A&A 421, 729-734 (2004)
DOI: 10.1051/0004-6361:20034131

A model to interpret the intensity-velocity and velocity-velocity phase differences from solar observations obtained with magneto-optical filters

P. F. Moretti - G. Severino

INAF, Osservatorio Astronomico di Capodimonte, via Moiarello 16, 80131 Napoli, Italy

Received 29 July 2003 / Accepted 16 February 2004

Abstract
The intensity (I) and velocity (V) signals obtained using magneto-optical filters (MOF), are not independent of each other. The induced spurious signals affect the intensity-velocity phase difference measurements and the effect is referred to as a I-V crosstalk (Moretti & Severino 2002). We show a new model to interpret the I-V phase measurements and, in particular, its application to the interpretation of the data obtained with sodium MOF systems. The model can also be applied to correct the velocity-velocity phase from multi layer observations.

Key words: Sun: oscillations - instrumentation: miscellaneous

1 Introduction

In solar physics, the velocity is mainly inferred through the Doppler effect. Magneto-Optical filters (MOF), used as narrow passband filters, can infer the velocity, sampling the intensity signal along the line profile. Depending on the number of samples and on the algorithm used to build the intensity (I) and velocity (V) signals, each of the two signals could be contaminated by the fluctuations of the other one. This is what has commonly been referred to as a I-V crosstalk (Moretti & Severino 2002).

In this paper a model to estimate the measured I-V phase differences through the dependence of the crosstalk on the velocity offset across the solar disk is presented. The results for the p-modes values observed in the Na I D lines are shown and compared with those obtained from the ${\ell -\nu }$ diagrams. An example of how the model can be applied to interpret the V-V phase difference from multi-layer observations is also shown.

2 The intensity-velocity crosstalk for MOF systems

To date, MOFs are operating in the sodium Na I D lines at 589.6 and 589.0 nm (like the systems at the Kanzelh $\ddot{\rm o}$he Solar Observatory, Cacciani et al. 1999 or the VAMOS up to 1999, Moretti et al. 1997) and in the potassium K I 769.9 nm line (like the current version of VAMOS and Low-$\ell$, Severino et al. 2001; Tomczyk et al. 1995).

The MOF transmission profile usually consists of two narrow passbands, each approximately 35 mÅ  wide, located in the blue, B, and red, R, wings of the solar line profile, whose distance from the resonance wavelength usually varies from $\simeq$50 mÅ  to $\simeq$100 mÅ. Depending on some MOF parameters, a central transmission peak can also be present (Cacciani et al. 1994).

Typically, the velocity signal is built as V⁰=(B-R)/(B+R) and the intensity signal as I⁰=B+R (Cacciani et al. 1997). The velocity signal is usually calibrated through the solar rotation, and the intensity variations are normalised to the mean solar disk signal as $\Delta I^{0}/I^{0}$, where $I^{0}={1 \over \pi R_{\odot}}\int_{0}^{2\pi}\int_{0}^{R_{\odot}}{I(r,\theta) r {\rm d}r {\rm d}\theta}$.

To first order, the measured intensity and velocity signals are written as follows:
${\frac{\Delta I}{I}}={\frac{\Delta I^{0}}{I^{0}}}+\alpha\Delta V^{0}$ and $\Delta V=\Delta V^{0}+\beta \frac{\Delta I^{0}}{I^{0}}$, where we considered the difference between the true and measured $\ell = 0$ values negligible (that is I=I⁰). The parameter $\alpha$ is the variation of the normalised intensity due to a velocity shift, and it is measured in ${\rm (m/s)^{-1}}$. The parameter $\beta$ is the variation of the measured velocity due to a temperature change measured as a $\frac{\Delta I^{0}}{I^{0}}$, and it is measured in m/s.

$\alpha$ and $\beta$ parameters describe the I-V crosstalk. In Moretti & Severino (2002), only the $\alpha$ parameter was used to correct the intensity images and obtain the I-V phase difference in the NaI D lines.

$\alpha$ and $\beta$ depend on the MOF transmission profile, on the solar line shape, and are functions of the velocity offset that shifts the MOF working wavelengths with respect to the solar line. This means that, once the instrumental parameters are chosen, at each time $\alpha$ and $\beta$ can be considered as masks on the solar disk according to the solar velocity rotation and the Earth-Sun relative velocity.

A convention has to be chosen for the velocity axis: we adopt a positive velocity when oriented out of the sun.

The MOF transmission profiles were simulated as in Cacciani et al. (1994). The solar line profiles measured at Kitt Peak at low resolution were used. The maximum changes along the line due to temperature fluctuations associated to the p-modes were estimated using the parameters in Severino et al. (1986) for the core of the NaI D lines (see Fig. 1).

