Forecasting (at least what you’re talking about ie days/week in advance) doesn’t really use radar but rather takes observations (temperature, pressure, atmospheric energy) at some distance from you and plugs those variables into equations that calculate what they will be in the future over your location. This is the basis for weather models.
Radars can tell you about what’s happening currently since precipitation scatters the energy from a radar beam. Depending on how strong the energy scatters, we know if it’s a strong rain/snow storm or just a drizzle or flurry.
A weather radar is a radar which use a wavelength which is well reflected in rain dropplets. So it is able to scan the sky for rain and clouds. A lot of them are also dopplar radars which means they can detect the direction of the dropplets as well as how many they are. Weather radars on their own is only able to dectect clouds within line of sight and only as they are just now. The short range means that you can only use their data to predict the weather in the very near future, usually only an hour into the future. However they are extremely accurate at this. Their usability is also limited in varied terrain both due to mountains obsucing their view but also because clouds and rain tends to form quickly and dissapear quickly in these types of terrain.
For longer forecast however you use completely different measurement instruments. There are measuring stations all over the world which record things like pressure, wind, temperature and humidity. We also send probes into the atmosphere on balloons to measure these parameters in the upper atmosphere. There are also weather satellites which use infrared cameras to record temperatures and humidity all over the world. All these parameters are put into weather models we have which we have made using the vast amount of data we have of how the weather on the globe behaves. Using these models we can calculate how the weather is going to behave over the next few days. The issue with this is that even small differences in the input may have vast changes to the output so even if we have perfectly exact weather models, which we do not, then we could still get vastly different weather forecasts depending on the error in our measurements and how we assume the weather are where we do not have measurements in place.
I am not so knowledgeable at forecasting but for current measurements over the ocean, there are some interesting processes. Using satellites researchers will measure the reflection off of the ocean waves of various wavelengths of radar beamed from the satellite. Based on the brightness, angle, polarization, fuzziness, etc they can tell how large the waves are, which way they are going what the wind speed is and where that is going etc.
I’m sure they use this for prediction somehow but I am more about looking at past data.
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