CONTRIBUTIONS TO

PROCEEDINGS FROM THE NATO ADVANCED RESEARCH

WORKSHOP ON THE SEISMOLOGY OF THE SUN AND

THE DISTANT STARS

CSSA-ASTRO-85-23

September 1985

At-

'

CENTER FOR SPACE SCIENCE AND ASTROPHYSICS

STANFORD UNIVERSITY

Stanford, California

(BASA-CE-176470) CONTEIBUTICK S TO N86-19275

PROCEEDINGS FBCU THE NATO ADVANCE D EESEABCH THBU

HOBKSHOP ON THE SEISMOLOGY OF THE SUN AND N86-19278

THE DISTANT S1AES (Stanford Univ.) 24 p Unclas

HC A02/MF A01 CSCL 03B G3/92 03696
CONTRIBUTIONS TO

PROCEEDINGS FROM THE NATO ADVANCED RESEARCH

WORKSHOP ON THE SEISMOLOGY OF THE SUN AND

THE DISTANT STARS

CSSA-ASTRO-85-23

September 1985

The Detection of Global Convective Wave Flows on the Sun by

Philip H. Scherrer, Richard Bogart, and J. Todd Hoeksema of the

Center for Space Science and Astrophysics, Stanford University,

and Hirokazu Yoshimura of the Department of Astronomy, University

of Tokyo.

Observations of Low-Degree P-Mode Oscillations in 1984 by

Harald M. Henning and Philip H. Scherrer

Comments on Techniques for Spectral Deconvolution by

Philip H. Scherrer

This work was supported in part by the Office of Naval Research under

Contract N00014-76-C-0207, by the National Aeronautics and Space

Administration under Grant NGR5-020-559. and by the Atmospheric

Sciences Section of the National Science Foundation under Grant ATM77-

20580.
N86-19276

THE DETECTION OF GLOBAL CONVECTIVE WAVE FLOWS ON THE SUN

Philip H. Scherrer,

Richard Bogart,

J. Todd HoeksemaT and,

Hirokazu Yoshimura

iCenter for Space Science and Astrophysics, Stanford University

^Department of Astronomy, University of Tokyo

ABSTRACT. Global convective flows in the solar convection zone have been

predicted by theoretical interpretations of the global-scale ordering of

magnetic fields and activity centers and by theoretical analyses of

rotating convection zones. Direct evidence of these flows in the photosphere has not previously been found despite several long-term efforts.

The signatures of such flows have now been detected by analyzing the

daily series of low-resolution Dopplergrams obtained at the Wilcox Solar

Observatory at Stanford University. The signatures are patterns of

alternating east and west flows with amplitudes on the order of 25 m/s

and longitudinal extent of about 30 degrees. The patterns move across

the disc at approximately the solar rotation rate and have lifetimes of

at least several rotations. Boundaries of the fast and slow flows are

often associated with large magnetic active regions.

The detection and study of global convective wave flows on the sun is of

great importance for understanding the basic mechanisms of the generation of magnetic fields and magnetic activity on the sun (Yoshimura,

1971). They have been considered as the most likely kind of fluid

motions to drive the solar dynamo which generates magnetic fields on the

sun (Yoshimura, 1972, Yoshimura, 1983). This global convection has

escaped detection despite a number of long-term observing and analysis

programs (Durney et al., 1985, Howard and Yoshimura, 1976, Howard and

LaBonte, 1980, LaBonte et al., 1981, Snodgrass and Howard, 198M).

Observations made at the Wilcox Solar Observatory at Stanford

University have been examined for the signature of global scale convective motions. This series of observations began in 1976 and has been

found to have lower noise in the detection of solar velocities than the

instruments used in previous analyses (Scherrer et al., 1980). We now

have a sufficient time span of observations to study the evidence for

global-scale horizontal motions superimposed on solar rotation. The

observations are Doppler-shift measurements which yield low-resolution

full-disc daily velocity maps. Clear velocity patterns with amplitudes
of at least 25 m/s have been found in zones near the equator. These

patterns can be seen to move across the disc with solar rotation. They

have a large latitudinal extent, crossing the equator, but are not

aligned on meridians. There are from U to 7 alternating patterns around

the sun (Scherrer and Yoshimura, 1985,-Yoshimura et al., 3-15).

There are several fundamental difficulties in detecting globalscale small-amplitude horizontal motions on the sun. One problem is

caused by the large contributions of solar rotation (about 2000 m/s at

the equator) and the limb shift (a "W-shaped red-shift profile across

the disc with an amplitude of 600 m/s at the limb and 30 m/s at the disk

center). Other sources of solar "noise" are the motions of 5-minute

oscillations and supergranulation. These noise velocities are larger in

amplitude but smaller in scale than the postulated horizontal flows.

