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Tue, July 29, 2014. Last week I have been discussing a dozen inverse problems with a crows of school kids. We started with discussing simulations. Assume you try to simulate the weather, for example.
Today, we are able to simulate quite complex problems. For example, we know many physical laws of the atmosphere. It is basically a fluid, which flows over a rotating sphere. But there are, of course, many physical processes influencing this motion. There is radiation, coming from the sun, all the time. It is heating up the atmosphere, which is then rising and leads to particular patterns of atmospheric motion. We all know the deep and high pressure areas, which strongly determine our weather all the time. Please look at the infrared radiation coming from the atmosphere, which is reflecting the temperature distribution all over the globe, compare the image on the right-hand side.
Standard models work globally, with grid points every 15 kilometers, and with heights from the ground to about 70km height, with 90 grid poins distributed vertically, compare for example Wikipedia. Please keep in mind that this is changing about every 18 month! All over the world there are teams which push this forward all the time. Assume you have a model which can calculate the next step in time, when it is given the current state.
You need to provide the state of the atmosphere with 90 grid points vertically and 15km spacing horizontally. The earth has a radius of about R = 6371km, c.f. http://en.wikipedia.org/wiki/Earth_radius. The earth surface A is calculated by $$ A = 4 \pi R^2 $$ which with $R = 6371km$ is approximately $$ A = 5.1 * 10^8 km^2, $$ compare for example Earth Surface. You will have approximately one point every $d = 1/(15km*15km) = 1/225 km^2$, i.e. $$ n = 90 * A * d = 2 * 10^8 points. $$
The task of data assimilation is to determine the state of the atmosphere at $n$ points globally every few hours. Today, this period for global weather models is usually 3 hours. But we are moving down, we might soon go to one hour “assimilation cycle”, which means that we determine the state of the atmosphere globally for each hour using a broad range of measurements which are available today.
If you want to know more about how the state of the atmosphere is calculated every three hours, then either look into current books on NWP = Numerical Weather Prediction, or come back this blog on a regular basis. Also, we will talk about various inverse problems, i.e. the reconstruction of physical quantities from measurements. It is a highly fascinating area, which touches many parts of science.
All the best,
Roland Potthast, University of Reading and DWD (German Meteorological Service) email@example.com