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2007). The larger the horizontal phase speed of the mode, the deeper the sen-
sitivity. Thus large patches give access to deeper regions in the Sun than small
patches. The dierences between the tted mode frequencies, !, and the mode
frequencies calculated from a standard solar model, !n(k), are used in one di-
mensional (depth) inversions to infer structural conditions under each patch (e.g.
Basu, Antia & Bogart 2004). Two independent structural quantities can be in-
verted at a time, e.g., sound speed and density, from which other quantities can
be inferred, such as the rst adiabatic exponent. Similarly, the depth dependence
of the horizontal
ows, uxanduy, can be inferred from a set of tted parameters
uxanduy(Hill 1989). Typically, ring analysis is used to probe the top 30 Mm of
the convection zone, with a maximum horizontal resolution of about 2near the
surface. Three-dimensional maps can then be obtained by combining neighboring
patches.14 Gizon, Birch & Spruit
3.2 The Cross-Covariance Function
Time-distance helioseismology is based on the measurement of the cross-covariance
between the Doppler signals at two points r1andr2on the solar surface,
C(r1;r2;t) =ZT
0dt0(r1;t0)(r2;t0+t); (5)
wheretis the correlation time-lag. Figure 6ashows a cross-covariance function
measured from 144 days of MDI medium-degree data. The cross-covariance has
been averaged over many pairs of points ( r2;r1) and is presented as a function
of the heliocentric angle between these two points. This diagram is known as
the "time-distance diagram." The cross-covariance is essentially a phase coherent
average of the random oscillations (Bogdan 1997, and Supplemental Movie 2 ).
It is a solar seismogram: it provides a way to measure wave travel times between
two surface locations.
A particular wave packet (consisting of set of modes with similar phase speeds)
is preferentially selected for each travel distance. Many applications involve much
less temporal and spatial averaging than was used in this gure. Typical cross-
covariances are therefore much noisier than the example shown here. The deeper
meaning of the cross-covariance was elucidated only recently in terms of Green's
functions (see Side Bar ).
An important tool for visualizing wave propagation in the Sun is the ray ap-
proximation (e.g. Kosovichev & Duvall 1997). In this approximation, the wave-
length is treated as if it were much smaller than the length scales associated
with the variations in the background solar model. The ray paths describe the
propagation of wave energy and are analogous to the rays in geometrical optics.
For discussions of the range of validity of the ray approximation see e.g. Hung,
Dahlen & Nolet (2000) and Birch et al. (2001).
Figure 6bshows some example ray paths computed from Model S. In this
gure, the rays all begin from the same point at the solar surface. Downward
propagating rays are refracted by the increase of the sound speed with depth
until they reach their lower turning where the horizontal phase speed matches
the sound speed. At frequencies below about 5 :3 mHz, upwards propagating
rays are re
ected from the solar surface by the sharp rise in the acoustic cuto
frequency. At higher frequencies, the waves escape into the solar atmosphere.
The main features in the time-distance diagram ( Figure 6a) are the ridges
which correspond to dierent paths that wave energy takes between pairs of ob-
servation points. For example, the blue line corresponds to "rst-bounce" arrivals
(i.e., waves that visit their lower turning points once between the two observation
points). The ne structure of the ridges in the time-dierence diagram re
ects
the band-limited nature of the power spectrum. The majority of the wave power
is near 3 mHz and as a result the cross-covariance shows ne structure that hasLocal Helioseismology 15
a period of about 5 min. The other ridges seen in the time-distance diagram cor-
respond to multiple bounces. One particularly important ray path is the third
bounce ray path (green 30) that travels to the farside of the Sun before returning
to the visible disk, this ray path plays a central role in farside imaging (Section 9).
As can be seen in the example ray paths in Figure 6b, the depth of the lower
turning point increases with the distance the ray travels in a single skip.
We note that the cross-covariance is directly related to the local power spectrum
in the case when the medium is assumed to be horizontally invariant over this
local area. In this case, the cross-covariance is simply given by the inverse Fourier
transform of the local power spectrum (Gizon & Birch 2002):
C(r1;r2;t) = const:Z
d2kZ
d!P(k;!)eik(r2r1)i!t: (6)
Changes in the local power spectrum, such as those discussed above in the context
of ring-diagram analysis, will aect the cross-covariance.
3.3 Time-Distance Helioseismology
Time-distance helioseismology (Duvall et al. 1993, 1997, Kosovichev 1996) con-
sists of (i) measuring wave travel times from the cross-covariance function, and
(ii) inverting the travel times to infer the solar subsurface structure and
ows.
As discussed in the Side Bar , the cross-covariance C(r1;r2;t) is closely re-
lated to a Green's function that gives the wave response at ( r2;t) to a source
located at ( r1;t= 0). Thus the cross-covariance is sensitive to the wave propa-
gation conditions (structure and
ows) between the two surface points r1andr2.
The sensitivity of the cross-covariance to any particular local change in the solar
interior is a non-trivial research topic, because the local wavelength of solar oscil-
lations is not necessarily small compared to the length scales of the heterogenities
in the Sun. In addition, this sensitivity depends strongly on the combination of
waves that contribute to the cross-covariance function. This will be discussed in
Section 4.
For the sake of simplicity, consider a constant horizontal
ow u. According
to Equations 6 and 4, the eect of such a
ow is a Galilean translation of the
unperturbed (no
ow) cross-covariance, C0, according to:
C(r1;r2;t) =C0(r1;r2ut;t): (7)
In practise, this result is only an approximation because the power spectrum is
often subject to additional ltering, which is not included in Equation 6 (Gizon
& Birch 2002). The waves travel faster along the
ow than against the
ow. If
uis directed from r1tor2, then thet >0 ridges of the time-distance diagram

2007). The larger the horizontal phase speed of the mode, the deeper the sen-

sitivity. Thus large patches give access to deeper regions in the Sun than small

patches. The dierences between the tted mode frequencies, !, and the mode

frequencies calculated from a standard solar model, !n(k), are used in one di-

mensional (depth) inversions to infer structural conditions under each patch (e.g.

