ANALYSIS OF CHARACTERISTIC PARAMETERS OF LARGE-SCALE CORONAL WAVES OBSERVED BY THE SOLAR-TERRESTRIAL RELATIONS OBSERVATORY/EXTREME ULTRAVIOLET IMAGER

Large-scale, large-amplitude disturbances propagating through the solar corona were first observed by Moses et al. (1997) and Thompson et al. (1998) in images recorded by the Extreme Ultraviolet Imaging Telescope (EIT; Delaboudinière et al. 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995) and are hereafter called "EIT waves." They were originally interpreted as the coronal counterparts to chromospheric Moreton waves (Moreton & Ramsey 1960), as suggested by Uchida (1968) in his coronal fast-mode magnetohydrodynamic (MHD) wave interpretation of Moreton waves. Though this interpretation has been confirmed for several case studies (e.g., Thompson et al. 1999; Warmuth et al. 2001; Pohjolainen et al. 2001; Vršnak et al. 2002, 2006; Khan & Aurass 2002; Veronig et al. 2006; Muhr et al. 2010), the statistical characteristics of extreme ultraviolet (EUV) and Moreton waves are quite different. In particular, EIT waves generally propagate at much lower velocities, v ≈ 200–400 km s⁻¹, than Moreton waves, v ≈ 1000 km s⁻¹ (e.g., Klassen et al. 2000; Biesecker et al. 2002; Thompson & Myers 2009). Thus, there is still an ongoing debate about whether EIT waves are really the coronal counterparts to Moreton waves. Additionally, it is still an open discussion whether they are caused by explosive energy releases of flares or the erupting coronal mass ejection (CME; e.g., Warmuth et al. 2001, 2004b; Zhukov & Auchère 2004; Cliver et al. 2005; Vršnak & Cliver 2008), and whether they are waves at all or rather propagating disturbances related to magnetic field line opening and restructuring associated with the CME liftoff (e.g., Delannée & Aulanier 1999; Chen et al. 2002; Attrill et al. 2007; Wills-Davey et al. 2007).

Numerical simulations of EIT waves as fast-mode MHD waves resulted in wave phenomena mimicking observational data quite closely (Wang 2000; Wu et al. 2001; Ofman & Thompson 2002; Ofman 2007). An interesting by-product of the simulations by Ofman & Thompson (2002) and Terradas & Ofman (2004) is stationary brightenings in the vicinity of the launch sites, in line with observations of such areas at the fronts of EIT waves (Delannée & Aulanier 1999; Delannée 2000; Attrill et al. 2007). Delannée et al. (2007) suggested a result of magnetic field restructuring during the CME liftoff generated either by Joule heating or an increase in density due to plasma compression. Simulations by Cohen et al. (2009) favored the latter interpretation, with some additional effects due to increased temperature resulting from plasma compression.

Until the launch of the Solar-Terrestrial Relations Observatory (STEREO; Kaiser et al. 2008) mission in 2006, with the Extreme Ultraviolet Imager (EUVI; Howard et al. 2008) onboard, observations of EIT waves were drastically limited by a ≈12 minute cadence of the EIT instrument in the 195 Å passband. The identically built instruments EUVI-A and EUVI-B onboard the twin STEREO spacecraft observe solar corona in four different EUV passbands with a high observing cadence (up to 75 s) and a large field of view (up to 1.7 R_☉) from two different vantage points.

Studies of large-scale waves using STEREO/EUVI reveal velocities ranging from ≈200–400 km s⁻¹ and a decelerating character consistent with freely propagating large-amplitude MHD fast-mode waves (Veronig et al. 2008, 2010; Long et al. 2008; Kienreich et al. 2009). From simultaneous observations by the two STEREO spacecraft, it was possible to analyze for the first time the three-dimensional (3D) nature of EIT waves using stereoscopic techniques (Kienreich et al. 2009; Patsourakos & Vourlidas 2009; Patsourakos et al. 2009; Ma et al. 2009; Temmer et al. 2011). Veronig et al. (2008), Long et al. (2008), and Gopalswamy et al. (2009) reported the refraction and reflection of a wave at the border of a coronal hole, providing strong evidence for the wave-like nature of the phenomenon. However, Attrill (2010) questioned these findings, favoring instead the hypothesis of two simultaneously launched waves from two distinctly separate sites. In the EUVI wave event of 2010 January 17, studied by Veronig et al. (2010), first observations of the full 3D wave dome and its lateral and radial expansions could be performed. In a recent study that became available during the revision process of the present paper, Long et al. (2011) analyzed the kinematical aspects and evolution of the full width at half-maximum (FWHM) and integral of the 2007 May 19 and 2009 February 13 events using similar methods. Therefore, we will discuss and compare their findings with our results in Section 5. For recent reviews regarding the kinematics of large-scale EUV waves and their morphology and relationship to associated solar phenomena, we refer to Wills-Davey & Attrill (2010) and Gallagher & Long (2010).

