Analysis of electron cyclotron emission with extended electron cyclotron forward modeling

At the ASDEX Upgrade tokamak the electron temperature (Te) profile is estimated from electron cyclotron emission (ECE) using radiation transport forward modeling within the integrated data analysis scheme. For the interpretation of ECE measurements in high T e > 5 keV plasmas, it was necessary to upgrade the forward model with a fully relativistic absorption coefficient including the relativistic Maxwell–Jüttner distribution. This model intrinsically enables the interpretation of ECE measurements affected by the so-called ‘pseudo radial displacement’ or by harmonic overlap. A numerically efficient implementation allows for the analysis of everyday ECE measurements at ASDEX Upgrade. Various ASDEX Upgrade plasma scenarios are discussed highlighting the benefits of the present radiation transport forward modeling for routine analysis.


Introduction
Electron cyclotron emission (ECE) is one of the primary diagnostics for estimating the electron temperature (T e ) profile in magnetically confined fusion research [1].
A calibrated ECE diagnostic measures the radiation temperatures (T rad ) for a set of measurement frequencies ω. Often it is possible to infer electron temperature (T e ) from T rad via the Rayleigh-Jeans law [2]. The position of the T e measurement is determined by ω and is usually mapped to the cold resonance position, where ω is either equal to the fundamental or an harmonic of the cyclotron frequency [2]. Most frequently the radiometer is optimized such that the second harmonic eXtraordinary mode (X-mode) is the main contributor to the observed T rad .
However, this ubiquitous approach to interpret the ECE measurements becomes inadequate if (I) emission from plasma layers other than the cold resonance position contributes either due to relativistically down-shifted emission or due to Dopplershifted emission in case of oblique lines of sight (LOS). The same applies if (II) the optical depth of the measurement is low, or if (III) harmonic overlap occurs. For low absorption near the cold resonance, emission from additional plasma layers can pass through the cold resonance layer resulting in a shine-through of down-shifted emission. This occurs typically near the plasma edge and in the near scrape-off layer (SOL) in high-confinement mode (H-mode) plasmas, where emission from the pedestal top and the gradient region is observed in channels with cold resonance positions in the near SOL region. The shine-through radiation causes the T rad profile to show a peak structure in the near SOL, which is called a shine-through peak [3,4]. Low optical depth at elevated temperatures (T 7 keV e > ) can result in shine-through of heavily down-shifted emission from relativistic electrons at the plasma core. Furthermore, even in the case of a large optical depth, a locally small absorption in the plasma core can result in a so-called 'pseudo radial displacement' [5] of the ECE measurements near the plasma core.
To overcome the density cut-off of the second harmonic X-mode, measurements of the third harmonic X-mode spectrum can be used. However, this poses the problem of harmonic overlap, i.e. in addition to the emission from the third harmonic resonance located at the low-field side (LFS), there can be also (at low torus aspect ratio) a contribution by the resonance with the second harmonic located at the high-field side (HFS). Finally, for oblique LOSs the Doppler effect can displace the origin of the observed radiation from the cold resonance position [6,7].
Radiation transport modeling [4,[7][8][9][10] in the framework of Integrated Data Analysis (IDA) [11] allows one to resolve all of these issues. A reliable reconstruction of T e profiles in H-mode from ECE measurements considering shine-through emission and Doppler broadening is obtained for second harmonic X-mode spectra at relatively large electron density (n e ) and moderate T e applying a previous electron cyclotron emission forward model (ECFM) [4]. In the present work the radiation transport model presented in [7] is applied, which extends the applicability of the radiation transport method to high T e and ECE measurements affected by third harmonic emission. This paper is structured as follows. In section 2 the ASDEX Upgrade profile radiometer is presented. In section 3 the radiation transport models described in [7] and [4] are compared. Section 4 shows new applications of the advanced forward model. Conclusions are drawn in section 5.

