Kinematic Extraction Methods

circumstance that the convolution show in Eq. 5.1 becomes a multiplication in Fourier do-main. The deconvolution therefore involves di-viding the Fourier transform of the galaxy spec-tra by the Fourier spec-transform of the stellar tem-plate. However, noise makes such a direct computation impractical. Noise dominates the Fourier transformed spectra at high frequencies, hence the ratio of the transforms at these fre-quency components will consist mainly of very amplified noise. Generally, the noise is white, i.e. its transform is essentially a constant out to high frequencies. To derive robust results, it

is therefore necessary to suppress the noise at high frequencies through filtering. For exam-ple, “Wiener” or “optimal filtering” (Brault &

White 1971) provides one of the best balancing between filtering insufficiently, resulting in more residual noise than necessary and filtering too severely, thereby losing important information.

Fig. 5.1 illustrates the effect of Wiener filtering of the Fourier transform. The Wiener filter is constructed by a least-square fitting to the com-puted power spectrum P_obs. The complex sig-nal and noise power spectra are approximated by a model of the signal power P_signal⁰ and a model of the noise power P_noise⁰ as P_obs(κ) = P_signal⁰ (κ) + P_noise⁰ (κ) which represents a good approximation of the observed power spectrum P_obs. The modelled signal and noise components are used to constructed the filter, which is in-dicated as the solid line in Fig. 5.1. Working in ‘Fourier space’ offers two great advantages.

On the one hand, the LOSVD is easier to ex-tract and to analyse. On the other hand, a fil-tering of residual low-order and high-order fre-quency components in the original spectrum is possible which arise from an imperfect contin-uum subtraction and Poisson noise. An alterna-tive approach to fit the LOSVD by a direct least-squares solution in ‘Real space’ and describing the velocity dispersion by a set of delta-functions in logarithmic wavelength space has been de-veloped by Rix & White (1992). To overcome the amplification of high frequency noise in the LOSVD, a method of weighting the eigenvalues is used. However, such computer codes (Rix &

White 1992; van der Marel & Franx 1993) will not be discussed here in greater detail.

• Classical Fourier Quotient Method

(Sargent et al. 1977): The algorithm adopts that the LOSVD F(v_los) has a Gaussian shape. Under this assumption, a Gaussian function is fitted to the quotient of the gal-axy spectra and the template spectra in

Fourier space as: dis-crete Fourier transforms of F(v_los), G(v) and S(v), respectively. ais a normalisation factor which measures the strength of the galaxy lines with respect to stellar lines (i.e., the mean relative line strength), σ denotes the velocity dispersion and v the mean ra-dial velocity, which is derived through the logarithmic redshift in the channels. The parameter N gives the number of pixels in the input spectra.

• Cross-Correlation Method

(Simkin 1974; Tonry & Davies 1979; Franx

& Illingworth 1988): Initiated by Simkin (1974) and improved by Tonry & Davies (1979), the Cross-Correlation approach is based on the fitting of a smooth symmet-ric function (quadratic polynomial) to the cross-correlation peak defined by the func-tions ˜G(κ) and ˜S(κ): rms of the template spectrum, respectively.

The star^∗ denotes the complex conjugation.

Cross-correlating the galaxy spectrum with the template spectrum produces a function C(κ), which has a peak at the redshift of the˜ galaxy and a width related to the dispersion of the galaxy. The resulting function is then fitted with a Gaussian in Fourier space.

• Fourier Correlation Quotient (FCQ) (Bender 1990): In contrast to the two pre-vious methods, which assume a Gaussian broadening function, the FCQ algorithm is an improvement of the former and a gener-alisation of the problem because it provides

Chapter 5: Kinematic Analysis 95

Figure 5.1: Illustration of the optimum (Wiener) filtering technique for the power spectrum of the galaxy WHDF # 437. The definition of the smooth power models of the signal P_signal⁰ and the noise P_noise⁰ are constructed by a least-square fitting (dot-dashed line) of the computed power spectrum Pobs. The cutoff frequencysc of the measurement, which is beyond the band-limit of the basic signal, and the characteristic frequency due to samplingsNy are also indicated. The Nyquist frequency or critical frequencysNy, is half the sampling rate for a signal (sNy= 0.5/∆λ). The sampling theorem is satisfied as long as the band-width of the measurement is less than the Nyquist frequency. However as noise is broadband, an band-limiting analog filter must be employed prior to sampling to avoid aliasing (i.e., contributions from undersampled high-frequencies) by the transform of the random noise signal in the spectrum.

the full information of the LOSVD function.