In this paper, we study high-cadence EUV observations of four well-defined EUV waves observed by STEREO/EUVI that occurred between 2007 May and 2010 April. These waves are special because of their pronounced amplitudes, although they occurred during the extreme current solar minimum. Our study has the following main aims.

• 1.  
  We apply and compare two different methods to derive wave kinematics: visual tracking of the foremost part of a wave front and a method based on the intensity profiles of the propagating perturbation.
• 2.  
  Based on the perturbation profiles, we derive quantities that provide insight into the physical character of the phenomenon in terms of Mach numbers, as well as the evolution of the amplitude, width, and integrated intensity of the disturbances.
• 3.  
  The perturbation profiles are further studied with respect to associated phenomena, such as stationary brightenings, coronal dimmings, and rarefaction regions behind wave fronts.

The four large-scale coronal waves under study were recorded by EUVI on 2007 May 19, 2009 February 13, 2010 January 17, and 2010 April 29. The EUVI instruments are part of the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) instrument suite on board the STEREO-A (Ahead; ST-A) and STEREO-B (Behind; ST-B) spacecraft. The separation angle between ST-A and ST-B steadily increases by ≈45° per year. EUVI observes the chromosphere and low corona in four spectral passbands (He ii 304 Å: T  ∼  0.07 MK; Fe ix 171 Å: T ∼1 MK; Fe xii 195 Å: T  ∼  1.5 MK; Fe xv 284 Å: T ∼2.25 MK) out to 1.7 R_☉, with a pixel-limited spatial resolution of 16 pixel⁻¹ (Wuelser et al. 2004). For the kinematical analysis, we used all passbands if possible, while for the perturbation profiles, only the 195 Å images were studied, where the wave signal is most prominent.

The EUVI wave event of 2007 May 19 was recorded by both STEREO spacecraft (ST-A and ST-B) and was observable from the Earth. High-cadence EUVI images in the 171 Å (75 s), 195 Å (600 s), 284 Å (300 s), and 304 Å (300 s) channels are available. This event was the first distinct EUV wave observed by EUVI and has already been studied in detail by several authors (Long et al. 2008, 2011; Veronig et al. 2008; Gopalswamy et al. 2009; Kerdraon et al. 2010), who revealed a propagation velocity decelerating from ≈450 to 200 km s⁻¹ as well as an associated type II burst that indicated shock formation in the corona.

The event of 2009 February 13 was observed by both STEREO spacecraft and occurred in perfect quadrature (separation angle of ≈90°). The wave was observed on the disk by ST-B and on the limb by ST-A. These observations provided a unique basis to study the 3D nature of the wave. Patsourakos & Vourlidas (2009) and Kienreich et al. (2009) determined a propagation height of the wave of ≈100 Mm, with a mean (on-disk) velocity of ≈230 km s⁻¹. For the analysis, 171 Å and 195 Å filtergrams of ST-B are available, with cadences of 300 s and 600 s, respectively.

On 2010 January 17, ST-A and ST-B were 134° apart from each other. The wave was observed in the eastern hemisphere of ST-B, which was situated 70° behind Earth on its orbit around the Sun. This event was unique because of the full 3D wave dome structure that was observed in the four different EUVI channels and had a lateral (on-disk) propagation velocity of ≈300 km s⁻¹ and an upward propagation velocity of ≈600 km s⁻¹ (Veronig et al. 2010). The EUVI-B imaging cadence is 2.5 minutes in the 171 Å, 5 minutes in the 195 Å, 2.5–5 minutes in the 284 Å, and 5 minutes in the 304 Å passbands.

The event of 2010 April 29 was the last and strongest of four homologous waves observed by ST-B within a period of eight hours at a position angle of 70° behind the solar limb, as seen from Earth (Kienreich et al. 2011). EUVI images in all four wavelengths are available during the event. However, the cadence of two hours in the 171 Å and 284 Å channels limited possible wave tracking to the 195 Å and 304 Å channels. Due to a lack of wave signatures in the 304 Å channel, only 195 Å filtergrams with a cadence of 300 s are used for the analysis.

All EUVI filtergrams are reduced using the SECCHI_PREP routines available within SolarSoft. Furthermore, we differentially rotate each data set to a common reference time. To enhance faint coronal wave signatures, we derive running-ratio (RR) images by dividing each image by a frame taken 10 minutes earlier, as well as base-ratio (BR) images by dividing each image by the last pre-event image. Finally, a median filter was applied to the images to remove small-scale variations. RR images are used for visual tracking of the coronal waves, because in RR images the signal of propagating disturbances is highest. The perturbation profiles are calculated from BR images, which provide better insight into the changes in physical parameters due to the passing wave front.