The ECE diagnostic at ASDEX Upgrade
At ASDEX Upgrade a 60-channel heterodyne radiometer is used for ECE measurements. The LOS of the ECE system are close to the mid-plane and the antennae are located at the LFS [4,12]. Figure 1 illustrates the optics of the ECE diagnostic. A quasi-optical system of three lenses focuses the radiation emitted by the plasma into a rectangularly arranged bundle of waveguides with three rows and four columns. These waveguides are illustrated by the small rectangles on the left side of figure 1. For clarity only the optical paths of two waveguides are shown in each of the figures. The axis of the quasi-optical system, which is shared with an ECE imaging diagnostic [13], is aligned with the center of the wave guide bundle. The waveguides are all parallel to each other and since there are four columns, none of the LOS is perfectly radial. For the two inner waveguides the toroidal viewing angle (i.e. the deviation from a perfectly radial view) is 0.7 tor f =   and for the two outer waveguides 2.2 tor f =  . The ASDEX Upgrade profile radiometer is designed to observe the emission of the second harmonic X-mode. In this paper the designation of X and O-mode refers to a propagation perpendicular to the magnetic field. The polarization filter is a wire grid aligned with the toroidal direction of the torus. Because of this alignment the filter is not 100% efficient, due to a small but non-zero pitch angle of the magnetic field and the deviation from a perfectly radial view of the diagnostic. Nevertheless, for this paper it is assumed that the contribution of O-mode radiation to the measurements is negligible. Accordingly all synthetic spectra shown in this paper are pure X-mode spectra. The validity of the assumption of 100% X-mode emission is discussed in section 3.7.
The radiometer covers a frequency range from 84.3 to 143.6 GHz. For ASDEX Upgrade's typical magnetic field strength ( B t » | | 2.5 T) the resonance positions are chosen such that they cover a region ranging from a few centimeters on the HFS close to the magnetic axis across the entire LFS and to the SOL. Thirty-six of the sixty channels feature a bandwidth of 300 MHz, which corresponds to a spatial resolution (disregarding frequency broadening effects) of 0.5 cm » at the plasma edge. The channels are distributed non-equidistantly with a typical spacing of 400 MHz. The other 24 channels have a wider bandwidth of 600 MHz and a frequency spacing of approximately 1 GHz. This translates into a spatial resolution of 1.2 cm » at the plasma core. The profile radiometer is absolutely calibrated with the hot-cold source technique [14]. The estimated systematic error of the calibration is 7%. The ECE measurements are sampled with a frequency of 1 MHz. The measurements shown in this work are averaged over 1 ms. The error bars are evaluated from the sum of squares of the systematic uncertainty and one standard deviation from the temporal average.

The improved radiation transport model
The previous ECFM [4] is compared with the present, improved radiation transport model [7] by using plasma scenarios with significant shine-through from the core to the edge due to heavily down-shifted emission of relativistic electrons. However, a straightforward comparison of the ECFM presented in [4] with the improved radiation transport model proposed in [7] is not possible. For the selected scenarios the model of [4] faces issues with numerical stability and validity limitations. It is emphasized that these problems arise only for the type of scenarios addressed in this paper, whereas the results presented in [4] are not affected. Three modification are made to the previous ECFM [4] for the benchmark against the improved radiation transport model of [7]: 1. In the model from [4] the cut-off density of the second harmonic X-mode was approximated with n 2 This approximation assumes that the measurement frequency ω equals twice the cyclotron frequency. While this approximation holds at the cold resonance position of the second harmonic, it is invalid for strongly down-shifted emission for which ω<2ω c,0 . Instead, the cold plasma refractive index is used to identify the regions of the LOS where the microwaves are evanescent. This is also consistent with the cold plasma raytracing performed in the improved radiation transport model of [7]. 2. The analytical solution of the emissivity integral presented in [4] is observed to be numerically unstable for strongly down-shifted emission ω<2ω c,0 . The stability issues are caused by the numerical implementation of the Dawson Integral which is required by the analytical formula. Replacing the analytical solution of the integral with a Gaussian quadrature scheme avoids this issue. 3. The forward Euler solver for the radiation transport differential equation is replaced with a 4th order Runge-Kutta integrator, which improves numerical stability.
From this point on the ECFM of [4] including the modifications explained above is denoted as model A. The model presented in [7] will be referred to as model B.

Similarities between the two models
Both models, A and B, first calculate the LOS and then solve the radiation transport equation along the LOS in a second step. Both models assume a thermal plasma and apply Kirchhoff's law relating the emissivity and the absorption coefficient. For the comparison only the second harmonic X-mode emission is considered, because unlike model B, model A was not designed for any other harmonic or wave polarization. Furthermore, an infinite reflection model is used in both models to include the effect of wall reflections [4,7,15]. The wall reflection coefficient is chosen to be R wall =0.9 for all calculated T rad profiles in this paper. This value has been proven to be reasonable for most ASDEX Upgrade plasmas that exhibit a shine-through peak.

