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A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
09(06.03.1; 06.15.1; 03.13.2; 03.13.6; 02.23.1)
ASTRONOMY
AND
ASTROPHYSICS
April 14, 1999
Exploring the dynamical nature of the lower solar
chromosphere
J.G. Doyle 1 , G.H.J. van den Oord 2 , E. O'Shea 1;3 , and D. Banerjee 1
1 Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland; e�mails: jgd@star.arm.ac.uk, dipu@star.arm.ac.uk
2 Sterrekundig Instituut, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands; e�mail: oord@astro.uu.nl
3 Dept. of Pure & Applied Phys., Queens University Belfast, Belfast BT7 1NN, N. Ireland email: E.Oshea@qub.ac.uk
Received date:April 14, 1999, accepted date:
Abstract. We examine spectral time�series of two lower�
chromospheric lines (N i 1319 � A and C ii 1335 � A) observed
with the SUMER instrument on the SOHO spacecraft.
We point out differences between (intensity and velocity)
power spectra of network and internetwork regions and ar�
gue that the behaviour resembles that of Ca ii power spec�
tra. No significant phase differences are found between the
intensities of both lines. However, when phase spectra are
averaged along the slit there is some evidence that the C ii
intensity lags that of N i by 16 sec near 3 mHz. Intensity
power spectra of C ii are affected at higher frequencies by
streams of emitting structures. Using contrast�enhanced
time slices we show that 1) there exists a grain�like pattern
which is found in both network and internetwork regions;
2) streams of supersonically moving structures probably
outline a wave interference pattern; 3) the sizes of struc�
tures observed in N i are smaller than when observed in
C ii. At various points our findings disagree with earlier
results from SUMER. A cookbook formalism is presented
to derive confidence levels for power, phase, gain and co�
herency spectra.
Key words: Sun: chromosphere -- oscillations -- Methods:
data analysis -- statistical -- Waves
1. Introduction
In the last few years the views on the solar transition
region and the chromosphere have changed dramatically
with the realization that these are very dynamic and not
static. Carlsson & Stein (1992, 1995, 1997) demonstrated
that the formation of Ca ii grains in the internetwork is
related to the presence of upward traveling shocks. MDI
observations (Schrijver et al. 1997) indicate that the net�
work magnetic field is constantly evolving (fragmentation
and merging of flux tubes in the lanes) and is being `fed' by
the emergence of ephemeral regions. This leads to flux an�
nihilation (reconnection) in the lower chromospheric net�
work but also to the creation of tangential discontinuities
Send offprint requests to: J.G. Doyle
in the overlying field. These again can be a source of en�
ergy release (Schrijver et al. 1998) as the field reconfig�
urations can result in non�periodic flare�like brightenings
and can also act as sources of waves which can become pe�
riodic when trapped. For example, Hansteen (1993) pro�
posed that the observed red�shifts in the transition region
are due to compressive waves generated in the corona by
`nano�flare' heating events.
In this paper we report on data for two lower�
chromospheric lines due to N i 1319 � A and C ii 1335 � A
observed with the SUMER instrument on�board SOHO.
Up till now only a limited number of analyses, based on
SUMER data, has been published. For the internetwork,
Carlsson et al. (1997) reported on oscillatory behaviour in
UV continua and in the lines of C i, N i and O i analogous
to the Ca ii grains. The 3 minute intensity oscillations are
also occasionally observed in C ii but never in C iii and
Si iii. Typically maximum intensity is first observed in the
continuum and C i, �12 sec later in C ii and �25 sec later
in N i. The phase differences between line and continuum
indicate the presence of upward propagating waves. The
line brightenings appear to be accompanied by blue shifts
of typically 5 km s \Gamma1 . The oscillations also show up in the
velocities of C i, N i and O i but the velocity oscillations in
He i, C iii and Si iii are not related to the 3 minute oscil�
lations. The 3 minute intensity oscillations appear to be
present in �50% of the areas studied and to be coherent
over 3--8 arc sec regions. Hansteen (1997) reported similar
(3 min.) bright grains in the wings of Ly ff but also showed
that in the Lyman continuum, formed much higher in the
chromosphere, the grains are absent. Betta et al. (1997)
report on an internetwork transition region oscillation in
velocity (C iii) at �7 mHz with a large (� 40 00 ) spatial
structure. Judge et al. (1997) found small amplitude (2--5
km s \Gamma1 ) coherent oscillations of 5--10 arc sec scale length
and � 130 sec (7.7 mHz) period in Doppler shifts of Si iii
between regions of bright network elements. More recently
Curdt & Heinzel (1998) analyzed the temporal evolution
of hydrogen Lyman lines observed by SUMER. In the cell
interiors they detected oscillations with periods of 3.3--3.5
minutes which they associate with unresolved clusters of

2 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
grains. For network regions they found slower oscillations
with frequencies in the range 2.4--2.7 mHz, in agreement
with the Judge et al. result that the network varies more
smoothly.
An overview of typical velocity power spectra of a
large number of lines observed in different sequences with
SUMER is presented by Steffens et al. (1997). All spectra
show considerable power below 10 mHz but only for the
lower ionization stages (i,ii) are distinct features present
in the power spectra in the range 2--5 mHz.
Doyle et al. (1998) analyzed upper transition region
data obtained with the CDS on SOHO. The O iii, O iv
and Ov data indicate that there is power below 4 mHz
everywhere along the slit but the clearest periods do not
always come from the most intense regions. In �40% of
the data statistically significant periods are observable in
the 2--5 mHz range with the largest power peaks found at
3.0 mHz. For the lines investigated there were no signifi�
cant time delays although this is not entirely unexpected
as the formation temperatures of O iii (�100,000 K) and
O v (�250,000 K) may be too close in order to result in
an observable phase shift. The sizes of the emitting fea�
tures were limited to �7 arc sec squared. Independent of
whether it was assumed that the 5 minute waves were
trapped below the transition region or in overlying loop
structures, a typical field strength of 1--2 Gauss was de�
rived, typical of quiet Sun regions. We note, however, that
basically no differences exist between the power spectra of
the lower chromosphere (see Doyle et al., 1997) and the
transition region (Doyle et al., 1998). This demonstrates
the inherent difficulty with studies of power spectra: many
different processes can result in power at comparable fre�
quencies.
