It is demonstrated that accurate values of the parameters, such as relaxation times, which determine the appearance of transient ESR spectra and, to some extent their time-dependence, can be obtained from analysis of the oscillation patterns observed at high microwave field strengths. A new methodology is introduced for this, which also allows more accurate determination of hyperfine coupling constants. This opens the route to precise simulation of polarized spectra and to the identification of the radical pairs from which the polarization results.