The kinematical analysis of large-scale propagating disturbances is usually based on visual tracking of the outer edge of the wave front. Thus, the results are severely influenced by the observer and his/her interpretation of the wave front. Therefore, more objective and reproducible—as well as automated— methods are desirable. Two alternative semi-automated methods have been applied to study EUV waves: the Huygens plotting method (Wills-Davey & Thompson 1999) and the perturbation profile method (Warmuth et al. 2004a; Podladchikova & Berghmans 2005; Muhr et al. 2010; Veronig et al. 2010).

In this study, we focus on the perturbation profile method for several reasons. First, this semi-automated method provides insight into important physical parameters of the EUVI wave (amplitude, width, Mach number). Second, it provides a semi-automated alternative to the reconstruction of wave kinematics by visual tracking. Third, detailed analysis of the perturbation profiles provides information on associated phenomena not observable by eye, which can give us important insight into the underlying physics of the events. Below, we describe two different methods, i.e., the visual tracking method and perturbation profiles, and their application to the study of EUV waves.

3.1. Kinematics via Visual Tracking

We visually tracked the wave fronts in a series of EUVI RR images. The eruption center was derived by applying circular fits to the earliest observed EUVI wave fronts on the 3D solar surface, which appear as ellipses in the two-dimensional projected image (Veronig et al. 2006). To enhance the statistical significance and ensure a realistic error estimate for the wave's central position, all available passbands were used to determine a mean value and standard deviation. For each event and for both methods, we focused on the same specific 45° propagation sector, in which the disturbance is most pronounced. The chosen sectors are restricted by two great circles (parts of them are shown as black curves in Figures 1–4) that pass through the derived wave center. To obtain the wave's kinematics, we calculated, for each point of the wave front within the selected propagation sector, the distance from the eruption center along great circles on the solar surface and then averaged the values.

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3.2. Kinematics via Perturbation Profiles

The second approach to analyze EUV wave propagation is based on perturbation profiles. The method starts at the eruption center determined from the visually tracked wave fronts (see Section 3.1) and sums the intensity values of all pixels between two constantly growing concentric circles (again in the deprojected heliospheric plane) defining annuli with a radial width of 1° within the selected propagation sector, which span over an angular width of 45° (for an illustration, see Figure 5 and the associated animation in the online version of the journal). Because the area over which we sum the pixel values is steadily growing by moving to greater radii, each annulus is averaged over its pixel sum. This results in a mean intensity that is a function of distance, measured along the solar surface, from the wave center. The procedure is repeated for each frame, i.e., time step, until the EUV wave fades away.

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In perturbation profiles, propagating disturbances can be identified as a distinct bump above the background level (intensity level of 1.0 in BR images). Modified Gaussian envelopes emphasizing the leading part of the wave bump are then fitted to the perturbation profiles for two reasons. First, perturbation profiles show a roughly Gaussian form (Wills-Davey 2006; Veronig et al. 2010; Kienreich et al. 2011); second, it is very useful for the robustness and automatization of the latter part of the profile algorithm, in which the positions of the leading edges of the waves are extracted. Downs et al. (2011) stated that peak intensity may be strongly affected by the emission of CME material, whereas the foremost front corresponds to the actual wave. The trailing edge of the wave is also strongly influenced by features such as stationary brightening, flare emission, CME, and dimming regions behind the wave front. Therefore, we use modified Gaussian envelopes different from the original data set. First, the original data are examined for peak intensity. Then, a simple Gaussian fit is applied to the first half of the wave bump from the maximum until the leading edge. Finally, the Gaussian envelope is mirrored at the position of the maximum. The result is a Gaussian fit that matches the foremost leading edge of the wave front much better than simple Gaussian fits, while for the rest of the wave bump a characteristic envelope shape is formed.

From the modified Gaussian fits to the perturbation profiles, we derive the position of the wave front. Starting at the maximum amplitude level of each wave bump, we define the leading edge of the wave fronts at the position where the Gaussian envelope of the profile falls below an intensity level of I/I₀ = 1.02, where I are the intensity entries of the current image and I₀ are the intensity entries of the base image. This value is reasonable because the human eye is able to identify intensity changes as small as 1%–2% above the background level. Thus, this should provide us with a good measure when comparing wave front positions from perturbation profiles and visual tracking.

3.3. Calculation of Perturbation Profile Parameters

It is possible to gain useful information on wave characteristics by extracting distinct wave parameters from the perturbation profiles. We derived the amplitude, FW, FWHM, and integral below the perturbation profiles (down to an intensity ratio of 1.02). The FW of the wave is defined as the width of the Gaussian envelope, whereas the FWHM is defined as the distance between the two points where the intensity level drops below 50% of the maximum value. The integral of the perturbation profiles is the area below the Gaussian fits.

