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\noindent
{\big SOI Investigation Proposal}
\bigskip

\noindent
{\bf Title:} Sunspot -- p-Mode Scattering Measurements Using Hankel
and Time-Distance Techniques
\medskip

\noindent
{\bf Lead Investigator:}  Douglas C. Braun
\medskip

\noindent
{\bf Other Team Members:} Thomas L. Duvall, Jr.
\medskip

\noindent
{\bf Abstract / Technical  Summary:}
\medskip

Our principal scientific objective is to measure the $p$-mode
scattering properties of a few selected sunspots and plage regions
using both the ``Hankel decomposition'' analysis and ``time-distance''
analysis. We will primary utilize the full disk Doppler images obtained
by MDI-SOI during the 2 month/year Dynamics program for both the 
Hankel scattering (phase shift) measurements and the 
time-distance (travel time)
mapping. Measurements of $p$-mode absorption by sunspots at very high
degrees will also be made using one or more of the 8 hour ``High
Resolution-Field'' scans made during campaign mode. The use
of both the Hankel and time-distance analyses will enable a 
direct quantitative comparison between the two methods, which is
necessary to understand and interpret the results obtained by
both procedures.

\bigskip
\noindent
{\bf Investigation  Plan:}
\medskip

Active region seismology is 
concerned with the determination and interpretation of the interaction
of the solar acoustic oscillations with near-surface target structures,
such as magnetic flux concentrations, sunspots, and plage.
Recent observations with high spatial resolution and long temporal
duration have enabled the measurement of the scattering matrix
for sunspots and solar active regions as a function
of the mode properties (wavenumber, frequency and azimuthal order).
 From this information one may determine the
amount of $p$-mode absorption, partial-wave phase shift, and mode
mixing introduced by the sunspot. In addition, new techniques
in the ``time-distance'' regime of helioseismology have 
been applied to studying local inhomogeneities, including sunspots.

There are two main parts of our proposed SOI investigation.
First, we will employ the recently developed techniques of 
Braun et al.\ (1992) and Braun (1995) which have 
enabled determination of phase shifts, absorption, and mode mixing 
of $p$-modes which are {\it not} averaged over azimuthal order, wavenumber or 
frequency.  The MDI-SOI data will allow measurements at higher spatial
frequencies and with higher signal-to-noise
levels, which will play a crucial role in the
identification of the absorption mechanism and allow the modeling
of wave speed perturbations below active regions. We will also measure
mode mixing (both in radial and azimuthal order) 
in sunspots and other active regions and construct maps of the 
spatial distribution of the scattering properties for quantitative
comparison with other active region parameters. 

Second, we will perform ``time-distance'' analyses of the same
active regions studied above. 
That travel times could be directly measured was only
discovered recently (Duvall et al. 1993).
Duvall (1995) found that travel times were reduced for waves
traveling through a sunspot with the effect being almost independent
of distance.  In this case the waves were not measured in the spot
at all but by looking at the second time-distance curve.  
Duvall et al. (1995) found travel times also reduced for waves 
traveling through spots and plage but in this case the measurements 
were made from locations in the spots and plage.  The time deficits
decreased with the distance for this study in contrast to the earlier
one.  The technique used in both of these studies will be applied
to the same regions in an attempt to reconcile the various results.

\bigskip
\noindent
{\bf a) Hankel Analysis of Sunspots and Active Regions}
\medskip

In this approach we consider the effect of the sunspot 
upon incident $p$-modes by a Fourier - Hankel decomposition of the acoustic 
waves in a polar coordinate system centered on the sunspot.  The amplitudes and
phases of the scattered $p$-modes provide the complete scattering properties of
the sunspot.  Typically, the Doppler signal 
is interpolated onto a spherical polar coordinate system $(\theta $, $\phi )$ 
with the sunspot situated at one pole $(\theta  = 0)$.  The analysis is 
confined to an annular region, defined by a range of polar angles 
$\theta _{min} < \theta  < \theta _{max}.$
For values of $\theta  \ll \pi $, we may employ Hankel functions as 
approximations to the Legendre function decomposition (Braun 1995).

