I am using the cross-correlation between an observed spectrum and doppler-shifted model templates to estimate a radial velocity of a star. The model template to be doppler-shifted is the best-fit template in broad-band typing. 

I did tests using simulated spectra made from AMBRE templates as an input observed spectrum. The result look good. This is a CCF in a case of Teff=7000K, Logg=4, Z=0, [alpha/Fe]=0. 10% Gaussian noise is added to the spectrum. The S/N of the broad band photometries is S/N=100. The peak of CCF is 200 km/s, which is consistent with the input velocity of 200 km/s. 

Now I am trying to estimate uncertainties of CCF to obtain an error of the estimated radial velocity. The errors will be used for spectral fitting. The ETC will be used for making simulated spectra with flux errors. 

Using the ETC/simulator output spectra, the radial velocities were estimated from its cross-correlation. These three examples are the results of cross-correlation for HSC r = 20 AB mag stars with Teff=6000K, 7000K, 8000K. The noisy near-IR spectra are now masked and also some strong atmospheric absorptions are masked. The radial velocity estimate works for these stars. 

The cross-correlation as a function of r-band magnitude.

As long as we detect absorption lines, we can obtain the reasonable estimate of radial velocity up to r=21 mag stars. in the ETC output spectra, there are spike pixels in NIR. These significantly make the estimate worse. The mask of NIR or clipping the low S/N pixels can improve it.

The uncertainties of our radial velocity estimate are measured as a function of HSC magnitude. The uncertainty, |delta v/v|, is ~1% r_AB = 13-18 mag with a large dispersion in both cases of 450 sec exposure and 900 sec exposure. At r_AB = 19, the uncertainty in the 900-sec exposure gets better than that in the 450-sec exposure. At r_AB >= 20, H-alpha is not detected and therefore we cannot obtain a reasonable estimate.
A question about how much uncertainty we tolerate for flux calibration will be addressed after we obtain flux calibration vector using this estimated radial velocity.
 
I used an AMBRE template set which includes 24 models and covers a wide range of stellar parameters:
Teff = [6000, 7000, 8000]
logg = [2, 4.5]
Z = [ -3., 0., 0.25]
[alpha/Fe] = [0., 0.4].
 
I have two estimated radial velocity. One is a peak of CCF (blue). The other is a peak of the best fit Gaussian (orange).
 

 

I found a systematic offset in the estimate. The estimated velocity is blue-shifted by 1%. I have not yet identify the cause.
 
Note: y-axis is now delta_v/|v|. It is different from |delta_v/v| in the above figures.
 

The systematic offset was fixed. That was caused by the rounded-off wavelength of the simulated spectra. The round-off error is 0.0045 nm which corresponds to ~2 km/s and ~1% of the input velocity of 200 km/s.
 
Now, the offsets disappear.
 

The estimate of radial velocity is calculated again. The input radial velocity is 200 km/s. 
abs(delta v/v) ~< 1 % at HSC r < 19 mag. 
 
450s exposure:

 
900s exposure:

 
 
 Statistics:
 

r_mag = [13 14 15 16 17 18 19 20 21 ]

45s exp time
median = [0.006 0.005 0.006 0.007 0.02 0.022 0.116 3.25 1.107]
mad*1.4826 = [0.006 0.004 0.006 0.007 0.017 0.02 0.077 0.221 0.943]
std = [0.01 0.007 0.009 0.014 0.019 0.021 0.12 0.299 1.155]

900s exp time
median = [0.006 0.003 0.007 0.006 0.006 0.012 0.035 1.249 2.569]
mad*1.4826 = [0.004 0.003 0.007 0.004 0.004 0.013 0.033 1.517 0.773]
std =  [0.007 0.004 0.008 0.01 0.004 0.014 0.048 1.211 0.981]

The results have been reported in the telecon. The systematic offsets are now fixed. We can close this ticket. 

I found that the x-axis of the figures is the total magnitude, not the fiber magnitude. This is the synthetic magnitude of an input spectrum to the ETC simulator. Let me correct it.