FIG 4 - uploaded by Tom Bogdan
Content may be subject to copyright.
ÈContour plot of the energy density E of the wave packet of at time t \ 27 min. The maximum value of the energy density has been scaled to ° 3 unity and the gray scale in the lower right corner of the plot indicates the relative variation of E over the Ðgure. The heavy solid line shows the projection of the WKBJ ray path onto the distance-depth plane, and the circled asterisk overplotted on this curve shows the expected location of the wave packet at this particular time (t \ 27 minutes).
Source publication
The relationship between the time-distance and modal-decomposition approaches to solar active region seismology is clarified through the consideration of the oscillations of a plane-parallel, isentropic polytrope. It is demonstrated by direct construction that a wave packet formed through the superposition of neighboring p-modes interferes construc...
Contexts in source publication
Context 1
... 1993a). To account for this behavior in a very crude fashion we 8 We have also veriÐed that Figs. are una †ected by using a smaller 3È8 and a factor of B4 additional modes in the direct construction of the dk M wave packet. except the wave packet has now been centered on an n \ 8, as opposed to an n \ 4, p-mode. To aid in facilitating comparisons Fig. 4, between this Ðgure and the overplotted WKBJ ray path is identical to that depicted in Fig. 4, Fig. ...
Context 2
... are una †ected by using a smaller 3È8 and a factor of B4 additional modes in the direct construction of the dk M wave packet. except the wave packet has now been centered on an n \ 8, as opposed to an n \ 4, p-mode. To aid in facilitating comparisons Fig. 4, between this Ðgure and the overplotted WKBJ ray path is identical to that depicted in Fig. 4, Fig. ...
Context 3
... una †ected by using a smaller 3È8 and a factor of B4 additional modes in the direct construction of the dk M wave packet. except the wave packet has now been centered on an n \ 8, as opposed to an n \ 4, p-mode. To aid in facilitating comparisons Fig. 4, between this Ðgure and the overplotted WKBJ ray path is identical to that depicted in Fig. 4, Fig. ...
Context 4
... sequentially, Figures do indeed illus- 4È7 trate, in a very gross sense, the propagation of a rather extended wave packet along the overplotted WKBJ ray path. The wave packet is of order 20 Mm across in both its vertical and horizontal dimensions and so during the course of its motion from to it basically sweeps Figure 4 Figure 7, out a ray bundle that is guided by the WKBJ ray path. The B20 Mm size of the ray bundle is set by the e †ective range of wavenumbers, Mm~1, spanned by the constit- *k M B 0.3 uent p-modes that comprise the wave packet (see Fig. 1). ...
Context 5
... the WKBJ ray path only penetrates to a depth of B30 Mm, the ray bundle samples a signiÐcant fraction of the p-mode acoustic cavity. It is worth mentioning that the pressure Ñuctuations generally contribute about 2/3 of the overall energy density, contoured in Figures 4È7, and that the kinetic energy density is partially responsible for Ðlling in the region between the WKBJ ray path and the surface of the polytrope. ...
Citations
... The ray approximation is expected to be accurate if the underlying flow field does not vary at length scales which are smaller than a wavelength (e.g., Birch et al. 2001). If the flow varies on smaller length scales, the Born approximation is considered to model the physical processes in the solar interior more accurately (e.g., Bogdan 1997, Birch and Felder 2004, Couvidat et al. 2006). ...
... The accuracy of the Born and ray approximations has been studied by, e.g., Bogdan (1997), Birch and Felder (2004), Couvidat et al. (2006), and . The ray approximation is expected to be valid when the length scale of variations of the flow is larger than the width of the first Fresnel zone (e.g., Hung et al. 2000;Birch et al. 2001). ...
... This pattern is due to the inclusion of higher-degree modes (roughly 90 l 170). This finding is in accordance with the result of Bogdan (1997), who showed that the wave path of a wave packet can be extended over a relatively large region from beneath the turning point of the corresponding ray path to the surface of the Sun. ...
Understanding the solar meridional flow is important for uncovering the origin of the solar activity cycle. Yet, recent helioseismic estimates of this flow have come to conflicting conclusions in deeper layers of the solar interior, i.e., at depths below about 0.9 solar radii.
