Tomography of fast-ion velocity-space distributions from synthetic CTS and FIDA measurements

PAPER Tomography of fast-ion velocity-space distributions from synthetic CTS and FIDA measurements To cite this article: M. Salewski et al 2012 Nucl. Fusion 52 103008 View the article online for updates and enhancements. Related content Combination of fast-ion diagnostics in velocity-space tomographies - Measurement of a 2D fast-ion velocity distribution function by tomographic inversion of fast-ion D-alpha spectra - On velocity space interrogation regions of fast-ion collective Thomson scattering at ITER - Recent citations Collective Thomson Scattering Diagnostic for Wendelstein 7-X at 175 GHz D. Moseev et al - Collective Thomson scattering with 77, 154, and 300 GHz sources in LHD M. Nishiura et al - Collective Thomson scattering diagnostic for the GDT open magnetic trap A G Shalashov et al - This content was downloaded from IP address 87.218.223.151 on 18/08/2020 at 11:22 IOP PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION Nucl. Fusion 52 (2012) 103008 (11pp) doi:10.1088/0029-5515/52/10/103008 Tomography of fast-ion velocity-space distributions from synthetic CTS and FIDA measurements M. Salewski1, B. Geiger2, S.K. Nielsen1, H. Bindslev3, M. Garc´ ıa-Mu˜ noz2, W.W. Heidbrink4, S.B. Korsholm1, F. Leipold1,F.Meo 1, P.K. Michelsen1, D. Moseev2,5, M. Stejner1, G. Tardini2and the ASDEX Upgrade team2 1Association Euratom-DTU, Technical University of Denmark, Department of Physics, DTU Risø Campus, DK-4000 Roskilde, Denmark 2Association Euratom-Max-Planck-Institut f¨ ur Plasmaphysik, D-85748 Garching, Germany 3Faculty of Sciences and Technology, Aarhus University, DK-8000 Aarhus C, Denmark 4Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA 5Association Euratom-FOM Institute DIFFER, 3430 BE Nieuwegein, The Netherlands E-mail: [email protected] Received 10 May 2012, accepted for publication 1 August 2012 Published 21 August 2012 Online at stacks.iop.org/NF/52/103008 Abstract We compute tomographies of 2D fast-ion velocity distribution functions from synthetic collective Thomson scattering (CTS) and fast-ion Dα(FIDA) 1D measurements using a new reconstruction prescription. Contradicting conventional wisdom we demonstrate that one single 1D CTS or FIDA view suffices to compute accurate tomographies of arbitrary 2D functions under idealized conditions. Under simulated experimental conditions, single-view tomographies do not resemble the original fast-ion velocity distribution functions but nevertheless show their coarsest features. For CTS or FIDA systems with many simultaneous views on the same measurement volume, the resemblance improves with the number of available views, even if the resolution in each view is varied inversely proportional to the number of views, so that the total number of measurements in all views is the same. With a realistic four-view system, tomographies of a beam ion velocity distribution function at ASDEX Upgrade reproduce the general shape of the function and the location of the maxima at full and half injection energy of the beam ions. By applying our method to real many-view CTS or FIDA measurements, one could determine tomographies of 2D fast-ion velocity distribution functions experimentally. (Some figures may appear in colour only in the online journal) 1. Introduction Fast ions play a key role in high performance plasmas: they mediate energy from external heating sources or fusion reactions to the bulk plasma and so maintain the high temperatures typical for fusion-relevant plasmas. The fastion orbits can be perturbed by fluctuations in the plasma, and the ions can then be prematurely ejected from the plasma, leading to undesired local heating of the first wall instead of plasma heating. Several types of modes selectively deplete or reorganize fast ions in particular velocity-space regions, for example sawteeth [1–3], Alfv´ en eigenmodes [4–6] and neoclassical tearing modes [7]. Turbulence also ejects ions selectively depending on their energy [8,9]. In particular, it is this selectivity of fast-ion depletion or reorganization in velocity space that can be quantified with velocity-space tomography. Additionally, velocity-space tomography could be used to monitor phase-space engineering of fast-ion velocity distribution functions which has enabled control of sawteeth and of neoclassical tearing modes [10]. We show velocityspace tomographies using parameters typical for