Basu, Antia & Bogart 2004). Two independent structural quantities can be in-

verted at a time, e.g., sound speed and density, from which other quantities can

be inferred, such as the rst adiabatic exponent. Similarly, the depth dependence

of the horizontal

ows, uxanduy, can be inferred from a set of tted parameters

uxanduy(Hill 1989). Typically, ring analysis is used to probe the top 30 Mm of

the convection zone, with a maximum horizontal resolution of about 2near the

surface. Three-dimensional maps can then be obtained by combining neighboring

patches.14 Gizon, Birch & Spruit

3.2 The Cross-Covariance Function

Time-distance helioseismology is based on the measurement of the cross-covariance

between the Doppler signals at two points r1andr2on the solar surface,

C(r1;r2;t) =ZT

0dt0(r1;t0)(r2;t0+t); (5)

wheretis the correlation time-lag. Figure 6ashows a cross-covariance function

measured from 144 days of MDI medium-degree data. The cross-covariance has

been averaged over many pairs of points ( r2;r1) and is presented as a function

of the heliocentric angle between these two points. This diagram is known as

the "time-distance diagram." The cross-covariance is essentially a phase coherent

average of the random oscillations (Bogdan 1997, and Supplemental Movie 2 ).

It is a solar seismogram: it provides a way to measure wave travel times between

two surface locations.

A particular wave packet (consisting of set of modes with similar phase speeds)

is preferentially selected for each travel distance. Many applications involve much

less temporal and spatial averaging than was used in this gure. Typical cross-

covariances are therefore much noisier than the example shown here. The deeper

meaning of the cross-covariance was elucidated only recently in terms of Green's

functions (see Side Bar ).

An important tool for visualizing wave propagation in the Sun is the ray ap-

proximation (e.g. Kosovichev & Duvall 1997). In this approximation, the wave-

length is treated as if it were much smaller than the length scales associated

with the variations in the background solar model. The ray paths describe the

propagation of wave energy and are analogous to the rays in geometrical optics.

For discussions of the range of validity of the ray approximation see e.g. Hung,

Dahlen & Nolet (2000) and Birch et al. (2001).

Figure 6bshows some example ray paths computed from Model S. In this

gure, the rays all begin from the same point at the solar surface. Downward

propagating rays are refracted by the increase of the sound speed with depth

until they reach their lower turning where the horizontal phase speed matches

the sound speed. At frequencies below about 5 :3 mHz, upwards propagating

rays are re

ected from the solar surface by the sharp rise in the acoustic cuto

frequency. At higher frequencies, the waves escape into the solar atmosphere.

The main features in the time-distance diagram ( Figure 6a) are the ridges

which correspond to dierent paths that wave energy takes between pairs of ob-

servation points. For example, the blue line corresponds to "rst-bounce" arrivals

(i.e., waves that visit their lower turning points once between the two observation

points). The ne structure of the ridges in the time-dierence diagram re

ects

the band-limited nature of the power spectrum. The majority of the wave power

is near 3 mHz and as a result the cross-covariance shows ne structure that hasLocal Helioseismology 15

a period of about 5 min. The other ridges seen in the time-distance diagram cor-

respond to multiple bounces. One particularly important ray path is the third

bounce ray path (green 30) that travels to the farside of the Sun before returning

to the visible disk, this ray path plays a central role in farside imaging (Section 9).

As can be seen in the example ray paths in Figure 6b, the depth of the lower

turning point increases with the distance the ray travels in a single skip.

We note that the cross-covariance is directly related to the local power spectrum

in the case when the medium is assumed to be horizontally invariant over this

local area. In this case, the cross-covariance is simply given by the inverse Fourier

transform of the local power spectrum (Gizon & Birch 2002):

C(r1;r2;t) = const:Z

d2kZ

d!P(k;!)eik(r2r1)i!t: (6)

Changes in the local power spectrum, such as those discussed above in the context

of ring-diagram analysis, will aect the cross-covariance.

3.3 Time-Distance Helioseismology

Time-distance helioseismology (Duvall et al. 1993, 1997, Kosovichev 1996) con-

sists of (i) measuring wave travel times from the cross-covariance function, and

(ii) inverting the travel times to infer the solar subsurface structure and

ows.

As discussed in the Side Bar , the cross-covariance C(r1;r2;t) is closely re-

lated to a Green's function that gives the wave response at ( r2;t) to a source

located at ( r1;t= 0). Thus the cross-covariance is sensitive to the wave propa-

gation conditions (structure and

ows) between the two surface points r1andr2.

The sensitivity of the cross-covariance to any particular local change in the solar

interior is a non-trivial research topic, because the local wavelength of solar oscil-

lations is not necessarily small compared to the length scales of the heterogenities

in the Sun. In addition, this sensitivity depends strongly on the combination of

waves that contribute to the cross-covariance function. This will be discussed in

Section 4.

For the sake of simplicity, consider a constant horizontal

ow u. According

to Equations 6 and 4, the eect of such a

ow is a Galilean translation of the

unperturbed (no

ow) cross-covariance, C0, according to:

C(r1;r2;t) =C0(r1;r2ut;t): (7)

In practise, this result is only an approximation because the power spectrum is

often subject to additional ltering, which is not included in Equation 6 (Gizon

& Birch 2002). The waves travel faster along the

ow than against the

ow. If

uis directed from r1tor2, then thet >0 ridges of the time-distance diagram