\thesischapter{Oscillations in {\cak} Observations of the Quiet-Sun Network}
\label{cak_longper}

\hrule height 1mm
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\noindent 
\\{\bf The spatial variation of chromospheric oscillations in network bright points (NBPs) is studied using high-resolution observations in \cak. Lightcurves, and hence power spectra, are created by isolating distinct regions of the NBP via a simple intensity thresholding technique. Using this technique, it is possible to identify peaks in the power spectra with particular spatial positions within each NBP. In particular, long-period oscillations (4--15~min; 1--4~mHz) were found in the central portions of each NBP, indicating that waves present are certainly not acoustic. These oscillations are possibly due to magneto-acoustic or magneto-gravity wave modes. We also show that spatially averaged or low spatial resolution power spectra can lead to an inability to detect such long-period waves.}
\\
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\section{Introduction}
\label{cak_intro}

The solar network, defined as the boundaries of supergranular flows, contains oscillation with periods longward of the acoustic cut-off and the Brunt-V\"{a}is\"{a}ll\"{a} cut-off (the low-frequency cut-off for gravity waves). These are not correlated with underlying fluctuations (Lites {\etal} 1993; Cauzzi, Falchi, \& Falciani 2000), and their origins have been attributed to magnetic stochastic processes (Kneer \& von Uexk\"{u}ll 1985; 1986; von Uexk\"{u}ll {\etal} 1989), or events spatially removed from the network, e.g., granular overshoot. Imaged in the light of {\cak} (3933.7\AA), the network is only partially defined, as brightenings at vertices of several supergranules. Due to the coincidence of these network bright points (NBPs) with high magnetic field concentrations, magnetohydrodynamic (MHD) waves (Kalkofen 1997; Hasan \& Kalkofen 1999) and magneto-gravity waves (Dam\'{e}, Gouttebroze, \& Malherbe 1984; Deubner \& Fleck 1990; Kneer \& von Uexk\"{u}ll 1993) seem to be the modes most likely present. The mechanism responsible for heating the network is still the topic of much debate, with a recent publication by Rosenthal {\etal} (2002) beginning to readdress the problem.

The existence of low-frequency (1--4~mHz) oscillations in the solar network of the quiet Sun is well known. Table~\ref{cak_summary} presents a summary of results from published literature (the frequency resolution term in Table~\ref{cak_summary} will be explained in Section~\ref{cak_four}). There are several points concerning these results which need to be addressed. Firstly, most studies simply integrate over all network pixels (normally defined as areas which have a higher intensity than some arbitrary value). This means that the lightcurve will be that of the entire network in the FOV, and any small-scale, localised oscillations may be washed out by stronger, global oscillations. There are some exceptions to this: Cauzzi {\etal} (2000) identify eleven separate NBPs, but then average over them; Kariyappa (1994) identifies three individual NBPs; Lites {\etal} (1993) and Curdt \& Heinzel (1998) both study only one. Other studies take a different approach, by creating a lightcurve for each and every pixel inside a network `mask'. However this method is very susceptible to errors due to misalignment and changes in atmospheric seeing, and normally any resulting Fourier power spectra will once again be averaged over the entire network.   

Secondly, few results identify specific Fourier peaks, instead preferring to refer to either `low-frequency power' below a specific value (e.g., Kneer \& von Uexk\"{u}ll 1993) or around the `acoustic band' (e.g., Bocchialini, Vial, \& Koutchmy 1994). Of course it must be recognised that this tendency to refer to a frequency band may simply be because of the requirements of the study (e.g., comparing oscillatory power between the network and internetwork). Other studies also prefer to use phase and coherence analysis (e.g., Deubner \& Fleck 1990), and do not concentrate on identifying peaks in Fourier power. In cases where specific peaks are identified, their validity may be compromised as they correspond to periods which may be due to insufficiently long observing windows. Through this argument, Kalkofen (1997) discounts all but the 2.7~mHz frequency in Dam\'{e} {\etal} (1984), and that at 2.5~mHz in Lites {\etal} (1993). A similar argument can be used to discount the 0.6~mHz period discovered by Cauzzi {\etal} (2000). Furthermore, the smoothing window used by Cauzzi {\etal} (2000) directly affects the 1.3~mHz oscillation found by these authors. 

