**FULL TITLE**
ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION**
**NAMES OF EDITORS**
A New Method for Comparing Numerical Simulations
with Spectroscopic Observations of the Solar Photosphere
J. Rybák,1 A. Kučera,1 H. Wöhl,2 S. Wedemeyer-Böhm,2 O. Steiner2
1 Astronomical
Institute, Slovak Academy of Sciences
Tatranská Lomnica, Slovakia
2 Kiepenheuer-Institut
für Sonnenphysik, Freiburg, Germany
Abstract.
A method for comparing high-resolution spectroscopic observations of the solar photosphere with numerical simulations of convection in the
solar photosphere is presented. It is based on the comparison of the granular
continuum contrast obtained from, both the observations and the synthetic spectra, when the latter are calculated from numerical simulations using a particular
type of data degradation. This method can be used post-facto when a minimum of auxiliary information on characteristics of the telescope/spectrograph
and on seeing conditions is available. Here, the method is applied to results of
numerical simulations computed with the CO5BOLD code and high-resolution
spectroscopic observations obtained with the VTT on Tenerife.
1.
Introduction
Results of numerical simulations (SIMs) and high-resolution long-slit spectroscopic observations (OBSs) of the solar photosphere still considerably differ in
their typical spatial sampling since the practical spatial resolution of OBSs is
spoiled by atmospheric seeing and instrumental effects. Different restoration
procedures are nowadays commonly applied to allow a comparison but only a
few attempts have been presented so far for the restoration of 1-D long-slit spectroscopy measurements (Keller, 1994, Keller & Johannesson, 1995, Rodrı́guez
Hidalgo & Ruiz Cobo, 2003). The aim of our contribution is to present a forward post-facto degradation method which could be applicable to the results
of numerical simulations of convection of the solar photosphere when almost no
special auxiliary data on seeing conditions and telescope/spectrograph characteristics are available. This approach is different from the previously applied
forward degradation procedures (e.g. Steffen & Freytag, 1991).
2.
Method
For degradation of SIM data, available in form of a 3-D (x,y,λ) data cube of
synthetic spectral profiles of a particular spectral line, a point spread function
(PSF) is needed. Even in the simplest possible form, the PSF must contain
information on both, the blurring and the scattering parts (Zwaan, 1965). Up
to now several parametric forms have been applied using single or combined
Gaussian and Lorentz profiles, usually derived from observations of eclipses or
1
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Rybák, Kučera, Wöhl, Wedemeyer-Böhm, Steiner
planet transits. Keeping our method as simple as possible the blurring part of
the PSF is deduced from its Fourier transform – the optical transfer function
(OTF) – using theoretical relations derived for the case of long exposures (>1s).
In this case the OTF can be factorized by a two component formula separating
the atmospheric and telescopic contributions using just 3 control parameters –
the Fried parameter r0 , the telescope aperture D, and the wavelength λ (von
der Lühe, 1992; Bonet, 1999). The scattering part of the PSF is in fact a rather
long-tail monotonic function seen in detailed limb/aureole measurements (e.g.
Deubner & Mattig, 1975). Having in mind that the typical photospheric granular
size is considerably smaller than the spatial extent of available SIMs, we prefer
to mimic the effect of scattering via a constant background, independent on the
scattering angle over the whole spatial domain of the SIMs. Hereafter, the PSF
will therefore be approximated as
P SF = P SFblur + P SFscatt = P SFlong−exp (D, r0 ) + C
(1)
where the term P SFlong−exp is given by the equations given in the work of Bonet
(1999). Two free parameters, r0 (alternatively, the radius of the seeing disk
λ/r0 ) and C, have to be determined to fully describe such an approximation
to the PSF. The granular contrast of the continuum intensity, δIrms , is used
as a criterion to determine the appropriate values of λ/r0 and C. In case of
OBSs, the value of the mean granular contrast, δIrms,obs , is derived from many
exposures of 1-D slit spectra of the quiet photosphere. For the SIMs, the mean
value of the granular contrast, δIrms,sim , from several snapshots is calculated
for several test PSFs with different combinations of the parameters λ/r0 and
C. Moreover, effects of sampling the 2-D spatial domain of the snapshots by a
’virtual slit’ of a width equal to that used in the OBSs, detector pixel length along
the slit, and estimation of the instrumental profile are introduced. Comparing
δIrms,obs , derived from OBSs, with the range of δIrms,sim , determined for test
degradation results of SIMs, the appropriate values of the parameters λ/r0 and
C can be deduced. Values of δIrms,obs and δIrms,sim should coincide for such
parameters. Finally, these values are used for the PSF to be applied to the SIMs
in each spectral plane together with additional degradation for the slit/detector
sampling.