The method is based on the deconvolution of the peak of the template-galaxy correla-tion funccorrela-tion with the peak of the autocor-relation function of a template star as:

F˜(κ) = G(κ)˜ ·S˜^∗(κ)

S(κ)˜ ·S˜^∗(κ) (5.6) In comparison to other procedures, the main advantage of the FCQ is that the most ap-propriate function to fit the LOSVD can be performed a posteriori rather than a pri-ori. Furthermore, the method is less sensi-tive to template mismatch due to the fact that the LOSVD is determined from the ra-tio of the template-galaxy correlara-tion and template autocorrelation functions, which are both smooth functions, instead of the division of the galaxy and the stellar

spec-tra (see next section 5.2for details).

• Unresolved Gaussian Decomposition (UGD) (Kuijken & Merrifield 1993): This tech-nique adopts that the deviations from a pure Gaussian are due to the composition of several Gaussians which are uniformly tributed in mean velocity and velocity dis-persion. However, the Gaussians are not ho-mogeneously distributed in amplitude:

F˜(κ)≈

N

X

κ=1

aκ·exp

−(vlos−vκ)² 2σκ

(5.7) where aκ indicates the amplitude, vκ the mean velocity andσκthe velocity dispersion of each Gaussian. The UGD method com-putes the different amplitudes aκ for each component that provide the best fit to the galaxy spectrum in the least-square sense.

Figure 5.2: LOSVDs F(R, vlos) in the galaxy UGC 12591. The surface height corresponds to the stellar density as a function of major axis radius and line-of-sight velocity. The surface is displayed from two opposite perspectives, demonstrating the symme-try between the LOSVDs from both sides of the gal-axy. It is clearly evident that the line profiles are not Gaussian distributed (Kuijken & Merrifield 1993).

However, this can only be achieved under certain physical constraints(i)The LOSVD must be non-negative everywhere, (ii) The LOSVD is expected to vary smoothly on small scales and (iii) The LOSVD is as-sumed to be non-zero over a fairly small range in velocity only. In addition, the Rayleigh criterion is required in order to be able to define the separation between two adjacent peaks. To avoid dips between two peaks, the separation should be less than 2σ.

Fig. 5.2 shows two different viewpoints of the LOSVDs for the local galaxy UGC 12591 (S0/Sa). This galaxy is mod-erately inclined to the line-of-sight (≈67^◦), with the azimuthal component being the main contribution to the 3D velocity

dis-persion along the line-of-sight on the ma-jor axis. UGC 12591 has a large rotation velocity vrot ≈ 500 km s⁻¹(Giovanelli et al. 1986) and a central velocity dispersion of σ₀ ≈ 780 km s⁻¹(Kuijken & Merrifield 1993). Fig. 5.2 demonstrates the symme-try between the LOSVDs from two different perspectives of the galaxy and clearly shows that the line profiles are not Gaussian.

• Gauss-Hermite Expansion

(van der Marel & Franx 1993; Gerhard 1993): The approach of this method as-sumes the line profile F(v_los) to be a Gaus-sian with free parameters of line strengtha, mean radial velocity v and velocity disper-sion σ as:

F(v_los) = aα(ω)

σ , ω≡ (v_los−v)

σ (5.8)

where α(y) defines the standard Gaussian function:

α(y) = 1

√2π e^−(1/2)y2 (5.9) The LOSVD is derived via the product of this normal Gaussian and a truncated ex-pansion of the Hermite polynomials in the form: The Hermite coefficients hj describe de-viations from the Gaussian shape of the LOSVD, h₃ measures asymmetries (mostly taking values of h3 <∼ 0) and h4 symme-tries (h4 ≥0) andHj characterise the Her-mite polynomials. Results for the parame-ters of (a, v, σ, h3, ..., hN) are found us-ing a simultaneousN+ 1 parameter fit that are based on a χ² residual minimisation of the galaxy spectrum and the convolution of the LOSVD (broadening function) with the template spectrum in the Fourier domain.

Chapter 5: Kinematic Analysis 97

In practice, the fits are restricted to the case N = 4, but extension to higher orders is straightforward. Note, that although physi-cally unmotivated, negative LOSVD are al-lowed (e.g., resulting from a non-zeroh₃).