Thirdly, the importance of the frequency resolution and other erroneous Fourier effects must be fully recognised. A poor frequency resolution may result in some peaks being omitted (Section~\ref{cak_filt}). The bottom half of Table~\ref{cak_summary} are results from {\sumer} (Solar Ultraviolet Measurements of Emitted Radiation, Wilhelm {\etal} 1995) and {\trace} data. Although the long duration observations, free of atmospheric distortions, makes these data very useful, it should be noted that most results from {\sumer} studies have been omitted. The reason for this is that in the vast majority of {\sumer} studies of the quiet Sun, solar rotation is not followed. Hence a new area of the Sun rotates into the {\sumer} slit every 380~s, introducing a low-frequency cutoff at 2.65~mHz. With only taking specific peaks into account, and addressing the frequency resolution problem, all non-discounted frequencies can be reduced to two broad peaks, one at 2.4$\pm$0.3~mHz, and a second at 3.0$\pm$0.3~mHz.

In this chapter I use a `contour \& contrast' technique to discover further peaks in NBP power spectra. Furthermore these peaks are identified with particular spatial positions in each NBP. The important improvements over previous work are, \\
\indent - The careful identification of seven individual NBPs. \\
\indent - The narrow-band {\cak} filter. \\
\indent - Long duration observations leading to good frequency resolution. \\
\indent - Excellent seeing and high spatial resolution. \\
\indent - Creation and implementation of a new filtering routine. \\

\begin{landscape}
\begin{table}
\begin{center}
\caption[Oscillations found in network regions.]{Summary of frequencies discovered in network regions in literature. The final column contains details of the full width half maximum (FWHM) of the passband, and whether the oscillations were found in intensity (I) or velocity  (V).}
\label{cak_summary}
\begin{tabular}{lllcccl} 
\hline
\hline
Reference       		& Wavelength 		& Frequencies  		& Dur.	  & Freq. Res.  & Cadence & Notes\\
				&			&    (mHz)      	& (mins)  &   (mHz)	& (secs)  &	  \\
\hline
This study			& \cak			& 1--4			& 150	  & 0.11	& 45	  & 0.3{\AA} FWHM bandpass\\
\hline
Dam\'{e} {\etal} 1984		& \cak			& 1.3,~2.0,~2.7,~3.2 	& 52	  & 0.32	& 10	  & 1.2{\AA} FWHM bandpass\\
Kneer \& von Uexk\"{u}ll 1986	& \ha, \ha$\pm$0.45\AA 	& $<$2.4,~$\sim$3.3	& 54	  & 0.31	& 20	  & I only\\
von Uexk\"{u}ll {\etal} 1989	& \ha			& 2.4,~(3.3$^a$)	& 64	  & 0.26	& 30.4	  & $^a$ V only\\
Kneer \& von Uexk\"{u}ll 1993   & \ha			& 3.5,~$<$2.4		& 128     & 0.13        & 15      & 0.25{\AA} FWHM bandpass\\
				& Ca~{\sc ii}~K 	& 3.2,~$<$2.4 		& 128	  & 0.13	& 15	  & 0.6{\AA} FWHM bandpass\\
				& Mg~{\sc i}~b$_2$ 	& $<$2.4		& 128	  & 0.13	& 15	  & 0.3{\AA} FWHM bandpass\\	
Deubner \& Fleck 1990		& Ca~{\sc ii}~8498\AA \& 8542\AA   & 3.3,~$<$2	& 275	  & 0.06	& 10	  & V\&I\\
Kulaczewski 1992		& Mg~{\sc i}	 	& 3.3--5		& 80	  & 0.21	& 15	  & V\&I\\
				& Ca~{\sc ii}~K$_3$ 	& $<$2			& 80	  & 0.21	& 15	  & V\&I\\
				& Ca~{\sc ii}~8542\AA	& $<$3			& 75	  & 0.22	& 15	  & V\&I\\	  
Lites {\etal} 1993		& Ca~{\sc ii}~H$_3$	& 0.9,~1.8,~2.5		& 62	  & 0.27	& 5	  & V only, 1 NBP\\	  
Kariyappa 1994			& Ca~{\sc ii}~H$_{2V}$	& 2.4			& 35	  & 0.48	& 12	  & I only, 3 NBPs\\
Bocchialini {\etal} 1994	& He~{\sc i},\cak	& $\sim$3.3		& 83	  & 0.20	& 5	  & V\&I, some low freq peaks\\
Bocchialini \& Baudin 1995	& He~{\sc i},Ca~{\sc ii}& $\sim$3.3		& 83	  & 0.20	& 5	  & V only, wavelet study\\
Baudin {\etal} 1996		& He~{\sc i},Si~{\sc i}	& $\sim$3.3		& 83	  & 0.20	& 5	  & V only, wavelet study\\
Cauzzi {\etal} 2000		& {\ha}	   		& 0.6, 1.3, 2.2		& 50	  & 0.33	& 12	  & 0.25{\AA} FWHM bandpass, 11 NBPs\\
\hline
Curdt \& Heinzel 1998		& {\lya}		& 2.2--2.4		& 33	  & 0.50	& 33.5    & {\sumer} \\
Krijger {\etal} 2001		& 1700,1600,1500	& ~3.5			& 90	  & 0.19	& 15.0523 & {\trace}, all network \\
				& 1700,1600,1500	& ~3.5			& 228	  & 0.07	& 21.8625 & {\trace}, all network \\
Judge {\etal} 2001		& JOP 72 data		& $<$5			&\multicolumn{3}{l}{multiple datasets}& {\trace}, {\sumer}, {\mdi}\\
Krijger {\etal} 2003		& 1216, 1600 		& $<2$, 3--4		& 48	  & 0.35	& 40	  & {\trace}  \\
				& Lyman series (5,9,15)	& 2--3			& 48	  & 0.35	& 29	  & {\sumer}\\		
				& Ca~{\sc ii}~K$_{2V}$	& 4-5			& 48	  & 0.35	& 6	  & 0.6{\AA} FWHM bandpass\\  	
\hline
\end{tabular}
\end{center}   
\end{table}
\end{landscape}
\newpage
\clearpage