The aim of this thesis is to contribute to a better understanding of the deep solar
meridional flow. Time-distance helioseismology is the major method for investigating this flow. In this method, travel times of waves propagating between pairs of locations on the solar surface are measured. Until now, the travel-time measurements have been modeled using the ray approximation, which assumes that waves travel along infinitely thin ray paths between these locations. In contrast, the scattering of the full wave field in the solar interior due to the flow is modeled in first order by the Born approximation. It is in general a more accurate model of the physics in the solar interior.
In a first step, an existing model for calculating the sensitivity of travel-time measu-
rements to solar interior flows using the Born approximation is extended from Cartesian to spherical geometry. The results are succesfully compared to the Cartesian ones and are tested for self-consistency.
In a second step, the newly developed model is validated using an existing numerical simulation of linear wave propagation in the Sun. An inversion of artificial travel times for meridional flow shows excellent agreement for noiseless data and reproduces many features in the input flow profile in the case of noisy data.
Finally, the new method is used to infer the deep meridional flow. I used Global Oscillation Network Group (GONG) data that were earlier analyzed using the ray approximation and I employed the same Substractive Optimized Local Averaging (SOLA) inversion technique as in the earlier study. Using an existing formula for the covariance of travel-time measurements, it is shown that the assumption of uncorrelated errors from earlier studies leads to errors in the inverted flows being underestimated by a factor of about two to four.
The inverted meridional flow above about 0.85 solar radii confirms the earlier results from ray theory regarding the general pattern of the flow, especially regarding a shallow return flow at about 0.9 solar radii, with some differences in the magnitude of the flow.
Below about 0.85 solar radii, the inversion result depends on the thresholds used in the singular value decomposition. One result is again similar to the original regarding its general single-cell shape. Other results show a multi-cell structure in the southern hemisphere with two or three cells stacked radially. However, both the single-cell and the multi-cell flow profiles are consistent with the measured travel times within the measurement errors.
To reach an unambiguous conclusion on the meridional flow below about 0.85 solar radii, the errors in the measured travel times have to be decreased considerably in future studies. For now, I conclude that the existing controversy of recent measurements of the deep meridional flow is relaxed by properly taking the associated errors into account.
... In the case of flows at the bottom of the convection zone, this scale was estimated to be of the order of 200 Mm (Böning et al. 2016 and references therein). If the flow varies on smaller length scales, the Born approximation is thought to be more accurate (e.g., Bogdan 1997, Birch and Felder 2004, Couvidat et al. 2006. ...
Accurate measurements of deep solar meridional flow are of vital interest for understanding the solar dynamo. In this paper, we validate a recently developed method for obtaining sensitivity functions (kernels) for travel-time measurements to solar interior flows using the Born approximation in spherical geometry, which is expected to be more accurate than the classical ray approximation. Furthermore, we develop a numerical approach to efficiently compute a large number of kernels based on the separability of the eigenfunctions into their horizontal and radial dependence. The validation is performed using a hydrodynamic simulation of linear wave propagation in the Sun, which includes a standard single-cell meridional flow profile. We show that, using the Born approximation, it is possible to accurately model observational quantities relevant for time-distance helioseismology such as the mean power spectrum, disc-averaged cross-covariance functions, and travel times in the presence of a flow field. In order to closely match the model to observations, we show that it is beneficial to use mode frequencies and damping rates which were extracted from the measured power spectrum. Furthermore, the contribution of the radial flow to the total travel time is found to reach 20% of the contribution of the horizontal flow at travel distances over $40^\circ$. Using the Born kernels and a 2D SOLA inversion of travel times, we can recover most features of the input meridional flow profile. The Born approximation is thus a promising method for inferring large-scale solar interior flows.
... It includes for example the distance between the points ∆ a such that τ a (r) represents the time it takes the wave packet to travel from the point r to r + ∆ a . It can also represent the type of filter used in the data analysis [1] or the type of averaging geometry [4]. Averaging is generally performed in order to reduce the noise level that is very high due to limited observation times. ...
An important problem in local helioseismology is the three-dimensional
reconstruction of flows from travel times obtained from measured dopplergrams.
The forward problem is approximately described by a large system of convolution
equations, and the data are random vectors with a known covariance matrix.
