Flow (phase) extraction

To obtain the column signal in a form suitable for phase and contrast demodula-tion, the extracted intensity is first factored out of the signal according to S⁰(y) = [2S(y)/I₀(y)] − 1. Using equation (4.2.2) we see that S⁰(y) = ζ(y) cos[φ_I(y) + φ_D(y)], i.e. it is a sinusoidal signal with zero mean, phase modulated by the Doppler phase φ_D and amplitude modulated by the fringe contrast. Phase and contrast extraction are then performed via the analytic signal representation of S⁰(y). The analytic signal was first introduced in Gabor [1946]. The general concept is that due to the Hermitian symmetry (i.e. f (−x) = −f (x)^∗) of the Fourier transform of a real-valued signal, the negative frequency components of such a transform can be discarded without losing any information about the original signal. The inverse Fourier transform then yields a complex representation of the original signal which makes certain properties, particularly the instantaneous phase and amplitude, more accessible. The analytic representation of the discrete signal S⁰ can be generated easily by taking the Discrete Fourier transform (DFT) of the image column to obtain S˜⁰[f ] and setting:

4.2. Interferogram Demodulation Technique 77 before performing the inverse transform [Lawrence Marple Jr., 1999]. The resulting analytic signal will be denoted S_a. The real part of S_a is equal to the original input signal, while the complex part is equal to its Hilbert transform [Gabor, 1946]. Since the effect of the Hilbert transform is a 90^◦ phase shift of all frequency components, the real and imaginary parts of S_a are in phase quadrature and can be used to recover the instantaneous phase and contrast according to (φ_I(y) + φ_D(y)) = arg(S_a(y)) and ζ(y) = |S_a(y)|.

Features in the brightness image with high spatial frequencies, which are not accurately recovered by the intensity extraction as noted in the previous section, cannot be accurately removed from the data when calculating S⁰ and cause ringing artefacts in the demodulated phase. High spatial frequency components of image noise also appear strongly in the resulting phase images. This is illustrated in fig-ure 4.5(b), which shows the φD image demodulated from noisy test data using the scheme described so far, alongside the ideal result in 4.5(a). The main features of concern in the demodulated image are the high level of noise on the image, and the ringing artefacts at the edges of the poloidal field coil structures. These

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Figure 4.5: Illustration of the effect of windowing when calculating the analytic sig-nal and of apodisation of the column sigsig-nal at sharp edges. (a) Ideal phase response for simulated noisy divertor data, (b) Demodulated phase without windowing or apodisation, (c) Demodulated phase with windowing and apodisation.

able effects are reduced (at the cost of fine detail in the image) by a combination of two techniques. The first is windowing in Fourier space when calculating the analytic signal. Using a window function centred at the carrier fringes’ spatial fre-quency, high spatial frequency contents of the recovered Doppler phase, including the ringing artefacts and some of the image noise, are attenuated. This operation is illustrated in Figure 4.6, which shows the magnitude of the DFT of S⁰ and a mul-tiplicative window function which is applied before the inverse transform to yield S_a. Indicated on the figure are f_c, the spatial frequency of the carrier fringes, and

4.2. Interferogram Demodulation Technique 78 W , the total width of the window function. Choice of the particular window

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Figure 4.6: Application of a window function in Fourier space when calculating the Analytic signal using the DFT of S⁰. The carrier fringe frequency f_c and total window width W are indicated.

tion and width is a compromise between removing high spatial frequency artefacts and noise while preserving desired high spatial frequency information (i.e. fine de-tails) in the signal. By visual inspection of Doppler phase images extratced from simulated test data, and comparison between the demodulated and ideal images, a Blackman-Harris window (as shown in figure 4.6) with width W = f_c was found to produce good robustness against noise and artefacts while preserving features of interest in the simulated data. The second technique, used to reduce the size of the ringing artefacts, is to apodise the column signal S⁰ around locations of sudden changes in I₀, before calculating the analytic signal. This removes the sharp jumps in the signal which give rise to the ringing. To do this, the gradient of I₀ down the image column is estimated with a 3-point central difference, and locations where the gradient has a numerical value greater than an empirically chosen fraction of I₀ are considered sharp edges. This fraction is usually set to 5%. Inverted Hann windows are then applied (in real space) to the column signal centred at these locations. This process is illustrated in figure 4.7, which illustrates the process of demodulating the phase from a single image column. Using both the windowing and apodisation, the demodulated Doppler phase from noisy test data is shown in figure 4.5(c).

The output of the phase demodulation for a given input image is the instanta-neous phase at each pixel, in the interval [−π, π]. An example section of a simulated (noiseless) image and corresponding demodulated phase are shown in figure 4.8. For

4.2. Interferogram Demodulation Technique 79

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Figure 4.7: Illustration of the steps involved in phase demodulation of a single image column. (a) Input noisy test data and extracted brightness profile. Multiple regions of large brightness gradient are seen. (b) Data with brightness profile factored out, referred to as S0 in the next. (c) S0 after apodisation at locations of large brightness gradient, (d) The calculated analytic signal corresponding to the signal in (c), from which the fringe phase is determined.

4.2. Interferogram Demodulation Technique 80

Figure 4.8: Illustration of fringe phase demodulation results for a small section of a noiseless simulated image: (a) Input image and (b) demodulated fringe phase.

calibration images taken in the lab to be used as an instrument phase reference, this phase image is unwrapped to reveal the shape of the instrument phase across the field of view, as will be described in section 5.3.2. For plasma data, the calibrated instrument phase φ_I is subtracted directly from the demodulation output and the result is wrapped into the interval [−π, π], to isolate the Doppler phase φ_D. Note that this limits the largest (line-average) flows which can be measured, without fur-ther processing to detect phase wrapping in the φ_Dimages, to ¯v_max = (c/2 ˆN ) (where c is the speed of light and ˆN is the instrument group delay). For the MAST system

¯

v_max ≈ 100km/s, around a factor of 3 larger than any observed flow speeds in the SOL and divertor.

As was derived in section 2.4.2, the measured Doppler phase is related to an emissivity weighted line average of the line-of-sight ion flow. This line-average flow is obtained straightforwardly from the demodulated φ_D according to:

¯

v = cφ_D

2π ˆN, (4.2.3)

and from equation 2.4.33 this is related to the local plasma flow by:

¯

While ¯v is not necessarily suitable for quantitative interpretation of the measure-ments directly, these images can already provide qualitative insight into flow patterns and dynamics. Before going on to discuss tomographic inversion of ¯v images to