three-dimensional inversion of sound speed below a ... - IOPscience

The Astrophysical Journal, 640:516–524, 2006 March 20 # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THREE-DIMENSIONAL INVERSION OF SOUND SPEED BELOW A SUNSPOT IN THE BORN APPROXIMATION S. Couvidat,1 A. C. Birch,2 and A. G. Kosovichev1 Received 2005 August 8; accepted 2005 November 28

ABSTRACT We revise the inversion of acoustic travel times for the three-dimensional sound-speed structure below the solar NOAA Active Region 8243 of 1998 June. We benefit from recent progress in time-distance helioseismology that provides us with more reliable tools to infer subsurface solar properties. Among the improvements we implement here are the use of Born approximation–based travel-time sensitivity kernels that take into account finite-wavelength effects and thus are more accurate than the previously employed ray-path kernels, the inclusion of solar noise statistical properties in the inversion procedure through the noise covariance matrix, and the use of the actual variance of the noise in the temporal cross-covariances in the travel-time fitting procedure. Of these three improvements, the most significant is the application of the Born approximation to time-distance helioseismology. This puts the results of this discipline at the same level of confidence as those of global helioseismology based on inversion of normal-mode frequencies. Also, we compare inversion results based on ray-path and Born approximation kernels. We show that both approximations return a similar two-region structure for sunspots. However, the depth of inverted structures may be offset by 1 or 2 Mm, and the spatial resolution of the results is more accurately estimated with the more realistic Born sensitivity kernels. Finally, using artificial realizations of Doppler velocities of the quiet Sun, we are now able to estimate the statistical uncertainties of these inversion results. Subject headings: methods: numerical — Sun: helioseismology — sunspots — waves

1. INTRODUCTION

over, we apply an improved inversion procedure: both the traveltime fitting code and the inversion code make use of the statistical properties of the solar noise to derive a more accurate and reliable sound-speed profile and to provide us with estimates of the error bars on this profile. The basic methodology for filtering the data and performing the inversion of measured travel times is described in detail in Couvidat et al. (2005) and references therein. Following the approach of Kosovichev et al. (2000) and more recently Hughes et al. (2005), we assume that the measured traveltime perturbations are due to a combination of subsurface soundspeed perturbations and solar realization noise. Numerous authors have demonstrated that effects other than changes in the local sound speed introduce travel-time perturbations around sunspots. For example: Bruggen & Spruit (2000) described the role of changes in the upper boundary condition in sunspots due to the Wilson depression; Lindsey & Braun (2005) argued that the effect of photospheric magnetic field on observed oscillation velocities, i.e., the ‘‘showerglass effect,’’ can be important; Woodard (1997) and Gizon & Birch (2002) demonstrated that increased wave damping in sunspots can introduce shifts in travel times; S. P. Rajaguru et al. (2006, in preparation) show that the reduced visibility of p-mode oscillations in sunspots can, in some cases, introduce substantial artifacts into time-distance measurements; finally, Cally et al. (2003) showed that models that include the effects of nonvertical magnetic field on wave propagation can essentially reproduce the observed Hankel analysis phase shifts (Braun 1997) around sunspots. A critical parameter that determines the role of magnetic field effects in the wave propagation is the plasma parameter
¼ 8P/B2, where P is the gas pressure and B is the magnetic field strength. When this parameter is much greater than unity, then the magnetic field affects the propagation speed of acoustic waves and travel times as described by Kosovichev & Duvall (1997) and Ryutova & Scherrer (1998) but does not cause wave conversion into other types of MHD waves,

