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model. A complete set of qincludes, e.g., pressure, density, sound speed ( c),

ow velocity ( u), and magnetic eld vector ( B). The integration variable xis a
three-dimensional position vector and runs over the entire reference solar model.
4.1.2 The ray approximation. The ray approximation, in which the
wavelength is approximated as small compared to the other length scales in the
problem (e.g., the scale heights of the reference model, and the length scales of
the perturbations q) has been used extensively in time-distance helioseismol-
ogy (e.g. Kosovichev 1996, Kosovichev & Duvall 1997) to compute the travel-time
sensitivity functions K. A ray sensitivity kernel for the travel-time perturbation
(r1;r2), is zero everywhere except along the ray, , that goes from r1tor2
(see Section3.2).
The starting point is the local dispersion relation:
(!ktu)2=c2k2
t+!2
ac; (10)
where!is the angular frequency, ktis the three-dimensional total wave vector, u
is the vector
ow, cis the sound speed, and !acis the acoustic cut-o frequency.
In the ray approximation the travel time is given by the path integral of the phase
slowness vector kt=!. Under the assumption that !is constant, the travel timeLocal Helioseismology 19
perturbation is given by
=1
!Z
kt^nds; (11)
where ^nis the unit vector along and ktis the perturbation to the total wave
vector caused by the
ow u, the change in the sound speed, c, and the change
in the acoustic cut-o frequency, !ac. Notice that the ray path, , is assumed
to be unchanged to rst order. Although the ray approximation has played an
important role in local helioseismology, it does not account for nite wavelength
eects and other complications (cf. Birch et al. 2009, Bogdan 1997).
4.1.3 The rst Born approximation. In the rst-order Born approxima-
tion, the perturbations qcause a perturbation to the wave eld that is due to
single scattering only. In this approximation, rst-order nite wavelength eects,
such as Fresnel zones and wavefront healing, are included (e.g., Hung, Dahlen &
Nolet 2001). The Born approximation has been used in the seismology of the
Earth (e.g., in the search for mantle plumes by Montelli et al. 2004). Gizon &
Birch (2002) give a detailed theoretical treatment of the application of the Born
approximation to problems in time-distance helioseismology.
Born kernels for the eects of sound-speed changes on travel times have been
obtained by, e.g., Birch & Kosovichev (2000) and Jensen, Jacobsen & Christensen-
Dalsgaard (2000). Born kernels for
ows have been computed for time-distance
helioseismology by Gizon, Duvall & Larsen (2000), Birch et al. (2007), and Jack-
iewicz et al. (2007). Three-dimensional sensitivity kernels for
ows and ring-
diagram analysis are given by Birch & Gizon (2007). Although magnetic pertur-
bations are expected to be small just a few hundred km below the surface (Gizon,
Hanasoge & Birch 2006), corresponding kernels have not been obtained yet.
Figure 8 shows slices through an example travel-time sensitivity kernel for
perturbations in the squared sound-speed ( di=andq=c2in Equation 9).
The main features are that the kernel is zero along the ray path, is maximum
(absolute value) in a shell around the ray path, and shows substantial ringing.
These kernels have been called \banana-doughnut kernels" in the context of seis-
mology of the Earth (Marquering, Dahlen & Nolet 1999).The zero along the ray
path is due to the lack of a geometrical delay (change in path length) for small
scatterers located on the reference ray path. In the solar case, there are additional
hyperbolic features across the ray path, due to the presence of distant sources
(Gizon & Birch 2002).
The rst-order Born approximation have been tested using exact solutions (e.g.
Gizon, Hanasoge & Birch 2006) and numerical simulations (e.g. Baig, Dahlen
& Hung 2003; Birch & Felder 2004; Birch et al. 2001; Hung, Dahlen & Nolet
2000) for simple cases. Duvall, Birch & Gizon (2006) used solar observations
to construct two-dimensional travel-time kernels using small (sub-wavelength)
magnetic features as point scatterers.20 Gizon, Birch & Spruit
4.1.4 Inversions and Resolution kernels. Linear inversions have been
developed for ring-diagram analysis and time-distance helioseismology, based
on experience gained from global helioseismology (e.g., Christensen-Dalsgaard,
Schou & Thompson 1990, and references therein). Two dierent inversion proce-
dures are commonly used.
Let us consider 1D depth inversions for the sake of simplicity. The rst in-
version method, called Regularized Least Squares (RLS, e.g., Haber et al. 2000,
Kosovichev 1996), is simply a t to the observational data diunder conditions
of smoothness. The second inversion method, called Optimally Localized Aver-
aging (OLA, Haber et al. 2004), looks for a linear combination of the kernels (an
averaging kernel) that is spatially localized around a target depth, z0. Both
methods build averaging kernels of the form
K(z;z0) =X
ici(z0)K
i(z) (12)
where theciare coecients to be determined. A regularization is applied to
ensure that the error on the inferred qis not too large, or that qis smooth.
In addition, the cross-talk between the inferred qand all other quantities q,
6=, should be avoided. Inversions require a good knowledge of the noise
covariance matrix of the measurements, which can be estimated directly form
the data (spatial averaging) or from a model (Gizon & Birch 2004).
The RLS and OLA methods give similar answers, although the RLS averag-
ing kernels are perhaps more likely to have undesired sidelobes near the surface
(Haber et al. 2004). Figure 9 shows example RLS averaging kernels for ring-
diagram analysis, in the near-surface layers. These particular kernels are used
to infer horizontal
ows. Very similar averaging kernels are obtained for time-
distance helioseismology (e.g. Jackiewicz, Gizon & Birch 2008).
Most time-distance inversions assume that the kernels are invariant by hori-
zontal translation, so that the horizontal convolution of the kernels with the q
becomes a multiplication in Fourier space. This property is used to speed-up the
inversions (Jacobsen et al. 1999).
Recent progress includes the inclusion of the Born kernels in the inversions
and the noise correlations (e.g. Couvidat et al. 2005). An OLA inversion for the

