The best data assimilation system in the world is the European Center for Medium Range Weather Forecasting (ECMWF) 4DVAR system. 4DVAR stands for four-dimensional variational assimilation.
4DVAR at ECMWF is an incredible marvel. According to their web site, ECMWF processes and uses 40 million weather observations daily, most from satellites. Incredibly, they have a web site where you can actually see the observations going into each forecast cycle. The slides below summarize everything that went into the 0600 UTC 3 April initialized ECMWF forecast cycle. You can click to enlarge, although for the purposes of this discussion, the details aren't important. The slides are mainly to provide a glimpse at the incredible collection and processing of data that occurs 24/7 to produce a global weather forecast.
An interesting thing about data assimilation is that all observations are not created equal. Some have a bigger impact than others and in some cases, bringing together multiple observation types gives more bang for the buck than if you added up the impact of each observing type independently.
We are currently running an inadvertent experiment on the value of aircraft observations, which have declined significantly in the past month due to COVID-19 travel restrictions and reductions. Below is the trendline for Europe through 24 March.
As of 23 March, the ECMWF reported a 65% reduction in European aircraft weather observations and a 42$ global reduction.
Based on early studies, the ECMWF suggests that removing all aircraft observations from their data assimilation system results in a degradation of short-range temperature and wind forecasts at jet-stream level of 15% and surface pressure forecasts of 3%. The former illustrates why it is advantageous for aviation companies to provide such observations for weather prediction, since the improvement in forecast skill reduces fuel consumption through better route planning and may also improve passenger comfort through better turbulence forecasts.
Modern data assimilation truly is an incredible scientific marvel. Without it, numerical weather prediction would not be possible and any great global forecast system has a great data assimilation system.