Improvements
For a general treatment of electron cyclotron waves the relativistic dispersion relation has to be solved for the complex refractive index N s, ¢ w [16]. Unfortunately, no analytical solution is available and numerical schemes [16] become unreliable if roots corresponding to electrostatic Bernstein waves [17] lie close to or coincide with the roots of the X-or O-mode [18].
A compromise between the general treatment and the approximations used in [4] is provided for model B by combining the absorption coefficient given by equation (7) of [19] with cold plasma geometrical optics raytracing. Cold plasma geometrical optics raytracing for ECE is a standard procedure [8,10], but, to our knowledge, the application of the absorption coefficient of [19] is new.
The approach of [19] inherently considers only the electromagnetic energy flux while the so-called sloshing flux, which is non-zero only if finite temperature effects are included in the dispersion relation, is neglected [19,20]. Furthermore, the refractive index N ω and the wave polarization vector are derived from the cold dielectric tensor. The advantage is that the absorption coefficient and the emissivity are expressed as an integral in momentum space that can easily and robustly be solved numerically. The details on the absorption coefficient and the emissivity can be found in appendix A. With these approximations the emissivity and absorption coefficient of model B are typically about 10% different from the values derived rigorously from the fully relativistic dispersion relation. Nevertheless, as shown in appendix B, the modeled ECE intensities do not differ significantly as long as the optical depth is not too small.
With the modified emissivity, absorption coefficient and the addition of geometrical optics raytracing, the four key improvements of model B relative to model A are:

Significance of improvements
In order to assess the significance of the various improvements implemented in model B, a hybrid model A¢ is introduced containing all improvements (1-3) except for the relativistic distribution function (4). Figure 2 compares the radiation temperature T rad evaluated with models A, A¢ and B (T rad,mod ) with the measured values T rad,ECE for three different scenarios. All three scenarios have strong, central electron cyclotron resonance heating (ECRH), a correspondingly large T e in the plasma core, and an on-axis magnetic field strength of about The T e profiles shown in figures 2(a), (c) and (e) are functions of the square root of the normalized poloidal flux, ρ pol . The T e -and n e profiles are estimated within the IDA framework combining measurements from ECE, interferometry [21] and lithium beam spectroscopy [22] for #31594 and #31539. Thomson scattering [23] measurements of n e replace the lithium beam spectroscopy measurement for #32740, because overlapping lithium and helium lines reduce the reliability of lithium beam spectroscopy in helium plasmas. The ECE measurements in figures 2(a), (c) and (e) are mapped to cold resonance positions. The T e profile is inferred from the ECE measurements with model B. Only ECE measurements within the confined region (ρ pol <1) were considered. Therefore, the comparison of the measured and forward modeled T rad in the SOL ( 1 pol  r ) allows one to validate the various models.
For channels with cold resonance positions in the SOL, model B provides the best agreement between the measured and modeled T rad even though these channels are not considered in the fit. Model A shows the worst agreement. Model A¢ including three out of the four improvements performs only little better compared to A. Although model B describes the relatively small measured ECE intensities at the outer edge reasonably well, there are residual discrepancies, especially in #31539 (see figure 2(e)). The residual discrepancy is discussed in sections 3.5 and 3.6.
In most plasmas there is no significant difference between the T rad of models A and B inside the confined region (see figures 2(c) and (e)). However, model A performs poorly if T e and n e are extremely small in the edge of the confined region (see figure 2(a)). Such conditions arise at ASDEX Upgrade routinely in ohmic discharges which necessitates the analysis with the extended model B.

Relativistic versus non-relativistic distribution function
For the scenarios studied in [4] with relatively large n e ( 5 10 m 19 3 >´-) and moderate T e ( 5 keV < ) the measurements and forward modeled T rad were consistent. This was confirmed with model B. The different performances of models A, A¢ and B for the present plasma scenarios mainly results from the different electron energy distribution functions considered and the shine-through of down-shifted emission from relativistic electrons in the plasma core. The origin of the radiation is given by the birthplace distribution of observed intensity (BPD) [6,7]. The BPD of ECE measurements corresponds to the power deposition profile of the ECRH being normalized to the total deposited power, whereas the BPD is normalized to one. All three cases show a significant contribution of the plasma core which decreases with increasing density as is expected due to the increased optical depth. This contribution was not observed for the scenarios discussed in [4]. The large SOL peaks in T rad predicted by models A and A¢ result from an overestimation of the strongly down-shifted radiation from electrons in the tail of the electron velocity distribution located in the plasma core. The Maxwell distribution does not account for the relativistic mass increase resulting in an overpopulation of the relativistic speeds. The contribution of down-shifted emission to ECE measurements is known since long [3,24,25], but associated with non-thermal electron velocity distributions. Studies of non-thermal distribution functions analyzing ECE data use, e.g., a non-relativistic Bi-Maxwellian [25][26][27]. However, strongly down-shifted ECE arises already from the thermal tail of the distribution function. With an on-axis magnetic field strength of B 2.5 T t = | | the second harmonic of the cyclotron frequency is  For each point on the LOS it is possible to derive the velocity β which contributes strongest to the observed down-shifted emission using the resonance condition and the integrand of the emissivity. With the BPD the position that has the strongest contribution of down-shifted emission can be identified. By combining these two pieces of information the velocity that contributes the most to the down-shifted emission can be calculated. This velocity is indicated by the vertical lines in figure 3. The variability of β from 0.55 to 0.60 is due to different Doppler shifts and BPDs. The scenario with the smallest (largest) n e shows the region with largest (smallest) frequency down-shift.