3.
Data
The described method has been tested using observational data described by
Rybák et al. (2004) and synthetic spectra based on the results of numerical
simulations by Wedemeyer et al. (2004). Briefly, spectroscopic observations
were performed using the Vacuum Tower Telescope (VTT) on Tenerife. The
Fe II 6458 Å line was taken with an exposure time of 1 s, while the slit width
was 0.5′′ and the detector pixel size along the slit 0.125′′ . 300 spectra were
acquired at center-to-limb position µ=0.65 with 600 spectral rows along the slit
giving an average value of the granular contrast δIrms,obs equal to 0.021. The
non-magnetic 3-D numerical simulations cover a spatial domain of 7.7′′ × 7.7′′
with a horizontal cell spacing of 0.055′′ . In total, 52 snapshots with the temporal
step of 3 min were calculated. The LTE radiative transfer code LINFOR3D was
Method for comparing simulations with observations
3
GRANULAR CONTRAST rms(I_c)
0.10
0.09
0.08
0.08
0.07
S/B PSF RATIO
0.06
0.06
0.05
0.04
0.04
0.03
0.02
0.02
0.01
0.00
0.00
0.2
0.4
0.6
0.8
PSF HALF-WIDTH [arcsec]
1.0
Figure 1. 2-D map of the simulated granular contras, δIrms,sim , as a function of the half-width of the blurring part of the PSF and the ratio between
the blurring and scattering parts of the PSF, using all 52 snapshots of the
SIMs. The white mainly vertical curve marks the granular contrast, derived
from the OBSs, δIrms,obs . The grey horizontal line corresponds to the estimated level of scattered light converted to the ratio between the scattering
and blurring part of the PSF.
used for calculating synthetic spectra of the Fe II 6456 Å line for the same centerto-limb position. From the simulations resulted an average value of the granular
contrast, δIrms,sim , equal to 0.146±0.006.
4.
Application
Diverse degradations were applied to the SIMs corresponding to values of the
PSF half-width of 0.14′′ to 1.05′′ and C of 0.0 to 5.5×10−5 (i.e. the relative ratio
between the scattering and the blurring parts of the PSF of 0.0 to 0.096). For
comparison, the theoretical resolution of the VTT is 0.23′′ at our wavelength
and a scattered light level of a few percent was reported for the VTT telescope.
The resulting 2-D map of the granular contrast derived from all snapshots is
shown in Fig.1. It shows that for a given granular contrast the PSF half-width
and the scattered light level are not uniquely determined as they slightly depend
on the applied scattered light level. The granular contrast of our observations,
δIrms,obs , is reached within the interval of the PSF half-width from 0.60′′ to 0.65′′
(Fig.1, white curve). An estimation of the scattered light level is inevitable to
solve the ambiguity. Data taken during the Mercury transit in 2003 (Soltau,
2005) were found to be the most appropriate for this purpose. Assuming that
exclusively scattering is causing the rest intensity at the Mercury disk center
(6 % of the nearby continuum at 5725±5 Å), a background scattered light level
of just 2.2×10−6 is derived for each pixel of the spatial domain of the SIMs.
The corresponding ratio between the scattering and blurring parts of the PSF
4
Rybák, Kučera, Wöhl, Wedemeyer-Böhm, Steiner
1.4
1.1
1.3
1.0
1.2
2
1.1
1.0
0.9
0
0.8
0.7
0.6
-2
SPATIAL SCALE X [arc sec]
SPATIAL SCALE X [arc sec]
2
0.9
0.8
0
0.7
0.6
-2
0.5
0.5
0.4
0.3
-0.2
-0.1
-0.0
0.1
WAVELENGTH [A]
0.2
0.4
-0.2
-0.1
-0.0
0.1
WAVELENGTH [A]
0.2
Figure 2. Example spectra of a spatial cut from a simulation snapshot
(intensities are normalized to the spatially averaged value of the continuum
intensity). The left panel shows the original synthetic spectrum of the Fe II
6456 Å spectral line. The right panel gives for comparison the same spectrum
degraded for all degradation effects taken into account.
is then 0.038 (Fig.1, grey line). The intersection of the white with the grey
curve (Fig.1) gives the optimal value of the PSF full-width equal to 1.25′′ . This
value corresponds to a Fried parameter r0 of 11 cm what is in agreement with
the typical values of r0 derived for the best observing sites like Canary Islands
(Brandt & Wöhl) and Sacramento Peak (Brandt, Mauter & Smartt, 1987).