Whereas for deterministic linear inverse problems several computationally
efficient optimal regularization methods exist, for statistical linear inverse
problem only one optimal linear estimator exists, the Pinsker estimator.
However, often it is computationally inefficient since it requires a singular
value decomposition of the forward operator or it is not applicable because of
an unknown covariance matrix, so it is rarely used for real-world problems.
Both reasons do not apply for the helioseismology problem above. We present a
simplified proof of the optimality properties of the Pinsker estimator and show
that it yields significantly better results than the state-of-the-art methods
of the field, Regularized Least Squares (Tikhonov regularization) and SOLA
(approximate inverse).
... Tested by numerical simulations (Parchevsky and Kosovichev, 2009). Validity of the ray-path theory; finite wavelength effects (Bogdan, 1997) Tested by using a Born-approximation (Birch et al., 2001; Couvidat, Birch, and Kosovichev, 2006a) Differences in travel-time definitions: Gabor wavelet (Kosovichev and Duvall, 1997), minimization of cross-covariance deviation (Gizon and Birch, 2002), and linearization of the deviation (Gizon and Birch, 2004) Tests using MDI data (Couvidat et al., 2010) found good agreement between the trave-time definitions of Kosovichev and Duvall (1997) and Gizon and Birch (2002), but strong systematic deviations of the linearized definition of Gizon and Birch (2004). Contributions of thermal and magnetic effects (Kosovichev and Duvall, 1997) Tested by numerical simulations (Olshevsky, Khomenko, and Shelyag et al., 2009b). ...
Mechanisms of the formation and stability of sunspots are among the
longest-standing and intriguing puzzles of solar physics and astrophysics.
Sunspots are controlled by subsurface dynamics hidden from direct observations.
Recently, substantial progress in our understanding of the physics of the
turbulent magnetized plasma in strong-field regions has been made by using
numerical simulations and local helioseismology. Both the simulations and
helioseismic measurements are extremely challenging, but it becomes clear that
the key to understanding the enigma of sunspots is a synergy between models and
observations. Recent observations and radiative MHD numerical models have
provided a convincing explanation to the Evershed flows in sunspot penumbrae.
Also, they lead to the understanding of sunspots as self-organized magnetic
structures in the turbulent plasma of the upper convection zone, which are
maintained by a large-scale dynamics. Local helioseismic diagnostics of
sunspots still have many uncertainties, some of which are discussed in this
review. However, there have been significant achievements in resolving these
uncertainties, verifying the basic results by new high-resolution observations,
testing the helioseismic techniques by numerical simulations, and comparing
results obtained by different methods. For instance, a recent analysis of
helioseismology data from the Hinode space mission has successfully resolved
several uncertainties and concerns (such as the inclined-field and phase-speed
filtering effects) that might affect the inferences of the subsurface
wave-speed structure of sunspots and the flow pattern. It becomes clear that
for the understanding of the phenomenon of sunspots it is important to further
improve the helioseismology methods and investigate the whole life cycle of
active regions, from magnetic-flux emergence to dissipation.
... L'approximation de rayon [40], [41], [42] est appliquée pour le cas où la longueur d'onde est petite par rapport à l'échelle caractéristique du Soleil. Cette approximation considère que les modes « p » sont sous forme de superposition d'ondes planes qui se propagent dans l'intérieur du Soleil. ...
C’est à partir de l’étude des oscillations confinées à l’intérieur du Soleil, que l’Héliosismologie est née. L’une des techniques de l’Héliosismologie qui s’intéresse aux phénomènes locaux est la technique « Temps–Distance ». Cette méthode inspirée de la sismologie terrestre est basée sur un modèle théorique reliant les temps de parcours des ondes acoustiques (modes p) qui parcourent le milieu solaire aux propriétés internes du soleil.
Notre travail consiste à construire le modèle théorique d’une part, qui met en relation les temps de parcours des modes « p » avec les propriétés internes du Soleil. Et d’autre part, de mettre en place les différentes procédures de traitement de données qui nous permettent de calculer les temps de parcours qui sont issus de l’observation à partir des données Dopplerogrammes du Soleil obtenues de l’instrument spatial MDI à bord de la sonde SOHO et du réseau terrestre GONG.