The outline of the paper is as follows. In Sect. 2 we dis�
cuss the observations and the data reduction. The results
of the analyses are presented in Sect. 3 and our conclu�
sions in Sect. 4. Although it is relatively easy to compute
power spectra and determine phase�lags between time av�
eraged spatial intensity variations, these have little value
unless statistically meaningful significance levels can be
assigned to them. An extensive discussion on the statis�
tics and methods of analyses can be found in the standard
reference Jenkins & Watts (1968) which is however not al�
ways easy to follow. Methods outlined in this book have
been used for QPO research (see van der Klis 1989). In the
Appendix we present in cookbook format uni� and bivari�
ate analysis techniques of time series and discuss how con�
fidence levels for power, phase, gain and coherency spectra
can be derived. Also we discuss how controlled smoothing
of auto� and cross�covariance estimates enables one to re�
duce the variance of the noise and improve the significance
of spectral features.
2. Observations and data reduction
The Solar Ultraviolet Measurements of Emitted Radia�
tion (SUMER) instrument is a stigmatic normal�incidence
spectrograph operating in the range 450 -- 1610 � A (Wil�
helm et al., 1995) The observations reported here were
obtained with the 1 x 300 arc sec slit (slit 2). The in�
strument was operated in a sit�and�stare mode with the
SUMER standard rotation compensation switched off. For
disk center pointing, this implies that it takes 353 seconds
(2.65 mHz) for a feature to transit the slit.
The datasets analyzed here were acquired from 22:47
UT to 23:49 UT on 31 July '96 at 84 arc sec West of
disk center, zero arc sec North, and from 18:38 to 19:09
UT on 30 July 1996 at 54 arc sec West and zero arc sec
North of disk center. Two spectral lines were observed
simultaneously, N i 1318.99 � A and C ii 1334.53/1335.71 � A,
covering 50 spectral pixels (�2.2 � A) and 300 arc sec in
the North�South direction. In order to increase the signal�
to�noise of the C ii signal, all analyses in this paper are
based on the sum of both C ii lines. The exposure time
was 30.0075 sec resulting in 124 temporal samples at each
slit position for the 31 July observations and 62 samples
for 30 July. In our analyses we restrict ourselves to slit
positions 40 -- 320 for which the data are of good quality.
We have compared the SUMER slit position with
several EIT images and with a synoptic map taken in
Na i 5896 � A on 31 July 1996 at 20:55 UT. Though the
region observed by SUMER was a typical quiet Sun area,
there are occasional concentrations of magnetic fields in
the synoptic map which correspond to the bright regions in
the SUMER data. These bright regions are therefore likely
to outline the chromospheric network. The morphology of
network and internetwork has also been checked against
Ca ii images obtained from Big Bear Solar Observatory.
For the dataset of 31 July N i and C ii velocities are
derived by fitting a Gaussian to the spectral data for each
spatial pixel at each temporal sample. We note that there
is no absolute calibration for the velocity scale so that
the derived velocities are merely indicative of the presence
of line shifts. For 30 July no velocity determinations are
available due to the reduced intensity of the spectral lines..
Details of the Fourier analysis can be found in the Ap�
pendix. Power and coherency spectra are obtained from
the Fourier transforms of the auto�covariance and cross�
covariance functions, respectively, multiplied by a window
function to reduce the variance of the noise. Power spectra
are normalized in such a way that the expected mean noise
level equals 2. For the 31 July data the `unsmoothed' (i.e.
no window applied) spectra contain 63 frequencies with
a resolution of \Delta� = 0:269 mHz and for 30 July 32 fre�
quencies with a resolution \Delta� = 0:538 mHz. Unsmoothed
spectra (formally) correspond to the use of a rectangu�
lar window function. For smoothed spectra we used the
Tukey window with 62 lags for 31 July data and 31 lags
for 30 July data. With this choice the number of frequen�

J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere 3
Fig. 1. Time slices of the N i (top panels) and C ii (bottom panels intensity observations of 31 July 1996. The panels in the left
column show the observed intensity. The panels in the second column show the smoothed intensity resulting from a convolution
of the original data with a Gaussian with oe = 150 sec. All frequencies above �6.7 mHz are suppressed in these graphs. The
third and fourth column show the contrast�enhanced time slices for oe = 150 sec and oe = 510 sec, respectively. This implies that
in the third column frequencies below � 6:7 mHz are suppressed and in the fourth column frequencies below � 2 mHz.
cies and the frequency resolution in the smoothed spectra
equal those in unsmoothed spectra but the number of de�
grees of freedom in the smoothed spectra is 16=3 and the
variance of the noise is reduced by a factor 3=8. For the 31
July data the effect of the Tukey window is a smoothing
in the frequency domain with a bandwidth of 0.717 mHz.
For the 30 July data this bandwidth is 1.433 mHz. Be�
cause the mean noise level and its variance are known we
are able to derive confidence limits for spectral features.
For intensity and velocity power spectra we use confidence
levels of 99.9% and 90%, respectively.
The relation between the N i and C ii intensity varia�
tions in a pixel is investigated using a model which checks
whether C ii intensity variations can be considered as re�
sulting from the convolution of the N i intensity varia�
tions with a response function. Gain and phase difference
spectra give the amplitude and the phase of this response
function at each frequency. In this way the phase of the

4 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
Fig. 2. Same as Fig. 1 but for the N i and C ii intensity data of 30 July 1996.
response function equals the phase difference between two
signals at each frequency.
