I want to study a case of heavy precipitation in a region. I considered SLP, 500 hPa geopoteintial height. However I want more quantities with their levels, but which quantities and levels are necessary and useful?
a paper published in 2014 entitled "Change in North American Atmospheric Conditions Associated with Deep Convection and Severe Weather using CRCM4 Climate Projections" from Paquin et al. might be useful. They studied meteorological quantities that possibly cause extreme weather like precipitation extremes.
I believe that stability indices like CAPE, LI etc are always a good indicator regarding convective precipitation. You could check their values during the previous hours (for example 1, 2, 6 hours prior to the event) during and after the event. Regarding their threshold values, a quick search may give you a lot of results. But be sure that these thresholds apply to your area of interest.
However, if you are thinking about analysis prolonged stratiform rainfall, these indices may not help you.
Integrated water vapour amount retrieved by GPS can be very useful to provide information about the water vapour field evolution, especially if you have several GPS in your area. (there are quite a few reference, just let me know if you'd like some).
there are quite a lot of papers on that topic ... depending on what exactly you are looking for.
on the basics of IWV retrieval from GPS :
Bevis, M., S. Businger, T. Herring, C. Rocken, R. Anthes, and R. Ware. “GPS Meteorology- Remote Sensing of Atmospheric Water Vapor Using the Global Positioning System.” Journal of Geophysical Research 97, no. D14 (1992): 15787–801
Solheim, Fredrick S., Jothiram Vivekanandan, Randolph H. Ware, and Christian Rocken. “Propagation Delays Induced in GPS Signals by Dry Air, Water Vapor, Hydrometeors, and Other Particulates.” Journal of Geophysical Research 104, no. D8 (1999): PP. 9663–70. doi:199910.1029/1999JD900095.
Concerning the use of GPS for case study :
Labbouz, L., J. Van Baelen, F. Tridon, M. Reverdy, M. Hagen, M. Bender, G. Dick, T. Gorgas, and C. Planche. “Precipitation on the Lee Side of the Vosges Mountains: Multi-Instrumental Study of One Case from the COPS Campaign.” Meteorologische Zeitschrift 22, no. 4 (August 1, 2013): 413–32. doi:10.1127/0941-2948/2013/0413.
Businger, Steven, Steven R. Chiswell, Michael Bevis, Jingping Duan, Richard A. Anthes, Christian Rocken, Randolph H. Ware, Michael Exner, T. VanHove, and Fredrick S. Solheim. “The Promise of GPS in Atmospheric Monitoring.” Bulletin of the American Meteorological Society 77 (January 1996): 5–18. doi:10.1175/1520-0477(1996)0772.0.CO;2.
Okamura, Osamu, and Fujio Kimura. “Behavior of GPS-Derived Precipitable Water Vapor in the Mountain Lee after the Passage of a Cold Front.” Geophysical Research Letters 30, no. 14 (July 23, 2003): 1746. doi:10.1029/2003GL017572.
Champollion, C., F. Masson, J. Van Baelen, A. Walpersdorf, J. Chéry, and E. Doerflinger. “GPS Monitoring of the Tropospheric Water Vapor Distribution and Variation during the 9 September 2002 Torrential Precipitation Episode in the Cévennes (southern France).” Journal of Geophysical Research 109, no. D24 (December 17, 2004): D24102. doi:10.1029/2004JD004897.
and many others, some using also tomographic techniques to retrieve the 4D water vapour field (instead of only the IWV) : Flores et al. 2001; Seko et al. 2004; Bastin et al. 2005; Bi et al. 2006; Champollion et al. 2009; Van Baelen et al. 2011
All above references are relevant. But the influencing parameters depend also on the problem of interest. If this is mountaneous region pressure difference on the line normal to topography ridges could be also relevant parameter. Also average temperature lapse rate between 850-500 mb surfaces and dew point deficit between ground and 500 mb are often found as relevant parameters in such cases.