the ASDEX Upgrade collective Thomson scattering (CTS) [11–15] and fast-ion Dα(FIDA) diagnostics [16]. CTS and FIDA diagnostics are sensitive to 1D functions gof local fast-ion velocity distribution functions fin magnetically confined plasmas. The spatial resolution of the CTS diagnostic at ASDEX Upgrade is about 10 cm, and the measurement location can be moved freely in the plasma core by means of steerable antennas. The time resolution has often been set to 4 ms. CTS diagnostics are sensitive to the 1D projection of fonto the wave vector kδ=ks−kiwhich is the difference between the wave vectors of scattered radiation ksand incident radiation ki. The most important angle to describe the pre-selected projection direction given by kδis 0029-5515/12/103008+11$33.00 1© 2012 IAEA, Vienna Printed in the UK & the USA Nucl. Fusion 52 (2012) 103008 M. Salewski et al the projection angle φCTS =(kδ,B)where Bis the magnetic field. In CTS experiments the ions leave spectral signatures in the scattered radiation. A frequency shift νδof scattered radiation can be related to an ion velocity vprojected onto kδ: νδ=νs−νi≈v·kδ/2π=ukδ/2π(1) where uis the projected velocity and kδ=|kδ|. We define here a CTS measurement as detection of the fast-ion phase-space density in a particular interval in uthat is related to an interval in νδvia equation (1). We define a view as a set of measurements taken in a projection direction described by φCTS. A second CTS receiver has been installed at ASDEX Upgrade in 2012, so that two simultaneous views with independently variable projection angles φCTS are available. The location of a FIDA measurement is determined by the intersection of the injected neutral beam (NBI) and the lineof-sight (LOS) of the optical head. The spatial resolution of the FIDA diagnostic at ASDEX Upgrade is about 7 cm, and the time resolution is 2 ms. Beam source S3 is observed in the plasma core at two different fixed angles φFIDA =(kLOS,B) where kLOS represents the wave vector along the LOS of the optical heads. The toroidal LOS has an angle of φFIDA =11◦, and the new poloidal LOS has φFIDA =64◦. The angles φCTS and φFIDA are analogue and will hereafter simply be called φ. FIDA diagnostics are also sensitive to 1D functions of fas the fast ions likewise leave a spectral signature in the detected light by Doppler shift and Stark splitting. For FIDA diagnostics no simple relation between the projected velocity uand the wavelength λexists, so we define here as FIDA measurement the detection of Dopplerand Stark-shifted light in a particular wavelength interval. Computed tomography in real space is used in many applications, for example in medical imaging in x-ray computed axial tomography (CAT or CT) scanners, positron emission tomography (PET) scanners or magnetic resonance imaging (MRI) scanners [17,18]. It is also widely used in nuclear fusion research [19,20]. We give a new prescription for tomographic reconstruction in velocity space that is analogue to those in real space. The prescription is based on CTS or FIDA weight functions [21–23] which were not available in previous work [24]. In [24] reconstructions from two and three synthetic CTS views have been shown to contain salient features of the underlying 2D fast-ion velocity distribution functions in idealized situations. It has since become conventional wisdom that a 2D velocity distribution function could not be found from one single 1D CTS or FIDA view and that at least two CTS or FIDA views with different projection directions would be necessary for that [12,22–32]. We demonstrate that in fact just one single 1D CTS or FIDA view theoretically suffices to compute tomographies of almost the entire discrete 2D velocity distribution function under idealized conditions. Nevertheless, in simulated tokamak experiments with many CTS or FIDA views, the resemblance of tomographies and the original functions improves with the number of available views. Several tokamaks have been equipped with multiple FIDA views, for example DIIID[33], NSTX [34], MAST or ASDEX Upgrade which is now also equipped with two CTS receivers. With our prescription we can compute tomographies for any set of fastion measurements, in particular those obtained with CTS or FIDA or other fast-ion charge exchange spectroscopy (FICXS) that detects other light than Dα. A mix of diagnostics would also be possible as will be relevant to the CTS/FIDA system at ASDEX Upgrade, the CTS/FICXS system at LHD [35,36] and the proposed two-view CTS system for ITER [37–40]in particular if it can be combined with FICXS [32]. However, only one of the two CTS views is an enabled ITER diagnostic. One could also include neutral particle analysers (NPAs) or other fast-ion diagnostics in such mixes. We will study tomographies from such diagnostic mixes elsewhere. In section 2we will argue that one single 1D set of CTS measurements at different frequencies in fact theoretically suffices to reconstruct the original 2D velocity distribution function under ideal conditions. As weight functions form the core of our tomographic reconstruction prescription to be presented in section 4, we briefly review their meaning and use in section 3. Tomographic reconstructions of a variety of functions from synthetic CTS measurements under idealized conditions are demonstrated in section 5and under simulated experimental conditions in section 6. In section 7we show that tomographies can likewise be computed from synthetic FIDA measurements. We discuss the analogy of velocityspace tomography to real-space tomography in section 8and draw conclusions in section 9. 2. Velocity-space tomography gedankenexperiment First we perform a gedankenexperiment to motivate how one single 1D projection can in fact contain enough information to reconstruct the underlying 2D velocity-space distribution function in discrete problems. Suppose that Alice has a way to construct a 2D velocity-space distribution function fion by ion and that Bob has a way to measure the 1D velocity distribution function gby CTS every time a new ion has been added. Bob will only know his own measurements obtained in a single CTS view. Alice adds an ion at some coordinate pair (v,v ⊥)of her choice, for example at the location chosen in figure 1(a). Bob then measures gwhich would have the characteristic hammock shape shown in figure 1(b)[23,41]. Bob can now work out the (v,v ⊥)-coordinates using u=vcos φ+v⊥sin φcos γ, (2) where γis the gyrophase of the ion [23]. Since cos γtakes values from −1 to 1, the width of the interval in which Bob detects the ion is 2v⊥sin φ. The centre of the interval is vcos φ. Knowing his projection angle φand the width and centre of his measured function g, he can tell at which coordinates (v,v ⊥)Alice has added the ion. Alice then adds a second ion at a velocity-space location of her choice, and Bob again measures gby CTS. Now the function glooks more complicated but Bob can subtract his previous function gand has again a simple hammock-shaped function from which he can deduce the location of the second ion. This procedure can be repeated until the entire 2D velocity distribution function is constructed ion by ion, and Bob will know the entire function exactly, looking just at his 1D measurements. Alice could also construct fby adding collections of ions with identical velocities instead of single ions. Bob could then tell how many 2 Nucl. Fusion 52 (2012) 103008 M. Salewski et al -3 -2 -1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 ( a ) f -4 -2 0 2 4 0 1 2 3 u [106 m/s] g [1012 s/m4] ( b ) g Figure 1. (a) Example function fconsisting of a single pixel in arbitrary units. (b) Projection gof the pixel function for a projection angle of φ=70◦. ions have been added since the integral over uis proportional to the number of ions: n=gdu=fdvdv⊥.(3) This gedankenexperiment shows that one single 1D CTS view can in fact contain enough information for accurate reconstruction firstly in simple situations and secondly also in arbitrarily complicated situations if the complexity is added step by step. In real experiments only the complicated situation can be generated, and it is not immediately obvious that the 1D function gcan contain enough information about the 2D function f. But we will demonstrate that we can compute accurate tomographies from one single CTS or FIDA view using our tomography reconstruction prescription if just enough information is available. 