3. Results
In the left columns of Figs. 1 and 2 we show time slices
of the observed N i and C ii intensity on 31 and 30 July,
respectively. An on�screen inspection of the time slices
indicates that there is more `structure' in the emission
of N i than in C ii. The brightest features, which relate
to network magnetic fields, are observable in both N i
and C ii. Often these features are observed for the whole
length of the observation but sometimes the in� or egress
of a feature in the slit is observed. Fluctuations in the
bright features are clearly visible and their appearance

J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere 5
Fig. 3. Power spectra (per pixel along the slit) of the N i intensity (top left) the N i velocity (top right), the C ii intensity
(bottom left) and the C ii velocity (bottom right). The spectra are based on the 31 July observations. In each group of three
panels the left panel shows the unsmoothed power spectrum and the middle panel the smoothed power spectrum for 62 lags.
Black indicates power above the detection level and white zero power. In the power spectra for the intensity the detection level
is based on the 99.9% confidence limit and in the power spectra of the velocity on the 90% confidence limit. The black vertical
line indicates 2.65 mHz. Also shown are the summed counts per pixel during the observation and the mean velocity per pixel
during the observation. In the graph of the summed counts of C ii the thick vertical lines indicate some internetwork regions
and the thin lines some network regions.

6 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
Fig. 4. Same as Fig. 3 but for the N i and C ii intensity data of 30 July 1996 (velocity data are not available due to the reduced
line intensities). The smoothed power spectra are based on 31 lags.
seems periodic. These structures are so bright that in a
gray�scale presentation it is difficult to identify weakly
emitting structures. To bring out details in the inten�
sity map we use a technique based on enhancing the con�
trast of these structures, thereby filtering out the bright
components in the brightness evolution displays. The in�
tensity maps I(y; t) are convolved in the time direction
with a Gaussian G(t). This results in a smoothed display
S(y; t) j I \Lambda G which contains no high frequencies. This
can be seen as follows. Let i(y; �) be the FT of the inten�
sity map and g(�) = (
p
2�) \Gamma1 exp(\Gammaoe 2 � 2 =2) be the FT
of a Gaussian with a FWHM equal to 2oe
p
2 ln 2. The FT
of the smoothed map is s(y; �) = i(y; �)g(�). The power
spectrum of the smoothed map equals the power spec�
trum of the intensity map multiplied by exp(\Gammaoe 2 � 2 ). This
implies that in the smoothed map higher frequencies are
suppressed by an amount depending on the choice of oe.
Next we construct a contrast�enhanced map by di�
viding the original intensity map by the smoothed map
C(y; t) = I(y; t)=S(y; t). The reciprocal of the smoothed
map acts as a low frequency filter so that the contrast map
C contains the high frequency components.
In the second columns of Figs. 1 and 2 we show time
slices resulting from smoothing with oe = 150 sec. In these
time slices frequencies above �6.7 mHz are suppressed.
The in� and egress of network elements is clearly visible
together with the fact that many elements show a move�
ment along the slit of a few arc sec corresponding to a
shift of �1 km s \Gamma1 .
In the third and fourth columns of the figures contrast�
enhanced time slices are shown for oe = 150 sec and
oe = 510 sec, respectively. In these time slices frequen�
cies below, respectively, �6.7 mHz and �2 mHz are sup�
pressed. The bright structures in these maps show a wave�
like patterns but closer inspection indicates that a also
granular�like pattern is present.
The contrast�enhanced N i time slices contain a mix�
ture of small isolated dots of emission and patches of emis�
sion. In C ii, these patches have a much larger dimension.
What is also clear is that groups of these patches form
`streams' which often show a displacement along the slit
of ten or more arc sec corresponding to supersonic veloci�
ties. It is therefore likely that these streams are not physi�
cal features but rather a wave interference pattern. These
streams often connect network and internetwork regions.
We emphasize that the structures in the contrast maps
are not an artifact of our procedure. All of them can be
found back as extremely weak structures in the original
intensity maps when viewed on a high resolution screen.
The fact that many structures show a movement along the
slit will affect power spectra of individual pixels, a point
we discuss below.
Next we consider power spectra of the observed in�
tensities and velocities. Because variations of these quan�

J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere 7
Fig. 5. Squared coherency (left panels) and phase difference (middle panels) spectra along the slit for the N i and C ii intensity
data of 31 July (left) and 30 July (right). For each day also the summed C ii counts are shown (right panels). In the squared
coherency plots black indicates a squared coherency exceeding the 80% confidence limit and white indicates zero coherency. In
the phase difference spectra black indicate a phase difference of +180 ffi and white \Gamma180 ffi . A positive phase difference implies
that the N I intensity lags the C ii intensity. The black vertical line indicates 2.65 mHz.
tities in neighbouring pixels are not in phase, we study
power spectra of individual pixels along the slit and do
not average over groups of pixels in order to increase the
signal�to�noise ratio. To check whether this choice is in�
deed correct we created artificial time series containing
a dominant frequency and added noise. When several of
these time series were randomly added in phase, the result�
ing power spectrum showed many (artificial) harmonics.
When a too large sample was added the periodic signal
disappeared (of course) completely. Therefore it is safer
to consider power spectra of individual pixels despite the
moderate signal�to�noise ratio.
Intensity and velocity power spectra for N i and C ii on
31 July are shown in Fig. 3 together with the total num�
ber of counts in a pixel (summed counts) during the ob�
servation and the average velocity in a pixel. The summed
counts are useful to identify network (NW) and internet�
work (IN) regions. Also shown are the smoothed power
spectra.
The N i intensity shows strong power in the range 0
-- 1 mHz, less power between 1 and 3 mHz and a diffuse
band of power in the range 3 -- 7 mHz. This diffuse band is
more pronounced in the smoothed power spectra in which
the variance of the noise is reduced. There is little power
above 7 mHz.
