Fourier Filtered Back Projection (FFBP) Techniques

Fourier Filtered Back Projection (FFBP) Techniques

Each spectrum of KU Cygni displays the relative flux at a given wavelength during a particular orbital phase. The radial velocity for a given wavelength displacement from the rest wavelength of a spectral feature can be calculated, via Eqn. 1.1. In the rotating frame of reference, the observed radial velocity of an emission source in a steady flow within the system is determined by its velocity components, ($V_x$, $V_y$), observed at a certain orbital phase, $\phi$, as described by the following equation:

$$V_{rad}(\phi) = -V_x \cos(2\pi\phi) + V_y \sin(2\pi\phi).$$ (5.13)

Looking at the trailed spectrogram of KU Cyg (Fig. 3.9, the S-wave reveals the existence of phase-dependent emission sources within the accretion disk. The shape of the S-wave is described by Eqn. 5.13. The emission-line profile of H$\alpha$ contains the radial-velocity information of emission sources within the disk. The emission-line profile of the H$\alpha$ line is assumed to be a result of solely Doppler effects, although turbulent motions within the stream may contribute to the overall line profile (Horne & Marsh 1986, Robinson, Marsh, & Smak 1993). In order to determine the intensity distribution of H$\alpha$ emission sources in the Doppler tomogram, a transformation from the relative flux in the spectra to relative intensity on the velocity map must be calculated. The average H$\alpha$ flux contribution of a gas parcel with a particular velocity coordinate can be calculated by summing and then averaging the flux contributions over all orbital phases. This calculation is performed for every velocity coordinate on the Doppler map. Ultimately, an intensity map of the emission sources in velocity space is acquired. This mathematical process defines the back-projection technique. The back-projection (BP) technique is described by the following equation (Horne 1991):

$$BP(V_x, V_y) = \frac{\int D( V(V_x, V_y, \phi), \phi) w(\phi) d\phi}{\int w(\phi) d\phi},$$ (5.14)

where $V (V_x, V_y, \phi)$ describes the shape of the S-wave, $D(v,\phi)$ is the trailed spectrogram data, that is, the observed radial velocity value at a given orbital phase for velocity coordinates ($V_x$, $V_y$), and $w(\phi)$ are phase dependent weights. In this thesis, equal weights are given to all spectroscopic data at a given orbital phase. Recall that two or more parcels of gas can have the same velocity coordinate ($V_x$, $V_y$). Thus, when viewing coordinates on the Doppler map, this similarity must be kept in perspective. The spatial location of these sources is not necessarily known nor can be determined from the velocity coordinates. The application of this back-projection technique to accretion disks assumes that the emitting regions are observable at every orbital phase. This condition is not satisfied in eclipsing binary systems, where the disk can be occulted by the secondary star. Thus, primary eclipse phases are omitted from the back-projection technique, as well as phases immediately prior to and after primary eclipse where the disk is occulted. In this accretion disk analysis of KU Cygni, orbital phases of $\pm0.09$ are excluded from the creation of the Doppler tomograms.

In the summation technique of back-projection, there are intensity contributions from gas parcels with radial velocities that exactly match the particular S-wave in question. On the Doppler tomogram, emission sources that coincide exactly with the observed S-wave will map into a bright spot. Intensity contributions of gas parcels with ``nearby'' radial velocities will add some intensity to the overall brightness of that bright spot. These contributions diminish, to form diminishing intensity wings, as the velocity distance increases away from the bright spot's location. So, the technique of back projection displays a Doppler map of the S-wave sources, but convolved with a point spread function (PSF) in the relative intensity distribution, which depends on the wavelength (or velocity) dispersion of the observations.

The PSF of the back-projection technique creates as artifacts broad, low-intensity wings around regions of high intensity that scale as $\frac{1}{V}$. In order to sharpen the PSF of this imaging technique, a Fourier filter can be applied to the data prior to back-projection. The Fourier-filtering technique entails passing each individual spectrum through a Fourier transform, multiplying the Fourier amplitudes by a Fourier filter, and then applying an inverse Fourier transform to the spectrum (Horne 1991). The Fourier-filtering technique uses a high-pass ramp filter, which amplifies high-frequency noise. Therefore, a Gaussian cut-off is added to the Fourier filter to reduce the high-frequency noise. Including the Gaussian cut-off to the Fourier filter reduces the background noise contained within the spectra, but at the expense of resolution in the Doppler map (Horne 1991).

Marsh & Horne (1988) and Robinson et al. (1993) thoroughly describe the linear-inversion techniques required to transform trailed spectrogram data to the Doppler tomogram map. In applying the Fourier filter to the spectral data, the high-pass ramp filter needed to sharpen the PSF of the back-projection is given by:

$$G(\omega) = \left(\frac{\omega}{\omega_N}\right),$$ (5.15)

where $\omega$ is the frequency equivalent for the given wavelength, $\lambda$, and $\omega_N$ is the Nyquist frequency. The Nyquist frequency, $\omega_N$, is defined as one-half of the sampling rate between data points. Since this analysis deals with spectroscopic data, the sampling rate is defined as the inverse of the wavelength dispersion of the spectrum. In order to curtail the $\frac{1}{V}$ intensity wings of the PSF, a Gaussion cut-off is applied to Eqn. 5.15, which results in the following Fourier filter:

$$G(\omega) = \left(\frac{\omega}{\omega_N}\right) \exp \left[\frac{ -(\omega/\omega_c)^2}{2} \right],$$ (5.16)

where $\omega_c$ is the cut-off frequency. Richards et al. (1995) and Richards & Albright (1996) used a Gaussian of FWHM = $3\sigma$ where $\sigma$ is the wavelength (or velocity) dispersion width. For the spectrocopic data used in the analysis of KU Cyg, the value of $\omega_c$ is given by $\frac{1}{3\sigma}$ = 0.833. (Note that the spectrograph at MLO yields a wavelength dispersion of approximately 0.4 Å/pixel, thus $\sigma=0.4$Å).

Idealistically, the back-projection technique requires equally spaced spectroscopic observations in orbital phase. The phase spacings between observations should be small enough to adequately and faithfully recreate features in the Doppler map. Realistically, undersampling and unequally spaced observations are occurrences that will produce artifacts in the Doppler tomogram. For example, Marsh & Horne (1988) discuss how a single flare in one spectroscopic observation will produce a streak through the Doppler map, located at an azimuth angle parallel to the orbital phase of that observation. Also, undersampling in orbital phase creates streaks or spokes, similar to flares, along the angle of phased observation. The streaks are produced as the emission-line flux is determined for the S-wave at a given orbital phase. The Doppler image, produced by back projection, can be considered as ``painting down'' probability strips onto the velocity map (Horne 1991). Emitting regions, corresponding to the S-wave data, would be located on the map where these probability strips would intersect consistently over all orbital phases. These intersection points will be described by a larger relative intensity, as compared to other areas in the map. With unequal phase sampling of observations, these ``strips'' do not adequately intersect at the velocity location of the S-wave. Thus, the streaks or spokes appear in the tomogram. Albright & Richards (1996) analyzed the long-period Algol system TT Hya, which should contain a permanent accretion structure. Using only 21 spectroscopic observations, Albright & Richards' analysis of TT Hya does reveal a permanent disk structure in their Doppler tomography. The streaking, as a result of this small sampling, is very apparent in their Doppler tomogram. This study of KU Cygni utilizes forty (40) spectroscopic observations to create a Doppler map of the accretion disk.