3. Discrete weight functions for CTS and FIDA Discrete weight functions will lead to the tomographic reconstruction prescription presented in section 4. The reconstruction prescription in [24] did not use weight functions and was made tractable by expansion of the 1D (synthetic) measurements as well as the 2D fast-ion velocity distribution functions into orthonormal sets of base functions. Bessel functions have been used but other choices would be possible [24]. Exploiting CTS or FIDA weight functions [21–23] we will give a simpler reconstruction prescription that is inherently tractable and obviates the use of such expansions. Weight functions have previously been used in an alternative reconstruction prescription where the tomography was found by iteration. This has the disadvantage that the solution depends on the arbitrary start conditions of the iteration [23]. The new prescription we present gives unique solutions. In this section we define weight functions in discrete form. Assuming fto be rotationally symmetric about the v- axis, weight functions describe the mapping from 2D velocityspace distribution functions fto 1D functions gthat are measured with CTS [23]orFIDA[22]. We here treat a discrete tomography problem and so also deal with discrete functions. The coordinates (u, φ, v,v ⊥)are discretized in (ui,φ j,v k,v ⊥l)where the subscripts i, j, k, l run from 1 to the corresponding upper case letter I,J,K,L.Iis the number of measurements at different uiin a CTS or FIDA view, Jis the number of available views, and (K, L) are the number of grid points in (v,v ⊥), respectively. gij =g(ui,φ j) is a matrix of discrete 1D functions in uifor each viewing angle φj.fkl =f(v k,v ⊥l)is the discrete 2D velocity-space distribution function. gij and fkl are related by discrete CTS or FIDA weight functions wij kl analogue to the continuous weight functions [23] so that gij = K  k=1 L  l=1 wij kl fklv⊥v.(4) Weight functions pick out and assign weights to the velocityspace interrogation region that is observed for a particular projection angle φjand a projected velocity range at ui(observed in a frequency range at fi) for CTS or a wavelength range at λi for FIDA. In (v,v ⊥)-coordinates CTS weight functions have a nearly triangular shape as shown in figure 2for ui=2× 106ms −1and four typical projection angles φj. Weight functions describing CTS measurements quantify the probability that a gyrating ion with velocity (v,v ⊥) is observed in a particular projected velocity range at uifor a given projection angle φj. The scattering must always originate from the coloured triangular region. A comprehensive discussion of weight functions for fast-ion CTS measurements is given elsewhere [23]. The weight functions describing FIDA measurements are more complicated and account for the charge exchange probability, the probability of photon emission from atomic level n=3 to n=2, Doppler shift of radiation originating from a gyrating particle, Stark splitting of the deuterium Balmer alpha line, and the instrument function of the FIDA spectrometer [16,21,22,26,29]. The Doppler shift part of FIDA weight functions is analogous to the CTS weight functions [23]. 4. Tomographic reconstruction prescription To find tomographies from CTS or FIDA measurements, we rewrite equation (4) to formulate a linear algebra problem of the form WmnFn=Gm.(5) The matrix elements Gm,Fnand Wmn are, respectively, obtained from the matrix elements gij ,fkl and wij kl by Gm=gij (6) Fn=fkl (7) Wmn =wij kl (8) 3 Nucl. Fusion 52 (2012) 103008 M. Salewski et al –4 –2 0 2 4 2 4 v|| [106 m/s] v⊥ [106 m/s] –1.5 –1 –0.5 –4 –2 0 2 4 2 4 v|| [106 m/s] v⊥ [106 m/s] –1.5 –1 –0.5 –4 –2 0 2 4 2 4 v|| [106 m/s] v⊥ [106 m/s] –2 –1 –4 –2 0 2 4 2 4 v|| [106 m/s] v⊥ [106 m/s] –2 –1.5 Figure 2. Gyromotion weight functions wfor u=2×106ms −1and various projection angles φ. The colourbar shows the base 10 logarithm. using the assignment rules m=(i −1)×J+j(9) n=(k −1)×L+l. (10) Fis a column matrix of size N×1 obtained from the discrete 2D fast-ion velocity distribution function described by N=K×Lpoints. Gis a column matrix of size M×1 obtained from the discrete 1D functions measured with CTS or FIDA. If Jviews are available and Imeasurements in ui (CTS) or λi(FIDA) are taken in each view, then the total number of measurements is M=I×J.Wis then a transfer matrix of size M×Ntaking Finto G. The prescription given here corresponds to stacking lines or rows on top of each other but the order of this reorganization of the matrices is arbitrary as long as we obey equation (4). The forward problem to determine gfrom for equivalently Gfrom Fis straightforward given that wand consequently Ware known. An example of the action of the transfer matrix Won a pixel function Fis illustrated in figure 1. The projection angle φjof this single-view example (J=1) is set to 70◦, and we compute a weight function for each uito obtain the value of Gfrom the inner product WF. The 1D function