The calculation of the magnetosonic Mach numbers is based on the peak perturbation amplitude values, A_max, of each wave event, derived from the peak of the Gaussian fits to the profiles. From the definition of BR images, we know that an increase in the perturbation amplitude above the background level can be interpreted as A = I/I₀, where the intensity I in an optically thin coronal spectral line is given as

$$\begin{equation} I=\int _h f(T, n_e) n_e^2dh\,, \end{equation} \tag{ 1 }$$

where the emission measure EM = ∫_hn²_edh along the line of sight and the contribution function f(T, n_e) depends on the plasma temperature T and electron density n_e (Phillips et al. 2008). To obtain an intensity value in a relatively broad EUV passband such as the EUVI 195 Å passband, the integral over several spectral lines must be conducted. Because there are temperature and density dependences for each spectral line, the following approximations are made: first, the integration is only along the pressure scale height H, with an average constant density n (Wills-Davey 2006). Thus, the intensity turns into I ≈ f(T, n_e)n²_eH, where n²_eH is the emission measure EM. Second, the quiet Sun in the 195 Å passband is dominated by its strongest line at 195.12 Å (Del Zanna et al. 2003), which is formed in a narrow temperature range of 1.2–1.8 MK, with the peak response at ≈1.4 MK (Feldman et al. 1999). In Figure 3 of Zhukov (2011), it is evident that the contribution function of the 195.12 Å spectral line only weakly depends on density, i.e., f(T, n_e) ≈ f(T), and on the temperature near its peak response, which is also the position where the differential emission measure peaks. Assuming that the temperature does not change considerably after the wave front passes, the relative intensity change I/I₀ (with I and I₀ referring to the initial and final states, respectively) can be used to estimate the density increase n/n₀ ∼ (I/I₀)^1/2. Zhukov (2011) gives a detailed discussion of this approach, which is under consideration by the CHIANTI atomic database.

For each wave, we estimate the density jump X_c = n/n₀ at the peak amplitude of its perturbation profile. Assuming the shock height is in the low solar corona (Veronig et al. 2010), we can conclude from typical EUV wave velocities (in the range of 200–500 km s⁻¹) that the assumption of a quasi-perpendicular fast-mode MHD shock is reasonable (Mann et al. 1999). Considering the Rankine–Hugoniot jump conditions at a perpendicular shock front (e.g., Priest 1982), the magnetosonic Mach number M_ms = v_c/v_ms (where v_c is the coronal shock velocity and v_ms is the magnetosonic speed) can be expressed as

$$\begin{equation} M_{\rm{ms}}= \sqrt{\frac{X_c (X_c +5+5\beta _c)}{(4-X_c)(2+5\beta _c/3)}}\,, \end{equation} \tag{ 2 }$$

where β_c is the ambient coronal plasma-to-magnetic-pressure ratio and X_c = n/n₀ is the density jump at the shock front. For the specific-heat ratio (the polytropic index), we substitute γ = 5/3. For the plasma beta in the quiet Sun, we assume a value of β_c = 0.1 and note that Equation (2) depends only weakly on β (Vršnak et al. 2002).

3.4. Coronal Dimmings, Stationary Brightenings, and Rarefaction Regions

4.1. Wave Kinematics

In Figures 1–4, the morphology and evolution of the four EUVI wave events under study are shown in BR images. We note that the wave fronts are not observed in close proximity to the eruption center but are first visible at distances of ≈240 Mm, 90 Mm, 150 Mm, and 260 Mm from the respective initiation centers of the 2007 May 19, 2009 February 13, 2010 January 17, and 2010 April 29 events, and can be followed up to distances of 600–900 Mm.

The top panels of Figures 6–9 show the distance–time plots of the events. We display the wave kinematics derived from both methods (visual tracking and perturbation profiles) with error bars, linear and quadratic least-squares fits to the visually tracked wave fronts, 95% confidence intervals, and prediction boundaries of the linear fits. The mean velocities derived for the four events under study lie in the range of 220–350 km s⁻¹.

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Three of the four events propagate with constant speed. The small deceleration values derived from the quadratic fits are in the range of −6 to −13 m s⁻², which corresponds to a deceleration of ≈5% with respect to the start velocity values derived from the quadratic fit to the wave kinematics. This lies within the confidence intervals, and thus it is valid to represent the kinematics of the three events by a propagation with constant velocity over the full propagation distance. We note that the kinematical analysis of the perturbation profiles of these three events reveals a somewhat stronger deceleration than the visual tracking method, but it is still only marginally significant, corresponding to a deceleration rate of ≈10% of the start velocity values. Only for the 2007 May 19 event do we obtain a significant deceleration (− 193 m s⁻²), which corresponds to 30% of the start velocity. This is also the fastest wave in our sample, with a mean velocity of ≈350 km s⁻¹. In Table 1, the derived information on propagation characteristics (velocity, deceleration) is summarized.

