3 Data errors

3.1 Photon noise

The first test considers image reconstructions from spectra with a large range of signal-to-noise (S/N) ratios. We add random noise to the profiles for both test cases to simulate S/N ratios of 3000, 900, 600, 300, 150, and 75:1, and then recover the image using Tikhonov and maximum-entropy inversion regularizations. The series of maps in Fig. 2 compare some of the results while Fig. 3 shows the "observations'' and the respective fits for both cases and for S/N=150:1. This S/N represents typical observations of a V=11th-magnitude star with a 4 m class telescope, a spectral resolution of R=100000 and a 50-min integration time (e.g. as achieved for the T Tauri star HDE 283572 with the 3.6-m CFHT; Strassmeier & Rice 1998b). Note that a case of S/N=75:1 has never been used for real applications so far although Barnes et al. (1998) used spectra with a S/N of as low as 45:1 but boosted this to a S/N of over 1000:1 for a co-added line profile using least-squares deconvolution. The maps in Fig. 4 are difference maps between the original input map and the recovery and emphasize the surface regions of increased sensitivity.

All inversions correctly recover the temperature and size of the polar spot and its appendage. The low-inclination case (Case 2) proves to be more of a challenge though than the high-inclination case (Case 1). This is because the low-to-moderate latitude spots, especially the two spots at $\ell=230$ $\hbox {$^\circ $ }$ , compete for recognition against the polar feature. This causes mild gaps in the otherwise homogeneous polar cap that mimic weak polar appendages. Except maybe for the highest S/N-ratio spectra of Case 2, the double spot is never resolved. None of the reconstructions reproduced the cool umbra of the large equatorial spot at $\ell=320$ $\hbox {$^\circ $ }$ nor the weak equatorial band between longitudes of 10-190 $\hbox {$^\circ $ }$ . On the other hand, all reconstructions sucessfully recovered the small hot spot at $\ell=120$ $\hbox {$^\circ $ }$ as well as the single, intermediate latitude feature at $\ell=270$ $\hbox {$^\circ $ }$ . Overall, we conclude that S/N ratio alone does not significantly improve the recovery once $S/N\approx300$:1 is surpassed if all other factors are perfect.

3.2 Constant continuum displacement

Spectroscopists define a continuum as the fit to the upper envelope of an absorption-line spectrum. The various astronomical data reduction packages offer many ways to achieve this fit, e.g. either a low-order polynomial or a bi-cubic spline and sigma-clipping might be used in IRAF, the NOAO image reduction and analysis facility (available at iraf.noao.edu). Fitting two-dimensional echelle spectra across many orders is usually more complex than a single-order coudé spectrum and systematic errors may creep into particular spectral regions. This is usually negligible for good S/N spectra of relatively normal stars but may amount to up to $\pm$1% for very cool stars with broad absorption lines and many blends.

In this and the following chapter, we test the effects of a systematic misfit of the continuum during the data reduction process. First, we recover our two test cases with a continuum offset, $\Delta I$, of +0.5% and -0.5%, respectively (Fig. 5, left two images). A positive $\Delta I$means a higher continuum and therefore the line depth is artificially increased. The effect on the recovered maps is very pronounced because the missing, or additional, line flux must be distributed across the stellar disk without adding or removing spots that are not in the data. It usually makes the average temperature lower by $\approx$200 K when the continuum is too high by 0.005, and 300 K warmer when the continuum is too low by 0.005. The broad-band V-I color can not be fitted in any of the two cases while the fit to the line profiles is of comparable $\chi ^2$. Then, we iteratively adjust the abundance so that it still matches the equivalent width of the spectral line but also fits the light and color-curve zero point (Fig. 5, right two images). This is what we usually do once the continuum offset is fixed (or went unnoticed). This allows a reasonably correct recovery of the average temperature ($\pm$50 K) and the individual feature's temperature contrast but still creates several artifacts, especially noticable at high latitudes in the low-inclination case (i.e. Case 2 in Fig. 6). However, errors in the adopted abundances are related to the adopted atomic line parameters, to the amount of synthesized blends, and to the uncertainty of the rotational velocity and are usually of the order of 10-50% (depending on spectral type and luminosity, pre-main sequence or post-main sequence, etc.).

Figure 7: The effects of a continuum slope of $\pm$1% for Case 1 (top left) and Case 2 (top right). S/N of the artificial data was chosen as high as possible (3000:1) in order to isolate the slope effect (bottom panels, respectively)

3.3 Continuum slope

Another type of continuum error appears if the continuum is overfitted, e.g. when a polynomial with too high a degree was used for setting the continuum. For a particular narrow wavelength region of only a few Ångströms, the continuum subtraction could thereby introduce a local slope of the spectrum and the shape of the line profile becomes slightly asymmetric (this is a small effect and usually unnoticable by visual inspection of the spectrum). It appears significant only for the extreme cases of very cool and rapidly rotating stars, e.g. for the pre-main sequence star V410 Tau (K4) or for the double-lined eclipsing binary YY Gem (dMe+dMe), where molecular TiO-bands contribute to obscure the true continuum.

Figure 7 compares the impact on the two testcases and also shows the actual input data and the fits. The S/N ratio was chosen as high as possible in order to make the effect clearly visible. Compared to "real'' data of limited S/N the test case is clearly exaggerated. The right panels in Fig. 7 show a recovery for Case 2, where we expect the larger effect due to the larger full width of the spectral line at continuum level. The input data were artificially changed to mimic a continuum slope of -1% to +1% from the blue side of the line profile to the red side, i.e. a total of 2% throughout the full width at continuum level. We find all features correctly recovered, even the small hot spot, but the average temperature of the polar spot appeared lowered by 150 K compared to the corresponding recovery without a continuum slope in Fig. 2. Case 1 did not have this problem. If the same test is made with $S/N\leq 150$, the polar brightening effect is buried in the noise.