The estimation of the sound-speed profile below NOAA AR 8243 of 1998 June 18 ( Kosovichev et al. 2000) was a major breakthrough in the ability of solar physicists to access the local properties of the plasma below the photosphere. This estimation was carried out by the inversion of time-distance travel times (Duvall et al. 1993) using the LSQR algorithm (Paige & Saunders 1982), ray-path approximation kernels (Kosovichev & Duvall 1997), and with no treatment of the correlations in the travel-time perturbation noise. Since then, helioseismic tools have been improved. Major changes have been made to the sensitivity kernels now derived in the Born approximation ( Birch et al. 2004): the sensitivity of wavepackets to sound-speed perturbations extends off the ray path because of finite-wavelength effects of solar oscillations. The accuracy of the Born approximation, in the context of time-distance helioseismology, has been studied by Birch et al. (2001). The ray-path kernels are still often used (e.g., the recent work by Hughes et al. [2005]); thus, it is important to assess how much the new kernels affect the inversion results. It is also important to assess whether or not the inversions previously carried out with the ray-path approximation remain valid. This question was also addressed by Couvidat et al. (2004), who compared ray-path kernels and Fresnel-zone kernels (Jensen 2001; Jensen et al. 2001) that were a first attempt to include finitewavelength effects. However, the Fresnel-zone kernels are not based on a solution to the wave equation. In this paper we use sound-speed sensitivity kernels that are derived in the Born approximation and also include several important aspects of the time-distance measurement procedure (especially phase-speed filtering; see Birch et al. 2004). More1 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085; [email protected]. 2 Colorado Research Associates Division, NorthWest Research Associates, Inc., 3380 Mitchell Lane, Boulder, CO 80301.

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SOUND-SPEED INVERSION BELOW SUNSPOT which could substantially affect the interpretation of the timedistance measurements. This mode conversion happens in the regions where
’ 1 (e.g., Bogdan et al. 2003). In the umbra of sunspots, where the magnetic field is the strongest, this parameter is close to unity but can be as low as 0.5 at the photospheric level ( Mathew et al. 2004). However, below the photosphere,
rapidly increases with depth because the pressure scale height in sunspots is about 200 km, and the sunspot structure becomes gas pressure–dominated (e.g., Pizzo 1986). Therefore we expect the magnetic effects to be relatively small at depths larger than ’1 Mm and to affect mainly the inversion results of the uppermost layer. However, as the focus of this paper is the comparison of ray- and Born kernel–based inversions, we will not consider any of these issues here. In x 2 we derive the cross-covariances and mean travel-time perturbations related to the 1998 June sunspot. In x 3 we present the inversion code we use to invert these travel times. It is based on a multichannel deconvolution algorithm ( MCD; Jensen et al. 1998) enhanced by the addition of horizontal regularization and the noise covariance matrix. In x 4 we comment on the inversion results. We conclude in x 5. 2. DERIVATION OF THE TRAVEL-TIME PERTURBATIONS We work in three-dimensional plane-parallel geometry, with r as the horizontal coordinate vector and z as the vertical coordinate (z increases upward and is negative below the solar surface, meaning that
z is the depth). The time coordinate is t. Time-distance helioseismology ( Duvall et al. 1993) uses the temporal cross-covariance C(r1 ; r2 ; t) of Doppler line-of-sight velocity signals between two points r1 and r2 at the solar surface to determine the travel times of wavepackets propagating between these two points. To increase the signal-to-noise ratio of this point-to-point cross-covariance, we first average C(r1 ; r2 ; t) over an annulus centered on r1 and with a radius
¼ jr1
r2 j. Thus, we produce a point-to-annulus cross-covariance C(r;
; t). From C(r;
; t) we derive the one-way travel times i/o (r;
) of wavepackets propagating inward i, from the annulus to its center, and outward o, by fitting a Gabor wavelet (Kosovichev & Duvall 1997). We define the mean travel times (r;
) as (r;
) ¼ ½i (r;
) þ o (r;
)/2. As a first approximation, perturbations (r;
) in these mean travel times are linearly related to the sound-speed perturbations c in the wave propagation region, Z Z (r;
) ¼

