Helioseismology of sunspots: defocusing, folding, and healing of wavefronts
^{1} Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan (R.O.C.)
^{2} MaxPlanckInstitut für Sonnensystemforschung, 37191 KatlenburgLindau, Germany
email:
gizon@mps.mpg.de
^{3} Institut für Astrophysik, GeorgAugustUniversität Göttingen, 37077 Göttingen, Germany
^{4} Université de Toulouse, ISAESUPAERO, 10 avenue Edouard Belin, 31055 Toulouse Cedex 4, France
Received: 15 March 2013
Accepted: 21 June 2013
We observe and characterize the scattering of acoustic wave packets by a sunspot in a regime where the wavelength is comparable to the size of the sunspot. Spatial maps of wave travel times and amplitudes are measured from the crosscovariance function of the random wave field observed by SOHO/MDI around the sunspot in active region NOAO 9787. We consider separately incoming plane wave packets consisting of f modes and p modes with radial orders up to four. Observations show that the traveltime perturbations diminish with distance far away from the sunspot – a finitewavelength phenomenon known as wavefront healing in scattering theory. Observations also show a reduction of the amplitude of the waves after their passage through the sunspot. We suggest that a significant fraction of this amplitude reduction is due to the defocusing of wave energy by the fast wavespeed perturbation introduced by the sunspot. This “geometrical attenuation” will contribute to the wave amplitude reduction in addition to the physical absorption of waves by sunspots. We also observe an enhancement of wave amplitude away from the central path: diffracted rays intersect with unperturbed rays (caustics) and wavefronts fold and triplicate. Wave amplitude measurements in timedistance helioseismology provide independent information that can be used in concert with traveltime measurements.
Key words: Sun: helioseismology / sunspots / Sun: activity / Sun: oscillations / Sun: interior / Sun: surface magnetism
© ESO, 2013
1. Introduction
The propagation of solar seismic waves is affected by sunspots. The scattering phase shifts and wave absorption coefficients can be measured by several techniques, such as FourierHankel analysis and timedistance analysis (see review by Gizon et al. 2010, and references therein).
Cameron et al. (2008) studied the interaction of fmode planewave packets with a sunspot using timedistance helioseismology. As we describe briefly below, the dataaveraging strategy used by Cameron et al. (2008) reduces the noise and enables a detailed study of the waveforms far from the scattering region. In this study, we extend the observations to modes with different radial orders and wavelengths that are closer to the sunspot radius, in order to investigate finitewavelength effects (e.g. Nolet & Dahlen 2000; Hung et al. 2001). We measure not only traveltime shifts but also wave amplitude perturbations with respect to the quiet Sun.
We use oneminute cadence Doppler velocity images measured by the Michelson Doppler Imager (Scherrer et al. 1995) onboard the Solar and Heliospheric Observatory (SOHO/MDI). The images are remapped using Postel’s azimuthal equidistant projection in order to track the motion of the sunspot in Active Region 9787 (Cameron et al. 2008; Gizon et al. 2009; Moradi et al. 2010). The map scale is 0.12 heliographic degrees per pixel. Nine data sets are obtained, one for each day of the period 20–28 January 2002. The selected sunspot is nearly circular and does not evolve significantly over the duration of the observations, T = 9 days. Its outerpenumbral radius is R = 20 Mm.
Characteristics of incoming wave packets.
Filters are applied to the Dopplergrams in 3D Fourier space to extract waves with the same radial order (ridge filters). Five different filters are applied, for modes f through p_{4} (corresponding to radial orders 0 ≤ n ≤ 4). Table 1 lists the characteristics of the wave packets for each radial order.The mean frequency, ν, and mean wavelength, λ, of each wave packet are defined as weighted averages over the filtered power spectrum: λ increases with n, as does ν due to the filtering. The finitewavelength effects of scattering by the sunspot are expected to be important in the regime where R/λ ≲ 1, a condition that is discussed further in Sect. 4. These effects should thus be easier to detect with increasing radial order n.
2. Crosscovariance function and wave packets
The crosscovariance of the filtered Doppler velocity is computed in the same way as in Cameron et al. (2008): (1)where (x,y) is a local Cartesian coordinate system with its origin at the center of the sunspot, t is the time, φ_{n} the filtered Doppler velocity for modes with radial order n, and the average of φ_{n} over the line L at x = −43.73 Mm. The spatial averaging over L is equivalent to filtering out all the wavevectors that are not perpendicular to L: only waves that propagate toward and away from L contribute to .In addition, to reduce noise, the crosscovariances are averaged over T = 9 days and over rotations about the sunspot center, as in Cameron et al. (2008) and Moradi et al. (2010).
As explained by Gizon et al. (2010), for t > 0, the crosscovariance is used to follow the propagation of two plane wave packets initially at L and propagating toward + x and −x respectively.The forward scattering of the waves by the sunspot occurs in the (x > 0,t > 0) quadrant.
3. Wave travel times and amplitudes
We define the traveltime shift, Δτ(x,y), as the time lag that maximizes the similarity between the measured crosscovariance, C_{n}(x,y,t), and a sliding quietSun crosscovariance, . More precisely, Δτ is the time τ that maximizes the function (2)The quietSun crosscovariance is constructed by averaging C_{n}(x,y,t) over  y > 100 Mm, far away from the region of wave scattering, and it is zeroed out for t < 0. The crosscorrelation traveltime shift defined here is also used in geophysics (e.g. Zaroli et al. 2010) and is analogous to the definition of Gizon & Birch (2002). The crosscorrelation F_{n}(x,y,τ) at each spatial point (x,y) as a function of τ is demodulated using the Hilbert transform to obtain its envelope. The maximum, A(x,y), of the envelope of F_{n}(x,y,τ) is a measure of the wave packet amplitude. By definition, A is equal to one in the quiet Sun. An example measurement of Δτ and A for C_{4} at (x,y) = (60,0) Mm (after interaction with the sunspot) is shown in Fig. 1.
Fig. 1 Crosscorrelation function F_{4}(x,y,τ) (solid line) between the crosscovariance C_{4}(x,y,t) at (x,y) = (60,0) Mm and the corresponding quietSun crosscovariance . The envelope (dashed line) is determined by demodulation. The traveltime shift, Δτ, and the wave packet amplitude, A, are indicated by arrows. In practice, we fit the peak nearest to τ = 0 with a parabola to measure the time shift. In this particular example, the time shift and reduced amplitude are Δτ = −57 s and A = 0.44 with respect to the quiet Sun. 