\section{Data Analysis}
\label{cak_data_anal}

The dataset used for this study consisted of a 150-minute sequence of 100{\arcsec}~$\times$~100{\arcsec} {\cak} (bandpass FWHM 0.3\AA) images centred on quiet Sun (Solar\_X = -25\arcsec, Solar\_Y = -35\arcsec) at a 45~s cadence. The sequence was part of the dataset described in Chapter~3 taken at the {\dst} (Sacramento Peak / National Solar Observatory), on 1998 September 22 under excellent seeing conditions. Co-alignment of the images with co-spatial and co-temporal {\trace} data led to a calculated spatial size for the {\cak} images of 0.33{\arcsec} per pixel. All images were initially flat fielded and dark-current subtracted. Each image was then aligned to a sub-pixel accuracy to the first image in the series by means of cross-correlation. 

\subsection{Image Segmentation}
\label{cak_imag_segm}

\begin{figure}
\begin{center}
\epsfig{figure=cak_image.eps,width=5in}
\caption[Time averaged \cak\ image]{Time averaged FOV of the \cak\ dataset. Each NBP which existed throughout the dataset is numbered}
\label{cak_fov}
\end{center}
\end{figure}

Seven NBPs were clearly identified (Figure~\ref{cak_fov}) which exist throughout the entire dataset and sub-fields of 30\arcsec\ $\times$ 30\arcsec\ centred on each NBP were extracted. This created seven 3-dimensional data cubes (two spatial and one temporal dimension) which were then analysed using custom-written programs in SolarSoftWare {\idl} ({\sc sswidl}; Freeland \& Handy 1998).

For each NBP subset, contours were drawn on each image in the sequence at a chosen intensity threshold value. This contouring ensures that each NBP is followed carefully over time (after Cauzzi {\etal} 2000). It also provides a means of separating the NBP brightenings from the internetwork contribution. Integrating over entire pixel subfields will include signal from both solar (internetwork) and non-solar (sky fluctuations) oscillatory components. Here we add a further improvement to the contour method of Cauzzi {\etal} (2000), that allows us to better isolate each individual NBP. A routine was used to ensure only pixels within a {\em closed} contour around the NBP were used. This ensures that erroneous pixels -- due to sudden fluctuations in the background, supergranular lane flow or other NBPs -- are ignored.  
\subsection{Contrast Lightcurves}
\label{cak_cont_ligh}
Lightcurves were created using the following method. For a single image $i$, a histogram of the pixel intensity is constructed, from which $I^{bck}$, the most common (modal) pixel value of the image, is selected to represent the background. A threshold pixel intensity, $I^{thr}$ is then chosen to contour the NBP. All pixels inside this contour are included in the NBP, and $I^{NBP}$ is calculated as the average intensity value of all these pixels. A further improvement on previous work is achieved by not choosing a constant {\em absolute} threshold for each image in the series. Instead a constant {\em relative} threshold is chosen throughout, as a number of standard deviations above the mean for the (invariably different) standard deviation and mean of each image. The contrast, $C_i$, of each image, i, is then defined as,
\begin{equation}
C_i = \frac{I^{\mathrm{NBP}}_i-I^{\mathrm{bck}}_i}{I^{\mathrm{NBP}}_i+I^{\mathrm{bck}}_i} \ . 
\label{contrast_eqn}
\end{equation}
This is repeated for every image in the sequence for a given threshold, such that the final lightcurve is then a series of $C_i$ values. 