dr S

0

Z

0

dzK(r
r 0 ; z;
)
d

c 2 0 (r ; z); c2

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2.1. Born Approximation Sensitivity Kernels Details on the sensitivity kernels K can be found in Birch et al. (2004). Here, we stress their strong dependence on the procedure applied to measure the travel times, especially the filtering applied to the data. There is a need for consistency that does not exist with less detailed kernels, such as the ray-path or Fresnelzone kernels, and that requires that we apply the same phasespeed filters to both the travel-time measurements and the kernel derivations. We also average the Born kernels in the same way that we average the cross-covariances. Unfortunately, the Born kernels are based on a definition of travel times introduced by Gizon & Birch (2004), which is different from the definition through a Gabor wavelet fit. The former works well for relatively small perturbations of travel times, but we must resort to the latter to measure travel times in sunspots. For the quiet Sun both definitions give very similar results. Even though this inconsistency may result in inaccuracy in the inversion results, it has no impact on the comparison of inversions made with the Born and ray-path approximations. An example of azimuthally averaged Born approximation kernel is shown in Figure 1. The original point-to-point kernel (top) exhibits rapid variations and fine details that are partly canceled out, especially near the surface, by the averaging and downsampling we apply to produce the pointto-annulus kernel (bottom). 2.2. Temporal Cross-Covariances We start from a 512 ; 512 ; 512 high-resolution datacube of Doppler velocities (r; t) obtained by the Michelson Doppler Imager instrument ( MDI; see Scherrer et al. 1995). The first step is to rebin this cube into a 128 ; 128 ; 512 datacube. We measure (r;
) from these rebinned data, related to the NOAA AR 8243 observed on 1998 June 18 from 15:36 to 0:06 UT. The temporal sampling of the datacube is ht ¼ 1 minute. The spatial sampling is hx ¼ 1:652 Mm. The computation domain is 211.5 Mm on the side, and the time duration is T ¼ 512 minutes. To select specific acoustic wavepackets, we multiply the Fourier transform of the datacube by Gaussian phase-speed filters F(k; !;
) (Duvall et al. 1997) to produce filtered datacubes (k; !), (k; !) ¼ F(k; !;
) (k; !);

ð2Þ

where k is the horizontal wavevector, ! is the angular frequency, and
is the annulus radius. The phase-speed filters F are of the form

ð1Þ

where S is the area of the region, and d is its depth. The sensitivity kernel for the relative squared sound-speed perturbations is given by K. Here a travel-time perturbation is defined as the difference between the measured travel time at a given location r on the solar surface and the average value of travel times in the quiet Sun. The sound-speed perturbation is relative to the solar model for which the kernels are calculated. When inverting the sound-speed profile below sunspots, it is important to note that equation (1) includes only the first-order sound-speed perturbations and thus may be inaccurate due to large values of the perturbations and also to the impact of the magnetic fields on the wavefield that has not been accounted for. As described in x 1, the interpretation that perturbations in the acoustic wave mean travel times arise only from sound-speed perturbations is approximate, to say the least, in presence of a strong magnetic field.

F(k; !;
) ¼ exp ½
(!=k
v)2 =2v2 ;

ð3Þ

where the mean phase speed v and the filter width v are listed in Table 1. The phase speed is derived from a solar model, while v corresponds to FWHM/½2(log 2)1/2 , where FWHM is the full width at half-maximum of the squared filter. FWHM is chosen to be the difference in phase speed between the rays that hit the inner and outer edges of the smallest and largest annuli that are used with that particular v. We compute the cross-covariances C(
; r; t) from (k; !) for 55 radii
listed in column (3) of Table 1. However, we only apply 11 phase-speed filters, which means that each one is used to produce five cross-covariances. These five cross-covariances are then shifted in time to locally straighten the first-bounce ridge on the time-distance diagram (see Fig. 2), and they are av¯ eraged to produce a ‘‘broad-annulus’’ cross-covariance C(
; r; t ) whose mean radius is listed in column (2) of Table 1. For a specific
-value, the temporal shift to apply to C(
; r; t) is

