## Appendix A: veiling errorFrom Eq. (2) and (4), we can easily compute the matrix and then the matrix . The veiling variance is written: and that on the scaling factor is given by: where the upper bar over stands
for the expected value, The value of is simply obtained by applying Eq. (2) to the continua: where is the expected value of the object continuum. In the following, we assume, for simplicity and whithout loss of generality, that the noise of the reference spectrum is much smaller than that of the object spectrum and can be neglected, . Now we consider constant additive and photon limited noises. ## A.1. Constant additive noiseSetting , inserting Eq. (A4) into Eq. (A1) and using the relation: it comes after some transformations: is the variance of the reference spectrum expected value , given by: is the signal to noise ratio on the object total spectrum flux, and is written: A rapid examination of Eq. (A8) shows that the coefficient is always close to 1. ## A.2. Photon limited noiseHere, . Inserting Eqs. (A4) into Eq. (A1) and using Eq. (A5), it comes: where is the signal to noise ratio on the object total spectrum flux and is given by: Numerical calculations using our K7V spectrum show that the coefficient is generally close to 1 for any veiling value. Note that using the relation , the relative veiling error for additive and photon noise takes the very simple form: where is the contrast of the object spectrum in noise units. ## Appendix B: bias calculation## B.1. Uncorrect noise estimateWe assume constant additive noises for the object and for the reference. Under these conditions, the estimated veiling value has the following analytical expression (see HHKHS): and
(resp.
and
) are defined in Appendix A,
and
is the input noise ratio
between the reference and the
object. The expressions and
are yet averaged quantities over
the number of points where is derived from where (resp.
) is defined in Appendix A, and
, with
. It can be verified that for
, there is no bias, i.e.
. We also checked the general
validity of Eq. (B4) through simulations, using the spectrum Two limiting cases are particularly interesting. The first one, , assumes that the noise associated with the reference is zero, in Eq. (2). The derived veiling expected value is maximum, the relative bias is positive and is given by: The second case corresponds to Figs. B1a and B1b show the relative bias as a function of ## B.2. Mismatches between the object and the reference spectraWe assume here that the measured object and reference spectra
with , where
represents the systematic error
function. If Let They are the sum of three terms and differ only through the third
term. The first term is due to the apparent veiling of the reference
To investigate further the third term in Eqs. (B10) and (B11), it is interesting to calculate the relative difference between and . It is given by: where is the correlation
coefficient between and © European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |