I'm a fisheries biologist and i'm scouting around for a robust way to determine what the principle environmental influences are on a number of response variables such as fish swimming speed or travel time between two points.
The environmental variables are daily mean water temperature, daily precipitation, barometric pressure, and river flow rate.
The challenge is that these variables are linked -- large rainfalls drop the water temperature and increase the river flow rate; high barometric pressure is associated with high water temperatures and low flow rates.
Another challenge is that most of the environmental variables are not normally-distributed, so parametric statistics may not be suitable without specific transformations. Are there recommended transformations for rainfall data, for example?
Finally, there is the temporal nature of the data -- with auto-correlation in all variates.
Is there a statistically-defensible process to decipher which predictor or predictors are key?
I am sorry, but your questions are much too general to be addressed here. You are basically asking for an entire course in empirical modeling. I suggest you find a consulting statistician who can work with you on this problem.