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determined by manually shifting Cref (
; t) [average of C(
; r; t) over r] until the location of its peaks corresponding to the firstbounce ridge matches that of Cref (
3 ; t), where
3 is the radius of the third—central—annulus in the group of five annuli. To reach a high enough resolution in time, we Fourier interpolate the cross-covariances. A better way of determining the shift to apply is to compute the group travel times of Cref (
; t) and Cref (
3 ; t), to define the shift as the difference in these travel times, and then to take advantage of the shift theorem in the Fourier domain. Even though this second approach is more precise, both methods of shifting give similar results. The first method is easier to apply at short
-values, for which fitting for the travel times is very delicate. 2.3. Gabor Wavelet Fit ¯ At each location r and for a given
, we fit C(
; r; t) by the following Gabor wavelet G:
   ! 2 2 (t
g ) cos !0 (t
p ) ; G(t; A; !; !0 ; g ; p ) ¼ A exp
4 ð4Þ where A is the amplitude of the wavelet, ! is the width of the envelope, !0 is the central frequency of the wavepacket, and p and g are respectively the phase and group travel times. Each average cross-covariance is fitted separately for the negative times, inward propagating wavepackets, and positive times, outward propagating. Only the phase travel times are used to compute the (r;
)-values. The reference travel times and an initial guess of the parameter values are obtained by fitting the refer¯ r; t) over r]. A ence cross-covariance C¯ ref (
; t) [average of C(
; rectangular window centered on the first-bounce ridge selects the timespan to fit. Here the width of this window is 14 minutes, and its locations are listed in columnP(6) of Table 1. The fit is done by minimizing the weighted 2 t G(t; A; !; !0 ; g; p )
¯ ¯ r; t ) C(
; r; t)2 /(t)2 . The weights  2 (t) are the variance of C(
; at each specific time t. This variance is derived from the temporal cross-covariances of 20 random realizations of Doppler velocity datacubes. The realizations of line-of-sight velocities in the quiet Sun are obtained following Gizon & Birch (2004) and are based on a MDI high-resolution power spectrum. This power spectrum is the average of two spectra obtained from 512 ; 512 ; 512

Fig. 1.—Example of a Born approximation kernel K(r; z;
). Top: Vertical cut at y ¼ 0 in the point-to-point Born kernel for
¼ 30:55 Mm. For comparison with the shape of a ray-path kernel, we superimposed the corresponding approximate ray path (solid line). Bottom: Corresponding azimuthally averaged (pointto-annulus) kernel. Five point-to-point kernels for slightly different
-values have been projected and then averaged on a coarser grid (see annulus index 6 in Table 1). The kernels have been multiplied by the value of the sound speed at each z.

TABLE 1 Annuli and Phase-Speed Filter Parameters

Index (1)

Mean
( Mm) (2)

1.................................. 2.................................. 3.................................. 4.................................. 5.................................. 6.................................. 7.................................. 8.................................. 9.................................. 10................................ 11................................

6.20 8.70 11.60 16.95 24.35 30.55 36.75 42.95 49.15 55.35 61.65


( Mm) (3) 03.7, 06.2, 08.7, 14.5, 19.4, 26.0, 31.8, 38.4, 44.2, 50.8, 56.6,

04.95, 07.45, 10.15, 15.72, 21.87, 28.27, 34.27, 40.67, 46.67, 53.07, 59.12,

06.20, 07.45, 08.7 08.70, 09.95, 11.2 11.60, 13.05, 14.5 16.95, 18.17, 19.4 24.35, 26.82, 29.3 30.55, 32.82, 35.1 36.75, 39.22, 41.7 42.95, 45.22, 47.5 49.15, 51.62, 54.1 55.35, 57.62, 59.9 61.65, 64.18, 66.7

v ( km s
1) (4)

v ( km s
1) (5)

t0 (min) (6)

12.77 14.87 17.49 25.82 35.46 39.71 43.29 47.67 52.26 57.16 61.13

2.63 2.63 2.63 3.86 5.25 3.05 3.15 3.57 4.46 3.78 3.41

11.67 14.53 19.55 28.17 33.50 33.55 35.90 38.15 40.55 42.22 43.95

Notes.—Eleven filters of mean phase-speed v and width v are used for different ranges of annulus radii
(v and v were provided by T. L. Duvall). Col. (1): Annulus index. Col. (6): Center t0 of the window function used to measure first-bounce travel times (see text).