The large aperture used for the Stanford observations greatly attenuates

these signals. A third solar source of velocity noise comes from the

red-shifts associated with regions of intense magnetic fields. These

"downflows" have been examined by a number of authors (Cavallini et al.,

1985, Labonte and Howard, 1982) and are generally believed to be due to

a local reduction in small scale convective motions in the vicinity of

magnetic fields.

A final difficulty is that we are restricted to measuring the

line-of-sight component of the motion. In order to infer individual

velocity components, one must assume either spatial continuity or temporal continuity during the disc passage of individual features. In the

present work, we have used two different methods in an attempt to understand and avoid the weaknesses of each. In both approaches, we assume

the rotation and limb shift signals are constant over several rotations,

and that the remaining horizontal motions are organized on a scale much

larger than supergranulation. In the first approach (Scherrer and

Yoshimura, 1985) we further restrict ourselves to horizontal velocity

structures with a large east-west extent. We assume that by averaging

over a large window (up to a solar radius) the components other than

east-west motions will cancel. In the second approach (Yoshimura et

al., 3-15), we do not assume such a large spatial extent, but we assume

that the structures are stable during their disc passage. The two

methods have different sensitivities to noise and changing solar velocity fields. We refer to the two methods as the slope method and the

decomposition method.

The Stanford velocity maps are a product of a 9-year synoptic

series of observations. The observations and instrument have been

described previously (Duvall, 1979, Scherrer et al., 1980) and will be

only briefly described here. The instrument is a scanning magnetograph

attached to a high dispersion spectrograph. The raw data consist of

intensity, magnetic field, spectral line position, position on the solar

image, and time. Daily scans are made with a -three-arc-minute square

aperture. The solar image is scanned in 21 steps of 90 arc-sec in the

solar east-west direction and stepped to 11 scan lines in the northsouth direction. In this way we obtain simultaneous magnetograms and

Dopplergrams on every suitable day. The observations are made in the

absorption line of neutral iron near 525.02nm. The calibration of the

line shift signal in m/s and removal of the earth's motion is done as
described previously. After excluding observations contaminated by

clouds we have 1494 daily Dopplergrams for the present study.

Previous analyses have removed the large rotation and limb-shift

signal by fitting and subtracting a particular functional form of differential rotation and limb-shift from each Dopplergram. To avoid any

spurious signal from an improper fit and to avoid the possibility of

removing some of the desired signal as rotation, we have adopted a new

procedure. The first step is to subtract the 54-day running mean of the

observed velocity at each point on the disc. This procedure removes any

signal dependent on disc position, such as rotation and limb shift,

without making any assumptions regarding its functional form. Next, we

subtract from each scan point residual the mean of all residuals within

the central 80% of that Dopplergram. This step is necessary because we

have no absolute velocity reference for our measurements. To reduce the

contamination of our horizontal velocity signal by active-region redshifts, we next discard all scan points which have a velocity residual

amplitude exceeding 30 m/s or a magnetic field exceeding 20 gauss. An

average of 27 percent of the scan points were discarded from each Dopplergram. Discarding these points may remove evidence of the larger

amplitude flows, but we believe it is necessary to remove most of the

contamination from the active regions.

The resulting series of residual Dopplergrams is then analyzed for

horizontal velocity components by fitting the east-west component of

motion. In the first approach (slope method) we fit the east-west component in a restricted portion of the solar disc with a large longitudinal window at a fixed latitude. This is done by finding (by least

squares) the east-west component of velocity (Vwest) such that:

Vobs

 = Vwestsin(L)cos(Bo) where L is the longitude measured from disc

center and Bo is the latitude of disc center. The regions included in

each fit were east-west strips 810 arc-seconds long by 180 arc-seconds

wide. These fits were computed for the nine central scan lines and all

possible east-west positions for each Dopplergram. From these fits we

can display daily maps of east-west motions. The relatively large area

used in these fits greatly attenuates the noise from oscillations and

supergranulation. It also acts as a spatial filter allowing only very

large-scale structures to produce a signal. The method is not sensitive

to day-to-day variability in instrumental zero or noise from changing

solar velocities, but it is sensitive to contamination from line shifts

not associated with east-west motion.