Gfor a pixel function has the characteristic hammock shape shown in figure 1. The inverse problem to determine ffrom gor equivalently Ffrom Gis more complicated: we have to find an optimum solution F+to the underor overdetermined system of linear equations (equation (5)) where Wand Gare known. We then also know f+because we know F+and the reorganization procedure. We find an optimum solution to WF =Gfor any size of Wfrom the Moore–Penrose pseudoinverse or generalized inverse W+under positivity constraint. W+is a unique N×M matrix [42–44]. It can be computed from the singular value decomposition (SVD) of W:anM×Nmatrix Wcan always be decomposed uniquely as W=UVT(11) where Uis the normalized eigenvector matrix of WWT(an orthogonal M×Mmatrix), Vis the normalized eigenvector matrix of WTW(an orthogonal N×Nmatrix), VTdenotes the transpose of V, and is a diagonal (but rectangular) M×N matrix [44]. The diagonal entries σ1,σ 2, ..., σRare the singular values of W, and Ris the rank of W. The other entries of  are zero. The Moore–Penrose pseudoinverse is then W+=V+UT(12) +is a diagonal (but also rectangular) N×Mmatrix, and the diagonal entries are 1/σ1,1/σ2, ..., 1/σR, i.e. the reciprocals of corresponding entries of . The other entries of +are zero. The computed tomography is then F+=W+G. (13) This is the equation from which we could determine F+from actual measurements. If Wis invertible, then W+is identical to the inverse W−1. But Wis generally a rectangular M×N matrix that cannot be inverted. If the system WF =Gis overdetermined, F+gives the minimum 2-norm of the residual |WF −G|2. If the system WF =Gis underdetermined, F+is the particular solution with minimum 2-norm |F|2 out of infinitely many solutions (the one with no nullspace component). 5. Tomographies under ideal conditions In this section we firstly demonstrate that our prescription for computed tomography in velocity space can reproduce a variety of functions—any function we tested—in an idealized situation. Secondly, we also demonstrate that just one single synthetic CTS or FIDA view on that function suffices to construct an accurate tomography. We assume that the function can be described accurately on a numerical 2D grid, i.e. the grid size is so fine that even features on the smallest scale are accurately described. We also assume that there is no 4 Nucl. Fusion 52 (2012) 103008 M. Salewski et al -3 -2 -1 0 1 2 3 0 1 2 3 v || [106 m/s] v⊥ [106 m/s] 0 0.5 1 Figure 3. The original checkerboard function shown here is digitized in N=30 ×61 pixels. Typical 1D projections are shown in figure 4. Tomographies are shown in figure 5. noise. The effects of insufficient resolution and noise will be discussed in section 6. Under these idealized conditions, we set the numerical grid of the tomography equal to that of the original function. As will be shown in section 6, these assumptions will not give a realistic picture of the recoverable information in real experiments. Nevertheless, previous work used identical grids for tomography and the original function [23,24], and the results found in this section give an upper limit of the quality that can be achieved and demonstrate that one single view is enough under ideal conditions. Our prescription immediately suggests that the tomography should be very accurate in this case if just M>N(more measurements than pixels). If the numerical grids of the original function and the tomography are equal, we can give a simple relation between Fand F+. One can substitute for Gand use the orthogonality of U. F+=W+G=W+WF =V+UTUVTF=V+VTF (14) +has as Rones on the diagonal and otherwise zeros. Therefore, only the first Rcolumns and rows of Vand VTwill be used in the reconstruction. In that sense the reconstruction for identical numerical grids is analogue to lossy data compression using SVD [44]. Under these assumptions, we reconstruct a checkerboard function (figure 3) and a pacman function (figure 6) using just one single view. We choose these test functions because it is easy to spot differences between the original function and the tomography. The checkerboard pattern in figure 3 covers the velocity-space region for −3.5×106ms −1< v<3.5×106ms −1and 0 <v ⊥<3.5×106ms −1 and is digitized in N=30 ×61 =1830 pixels. This resolution is typical for simulated fast-ion velocity distribution functions today. We distribute Mmeasurements evenly in the interval −5×106ms −1<u<5×106ms −1to ensure complete coverage of the velocity-space