Analysis of electron cyclotron emission with extended electron cyclotron forward modeling

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.

For a general treatment of electron cyclotron waves the relativistic dispersion relation has to be solved for the complex refractive index ${N}_{{\rm{s}},\omega }^{{\prime} }{}_{}$ [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:

• 1.  
  Cold plasma refractive index and wave polarization in model B [7, 19] instead of N_ω = 1 in model A [2, 4].
• 2.  
  Arbitrary propagation direction of the wave in model B [7, 19] instead of quasi-perpendicular propagation in model A [2, 4].
• 3.  
  Fully relativistic single electron emissivity [7, 17, 19] instead of non-relativistic single electron emissivity [2, 4].
• 4.  
  Fully relativistic Maxwell–Jüttner distribution for the emissivity/absorption coefficient in model B [7, 19] instead of a non-relativistic Maxwellian in model A [2, 4].

In order to assess the significance of the various improvements implemented in model B, a hybrid model $A^{\prime}$ 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, $A^{\prime}$ 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 ${B}_{{\rm{t}}}=-2.5\,{\rm{T}}$. The main distinction is given by different on-axis electron densities n_e: (a) #31594 at $t=1.30\,{\rm{s}}$ has a very low plasma core density of ${n}_{{\rm{e}}}=1.8\times {10}^{19}\,{{\rm{m}}}^{-3};$ (b) #32740 at $t=5.06\,{\rm{s}}$ is in mainly helium with a plasma core density of ${n}_{{\rm{e}}}=3.0\times {10}^{19}\,{{\rm{m}}}^{-3};$ (c) #31539 at $t=3.29\,{\rm{s}}$ has a comparatively large plasma core density of ${n}_{{\rm{e}}}=4.7\times {10}^{19}\,{{\rm{m}}}^{-3}$. The n_e values are summarized in table 1.

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Table 1.  The core electron density values of the three benchmark scenarios.

Shotnumber	ne
#31594, $t=1.30\,{\rm{s}}$	${n}_{{\rm{e}}}=1.8\times {10}^{19}\,{{\rm{m}}}^{-3}$
#32740 $t=5.06\,{\rm{s}}$	${n}_{{\rm{e}}}=3.0\times {10}^{19}\,{{\rm{m}}}^{-3}$
#31539 $t=3.29\,{\rm{s}}$	${n}_{{\rm{e}}}=4.7\times {10}^{19}\,{{\rm{m}}}^{-3}$

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 (${\rho }_{\mathrm{pol}}\geqslant 1$) 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 $A^{\prime}$ 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.

For the scenarios studied in [4] with relatively large n_e ($\gt 5\times {10}^{19}\,{{\rm{m}}}^{-3}$) and moderate T_e ($\lt 5\,\mathrm{keV}$) the measurements and forward modeled T_rad were consistent. This was confirmed with model B. The different performances of models A, $A^{\prime}$ 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. Figures 2(b), (d) and (f) compare the BPDs from models A and B for the cases of figures 2(a), (c) and (e), respectively. Negative (positive) values of ρ_pol correspond to positions on the HFS (LFS), respectively. The gap in the BPDs close to the plasma center arise from the LOS not going exactly through the plasma center. The ECE channel was chosen such that its cold resonance position lies at ρ_pol ≈ 1.04 (dotted line).

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 $A^{\prime}$ 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 over-population 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 [25–27]. However, strongly down-shifted ECE arises already from the thermal tail of the distribution function. With an on-axis magnetic field strength of $| {B}_{{\rm{t}}}| =2.5\,{\rm{T}}$ the second harmonic of the cyclotron frequency is $2{f}_{{\rm{c}}}=140\,\mathrm{GHz}$ in the plasma core. In contrast, the measurement frequency f_ECE of the channels for which the cold resonance positions lie in the SOL is only about 105 GHz. If the down-shift is attributed to the relativistic mass increase alone (i.e. if the Doppler shift is neglected), a Lorentz factor of γ = 1.4 is required. This corresponds to an electron velocity $\beta =v/{c}_{0}=0.7$, where c₀ is the vacuum speed of light, and a kinetic energy of about $200\,\mathrm{keV}$. Figure 3 compares the Maxwellian with the Maxwell–Jüttner distribution for ${T}_{{\rm{e}}}=8\,\mathrm{keV}$. For $\beta \gt 0.3$ the Maxwellian is significantly larger than its relativistic counterpart. The position on the LOS contributing the most to the observed, strongly down-shifted emission is given by the local maxima closest to the magnetic axis of the BPDs shown in figures 2(b), (d) and (f). 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.