The Abraham-Lorentz force
Model B describes small T rad values in the SOL region reasonably well for plasmas with either low T e or low n e , e.g., for the discharges #31594 and #32740. But it overestimates T rad for plasmas with large core T e and n 4.0 10 m e 19 3 >´-, e.g., for #31539. Reference [28] shows that the radiation drag can reduce the efficiency of electron cyclotron current drive in future fusion devices by depleting the high-energy tail of a thermal distribution. Therefore, the omission of the Abraham-Lorentz force [29] is investigated as a possible reason for an overestimation of T rad . Although the radiation drag was found to be negligible for the presented ECE measurements, the approach is described to allow for further evaluation of ECE measurements in future plasma scenarios.
For estimating the effect of the Abraham-Lorentz force on ECE measurements the radiation drag has to be balanced with thermalizing collisions. For small velocities the radiation reaction force is expected to be negligible as the collision frequency is large. For relativistic velocities the radiation reaction force is expected to alter the thermal distribution due to a relatively small collision balancing term. To estimate the significance of the radiation reaction force on ECE measurements the linear solution for the steady-state electron distribution function resulting from the Abraham-Lorentz force [30,31] and relativistic collisions [32] was calculated analytically assuming a homogeneous plasma and a dimensionless momentum u 1 . The details on the calculation can be found in appendix C. The steady-state distribution Taking into account the radiation reaction force, is expressed as a sum of Legendre polynomials P n with the pitch angle The pitch angle is given by the fraction of u P the dimensionless momentum parallel to the magnetic field over the dimensionless, total momentum u. The coefficients of the Legendre polynomials g n are given by: It can be shown that all other coefficients (g 1 , g n>2 ) are zero.  The following variables were introduced in equations (2) and (3): where Z the effective charge and lnL the Coulomb logarithm. For the calculations the mean value of B on the flux surface and Z=1.5 is considered. Note that the distribution function given by equation (1) has to be multiplied by a factor to ensure normalization. However, the normalization factor differs only very slightly from one, because only f 0 g 0 , which is of the order of 1.0×10 −5 , contributes to the normalization. This analytical solution was compared with the numerical solution obtained with the Fokker-Planck code RELAX [33], which was extended to include the Abraham-Lorentz force. The deviation of the two non-thermal distributions from the thermal distribution was computed and normalized by the value of the thermal distribution. These normalized deviations are shown in figure 4(a) as functions of the dimensionless momentum perpendicular to the magnetic field u^for u P =0 and ρ pol =0.2. Both distributions show a depletion of highenergetic electrons of the order of 5%-20% in the relevant range of u ⊥ =0.6-0.8.
The expressions for the emissivity and the absorption coefficient given in appendix A allow T rad to be computed with model B even in case of non-thermal distributions. In figure 4(b) T rad derived from the analytical distribution function profile and the distribution function profile of RELAX are compared to the thermal T rad profile for the ECE channels with resonance positions in the SOL. The reduction of T rad compared to the T rad evaluated with a thermal distribution function is only in the order of a few %. This clearly shows that the effect of the Abraham-Lorentz force is too small to be responsible for the overestimation of T rad . The observed discrepancy between the modeled and the measured T rad must therefore have a different reason.

