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will be shifted to smaller time-lags tand thet<0 ridges will be shifted to more
negativet. Thus a
ow will break the symmetry between the t >0 andt <016 Gizon, Birch & Spruit
parts of the cross-covariance. In contrast, a horizontally uniform change in, e.g.,
sound speed would introduce a time-symmetric change in the cross-covariance.
Several techniques have been proposed to measure travel times from the cross-
covariance. The (phase) travel times for inward- and outward-going waves are
measured by tting a Gabor wavelet to the two branches of the cross-covariance
(Duvall et al. 1997, Kosovichev & Duvall 1997). The travel times can also be
measured with a simple one-parameter t (Gizon & Birch 2002, 2004), as is done
in geophysics (e.g. Marquering, Dahlen & Nolet 1999).
The cross-covariance function computed between two spatial points is in general
very noisy. Spatial averaging is a useful tool to reducing random noise. Duvall
et al. (1997) considered an averaging scheme whereby the cross-covariance is
computed between a point and a concentric annulus or quadrants of arc. For
example, the cross-covariance between a point and an annulus is used to study
waves that propagate outward from the central point to the annulus (positive
time-lag) and inward (negative time-lag). The dierence between inward and
outward travel times is sensitive to the horizontal divergence of the local
ow
or to a local vertical
ow, while the average travel time is sensitive to the local
wave speed. Similarly, the covariance between a point and a north quadrant is
used to study waves that propagate either northward or southward. The various
combinations of travel times are shown in Figure 7 for annulus radii ranging
from 12 Mm to 27 Mm. Each travel-time map is obtained by translating the
central point of the annulus. With only a few hours of averaging, it is clear that
the travel-times dierences are sensitive to the supergranulation
ows.
Finally, the travel-time maps must be inverted (the inverse problem). This
requires a model for the relationship between the travel-time perturbations and
perturbations in solar properties (the forward problem). Recent progress regard-
ing the interpretation of traveltimes is described in Section 4.
3.4 Helioseismic Holography
Helioseismic holography (Lindsey & Braun 1997) and acoustic imaging (Chang
et al. 1997), which are virtually indistinguishable, are closely related to the time-
distance method. In both of these methods, observations of the waveeld (e.g.
Doppler velocity) at the solar surface are used to estimate the waveeld in the so-
lar interior. Separate estimates are constructed by computationally evolving the
observed waveeld either forwards or backwards in time. In helioseismic holog-
raphy, these two estimates are called the ingression ( H, propagates forwards in
time) and egression ( H+, propagates backwards in time) and are computed as
H(r;z;!) =Z
Pd2r0G(kr0rk;z;!)(r0;!); (8)Local Helioseismology 17
where ris the horizontal focus position, zis the focus depth, and the integration
over surface positions r0is carried out over the region described by the "pupil"
P. The functions Gare causal (subscript ) and anti-causal (subscript +)
Green's functions. These Green's functions can be thought of as propagators
which (approximately) evolve the waveeld either forwards or backwards in time.
The amplitude and phase of the correlation between the two estimates H
contain information regarding wave propagation in the Sun (e.g. Lindsey &
Braun 1997, 2000). For example, in farside imaging (Section 9), the phase of
the ingression-egression correlation is used to detect active regions.
3.5 Direct Modeling
Direct modeling (Woodard 2002, 2007) is a method for interpreting correlations in
the waveeld. These correlations, however, are not measured in real space but in
the Fourier domain. For example, any steady heterogeneity in the Sun is expected
to introduce correlations between incoming and scattered waves with dierent
wave vectors but the same frequency. Unlike time-distance helioseismology, direct
modeling can also treat time-varying perturbations, which couple waves with
dierent frequencies. One of the main characteristics of direct modeling, as its
name suggests, is that it does not produce any intermediate data products (e.g.
no travel time map) as the inversions are carried out directly on the correlations.
3.6 Fourier-Hankel Analysis
Fourier-Hankel analysis (Braun, Duvall & Labonte 1987) was specically designed
to study the waveeld around sunspots. The analysis is carried out in a cylindrical
coordinate system with origin centered on the sunspot. The waveeld observed
in an annular region around the sunspot is decomposed into inward and outward
propagating components, using a Fourier-Hankel transform. The amplitudes and
the phases of the incoming and outgoing waves can be compared in order to
characterize the interaction of the waves with the sunspot. In particular Fourier-
Hankel analysis was the rst method to measure the absorption coecient of
incoming waves by a sunspot (Braun, Duvall & Labonte 1987), dened by ( Pin
Pout)=Pin, wherePinandPoutare respectively the incoming and outgoing powers.
In addition, the Fourier-Hankel method has been used to measure the phase shift
between the incoming and outgoing waves, as well as the scattering from one
mode to another (Braun 1995).18 Gizon, Birch & Spruit
4 THE FORWARD AND INVERSE PROBLEMS
4.1 Weak Perturbation Approximation
Many important solar features can be reasonably approximated, in the context
of wave propagation, as small deviations from a horizontally uniform reference
model. Examples include the supergranulation, meridional
ow, torsional oscil-
lations, and subsurface (but not surface) magnetic elds. For this class of solar
features, linear forward modeling can be employed. The main advantage of lin-
ear forward models is that they lead to linear inverse problems, which is the only
class of inverse problems that can be solved easily.
4.1.1 Sensitivity functions. In linear forward models, the helioseismic
measurements (e.g., travel-time shifts) can be related to weak, steady, perturba-
tions to a reference model through sensitivity functions (also called kernels) by
equations of the form
di=X
Z
K
i(x)q(x)d3x; (9)
where thedistand for an arbitrary set of helioseismic measurements (for ex-
ample travel-time perturbations i), the functions qdescribe the deviations
from the reference solar model, and the functions K
iare the corresponding ker-
nel functions. The sum over is over all types of perturbations to the solar