model. A complete set of qincludes, e.g., pressure, density, sound speed ( c),

ow velocity ( u), and magnetic eld vector ( B). The integration variable xis a

three-dimensional position vector and runs over the entire reference solar model.

4.1.2 The ray approximation. The ray approximation, in which the

wavelength is approximated as small compared to the other length scales in the

problem (e.g., the scale heights of the reference model, and the length scales of

the perturbations q) has been used extensively in time-distance helioseismol-

ogy (e.g. Kosovichev 1996, Kosovichev & Duvall 1997) to compute the travel-time

sensitivity functions K. A ray sensitivity kernel for the travel-time perturbation

(r1;r2), is zero everywhere except along the ray, , that goes from r1tor2

(see Section3.2).

The starting point is the local dispersion relation:

(!ktu)2=c2k2

t+!2

ac; (10)

where!is the angular frequency, ktis the three-dimensional total wave vector, u

is the vector

ow, cis the sound speed, and !acis the acoustic cut-o frequency.

In the ray approximation the travel time is given by the path integral of the phase

slowness vector kt=!. Under the assumption that !is constant, the travel timeLocal Helioseismology 19

perturbation is given by

=1

!Z

kt^nds; (11)

where ^nis the unit vector along and ktis the perturbation to the total wave

vector caused by the

ow u, the change in the sound speed, c, and the change

in the acoustic cut-o frequency, !ac. Notice that the ray path, , is assumed

to be unchanged to rst order. Although the ray approximation has played an

important role in local helioseismology, it does not account for nite wavelength

eects and other complications (cf. Birch et al. 2009, Bogdan 1997).

4.1.3 The rst Born approximation. In the rst-order Born approxima-

tion, the perturbations qcause a perturbation to the wave eld that is due to

single scattering only. In this approximation, rst-order nite wavelength eects,

such as Fresnel zones and wavefront healing, are included (e.g., Hung, Dahlen &

Nolet 2001). The Born approximation has been used in the seismology of the

Earth (e.g., in the search for mantle plumes by Montelli et al. 2004). Gizon &

Birch (2002) give a detailed theoretical treatment of the application of the Born

approximation to problems in time-distance helioseismology.