We also derived east-west motions by a second approach (decomposition method) in which we examine the Doppler-shift variation of each

location on the sun as it crosses the disc. This is accomplished by

using all Dopplergrams having a given Carrington coordinate within 0.8

R0 of disc center to find the inferred components U, V, and W that best

fit (least squares) the observed velocity residual VODS:
20 SEPT 79 21SEPT79 22 SEPT 79 23 SEPT 79

Figure 1: Velocity residual maps for September 19 to 23, 1979. The top

row is the line-of-sight "velocity" with limb-shift and rotation removed

(positive away from observer). The middle row if the computed east-west

velocity (positive to the west) and the bottom row is the magnetic

field. The contour levels are 10 m/s for the velocity maps and 0, +.1,

2, 5, 10 ... gaus for the matnetograms. The regions excluded from the

analysis are shaded on the magnetograms.

vobs = Ucos(B0)sin(L)

+V(cos(B0)sin(B)cos(L) - sin(B0)sin(B))

-W(cos(B0)cos(B)cos(L) + sin(Bo)cos(B))

where B is the latitude and U, V, W are the westward, northward, and

upward components respectively. This method has the advantage that it

cleanly separates the westward velocity from active-region "downdrafts"
but has the disadvantage that it assumes a static velocity structure, no

day-to-day zero level errors, and the Carrington rotation rate.

Using the slope method, Scherrer and Yoshimura (1985) have shown

that there are global-scale velocity features that are stable for their

transit across the disk. This is the first requirement for detecting

large scale flows. Figure 1 shows data from 5 consecutive days, September 19 to 23, 1981. The top row shows residual line-of-sight signals

(before excluding points with large residuals or fields), the center row

shows vweat as described above, and the bottom row shows the corresponding magnetograms for comparison. The Vwest signal can easily be seen to

move across the disc as the sun rotates. The Vwest signal is shown as

positive values with solid contours in regions of westward motion (i.e.

motion in the same direction as rotation).

To study the morphology of the horizontal flows we have constructed synoptic charts of both Vwest and U in the same way as is usually done for magnetic fields. These charts show a map of the solar

surface made by averaging the signals from each day's observations for

each latitude-longitude bin. The synoptic charts are computed on a grid

of 5-degrees in longitude by 30 equal steps in sine-latitude. Some

additional temporal and spatial averaging results from this process. To

reduce the small-scale noise a further averaging with a 40-degree-inlongitude low-pass filter was applied to each latitude strip.

Figure 2a shows Vwest for three consecutive rotations from the

fall of 1979, Figure 2b shows three rotations from the fall of 1981, and

Figure 2c shows the fall of 198M. The velocity pattern shows 4 or 5

structures of alternating eastward and westward motion per rotation.

These structures are most clearly seen in low northern latitudes. They

can be followed for several rotations.

Synoptic maps computed with the decomposition method have roughly

the same character but differ in detail. They differ most in the vicinity of active regions, where the slope method could be contaminated by

"downflows" if their spatial extend is large, and the decomposition

method would be contaminated if they are changing.

This signal appears to persist for more than one rotation. Flows

of about the same magnitude are generally present during the past six

years during periods of both intense and low magnetic activity. For

instance, the signal is present in the fall of 1984 when activity was

quite low indicating that the while there is a correlation with the magnetic structures, we really are measuring different physical quantities.

A full understanding of the relation between the flow patterns and magnetic activity will require more analysis and more complete observations

Figure 2: Synoptic maps of east-west velocity Vweg^ for three consecutive rotations in 1979 (part a), in 1981 (part b), and in 1981 (part c).