The N i and C ii velocity power spectra show power
at low frequencies ( ! � 1.5 mHz) in NW regions and little
power in IN regions. At frequencies above 3 mHz the situ�
ation is reversed. IN regions stand out as diffuse bands of
enhanced power at all frequencies while NW regions are
devoid of power above 3 mHz. In fact, based on velocity
power spectra like those shown in Fig. 3, one could in prin�
ciple identify NW and IN regions while this is not possible
with intensity power spectra.
The C ii intensity power spectra are more difficult to
classify. In IN regions the power is concentrated at fre�
quencies below 2 mHz and in NW regions below 5 mHz.
Strong power above 5 mHz, like, e.g., near slit position
260, is caused by the in� or egress of NW elements which
show a motion along the slit and are, therefore, observed
for a limited time in a pixel (see also Fig. 1).
A similar picture arises from the N i and C ii intensity
power spectra of the 30 July data shown in Fig. 4. How�
ever, on that day the intensity of the network was less
than on 31 July.
It seems that frequencies below 2.65 mHz are not
strongly affected by the in� and egress of structures. The

8 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
Fig. 6. For the smoothed and unsmoothed intensity power spectra, per pixel along the slit, we determined the frequencies above
2.65 mHz at which the power exceeds the 99.9% detection limit. These frequencies are plotted against the total counts per pixel
during an observation. Whenever the power at a specific frequency is the maximum in a power�spectrum it is plotted as a small
vertical line. Whenever the power at a specific frequency is not the maximum power but merely exceeds the detection limit, it
is plotted as a dot. The different panels show the results for N i and C ii on 30 and 31 July and for smoothed and unsmoothed
power spectra.
effect on power spectra of structures moving through the
slit has been discussed by Doyle et al. (1998, see their
Appendix). Suppose that a structure shows intensity os�
cillations at frequency � 0 . These authors show that a finite
transit time results in the power at frequencies � = � 0 and
� = 0 being spread over at most a frequency range v=\Delta,
where v is the solar rotation velocity and \Delta the projected
width of the slit. This spreading follows the sinc function.
For structures with dimensions larger than \Delta the band�
width of the spreading is even smaller. For the present
observations we have that v=\Delta = 2:65 mHz. The fact that
the power near zero frequency often drops faster, creating
the `gap' between 1 and 3 mHz, suggests that the struc�
tures have lengths of at least several arc sec.
We calculated coherency and phase spectra for the N i
and C ii intensity for both observing dates. These spec�
tra are shown in Fig. 5 together with the summed C ii
intensity. We analyzed the phase difference spectra exten�
sively taking the values of the squared coherency and the
errors in the phase angles into account. No systematic be�
haviour in the phase difference was found. At several slit
positions a significant positive or negative phase differ�
ence was found at a specific frequency but these can be
the result of statistical fluctuations. We consider 281 pix�
els along the slit \Theta 62 (31) frequencies so that we, in fact,
consider 17,422 (8,711) samples. For a confidence limit
of 80% one can then expect the detection of tens of sig�
nificant phase differences, even from random signals. At
frequencies below, say, 7 mHz the (intensity) power is rel�
atively strong and the phase differences cluster around 0 ffi .
At higher frequencies, where there is less power, the phase
differences vary randomly between +180 ffi and \Gamma180 ffi , as
expected for uncorrelated signals.
Network regions are brighter than IN regions so that
the summed counts of a time slice provide some infor�
mation on the NW�IN distribution along the slit. From
the intensity power spectra in Figs. 3 and 4 it is difficult
to see whether there is a relation between the location
of the peak power in a power spectrum above 2.65 mHz
and the presence of NW or IN. Therefore we determined
for all intensity power spectra, at frequencies above 2.65
mHz, the frequencies at which the power exceeds the de�

J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere 9
Fig. 7. Average power spectra, based on the 31 July data, for some internetwork (IN) and network (NW) regions indicated in
Fig. 3. Also shown are the average power spectra along the whole slit (!slit?). The top row shows the N i and C ii intensity
power spectra. The middle row the velocity power spectra. The bottom row shows on the left the squared coherency spectra
and on the right the phase difference spectra for the N i and C ii intensity. The dotted line in the intensity power spectra, the
squared coherence and phase spectra (along the slit), are the results for the 30 July data.
tection limit. These frequencies are then plotted against
the summed counts of the corresponding pixel. The re�
sults are shown in Fig. 6 for N i, C ii, smoothed and un�
smoothed power spectra, and for both days. Whenever a
power is significant and is also the maximum power in a
power spectrum, it is plotted as a vertical dash. Whenever
it is merely significant it is plotted as a dot. The pattern
which arises for N i is that for high count rates (NW) the
maximum power is found in the range 3 -- 4 mHz while
for low count rates (IN) it is distributed over the range 3
-- 6 mHz. For C ii a similar pattern is visible but for this
line there is also significant maximum power below 3 mHz
together with many significant detections in the range 3 --
7 mHz (especially in the smoothed power spectra).
To investigate this further we averaged the intensity
and velocity power spectra, and the phase and squared
coherency spectra, of some IN and NW regions and also
took averages along the whole slit. The selected IN and
NW regions are indicated by thick and thin vertical lines,
respectively, in Fig. 3. The result is shown in Fig. 7. The
columns indicate IN, NW and slit�averaged results. The
first row shows the N i and C ii intensity power spectra, the
second row the velocity power spectra and the third row
the squared coherency and phase difference spectra. For
the slit�averaged intensity, squared coherency and phase
spectra we have also plotted the 30 July results (dotted
lines). In the power spectra the expected noise level equals
2. Most power spectra indeed converge towards this value
at high frequencies except for the NW velocity power spec�
trum of C ii. We have no explanation for this behaviour.
