Astron. Astrophys. 342, 763-772 (1999)

Appendix A: veiling error

From 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, r is the veiling of the object and , where and represent the variances of and , respectively. Making the approximation , the relative error on the scaling factor is written:

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 noise

Setting , 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 noise

Here, . 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 estimate

We 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):

with:

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 m and so, are close to their expected values. For example, and . Hence, a good approximate of the expected value of can be obtained by taking the expected value of the series development to the second order of around the expected values of the 's and the 's (see Papoulis 1965). We find that, within a relative precision of , is given by:

where is derived from C by replacing the 's by their expected values. Let r be the correct veiling, we define the bias by . From Eq. (B3), using the relations , and , and after some simple transformations, the relative bias is written:

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 S of Sect. 2.4 with R=40000, centered at 5200 Å and of width 40 Å.

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 f infinite, it assumes that the noise associated with the object is zero, in Eq. (2). The derived veiling expected value is minimum, the relative bias is negative and is given by:

Figs. B1a and B1b show the relative bias as a function of f for various values of and .

Fig. B1a and b. Relative bias on the estimated veiling as a function of the input error ratio between the reference and the object, normalized to the true error ratio, : a for and various values of , b  for .

B.2. Mismatches between the object and the reference spectra

We assume here that the measured object and reference spectra O and S are not noisy, i.e. , and for simplicity that the object spectrum is unbiased. Let us study for example the problem of mismatches between the reference spectrum S and the underlying object stellar spectrum T. In the absence of systematic errors, we define as the apparent veiling of the reference with respect to T (calculated for example for the limit ) and the residual function of the veiling equation. S can be written as follows:

with , where represents the systematic error function. If S is an exact veiled version of T, i.e. , then the object veiling with respect to S must be independent of any input noise ratio . Let us examine the extreme object veiling values and obtained for the limits and infinite, respectively. From Eq. (B1) and (B2), and have very simple analytical expressions:

and

Let r be the correct object veiling with respect to T, obtained for example by replacing S by T in Eq. (B8). We define the biasses and by and . Combining Eqs. (B7), (B8) and (B9) and after some transformations, the relative veiling biasses are written:

and

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 S with respect to T. Assuming and , it can be approximated by . The second and the third terms are due to real spectral mismatches and to systematic errors. The second term is in general negligible, because we expect the mean value of to be close to zero for local mismatches of the order of a few% only. The third term is scaled by the ratios and and is also negligible if the spectrum contrast is large compared to the residual function . Under these conditions, the bias due to mismatches between the object and the reference is dominated by the apparent veiling of the reference. Setting and making the approximation , the relative bias is simply given by:

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 T. Eq. (B13) clearly shows that the two veiling values and are equal only if the is the null function or if it is a linear function of T. As the latter possibility is unlikely, the equality between and implies necessarily that the function is null and consequently that Eq. (B12) is valid. On the other hand, if and are distinct, the third term in Eqs. (B10) and (B11) cannot either be neglected. It introduces another additive bias which depends on the input value. The distance between and is a good indicator to evaluate the importance of mismatches between S and T which cannot be interpreted in terms of veiling, and probably gives also a good order of magnitude of the resulting bias.

© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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