region we show here for any φ. Synthetic measurements in one single view for φ=30◦and M=101 to M=2501 are illustrated in figure 4. By increasing the resolution one can capture an increasingly more fine-grained structure of gthat contains recoverable information about the 2D function f. We stress that the noisy looking curve (M=2501) is the accurate one whereas the smooth looking curve (M=101) contains the least information. Actually the smooth curve has a large noise level originating from the discretization. The resolution in the ucoordinate for M=101 corresponds roughly to the resolution of most of the channels of the ASDEX Upgrade CTS receivers. Over 2500 measurements in one view seem possible in high-frequency resolution measurements that were demonstrated at TEXTOR [45–47]. Single-view tomographies computed from Msynthetic measurements such as those in figure 4are presented in figure 5. For any resolution they contain a fine-grained structure that is similar to that in the original in figure 3. Even for M∼N/20 (M=101), the tomography contains evenly distributed smallscale structures but they are larger than those in the original by a factor of two. The checkerboard pattern at a correct scale begins to emerge when M∼N/2(M=1001). For M∼Nthe tomography closely resembles the original with minor defects, and for M∼4N/3 they are indistinguishable. The reconstruction prescription in previous work [24] failed to reconstruct the original function for low v⊥corresponding to about v⊥<106ms −1in our graphs. The checkerboard patterns in figure 5demonstrate that our prescription works for all v⊥about evenly. Figure 7shows single-view tomographies of the pacman function (figure 6), which we consider to be quite complex, for a various number of measurements M. From here on we do not use measurements in the interval −0.7×106<u<0.7× 106ms −1. CTS due to bulk ions makes unambigous detection offastions verydifficult ifnotimpossible inthisinterval, and so we block it in the synthetic diagnostic. This loss of information results in the appearance of triangular regions that are not experimentally accessible (figures 7(a)–(d)). The shape of such triangles depends on the projection angle φ. The sides of these triangles are given by v⊥=(const×vth ±vcos φ)/sin φ and v⊥=0[23]. The original pacman function contains complicated structures with a scale separation of one order of magnitude between the large-scale structures (pacman head, spook) and the small-scale fine details (eyes and mouth, zick zack pattern of the spook fringe). The tomography of the pacman function is also an accurate reproduction of the original function if Mis large enough. The required number of measurements Mfor accurate tomographies is similar to the required Mfor the checkerboard—and in fact for any function we tested—and does not significantly depend on whether an interval in uhas been blocked. Lastly, we note that the projection angle φis not very important in the idealized situation except for at φ=90◦when all information about vis lost and at φ=0◦where the weight functions are singular. No advantage is gained from having many views for an equal total number of measurements Min the idealized situation. For example a tomography from two views each with 1000 measurements (M=2×1000 =2000) roughly resembles the original as much as the tomography from one view with M=2000 measurements. Likewise, the angles are not important for many-view systems either if just the resolution of the measurements is high enough. The quality of the tomography for any number of views depends mostly on the total number of measurements Min the idealized situation. 6. Tomographies for heavily under-diagnosed fast-ion distribution functions The previous section demonstrated that our tomography prescription will work in an idealized situation. The original function had the same number of grid points as 5 Nucl. Fusion 52 (2012) 103008 M. Salewski et al –4 –2 0 2 4 0 1 2 3 u [106 m/s] g [1012 s/m4] 101 501 2501 1.3 1.4 1.5 1.6 1.8 2 2.2 2.4 2.6 u [106 m/s] g [1012 s/m4] 101 501 2501 Figure 4. Projections of the checkerboard function (figure 3) for φ=30◦with M=101, 501 and 2501 measurements in one view g.(a) Zoomed out showing the entire functions g.