TRMM rainfall measurement for synoptic systems I have gone through the website of TRMM that explains how the TRMM satellite gather precipitation data- http://pmm.nasa.gov/node/158.  During the Indian Summer Monsoon (ISM), we have synoptic systems that deliver rainfall during the months of June through September. These systems are spread over days and often provide continuous rainfall i.e. they are continuous atmospheric processes. How does the TRMM satellite, which I do not believe is a geostationary satellite, 'continuously' measure the rainfall over a wide region of space? 

Now one way they could overcome that is they mention they have a dozen such satellites that measure the precipitation but do those dozen satellites coordinate their work as in a A-train satellite constellation? If they do not coordinate their work they probably return to the same location every few hours to make a measurement. 

If that is true what sort of assumptions are made  about what happens between these observations from a algorithms perspective to derive precipitation values?
The Tropical Rainfall Measuring Mission (TRMM) satellite is a single satellite, carrying multiple instruments on board, as described here. It makes a total of 16 orbits/day at a low altitude of 400 km, with different swath widths for each instrument. Thus, the measurement product that you get is not a "continuous" measurement of a single atmospheric phenomenon, but a composite of multiple orbital passes and instruments, where the density of coverage will depend on the time period over which the composite is made.

If you take a look at TRMM data access page, there are 3 levels of data, each level depending on the amount of processing and quality control:


Level 1: Precipitation radar (PR) reflectivity
Level 2: Combined rainfall profile - PR + Microwave interferometer (TMI)
Level 3: 3-hourly composite of the above + additional data from other instruments (satellites, ground measurements, ground radars), with extended range to 60S - 60N. 


Level 1 and 2 data are nearly raw, and are not composites - they give you data from all 16 orbits/day, and each swath cross-section along the orbit will have a timestamp associated with it. Level 3 data will be the most spatially "continuous" of the three, but one must be aware that this is a composite of multiple orbital swaths and multiple instruments, so it is more suitable for slowly evolving (larger scale) systems.

Related: Is there a database of global rainfall time series?