Table 1. EUVI Wave Properties of the Events under Study

Event	S/C	vlin (km s−1)		vquad, 0 (km s−1)		a0 (m s−2)		Imax	Xc	Mms
		Vis.	Prof.	Vis.	Prof.	Vis.	Prof.
2007 May 19	ST-A	348 ± 29	338 ± 18	483 ± 73	429 ± 58	−193.2 ± 87	−84.7 ± 60	1.61	1.27	1.21
2009 Feb 13	ST-B	228 ± 17	221 ± 36	237 ± 72	261 ± 72	    −6.9 ± 25	−23.4 ± 30	1.39	1.18	1.14
2010 Jan 17	ST-B	286 ± 7	275 ± 17	300 ± 24	314 ± 53	−13.4 ± 38	−39.4 ± 49	1.59	1.26	1.20
2010 Apr 29	ST-B	337 ± 27	321 ± 23	342 ± 88	361 ± 76	−7.6 ± 52	−51.7 ± 93	1.22	1.12	1.08

Notes. We list the mean velocity values v_lin derived from linear fits, as well as the start velocity values v_{quad, 0} and acceleration values a₀ derived from the quadratic fits for both methods, i.e., visual tracking and perturbations profiles. Additionally, the peak intensity amplitudes extracted from the perturbation profiles, the density jump values X_c, and the calculated Mach numbers M_ms are listed.

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We note that the position measurements of both methods match each other quite well: the maximum differences lie within an average of 20–40 Mm. As a consequence, the velocity values derived via visual tracking correspond to those obtained via perturbation profiles, and all values agree within the error limits (see Table 1).

4.2. Overall Characteristics of Perturbation Profiles

The consistent kinematical results obtained from both methods suggest that the perturbation profile method is an adequate alternative to the visual tracking method. In particular, the former allows one to analyze large-scale disturbances in the solar atmosphere by producing kinematical results unburdened by the subjective judgments of observers. Thus, the perturbation profile results are easier to reproduce than those derived from the visual tracking method.

Moreover, we can quantify various important wave parameters from the perturbation profiles. Figures 10–13 show the evolution of the perturbation profiles for the four wave events under study. In all four cases, we observe a clear intensification in the early phase, reaching a peak amplitude A_max value around 10–20 minutes (second or third panel of Figures 10–13) after the first remarkable wave bump can be observed. The evolution of the intensity amplitudes of all events under study is displayed in the middle panels of Figures 6–9. The peak perturbation amplitudes A_max lie in the range of 1.2–1.6. The calculated density jump values, X_c, therefore range from 1.1–1.3, and the peak magnetosonic Mach numbers, M_ms, range from 1.08–1.21 (Table 1).

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In addition, we derived the full width of the wave as well as the FWHM. The evolution of the pulse width is plotted in the bottom panel of Figures 6–9 for the four wave events under study. The width of the wave pulse, as well as the FWHM, increases during its evolution by a factor of two to three, whereas the integral below the perturbation profile remains basically constant for the events of 2007 May 19 and 2010 April 29 but decreases for the events of 2009 February 13 and 2010 January 17 by factors of four and three, respectively.

4.3. Distinct Features Observable in Perturbation Profiles after Wave Passage

Inspection of the derived BR perturbation profiles reveals a diversity of distinct features after wave passage, including coronal dimmings, stationary brightenings, and rarefaction regions. Below, these are discussed individually for the four events in our sample.

4.3.1. Event of 2007 May 19

The profiles of 2007 May 19 show three distinct phases: a leading wave pulse (enveloped by Gaussian fits), a reversal point (where the intensity level switches from values greater than 1.0 to less than 1.0), and a trailing rarefaction region. Additionally, in the rear section of the perturbation profiles, we observe a deep core dimming extending from the initiation site to ≈300 Mm, with an intensity decrease of ≈70% lasting for several hours (Figure 10).

The rarefaction region and its propagation are of special interest. This phenomenon is theoretically predicted to be located behind the propagating front of a compression wave, but observations are barely reported (Cohen et al. 2009). From 12:52 UT until 13:02 UT, we observe the formation of a deep coronal dimming behind the wave pulse. During its early evolution, it becomes more pronounced and reaches its maximal spatial extension at 13:02 UT. After 13:02 UT, it is quasi-stationary. However, in the frontal part of the dimming, from 300 Mm to the trailing part of the wave pulse, a smaller, propagating dip is clearly visible. We interpret this to be the rarefaction region that follows the wave front. Its formation is a combination of two aspects: (1) the driver stops at a distance of ≈300 Mm (where the outermost point of the deep-dimming region is located) and (2) behind the wave pulse, the wave forms a region where the density and pressure fall below equilibrium.

This event was associated with a metric type II burst, indicating shock formation in the corona. We use the observations of the type II burst to obtain an alternative estimate of the shock Mach number. Type II bursts usually show the fundamental and harmonic emission bands, which are both frequently split in two parallel lanes, the so-called band split (Nelson & Melrose 1985). The interpretation of the band split as the plasma emission from the upstream and downstream shock regions was affirmed by Vršnak et al. (2001). Thus, the band split can be used to obtain an estimate of the density jump at the shock front and thus the Mach number of the ambient plasma (Vršnak et al. 2002).

