# Temporal Variance Spectrum

##### Temporal variance spectrum

My description follows roughly the detailed explanations of how to create and how to explain the temporal variance spectrum (TVS) in the elaborate article of Fullerton, Gies and Bolton Fullerton 1996.

To search for variations in the profile of a line over time one can overplot different spectra as you see here or create a dynamic spectrum as gray scale plot (to come) or most easy calculate the variations of each pixel at a each wave length in your time series. This is quite easy with modern software.

All we need is love. Eh, all we need are spectra covering a common wave length range. $S_1 ... S_N$ We use normalised spectra. We do not care about the different noise sources in our equipment if the SNR is large enough, say 200. The we will only consider the photon noise in our spectra.

From this sequence we calculate the mean spectrum $$\overline{S}$$. $\overline{S} = \frac{1}{N}\sum_{i=1}^N S_i$ Your software does this on a pixel basis.

The variations are the deviations of the single spectra from the mean spectrum. The mean sum of the quadratic deviations is the temporal variance spectrum. $\mathrm{TVS} = \frac{1}{N-1}\sum_{i=1}^{N}(S_i - \overline{S})^2$

We are nearly done. Naturally the variance is proportional to the amplitude of the signal. A larger signal consists of more photons and more photon noise. Noise rises with the square root of the number of photons. We do not want to read a higher variance due to a better signal. We only want to see true line variations. We divide the VAR by the mean spectrum to remove this unwanted enhancement. This is the normalised TVS. $\mathrm{TVSn} = \frac{\mathrm{TVS}}{\overline{S}}$

In practice, we find that it more convenient to plot (TVS)1/2 instead of the TVS itself, because the (TVS)1/2 scales linearly with the size of the spectral deviations and gives a more accurate impression of the relative amplitudes of the line profile variations from star to star. (This sentence is literally stolen from Fullerton, Gies and Bolton). In short: $\mathrm{TVS\sigma} = \sqrt{\mathrm{TVSn}}$

If you like a percant scale at the ordinate just multiply with 100: $\mathrm{TVS\%} = 100\cdot\mathrm{TVS\sigma}$

I'm pretty shure your software does this for you. Find an example plot here.