 From the set of measured amplitudes of the incoming and outgoing modes we
can deduce the scattering properties of the feature (e.g., sunspot) selected.
As in quantum or classical theory, it is customary to characterize the
scattering properties of an object by means of the scattering matrix.
For each temporal 
frequency $\omega$, we define a matrix $\cal S$ so that 
the amplitude $B_i$ of an outgoing mode $i$ (with 
azimuthal order $m_i$ and radial order $n_i$)
depends upon the complete set of incident mode amplitudes $A_j$ as:
$$
B_i(\omega) = \sum_{j} A_{j}(\omega) {\cal S}_{j i}(\omega).
$$
The partial wave phase shifts are simply the arguments
of the elements of $\cal S$:
$$
\delta_{j i} = \arg({\cal S}_{j i}).
$$
If the different incoming wave amplitudes $A_j$ are uncorrelated
then the above phase shift can be determined from the observed amplitudes
by simply taking the average of $\arg (B_i / A_j)$ over some small
range of modes (i.e., the frequency width of a $p$-mode ridge at a
measured wavenumber). 
The arguments of the diagonal components (${\cal S}_{i i}$)
represent the scattering phase shifts presented in Braun et al.\ (1992)
and Braun (1995).
The off-diagonal components of $\cal S$ characterize the amount of
{\it mode mixing} introduced by the scatterer.
For example, depth or azimuthal asymmetries of the spots 
(combined with the stratification of the solar atmosphere itself) 
will couple an incoming mode of given azimuthal order $m$ and
radial order $n$ to outgoing modes with different $m$ 
and $n$ having the same temporal frequency. 

The spatial and temporal resolution of the MDI-SOI instrument
will allow measurements of $p$-mode scattering, mode mixing and
absorption for a wide range of frequency and wavenumber.
We will use observing sequences with the High-Resolution Field
of a few sunspots to measure p-mode absorption 
up to very high degree ($l > 1000$).  
Detailed measurements of the absorption of
$p$-modes by sunspots is very likely to be a crucial element in
identifying the absorption mechanism. Spruit and Bogdan (1992) and
Cally and Bogdan (1993), for example, predict {\it multiple} absorption 
minima along the $p$-mode ridges. As noted by Braun (1995) the observations
tentatively confirm what may be the first minimum at 5 mHz.  Further
analysis of better quality data is required.

As was first noted by Bogdan et al.\ (1993)
it is important to separate the effects of
finite p-mode lifetimes from the absorption measurements. We propose
to do this in the following manner.
Incoming $p$-modes
entering the outer boundary of the annulus most likely have ranges
which in general decrease with increasing degree (we define the range
to be the distance traveled by the mode during its lifetime).
Modes with ranges comparable or smaller than the annulus dimensions
should show smaller values of measured absorption, since their ``local''
nature implies they interact more with the convection in their immediate
environment than with a distant sunspot.
The full disk images obtained during the Dynamics Program will
allow the measurement of $p$-mode absorption as a direct function
of annulus size.  In a preliminary test using the 1988 South Pole data
we repeated the scattering
analysis performed for sunspot group NOAA 5254 with a variety of
different annulus dimensions (Bogdan and Braun 1995). 
It is quite
striking that while $p$-modes with degrees less than about 300 show
very nearly the same levels of absorption over the range of annulus
sizes, this is not the case for higher degree modes which show
a marked decrease in the absorption with mean annulus radius.
The different behavior for modes above and below $l$=300
is consistent with the picture put forward by Bogdan, et al.\ (1993)
which considered $p$-modes below 300 to be ``global'' in nature while
modes with $l$ considerably greater than 300 to be ``local''.
Employing an ``acoustic transfer equation'' (along the lines suggested
by Cally, Bogdan and Zweibel 1994)
it is possible to model
the decrease in $\alpha$ to yield both the true 
absorption (${\alpha}_o)$ produced
by the sunspot and the $p$-mode range, or lifetime. For example, if $I_-$
is the (steady state) acoustic intensity of an incoming mode with 
a range $\Delta$ it is straightforward to show that the outgoing
mode intensity at a given angular distance $\theta$ from the spot
is $I_+(\theta) = I_- (1 - {\alpha}_o e^{{-\theta}/\Delta})$. The measured
absorption as determined by integrating the intensities over an annulus
with mean radius $\overline{\theta} = (\theta _{max} + \theta _{min})/2$ 
has the form $\alpha(\overline{\theta}) =
{\alpha}_o e^{-\overline{\theta}/\Delta}$. 
We stress that in addition to determining the true sunspot absorption,
required for comparisons with models of the absorption mechanism,
the technique will yield measurements of $p$-mode ranges (and knowing
their group velocities, the lifetimes as well) in a manner independent
of interpreting mode line profiles in observed power spectra.
Knowledge of mode amplitudes and lifetimes are
required for a complete understanding of the interaction
of p-modes with convection and its relation to acoustic excitation
in the Sun.