The composite dynamic radio spectrum (Figure 14) shows complex and intense radio emission, consisting of a group of type III bursts and a type II burst. For a detailed dynamic spectrum analysis covering the frequency range 0.4–300 MHz, we refer to Figure 4 in Kerdraon et al. (2010). In the present study, we focus on the type II burst that started around 12:51:30 UT at a frequency of 160 MHz. In the period 12:51:30 UT to 12:54:00 UT, we recognize a band split pattern (indicated in Figure 14 by two black lines), with a relative bandwidth of BDW = Δf/f ≈ 0.19–0.22. The relative bandwidth BDW is determined by the density jump at the coronal shock front X_c = ρ_2c/ρ_1c, where ρ_1c and ρ_2c are the densities upstream and downstream of the shock front, respectively (for details, see Vršnak et al. 2001). Since BDW $\equiv (f_2-f_1)/f_1=\sqrt{(\rho _{2c}/\rho _{1c})}-1$, we find X_c = 1.41–1.48. Using a five-fold Saito coronal density model (Saito 1970), we find a Mach number of 1.32–1.40 from the derived values of the density jump X_c.

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From the peak amplitude of the EUVI perturbation profile, we derive M_ms ≈ 1.21. A possible explanation for the smaller estimate is that, due to the width of 45° over which the perturbation profiles are derived, eventual higher values are averaged (see Figure 10). The curve displayed by small black crosses shows the overall 45° sector we use for our analysis, while the gray line profile displays a small 2° sub-sector in the central part of the overall 45° sector. By shrinking the sector to a narrower width around the most intense propagation direction, we obtain a considerably higher peak due to lower signal averaging, and the Mach number increases to values of at least M_ms = 1.3. Note that sector narrowing and the subsequent increased Mach number imply that the extracted values are (in fact always) underestimated and that there is a lower limit for the Mach number estimate. Comparing the results of both approaches, we find the Mach number of the ambient plasma to be M_ms = 1.2–1.4, a reasonable range of values for a coronal shock wave (Vršnak et al. 2001). Since EUV waves and type II bursts are generated at different heights in the solar atmosphere, the Mach numbers are not necessarily the same. In particular, type II bursts are point-like sources formed where the disturbance is most intense, corresponding to higher Mach numbers. Shocks with Mach numbers ranging from 1.2–1.4, as determined for the events under study, are usually considered weak shocks.

4.3.2. Event of 2010 January 17

The perturbation profiles of 2010 January 17 show persistent stationary brightenings (Figure 11). They appear after the wave front passes and only slowly fade away (Delannée & Aulanier 1999; Attrill et al. 2007; Delannée et al. 2007). The EUV wave starts at 03:56 UT as a relatively small wave pulse that intensifies until 04:01 UT. In the subsequent panels of Figure 11, after the wave has intensified to its maximum amplitude of 1.6 at 04:01 UT, a strong, long-lasting, quasi-stationary bright feature becomes clearly visible. We find the first evidence of this stationary part at 04:06 UT at a distance of 100–300 Mm from the source location, i.e., at the distance where the first wave front appears (see Figure 11). It is well pronounced and exceeds an amplitude level of 1.25, lasting for at least 30 minutes after wave passage. Compared to the timescale of the propagating disturbance, it only slowly fades away. This evolution can also be seen in the BR images in Figure 3, where we overplot the positions of the stationary brightenings using gray lines.

4.3.3. Event of 2009 February 13

The perturbation profiles of the quadrature event of 2009 February 13 are shown in Figure 12. The evolution of the most enhanced direction of 45° ± 225 is displayed in the perturbation profiles as crosses. The propagating wave pulse can be clearly identified. The gray line profile represents a different sector with a main direction of 270° ± 225. The intensity profile is characterized by an intensity enhancement that can be interpreted as a stationary brightening showing a similar evolution to the one of the 2010 January 17 event. The maximum intensity of these stationary brightenings is even more pronounced than that of the 2010 January 17 event and exceeds a level of 1.35 at 05:55 UT at 100–300 Mm from the source location (see Figure 12). It is again induced by the wave front passage and only slowly fades away. The propagating wave pulse ahead of the stationary brightening is not as pronounced as that in the main direction but is still detectable, although it is considerably smaller than the stationary brightening. In both directions, the coronal dimming is clearly visible. Its location is restricted to the area behind the stationary brightening at a distance up to ≈100 Mm.

4.3.4. Event of 2010 April 29

The wave event of 2010 April 29 is the fourth and strongest of four homologous waves that occurred within a time span of eight hours (Kienreich et al. 2011). The profile evolution of the event is displayed in Figure 13. We can identify three different parts in the rear section of this perturbation profile: a dimming region, a stationary brightening, and a rarefaction region, all visible after the waves' passage. The dimming region is prominent from the first image at 06:20 UT to the last profile shown at 06:40 UT, occurring at 0–100 Mm from the source location. Just in front of it, an intensity enhancement is present. A definite identification with a stationary brightening is difficult due to its relatively weak appearance compared to the background level of 1.0. This brightening lasts for at least 40 minutes after the waves passage and only slowly fades away. In front of this intensity enhancement, a dip in the profile propagates away from the initiation center, which we interpret as the rarefaction region, similar to the event of 2007 May 19.