The power spectra shown in Fig. 7 demonstrate that
at near zero frequency the NW contains more power than
the IN. In the N i intensity power spectra the NW has
a distinct peak at 3 mHz while in the IN the power is
distributed over a broader interval between 3 and 6 mHz.
This is also found in the N i IN velocity power spectra.
The N i NW velocity power spectra show a weak peak
near 3 mHz. Comparison of the N i NW and IN velocity
power spectra explains why in Fig. 3 the IN shows the
typical bands of power towards higher frequencies. The
C ii power spectra are characterized by decreasing tails
towards higher frequencies. In the NW power spectra there
is a weak indication of a feature near 3 mHz.
The squared coherency is low at all frequencies and
equals on average �0.3. This implies that the errors on the
phase angles are about \Sigma16 ffi for a 95% confidence limit.
From this one can not conclude that there are statistically
significant phase differences other than zero since at the
frequencies where these occur, the squared coherency is
small. What is interesting is that the slit�averaged power,

10 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
squared coherency and phase spectra of both observing
dates do correspond very well and show the same overall
behaviour. In the slit�averaged phase spectrum all phase
differences below 2 mHz and above 7 mHz average out to
zero degrees. However, near 3 -- 4 mHz the data of both
days suggest a phase difference of � 20 ffi . This suggests,
but not more than that, that C ii lags N i by �16 seconds.
4. Discussion and conclusions
In this paper we have analyzed two datasets obtained with
SUMER on 30 and 31 July 1996. The observed lines are
N i 1318.99 � A and C ii 1334.53/1335.71 � A. We have looked
for differences between the power spectra of network and
internetwork regions, and searched for evidence of propa�
gating waves by considering phase differences in the emis�
sion of the above lines which are formed at different tem�
peratures.
The SUMER datasets contain relatively few temporal
samples, e.g., 62 and 124 in our case, while the data are
susceptible to photon noise. This makes it necessary to
apply controlled smoothing of the data in order to reduce
the variance of the noise. We describe in the Appendix a
procedure, based on Jenkins & Watts (1968), which makes
it possible to assign confidence levels to the power spectra.
For the cross�correlation studies of intensity variations we
show how the squared coherency spectrum can be used to
derive confidence limits for the phase difference (and the
gain) between two intensity time series.
Our results can be summarized as follows. In the spa�
tially resolved intensity power spectra (Fig. 3) there is
little difference between NW and IN regions. The power
spectra resemble those of the Ca ii H�index (Lites et al.,
1993, their Fig. 5). Average power spectra of IN and NW
regions show that NW regions do have a distinct peak at
3 mHz while in IN regions the power is enhanced between
3 and 7 mHz with a flat maximum at 3 -- 5 mHz.
In the spatially�resolved N i velocity power spectra
shown in Fig. 3, the NW regions have strong power below
2 mHz and are relatively featureless at higher frequen�
cies. IN regions are characterized by less power below 2
mHz (they are less bright) and the presence of (feature�
less) power in the range 2 -- 15 mHz. This power shows
up as darker strips which are reminiscent of Ca ii veloc�
ity spectra of Lites (et al., 1993, their Fig. 4). Averaged
IN and NW N i velocity power spectra show some resem�
blance with the averaged intensity power spectra although
the distinct features are less pronounced. The spatially�
resolved velocity power spectra of C ii show the same pat�
terns as the N i velocity power although in averaged power
spectra less distinct features are present and the power
drops more rapidly towards higher frequencies.
C ii intensity power spectra simply behave as decreas�
ing tails towards high frequencies with perhaps a weak
feature near 3 mHz.
There is no straightforward explanation for the differ�
ent behaviour of intensity and velocity power of a specific
line. R.J. Rutten (private communication) has suggested
that both lines are optically thick. The behaviour of the
intensity is then the result of a complicated interplay be�
tween the dynamics of the regions and radiative transfer
effects. The line shift basically contains information of the
location of the last scattering and is therefore more sen�
sitive to local conditions than the line intensity. However,
for optically thick formation it is questionable whether the
line profiles are really Gaussians. Our Gaussian fits then
merely describe some average changes in the line profile.
Our results do not really confirm the results by Carls�
son et al. (1997) that 3 minute oscillations are present in
the grain�like pattern exhibited by N i in IN regions. Eye
estimates by Carlsson et al. of the UV continuum vari�
ations resulted in a typical time scale of 200 seconds (5
mHz) for the brightenings. The power spectra of N i an�
alyzed here for the IN, indicate that the spectra do not
have a distinct peak at 5 mHz but a broad maximum. The
power gap between 1 and 3 mHz seems to be character�
istic of intensity measurements in the upper photosphere
and was also present in Fleck & Deubner (1989) Ca ii IR
line power spectra. These authors suggested the presence
of gravity waves in the chromosphere.
We found no clear pattern of phase differences between
N i and C ii intensity. This is not surprising because al�
ready the C i versus 160 nm intensities are disconnected
in phase (Hoekzema et al. 1997). The fact that N i and C ii
are disconnected in phase also follows from the low values
of the squared coherency. For both the 30 and 31 July ob�
servations we found that if, the phase spectra are averaged
along the slit, there is the suggestion that the C ii inten�
sity lags the N i intensity by �16 seconds. If true, this re�
sult contradicts the findings by Carlsson et al. (1997) who
found that the brightenings are first observed in the UV
continuum and C i, then in C ii and thereafter in N i. The
sequence with C i brightening first and then C ii suggests
upward propagating (shock) waves. A possible explanation
to make the N i behaviour agree with the models for the
internetwork chromosphere proposed by Carlsson & Stein
is that the upward propagating shocks ionize N i and that
the line only becomes visible again after recombination.