(b) Zoomed in showing that fine-scale structure can be resolved with M=2501 measurements, which is sufficient to compute an accurate tomography. –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 Figure 5. Single-view tomographies of a checkerboard function in N=30 ×61 =1830 pixels giving only Mmeasurements at φ=30◦. The Mmeasurements are evenly spaced in −5×106ms −1<u<5×106ms −1. The number of measurements is varied from M=101 to M=2501. The axes and colourbars are identical to those for the original in figure 3. –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 1 2 3 Figure 6. The original pacman function shown here is digitized in N=30 ×61 =1830 pixels. Tomographies are shown in figure 7. the reconstructions as in previous work [23,24]. The 2D velocity distribution function in an actual tokamak experiment will have a fine-grained structure. It is then practically impossible to make enough CTS or FIDA measurements to carry all information about the fine-grained f. To simulate experimental conditions, we first construct a 1D projection gwith Msynthetic measurements from a finely resolved 2D distribution: G=W1F. (15) Here we discretize fin N1=350 ×701 ∼250 000 grid points and take M∼3400 or M∼340 measurements leaving funder-diagnosed by a factor on the order of 100 or 1000, respectively. Accurate tomographies are impossible for such under-diagnosed f. Then we compute a tomography with a much lower number of grid points than N1:N2=30 ×61 = 1830 N1. F+=W+ 2G. (16) Substitution of Gnow gives F+=V2+ 2UT 2W1F. (17) Substitution of the SVD of W1=U11VT 1does not lead to simplifications as in equation (14) since UT 2U1does not disappear. If the grids for the tomography and the original function are not identical, it is necessary to truncate the SVD and use only singular values above a selected level. This is effectively also a lossy data compression technique since we find a lower rank approximation of the transfer matrix W2that has about rank R∼1700 in our example. Reference [24] used such a lossy data compression technique to simulate the effects of noise, noting that noise decreases the information content of the smallest singular values. The effect of noise and of underdiagnosing, i.e. computing the tomography on a much coarser grid than the original, are similar. G=W1Fis different from G+=W2F+, and this difference can be interpreted as noise originating from the discretization. Figure 8shows a typical beam ion velocity distribution function at ASDEX Upgrade resolved on N1=350 ×701 grid points for which we present tomographies from CTS 6 Nucl. Fusion 52 (2012) 103008 M. Salewski et al –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 –3 –2 –1 0 1 2 3 0 1 2 3 Figure 7. Single-view tomographies of a pacman function in 1830 pixels giving only Mmeasurements at φ=30◦. The Mmeasurements are evenly spaced in −5×106ms −1<u<−0.7×106ms −1and 0.7×106ms −1<u<5×106ms −1. The number of measurements is varied from M=88 to M=2868. The axes and colourbars are identical to those in figure 6. -3 -2 -1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 105 Figure 8. Typical beam ion distribution function for beam S3 at ASDEX Upgrade computed with TRANSP/NUBEAM. The distribution function is shown on a grid with 350 ×701 pixels. measurements here. The original function has peaks at full and half injection energy of 60 keV in deuterium. We plot tomographies (N2=30 ×61 =1830 grid points) of the original function in figure 9for a various number of available views Jand measurements M. The three-view and fourview CTS tomographies are proxies for mixed CTS/FIDA tomographies that can be reconstructed from the two available CTS views and the two available FIDA views at ASDEX Upgrade. The combination of different diagnostics in our method will be discussed elsewhere. We set the number of measurements per view inversely proportional to the number of views so that the total number of measurements Mis almost the same in each column of figure 9. The left column shows tomographies for one to four views with about M∼340 measurements in total (M<N), and the right column with about M∼3400 measurements in total (M>N). In the idealized situation the number of views Jis unimportant; only the number of measurements Mmatters. Therefore, just one view suffices for accurate tomographies in the idealized situation. However, under simulated experimental conditions, the number of views Jis highly important for the relevance of the tomography to the original function. The singleview tomographies do not resemble the original function but they resemble rather the weight functions, and taking more measurements Min that one view does not help significantly. Nevertheless, the