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Fig. 2.—Example of time-distance diagram for the 55 annulus radii
listed in column (3) of Table 1. Left: Original reference cross-covariances Cref (
; t). Right: Shifted reference cross-covariances. By groups of five consecutive distances
, the cross-covariances have been shifted in time to obtain a locally straight timedistance diagram. The vertical solid lines delimit these groups of five distances.

datacubes of the quiet Sun, rebinned into 128 ; 128 ; 512 datacubes. We are using the approximation that the presence of the sound-speed perturbations does not significantly change the noise variance on the cross-correlations. Taking into account the ac¯ tual variance on C(
; r; t) in the fitting procedure is already an improvement compared to the standard procedure of travel-time measurements (prior works assumed that 2 ¼ 1). It has an impact on the travel-time determination, especially for the shortest distances
. However, an extra step could be made by taking into ¯ account the correlations between the values of C(
; r; t) at different times t and, for a given time t, between the values of the cross-covariances obtained for different annulus radii
and centers r. This implies a more complex measurement procedure ¯ that fits all of the cross-covariances C(
; r; t) at the same time, and this exceeds the scope of this paper. When the fitting code does not converge for a specific r, we set (r;
) ¼ 0 (this happens for less than 0.1% of the pixels). For the shortest
distances (the first three annuli), the Gabor wavelet fit is very sensitive to the time window location and width: the phase-speed filter artifact and second-bounce ridge both overlap with the first-bounce ridge (see Fig. 2), making it crucial to carefully select the peaks to fit. For the other annuli, the fit is, to a certain extent, independent of the time window location and width. 3. INVERSION CODE The inversion of the mean travel-time perturbation maps is done with a regularized least-squares method through a modified MCD algorithm (see Couvidat et al. [2005] for more details). Equation (1) is discretized by applying a piecewise constant scheme. This equation is a convolution product due to the translational invariance of the sensitivity kernels. Therefore to speed up the inversion procedure, computations are done in Fourier

space. In order to formulate the problem in the Fourier domain we introduce the following vectors (Jensen 2001): di ¼ (k;
i ); Gi ¼ K(k; z ;
i ); m ¼

c2 (k; z ): c2

ð5Þ ð6Þ ð7Þ

For each horizontal wavevector k we find the vector m that solves h i min jj
1 (d
Gm)jj22 þ k2 (k)jjLmjj22 ; ð8Þ where L is a regularization operator, k(k) is the regularization parameter, and  is obtained by Cholesky decomposition of the travel-time perturbation noise covariance matrix. Thus the statistical properties of the noise are taken into account through (k). These (k) matrices are derived from the 20 realizations of simulated Doppler velocities. Here we set k2 (k) ¼ k2v þ k2h k 4 , where kv and kh are constant. The operator L is chosen so that our regularization scheme is a discrete approximation of the norm of the second derivatives of c 2 /c 2 in the horizontal direction and of the norm of c 2 /c 2 (weighted by the inverse of the spatial sampling hz at each z) in the vertical direction. The traditional MCD algorithm has been modified to include horizontal regularization. We invert for 12 layers in depth, whose locations are listed in column (1) of Table 2. 4. RESULTS 4.1. Travel-Time Maps Some of the resulting 11 travel-time maps, (r;
), are shown in Figure 3. We compared these maps with the original

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COUVIDAT, BIRCH, & KOSOVICHEV TABLE 2 Uncertainty in the Inverted c 2 /c 2 Resulting from Noise Propagation through the Inversion Algorithm Layer Depth ( Mm) (1) 0–0.62 .............................. 0.62–1.42 ......................... 1.42–2.38 ......................... 2.38–3.51 ......................... 3.51–4.88 ......................... 4.88–6.52 ......................... 6.52–8.54 ......................... 8.54–10.97 ....................... 10.97–13.77 ..................... 13.77–16.95 ..................... 16.95–24.6 ....................... 24.6–33.8 .........................

Born () (2) 4.95 4.91 5.56 4.83 5.18 4.21 3.93 3.35 2.59 2.10 2.17 2.19

; ; ; ; ; ; ; ; ; ; ; ;

10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
4

Ray Path () (3) 4.61 3.76 5.74 5.14 4.97 5.31 4.19 4.04 3.65 2.63 2.03 1.42

; ; ; ; ; ; ; ; ; ; ; ;

10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
3 10
4

Notes.—Col. (1): Location of the inverted layer. Col. (2): Uncertainty on c 2 /c 2 (as a value of the rms noise dispersion ) when the inversion is carried out with the Born approximation. Col. (3): Uncertainty when the inversion is carried out with the ray-path approximation.