Born kernels for the eects of sound-speed changes on travel times have been

obtained by, e.g., Birch & Kosovichev (2000) and Jensen, Jacobsen & Christensen-

Dalsgaard (2000). Born kernels for

ows have been computed for time-distance

helioseismology by Gizon, Duvall & Larsen (2000), Birch et al. (2007), and Jack-

iewicz et al. (2007). Three-dimensional sensitivity kernels for

ows and ring-

diagram analysis are given by Birch & Gizon (2007). Although magnetic pertur-

bations are expected to be small just a few hundred km below the surface (Gizon,

Hanasoge & Birch 2006), corresponding kernels have not been obtained yet.

Figure 8 shows slices through an example travel-time sensitivity kernel for

perturbations in the squared sound-speed ( di=andq=c2in Equation 9).

The main features are that the kernel is zero along the ray path, is maximum

(absolute value) in a shell around the ray path, and shows substantial ringing.

These kernels have been called \banana-doughnut kernels" in the context of seis-

mology of the Earth (Marquering, Dahlen & Nolet 1999).The zero along the ray

path is due to the lack of a geometrical delay (change in path length) for small

scatterers located on the reference ray path. In the solar case, there are additional

hyperbolic features across the ray path, due to the presence of distant sources

(Gizon & Birch 2002).

The rst-order Born approximation have been tested using exact solutions (e.g.

Gizon, Hanasoge & Birch 2006) and numerical simulations (e.g. Baig, Dahlen

& Hung 2003; Birch & Felder 2004; Birch et al. 2001; Hung, Dahlen & Nolet

2000) for simple cases. Duvall, Birch & Gizon (2006) used solar observations

to construct two-dimensional travel-time kernels using small (sub-wavelength)

magnetic features as point scatterers.20 Gizon, Birch & Spruit

4.1.4 Inversions and Resolution kernels. Linear inversions have been

developed for ring-diagram analysis and time-distance helioseismology, based

on experience gained from global helioseismology (e.g., Christensen-Dalsgaard,

Schou & Thompson 1990, and references therein). Two dierent inversion proce-

dures are commonly used.

Let us consider 1D depth inversions for the sake of simplicity. The rst in-

version method, called Regularized Least Squares (RLS, e.g., Haber et al. 2000,

Kosovichev 1996), is simply a t to the observational data diunder conditions

of smoothness. The second inversion method, called Optimally Localized Aver-

aging (OLA, Haber et al. 2004), looks for a linear combination of the kernels (an

averaging kernel) that is spatially localized around a target depth, z0. Both

methods build averaging kernels of the form

K(z;z0) =X

ici(z0)K

i(z) (12)

where theciare coecients to be determined. A regularization is applied to

ensure that the error on the inferred qis not too large, or that qis smooth.

In addition, the cross-talk between the inferred qand all other quantities q,

6=, should be avoided. Inversions require a good knowledge of the noise

covariance matrix of the measurements, which can be estimated directly form

the data (spatial averaging) or from a model (Gizon & Birch 2004).

The RLS and OLA methods give similar answers, although the RLS averag-

ing kernels are perhaps more likely to have undesired sidelobes near the surface

(Haber et al. 2004). Figure 9 shows example RLS averaging kernels for ring-

diagram analysis, in the near-surface layers. These particular kernels are used

to infer horizontal

ows. Very similar averaging kernels are obtained for time-

distance helioseismology (e.g. Jackiewicz, Gizon & Birch 2008).

Most time-distance inversions assume that the kernels are invariant by hori-

zontal translation, so that the horizontal convolution of the kernels with the q

becomes a multiplication in Fourier space. This property is used to speed-up the

inversions (Jacobsen et al. 1999).

Recent progress includes the inclusion of the Born kernels in the inversions

and the noise correlations (e.g. Couvidat et al. 2005). An OLA inversion for the