5.
Results
Fig.2 shows an example of an original and a degraded SIM spectrum. Besides
a decrease of the intensity range from [0.3,1.4] to [0.4,1.1], a remarkable degradation of the spectral profiles is found. For example, the range of the line-ofsight velocities, derived from the Doppler shifts, changes from ±5 km/s to only
±1 km/s. Almost all (99.6%) of our observed Doppler shifts are within the latter
velocity interval.
The effects of the degradation procedure on distributions of two spectral line
characteristics – the continuum intensity and the full-width at half minimum of
the line (FWHM) – are shown in Fig.3, where histograms of these quantities
are displayed. They have been calculated for the position µ=0.65 taking each
snapshot individually. The histograms of the continuum intensity of the original
SIM data show a wide range of values with an almost flat central part in the
interval 0.8 to 1.2. The degradation procedure significantly narrows these distributions to the interval [0.9,1.1] but the significant part of the data are within
the interval [0.94,1.06]. Generally, individual distributions keep their significant
statistical differences also after degradation.
The distribution of the line width of the original SIM data is very asymmetric. Therefore, the most frequent value of this quantity is shifted by 10 %
5
Method for comparing simulations with observations
CONTINUUM INTENSITY DISTRIBUTION
FWHM DISTRIBUTION
1.000
1.000
0.100
0.100
Synthetic spectra
0.010
Synthetic spectra
0.010
0.001
0.001
1.000
1.000
Synthetic spectra + PSF + slit + IP
Synthetic spectra + PSF + slit + IP
0.100
0.100
0.010
0.010
0.001
0.001
0.6
0.8
1.0
1.2
NORMALIZED CONTINUUM INTENSITY Ic
1.4
0.5
1.0
1.5
2.0
NORMALIZED FWHM
2.5
3.0
Figure 3. Distributions of the continuum intensity (left column) and the line
width (right column) for all individual snapshots from the original simulation
data (top row) and for the same data degraded for all effects including the
PSF, the slit width, detector pixel sampling, and the estimated instrumental
profile of the VTT echelle spectrograph (bottom row).
from the mean value. Again, the degradation is removing the far wings of the
distributions below 0.8 and above 1.4. The distributions are still asymmetric
and significant variations between individual distributions are present.
A comparison of the average distributions of the normalized FWHM coming
from OBSs and from degraded SIMs for µ=0.65 is given in Fig.4. Both distributions show similar asymmetry but the distribution from the SIMs is broader.
The difference is more pronounced for the tail of large values (∼20 %). For
checking purposes, we have calculated a similar distribution for the continuum
intensity (Fig.4). It advices that the PSF might be still a bit underestimated
as the distribution of the SIMs is still broader than that of the OBSs. These
preliminary results demonstrate the possible usage of this method for statistical comparison of high-resolution spectroscopic observations with synthetic data
obtained from numerical simulations.
6
Rybák, Kučera, Wöhl, Wedemeyer-Böhm, Steiner
CONTINUUM INTENSITY DISTRIBUTION
FWHM DISTRIBUTION
1.0000
1.0000
0.1000
0.1000
SIMs
0.0100
0.0100
OBSs
0.0010
0.0001
0.8
SIMs
OBSs
0.0010
0.0001
0.9
1.0
NORMALIZED Ic
1.1
1.2
0.8
1.0
1.2
NORMALIZED FWHM
1.4
1.6
Figure 4. Average distributions of the continuum intensity (left panel) and
the full-width at half minimum (right panel) for the degraded simulation data
(thin line) and for the observational data (thick line) for a a heliocentric angle
of µ=0.65. Distributions are normalized to the mean value of the particular
spectral characteristic.
Acknowledgments. The VTT is operated by the Kiepenheuer-Institut für
Sonnenphysik, Freiburg, at the Observatorio del Teide. This research is part of
the European Solar Magnetism Network (EC/RTN contract HPRN-CT-200200313). This work was supported by the Slovak grant agency VEGA(2/6195/26)
and by the Deutsche Forschungsgemeinschaft (DFG 436 SLK 113/7). The authors thank D. Soltau for the Mercury transit data used in this work and the
NSO/SP workshop organizers for kindly providing valuable support.
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