For example, using the expression for radiative recombi�
nation by Mewe et al. (1980) we find a radiative recom�
bination time for N ii!N i of 26:6 T 0:68
f =(n=10 11 ) seconds
were T f is the formation temperature of the N i line and
n the density. This number is in good agreement with
the time difference of 25 seconds between the continuum
brightenings and N i found by Carlsson et al. This result
suggests that transient ionization may be an important
ingredient. If the shocks would also ionize C ii then the
above argument may not be valid. However, the recombi�
nation time for C iii!C ii is only 9 T 0:7
f =(n=10 11 ) seconds
which is much shorter, even shorter than the exposure
times we used. For lower densities the recombination time

J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere 11
increases strongly. On the other hand, the result of the
present observations (N i brightening first, then C ii) can
be explained by upward propagating waves without invok�
ing ionization effects.
Using a contrast enhancement technique we found that
the grain�like pattern present in internetwork regions is
also present in network regions. We argue that the power
spectra can be affected by the fact that time sequences
contain only a limited number of grains and by the fact
that neighbouring streams of grains are not in phase. The
fluctuations in individual grains around the average grain
properties can generate spurious power peaks. Sometimes
streams of brightenings are observed with relatively large
velocities along the slit. We suggest that these are not
physical features which move but instead are signatures
of a wave interference pattern like in the Ca ii V/R plot
by Cram & Dam'e (1983, their Fig. 4).
Previous authors have argued strongly in favor of the
dominance of a particular power peak in different regions,
e.g., network versus internetwork. Steffens et al. (1999)
have analyzed velocity oscillations of 15 spectral lines
observed with SUMER. For internetwork regions, they
found that the chromospheric 3 min (5.5 mHz) oscilla�
tions increases with respect to the 5 min (3.3 mHz) oscil�
lations with increasing formation height until a maximum
is reached in lines formed at about 50,000 K. The absence
of a 5 min or 3 min peak in their power spectra, for lines
formed at coronal heights lends supports to the assump�
tion that the transition layer acts as a reflecting barrier
for the internetwork waves, or that they are dissipated.
Furthermore, Gouttebroze et al. (1999) did not find any
evidence of oscillations for lines formed at temperatures
higher than 5 10 5 K. They conclude that the waves in
the frequency range 2:5 \Gamma 7 mHz are entirely reflected (or
dissipated) by the chromosphere�corona transition region.
Our results show that (power) spectral features only be�
come apparent when a sufficient number of power spectra
is averaged. At the formation height of N i distinct fea�
tures are clearly recognizable but at the formation height
of C ii these have almost disappeared.
Acknowledgements. Research at Armagh Observatory is grant�
aided by the Dept. of Education for N. Ireland while par�
tial support for software and hardware is provided by the
STARLINK Project which is funded by the UK PPARC. This
work was supported by PPARC grant GR/K43315. G.H.J van
den Oord acknowledges financial support from the Dutch Or�
ganization for Scientific Research (NWO). We like to thank
the SUMER and EIT teams at Goddard Space Flight Cen�
ter for their help in obtaining the present data. The SUMER
project is financially supported by DLR, CNES, NASA, and
the PRODEX programme (Swiss contribution). SUMER is
part of SOHO, the Solar and Heliospheric Observatory of ESA
and NASA. We would like to thank the referee Rob Rutten
for his wide�ranging comments which resulted in substantial
improvements.
Appendix A: Time series analysis
In this section we discuss the uni� and bivariate analysis of
observed time series. The noise component in the signals
has, in general, a relatively large variance which affects
the significance of features in the power spectra. In the
following we discuss how controlled smoothing of the time
series leads to a reduction of the variance and makes it
possible to derive the required information from a signal.
The smoothing is achieved by windowing in the time do�
main which corresponds to a convolution in the frequency
domain. We closely follow the treatment by Jenkins &
Watts (1968). We start by summarizing various relevant
quantities.
Consider two time series x p and y p , with p = 0; : : : ; N \Gamma
1, sampled at intervals of \Delta seconds. These time series
are considered as realizations of two processes X(t) and
Y (t). A statistical description requires knowledge of the
following quantities:
ffl the mean values of the series
x = 1
N
N \Gamma1
X
p=0
x p ; y = 1
N
N \Gamma1
X
p=0
y p . (A1)
ffl the auto�covariance function (acvf) estimates
c xx (`) = 1
N
N \Gamma1\Gamma`
X
p=0
(x p \Gamma x)((x p+` \Gamma x) , (A2)
c yy (`) = 1
N
N \Gamma1\Gamma`
X
p=0
(y p \Gamma y)((y p+` \Gamma y) . (A3)
Note that c xx (0) = oe 2
x and c yy (0) = oe 2
y are the variances
of the time series. The functions c xx (`) and c yy (`) are even
functions of lag `.
ffl the cross�covariance function (ccvf) estimates
c xy (`) = 1
N
N \Gamma1\Gamma`
X
p=0
(x p \Gamma x)((y p+` \Gamma y) , (A4)
c yx (`) = 1
N
N \Gamma1\Gamma`
X
p=0
(x p+` \Gamma x)((y p \Gamma y) . (A5)
In this case we have that c yx (`) = c xy (\Gamma`).
ffl the even and odd ccvf estimates
l xy (`) = (c xy (`) + c xy (\Gamma`)) =2 , (A6)
q xy (`) = (c xy (`) \Gamma c xy (\Gamma`)) =2 . (A7)
The function l xy (`) = l xy (\Gamma`) is even and the func�
tion q xy (`) = \Gammaq xy (\Gamma`) is odd. Related quantities are
the autocorrelation functions r xx (`) = c xx (`)=c xx (0) and
r yy (`) = c yy (`)=c yy (0), and the crosscorrelation function
r xy (`) = c xy (`)=
p
c xx (0)c yy (0).