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Fig. 2 Spatial maps of traveltime shifts Δτ(x,y) (top) and wave packet amplitudes A(x,y) (bottom) with respect to the quiet Sun. From left to right are results for wave packets p_{1}, p_{2}, p_{3}, and p_{4}. Waves propagate from L (red lines) in the + x direction. The black circles denote the outer boundary of the penumbra of the sunspot. The parabolic curves (white lines with equation (y/R)^{2} + x/(x_{F} + λ/8) = 1) depict the boundaries of the first planewave Fresnel zones with focus at (x,y) = (x_{F},0), chosen such that the sunspot fills the width of the Fresnel zone. The focus position x_{F} depends on radial order and is taken from Table 1. We do not display the results in the near fields (gray regions where distances from L are less than 3λ). 

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For each wave packet, we obtain spatial maps of the traveltime shift Δτ(x,y) and wave amplitude A(x,y). Figure 2 shows the maps of Δτ(x,y) and A(x,y) for modes p_{1} through p_{4}. The maps of Δτ clearly show negative traveltime anomalies behind the sunspot along y = 0 as a result of the increase in the wave speed in the sunspot (Cameron et al. 2008), while the maps of A show the amplitude reductions.
We only display the results for distances for L larger than three wavelengths in each panel in order to exclude the near field where the t < 0 and t > 0 branches of C_{n} are not well separated. This is not a limitation of this work since we are interested in wave scattering in the far field.
For p_{1} modes, it is interesting to note the positive traveltime anomalies on the left side of the penumbra (x < 0), caused by the outwarddirected flows in the penumbra and the moat. The effect of the moat flow is not seen in the far field: waves propagate against the moat flow for x < 0 and with the moat flow for x > 0.
4. Wavefront healing and finitewavelength effects
Fig. 3 Observed traveltime shifts, Δτ. The left panel shows the traveltime shifts along y = 0 for modes f through p_{4}, which heal as the distance from the sunspot, x, increases. The right panel shows the traveltime shifts of p_{4} modes as functions of y at four different values of x. Notice the spread in the transverse direction as x increases. The hatched and crosshatched area indicate the locations of the penumbra and umbra, respectively.The data is averaged over patches of size 16 × 16 Mm^{2} to filter out spatial scales that are dominated by noise. Error bars give the standard deviation of the mean. 