\begin{figure}
\begin{center}
\epsfig{figure=threshold.eps,width=4.5in}
\caption[Choice of threshold for an NBP]{Choice of threshold for an NBP. Top row (Image A) - Left plot:  Histogram of pixel intensities. The solid line is at the mean, and the dashed line is at 2$\sigma$ above the mean. Right image: The dashed contour is at the intensity value given by the dashed line of the plot on the left. Bottom row (Image B) -  Left plot: Histogram of pixel intensities resulting from the same image as the top row, but with 10\% less intensity. The solid line is at the mean, the dashed line is at 2$\sigma$ above the mean, and the dash-dot line is at the same absolute threshold as the top row. Right image: Both the dashed and dash-dot lines are contoured at the intensities given by the plot on the left. The dash contour on the bottom image contains the same pixels as the contour on top image. The dash-dot contour contains fewer pixels.}
\label{threshold}
\end{center}
\end{figure}

This lightcurve construction removes any long-term trend due to sky fluctuations. For example, assume an NBP exhibits no change in intensity but a thin cloud results in a decreased intensity of 10\% between two sequential images, A~\&~B (Figure~\ref{threshold}). Hence $I^{\mathrm{Mod}}$, the mean ($\overline I$), and the standard deviation ($\sigma$) will all decrease by 10\% in the second image. If the same absolute threshold intensity is used in both images, fewer pixels will be contoured in image B (Figure~\ref{threshold} dash-dot line), which will alter $I^{\mathrm{NBP}}$ erroneously. However if a constant relative threshold (e.g., 2$\sigma$ above the mean) is used, the same exact pixels will be contoured in each image (Figure~\ref{threshold} dash line). As this results in a decrease of 10\% in both $I^{\mathrm{Mod}}$ and $I^{\mathrm{NBP}}$ the contrast value of each image will be the same. 

\begin{figure}
\begin{center}
\epsfig{figure=f1.eps,width=4.5in}
\caption[Description of thresholding contours on an NBP]{Description of thresholding contours on an NBP. (a) Sample NBP. The dark lines show the contour for all pixels in the image which are $1.0\sigma$ (thickest line), $2.0\sigma$ and $3.0\sigma$ (thinnest line) above the mean. Only pixels inside the three centre-most contours are included. The white line contours the background level.(b) Corresponding histogram of the sample image. The background level is the modal value. The three vertical lines refer to the contour levels of (a).}
\label{cak_net1}
\end{center}
\end{figure}

After the creation of the contrast lightcurve the threshold value was increased and a new lightcurve created. The threshold was chosen to increase from a minimum of $1\sigma$ above the mean, in steps of $0.05\sigma$, 
\begin{equation}
I^{thr} = {\overline I} + (n{\sigma}) \ ,
\end{equation}
$n \in (1.0, 1.05, 1.10, .... N) $,
where N is chosen so that the contour confines at least ten pixels in all images in the sequence (this normally occurred in only one image per sequence, such that the average number of pixels in an image at maximum threshold was around one hundred). With this method, increasing the threshold corresponds to better isolation of the very centre of the NBP (Figure~\ref{cak_net1}a), and provides a means of probing and comparing different parts of the NBP. Essentially this increasing threshold creates a series of nested contours in the NBP.  

This `contour \& contrast' method can only be applied to data with a high spatial resolution in two simultaneous spatial dimensions and with good seeing conditions. A low spatial resolution means few pixels will be contoured at high threshold. Spectrometers normally only have one spatial dimension so require an image to be built up by scanning the slit across a region. The final image will therefore contain contributions from different points in time. In 'sit and stare' mode the slit may be too small to contour the NBP correctly, and causes an extra low-frequency cutoff (Section~\ref{cak_intro}). Finally, poor seeing will lead to poor image contrast, increasing the difficulty of separating the NBP from the background intensity. 