When the number of temporal samples N in a series is
infinite the lag index ` is in the range \Gamma1 � ` � 1. For
finite N only a finite number of lags are considered in the

12 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
range ` = \GammaL; : : : ; L \Gamma 1 with L at most, approximately,
half times N . This can be achieved by multiplying the
functions c xx , c yy , l xy and q xy with a window function w(`)
satisfying w(\GammaL) = w(L) = 0. The resulting functions are
considered to be periodic with periodicity 2L. L is referred
to as the truncation point of the window. Next we consider
the discrete Fourier transforms (DFT) of these functions.
Because c xx , c yy and l xy are even, and q xy is odd,
the resulting DFT can be written in terms of cosine and
sine transforms. The index j corresponds to frequency
� j = j=(2L\Delta) with j = 0; : : : ; L. The following trans�
forms result
ffl the smoothed spectral estimates
C xx (j) = c xx (0) + 2
L\Gamma1 X
k=1
c xx (k)w(k) cos �jk
L
(A8)
C yy (j) = c yy (0) + 2
L\Gamma1 X
k=1
c yy (k)w(k) cos �jk
L
(A9)
ffl the smoothed co� and quadrature spectral estimates
L xy (j) = l xy (0) + 2
L\Gamma1 X
k=1
l xy (k)w(k) cos �jk
L
, (A10)
Q xy (j) = 2
L\Gamma1 X
k=1
q xy (k)w(k) sin �jk
L
, (A11)
with Q xy (0) = Q xy (L) = 0.
From these (smoothed) estimates we can define, re�
spectively, the cross amplitude spectral estimate A xy , the
gain G xy , the phase spectral estimate F xy and the squared
coherency spectral estimate K 2
xy
A xy (j) =
q
L 2
xy (j) +Q 2
xy (j) , (A12)
G xy (j) = A xy (j)=C xx (j) , (A13)
F xy (j) = arctan
/
\Gamma Q xy (j)
L xy (j)
!
, (A14)
K 2
xy (j) = A 2
xy (j)
C xx (j)C yy (j)
. (A15)
The phase difference between x p and y p at frequency � j is
given by F xy (j). A positive phase implies that x p lags y p
by F xy (j)=(2�� j ) seconds.
Multiplication of the quantities c xx (k), c yy (k) and
c xy (k) with the window function w(k) in real space cor�
responds to a convolution of the FT of these quantities
with the FT of the window function in frequency space:
F [v(`)w(`)] = F [v] \Lambda F [w]. As such the effect of multiplica�
tion with a window function corresponds to a smoothing
operation in frequency space. The amount of smoothing
depends on the width of the window function which is
determined by its truncation point L (therefore the sum�
mations in the expressions above end at L \Gamma 1). The ef�
fective bandwidth of a window depends on its shape and
is given by fi=(L\Delta) Hz where fi is the standardized band�
width. Several windows and their properties are given in
Table A1.
Unsmoothed spectral estimators result in N=2 esti�
mates in the range zero�frequency -- Nyquist frequency.
These estimates are distributed according to � 2
2 except
at zero frequency and at the Nyquist frequency which are
distributed according to � 2
1 . Here � 2
� is the chi�squared dis�
tribution with � degrees of freedom (dof). For smoothed
estimates the number of dof equals � = 2(N=2)=(L=2fi) =
2fiN=L. The case of unsmoothed estimators is recovered
by applying a rectangular window with a total width
2L = N resulting in � = 2 as stated above.
To appreciate the idea behind smoothing con�
sider a process X(t) with a spectrum \Gamma XX (f) =
lim N!1 E[CXX (f)] where E[: : :] indicates the expecta�
tion value. The relation between the spectrum and the
smoothed spectrum is given by the bias B(f) defined as
B(f) = \Gamma XX (f) \Gamma \Gamma XX (f) .
The variance of the smoothed spectral estimate is approx�
imately
Var[CXX (f)] � \Gamma 2
XX (f)L
N fi
= 2 \Gamma 2
XX (f)
�
.
A good spectral estimate requires the bias to be as small
as possible at each frequency. The above expression shows
that this occurs when the amount of smoothing is small
(zero) which requires L to be as large as possible. On
the other hand, large values of L correspond to large val�
ues of the variance of the smoothed spectral estimate. A
small variance requires small values of L implying consid�
erable smoothing at the expense of a considerable bias.
A good estimate corresponds to minimizing the quantity
Var[CXX (f)] +B 2 (f).
A problem one often encounters in, e.g., solar physics is
that the true process X(t) and its true spectrum \Gamma XX (f)
are not known so that an estimate of the bias is not possi�
ble. One approach to this problem is that one can assume
that the process X(t) is some type of noise with known
properties and then derive confidence (detection) levels
for any superimposed signal. For Normally distributed
white noise \Gamma XX (f) = oe 2
x and �CXX (f)=oe 2
x is exactly
distributed as � 2
� while for non�Normal processes it is ap�
proximately distributed as � 2
� provided N is large. We use
this property to derive confidence levels for the estimates.
A.1. Confidence levels
The quantity �C xx =c xx (0) (or �C yy =c yy (0)) is distributed
as � 2
� except at zero frequency and the Nyquist frequency.
For Poisson noise the variance c xx (0) equals the mean
x. The total number of photon counts received equals
N ph;x = Nx. We define the power spectrum estimate as
P xx = 2NC xx =N ph;x . (A16)

J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere 13
Table A1. Several window functions and their properties.
window w(k) variance dof fi
jkj � L ratio �
rectangular 1 2 L
N
N
L
1
2
Bartlett 1 \Gamma jkj=L 2
3
L
N 3 N
L
3
2
Tukey 1
2
\Gamma 1 + cos �k
L
\Delta 3
4
L
N
8
3
N
L
4
3
It is straightforward to show that the expectation
value and variance of this estimator equal, respectively,
E[P xx ] = 2 and Var[P xx ] = 8=�. The unsmoothed es�
timate (applying a rectangular window so that � = 2)
has an expectation value 2 and a variance of 4 which is
rather large. Table A1 shows that other windows have a
larger number of dof and hence result in a reduction of the
variance. The estimator P xx corresponds to the so�called
Leahy normalization in QPO research (Leahy et al. 1983,
van der Klis, 1989) and was used in Doyle et al. 1997,
1998).