Westward moving regions are shown in unshaded. Eastward moving regions

are shaded dark. The contour levels are ±10, 20, 50, ... m/s.
J U L I 97 9

1 2,1 1,1 0| 9 | 6 | 7 | 6 | 5 | 4 | 3 , 2 , I ,3 1,3 0, 2 9,2 8,2 7, 2 6, 2 5, 2 4, 2 3,2 2,2 1,2 0,1 9,1 6,

6 0 N

3 0 N

305

603-

30 60 90 120 150 160 210 240 270 300 330

60N

3 0 N

303

60S

168 4

flU G 1979

, 6, 7 | 6, 5 | 4 | 3 | 2 | 1 I 3 I I 30|29 I 28|27|26 I 25|24|23 I 2 2 I 2 1L2_0, I 9, I 8, 1 7, I 6,J_5, I 4,

6 0 M3 0 N

303

603-

i

30 60 90 120 150 180 210 240 270 300 330

6 0 N

3 0 N

303

-60S

6 0 NSON -

SOS6 0 S168 5

S E P 197 9

6,5 , 4 | 3 .2 I 1 ,3 0|2 9,2 8,2 7,2 6,2 5,2 4,2 3,2 2,2 1,2 0,1 9, 1 8, I 7, 1 6, I 5, 1 4, I 3, 1 2, 1 I, 110,

I 1 i 1 i 1 1 1 1 1

30 60 90 120 150 180 210 240 270 300 330

60N

3 0 N

-303

-60S

1686

Figure 2a

6
ft UG 1 98 1

60N3 ON

60 M

3 0 N

303-

605-

303

6 0 M3 0 N

303-

603-

,1 7,1 6,1 5,1 4,1 3,1 2,1 1,1 0, 9 , 8 i 7 , 6, 5 i 4 , 3 i 2 , 1 ,3

3 0 6 0 9012 0 15 0 18 0 21 0 24 0 27 0 30 0 33 0

I 4. 1 3, I 2, 1 1, I 0, 9 | 8 , 7 | 6 | 5 , 4 , 3, 2 , I ,3 1,3 0,2 9, 2 8, 2 7, 2 6,2 5,2 4,2 3,2 2,2 1,2 0, I 9

30 60 90 120 150 180 210 240 270 300 330

-303

-60S

171 2

S E P 198 1

, 1 1,10 , 9 | 8 , 7 | 6 | 5 | 4 | 3 | 2 , 1 ,3 0, 2 9, 2 8, 2 7, 2 6, 2 5, 2 4, 2 3, 2 2, 2 1, 2 0, 1 9, I 8, I 7, 1 6

6 0 N3 0 N

3 0 3 -

60 SH

3 0 6 0 9 0 12 0 15 0 18 0 21 0 24 0 27 0 30 0 33 0

-6 0 M

>-3 0 M

O J-30S

60S

171 3

Figure 2b

7
J U L 198 4

ORIGINAL PAGE IS

OF POOR QUALITY

1 2,1 1,1 0, 9 | 8 | 7 | 6 | 5 | 4 | 3 , 2 , 1 ,3 1,3 0,2 9,2 8,2 7,2 6,2 5,2 4, 2 3, 2 2, 2 1) 2 0, I 9, I 8,

6 0 N3 0 N6 0 N

3 0 H

303

603-

303

60S

6 0 NJON -6 0 N

-3 0 N

JOS603-

60 N

3 0 IH30 S60 S

W >

30 60 90 120 150 180 210 240 270 300 330

ftU G 198 4

9 , 8 . 7 . 6 , 5 , 4 , 3 , 2 , 1 .3 1,3 0. 2 9,2 8,2 7,2 6, 2 5.2 4,2 3,2 2.2 1,2 0,I 9, 1 8,1 7, 1 6. ! 5.1 4

30 60 90 120 150 180 210 240 270 300 330

S EP 198 4

,30,2 9,2 8,2 7,2 6, 2 5,2 4,2 3, 2 2, 2 1, 2 0, 1 9, 1 8, 1 7, I 6, 1 5, I 4, I 3, I 2, I 1

1 2 0 15 0 18 0 21 0 24 0 27 0 30 0 33 0

-60S
sequences than are presently available.

The intervals presented in Figure 2 were chosen both because of

sufficient data coverage to show nearly complete maps and because an

initial survey had identified these intervals as times of simple modal

structure. This survey was done by computing power spectra of Vwest for

intervals of nearly continuous data coverage. It is interesting to note

that the day-to-day variation in the whole-disc fit equatorial rotation

rate (Scherrer et al., 1980) is very similar to the meridional average

of Vwest for the intervals examined. This was examined both by comparing power spectra and by inspection of the raw data. Essentially all of

the day-to-day variation in the full disc fit of rotation comes from the

velocity structures described here.

These velocity patterns have a large amplitude compared to the

upper limits reported in previous analyses of observations from Mt. Wilson (Snodgrass and Howard, 19810. However, we note that the present

analysis of Stanford observations finds horizontal flow patterns which

are different in structure, and larger in scale than those searched for

in the Mt. Wilson studies.

The complete understanding of the flow patterns reported here will

require an extended set of high quality continuous observations

throughout a solar cycle. The present results have been obtained with

the instrument at Stanford, This instrument at present is limited to

low resolution measurements and seldom has more than 30 or 40 days

without interruption of a day or two. We are taking steps to increase

both the resolution and the Doppler zero stability for future observations. We are also examining the comparable data of improved quality

available from Mt. Wilson since 1982. We hope to be able to test our

interpretation of the Stanford observations as signatures of the global

convection and to begin the process of examining the morphology of these

flows by comparisons with other data and further observations.

Acknowledgements; We thank Harald Henning for useful comments.

This work was supported in part by the Office of Naval Research under

Contract N00014-76-C-0207, by the National Aeronautics and Space

Administration under Grant NGR5-020-559, and by the Atmospheric Sciences

Section of the National Science Foundation under Grant ATM77-20580.

References

F. Cavallini, G. Ceppatelli, and A. Righini, Meridional and Equatorial

Center to Limb Variation of the Asymmetry and Shift of three Fe I Solar

Photospheric Lines Around 6300A, Astron. and Astroohys. in

B. R. Durney, L. E. Cram, D. B. Guenther, S. L. Keil, and D. M. Lytle, A

Search for Long-Lived Velocity Fields at the Solar Poles, An. J..