Wall reflections
Another possible reason for the overestimation of T rad is given by the simplicity of the infinite wall reflection model. Figure 5 compares T rad,mod computed with the wall reflection coefficients R wall =0.9 and R wall =0 (no reflection). The much better agreement of the measurements with the modeling without wall reflections indicates that for this discharge wall reflection might be overestimated.
This appears to be plausible since the infinite wall reflection model is known to be inappropriate if the optical depth is very small [15]. In #31539 the optical depth of the plasma for channels with resonance position ρ pol,res >1.04 is τ ω <0.2. For this situation and assuming an idealized case of a perfectly reflecting wall (R wall =1) more than 15 direct reflections are needed to provide a total optical depth τ ω >3. For these cases, it is expected that the entire plasma volume contributes to the ECE measurement [34]. Reference [15] suggests to assume that the radiation from the plasma is in thermal equilibrium with the wall. The wall then provides an initial radiation temperature to the radiation transport equation resulting in a significantly smaller intensity at the ECE antenna compared to the infinite wall reflection model. However, the formalism of [15] is not directly usable for radiation transport modeling, since it is assumes that only emission from cold resonance positions contributes to the measurements. This is clearly not valid for the SOL ECE channels of, e.g., scenario #31539 (see figure 2(f)), where there is no significant contribution from the cold resonance position. For ECE measurements in the mid-plane it is expected that a generalized form of the reflection model of [15] results in a significantly smaller enhancement of T rad than the infinite reflection model predicts. A detailed comparison between the infinite wall reflection model and an appropriately extended version of the reflection model of [15] will be subject of future work.
To conclude, for optical depths below about τ ω <0.5 the infinite reflection model does not provide reasonable values for T rad . Additionally, for low density, primarily electron heated plasmas (e.g. #31594 and #32740) the situation complicates, because the ECRH might give rise to nonthermal distribution functions. Fast electrons are known to be able to affect the ECE measurements of the SOL significantly [35]. In #31539 a significant influence of fast electrons is not expected due to the relatively large core density.

O-mode
Due to the toroidal alignment of the polarizing beam splitter a small fraction of the X-mode ECE is reflected and a corresponding fraction of O-mode emission can pass through. In routine analysis it is assumed that only X-mode emission contributes but with 100% of its intensity. The X-mode T rad is typically much larger than the O-mode T rad because of the rather low optical depth of the second harmonic O-mode ECE in ASDEX Upgrade plasmas. Hence, assuming 100% X-mode erroneously results in an overestimation of T rad . To investigate the relevance of this overestimation, the radiation transport model presented in [7] was extended to include the polarization filter: The modeling of the polarization filter shows that 5%-10% of the O-mode ECE can pass through the filter. The fraction depends on the toroidal angle of the antenna and the ratio between the poloidal and the toroidal magnetic field strength in the mid-plane. Correspondingly, the X-mode contribution is reduced by the same fraction. Wall reflections are expected to mitigate the rather small optical depth of the O-mode emission (τ ω <0.2). Assuming the infinite reflection model the calculated T rad spectra superposed by the O-and X-mode are at most 5% smaller than the pure X-mode spectra. Neglecting reflection completely, results in T rad of the combined O-and X-mode spectrum being 5%-10% smaller than the pure X-mode spectrum.
Although 5%-10% overestimation of T rad might be significant in special applications, for routine analysis it is considered to be negligible. The code allows one to optionally include (switch on) the filter effect and the O-mode contribution but at the cost of increased numerical effort.

Applications of electron cyclotron radiation transport modeling
In standard plasma scenarios with relatively large n e and moderate T e , radiation transport modeling describes the shinethrough peak at the plasma edge as down-shifted emission from the pedestal top through the steep H-mode pedestal gradient without the need for non-thermal electrons [4]. For routine evaluation of ECE measurements in a broad operational space, radiation transport modeling has to reliably describe the effects observed in all plasma scenarios. This includes the two cases: 'pseudo radial displacement' in the plasma core and measurements affected by harmonic overlap.