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The lefthand panel of Fig. 3 shows the traveltime shifts as a function of x along y = 0. For all the modes, the magnitude of the traveltime shift gradually decreases as waves propagate away from the sunspot.This phenomenon, called wavefront healing (Nolet & Dahlen 2000; Hung et al. 2001), is inevitable because the amplitude of the scattered wave (a “circular” wave) decreases with distance from the sunspot more rapidly than the amplitude of the direct wave (a plain wave) by a factor proportional to the square root of distance. Wavefront healing is a finitewavelength effect (Nolet & Dahlen 2000). In particular, singleray theory cannot explain wavefront healing but instead gives a constant traveltime perturbation for x > R (Gizon et al. 2006).
The occurrence of finitewavelength effects at a measurement point depends on whether the diameter of the sunspot is smaller than the width of the first Fresnel zone of that point. For any observation point (x,y) that enters the computation of the crosscorrelation (Eq. (1)), the corresponding first planewave Fresnel zone is bounded by a parabola with focus at this point (Gudmundsson 1996). Assuming a path length perturbation of λ/4, the width of this Fresnel zone measured along x = 0 is given by (Gudmundsson 1996). We define the characteristic observation point (x_{F},0) such that the sunspot fills the first Fresnelzone width, i.e. Δy_{F}(x_{F}) = 2R. For each wave packet, the value of x_{F} is provided in Table 1 and the corresponding Fresnel zone is the white curve in the top panels of Fig. 2. In the regime x > x_{F}, the first Fresnel zone is always larger than the sunspot: finitewavelength effects dominate and the ray approximation is no longer valid.
The righthand panel of Fig. 3 shows p_{4}mode traveltime shifts as a function of y, at four different values of x. As the distance away from the sunspot increases, we see wavefront healing around y = 0. In addition, we see that the width of the traveltime perturbation in the transverse direction grows with x. This observation cannot be explained in the context of linearized ray theory, according to which traveltime perturbations are computed along unperturbed paths (y = const.) by using Fermat’s principle (Hung et al. 2001). This is another warning against using linearized ray theory in sunspot seismology.
As we see in the next section, a simple raytracing calculation (geometrical optics) indicates that a fast wavespeed anomaly, like a sunspot, would defocus the rays and lead to traveltime perturbations that spread in the transverse direction as x increases. However, ray tracing will still overestimate the traveltime shifts in the far field and would be incapable of explaining wavefront healing along y = 0.
5. Amplitude enhancements and caustics
The bottom panels of Fig. 2 show that the transmitted wave packets have a reduced amplitude around y = 0 compared to the quietSun value, for all radial orders. It is known that wave absorption by sunspots causes a reduction in outgoing wave amplitude (Braun et al. 1987, 1988), as the result of partial mode conversion of incoming waves into slow magnetoacoustic waves that propagate down the sunspot (e.g. Spruit & Bogdan 1992; Cally & Bogdan 1997).
In our case (plane wave geometry), a significant fraction of the amplitude reduction that is observed onaxis behind the sunspot is due to the defocusing of wave energy by the fast wavespeed perturbation introduced by the sunspot. This can be illustrated by a simple 2D ray tracing experiment shown in the lefthand panel of Fig. 4, where the sunspot is approximated by a 10% enhancement of the wave speed. This “geometrical attenuation” will contribute to the wave amplitude reduction in addition to the physical absorption of waves. Their respective contributions would require better modeling since ray tracing cannot take wavefront healing into account.
Fig. 4 Wave amplitude perturbations for p_{4} caused by the sunspot. Left: map of observed amplitudes A (color scale). The white lines are 2D rays traced through a sunspot model, such that the wave speed is enhanced by 10% inside the circle and transitions smoothly to the background value (53.28 km s^{1}) at the dashed circle. The wavefronts are given by the black lines. The thick black lines indicate the boundaries of the caustics where rays intersect and wavefronts fold. Right: observed amplitudes A as a function of y at fixed distances from the sunspot ranging from x = 50 to 200 Mm. Notice the amplitude enhancements around y ~ ± 60 Mm. 