3 The model for the measured I - V phase

To estimate the error induced in the I-V phase difference, the intensity and velocity signals are described as vectors. The I-V phase is the angle between them and crosstalk causes both a rotation of the vectors and a consequent phase change.

Since the $\alpha$ and $\beta$ parameters change their signs crossing the zero velocity offset, the induced signals are parallel or anti-parallel to their fluctuations depending on the velocity offset.

We choose an x axis parallel to the unperturbed velocity fluctuations $\Delta V^{0}$.

The measured intensity and velocity signals can be written as

${\frac{\displaystyle \Delta I}{\displaystyle I}}=\left( \begin{array}{c} x \\ y... ...\ y \end{array} \right){\frac{\displaystyle \Delta I^{0}}{\displaystyle I^{0}}}$ and
$\Delta V=\left( \begin{array}{c} 1 \\ 0 \end{array} \right) \Delta V_{0} +\beta... ...ray}{c} 1 + \beta x/ \gamma \\ \beta y/ \gamma \end{array} \right) \Delta V_{0}$

where $x \Delta I^{0}/I^{0}$ and $y\Delta I^{0}/I^{0}$ are the projections of the intensity along the x and y axis (x²+y²=1) and the true I-V phase difference is $\phi^{0}_{I-V}={\rm atan}(y/x)$.

The parameter $\gamma =\Delta V^{0} / {\frac{\Delta I^{0}}{I^{0}}}$ is the ratio between the solar velocity and the intensity fluctuations. It is a crucial parameter since if one of the signals, the velocity or the intensity, was dominant on the Sun, the crosstalk would cause the I and V vectors to be parallel (or antiparallel, depending on the convention adopted for the sign of the velocity).

The rotation angle the measured intensity and velocity vectors are rotated by depends on $\alpha , \beta , \gamma$, and $\phi^{0}_{I-V}$. The measured phase difference will be
$\phi_{I-V} = {\rm atan} \left( {\frac{\displaystyle y}{\displaystyle x + \alpha... ...frac{\displaystyle \beta y/ \gamma}{\displaystyle 1 + \beta x/ \gamma}} \right)$.

	Figure 1: Effects of oscillations on the NA I D1 line, based on the simulation from Severino et al. (1986) and the Kitt Peak solar profiles. From top to bottom: the residual intensity for the hotter and cooler lines during an acoustic wave; their slope, the ratio between the slopes and the normalised intensity variation.
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4 The data

The data we show consist of sets of dopplergrams and intensity images obtained with a sodium MOF at Kanzelh $\ddot{\rm o}$he on January 30th, 1998. The images were acquired every minute, and 256 min were selected for the analysis. The spatial resolution is $\simeq$4 $^{\prime \prime }$ (Cacciani et al. 1999).

All the images were calibrated (Moretti & the MOF Development Group 2000) and registered. The differential rotation was not removed and produced a maximum sweep at disk center corresponding to 5 pixels (that is $\simeq$21 $^{\prime \prime }$).

The images were fast Fourier transformed maintaining the spatial resolution, that is, performing a pixel-by-pixel analysis to obtain the frequency dependence of the maps of the velocity and intensity power and of the I-V phase difference.

5 The data analysis and the results

The $\gamma$ parameter has been estimated as follows. For any couple of I and V power maps at each frequency, the mean $\gamma$ values and their sigmas have been computed from the spatial distribution over the solar disk (see Fig. 2). The p-modes show the larger amplitudes around 3.3 mHz, where also the I-V coherence is maximum. For this reason we selected the interval $\simeq$3-4 mHz for the analysis. In this frequency range $\gamma$ values between 3000 and 12 000 m/s are obtained.

	Figure 2: The $\gamma$ parameter versus frequency for 4.3 $^{\prime \prime }$ spatial resolution sodium data obtained from the velocity and $\Delta I/I$ power maps. The mean and sigma values at each frequency have been computed from the spatial distribution over the solar disk.
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From Fig. 1 we expect intensity fluctuations of the order of 0.5 at those wavelengths typically tuned by the MOFs ($\pm$ $100~\rm m\AA~$ relative to the line central wavelength). If we compare this value with the measured intensity fluctuations (see Fig. 3), the latter show a value $\simeq$20 times smaller. The reduced amplitude of the oscillations is mainly due to the average of the many oscillation modes (whose number increases as $2 \ell +1$) by the integration over the 4 $^{\prime \prime }$ pixel, while the model assumes very high spatial resolution measurements. This effect is present in both V and I signals and has no influence on $\gamma$. Nevertheless, the estimate of $\gamma$ is affected by the signal-to-noise ratio (S/N) of the I and V signals. Our data show S/N values equal to $\simeq$50 and $\simeq$8 for $\Delta V$ and $\Delta I$ respectively, so that we expect the measured $\gamma$ values to be overestimated by at least a factor of $\simeq$6.