Data taken during the SOI-MDI 2 month/year Dynamics Program
will allow substantial improvement in the measurement of 
$p$-mode scattering (phase shifts and mode mixing) from sunspots.
This database will provide the temporal continuity required
for the phase shift measurements as well as allow a 
range of active region targets.
Scattering phase shifts measured using the Hankel techniques
will also be compared directly with the phase shift inferred
from travel time measurements of acoustic rays isolated by
time-distance analysis (see the next section).
Improved measurements of scattering phase shift by active regions
will be used to model the subsurface structure of the perturbations
in the acoustic properties of the medium introduced by the local
magnetic fields. Initial modeling efforts, which 
solve the inhomogeneous wave equation for the
scattered waves using the standard Green's function method
under the simplification of the Born approximation, have been
highly successful (Fan, Braun, and Chou 1995). It is found that
the variation of scattering phase shifts with degree and radial order
yields direct information about the depth variation of the inhomogeneities
in pressure and wave speed induced by the sunspot.

The discovery of mode mixing in the observed $p$-mode scattering
(Braun 1995a,b)
was an unexpected result and we intend to devote more time to
exploring this interesting phenomenon. We intend to improve the
data reduction techniques to measure the strength of the mode
mixing (rather than just the mode mixing phase shifts). One method
will be to perform a least squares fit of outgoing mode amplitudes
to the appropriate sum of incoming amplitudes, in order to
directly measure the scattering matrix components.
The absence in the SOI data of the degrading effects of the 
terrestrial atmosphere 
will be crucial in performing these
mode mixing measurements, since
it is necessary to know the the true $p$-mode amplitudes 
as a function of degree (seeing effects will reduce the amplitudes
of higher degree modes relative to those at lower degree).
Our models of $p$-mode scattering using the Green's function
solution can also be readily extended to compute the off-diagonal
elements of the scattering matrix for a model sunspot. Thus, by
a comparison of the observed $\cal S$ matrix with those computed
by our ``forward'' modeling effort, we hope to deduce further
constraints on the internal and subsurface conditions of
sunspots. 

It is our goal to measure the absorption and scattering properties
of a variety of solar activity, ranging from plage to pores and
sunspots. A plage observed by Braun (1995) in 1988 South Pole data
showed absorption but no measurable phase shifts. We wish to quantitatively
relate the amount of scattering and absorption to 
parameters of the active regions, most notably mean magnetic flux
(from MDI-SOI magnetograms), and the presence and size
of sunspots.  For one or more complex active regions we plan on spatially
mapping both the absorption and phase shifts (which to first order
may be considered a measure of the sound speed perturbation) using a
procedure developed by Braun, LaBonte and Duvall (1990) which
will be modified to allow phase shift measurements.
In addition to providing a quantitative comparison between the
scattering properties and magnetic field, the
mapping technique can reveal scattering sources which may be associated
with the active regions but do not have a visible surface manifestation.
These maps will also be compared with wave speed maps, constructed from 
travel time measurements discussed in the next section.

\bigskip
\noindent
{\bf b) Time-Distance Analysis}
\medskip

Most work in helioseismology to date uses the measurement and
prediction of the frequencies of the acoustic normal modes
to decipher the solar interior structure.
It has been known for several years that solar activity affects these
frequencies (Woodard and Libbrecht, 1993).
Increased solar activity leads to an increase of the mode frequencies.
One might make the simplification that an increase in the wave speed
in the mode's trapping cavity would cause the normal mode frequency to
increase.
The increase of the normal mode frequencies is a strong function of
frequency but only weakly dependent on the spatial scale of the modes.
This signature is that of a near-surface effect, as the height
in the atmosphere of the reflection of waves is also a function
mostly of frequency.
One possible effect is a change in the depth of the reflecting surface. If the
reflecting surface moves down, the trapping cavity becomes smaller, and the
mode frequency increases. Another way of saying that the reflecting surface
is moving down is to say that the acoustic cutoff frequency is increasing at
a given depth.

