Abel inversion

Fig. 4. Abel inversion procedure used for the determination of radial distributions in plasma sources, x lateral position, y direction of observation, r radial distance.

From these integral values, however, the values at a well-defined radial distance away from the plasma center can be calculated. This necessitates side-on observation along several parallel axes, which are equidistant with respect to each other and an Abel-inversion. Indeed, when I (x) is the first derivative of the function I(x), describing the variation of a measured value as a function of the lateral position (x) during side-on observation (Fig. 4), the radial values at a distance r are given by ... 

Fig. 12. Spatial profile of the absolute CH density resulting from an Abel inversion of the measured CH column densities. The column densities were smoothed prior to the Abel inversion. (Reprinted with petmission from Menningen et al, 1995a, Contrib. Plasma Phys. 35, 359, 1995 Wiley-VCH, Inc.)...

It can be stated with reasonable assurance that the particle content is conserved in the process though the dissymmetry of the density profile in the blobs renders any Abel inversion strictly speaking unpracticable. Relaxation sets in, in pinch discharges, provided there is a small dissipative mechanism related to some level of microscopic activity. Then lines of force can break and rejoin on a time scale close to the resistive diffusion time. It is widely accepted that such states evolve towards a minimum energy state which obeys a force-free configuration in which V X -B = (jl-B with (ji a constant for all field lines, and B is the magnetic field. 

The Abel inversion described by Eq. A4 is commonly used in many areas of science and engineering [65-67] and several methods for performing this task are available [68-71] One method involves least squares fitting a functional form to... 

For each row of data, we have shown that the inverse Abel inversion yields the intensity profile along a diameter of the 2D ring distribution. Hence the total intensity per ring R for a given distance from the symmetry axis r, R(r), is given by... 

For an estimation of the local spray drop size, Schmidt and Mewes [38] used a mathematical operation the Abel Inversion. With this operation, integral measurements of spray drop sizes may be converted into local spray drop sizes. The spray drops are assumed to be symmetrically distributed throughout the spray cone for the application of Abel Inversion on the data. 

For this operation, software by Hipp et al. [40] was used, which was designed for Interferometric Data Evaluation (IDEA). Since the spray has to be axially symmetric for the application of the Abel Inversion, divergences within the spray may be compensated within the used program or via a manual horizontal flip. These adjustments should only be performed if the divergences are caused by inaccuracy of the measurements and are only minor. 

Pretzier, G., Jager, H., Neger, T., Philipp, H., Woisetschlager, J. (1992). Comparison of different methods of Abel inversion using computer simulated and experimental side-on data. Zeitschrift fiir Naturforschung A, 47(9). 

This three-dimensional distribution of the fluid density can be resolved by recording with the optical setup several projections in different directions through the flow and processing the obtained data with the methods of computer tomography. If the flow has rotational symmetry, one projection only is sufficient, and the axisymmetrically distributed fluid density can be determined by applying to the data an inversion scheme Abel inversion). 

---