\subsection{Fourier Analysis}
\label{cak_four}

A Fourier analysis was carried on each lightcurve using routines written in {\sc sswidl}. For a continuous signal, $f(t)$, sampled at $N$ discrete values, $f_k$ ($k=0, 1, 2, ...., N-1$), the discrete Fourier transform (DFT), $F_j$ ($j=N/2, N/2-1, ..., -N/2+1$) is given by,
\begin{equation}
F_j = \sum_{k=0}^{N-1}f_{k}e^{\frac{-2{\pi}ijk}{N}} \ .
\end{equation}
Each sample is separated by a time interval, ${\delta}t$, such that the entire signal lasts for time $T={\delta}t{\times}(N-1)$. Each value of $F_j$ corresponds to the Fourier transform at a frequency $\nu_j = j/N{\delta}t$, hence the maximum detectable frequency, termed the Nyquist frequency is given by,
\begin{equation}
\nu_{Ny} = \frac{N/2}{N{\delta}t} = \frac{1}{2{\delta}t} \ ,
\end{equation}
and the frequency resolution, is given by,
\begin{equation}
\delta\nu = \frac{1}{N{\delta}t} \ .
\label{freq_res}
\end{equation} 
A poor frequency resolution will blend peaks in the Fourier spectrum, and the Fourier analysis will be susceptible to the picket fence effect (Section~\ref{cak_filt}). 

Where the DFT will be complex and of the form~$a(j)+ib(j)$, the Fourier amplitude, $A_j$, phase, $\phi_j$, and power, $P_j$, can be determined from,
\begin{equation}
A_j = \sqrt{a^2+b^2} \ ,
\end{equation}
\begin{equation}
\phi_j = tan^{-1} \left( \frac{b}{a} \right) \ ,
\end{equation}
\begin{equation}
P_j = {A_j}^2 = a^2+b^2 .
\end{equation} 
A useful check at this stage is to ensure the resulting Power spectrum obeys Parseval's theorem, whereby the total Fourier power should be equal to the variance of the original signal, i.e.,
\begin{equation}
\sum_j P_j = \sigma^2 \ .
\end{equation} 

A study of the statistical significance of peaks in the power spectrum can be made by assuming a Gaussian distribution for each point in the signal. Hence each point in the power spectrum will have a {\large$\chi^2$} distribution, with two degrees of freedom (DOF). For a significance level, $sig$, (where $0<sig<1$), the corresponding value for the power spectrum, $P_{sig}$, will be
\begin{equation}
P_{sig} = \frac{\sigma ^2 {\mathbf\chi ^2}(1-sig, DOF)}{\frac{N}{2} DOF} \ .
\label{cak_signifcancelevel}
\end{equation}  
This equation is used throughout at both the 99\% ($sig$ = $0.99$, {\large$\chi^2$}(0.01,2) = 9.21) and 95\% ($sig$ = $0.95$, {\large$\chi^2$}(0.05,2) = 5.99) level. Essentially this means that if one hundred Fourier peaks were chosen at random, five would have power above the 95\% level, and one would have power above the 99\% level.

\subsection{Filtering}
\label{cak_filt}

\begin{figure}
\begin{center}
\epsfig{figure=filter_effect.eps,width=5in}
\caption[The effect or removing low-frequency power]{The effect of filtering the timeseries and power spectra of the \cak\ lightcurve for NBP2. The top row (panels a and b) gives the raw timeseries and its corresponding power spectrum, while the bottom row (panels c and d) shows the effect of the low-pass filter. (a) and (c) share the same x axis as do (b) and (d). The thick and thin horizontal lines on the power spectra refer to the 99\% and 95\% significance levels respectively.}
\label{cak_filtered}
\end{center}
\end{figure}

If the original continuous signal contains oscillatory power at frequencies greater than the Nyquist frequency (i.e., signal is under-sampled), aliasing may occur. Aliasing results in power at these higher frequencies showing up as power at low frequencies in the resulting power spectrum. Hence the first few components of the power spectrum may contain a lot of false power swamping any real peaks in the power spectrum. Figure~\ref{cak_filtered} shows the effect of filtering a lightcurve to remove power at low frequencies. The top row contains the original lightcurve and resulting power spectrum. The slowly varying nature of the lightcurve results in the large peaks at $<$~1~mHz. The bottom row displays the filtered lightcurve and resulting power spectrum. As a result of removing the slowly varying component, the standard deviation of the lightcurve is lower, decreasing the 99\% and 95\% siginificance levels (Eq.~\ref{cak_signifcancelevel}). Hence the peaks at around 3~mHz become 99\% significant. 