'Pseudo radial displacement' in the plasma core
At high T e and low n e ECE measurements are known to show a 'pseudo radial displacement' [5], which was observed, e.g., at ASDEX Upgrade [36,37], JET [38], DIII-D [39] and TORE-SUPRA [40]. The 'pseudo radial displacement' is observed for large T e gradients in the plasma core for discharges with large T e and small n e . It results from reduced absorption close to the cold resonance position and a corresponding shining of down-shifted emission through the cold resonance. The 'pseudo radial displacement' is already intrinsically included in the radiation transport model of [4]. However, 'pseudo radial displacement' is observed at ASDEX Upgrade only in high T e discharges where the usage of a non-relativistic Maxwellian is inappropriate for the interpretation of the plasma edge ECE measurements. The improved model allows the consistent description of ECE measurements in scenarios with 'pseudo radial displacement'. The 'pseudo radial displacement' can best be seen if the ECE channels are mapped to magnetic coordinates, where a loop structure in T rad appears even though T e is expected to be constant on flux surfaces. A similar loop structure might also occur (and is observed) when the flux surfaces of the magnetic equilibrium are inaccurately estimated. The two possible sources of the loop in T rad,ECE (ρ pol ) can be distinguished by radiation transport modeling. The loop structure by 'pseudo radial displacement' can be described by radiation transport modeling whereas a residual loop not modeled properly with radiation transport could be explained by an erroneous equilibrium. This residual loop structure provides valuable information to improve the equilibrium reconstruction employing iso-flux constraints [41]. Figure 6 shows ECE measurements from discharge #30907 at t 0.73 s = with a large T e gradient and a relatively small electron density of n 1.2 10 m e 19 3 »´in the plasma core. Negative (positive) values of ρ pol correspond to cold resonance positions on the HFS (LFS), respectively. T rad on the HFS (LFS) is smaller (larger) than T e at the cold resonance position. This displacement is due to the low absorption at the cold resonance position and the shine of down-shifted radiation through the cold resonance [37]. The corresponding modeled T rad,mod values describe the measurements T rad,ECE reasonably well. This indicates that for the present case the equilibrium is not responsible for the displacement. In the chosen scenario the 'pseudo radial displacement' is large compared to the uncertainty of the equilibrium. The equilibrium was validated using tomographic reconstruction of the soft x-ray measurements [42]. For this case the soft x-ray reconstruction allows one to determine the magnetic axis with an upper uncertainty margin of 1 cm. Shifting the magnetic axis by 1 cm in any direction affects the HFS-LFS asymmetry of the ECE measurements negligibly. Radiation transport modeling allows the reconstruction of T e without the need for extra displacement of the emission location as proposed e.g.
by [43]. This is very convenient for routine analysis of large data sets.

Third harmonic and harmonic overlap
The application of ECE for high n e operation is limited by cut-offs, which is expected to hamper the use of the ECE diagnostic in future fusion devices. One solution to this problem is to measure the emission of a higher harmonic. In the case of ASDEX Upgrade one can measure third harmonic X-mode instead of the second harmonic, which increases the cut-off density by a factor of 1.5 compared to the X2-mode. However, this approach is of limited applicability for two reasons. The first is that the X3 absorption coefficient in medium size devices like ASDEX Upgrade is small. This broadens the BPD significantly compared to the BPD of the second harmonic emission. The second challenge is given by the harmonic overlap. Typically, the cold resonance position of third harmonic ECE lies on the LFS and it can be accompanied by an additional resonance with the second harmonic on the HFS. The combination of the low optical depth of the third harmonic resonance with the harmonic overlap can cause the second harmonic emission from the HFS to shine through the absorption layer of the third harmonic [44]. Hence, in order to estimate the T e profile the mixture of X2 and X3 emission needs to be modeled properly.
An example of harmonic overlap is illustrated in figure 7 for the ASDEX Upgrade discharge #32934 at t=3.298 s with a magnetic field of B 1.8 T t =and a plasma core density of 7.4 10 m 19 3 »´-. Figure 7 shows the corresponding plasma frequency f p , the right-hand cut-off frequency f R , and the fundamental, second and third harmonic of the cyclotron frequency f c . For a measurement frequency of f 105 GHz ECE < the emission of the X2-mode is inaccessible for the entire LFS (R>1.65 m), because the right-hand cutoff frequency f R >2f c (see figure 7). In contrast, an increased is not in cut-off, but introduces the problem of harmonic overlap.
At DIII-D the harmonic overlap has been addressed by calculating the optical depth of the X3-resonance and by evaluating T rad from the mixture of X2 and X3 radiation under the assumption of T rad =T e at the positions of both cold resonances [44]. Compared to the rigorous treatment employing radiation transport modeling, the DIII-D approach has two disadvantages. The first is that the method of [44] inherently requires Thomson scattering measurements to determine T e at the cold resonance position of the second harmonic. The second disadvantage is that any relativistic broadening of either resonance is neglected.
The radiation transport model in the IDA framework allows one to determine the T e profile by only considering T e information from ECE measurements. This works for any plasma region provided that significant local T e information is supplied by the ECE measurements. Figure 8 shows discharge #32934 at t= 3.298 s where harmonic overlap has to be considered for nearly all channels. In contrast to the previous figures, the measured T rad,ECE are mapped to the cold resonance position of the third harmonic. The black line depicts the T e profile estimated from ECE measurements only. The black dashed lines indicate the upper and lower error band of the T e profile. Although T rad,ECE is matched very well for all measurements, T e is only reliable in the region of ρ pol <0.8.
The T e profile has large upper and lower uncertainties for ρ pol >0.8, because the ECE measurements do not provide significant information on the T e profile in this region. Figure 9 shows the BPDs for three selected channels. The black line corresponds to a channel with X2-resonance at 0.88 pol r = | | on the HFS and with X3-resonance at ρ pol =0.48 on the LFS. Only the X3-mode contributes significantly to the measured intensity. The X3-mode emission is broad with a significant contribution of down-shifted ECE.    Although the method intrinsically works without additional diagnostics, improved results in regions with poor ECE coverage can be obtained if the ECE measurements are supplemented with Thomson scattering data. The red line in figure 8 depicts the T e profile of the combined analysis T e [ECE+TS] and the red dashed line the corresponding upper and lower error margins. For ρ pol <0.8 Thomson scattering does not provide significant additional information. It only confirms the ECE measurements. For ρ pol >0.8 the lack of information from ECE is compensated by Thomson scattering. The modeled values T rad,mod [ECE] and T rad,mod [ECE+TS] agree very much indicating that the large error bars for ρ pol >0.8 in the T e profile considering ECE only is indeed resulting from missing information.