, 752-762.

Thomas L. Duvall, Jr., Large-Scale Solar Velocity Fields, Sol. Phys.

£1(1979), 3-15.

Robert Howard and Hirokazu Yoshimura, Differential Rotation and Global-
J3

d

•3

4-

\>

I

(9

^^

I*

I

PL,

1000

 Frequency

Figure 1

The power spectrum of the full data, from 1000 jjHz to 5000 jiHz.

5000

Iw I

2030

Frequency (//Hz)

Figure 2

The window function: power spectrum of a 3000 uHz sine wave put into

the actual data window.

3070

to the main peaks greatly affect the fine scale structure of the spectrum and make the identification of individual peaks difficult.

In addition to the complications introduced by the window, the

effects of splitting due to rotation and lifetimes, each producing multiple peaks with sidelobes, as well as noise make the spectrum very complex. Several computerized deconvolution and peak-subtracting techniques failed to make significant inroads on the problem (Scherrer

1985). A scheme combining computed power spectra and manual pattern

recognition was decided upon as the best method to identify real solar

peaks. Figures 3 and 4 are typical examples of the graphs used. These

figures each show a section of the spectrum computed with several resolutions. Frequency runs along the abscissa, with each panel covering

the same span, from 1432 uHz to 1572 uHz in figure 3, and from 3200 uHz

to 3340 uHz in figure 4. We realized that the full resolution spectrum

(panel E) is too complex to enable recognition of any given peak as a

12
ORIGINAL PAQE fs

°f POOR QUALITY

A

B

C

D

1432 1572

Frequency (//Hz)

Figure 3

Plot of data in the frequency range from 1432 pHz to 1572 jiHz :

A) average power spectrum of 4 day chunks of data, B) peak search on A,

C) average power spectrum of 8 day chunks of data, D) peak search on C,

E) full resolution power spectrum, F) peak search on E.

solar oscillation mode. At lower resolution, however, it becomes much

clearer to see where there is power in the spectrum. Thus, the average

of 6 spectra of 4-day segments of the data is plotted in panel A. Panel

C shows the average of 3 spectra of 8-day segments of the data. To

further aid in locating modes a simple computer program was used to scan

the spectra, looking for structures which correspond to the main

features of the window function, i.e. a central peak with two sidelobes

at 1/day distance and of about 50% magnitude. The scan allowed the

sidelobes to be as large as S0% or as small as 20$ of the central peak.

The program took the value of the spectrum at a frequency if it

corresponded to a possible central peak and zero otherwise. The results

of this search are plotted below the corresponding spectrum, in panels

B, D and F. This composite graph proves to be a very useful tool in

determining the frequencies of the modes. The first four panels, A-D,

serve to locate and emphasize small frequency ranges where there is

power. Using this information and a transparency with the window function plotted on it, one can then pinpoint the group of peaks (and its

sidelobes) in the high-resolution full spectrum, panels E and F, that

correspond to a solar p-mode. The frequency recorded for each mode was

13
1 1 1 'i

I I'M 1 1 M—•—I « 1 1

Mil mil . <i .. ii JlniiJ K li .

32°°

 334 Frequency (/^Hz) °

Figure 4

As figure 3, but for the range from 3200 jaHz to 3340

that of the central peak or approximate center of the group of peaks. A

conservative estimate of the error in the frequency of any given mode is

2 jaHz.

Analyzing the spectrum, we located modes ranging from 1035 nHz up

to 5008 jiHz. Identification of the modes was accomplished using previously published frequencies (Duvall and Harvey 1983, Duvall - private

communication) for modes in the range from about 2000 to 4000 jiHz as

"anchors" for the determination of order n and degree 1. The fact that

the Stanford instrument is most sensitive to modes of degree 2 through 5

prevents confusion with higher degrees, although those modes certainly

contribute to the "noise". Modes with frequencies outside of this range

were then identified by making an echelle diagram (figure 5) and following the curve described by different n values of a given 1. It is

important to note that although there certainly is some bias in the

location and identification of the modes with previously published frequencies, there is no such bias in the frequencies of modes of lower and

higher order, since there was no preconception of where they should be.

Table I lists the frequencies of the identified modes.

In addition to identification of the modes, we have also examined

the data for evidence of rotational splitting. As mentioned before, the

power of each mode is split into a multitude of peaks resulting from a

combination of mode lifetimes, window function, and noise, as well as

rotational splitting. This confusion effectively prevents one from

14
1024

1160

1296

1432

1568

1704

1840

1976

2112

2248

2384

2520

2656

2792

2928

3064

3200

3336

3472

3608

3744

3880

4016

4152

4288

4424

4560

4696

4832

4968

Echel

1 1 1 1

............ . T . .

