˜second bounce™ ridge). The upper ridges correspond to waves with multi-

ple re¬‚ections from the surface. The cross-covariance function represents a

˜helioseismogram™.

For an unperturbed solar model the cross-covariance function can be rep-

resented by a superposition of wave packets (Kosovichev & Duvall 1997):

2

δω 2

∆ ∆

Ψ(„, ∆) ∝ „’ exp ’ „’

cos ω0 . (19.2)

v 4 u

δv

Here δv is a narrow interval of the phase speed, v = ωnl /L, where L =

l(l + 1), u ≡ (‚ω/‚kh ) is the horizontal component of the group velocity,

kh = 1/L is the angular component of the wave vector, and ω0 is the central

frequency of a Gaussian frequency ¬lter applied to the oscillation data, and

δω is the characteristic width of the ¬lter. The phase and group travel

times are measured by ¬tting individual terms of eq. (19.2), represented

by a Gabor-type wavelet, to the observed cross-covariance function using a

least-squares technique.

Usually the phase travel time is measured more accurately and, thus, is

more commonly used for inferring internal properties of the Sun. Typically

the measured variations of the travel time do not exceed 5%. These vari-

ations can be related to perturbations of the sound speed δc/c and ¬‚ow

velocity U via a linear integral equation:

δc

δ„ = Kc dV + (K u U )dV, (19.3)

c

V V

where Kc (r) and Ku (r) are sensitivity functions for δc/c and U ; the inte-

gration is over the interior volume, V .

The sensitivity kernels are derived by using some approximations, e.g.

the ray theory or the ¬rst Born approximation. In the Born approxima-

tion, the sensitivity kernels can be expressed in terms of the unperturbed

eigenfunctions of solar oscillation modes (Birch & Kosovichev 2000). Exam-

ples of the Born sensitivity kernels for sound-speed perturbations are shown

in Fig. 19.2. Calculations of these kernels involve double summation over

a large set of normal modes, and represent a signi¬cant computing task.

For su¬ciently large distances the kernels are relatively simple (Fig. 19.2a)

and can be approximated by a semi-analytical formula (Jensen et al. 2000).

However, for shorter distances and multiple bounces (Fig. 19.2b) the Born

kernels are quite complicated.

One unexpected feature of the ¬rst-bounce kernels calculated in the Born

Telechronohelioseismology 283

Fig. 19.2. Samples of the travel time sensitivity function in the Born approximation

for (a) a direct waves and (b) a second-bounce waves. The solid curves show the

corresponding ray paths.

approximation is that these kernels have zero values along the ray path. Such

kernels are called ˜banana-doughnut kernels™ by Marquering et al. (1999) and

Dahlen et al. (2000). This curious property is related to a ˜wave-front heal-

ing™ e¬ect (Hung et al. 2001) illustrated in Fig. 19.3 which shows the results

of numerical simulations of propagation of a spherical wave from a point

source through a localized positive sound-speed perturbation. After scat-

tering on the perturbation, the wave front is split into an accelerated direct

wave and a retarted di¬racted wave. However, later the structure of the

wave front is gradually restored and becomes close to the original spherical

shape. The wave healing can be explained by the Huygens principle accord-

ing to which the sharp edges of the perturbed wave front generate secondary

waves that ¬ll in the break in the wave front caused by the perturbation.

The simple numerical model of Hung et al. was used by Birch et al. (2001)

to test the accuracy of the Born and ray approximation (Fig. 19.4). The

results (Fig. 19.4) show that for typical perturbations in the solar interior

the Born approximation is su¬ciently accurate, while the ray approximation

signi¬cantly overestimates the travel times for perturbations smaller than

the size of the ¬rst Fresnel zone. That means that the inversion results based

on the ray theory may signi¬cantly underestimate the strength of the small-

scale perturbations. Recently, Gizon & Birch (2002) obtained sensitivity

kernels for distributed sources.

Kosovichev

284

Fig. 19.3. Numerical simulation illustrating the wave-front healing e¬ect. A circular

wave initiated by a point source at the left boundary propagates through a strong

(50%) localized sound-speed spherical perturbation (the white object in the middle).

The perturbed wave front consists of two parts, an accelerated direct wave and a

di¬racted wave which lags behind. At larger distances the wave becomes close to

the original spherical shape again.

Fig. 19.4. Tests of the ray and Born approximations: travel-time perturbations

due to spherically symmetric sound-speed perturbations, as functions of the radius

of the perturbing region. The sound-speed perturbation has a cosine-bell pro¬le,

with maximum amplitude A and half-width-at-half-maximimum R. The solid lines

are the numerical results. The dashed curves are the travel-time perturbations

calculated in the Born approximation and the dotted lines are those calculated in

the ¬rst-order ray approximation. The left panel is for two perturbations of the

relative amplitude A = ±0.05. The right panel is for A = ±0.1. (Birch et al. 2001).

Telechronohelioseismology 285

Fig. 19.5. Samples of travel-time sensitivity functions in the ray approximation for

two schemes of calculations of the cross-covariance function: (a) “surface-focusing”

scheme in which the central-area oscillation signal is the cross-correlated with the

signals of surrounding annuli; (b) the “deep-focusing” scheme in which the signals

of the opposite sides of an annulus are cross-correlated. The light colour of the

sensitivity functions corresponds to high absolute values of the sensitivity functions,

and darker colour corresponds to the low values. Each picture shows the sensitivity

functions for two distances.

In the ray-theoretical approximation, which is still often used for helio-

seismic inferences, the ¬rst-order perturbations to the phase travel time is

given by (Kosovichev & Duvall 1997):

(n · U ) δc (k · cA)2

c2

2

δωc ωc 1

δ„ = ’ ’

A

+ S+ + S ds,

c2 ω 2 c2 S 2 c2 k2 c2

c ωc

“

(19.4)

where n is a unit vector tangent to the ray, S = k/ω is the phase slowness, k

is the wave vector and k its magnitude, ωc is the acoustic cut-o¬ frequency,

cA = B/ 4πρ is the vector Alfv´n velocity, B is the magnetic ¬eld

e

strength, c is the adiabatic sound speed, and ρ is the plasma density. The

integration is performed along the unperturbed ray path “ according to

Fermat™s principle (e.g. Gough 1993). The e¬ects of ¬‚ows and structural

perturbations are separated by taking the di¬erence and the mean of the

reciprocal travel times. Magnetic ¬eld causes anisotropy of the mean travel

times, which allows us to separate, in principle, the magnetic e¬ects from

the variations of the sound speed (or temperature).

The travel time measured at a point on the solar surface is the result of the

cumulative e¬ects of the perturbations in each of the traversed rays of the

3D ray systems. This pattern is then translated for di¬erent surface points

in the observed area, so that overall the travel times are sensitive to all

subsurface points in some depth range. In a “surface-focusing” scheme the

travel time is measured between a small central area and a set of surrounding

Kosovichev

286