• 1.  
  We analyzed four well-pronounced EUV wave events observed by the STEREO EUVI telescopes to compare two different kinematical analysis techniques: the generally used visual tracking method and semi-automated perturbation profiles. The differences in the determined positions of the leading edge of the wave fronts using both methods are no more than 40 Mm. Thus, we conclude that perturbation profiles are a suitable method to analyze large-scale waves. The big advantage of the perturbation profile method is the higher degree of automatization and reproducibility due to objective measurements of the wave location. Nevertheless, there are some restrictions. In the later evolution phase, wave fronts become more irregular, showing changes in shapes and propagation direction. These effects are not considered in the perturbation profile method. Thus, due to lower amplitude and smearing of irregular fronts, the wave profile is no longer distinct against the background. These properties lead to a systematic underestimation of the distance of the wave compared to the visual tracking method. Additionally, diffuse bright fronts are easier to identify in RR images because they have higher contrast than BR images.
• 2.  
  The determined propagation velocities derived from both methods match within the error limits. For the four wave events under study, we obtained mean velocity values in the range of 220–350 km s⁻¹, i.e., within the velocity range for fast magnetosonic waves under quiet-Sun conditions (e.g., Mann et al. 1999). Three of the four kinematical curves show only small deceleration values of −6 to −13 m s⁻², which lie within the error range. For the fastest event, 2007 May 19, the obtained deceleration of −193 m s⁻² is significant.Long et al. (2011) analyzed the kinematics of the 2007 May 19 and 2009 February 13 events using a similar perturbation profile method. They found the following values for the 2007 May 19 event: start velocity v₀ = 447 ± 87 km s⁻¹, acceleration a = −256 ± 134 m s⁻², a broadening of the pulse width from 50 to 200 Mm, and a decrease in the peak amplitude intensity from 60% to 15% of the background values. Our findings are v₀ = 429 ± 58 km s⁻¹, a = −85 ± 60 m s⁻², and similar results for the broadening of the pulse and the intensity evolution. For the second event of 2009 February 13, Long et al. (2011) found v₀ = 274 ± 53 km s⁻¹ and a = −49 ± 34 m s⁻², while our results are v₀ = 261 ± 72 km s⁻¹ and a = −24 ± 30 m s⁻². The profile broadening and the intensity evolution are again similar. For the 2009 February 13 event, the outcomes of both studies are consistent within the error limits. The results of 2007 May 19 show differences in the acceleration values, which can be explained by noting that the method used by Long et al. (2011) is not identical to ours. (1) While they extracted the position of maximum intensity, we used the leading edge of the wave fronts for our analysis. The difference in the acceleration values thus results from the usage of different features for the position measurements. The maximum intensity peaks are propagating at a slower speed compared to the leading edges, an effect that is expected for a feature that is broadening during its evolution. Because of that, the deceleration values are smaller for the leading edge measurements, while the starting velocity values are not affected. (2) The overall sector width used in Long et al. (2011) varied from event to event, while we used a constant angular width of 45° in each event. (3) The main direction of the calculated sector as well as the initiation centers are not exactly the same (the uncertainty is about 20 Mm).
• 3.  
  Considering the events under study as low-amplitude MHD fast-mode waves, the Mach numbers and wave velocity values are proportional to each other, M_ms = v/v_ms. Thus, from the derived Mach number evolution (from the maximum value down to M_ms ≈ 1), we expect a decrease in the propagation velocity by Δv ≈ 100 km s⁻¹ (2007 May 19), ≈35 km s⁻¹ (2009 February 13), ≈60 km s⁻¹ (2010 January 17), and ≈30 km s⁻¹ (2010 April 29). The velocity changes we derive from the least-squares quadratic fits reveal ≈200 km s⁻¹ (2007 May 19), ≈40 km s⁻¹ (2009 February 13), ≈20 km s⁻¹ (2010 January 17), and ≈30 km s⁻¹ (2010 April 29). For the 2009 February 13 and 2010 April 29 events, the derived velocity changes are in the same order of ≈30 km s⁻¹ as the error on the velocity determination. Hence, the weak deceleration is hidden in the measurement uncertainties. For the 2007 May 19 event, the result of 200 km s⁻¹ clearly exceeds the error limit in velocity of ≈70 km s⁻¹, and thus the deceleration is observable and significant. According to the Mach number evolution for the 2010 January 17 event, an observable deceleration in the propagating wave is expected. The determined velocity change of ≈60 km s⁻¹ exceeds the velocity error of 20 km s⁻¹; masking of the deceleration from data scatter is therefore unlikely. Nevertheless, from the kinematical measurements obtained via visual tracking and perturbation profiles, we do not derive a significant deceleration, as would be expected from the peak Mach number estimated for this event.
• 4.  
  Each event can be followed over a period of at least 30 minutes, during which we first observe an intensification of the wave, which is followed by a steady decrease in and broadening of wave pulses, as extracted from the perturbation profiles. From this, we derive useful information on the characteristics of wave pulses, such as the peak amplitude, width, and integral of a wave pulse. The evolution of a pulse width shows a clear broadening over time by a factor of two to three, similar to the broadening of the FWHM values. These values are typical for large-scale waves (Warmuth et al. 2004b; Wills-Davey 2006; Warmuth 2010; Veronig et al. 2010). The integral below the perturbation profile remains mostly constant (2007 May 19 and 2010 April 29) or decreases by a factor of three to four (2009 February 13 and 2010 January 17). We stress that the combination of profile broadening and amplitude decay, leading to a constant integral below the disturbance profile, is consistent with the characteristics of a freely propagating wave (Landau & Lifshitz 1987).
• 5.  
  The peak magnetosonic Mach numbers derived for the events under study range from M_ms = 1.08–1.21, indicating the evolution of weak coronal shocks. The magnetosonic Mach number for the 2007 May 19 event was determined via two different approaches, leading to similar results ranging from M_ms = 1.2–1.4. Thus, using the perturbation maximum amplitude for determining the magnetosonic Mach number seems reasonable. Grechnev et al. (2011) discussed the problem of projection effects in measurements of EUV waves. Because we do not observe the perturbation from a viewpoint that is located directly above the propagating disturbance but rather from an inclined position, the intensity values are always underestimated. This is probably also the reason why we observe Gaussian wave pulse profiles instead of a sharp edge, as would be expected in the case of a shocked disturbance.
• 6.  
  We observe three different distinct features in the perturbation profiles occurring after the wave passage. In three of four events, deep coronal dimmings are prominent at small distances from the eruption center. Only for the 2010 January 17 event the dimming is hardly present in the perturbation profiles of the used sector. However, an inspection of Figure 3 reveals the existence of coronal dimmings near the initiation site that are most prominent in the western sectors close to the limb and off-limb.The second distinct feature in the wake of three of the four EUV waves is the presence of stationary brightenings. These stationary parts appear after the waves' passage at distances of 100–300 Mm and slowly fade. Because the most intense wave profile always forms one frame before the stationary brightening appears, these parts seem to be an obstacle to the wave front. These stationary brightenings are not circumferential around the eruption center but rather confined to specific sectors, which is best observed in the event of 2009 February 13. Cohen et al. (2009) did numerical simulations on this event and showed that stationary brightenings are developing at the outer edges of core coronal dimmings. Observations by Delannée & Aulanier (1999), Delannée (2000), and Delannée et al. (2008), as well as simulations by Chen et al. (2005), indicate that these stationary brightenings appear at the locations where the connectivity of the magnetic field lines changes. Since these stationary brightenings are located in front of the deep core dimming regions (interpreted as the footprints of the expanding CME), are restricted in their expansion (their maximal distance is given by the first wave front's location), and appear only in confined sectors after the waves' passage, they can be interpreted as a signature of the CME's expanding flanks.The third distinct feature is the presence of rarefaction regions observable in two of the four wave events, on 2007 May 19 and 2010 April 29. A rarefaction region behind a wave pulse is a prospective feature that develops while trailing a large-amplitude perturbation (Landau & Lifshitz 1987). Indeed, simulations by Cohen et al. (2009) revealed regions that are dim in intensity and propagate after the bright wave front.