• •

1• I I • lif t 1 I • il l 1 1 li t

: : 1: : : : i: : :i : : i

•t m [•••• t • -i i • •• i • • • • • • i •

i t ••

1 1 • i t i i i • i • I I

i il l i • i • • i il l il l

• . i|i i • i • i • ••• • 1 •

: :i : : : i : : i : i :

: : i: : : : : i: : :| : : i : I : : : : 1 : : i ] : \ 1 i

• ii1i i I • i ii i i 1 . 1 1 i ii i

S

I 1 1

: : : : : i : i : : :i : i • i r

• ii i ^ • f i •• • ii i • ii i

1 1

• ii i i • i ii i i i i • ii i

: : : i: : : : i: : : : : :

: : i ::::::::: :

t i iii i i t • i i t • i i

i i • ill i i i i • • i i

1 i • i iii i • i i f i i i i

Figure 5

le diagram of the identified modes, each strip covers 136 pHz.

15
n

5

6

7

B

9

10

11

12

13

14

15

16

17

18

19

frequency

1=2

1083

1227

1375

1522

1659

1802

2217

2352

2486

2621

2754

2889

1=3

1144

1292

1446

1585

1726

1859

2001

2139

2274

2411

2543

2677

2812

2948

1=4

1035

1187

1338

1488

1630

1768

1916

2052

2187

2052

2457

2591

2727

2864

3001

1=5

1115

1263

1412

1552

1691

1830

1963

2105

2236

2372

2505

2640

2779

2911

3050

n

20

21

22

23

24

25

26

27

26

29

30

31

32

33

34

frequency

1=2

3026

3161

3295

3430

3570

3705

1=3

30B3

3217

3353

3490

3627

3763

3902

4037

4177

4312

4457

4591

4727

4875

5008

1=4

3136

3271

3409

3544

3682

3819

3958

4088

4231

4373

4497

1=5

3187

3320

3462

3597

3733

3870

4007

4147

4285

Table I - Frequencies of identified modes

assigning m-values to given peaks and directly measuring the splitting.

Rather, we have analyzed the spectrum using autocorrelations. Using the

determined frequencies we calculated the autocorrelation of a small section of the full spectrum around each mode. All autocorrelations with a

given degree 1, but different order n, were then averaged together. The

assumption is that any rotational splitting will not change much with

order n. The results are plotted in the first four panels of figure 6.

An autocorrelation of the window function is shown in the last panel of

figure 6. It can be seen that the window is responsible for much of the

systematic splitting, as evidenced by the peaks in its autocorrelation

which are repeated and actually dominate the autocorrelations of the

modes. However, all degrees show a peak around 0.8 uHz, which is close

to twice the rotational splitting as measured by Duvall and Harvey

(198H). Since the Stanford instrument is sensitive only to modes with

1+m even, this peak can be identified as the result of rotational splitting.

2. CONCLUSION

We have determined the frequencies of over 20 modes which have not been

identified before. The majority of these modes lies in the frequency

range from 1035 uHz to 1800 uHz. The echelle diagram of the modes shows

clearly the change in slope at low n, as the modes become spread further

apart than 136 uHz. Also, we have been able to see a characteristic

splitting which is consistent with the rotational splittings measured by

16
0.5

0.0

1=2

r\

0.5

0.0

0.0

1=4 :

(mtoroHz)

0.5

0.0

0.0

1=3

—I—I—I—I—I—I—I—I—I—I—I—I I—I—I—I—I—I—I—I—I—l_

0 lae (mtoroHz) 3 Q lae (mtoroHz)

0.5ii- ......... . |Q.5

1=5 :

J f^^^^^^*^ t ? * * 1 T * I I

3 0 lag (mtoroHz) 3

window .

0 lag (mtoroHz) 3

Figure 6

Autocorrelations of the power spectrum around different degrees 1 and

the window function.

Duvall and Harvey (1984). Future examination of other Stanford data

will hopefully refine the values, although no other year has had data as

complete as 1984.

3. REFERENCES

Duvall Jr., T. L.f and J. W. Harvey, 'Frequency spectrum of solar 5-

minute oscillations with degree less than 140', Nature, 102f 24,

1983.

17
Duvall Jr., T. L., and J. W. Harvey, 'Rotational frequency splitting of

solar oscillations', Hatujce., .H0_, 19, 1984.

Scherrer, P. H., J. M. Wilcox, J. Christensen-Dalsgaard, and D. 0.

Gough, 'Detection of solar five-minute oscillations of low degree',

Solar Phvsics. 82, 75, 1983.

Scherrer, P. H., 'Comments on techniques for spectral deconvolution',

these proceedings, 1985.