ones obtained by T. L. Duvall. Both sets of maps are similar and display the same features, but the range of the mean travel-time perturbations is larger on the new maps, especially for the second and third annuli (the difference is significant, as it is larger than the noise dispersion). These differences in the amplitudes seem to be due to the additional weights in our fitting procedure and to a different window function (which is only 14 minutes wide instead of the 20 minutes used by T. L. Duvall). We used a narrower window to make sure that we did not fit some peaks that belong to the filter artifact or to the second-bounce ridge. 4.2. Inversion with the Born Kernels We choose the two regularization parameters kv and kh in an empirical way by increasing them until the inversion results become smooth enough. There is a rapid transition from noisy inversion results with no apparent discernible structures to a sunspot-like structure, which makes it easy to determine a lower value for the inversion parameters. Using T. L. Duvall’s original travel-time maps, the maps computed for this paper, and different regularization parameters, we can point out the actual features of the sunspot and discard some of the numerical artifacts. The results are shown in Figures 4 (left) and 5. The conspicuous characteristic of the inversion results is that beneath the spot the relative sound-speed perturbation is negative and becomes positive deeper, as was first demonstrated by Kosovichev et al.

Fig. 3.—Mean travel-time perturbation (r;
) maps for
¼ 6:2 Mm (top left),
¼ 11:6 Mm (top right),
¼ 30:55 Mm (bottom left), and
¼ 49:15 Mm (bottom right).

Fig. 4.—Vertical cut in the inversion results around y ¼ 97 Mm. Left: Inversion using Born approximation kernels. Right: Inversion using ray-path kernels.

Fig. 5.—Top: Two views of the NOAA AR 8243 in the solar continuum from the MDI instrument. The top left panel shows the sunspot on 1998 June 18, and the top right panel shows the spot on 1998 June 20 (some data are missing because the spot has reached the western edge of the MDI high-resolution field). Bottom: Cuts in the inversion results for the sunspot on 1998 June 18. The x and y ranges are not the same in the top and bottom panels. The bottom left panel is a vertical cut along the dashed line visible on the top left figure. A tubular structure (arrow) appears at the top left edge of the spot. The bottom right panel is a horizontal cut at the layer ranging from z ¼
1:42 to 0.62 Mm. The top of the tubular structure (arrow) appears as a pale white spot at the bottom right edge of the sunspot.

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(2000). The transition between these two regions occurs a bit deeper than 2.4 Mm. In the deep region, the absolute value of c 2 /c 2 peaks around 4.3 Mm and then decreases. The real extent of the deeper region is unknown: the very small values of c 2 /c 2 in the deepest layers may be due to the combination of the loss of sensitivity our set of kernels experience below about 16 Mm and of the vertical regularization scheme. The amplitudes of the inverted sound-speed perturbations depend on the set of traveltime maps, because of the differences in (r;
), and regularization parameters we use. For the zone of decreased sound speed, jc 2 /c 2 j is about 0.12 at the most. For the zone of increased sound speed the maximum sound-speed perturbation is c2 /c2  0:16. In order to estimate the statistical noise in the inversion results, we produced 20 artificial realizations of simulated Doppler velocity datacubes. From these artificial datacubes, we produced 20 sets of travel-time maps with the procedure described in previous sections. These maps show the noise on the mean traveltime perturbations due to the stochastic nature of solar oscillations. We invert them with the same regularization parameters as those used to invert the maps related to the June 18 sunspot. The rms variation at each z of the inversion results provides us with an estimate of the uncertainty due to noise propagation through the inversion procedure (see Table 2). Such an estimate was first computed by Jensen et al. (2003), but their work was based on simpler realizations of the noise on travel-time maps, ignoring the correlations between maps for different
-values. Here, such correlations are accounted for. However, our estimate of the error bars on the inversion results very likely underestimates the actual errors, since we do not account for the systematic errors. In Couvidat et al. (2005) it was possible to observe these sytematics because we were inverting artificial data for which we knew the exact solution. In particular, it appeared that the maximum amplitude of the sound-speed perturbation was systematically underestimated in the sunspots, because of the smoothing properties of the sensitivity kernels and regularization scheme, and that sharp spatial transitions were spread out in the inversion results. Therefore, more realistic error bars can be obtained by multiplying the noise propagation 1  uncertainties of Table 2 by a factor of 2 or 3. In Table 2 we note that deeper than 4.88 Mm the uncertainty seems to decrease. This is likely due to the decrease of sensitivity at high k that the kernels exhibit with depth: the deeper a kernel goes, the less it is sensitive to the spatial high-frequency perturbations. Therefore the propagation of noise through the inversion procedure is less pronounced in deep layers because the regularization scheme—that constrains the norm of the inversion results to be minimal—forces c 2 /c 2 to be zero at frequencies where the kernels have no sensitivity. However, the systematic errors are probably larger when the depth increases. In Figure 5, we emphasize the presence of a tubular structure of about 10 Mm diameter that extends from the deeper region of the sunspot, c 2 /c 2 > 0, toward the solar surface. The top of this structure is located about 29 Mm southwest from the spot center and at a depth of 0.62–1.42 Mm. Its maximum amplitude is c 2 /c 2 ¼ 0:057, which is significantly larger than the 1  uncertainty—due to noise propagation—listed in Table 2 for this layer. Two days later, on 1998 June 20, two finger-like structures extending from the deeper region of the same sunspot toward two pores located about 30 Mm south of the spot center are visible, as was first mentioned in Kosovichev et al. (2000) and later confirmed by Couvidat et al. (2004). It might be interesting to know whether or not the tubular structure of late June 18 is related to the finger-like structures of the beginning of June 20.