It is vital to remove these low frequencies without introducing erroneous effects. The most common methods of filtering are applied in the time domain (i.e., simply subtract the trend from the lightcurve, or perform a running average.) However these methods can change the overall shape of the lightcurve. It is statistically preferable to carry out the filtering in the frequency domain by convolving the lightcurve with a suitable Bessel function. The convolution theorem states that the Fourier transform of a convolution is simply given by the product of the individual transforms, i.e., for a convolution, $h(t)$, of two signals, $f(t)$, and $g(t)$, which have Fourier transforms, $H(\nu)$, $F(\nu$), and $G(\nu)$, respectively,
\begin{equation}
h(t)=f(t){\otimes}g(t) \ ,
\end{equation}
\begin{equation}
H(\nu)=F(\nu)G(\nu) \ .
\end{equation}
Let the lightcurve be $f(t)$, and the Bessel function be $g(t)$. Their Fourier power spectra, $F(\nu)$, and $G(\nu)$, are displayed in Figure~\ref{cak_filtered}b, and Figure~\ref{cak_digifilt}, respectively. Note the power spectrum of a Bessel function is simply a high-pass filter with a smooth cutoff. A sharp cutoff would introduce ringing effects into the Fourier analysis. Multiplying the two Fourier spectrum results in a high-pass filtered power spectrum, $H(\nu)$ (Figure~\ref{cak_filtered}d). An inverse Fourier transform of $H(\nu)$ then reproduces the high-pass filtered lightcurve $h(t)$ (Figure~\ref{cak_filtered}c).     

\begin{figure}
\begin{center}
\epsfig{figure=filter_atten.eps,width=5in}
\caption[High-pass filter frequency response curve]{The Fourier power spectrum of the chosen Bessel function. This acts
as the frequency response curve for the high pass filter and is used to remove low frequency variations. Only power under 1~mHz is attenuated.}
\label{cak_digifilt}
\end{center}
\end{figure}

For the data presented here, the filter profile (Figure~\ref{cak_digifilt}) was chosen as to have no effect on frequencies above 1.5~mHz, and only reduce the power at 1.0~mHz to 90\% of the unfiltered lightcurve. As well as aliasing, other factors of non-solar origin (e.g., atmospheric effects, edge effects) are also strong in this low frequency region.

There are two other important effects which need to be considered when using the DFT, namely {\em spectral leakage} and the {\em picket-fence} effect. Spectral leakage is a spreading, or leaking, of spectral components away from the correct frequency. This is a direct result of the finite length of the signal and hence the length is never an exact integral number of any periodicity which may be present. This leads directly into the picket fence effect whereby the actual Fourier peak may lie between two values of $\nu_j$. This effect can be reduced by padding the input signal with zeros, hence increasing the length of the signal and shifting the values of $\nu_j$. This shift may discover spectral components which were initially hidden. In the analysis of this dataset we incorporate an error value in each peak of $\pm\delta\nu$ (the frequency resolution). Hence a peak at $\nu_j = a$Hz refers to power in the range ($a\!-\!\delta\nu$) -- ($a\!+\!\delta\nu$)~Hz. This reduces the combined effects of spectral leakage and the picket-fence effect.  

\newpage
\section{Results}
\label{cak_resu}

\begin{figure}
\begin{center}
\epsfig{figure=f2.eps,width=5in}
\caption[Power spectra of a single NBP at different threshold values]{Power spectra of a single NBP at different threshold values where the solid and dotted lines refer to a threshold of 1.0 and 2.5 standard deviations above the mean, respectively. The 99\% and 95\% significance levels are shown, corresponding to the levels at which 1\% and 5\%  of data points, respectively, are expected to be due to random fluctuations.}
\label{cak_ft}
\end{center}
\end{figure}

All lightcurves at each threshold for all the NBPs were filtered, subjected to a Fourier analysis and the power spectrum inspected for significant peaks. In Figure~\ref{cak_ft} we show an example of how the power spectrum changes with increasing threshold. At high frequencies, the power spectra are very similar for both thresholds, showing no peaks above 5~mHz. At low frequencies the power spectra are markedly different. The power of the two largest peaks, at 3.11~mHz and 3.33~mHz, has increased by a factor of $\sim$2 when the higher threshold is chosen. Furthermore, a peak at 2.67~mHz has increased by a factor of $\sim$2.5 to over 95\% significance, and a peak at 2.11~mHz has increased by a factor of $\sim$5, to well above the 99\% significance level. 