will be shifted to smaller time-lags tand thet<0 ridges will be shifted to more

negativet. Thus a

ow will break the symmetry between the t >0 andt <016 Gizon, Birch & Spruit

parts of the cross-covariance. In contrast, a horizontally uniform change in, e.g.,

sound speed would introduce a time-symmetric change in the cross-covariance.

Several techniques have been proposed to measure travel times from the cross-

covariance. The (phase) travel times for inward- and outward-going waves are

measured by tting a Gabor wavelet to the two branches of the cross-covariance

(Duvall et al. 1997, Kosovichev & Duvall 1997). The travel times can also be

measured with a simple one-parameter t (Gizon & Birch 2002, 2004), as is done

in geophysics (e.g. Marquering, Dahlen & Nolet 1999).

The cross-covariance function computed between two spatial points is in general

very noisy. Spatial averaging is a useful tool to reducing random noise. Duvall

et al. (1997) considered an averaging scheme whereby the cross-covariance is

computed between a point and a concentric annulus or quadrants of arc. For

example, the cross-covariance between a point and an annulus is used to study

waves that propagate outward from the central point to the annulus (positive

time-lag) and inward (negative time-lag). The dierence between inward and

outward travel times is sensitive to the horizontal divergence of the local

ow

or to a local vertical

ow, while the average travel time is sensitive to the local

wave speed. Similarly, the covariance between a point and a north quadrant is

used to study waves that propagate either northward or southward. The various

combinations of travel times are shown in Figure 7 for annulus radii ranging

from 12 Mm to 27 Mm. Each travel-time map is obtained by translating the

central point of the annulus. With only a few hours of averaging, it is clear that

the travel-times dierences are sensitive to the supergranulation

ows.

Finally, the travel-time maps must be inverted (the inverse problem). This

requires a model for the relationship between the travel-time perturbations and

perturbations in solar properties (the forward problem). Recent progress regard-

ing the interpretation of traveltimes is described in Section 4.