4. ACKNOWLEDGEMENTS

This work was supported in part by the Office of Naval Research under

Contract N00014-76-C-0207, by the National Aeronautics and Space

Administration under Grant NGR5-020-559, and by the Atmospheric Sciences Section of the National Science Foundation under Grant ATM77-

20580.

18
N86-19278

COMMENTS ON TECHNIQUES FOR SPECTRAL DECONVOLUTION

Philip H. Scherrer,

Center for Space Science and Astrophysics

Stanford University

Stanford, California

U.S.A.

ABSTRACT. Current observational questions in asteroseismology require

high spectral resolution that can only be obtained with observations

spanning many days or months. The primary constraint in the full utilization of single mid-latitude observing sites is the presence of diurnal

data gaps. Several methods for removing the effect of these gaps in the

spectra obtained from velocity observations have been suggested. In the

case of data coverage of less than 50%, none of the methods considered

has been successful at unambiguously recovering the true spectrum from

test spectra with realistic complexity. The limitations of these

methods and their applicability to the helioseismology problem is discussed.

The presence of data gaps is particularly troublesome when computing

power spectra of multi-day observations of solar or stellar oscillations. If the expected lifetime of the observed modes is longer than

individual uninterrupted observing sequences, one naturally wants to

combine many days of data to obtain higher spectral resolution. The

gaps from nights or poor weather produce a complicated structure of

sidebands in the power spectrum unless the gaps are filled with

estimated data. Since the power spectrum is computed from the Fourier

transform, and the observed transform can be considered as the convolution of the true spectrum and the transform of the observing window,

filling the gaps is essentially a deconvolution process. Since there

are an unlimited number of ways to fill the gaps with data, most of them

non-physical, we must have some guidance from our a priori knowledge of

the sun or star in question and we must have some method for incorporating this knowledge in an analytical procedure. Thus the whole problem

of deconvolution or gap-filling is one of explicitly recognizing the

assumptions about the star and finding objective ways to incorporate

those assumptions in the analysis.

This problem is not unique to asteroseismology. In fact, there is

a significant body of literature dealing with the deconvolution problem.

The specific problems of asteroseismology are different from many of the

previous applications however, so care must be exercised in applying

previously developed methods. Some of the particular problems of solar

19
' . «*1 1}-, ll ,

and stellar observations are low filling factors (i.e. large gaps relative to intervals of continuous data), very complex spectra with

thousands of modes allowed over several decades in frequency, line

separation on the same order as the main contributions from typical

observing windows, and low signal to noise in frequency bands that contain important oscillation modes. A number of gap-filling methods have

been suggested for the asteroseismology problem. These include maximum

entropy methods of "predicting" the data in the gaps (Fahlman and

Ulrych, 1982), variations of the CLEAN algorithm of iterative peak subtraction (Scherrer, 1984), rearranging the data to fill the gaps at the

expense of resolution (Kuhn, 1982), and iterative multiplicative deconvolution (Connes and Connes, 1984).

Brown and Christensen-Dalsgaard (1985) have completed a thorough

study of the applicability of the maximum entropy method to the problem.

They found that complex spectra can be correctly recovered provided the

filling factor is high (greater than 0.8) and the signal-to-noise ratio

is high (greater than 100). Fortunately, these conditions are expected

to be met for p-mode observations from an earth-bound network They found

that as the filling factor dropped below 0.5 or the s/n dropped below

about 30 the error rate increased rapidly. Kuhn found similar results

when applying the data-swapping method to data with a low filling factor

(Kuhn et al., 1985).

Delache and Scherrer (1983) successfully applied the CLEAN-like

iterative peak subtraction method to low-filling factor observations in

the long-period spectral region but that method was not successful when

applied to p-mode spectra (Henning and Scherrer, 1985). Experiments

with linear iterative deconvolution methods have not been as successful

as the CLEAN technique. There is a tendency for noise peaks to grow

rather than to be simply extracted unchanged as in the CLEAN method.

This is a characteristic of linear iterative methods (Jansson, 1984).

In order to push these procedures to the regime of low-filling

factor, the methodology of inserting a. priori knowledge into the

analysis must be explicitly examined. There are different assumptions

implicitly made by the various methods. The maximum entropy method

assumes that the signal is locally stationary. This is a fairly weak

assumption and leads to failure when the gaps are large. The stronger

assumption of completely stationary signals made in the CLEAN method can

do a better job of recovering the true spectrum with low filling factors, but only in cases where the assumption is valid. Thus, the long

period success and the p-mode failure of the method.

Particularly strong claims have been made for the advantages of

the non-linear spectral deconvolution method (Connes and Connes, 1984).