In Figs. 4-6 some examples of the I-V crosstalk parameters are shown for two extreme cases of MOF transmission profiles. The exact transmission profile knowledge is not crucial when the errors in both the intensity and velocity are computed to estimate the measured I-V phase, while it can be important when only the intensity error is measured, as in the Moretti & Severino (2002). This is true for the measurements we show in the sodium Na I D lines, while for other solar lines the effects depend on the slope of the line and on the particular MOF transmission profiles.

In contrast, the dependence of the phase error on the velocity offset is always very sensitive to changes in $\gamma$ (see Fig. 7).

An estimate of the true I-V phase difference can be obtained from the value measured in that area of the solar disk where the velocity offset is close to zero and the I-V crosstalk negligible (Fig. 8). The wavelengths tuned by the filter in that area select a particular formation layer on the sun (see Moretti & Severino 2002) and the obtained I-V phases cannot be attributed to other layers.

The measured I-V phase differences in the p-mode pixels between 2.9 and 4.2 mHz are displayed as function of the velocity offset in Fig. 9. This dependence on the velocity offset can be reproduced by the model using different $\gamma$ parameters and the constraint of the value at the zero velocity offset.

In Fig. 9 we show the curves obtained with $\gamma$ equal 100 and 1000 m/s for $\phi^{0}_{I-V} = 115^{\circ}$ and $\phi^{0}_{I-V} = 125^{\circ}$. The area bounded by these curves includes most of the measured I-V phases, but different choices of the parameters select different areas with the same number of points. The range 100-1000 m/s is in reasonable agreement with the measured range for $\gamma$ taking into account the effect of different S/N for the I and V signals.

The curve obtained with $\gamma = 300$ m/s and $\phi^{0}_{I-V} = 118^{\circ}$ well reproduces a 4th polynomial fit of the measured I-V phase values (same figure).

Taking into account the errors in the $\gamma$ and in the I-V phases, we can affirm that the results in the NaI D lines are consistent with a constant I-V phase at the p-modes peaks equal $\phi_{\rm Na}=120^{\circ}\pm 10^{\circ}$.

	Figure 3: The sodium intensity oscillations at a single pixel (corresponding to a $4.3 \hbox {$^{\prime \prime }$ }\times 4.3 \hbox {$^{\prime \prime }$ }$ solar area) at disk center.
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	Figure 4: Crosstalk in the case of a two peak NaI D2 MOF profile. Top four panels: the solar lines, MOF blue wing transmission profile, intensity (B+R) and velocity (R-B)/(B+R) raw signals. Bottom four panels: the velocity sensitivity, $\alpha$, the changes in the signals due to the maximum temperature variation during an acoustic wave, $\beta$.
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	Figure 5: Crosstalk for a strong central peak NaI D2 MOF profile. Top four panels: the solar lines, MOF blue wing transmission profile, intensity (B+R) and velocity (R-B)/(B+R) raw signals. Bottom four panels: the velocity sensitivity, $\alpha$, the changes in the signals due to the maximum temperature variation during an acoustic wave, $\beta$.
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	Figure 6: The computed measured I-V phase difference vs. the offset velocity for the two very different MOF transmission profiles in Figs. 4 and 5: with a strong central peak (solid) an in the absence of the central peak (dashed). The true I-V phase was set to  $110^{\circ }$.
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	Figure 7: The computed measured I-V phase difference for two very different $\gamma$ values: 600 m/s (solid) an 6000 m/s (dashed). The true I-V phase was set to $110^{\circ }$.
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	Figure 8: The I-V phase differences corresponding to the highest power pixels (corresponding also to the highest coherence pixels) were selected in that area of the solar disk where the I-V contamination is negligible (that is at zero velocity offset, left) and averaged over the disk. The resulting values and the spread over the disk are shown for the sodium data ( right).
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5.1 The I - V phase in the ${\ell -\nu }$ diagram

Using MOF data in the sodium D lines, the I-V phase difference measured on the p-mode peaks in the ${\ell -\nu }$ diagram is $\phi_{\rm Na}=155^{\circ} \pm 15^{\circ}$ (Moretti et al. 2001; Oliviero et al. 1998).

