Most of the methods re geophysical but the question appears about hydro-chemical data test. Very simple to say please look to the groundwater properties in the karst reservoir, you will find number of standard test methods explained .
Your question is a bit broad, I shall assume that by "karst detection" you mean detecting underground drainage or caves systems.
1: identifying springs
Not all springs are obvious, e.g. if they are rising in the bottom of a lagoon. ALso not all springs in an area will be karstic, some may be coming from local shallow flows through the regolith. The chemistry of the water, and other properties (eg temperature) may help distinguish these 2 types.
To detect hidden springs, sample the surface stream waters systematically along the valley and look for sudden changes in chemistry (especially Ca++ & HCO3-) and also in temperature. Also look for areas of tufa or flowstone deposition along the channel or in the adjoining alluvium - these will indicate lime rich waters. See the link to a paper by S.A. Lucas (2001) for examples. Temperature differences are best detected in mid-winter or mid-summer when the underground water shows most difference from the surface waters.
See link to a paper by S.A. Lucas at my Dropbox site. https://www.dropbox.com/s/lurnhhfsvizmtd9/Lucas.2001.DavysCkHydrology.pdf?dl=0
2: Water tracing
This is a separate issue - you are now trying to determine the underground flow route, and the possible distant entry points (including pollution sources, etc).