieved strategic surprise over the German forces. Had Eisenhower decided on the several-week delay, Allied invasion forces would have had to contend with the worst weather over the English Channel in twenty years.   

Numerical Weather Prediction

Numerical weather prediction is the science of forecasting weather using computer simulations built from mathematical models. In this process, the atmosphere is divided into a three-dimensional lattice of grid points, and at each point the various atmospheric variables of interest are represented. These values are initialized with a state determined through analysis of past and present conditions. This state is then evolved forward into the future by solving, at each grid point, the classical laws of fluid mechanics and thermodynamics, which are known to accurately approximate the behavior of the atmosphere. The output from the model provides the basis of the weather forecast.

The equations that govern how the state of a fluid changes with time contain many variables and require a great deal of computer processing resources to solve. Weather prediction centers have access to supercomputers containing thousands of processors on which to run a forecasting model. The required calculations are shared among the processors and computed simultaneously to produce a complete forecast in a fraction of the time possible with a single computer. This system is essential to ensure that an accurate prediction can be made within a useful time frame.

Good weather forecasts depend upon an accurate knowledge of the current state of the weather system, called the “starting point” or “initial condition.” The initial conditions are determined from global measurements of the state of the atmosphere. Surface weather observations of atmospheric pressure, temperature, wind speed, wind direction, humidity, and precipitation are made near the Earth’s surface by trained observers, automatic weather stations, or buoys. The initial state has a degree of uncertainty since there are an insufficient number of measurements to initialize all meteorological variables at every grid point. Furthermore, the locations of the measurements do not usually coincide with the numerical grid points and there is also a degree of error in the actual measurement. The problem of determining the initial conditions for a forecast model is very important, highly complex, and has become a science in itselfknown as “data assimilation.”

The atmosphere is an incredibly complex dynamical system and the approximation of its behavior is only compounded by the inability to measure its state at each and every grid point in the model. The limit on useful weather forecasts using present technology is typically one week. The forecast errors are initially localized, leading to incorrect predictions in small regions, but are generally accurate enough to be useful in most of the forecast area. The longer the simulation is run, the more the measurement and model approximation errors begin to dominate the calculation. However, steady improvements in computer power and prediction models in the twenty-first century have led to a three-day forecast being as accurate as a two-day forecast from the 1990s. Weather forecasting centers are constantly reviewing the accuracy of their forecasts and set themselves annual targets for accuracy improvements.

The raw output from the simulation is often modified before being presented as a forecast. Modifications include either the use of statistical techniques to remove known biases in the model or adjustments to take into account consensus among other numerical weather predictions. Accurate forecasts of precipitation for a specific location are particularly challenging because of the chance that the rainfall may fall in a slightly different placesuch as several kilometers awayor at a slightly different time than the model forecasts, even if the overall quantity of precipitation is correct. Therefore, daily forecasts give fairly precise temperatures but put probabilistic values on quantities such as rain, based on knowledge of the uncertainty factors in the forecast.

Probability of Precipitation

A Probability of Precipitation (PoP) is a formal measure of the likelihood of precipitation that is often published from weather forecasting models, although its definition varies. In U.S. weather forecasting, PoP is the probability that greater than 1/100th of an inch of precipitation will fall in a single spot, averaged over the forecast area. For instance, if there is a 100 percent probability of rain covering one side of a city and a zero percent probability of rain on the other side of the city, the PoP would be 50 percent. A 50 percent chance of a rainstorm covering the entire city would also lead to a PoP of 50 percent. The mathematical definition of PoP is defined as PoP=C × A × 100, where C is the confidence that precipitation will occur somewhere in the forecast area, and A is the percent of the area that will receive measurable precipitation, if it occurs at all.

For example, a forecaster may be 40 percent confident that precipitation will occur and that, should rain happen to occur, it will happen over 80 percent of the area. This results in a PoP of 32 percent0.4 × 0.8 × 100 = 32.

The Future

Global climatic changes have altered many historical weather patterns. The ability to predict weather and coming atmospheric conditions, nonetheless, continued to grow more accurate over time. Over the years, the quality of the models and methods for integrating atmospheric observations improved continuously, resulting in major forecasting improvements. The power of supercomputers increased dramatically, allowing for the use of much more detailed numerical grids and fewer approximations in the operational atmospheric models. Small-scale physical processes like clouds, precipitation, turbulent transfers of heat, moisture, momentum, and radiation have been more accurately represented within the model. Finally, the use of increasingly accurate methods of data assimilation and the integration of satellite and aircraft observations resulted in improved initial conditions for the models, which ultimately lead to better forecasts. Weather assessments can be done in real-time. Certain events, such as winter storms, can be accurately done such that a current seven-day forecast can be as accurate as a three-day forecast done in the 1980s. Other climatic events like hurricanes are different and more difficult to predict because rising earth temperatures can cause hurricanes to rise or fall in intensity more quickly.

Bibliography

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