Upgrade of the relative calibration methods and Bayesian inference processing for electron cyclotron emission radiometry

. An upgraded local oscillator (LO) hopping calibration method based on a blackbody hot source and a perturbation analysis of the magnetic field difference method are introduced in this work. The blackbody hot source is used to evaluate the difference in the relative coefficients between the two LO hopping frequencies in the same channels. Then the coefficients are obtained by multiplying the LO hopping frequencies coefficients by LO hopping calibration coefficients. In this way, it is more flexible and stable than the in-situ calibration. The magnetic field difference method provides another calibration method to obtain the relative calibration coefficients of the electron cyclotron emission radiometers (ECE). In general, the magnetic field difference method needs two similar shots but with a difference of 2.1% (for HL-2M) in the magnetic field. Meanwhile, there are some errors because of the deviation of detection positions in the same channels between the two shots. For evaluating the calibration errors, the impact of the displacement, T e perturbation of the core region, and magnetic field difference has been discussed. The result shows that a larger magnetic field difference can improve the accuracy of the calibration. In the end, Bayesian inference has been utilized to evaluate the calibration coefficients and get the most probable calibration coefficients along with its the confidence interval.


Introduction
ECE radiometry is a powerful Te and Te perturbation (δTe) diagnostic with high spatial-temporal resolution and it has been established on many tokamaks [1][2][3][4]. The calibration of the ECE radiometry is the key step in obtaining the absolute or relative Te profile. The conventional calibration method is the in-situ calibration using the blackbody hot source with known temperature and it is utilized on many devices such as JET, DⅢ-D, EAST, and ITER (in the future) [5][6][7][8]. However, the blackbody hot source method is complex and inflexible because it requires a long time operation (tens of minutes or even hours) and a large space to operate. Therefore, handy and flexible relative calibration methods like the LO hopping and magnetic field difference method are required [9].
Following the introduction section, an upgraded LO hopping calibration method will be described in Section 2. The perturbation analysis for the magnetic field difference method will be discussed in Section 3. In Section 4, Bayesian inference for the relative calibration methods will be introduced. Finally, Section 5 presents the summary.

Upgraded LO hopping method
The principle of the LO hopping method is to compare the signal amplitude between the adjacent channels in the same measurement position by hopping the LO frequency. However, the mixer efficiency varies with the LO frequency which results in different initial bias voltages at different LO frequencies. In the strict sense, there are many elements varying with the different LO frequencies such as the antenna gain, transmission loss, and electronic response (including the mixer's efficiency). Therefore, merely comparing the signal amplitude is not complete which will cause prominent calibration errors. The upgraded LO hopping calibration method uses a blackbody hot source to evaluate the response of the measurement channels to different LO frequencies. With this method, the relative Te can be derived as below: where I, C, f, Ib, Tb, S, and i are the plasma ECE signal amplitude, calibration coefficients, radiation frequency, radiation amplitude of the blackbody hot source, radiation temperature of the blackbody hot source, LO hopping calibration coefficients ratio, and the channel number, respectively. Setting the C1 to 1, the relative calibration coefficients can be derived with Eq. 7. Normally, the blackbody hot source's in-situ absolute calibration is the conventional calibration method for ECE radiometry. However, the absolute calibration coefficients change slowly over time because of the change in the response of the radiation frequency (RF) amplifier, intermediate frequency (IF) amplifier (mainly caused by active devices), the loss of the transmission line, window coating, and so on. It can be approximatively described as follow: In the above, the Ci(0) is the initial coefficient. The t, RFA, RFM, IFA, IFM, and L are the time, the relative variation coefficients of the RF amplifier, RF mixer, IF amplifier, IF mixer, and the loss of the transmission, respectively. Generally, the variation of the response of those active microwave devices is independent so that the response of the whole working band enhances or abates independently. For a frequency interval of 1 GHz (the interval of IF), the response variable of the active microwave devices and transmission loss can be approximatively considered to be linear. Therefore, Eq. 8 can be simplified as: The relative variation ratio of the calibration coefficients in the two hopping LO frequencies (Si) can be derived as below: This means Si does not vary with time or vary slowly. It should be noted that IFAi and IFMi can vary differently in different channels which induces a difference in the relative coefficients between different channels: where the j represents another channel. Therefore, the blackbody hot source's in-situ absolute calibration needs to be conducted once in a while, but the upgraded LO hopping relative calibration uses the blackbody hot source just once. As shown in Fig. 1, a mirror is utilized to transmit the radiation generated by the blackbody hot source to the radiometry from the transmission line. The radiation signal is hopping by the chopper. Subsequently, by comparing the signal intensity in different LO hopping frequencies, the Si can be obtained by Eq. 6. The schematic diagram of the upgraded LO hopping calibration is shown in Fig. 2. The measurement positions are constant between two adjacent channels in the two LO hopping frequencies (LO1, LO2). Therefore, the relative calibration coefficients can be derived by Eq. 7 and the Si can be used to calibrate several times without blackbody hot source evaluation.