Seismology is a highly effective tool for investigating the internal structure of the Earth. Similar techniques have also successfully been used to study other planetary bodies (planetary seismology), the Sun (helioseismology), and other stars (asteroseismology). Despite obvious differences between stars and planetary bodies, these disciplines share many similarities and together form a coherent field of scientific research. This unique book takes a transdisciplinary approach to seismology and seismic imaging, reviewing the most recent developments in these extraterrestrial contexts. With contributions from leading scientists, this timely volume systematically outlines the techniques used in observation, data processing, and modelling for asteroseismology, helioseismology, and planetary seismology, drawing comparisons with seismic methods used in geophysics. Important recent discoveries in each discipline are presented. With an emphasis on transcending the traditional boundaries of astronomy, solar, planetary and Earth sciences, this novel book is an invaluable resource and reference for undergraduates, postgraduates and academics.
Context. Previous helioseismology of sunspots has been sensitive to both the structural and magnetic aspects of sunspot structure.
Aims. We aim to develop a technique that is insensitive to the magnetic component so the two aspects can be more readily separated.
Methods. We study waves reflected almost vertically from the underside of a sunspot. Time–distance helioseismology was used to measure travel times for the waves. Ray theory and a detailed sunspot model were used to calculate travel times for comparison.
Results. It is shown that these large distance waves are insensitive to the magnetic field in the sunspot. The largest travel time differences for any solar phenomena are observed.
Conclusions. With sufficient modeling effort, these should lead to better understanding of sunspot structure.
What is waveform tomography? Seismic tomography – in which we construct images of a body's interior using seismic waves – is an inverse problem; that is, our goal is to find a model that fits a set of existing data observations. This is much less straightforward than the reverse, forward problem (i.e., generating synthetic data from an existing model) due to the fact that multiple models can fit the same data or, in other words, the solution is non-unique. Furthermore, there may be parts of the model to which the data have no sensitivity, and small errors in the data can propagate into significant errors in the model (e.g., Trampert, 1998). In order to transform data space into model space, a seismic modeling algorithm is required that can generate synthetic data from an initial model, and then update the initial model to minimize the misfit between synthetic and observed data. There is therefore always a trade-off between the computational efficiency of the modeling algorithm and the accuracy or resolution with which it can represent the real seismic structure. Which kind of modeling algorithm is employed in a given situation depends very much on the nature of the structure being imaged, the quality of the data, and the available computational resources. Traditionally, seismic tomography has used the travel times of wave phases between a seismic source and receiver to infer the sound-speed structure along the path between them. This so-called travel-time tomography, more typically referred to as “time–distance helioseismology” when applied to the Sun, is based upon ray theory, which assumes that waves travel with an infinitely high frequency, in much the same way as light rays propagating through a medium with smoothly varying refractive index, occasionally encountering a sharp interface. Under this approximation the seismic energy propagates along infinitesimally narrow geometric “ray paths.”
The beginning of the seismological inv estigation of the Moon dates back to the beginning of space flight: A working group implemented by NASA in 1959 (Hall, 1977) suggested the development of a seismometer for a hard landing on the Moon. This resulted in Ranger missions 3 to 5, which all unfortunately failed for technical reasons (Hall, 1977). The first measurement of elastic properties of lunar soil was conducted by the Surveyor landers a few years later (Christensen et al., 1968). Besides these early attempts, seismological studies of the Moon divide into two phases: The first one saw the installation of a seismometer network on the Moon, starting with Apollo 11 on July 20, 1969 (Apollo 11 Mission Report, 1969), followed by the collection of continuous data until network shutdown on September 30, 1977 (Bates et al., 1979), and, in parallel and ongoing until the early 1990s, the analysis of the data. The second phase began in the late 1990s, when cheap computer power allowed for massive data processing on desktop workstations and the application of new methods. This chapter aims to give a sketch of the Moon as it results from these two phases. The following sections will first describe the different types of seismic events observed on the Moon, and then detail the structure of the main layers of the lunar interior, i.e., the crust, mantle, and core. A summary section finally gives an overview of the present-day concept of the interior structure of the Moon. Seismic sources and seismicity Both endogenous and exogenous sources create seismic waves on the Moon. It is thus common to speak of “events” rather than “quakes,” unless the type of source has been identified. However, analysis of the spatial and chronological distribution of events, and of seismogram characteristics, leads to the identification of several classes of sources.