Suppose that we consider a spectrum determined at
n t frequencies. The powers of P xx (j) can be considered
as n t trials of the distribution. For each individual power
there is a probability equal to ff that it is due to noise
and (1 \Gamma ff) that it is not due to noise. For n t powers the
joint probability distribution has to be evaluated. The n t
powers are considered to be independent and obeying the
same probability distribution. Then the probability for n t
powers not to be caused by noise is (1 \Gamma ff) 1=n t and to be
caused by noise is 1 \Gamma (1 \Gamma ff) 1=n t � ff=n t . Associated with
ff is a detection level P det so that
ff
n t
= Q
` �P det
2 j�
'
(A17)
with Q(� 2 j�) the integral probability of the � 2
� distri�
bution. For unsmoothed estimators � = 2 and P det =
2 ln(2n t =ff). A higher confidence level corresponds to a
smaller value of ff and therefore a higher detection level.
Summarizing, in the above outlined approach one assumes
that a pure noise process is observed and uses the proper�
ties of the noise to derive a confidence level for the spectral
estimates P xx (j) and, similarly, P yy (j). Because we ex�
clude zero frequency and the Nyquist frequency we have,
in general, that n t = L \Gamma 2.
Next we consider confidence levels for the parameters
of the bivariate analysis. For the cross amplitude squared
A 2
xy it is not straightforward to derive confidence levels.
This can be seen as follows. Since A 2
xy = jC 2
xy j = C xx C yy
it follows that
� 2 A
2
xy
oe 2
x oe 2
y
=
`
�C xx
oe 2
x
'`
�C yy
oe 2
x
'
is distributed as the product of two chi�square distribu�
tions each with � dof. The expectation value equals � 2
and the variance 4(� 3 + � 2 ) provided X(t) and Y (t) are
uncorrelated. This expectation value and variance already
indicate that � 2 A 2
xy =oe 2
x oe 2
y does not obey a simple distri�
bution. The physical information in the cross amplitude
squared is however limited. Therefore we do not elaborate
on the statistical properties of this estimator but concen�
trate instead on the gain and the phase. The basic idea
behind confidence levels for the gain and the phase is to
consider the case that two signals are completely uncorre�
lated so that any apparent correlation is due to noise.
Consider the following model for the relation between
two processes Y (t) and X(t) from which we have observed
some finite length time series
y(t) =
Z 1
\Gamma1
h(u)x(t \Gamma u)du + z(t) (A18)
with h(u) the response function and z(t) a white noise pro�
cess. For finite length records one can make a least squares
estimate for the response function ( � h(u)) based on the acvf
and ccvf of the data sets (c xy (u) =
R � h(u)c xx (u \Gamma v)du).
The FT of the least squares estimate satisfies �
H(f) =
C xy (f)=C xx (f). The FT of Eq. (A18) equals approxi�
mately Y (f) � H(f)X(f) + Z(f) which can be written
as
Z(f) � fY (f) \Gamma X(f) �
H(f)g + X(f)f �
H(f) \Gamma H(f)g
so that
jZ(f)j 2 � jY (f) \Gamma X(f) �
H (f)j 2 + jX(f)j 2 j �
H(f) \Gamma H(f)j 2 .
The estimate of the noise at frequency f is given by
�
Z(f) = Y (f)\GammaX(f) �
H (f) so that the above expression can
be written as C zz (f) � C �
z�z (f) + C xx (f)j �
H(f) \Gamma H(f)j 2 .
For smoothed estimates we can replace this expression by
�C zz (f)
\Gamma zz (f) = �C �
z�z (f)
\Gamma zz (f) + �C xx (f)
\Gamma zz (f) jH(f) \Gamma H(f)j 2
where H(f) = G xy (f) exp(\GammaiF xy (f). The term on the
left�hand side is distributed according to � 2
� and the terms
on the right�hand side according to � 2
� \Gamma2 and � 2
2 , respec�
tively. Suppose now that the time series are completely un�
correlated in which case H(f) = 0 and in the last term on
the right we can write C xx (f)jH(f)j 2 = C xx (f)G 2
xy (f) =
C yy (f)K 2
xy (f). The first term on the right can be written
as C � z�z (f) = C yy (f)f1 \Gamma K 2
xy (f)g. Using the � 2 properties
of the terms on the right implies that the random variable
constructed from their ratio
Z(j) j
(� \Gamma 2)K 2
xy (j)
2(1 \Gamma K 2
xy (j))
(A19)
is distributed according to the F 2;� \Gamma2 distribution. Let
f 2;� \Gamma2 (ff) indicate the 100(1 \Gamma ff)% confidence level of this
distribution then
f 2;� \Gamma2 (ff) = � \Gamma 2
2
` 1
ff 2=(� \Gamma2) \Gamma 1
'
. (A20)

14 J.G. Doyle et al.: Exploring the dynamical nature of the lower solar chromosphere
Combining Eqs. (A19) and (A20) results in a confidence
value for K 2
xy given by
K 2
xy = 1 \Gamma ff 2=(� \Gamma2) . (A21)
So all K 2
xy (j) � K 2
xy satisfy the confidence criterium.
Without smoothing (� = 2) K 2
xy = 1. Using Eqs. (A12)--
(A15) it follows that the approximate 100(1 \Gamma ff)% confi�
dence levels for the gain and the phase at frequency j are
given by
G xy (j)(1 \Sigma
q
f 2;� \Gamma2 (ff)=Z(j)) (A22)
and
F xy (j) \Sigma arcsin
q
f 2;� \Gamma2 (ff)=Z(j) . (A23)
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