Perturbation analysis of the magnetic field difference method on HL-2M
The magnetic field difference method is another relative calibration method for the ECE which is very handy and flexible. Generally, the calibration coefficients are obtained by comparing the signal intensity in the same discharge parameters with an appropriate magnetic field difference. Normally, the appropriate magnetic field difference is dependent on the minimum of the mean of the position deviation (δRm) between the adjacent channels with the change of the magnetic field in two shots as shown in Fig. 3. The Bt00 and Bt01 are the calibration magnetic fields on the magnetic axis. The relative calibration coefficients can be obtained below： In the above, the IA, IB, and RA, RB are the ECE signal intensities and the measurement positions in the two shots, respectively. Unfortunately, the position deviation always exists since the variation of the magnetic field causes different changes in the detected location for the different radial channels. Consequently, the calibration coefficients obtained by this method are not very accurate. Theoretically, by sweeping the magnetic field rapidly, the radiometry can measure the same positions in adjacent channels. In fact, it is difficult for the toroidal magnetic coils, especially for superconducting toroidal magnetic coils.  However, the perturbation analysis shows that the magnetic field difference in which the δRm reaches up to the minimum is not the most appropriate. FIGURE. 4. (a) shows the signal intensities obtained from the preset Te profiles (TeA, TeB) in Fig. 4. (b). The IA is set as a random series and the IB is obtained by About the preset profiles, they are fitted using the standard temperature profile model: where r is the minor radius, a = 65 cm, and α is set up to 2.7. The core Te (Te0) and the magnetic axis positions (R0) are set as 1 keV, 178 cm for the TeA, and 1.05 keV, 179 cm for the TeB, presumably induced by the plasma displacement or other physical problems. The dashed line shown in Fig. 4. (b) is the calibrated Te (TeAcal) profile regarding the TeA. The shadowed area is the deviation between TeA and TeAcal. Therefore, the error is defined as the ratio of the deviation area to the Te profile area (Sde/STe as shown in Fig. 4. (b)) and the accuracy is defined as 1-Sde/STe. By setting up the Te0A = 1 keV, R0A = 178 cm, and Bt0A = 1.5 T, scanning the R0B, Te0B, and Bt0B, the calibration accuracy depending on the R0B, Te0B, and Bt0B can be obtained as shown in     show the dependence of the calibration accuracy on the R0B and Te0B when Bt0B equals 1.532 T and 1.75T. Apparently, 1.532 T is not the most appropriate Bt0B for this calibration method because the allowable range is much more narrow than the case of Bt0B = 1.75 T compared to Fig. 6. (b).
FIGURE 6(b) shows that for the calibration accuracy of 0.95, the R0B and Te0B can vary in the range of 173 -183 cm and 0.95 -1.05 keV approximately. In other words, the position deviation and variation of the Te0 between the two calibration shots caused by the plasma displacement or the difference of the discharges are allowed to deviate by 5 cm on R0 and 5% on Te0 for 95% calibration accuracy, which is acceptable for the control on the tokamak plasma.
The perturbation analysis shows the feasibility of accurate calibration of the Te profile using this method. It's worth noting that, the accurate calibration range with this method depends on many elements, such as the geometric dimensioning of the plasma, the working magnetic field, and the working frequency band of the ECE radiometry. Therefore, the perturbation analysis needs to be performed before the calibration to get an appropriate magnetic field difference. Due to the convenience and flexibility, the magnetic field difference method is chosen as one of the calibration methods for ECE radiometry on HL-2M to access the calibration coefficients.

Bayesian inference processing
In order to evaluate the calibration coefficients obtained by the upgraded LO hopping method or magnetic field difference method, the Bayesian inference [10] is utilized to process the calibration data. Normally, the calibration coefficient Cim depends on the measured temperature Tei, signal intensity Iim, and the systematic and statistical errors εi (the i and m denote the channel number and calibration number): As mentioned above, it can be solved by the Bayesian theory: In the above equation, the n and μTe are the number of the channels and expected temperature, respectively. The Σc and ΣTe are the calibration uncertainty and inference uncertainty covariance matrixes, respectively. They are denoted as: ( , ) = 2 exp (− 2 2 2 ) , = | − | (21) In the above, the Cei, and Ri, Rj are the calibration errors and the measured positions, respectively. The 2 and 2 are the hyperparameters obtained by Occam's razor optimization criterion. Consequently, the combination of the prior and likelihood is derived as below: The expected number ( * ) and covariance matrix (Σ * ) of the prior can be derived as: Experimentally, a magnetic field difference relative calibration is processed with Bayesian inference on HL-2A. The data points are 100 times calibration in a stable discharge period of 200 ms. FIGURE 7 shows the processing result. The shadowed area is the 95% confidence interval and the solid line is the expected value of the highest probability. It can be seen that the maximum probability profile is different from the mean of the original samples and it is more smooth. The maximum uncertainty comes from the core region of the plasma. Bayesian inference provides a method with a confidence interval to determine the calibration coefficients and is helpful to evaluate the calibration accuracy. An upgraded LO hopping calibration method utilizes the blackbody hot source to evaluate the relative variation ratio of the calibration coefficients in the two hopping LO frequencies. The absolute coefficients can be obtained by simple ratio calculation. The advantage of this method is more flexible and stable. The perturbation analysis of the magnetic field difference method shows that the magnetic field when the mean of the position deviation reaches the minimum is not the most appropriate. By perturbation analysis, the most appropriate magnetic field and the allowable range of errors are obtained. In this analysis, the position deviation and variation of the Te0 between the two calibration shots caused by the plasma displacement or the difference of the discharges are allowed to deviate by 5 cm on R0 and 5% on Te0 for 95% calibration accuracy. It is acceptable for the control of the tokamak plasma. In addition, the Bayesian inference processing has been simply utilized to evaluate the calibration coefficients. The result shows that the maximum probability profile is different from the mean of the original samples and it is more smooth. The maximum uncertainty comes from the core region of the plasma. Therefore, it can provide a method with a confidence interval to determine the calibration coefficients and helps us to evaluate the calibration accuracy. The experimental verification will be carried out at the end of 2022.