Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals

We present an extension of the theory of the time-frequency (TF) misfit criteria for quantitative comparison of time signals. We define TF misfit criteria for quantification and characterization of disagreement between two three-component signals. We distinguish two cases—with and without having one signal as reference. We define locally and globally normalized TF criteria. The locally normalized misfits can be used if it is important to investigate relatively small parts of the signal (e.g. wave groups, pulses, transients, spikes, so-called seismic phases) no matter how large amplitudes of those parts are with respect to the maximum amplitude of the signal. They provide a detailed TF anatomy of the disagreement between two entire signals. The globally normalized misfits can be used for quantifying an overall level of disagreement. They allow accounting for both the envelope/phase difference at a TF point and the significance of the envelope at that point with respect to the maximum envelope of the signal. We also introduce the TF envelope and phase goodness-of-fit criteria based on the complete signal representation, and thus suitable for comparing arbitrary time signals in their entire TF complexity. The TF goodness-of-fit criteria quantify the level of agreement and are most suitable in the case of larger differences between the signals. We numerically demonstrate the capability and important features of the TF misfit and goodness-of-fit criteria in the methodologically important examples.