Figure 8: The effect of unnoticed scattered light in the spectrograph. Left column: Recovery of Case 1, right column: Case 2. The lower panels are the difference maps in the sense input minus recovery. The total amount of scattered light was 4%, which is a common number for cross-dispersed echelle spectrographs. The two top images should also be compared with the input map in Fig. 1. Notice the artificial polar brightening for Case 2 and the artificial bright and dark bands at low and intermediate latitudes for Case 1

3.4 Scattered light in the spectrograph

Scattered light in our artificial data is created by adding in 4% of white light before the recovery in order to simulate the most common form of scattered light. The recovery is then made with the initial stellar parameters, just as if the scattered light went unnoticed in the data reduction.

Figure 8 compares the reconstructed maps for both cases. A Tikhonov regularisation with a low smoothing factor was adopted and the S/N of the artificial data was 3000:1. A small change of about 0.03 in the logarithm of the abundance was adopted to allow for the small change in the line strength. All main features are correctly recovered but we find that the polar region of the low-inclination case (Case 2) is more prone to artifacts than in the high-inclination case (Case 1). On the other hand, high-to-moderate latitudes and the equatorial regions are more affected in Case 1 than in Case 2. Bright bands with $\Delta T\approx100$ K above photospheric or, for Case 2, a polar brightening are the results. The temperature of individual features can increase with respect to the input image by up to 700 K. The largest increase is seen, as expected, for the two smallest spots at $\ell\approx 225$ $\hbox {$^\circ $ }$ that are barely recovered. This behaviour is due to the effective change in the observed profile when scattered light remains uncorrected. Note that the behaviour in Case 1 is quite similar to what we see when the macroturbulence is modified substantially as in Fig. 12a. It is interesting that when more modest amounts of scattered light as well as macroturbulent errors are all introduced together as in our final test case with realistic amounts of error for many parameters all introduced at once, the banding effect is far less pronounced.

Figure 9: Recoveries with a spectroscopic phase gap but with continuous photometric coverage. The input spectra have S/N=300:1. Maps in the left column are for Case 1, in the right column for Case 2. From top to bottom: phase gaps of 60 $\hbox {$^\circ $ }$ (0 $.\!\!^{\scriptscriptstyle\rm p}$167) from 305 $\hbox {$^\circ $ }$ to 5 $\hbox {$^\circ $ }$ , 80 $\hbox {$^\circ $ }$ (0 $.\!\!^{\scriptscriptstyle\rm p}$222) from 285 $\hbox {$^\circ $ }$ to 5 $\hbox {$^\circ $ }$ , and 100 $\hbox {$^\circ $ }$ (0 $.\!\!^{\scriptscriptstyle\rm p}$278) from 265 $\hbox {$^\circ $ }$ to 5 $\hbox {$^\circ $ }$

3.5 Phase gaps

All previous recoveries in this paper were made with optimal phase coverage, we used 18 line profiles equidistantly distributed. This is only seldomly achieved with real observations given interruptions due to bad weather, telescope-time limitations etc. In this section, we will successively remove more and more line profiles from the input and investigate the increasing detoriation of the recovered images. However, all maps were obtained with full phase coverage for the photometry.

In the first test only two successive phases are removed, according to a phase gap of 0 $.\!\!^{\scriptscriptstyle\rm p}$167 or 60 $\hbox {$^\circ $ }$ in stellar longitude. Then, we remove three successive phases (a gap of 0 $.\!\!^{\scriptscriptstyle\rm p}$222 or 80 $\hbox {$^\circ $ }$ ) and four successive phases (a gap of 0 $.\!\!^{\scriptscriptstyle\rm p}$278 or 100 $\hbox {$^\circ $ }$ ). Figure 9 compares the obtainable image quality for input data with S/N=300:1 and no external errors. In such a situation the recoveries are better than intuition would suggest, even directly at the phases that are not covered by spectroscopic observations. Note that TEMPMAP shifts more weight from the profile data to the photometric data when a (to be specified) gap occurs. In the present case this was only relevant for the 100 $\hbox {$^\circ $ }$ -gap tests.

3.6 Phase gaps without photometry

The same series of inversions from the previous section is now made without any photometry as input. For real applications, we shift more weight to the continuum once a preset phase gap of approximately 40-50 $\hbox {$^\circ $ }$ in the spectrum coverage occurs. The recovery of the missed regions on the star is then dominated by the light and color curve information rather than the spectral line profile. A worst case scenario in real observations occurs when we are left with unevenly distributed spectra with large phase gaps and no photometry. The result of such a test is shown in Fig. 10.

Figure 10: Two recoveries with a large phase gap and without photometric data: a) for high-S/N input spectra (3000:1), b) for realistic S/N spectra (300:1). Only Case 1 is shown. A correct recovery was possible for both cases even within the phase gap between $\ell =265$ $\hbox {$^\circ $ }$ to 5 $\hbox {$^\circ $ }$

We find that the recovery is essentially independent of the S/Nratio of the data once S/N is greater than $\approx$300:1. Even a phase gap of as large as 100 $\hbox {$^\circ $ }$ does not significantly affect the recovered features. Of course, this test assumes randomly distributed photon noise and that no external errors are present; a situation probably never achieved with real data. However, the test clearly demonstrates the robustness of our code.