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Zhao et al. (2001) showed the presence of horizontal eastward flows with a maximum speed of about 1 km s
1 on June 19, which make it possible for the tubular structure to migrate to the region where the pores will appear 26 hr later at the most. Meanwhile the tube would have to divide into two flux tubes that pierce the surface, producing the pores. The analysis of intermediate data is required to determine the fate of the tubular structure. Information on the lifetime and dynamic of magnetic flux tubes could be gathered from such an analysis. However, we have to be cautious not to overinterpret these inversion results because of limitations already mentioned in x 1. Some averaging kernels for these results are shown in the left panels of Figure 6; the averaging kernels tell us the spatial resolution of the inversion results. We see that near the surface we have better vertical resolution than in deeper layers, which is consistent with the shape of the Born kernels. The same appears to be true for the horizontal resolution, even though this is less obvious from the figure. 4.3. Comparison with the Ray-Path Kernels: The Born Approximation Makes a Difference J. Zhao produced a set of ray-path kernels for the same
-values (using the method described in Kosovichev & Duvall [1997]) and with the same horizontal and vertical grids as the Born approximation kernels. We paid attention to the consistency of the analysis, to make sure that the difference between inversion results is entirely due to the Born and ray-path approximations. Comparison of these inversion results stresses several differences despite qualitatively remarkably similar results (Fig. 4, right): First, the transition between c2 /c2 < 0 and c2 /c2 > 0 occurs deeper with the ray-path kernels (at a depth of about 3.5 Mm instead of 2.4 Mm). Second, the maximum value of c2 /c2 is at a depth of 6 Mm instead of about 4.3 Mm. Finally, with the Born kernels c 2 /c 2 is still slightly positive down to z 
25 Mm (a feature difficult to see on the figure), while with the ray-path kernels c 2 /c 2 ¼ 0 at a shallower z 
17 Mm. The results we obtain with the ray-path approximation corroborate the ones published by Kosovichev et al. (2000). Even though the raypath approximation produces a two-region structure for the sunspot underneath the solar surface with sound-speed perturbation amplitudes very similar to those obtained with the Born approximation, the vertical location of the structures differs. Another element of comparison is the averaging kernels: they seem better localized with the ray-path approximation, at least for the deep target locations ( Fig. 6, right). This pattern is reminiscent of the comparison between ray-path and Fresnel-zone kernels (Couvidat et al. 2004). However, the ray-path kernels are intrinsically better localized because they do not account for the finite wavelength of wavepackets and thus overestimate the spatial resolution of the inversion results. In Table 2, we note that the uncertainties on the inversion results are very similar to the Born approximation ones. The ray-path approximation produces inversion results with a location in depth offset by 1–2 Mm, and this discrepancy seems to increase with depth. The complete lack of sensitivity of raypath kernels to perturbations outside the ray path gives the impression that the sunspot ends abruptly at z ¼
17 Mm (at the most), while the Born kernels seem to favor a deeper structure. Recovering the deeper structure of sunspots with the ray-path kernels requires measurements of the travel times for distances
larger than the distances used in this paper. However, the overall difference between the inversion results is surprisingly not as significant as could be expected (e.g., Bogdan 1997) considering the complete dissimilarity between point-to-point