Conclusions
An improved radiation transport model for ECE is compared to a previous model used for standard plasma scenarios at ASDEX Upgrade. Only the improved model is capable of describing the ECE measurements in the plasma edge and SOL region correctly for plasmas with a core T 7 keV e > . The benefits of the extended validity of the new model were highlighted for two plasma scenarios. The improved model extends the pool of plasma scenarios that can be evaluated reliably within the routine analysis.
The improved model includes cold plasma geometrical optics raytracing and a fully relativistic absorption coefficient considering cold plasma dispersion for wave polarization and refractive index. The relativistic Maxwell-Jüttner distribution is the most important new ingredient for describing properly the shine-through of heavily down-shifted emission from relativistic electrons in the plasma core. In addition, the effect of the radiation reaction force on the high-energy tail of the electron distribution function and its consequence for the ECE measurement was investigated analytically and numerically. No significant effect on the modeled T rad was found for the scenarios discussed. The influence of wall reflections was identified as a plausible explanation for the residual discrepancy which remains between measured and modeled T rad for measurements with extremely low optical depth. The radiation transport model of [7] was extended to include the polarization filter, which allowed the estimation of the influence of O-mode emission on the ECE measurements at ASDEX Upgrade. Assuming that purely X-mode emission contributes to the ECE spectra overestimates the modeled T rad by 5%-10% at ASDEX Upgrade.
The improved radiation transport model is applied to two plasma scenarios which constitute special cases from the perspective of ECE. The 'pseudo radial displacement' observed for large core T e gradients is correctly accounted for by the model. The successful reconstruction of T e profiles from ECE measurements containing a mixture of X2 and X3 emission is demonstrated. Supplementing the ECE data with Thomson scattering data in the IDA framework helps to recover regions which are not covered properly by the ECE data alone.
In summary, the IDA approach including the advanced radiation transport forward model considerably improves the accuracy of T e profile reconstructions and extends the operational space of the ECE diagnostic. The ECE forward model is applicable for routine analysis in everyday ECE data interpretation.

Acknowledgments
One of the authors (S S Denk) acknowledges Daniela Farina and Lorenzo Figini, who provided the numerical implementation of the absorption coefficient rigorously derived from the warm dispersion relation [16].
The computations have been performed in part on the EUROfusion Gateway Cluster, using the framework developed and maintained by the EUROfusion-IM Team 6 . This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the The Lorentz factor is denoted as u 1 2 1 2 g = + ( ) and u P and u ⊥ are the components of the dimensionless momentum u parallel and perpendicular to the equilibrium magnetic field, respectively. The momentum distribution function u f ( ) can be arbitrarily chosen but the implementation of model B is limited to gyrotropic distributions. The Dirac's δ-function accounts for the cyclotron resonance, with n the harmonic number. At last [19] where J n is the Bessel function of first kind with argument b u N c,0 w w =^^. The components of the refractive index N ω perpendicular/parallel to the ambient magnetic field are denoted as N ⊥/P . If the distribution function f is chosen to be the Maxwell-Jüttner distribution it is possible to reproduce Kirchhoff's law with expressions given by equations (A1) and (A2).
The integrals in equations (A1) and (A2) can be solved numerically if the cold dielectric tensor is considered in the derivation of the refractive index N ω and the normalized polarization vector e as described in [19].