The EUV wave velocities are in the range of fast magnetosonic waves in the quiet solar corona and show either a decelerating characteristic or a constant velocity (within the measurement uncertainties) during propagation. The perturbation profiles reveal intensification, broadening, and decay of the propagating wave pulse, while the integral evolution is either constant or decreasing. The small Mach numbers are indicative of a weak coronal shock. Stationary brightenings formed in confined sectors at the outer edge of the core coronal dimming can be interpreted as signatures of the expanding CME flanks. The first observable wave fronts are always located just in front of them. Thus, we conclude that the EUV wave events can be interpreted as freely propagating, weak-shock fast-mode MHD waves initiated by the CME's lateral expansion.

The STEREO/SECCHI data are produced by an international consortium of the Naval Research Laboratory (USA), Lockheed Martin Solar and Astrophysics Lab (USA), NASA Goddard Space Flight Center (USA), Rutherford Appleton Laboratory (UK), University of Birmingham (UK), Max-Planck-Institut für Sonnensystemforschung (Germany), Centre Spatiale de Liège (Belgium), Institut d'Optique Théorique et Appliquée (France), and Institut d'Astrophysique Spatiale (France). N.M., I.W.K., and A.M.V. acknowledge the Austrian Science Fund (FWF): P20867-N16. The European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 218816 (SOTERIA) is acknowledged by B.V. and M.T.