Are there measurement techniques that can help determine the exact geometry of a volcanic vent?
Interferometric synthetic aperture radar (or InSAR) is a remote sensing (airplane or, more commonly, satellite) radar techinque used to measure surface deformation. Repeated measurements of radar amplitudes over a common area are used to estimate changes to surface topography. InSAR measurements of surface changes have been used with numerical models of dyke propagation to model magma chambers in the East African Rift System. 

Other geophysical methods that can be used to image/model active magma chambers include seismic interferometry, magnetic and gravity methods. Here is a recently published example from Japan which used seismic interferometry to image an active magma chamber. Magnetic and gravity measurements can be taken from airplanes, which may be a bit safer than being on the ground. If  the magma chamber has magnetic/density properties different from the surrounding country rock, then mag/grav forward/inverse modelling could be used to create a model of the magma chamber geometry.