A more detailed graph showing the development of peaks with increasing threshold is given in Figure~\ref{cak_thr}. This shows quite clearly the appearance of a 2.11~mHz oscillation at $\sim$1.5$\sigma$, peaking in power at $\sim$2.5$\sigma$ before levelling off around the 99\% significance level. The 3.33~mHz and 3.11~mHz frequencies more than double in power from minimum to maximum threshold. A 2.67~mHz frequency also appears, but only to the 95\% signficance level, whilst the higher value of 5.89~mHz (although staying at a similar power) becomes less statistically significant. Similar behaviour is repeated in all the other NBPs, but for different frequencies in the 1--4~mHz range. The fact that no single specific frequency dominates throughout the network, combined with the lack of this behaviour in internetwork regions, suggests that this is a real effect.  

\begin{figure}
\begin{center}
\epsfig{figure=f3.eps,width=5in}
\caption[Power against threshold for an example NBP]{Power against threshold for the NBP of Figure~\ref{cak_net1}. The $x$-axis gives the threshold in terms of $\sigma$ above the mean. The 99\% and 95\% levels are also shown for each threshold}
\label{cak_thr}
\end{center}
\end{figure}

A summary of the frequencies found in each NBP is shown in Table~\ref{cak_restab}. In this table the error is taken to be the frequency resolution (i.e., $\pm$0.11~mHz). There are two distinct types of behaviour described in this table. Some frequencies show a steady increase in power with increasing threshold and, in many cases, are confined to the centre of the NBP. These are denoted `{\it s}'. Other frequencies showed a steady increase  up to a threshold value, after which there is a plateau in power (or slight decrease). These are denoted as `{\it p}'. The `99' and `95' refer to the fact that the power of the frequency reached higher than the 99\% and 95\% significance levels, respectively. Frequencies at the 95\% significance level which only exist for a small range of threshold values are not included, as they may be random effects, although  we note the possibility of frequencies only existing in a very small part of a NBP. It also important to note the long duration of our observations means that even the lowest frequencies are trustworthy 

\begin{table}
\begin{center}
\caption[Frequencies detected in seven NBPs]{Summary of frequencies found in the seven NBPs, their significance (above {\bf 99}\% or 95\%), and behaviour as a function of thresholding intensity (steady increase --s, plateau --p). }
\label{cak_restab}
\begin{tabular}{cllllllll} 
\hline
\hline
	&\multicolumn{8}{c}{Frequency (mHz)}\\
NBP	&1.11	  	&1.56	    	&1.89     	&2.22	 	&2.78	    	&3.11      	&3.33	  	&3.78    \\
\hline
1	&    	  	&	    	&	      	&\bf{99}~\it{p} &95~\it{s} 	&\bf{99}~\it{s} &\bf{99}~\it{s} &	 	\\
2	&    	  	&95~\it{p}	&	      	&95~\it{s} 	&\bf{99}~\it{s} &\bf{99}~\it{s} &\bf{99}~\it{s} &\bf{99}~\it{p}	\\
3	&95~\it{s}	&\bf{99}~\it{s}	&	      	& 	 	&\bf{99}~\it{s} &95~\it{s} 	&\bf{99}~\it{s} &	 	\\
4	&    	  	&	    	&         	&95~\it{s} 	&95~\it{s} 	&95~\it{p} 	&          	&\bf{99}~\it{p}	\\
5	&\bf{99}~\it{p}	&	    	&         	&\bf{99}~\it{p} & 	    	&          	&\bf{99}~\it{p} &95~\it{p}	\\
6	&\bf{99}~\it{p}	&	    	&         	&95~\it{p} 	&\bf{99}~\it{s} &          	&\bf{99}~\it{p} &95~\it{s}	\\
7	&\bf{99}~\it{p}	&	    	&\bf{99}~\it{p}	&\bf{99}~\it{s} &	    	&          	&          	&	     	\\
\hline
\end{tabular}
\end{center}   
\end{table}


The most striking feature of Table~\ref{cak_restab} is that each NBP exhibits a different oscillatory signal. Some NBPs only show power at the higher (i.e., NBP~1, NBP~4) or lower (i.e., NBP~7) frequencies. Others exhibit strong oscillatory power throughout the entire frequency range (e.g., NBP~5). In terms of frequency occurrence, the 2.22~mHz, 2.78~mHz and 3.33~mHz oscillations are most popular (6 NBPs, with 3.33~mHz always at 99\% significance), followed by 1.11~mHz, 3.11~mHz and 3.78~mHz (all in 4 NBPs), 1.56~mHz (2 NBPs) and 1.89~mHz (only 1 NBP). There is no oscillatory power above 4~mHz in any NBP. Concentrating on any one type of behaviour (`{\it s}' or `{\it p}'), only the 2.78~mHz oscillation is constant. Power at this frequency always seems to steadily increase with increasing threshold.        