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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 ${n}_{{\rm{e}}}\gt 4.0\times {10}^{19}\,{{\rm{m}}}^{-3}$, 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 $u=\gamma \cdot \beta =\tfrac{\beta }{\sqrt{1-{\beta }^{2}}}\approx 1$. The details on the calculation can be found in appendix C. The steady-state distribution

$$\begin{eqnarray}&&f={f}_{\mathrm{MJ}}(u)\left(1+\displaystyle \sum _{n=0}^{\infty }{g}_{n}(u){P}_{n}(\zeta )\right).\end{eqnarray} \tag{ 1 }$$

Taking into account the radiation reaction force, is expressed as a sum of Legendre polynomials P_n with the pitch angle $\zeta =\left(\tfrac{{u}_{\parallel }}{u}\right)$ as argument. The pitch angle is given by the fraction of u_∥ 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:

$$\begin{eqnarray}&&{g}_{0}=-\alpha \left(\arctan (u)-u+\displaystyle \frac{{u}^{3}}{3}\right)\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{g}_{2}=\alpha {\left(\displaystyle \frac{\gamma +1}{u}\right)}^{3(Z+1)}{\int }_{0}^{u}\displaystyle \frac{u{{\prime} }^{4}}{\gamma {{\prime} }^{2}}{\left(\displaystyle \frac{u^{\prime} }{\gamma ^{\prime} +1}\right)}^{3(Z+1)}{\rm{d}}u^{\prime} .\end{eqnarray} \tag{ 3 }$$

It can be shown that all other coefficients (g₁, g_n > 2) are zero. The following variables were introduced in equations (2) and (3):

$$\begin{eqnarray*}\gamma & = & \sqrt{1+{u}^{2}},\alpha \equiv \displaystyle \frac{2\mu \tau }{3{\tau }_{{\rm{r}}}},\mu =\displaystyle \frac{{m}_{{\rm{e}},0}{c}_{0}^{2}}{{T}_{{\rm{e}}}},\\ \tau & = & \displaystyle \frac{4\pi {\epsilon }_{0}^{2}{m}_{{\rm{e}},0}^{2}{c}_{0}^{3}}{{n}_{{\rm{e}}}{e}^{4}\mathrm{ln}{\rm{\Lambda }}},{\tau }_{{\rm{r}}}=\displaystyle \frac{6\pi {\epsilon }_{0}{m}_{{\rm{e}},0}^{3}{c}_{0}^{3}}{{e}^{4}{B}^{2}},\end{eqnarray*}$$

where Z the effective charge and $\mathrm{ln}{\rm{\Lambda }}$ 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 f₀g₀, 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 ${u}_{\perp }$ for u_∥ = 0 and ρ_pol = 0.2. Both distributions show a depletion of high-energetic electrons of the order of 5%–20% in the relevant range of u_⊥ = 0.6–0.8.

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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.

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.

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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 non-thermal 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.

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:

$$\begin{eqnarray}&&{T}_{\mathrm{rad},\mathrm{mod}}={({\overrightarrow{e}}_{{\rm{X}}}\cdot \overrightarrow{p})}^{2}{T}_{\mathrm{rad},\mathrm{mod},{\rm{X}}}+{({\overrightarrow{e}}_{{\rm{O}}}\cdot \overrightarrow{p})}^{2}{T}_{\mathrm{rad},\mathrm{mod},{\rm{O}}}.\end{eqnarray} \tag{ 4 }$$

Hence, the synthetic radiation temperature becomes a sum of the O-/X-mode radiation temperature ${T}_{\mathrm{rad},\mathrm{mod},{\rm{O}}/{\rm{X}}}$ which is weighted by the transmissivity of the polarizer. The transmissivities of the polarizer for each mode is computed by projecting the normalized polarization vector of the O-/X-mode ${\overrightarrow{e}}_{{\rm{O}}/{\rm{X}}}$ onto the polarization axis of the filter $\vec{p}$ .

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.

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 $t=0.73\,{\rm{s}}$ with a large T_e gradient and a relatively small electron density of ${n}_{{\rm{e}}}\approx 1.2\times {10}^{19}\,{{\rm{m}}}^{-3}$ 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.

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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 ${B}_{{\rm{t}}}=-1.8\,{\rm{T}}$ and a plasma core density of $\approx 7.4\times {10}^{19}\,{{\rm{m}}}^{-3}$. 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 ${f}_{\mathrm{ECE}}\lt 105\,\mathrm{GHz}$ the emission of the X2-mode is inaccessible for the entire LFS (R > 1.65 m), because the right-hand cut-off frequency f_R > 2f_c (see figure 7). In contrast, an increased measurement frequency of ${f}_{\mathrm{ECE}}\gt 129\,\mathrm{GHz}$ is not in cut-off, but introduces the problem of harmonic overlap.

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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 ${T}_{{\rm{rad}},{\rm{ECE}}}$ 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 $| {\rho }_{\mathrm{pol}}| =0.88$ 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. The red line corresponds to a channel with X2-resonance at $| {\rho }_{\mathrm{pol}}| =0.71$ on the HFS and with X3-resonance at ρ_pol = 0.90 on the LFS. Both harmonics contribute to the measured intensity. The cyan line corresponds to a channel with X2-resonance at $| {\rho }_{\mathrm{pol}}| =0.66$ on the HFS and with X3-resonance in the SOL (ρ_pol = 1.01). Although mapped in figure 8 to the third harmonic resonance position only X2-mode emission contributes to this channel. The combination of a decreasing contribution from the X3-mode in the LFS region $| {\rho }_{\mathrm{pol}}| \gt 0.90$ and the small X2-mode emission on the HFS for $| {\rho }_{\mathrm{pol}}| \gt 0.90$ results in missing information about T_e in this region.

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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.