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Goodness-of-Fit Tests DIFF1 and DIFF2 for Locally Normalized Supernova Spectra

Two quantitative tests DIFF1 and DIFF2 for measuring goodness of fit between two locally normalized supernova spectra are presented. Locally normalized spectra are obtained by dividing a spectrum by the same spectrum smoothed over a wavelength interval relatively large compared to line features, but relatively small compared to continuum features. DIFF1 essentially measures the mean relative difference between the line patterns of locally normalized spectra, and DIFF2 is DIFF1 minimized by a relative logarithmic wavelength shift between the spectra: the shift is effectively an artificial relative Doppler shift. Both DIFF1 and DIFF2 measure the physical similarity of line formation and thus of supernovae. DIFF1 puts more weight on overall physical similarity of the supernovae than DIFF2 because the DIFF2 shift compensates somewhat for some physical distinction in the supernovae. Both tests are useful in ordering supernovae into empirical groupings for further analysis. We present some examples of locally normalized spectra for Type IIb supernova SN 1993J with some analysis of these spectra. The UV parts of two of the SN 1993J spectra are HST spectra that have not been published before. We also give an example of fitted locally normalized spectra and, as an example of the utility of DIFF1 and DIFF2, some preliminary statistical results for hydrogen-deficient core-collapse (HDCC) supernova spectra. This paper makes use of and refers to material to found at the first author's online supernova spectrum database SUSPEND (Supernovae Spectra Pending further analysis).