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Fig. 6.—Example of vertical cuts at y ¼ 0 in the averaging kernels obtained with the Born approximation (left) and with the ray-path approximation (right) and for target locations (x; y; z) ¼ (0; 0;
0:62 to 0) Mm (top), (0, 0,
3.51 to
2.38) Mm (middle), and (0, 0,
13.77 to
10.97) Mm (bottom). The averaging kernels have been normalized by their maximum value.

ray-path and Born kernels: the large amount of averaging we perform on both sets of kernels (first azimuthal averaging, then averaging of groups of five kernels) partly cancels out these initial differences between point-to-point kernels. It is likely that if we manage to increase the signal-to-noise ratio on travel-time maps without averaging that much, we can preserve the fine structure of Born kernels and thus better reveal the difference in the inversion results between ray-path and Born approximations. 5. CONCLUSION For the first time, we have obtained the sound-speed profile below a solar active region using Born approximation sensitiv-

ity kernels, which take into account finite-wavelength effects. These kernels are highly dependent on the details of the traveltime computations and are arguably more realistic than the raypath and Fresnel-zone kernels. We showed that the ray-path and Born approximations provide qualitatively similar results showing the two-region structure of sunspots. However, the Born kernels favor a shallower transition between these two regions and an overall deeper extent of the sound-speed perturbations. The location in depth of the structures inverted with ray-path kernels differs, even though the amplitudes of the sound-speed perturbations remain the same. This corroborates the results of a previous comparison between ray-path and Fresnel-zone approximations:

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a precise vertical localization of the inverted structures requires sensitivity kernels that take into account finite-wavelength effects. Moreover, the apparent improved localization of ray-path averaging kernels is misleading and is due to the concentration of sensitivity along the ray path: the Born approximation returns more realistic averaging kernels. However, the averaging performed in the standard time-distance measurements ( Duvall et al. 1997) to increase the signal-to-noise ratio on the traveltime maps reduces the differences between Born and ray-path kernels, making the latter qualitatively correct. The inversion procedure we use has been largely improved compared to previous works (e.g., Couvidat et al. 2004) or to what is usually done in time-distance helioseismology: the traveltime noise covariance matrix is included in the inversion code, we regularize both in the vertical and horizontal directions, and the variance in the noise on the temporal cross-covariance is accounted for in the travel-time computations. All these advances make the entire inversion procedure more reliable, robust, and also complete: along with the spatial resolution of the inversion results, we can now provide estimates of the uncertainties of these results. Even though only the error due to noise propagation can be computed, our ability to estimate them based on realistic

Doppler velocity simulations of the quiet Sun is another step toward comprehensive time-distance analysis. We presented the first time-distance helioseismology inferences based on a realistic model of solar oscillations, which included such important properties as the finite-wavelength effects and stochastic excitation. Still further substantial improvements have to be made in the time-distance measurement and inversion procedures to account for the complicated physics of sunspots and oscillations. In particular, the next steps are to modify the Gizon & Birch (2004) definition of travel times to allow for a consistent use of the Born kernels in the active regions, and not only in the quiet Sun, to apply these kernels in the inversion of material flows beneath the solar surface, and to include the magnetic field effects. This work was supported by NASA grants NAG 5-12452 and NAG 5-13261. Aaron Birch is supported by NASA contract NNH04CC05C. We are very grateful to Thomas L. Duvall for providing us with the MDI datacube and his original set of travel-time maps, and to Junwei Zhao for providing us with a set of ray-path kernels. We also thank the anonymous referee for useful comments that helped improve a first draft of this paper.

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