When spherical harmonics are applied to the images to produce the ${\ell -\nu }$ diagram, the main contribution to the signal comes from the central part of the solar disk. According to the model we showed, the measured phase is the one that corresponds to the mean velocity offset intercepted by the disk center. In a given region, there is a diurnal and an annual phase change due to the velocity offsets, and a crosstalk discussion is needed before interpreting the phase changes.

As can be seen from Fig. 9, the model predicts a I-V phase at the central part of the disk consistent with the one measured in the ${\ell -\nu }$ diagram.

Figure 9: From the pixel-by-pixel analysis: the measured I-V phase differences on the p-modes for the sodium data averaged between 2.9 and 4.2 mHz (grey points). The region is bounded by two solid curves corresponding to the computed I-V phase differences for $\gamma$ equal to 100 and 1000 m/s. These solid curves refer to a true I-V phase equal to  $115^{\circ }$ and  $125^{\circ }$. The black crosses refer to a 4th degree polynomial fit of the data, and the white diamonds to the computed I-V phase differences for $\gamma$ equal to 300 m/s and a true I-V phase equal to  $118^{\circ }$.

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6 The application to the V - V phase difference

When two different spectral lines are observed simultaneously, the velocity-velocity phase difference can provide important information of the propagation properties of the waves. If narrow passbands as the MOFs are used, the error in the V-V phase difference depends on the angles by which the velocity vectors are rotated, due to the I-V crosstalk in each solar line.

The V-V phase error can be written as
$\Delta \phi_{\rm Na-K} = {\rm atan} \frac{y_{\rm Na} \beta _{\rm Na} /\gamma _... ...} {1+\alpha _{\rm K}\beta _{\rm K} + x_{\rm K} \beta _{\rm K} /\gamma _{\rm K}}$, where the two terms refer to the rotations of the velocity vector in each line, and x and y are such that $\tan(\phi_{I-V})=y/x$, for each line. In order to compute the V-V phase errors, the true I-V phases for the two lines are needed. As an example, when sodium and potassium MOFs are used, we adopt a constant $\phi^{I-V}_{\rm Na} = 120^{\circ}$ and $\phi^{I-V}_{\rm K}=110^{\circ}$. A $\pm$ $20^{\circ}$ error on these values should usually be assumed, due to the spread of the values for the different m when the spherical harmonic decomposition is used, or the spread for different velocity offsets when the pixel by pixel analysis is used.

We computed the V-V phase error at any position on the disk using its velocity offset.

If a local time-distance analysis is performed using different diameter annuli centered on any position on the disk (Duvall et al. 1993; Chou & Duvall 2000), the V-V phase errors has to be computed taking into account the geometry of the analysis (see Fig. 10). For the dimension of any annulus we obtained a V-V phase map, where the spatial coordinates correspond to the centers of the annuli (see an example in Fig. 11).

	Figure 10: Left: the velocity distribution map over the solar disk as seen on a $512 \times 494$ chip (velocity offset = -700 m/s). The boundaries of a typical annulus are superposed. Right: the $V_{\rm NaI D2}-V_{\rm K}$ phase error over the annulus. $\gamma =500$ m/s for both lines, $\phi _{\rm NaI D2}^{0}=120^{\circ }$ and $\phi _ {\rm K}^{0}=110^{\circ }$.
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	Figure 11: The $V_{\rm NaI D2}-V_{\rm K}$ phase error averaged over a $10^{\circ }$ annulus across the solar equator (that is at different velocity offsets) at two different Earth-Sun relative velocity (-700 m/s for crosses and zero for points). $\gamma =500$ m/s for both lines, $\phi _{\rm NaI D2}^{0}=120^{\circ }$ and $\phi _ {\rm K}^{0}=110^{\circ }$.
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We remark that the V-V error maps depend on the following quantities:

• the line changes due to temperature fluctuations (assumed to follow Fig. 1) or magnetic changes (Cacciani & Moretti 1997);
• MOF transmission profiles in both lines;
• Earth-Sun disk center velocity offset (changing with the date and time of acquisition);
• true I-V phases for both lines (they depend on frequency);
• $\gamma$ for both lines;
• the dimension and width of the annulus (in the case of time-distance analysis).

7 Conclusions

A new model for a correct interpretation of the intensity and velocity signals from magneto-optical filters has been shown.

The results of the phase difference between the intensity and velocity signals from a sodium MOF are well reproduced by the model in the pixel-by-pixel analysis and in the ${\ell -\nu }$ diagram too.

The data are consistent with a constant $\phi_{\rm Na}=120^{\circ}\pm 10^{\circ}$ on the p-mode peaks.

Acknowledgements

P.F.M. thanks Federica Brandizzi. We also thank S. M. Jefferies and the MOF Development Group.

References

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