\section{Discussion}

\begin{figure}
\begin{center}
\epsfig{figure=f4.eps,width=5in}
\caption[Change of frequency with increasing threshold]{Power against threshold for NBP 2 in Table~\ref{cak_restab}. Labels A, B, C and D refer to the points in the diagram where different frequencies dominate.}
\label{cak_difffreq}
\end{center}
\end{figure}

It can clearly be seen from Figure~\ref{cak_difffreq} that particular frequencies may dominate in different areas of a NBP. At maximum threshold, the 3.33~mHz frequency dominates (point D). With smaller threshold, hence including more of the edge of the NBP, 2.89~mHz (point C), 3.89~mHz (point B) and 3.11~mHz (point A) dominate in turn. As these incremented thresholds refer to a series of nested contours, this highlights how the power spectrum is highly dependent on the chosen threshold value. Hence all frequencies found in the network are spatially localised at both large scale (i.e., the entire network cannot be assumed to have a global mode) and small scale (i.e., even within a NBP different frequencies may dominate in different sections). When the entire 100\arcsec\ $\times$ 100\arcsec\ image is used to create lightcurves using a similar method, (i.e., summing over all NBPs) it is the $\sim$2.22 ~mHz (steadily increases) and $\sim$1.00~mHz (plateaus) frequencies  which dominate. This agrees with the results of Lites {\etal} (1993), who only use one NBP, and Cauzzi {\etal} (2000) who used the average of eleven. 

The 3.33~mHz frequency (5-minute) has been previously reported in solar network regions (Kneer \& von Uexk\"{u}ll 1993; Al, Bendlin, \& Kneer 1998; Bocchialini {\etal} 1994; Baudin {\etal} 1996). However, it does not appear in all the NBPs, and in several cases is not the dominant frequency. Hence it does not appear to be connected to the well known 5-minute oscillation. Instead the results presented here suggest it localised to specific NBPs.

Despite the seemingly random selection of oscillatory power in Table~\ref{cak_restab}, a connection to existing theories can be drawn. In most cases, the onset of low-frequency oscillations coincides with the disappearance of higher frequencies (Figure~\ref{cak_thr}), suggesting that gravity or magnetism is the dominant restoring force in the centre of NBPs, at the expense of weaker acoustic waves ($\nu_{ac}\sim5.55$~mHz). Lou (1995) suggested that diverging magnetic flux tubes in the chromosphere and lower corona forms a magnetic bottle that can trap magneto-gravity modes. This scenario can create long-period oscillations in the range 1200--200~s (0.8--5.0~mHz) in the chromospheric network. All frequencies found here fall within this range, 

Kalkofen (1997) shows how fast-mode MHD waves at frequencies greater than $\sim$1.3~mHz can be excited by granular buffeting (Hasan \& Kalkofen 1999). These propagate upward and couple with weaker slow-mode MHD waves at a higher frequency of $\sim$3.4~mHz, thereby transferring power to the slow-mode waves. These longitudinal slow-mode waves become shocks, thereby heating the upper chromosphere. Ulmschneider, Z\"{a}hringer, \& Musielak (1991) show that during coupling a fast-mode wave at frequency $\nu$, will transfer most its energy to a slow-mode wave at frequency  2$\nu$, although some energy will be left in waves at the original frequency  $\nu$. This agrees with Table~\ref{cak_restab} where the weaker, less common, lower frequencies may be the signature of the fast-mode waves, and the stronger, more frequent, higher frequencies may be the onset of the slow-mode waves   

The tendency of NBPs to exhibit stronger oscillatory power in the centre of NBPs has also been pointed out by Judge {\etal} (2001). They use the phrase `magnetic shadows' to describe the decrease in both intensity and oscillatory power in the 3--8~mHz range near but not directly above magnetic elements. This is discussed in terms of upwardly propagating acoustic waves reaching the magnetic canopy and undergoing mode-conversion to MHD waves suggesting `magnetic heating occurs only close to the center of expanding flux tubes, not throughout them'. 

In this chapter I have presented a contour \& contrast method to identify frequency peaks with particular spatial position inside NBPs. Frequencies detected confirm the $\sim$2.2~mHz and $\sim$3.3~mHz values found previously. New frequencies are also detected at 3.78~mHz, 3.11~mHz, 2.78~mHz and 1.11~mHz in several NBPs, whereas 1.56~mHz only occurs in two NBPs and 1.89~mHz only one. The lack of detection of these frequencies in previous studies may be due to the improved method of isolation of each NBP and a higher frequency resolution. The frequencies found agree with the existence of a magnetic component in the waves, either as magneto-gravity waves or magneto-acoustic waves. In the next chapter I will extend this search from one dimension (i.e., distance from NBP centre, r), into 2 extra dimensions (height in solar atmosphere, z, and localised time domain, t) in order to look for waves propagating throughout the chromosphere. 