A new approach to the study of solar activity uses
measurements of travel times between different surface locations
to attempt to isolate subsurface inhomogeneities.
At first this promising new technique for studying local
effects was used to study global phenomena, in particular
it was found that waves of sufficiently high frequency were
not reflected by the solar atmosphere (Duvall et al. 1993) and
that it was possible to uncover the near-surface atmospheric
structure by measuring the frequency dependence of the
travel times (Jefferies et al. 1994).
The first uses of the technique for local problems were to measure that
sunspots caused a decrease in the travel time through the region
(Duvall 1995),
an inconclusive study of the travel times through active regions
(Korzennik et al. 1995), and
a rudimentary search for subsurface convective motions
(Duvall 1995).

In the present work, we would like to tie up a few of the ``loose ends''
of the previous time-distance work.  Described above, phase shifts
were not observed in a plage region, but travel time deficits are
detected in plage regions (Duvall et al. 1995).  A naive first
thought might be that phase shifts would be related to travel times
through a simple relation like $\phi=\omega t$, where $\phi$ is the
phase shift, $\omega$ is the frequency of the wave, and $t$ is a
travel time.  If one puts in $\phi=90\deg$
(a typical phase shift observed in a sunspot), $\omega=2\pi(3mHz)$,
one finds a time of 83 seconds which is of the order of the travel 
time deficits measured.  By using the different analysis procedures on
the same regions, we could hope to eliminate different regions as
the source of a difference.  
\bigskip

\noindent
{\bf References}
\bigskip

\noindent
Bogdan, T.~J. and Braun, D.~C. 1995 
in {\it Proc.~4th SOHO Workshop: Helioseismology}, 
\indent
ESA SP-376, eds: Domingo, V. et al., (Noordwijk: ESTEC), in press.

\noindent
Bogdan, T.~J., Brown, T.~M., Lites, B.~W., and
Thomas, J.~H. 1993, {\it Ap.~J.}, {\bf 406}, 723.

\noindent
Braun, D.~C. 1995, {\it Ap.~J.}, October 1.

\noindent
Braun, D.~C., LaBonte, B.~J., and Duvall, T.~L.~Jr. 1990,
{\it Ap.~J.}, {\bf 354}, 372.

\noindent
Braun, D.~C., Duvall, T.~L. Jr., LaBonte, B.~J., Jefferies, S.~M., 
Harvey, J.~W., and 
\indent Pomerantz, M.~A. 1992, {\it Ap.~J.}, {\bf 391}, L113.

\noindent
Cally, P.~S., and Bogdan, T.~J. 1993, {\it Ap.~J.}, 
{\bf 402}, 721.

\noindent
Cally, P.~S., Bogdan, T.~J., and Zweibel, E.~G. 1994, {\it Ap.~J.}, 
{\bf 437}, 505.

\noindent
Duvall, T.~L.~Jr., Jefferies, S.~M., Harvey, J.~W. \& Pomerantz, M.~A.
1993, {\it Nature}, {\bf 362}, 
\indent 
430.

\noindent
Duvall, T.~L.~Jr. 1995,
{\it GONG 1994: Helio- and Astero-Seismology from the Earth and
\indent
Space} (eds R. Ulrich, E. Rhodes \& W. D\"{a}ppen),
ASP Conference Series, 465.

\noindent
Duvall, T.~L.~Jr., Jefferies, S.~M., Harvey, J.~W.,
D'Silva, S. \& Schou, J. 1995, {\it Nature}, 
\indent 
submitted.

\noindent
Fan, Y., Braun, D.~C., and Chou, D.-Y. 1995, {\it Ap.~J.}, October 1.

\noindent
Jefferies, S.~M., Osaki, Y., Shibahashi, H.,
Duvall, T.~L.~Jr., Harvey, J.~W. \& Pomerantz, 
\indent 
M.~A. 1994, {\it Ap.~J.}, {\bf 434}, 795.

\noindent
Korzennik, S.~G., Noyes, R.~W. \& Ziskin, V. 1995,
{\it GONG 1994: Helio- and Astero-
\indent 
Seismology from the Earth and
Space} (eds R. Ulrich, E. Rhodes \& W. D\"{a}ppen),
\indent
ASP Conference Series, 268.

\noindent
Spruit, H.~C. and Bogdan, T.~J. 1992 {\it Ap.~J.} {\bf 391}, L109.

\noindent
Woodard, M.~F. \& Libbrecht, K.~G. 1993,
{\it Ap.~J.} {\bf 402}, L77.
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