This has lead to a careful examination of the applicability of that

method to the problem. The method can be summarized as the iterative

task of finding a spectrum that when convolved with the window function

yields the observed spectrum. The iteration is done under some constraints that provide the mechanism for including prior knowledge. This

method has the advantage of explicitly specifying the assumptions. It

has been applied for some time to the deconvolution of instrumental profiles from spectra and has been described by (Blass and Halsey, 1981)

for that application. The general method has been examined by a number

20
of authors in other applications and it has been shown, that convergence

can be guaranteed in many cases (Sanz and Huang, 1983, Schafer et al.,

1981). A general discussions of both subtractive and multiplicative

deconvolution techniques have been presented by Bates and colleagues

(Bates et al., 1982a, 1982b, 1982c) and by Jansson (1984).

Connes and Connes suggested applying the multiplicative deconvolution method independently to the real and imaginary parts of the Fourier

transform of oscillation observations. It turns out that constraints

suggested do not quite match the problem. In particular, the method

assumes that the true signal is band-limited, and that power outside the

known band comes from the gaps. Unfortunately, it is not possible to

arrange the observations of the sun (or many stars) to match this criterion. The p-mode frequency band, for example, extends from possibly

300 micro-Hz through 4500 micro-Hz. The sidebands from nightly gaps are

spaced at about 11.6 micro-Hz. Thus most of the sidebands are within

the desired frequency band. The method probably works but is expected

to be rather inefficient. Thus, this method at present does not solve

our problem (J. Connes, private communication, 1985).

What has yet to be done, is to find a methodology for tuning an

analysis to specific prior knowledge or assumptions. For example, if we

were asking what is the rotational splitting for solar p-modes, then

since we know the central frequencies for each multiplet (to a few

micro-Hz) we should be able to tell our analysis procedure just where

power is from gaps and where it is probably of solar origin. If we

believe that the lifetime of low-frequency p-modes is several months but

the lifetime of high-frequency modes may be only a few days, we should

be able to find peak centroids using varying resolution across the pmode band. Perhaps the iterative methods can be adapted to such more

problem-specific constraints. In any case, we must always be aware of

the explicit and implicit assumptions in our procedures. In summary,

two things are clear: First, if we do not attempt to account for the

sidebands introduced by the data gaps we are making the implicit assumption that the sun does not oscillate at night and second, we do not yet

have a reliable, objective method for filling the gaps in the case of

low filling factor.

Acknowledgements: This work was supported in part by the Office of

Naval Research under Contract N00014-76-C-0207, by the National Aeronautics and Space Administration under Grant NGR5-020-559, and by the

Atmospheric Sciences Section of the National Science Foundation under

Grant ATM77-20580.

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Norton, A.E. McKinnon, and R.H.T. Bates, Subtractive Image restoration.

III. Some practical Applications, £fi£ik £2/1982), 333-346.

J.H.T. Bates, A.E. McKinnon, and R.H.T. Bates, Subtractive Image

Restoration. II): Comparison with Multiplicative Deconvolution, Optik

£2(1982), 1-14.

21
J.H.T. Bates, A.E. McKinnori, and R.H.T. Bates, Subtractive Image

Restoration. I: Basic Theory, Qotik 61(1982). 349-364.

William E. Blass and George W. Halsey, Deconvolution of Absorption

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Timothy M. Brown and Jorgen Christensen-Dalsgaard, A Technique for

Filling Gaps in Time Series with Complicated Power Spectra, preprint,

High Altitude Observatory, NCAR, 1985.

Janine Connes and Pierre Connes, A Numerical Solution of the Data-Gaps

Problem, Space Research Prospects in Stellar Activity and Variability,

Meudon Observatory, February 1984.

Philippe Delache and Philip H. Scherrer, Detection of Solar Gravity Mode

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G.G. Fahlman and T.J. Ulrych, A New Method for Estimating the Power

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Harald M. Henning and Philip H. Scherrer, Observations of Low-Degree PMode Oscillations in 1984, These Proceedings, 1985.

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Academic Press, Inc., 1984.

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Two Machine-Efficient Solutions, The Astronomical Journal 87 , 1 (January

1982), 196-202.

J.R. Kuhn, K.G. Libbrecht, and R.H. Dicke, Solar Ellipticity

Fluctuations Yield No Evidence of g-Modes, submitted to Nature, 1985.

Jorge L.C. Sanz and Thomas S. Huang, Iterative Time-Limited Signal

Restoration, IEEE Transactions on Acoustics, Speech, and Signal

Processing ASSP-31.3 (June 1983), 643-650.

Ronald W. Schafer, Russell M. Mersereau, and Mark A. Richards,

Constrained Iterative Restoration Algorithms, Proceedings of the IEEE

.l (April 1981), 432-450.

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Solar Seismology From Space, A Conference at Snowmass, Colorado, R. K.

Ulrich, J. Harvey, E. J. Rhodes, Jr., and J. Toomre (editor), NASA,

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22