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Apart from the onaxis amplitude reduction, an enhancement of wave amplitude is seen away from the axis y = 0 for modes p_{2}, p_{3}, and p_{4} (bottom panels in Fig. 2). This offpath enhancement is more pronounced for the p_{4} wave packet (right panel of Fig. 4). As above, we suggest that this offpath feature is caused by sunspotinduced refraction. Since rays tend to bend away from the higherspeed medium (the sunspot), diffracted rays will cross over the unperturbed rays on both sides off the central axis. The envelope of intersecting rays is called a caustic in optics. The higher density of rays inside the caustics explains the enhancement of wave amplitude. The wavefronts fold and triplicate as they pass through the caustics (e.g. Dahlen & Tromp 1998). Inside the caustics, multiple arrivals contribute to the crosscorrelation traveltime measurements and the interpretation of travel time becomes ambiguous under the raytheoretical picture (Nolet & Dahlen 2000; Hung et al. 2001). The location of the caustics depends on the perturbation strength and the sunspot geometry. In the case of p_{4} modes, a 10% wavespeed perturbation provides a rather satisfying explanation for the onaxis power deficit and offpath enhancement (left panel of Fig. 4).
6. Conclusion
We have measured the traveltime shifts and amplitudes of plane wave packets as they traverse a sunspot using a crosscovariance technique. Observations show onaxis wavefront healing and amplitude reduction. Traveltime anomalies spread in the transverse direction with distance and amplitude enhancements are seen away from the central ray. Wavefront healing is a finitewavelength phenomenon, which cannot be modeled by ray theory. To the best of our knowledge, the only other direct observational evidence of finitewavelength effects in timedistance helioseismology is reported by Duvall et al. (2006), who studied the interaction of waves with subwavelength magnetic features.
We saw that ray tracing (unlike linearized ray theory) is quite useful for studying the geometrical attenuation caused by the defocusing of wave energy by a fast wavespeed anomaly. Physical wave absorption by the sunspot further enhances this reduction in the wave amplitude. The method of ray tracing is also useful for interpreting offpath amplitude enhancements.
The observations of traveltime shifts and wave amplitude that have been presented in this paper contain a wealth
of information about the seismic signature of sunspots. Their interpretation requires more sophisticated methods of analysis than linearized ray theory (which is still often used in sunspot seismology today). A promising approach is to solve the wave equation numerically (Schunker et al. 2013, Paper II), in combination with iterative inversion methods (Hanasoge et al. 2011, 2012).
Acknowledgments
L.G. and H.S. acknowledge support from Deutsche Forschungsgemeinschaft SFB 963 “Astrophysical Flow Instabilities and Turbulence” (Project A18). Z.C.L. thanks the MaxPlanckInstitut für Sonnensystemforschung for their hospitality. Z.C.L. was supported by the NSC of R.O.C. under the Study Abroad Program grant NSC982917I007121. Partial support was also provided by ERC Starting Grant #210949 “Seismic Imaging of the Solar Interior” to PI Gizon. The German Data Center for SDO, funded by the German Aerospace Center (DLR), provided the IT infrastructure required to carry out this work. SOHO is a project of international cooperation between ESA and NASA.
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All Tables
All Figures
Fig. 1 Crosscorrelation function F_{4}(x,y,τ) (solid line) between the crosscovariance C_{4}(x,y,t) at (x,y) = (60,0) Mm and the corresponding quietSun crosscovariance . The envelope (dashed line) is determined by demodulation. The traveltime shift, Δτ, and the wave packet amplitude, A, are indicated by arrows. In practice, we fit the peak nearest to τ = 0 with a parabola to measure the time shift. In this particular example, the time shift and reduced amplitude are Δτ = −57 s and A = 0.44 with respect to the quiet Sun. 

Open with DEXTER  
In the text 
Fig. 2 Spatial maps of traveltime shifts Δτ(x,y) (top) and wave packet amplitudes A(x,y) (bottom) with respect to the quiet Sun. From left to right are results for wave packets p_{1}, p_{2}, p_{3}, and p_{4}. Waves propagate from L (red lines) in the + x direction. The black circles denote the outer boundary of the penumbra of the sunspot. The parabolic curves (white lines with equation (y/R)^{2} + x/(x_{F} + λ/8) = 1) depict the boundaries of the first planewave Fresnel zones with focus at (x,y) = (x_{F},0), chosen such that the sunspot fills the width of the Fresnel zone. The focus position x_{F} depends on radial order and is taken from Table 1. We do not display the results in the near fields (gray regions where distances from L are less than 3λ). 

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In the text 
Fig. 3 Observed traveltime shifts, Δτ. The left panel shows the traveltime shifts along y = 0 for modes f through p_{4}, which heal as the distance from the sunspot, x, increases. The right panel shows the traveltime shifts of p_{4} modes as functions of y at four different values of x. Notice the spread in the transverse direction as x increases. The hatched and crosshatched area indicate the locations of the penumbra and umbra, respectively.The data is averaged over patches of size 16 × 16 Mm^{2} to filter out spatial scales that are dominated by noise. Error bars give the standard deviation of the mean. 

Open with DEXTER  
In the text 
Fig. 4 Wave amplitude perturbations for p_{4} caused by the sunspot. Left: map of observed amplitudes A (color scale). The white lines are 2D rays traced through a sunspot model, such that the wave speed is enhanced by 10% inside the circle and transitions smoothly to the background value (53.28 km s^{1}) at the dashed circle. The wavefronts are given by the black lines. The thick black lines indicate the boundaries of the caustics where rays intersect and wavefronts fold. Right: observed amplitudes A as a function of y at fixed distances from the sunspot ranging from x = 50 to 200 Mm. Notice the amplitude enhancements around y ~ ± 60 Mm. 

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In the text 