lopsidedness towards negative velocities is correctly reconstructed in the tomographies. For two views the region of the beam ions is roughly identifiable, and two maxima emerge. The four-view tomographies resemble the original function best. To quantify the difference between the original function and the tomography, we define an error measure as Qtom =1 nfast |f−f+|dvdv⊥(18) nfast =fdvdv⊥(19) which is a single number quantifying the resemblance of the tomography with the original function. Qtom =0 means that the match is perfect, and Qtom ∼1 means that f+does not resemble the original function f. In this example Qtom =1 for one view, Qtom =0.8 for two views, Qtom =0.7 for three views and Qtom =0.5 for four views, but the particular values depend on the particular distribution function and the diagnostic setup. This measure should be useful for future optimization studies. Comparing the low resolution column (M<N) with the high resolution column (M>N), we find that taking ten times more measurements per view does not help improving the tomographies much whereas adding extra views does. For the high resolution cases with M∼3400 only 340 singular values are useful whereas about 300 are useful in the low resolution cases with M∼340. We now illustrate tomographies of a simpler function from synthetic CTS measurements. Figure 11 shows tomographies of a drifting Maxwellian function with N1=350 ×701 pixels (figure 10) that we then diagnose in one to four views, and we seek tomographies with N2=30 ×61 =1830 pixels. Even though the number of measurements M∼2000 is again almost the same for the four cases, the tomographies improve with the number of views. One view is not enough to give tomographies that resemble the original. Nevertheless, the tomography for just one single view correctly identifies the location of the Maxwellian peak, so we can conclude that measurements in one single view contain relevant information about feven under simulated experimental conditions. 7. Tomographies from FIDA measurements So far we have built the transfer matrix Wfrom gyromotion weight functions, and these are sufficient to describe CTS measurements. Analytic expressions for these CTS weight functions are available [23]. Weight functions relevant to FIDA measurements are more complicated and are calculated 7 Nucl. Fusion 52 (2012) 103008 M. Salewski et al –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 105 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 10 5 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 105 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 10 5 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 105 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 10 5 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 10 –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 0 5 10 15 x 10 5 Figure 9. Tomographies (N2=30 ×61 pixels) of a typical beam ion distribution function (N1=350 ×701 pixels) for various numbers of views and measurements M. The total number of measurements Mis similar in each column. In the left column (a), (c), (e), (g)M∼340 whereas in the right column (b), (d), (f), (h)M∼3400. The viewing angles are φ=20◦for one view, φ=(10◦,80◦)for two views, φ=(10◦,40◦,80◦)for three views and φ=(10◦,30◦,60◦,80◦)for four views. –3 –2 –1 0 1 2 3 0 1 2 3 v|| [106 m/s] v⊥ [106 m/s] 2 4 6 8 Figure 10. A drifting Maxwellian resolved in 350 ×701 pixels. by counting photons in the different wavelength intervals. FIDA weight functions therefore contain numerical noise that decreases with the square root of the computer time allowed for their computation. We use φ=(11◦,64◦)for the two FIDA views available at ASDEX Upgrade. The measurements M are evenly distributed in the wavelength intervals 649 nm < λ<654 nm and 659 nm <λ<663 nm. FIDA light cannot be observed in the wavelength interval 654 nm <λ< 659 nm due to beam emission and halo neutrals [16], and so we exclude this wavelength range also in the synthetic measurements. Figure 12 shows a tomography of the original function (figure 8) from synthetic measurements for the twoview FIDA system at ASDEX Upgrade and demonstrates that our prescription also works for FIDA measurements. The original has N1=350 ×701 grid points which was here diagnosed by M=2×90 =180 measurements, and the tomography in figure 12 has N2=30×61 =1830 grid points. We here use the largest 80 singular values for the computation of the Moore–Penrose pseudoinverse. 8