Doppler radar and mathematics

Summary: Doppler radar uses the mathematical characteristics of waves to track and predict weather patterns.

Radio detection and ranging, commonly known by the acronym “radar,” was initially developed to detect and determine the distance of enemy aircraft when visual methods were insufficient, such as in poor weather or at night. It is commonly traced to the nineteenth century work of physicist Heinrich Hertz, who investigated the reflection of radio waves from metallic objects. Doppler radar is a type of radar that uses the Doppler effect to judge the speed and direction of distant objects. The Doppler effect (also known as “Doppler shift”) is a physical property that applies to all types of waves, including sound and light. Mathematician and physicist Christian Doppler presented a paper on this effect in 1842, describing how frequencies of waves change in correspondence to the relative movement between source and observer. In 1948, Hippolyte Fizeau independently discussed the shift in the wavelength of light coming from a star in similar terms. Doppler radar has applications in many fields including aviation, meteorology, sports, and traffic control. For example, Doppler radar is widely used for detecting severe weather, and it is a critical component in wind-shear detection and warning systems for airports.

Mathematics of Waves

The Doppler effect relies on the mathematical properties of waves. Transverse waves, which disturb a medium perpendicular to the direction the wave is traveling, are described in terms of their wavelength and amplitude. Wavelength is the distance between two wave crests or troughs, while amplitude is the height of the wave. An example of this is light. Longitudinal waves produce a series of compressions and rarefactions in a medium and are described by their amplitude and frequency. An example is sound, where amplitude corresponds to intensity (or “loudness”) and frequency corresponds to pitch.

A car with a siren emits a series of sound waves of constant frequency. If the car moves toward a stationary observer, the waves will seem to be “bunched up” (to have greater frequency), thus a higher pitch. The same siren moving away will have waves that appear “stretched out,” with lower frequency and pitch. Similarly, an oncoming light source will appear more blue, while one moving away will appear more red, corresponding to higher and lower frequencies on the electromagnetic spectrum. The amount of change in frequency is relative to both speed and direction of the moving object. The speed of a moving object can be measured by shooting waves of a known frequency at the object, and then observing the frequency of the waves that bounce from the object to the source. The difference between the outgoing and incoming frequencies is used to calculate speed. Common examples are the handheld radar guns used to measure the speed of automobiles or a thrown baseball. Edwin Hubble, for whom the Hubble Space Telescope was named, used the Doppler effect to help measure the distances to other galaxies. Light from other galaxies looks more red, indicating they are moving away. This “redshift” is commonly used as evidence in favor of the Big Bang theory of the origin of the universe.

Weather Detection

Many consider Doppler radar to be the best tool available for detecting tornadoes, hurricanes, and other extreme weather in the twenty-first century. Weather stations commonly emit radio waves that strike objects like clouds or heavy rain, and reflect back. Meteorologists use this data to determine the speed and direction of a weather system, as well as for probabilistic models to predict the path and potential severity of a storm in a given geographic area. Mathematical algorithms produce color-coded weather maps, weather animations, and other visualizations for new programs or Web sites, indicating how a storm system is predicted to move through a geographic area. Some researchers have used input data from a single radar station and knowledge of the mathematical structure of hurricanes to construct three-dimensional maps.

In the twenty-first century, a system of 21 Atlantic and Gulf coast radar stations, starting in Maine and ending in Texas, gathers real-time data to mathematically estimate the characteristics of hurricanes within 120 miles of the coast. Previously, forecasters had to fly aircraft into oncoming hurricanes and throw instruments overboard to collect data, giving them a lead time of about half a day before hurricane landfall. Other mathematicians have explored numerical weather prediction using Doppler radar and a technique known as “four-dimensional variational data assimilation,” which estimates model parameters by optimizing the fit between the solution of a given model and a set of observations the model is intended to predict.

Bibliography

Harris, William. “How the Doppler Effect Works.” http://www.howstuffworks.com/science-vs-myth/everyday-myths/doppler-effect.htm.

O’Connor, J. J., and E. F. Robertson. “Mactutor History of Mathematics Archive: Doppler Biography.” http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Doppler.html.

Schetzen, Martin. Airborne Doppler Radar: Applications, Theory and Philosophy. Reston, VA: American Institute of Aeronautics and Astronautics, 2006.