Appendix B. Validation of the absorption coefficient
For the approximated absorption coefficient given by equation (A1) the refractive index, the polarization vector and the energy flux are derived from the cold plasma dispersion relation. This approximation is valid for any harmonic n>2 and for the second harmonic (n=2) if [17,20]: But for, e.g., B 2.5 T t =which is typical for ASDEX Upgrade, ω c,0 ≈ω p,0 for n 6.0 10 m e 19 3 »´-. For routine data analysis at ASDEX Upgrade the radiation transport model has to perform reasonably well for densities up to the cut-off density n 1.0 10 m e,cut off,AUG 20 3 »´--. Nevertheless, for the analysis of ASDEX Upgrade ECE measurements the approximated absorption coefficient is a viable approximation even if the condition given by equation (B1) is violated. The approximated absorption coefficient is benchmarked against the absorption coefficient derived self-consistently from the solution of the fully relativistic (or warm) plasma dispersion relation (see [16]).
The two absorption coefficients are compared in figure B1(a) for three densities n e = 1.0 10 m 20 3 , n e = 8.0 10 m 19 3 and n e = 6.0 10 m 19 3 and a measurement frequency f 140 GHz 2 = = w p . In figure B1(b) f 110 GHz = is chosen and, to avoid cut-off, lower densities n e = 6.0 10 m 19 3 , n e = 4.0 10 m 19 3 and n e = 2.0 10 m 19 3 are selected. In both cases the angle between the magnetic field lines and the wave vector is 85°and T e is 8 keV. For the selected frequencies, all densities and almost for the entire range of c,0 w w , there are deviations in the order of 5%-15%. Larger deviations occur for 1.75 c,0 < w w in case of the largest n e respectively in figures B1(a) and (b). In contrast to the absorption coefficient from the warm plasma dispersion relation the approximated absorption coefficient does not allow for wave propagation for small c,0 w w and large n e . Figure B1 shows, as expected, that the cold plasma dispersion produces the largest errors close to cut-off. Although the relative deviations are in the order of ten percent in the full range of c,0 w w and all considered n e and frequency combinations, in practical applications, the cold plasma dispersion for the absorption coefficient is sufficient to provide good estimates for T rad for the ASDEX Upgrade ECE diagnostic. To exemplify this, discharge #33596 t 3.48 s = is selected for the benchmark of T rad . The peak T e is about 11 keV and n e is about 6.0 10 m 19 3 -, which is similar to the parameters of the benchmark of the absorption coefficients. The magnetic field of the discharge is on-axis −2.5 T and the for data analysis useful channels of the ECE diagnostic cover a frequency range from 110 to 143 GHz. The result of the benchmark of T rad is shown in figure B2. The T e profile used for the benchmark and shown in figure B2 was inferred from the ECE measurements with model B. The ECE measurements and the forward modeled T rad computed with each of the two absorption coefficients correspond to the left y-axis.
The right y-axis shows the relative deviation between the two forward modeled T rad . The measurements shown in the red shaded region have optical depths below τ ω <0.5 and are not considered in the analysis. Measurements with optical depth τ ω <0.1 were removed from the figure. The relative deviations between the two calculated T rad profiles are below 5% for all measurements that enter the data analysis. Only channels with extremely low optical depth τ ω <0.5 show a significant deviation between the two forward modeled T rad profiles. For these channels, however, the high sensitivity on the empirical wall reflection coefficient renders the T e information stored in the measurement irrecoverable by the IDA method regardless of which absorption coefficient is used. where we have defined 2 3 r a t te º ( ). For g 0 not to grow exponentially we have to demand that the integration constant  ~¢~. Note also that α∼1 in our ASDEX Upgrade case. To zeroth order in ò, we obtain where D is an integration constant. Since equation (C10) has a regular singularity at u 0 = , we expect the derivatives g 2  and g 2 ¢ to become large at low velocities. The expression (C12) has the non-relativistic limit There are two cases depending on the magnitude of g 2 . Either the left hand side terms balance, giving g Ax 2 = b -(which matches the first term in (C13)), or all terms of equation (C15) must be kept. In the latter case, we get g x 5 2 5 5 2 a e b = + ( ), which matches the second term in (C13). One can continue to look at what the corresponding two solutions to equation (C14) are, but since we are mainly interested in u 1 , we will neglect the term involving D in (C12) and (C13). As an interesting example, we take Z=1, i.e., β=6.