3.4 Helioseismic Holography

Helioseismic holography (Lindsey & Braun 1997) and acoustic imaging (Chang

et al. 1997), which are virtually indistinguishable, are closely related to the time-

distance method. In both of these methods, observations of the waveeld (e.g.

Doppler velocity) at the solar surface are used to estimate the waveeld in the so-

lar interior. Separate estimates are constructed by computationally evolving the

observed waveeld either forwards or backwards in time. In helioseismic holog-

raphy, these two estimates are called the ingression ( H, propagates forwards in

time) and egression ( H+, propagates backwards in time) and are computed as

H(r;z;!) =Z

Pd2r0G(kr0rk;z;!)(r0;!); (8)Local Helioseismology 17

where ris the horizontal focus position, zis the focus depth, and the integration

over surface positions r0is carried out over the region described by the "pupil"

P. The functions Gare causal (subscript ) and anti-causal (subscript +)

Green's functions. These Green's functions can be thought of as propagators

which (approximately) evolve the waveeld either forwards or backwards in time.

The amplitude and phase of the correlation between the two estimates H

contain information regarding wave propagation in the Sun (e.g. Lindsey &

Braun 1997, 2000). For example, in farside imaging (Section 9), the phase of

the ingression-egression correlation is used to detect active regions.

3.5 Direct Modeling

Direct modeling (Woodard 2002, 2007) is a method for interpreting correlations in

the waveeld. These correlations, however, are not measured in real space but in

the Fourier domain. For example, any steady heterogeneity in the Sun is expected

to introduce correlations between incoming and scattered waves with dierent

wave vectors but the same frequency. Unlike time-distance helioseismology, direct

modeling can also treat time-varying perturbations, which couple waves with

dierent frequencies. One of the main characteristics of direct modeling, as its

name suggests, is that it does not produce any intermediate data products (e.g.

no travel time map) as the inversions are carried out directly on the correlations.

3.6 Fourier-Hankel Analysis

Fourier-Hankel analysis (Braun, Duvall & Labonte 1987) was specically designed

to study the waveeld around sunspots. The analysis is carried out in a cylindrical

coordinate system with origin centered on the sunspot. The waveeld observed

in an annular region around the sunspot is decomposed into inward and outward

propagating components, using a Fourier-Hankel transform. The amplitudes and

the phases of the incoming and outgoing waves can be compared in order to

characterize the interaction of the waves with the sunspot. In particular Fourier-

Hankel analysis was the rst method to measure the absorption coecient of

incoming waves by a sunspot (Braun, Duvall & Labonte 1987), dened by ( Pin

Pout)=Pin, wherePinandPoutare respectively the incoming and outgoing powers.

In addition, the Fourier-Hankel method has been used to measure the phase shift

between the incoming and outgoing waves, as well as the scattering from one

mode to another (Braun 1995).18 Gizon, Birch & Spruit

4 THE FORWARD AND INVERSE PROBLEMS

4.1 Weak Perturbation Approximation

Many important solar features can be reasonably approximated, in the context

of wave propagation, as small deviations from a horizontally uniform reference

model. Examples include the supergranulation, meridional

ow, torsional oscil-

lations, and subsurface (but not surface) magnetic elds. For this class of solar

features, linear forward modeling can be employed. The main advantage of lin-

ear forward models is that they lead to linear inverse problems, which is the only

class of inverse problems that can be solved easily.

4.1.1 Sensitivity functions. In linear forward models, the helioseismic

measurements (e.g., travel-time shifts) can be related to weak, steady, perturba-

tions to a reference model through sensitivity functions (also called kernels) by

equations of the form

di=X

Z

K

i(x)q(x)d3x; (9)

where thedistand for an arbitrary set of helioseismic measurements (for ex-

ample travel-time perturbations i), the functions qdescribe the deviations

from the reference solar model, and the functions K

iare the corresponding ker-

nel functions. The sum over is over all types of perturbations to the solar