Advanced Residual Analysis for JET Polarimetric Measurements Validation

1. Validation of Polarimetric measurements on JET using advanced statistical analysis of the residuals M. Gelfusa, A. Murari, D. Patanè, P. Gaudio, A. Boboc Frascati 26-28 March 2012

2. Outline • Introduction to the correlation methods for model validation • Example: nonlinear pendulum • Application to the new calibration code • Application to the propagation code of polarimetry (chord 3) • Conclusions M.Gelfusa, A.Murari, et al, Validation of JET polarimetric measurements with residual analysis, Meas. Sci. Technol., 21, 115704 (2010).

3. New model validation method The main idea is that, if a model is perfectly adequate to describe certain measurements and the noise is white, the residuals, defined as measurements minus the model estimates, should present the statistics of white noise. This idea can be formalised as a statistical hypothesis testing problem. You can determine whether the residuals have the statistical properties of white noise within a certain confidence interval. It requires some careful considerations when, as in our case, non linear MIMO systems are to be studied. The methods, originally devised for control, have been adapted to investigate dynamic systems (such as propagation codes).

4. New model validation method A complete and adequate set of tests for a nonlinear, MIMO system is provided by the higher order correlations between the residuals and input vectors given by the following relations (e residuals, u inputs, y outputs): where q is the number of the dependent variables and r is the number of the independent variables. If the non linear model is an adequate representation of the system, in the ideal case, should be:

5. Example: Pendulum Example: pendulum. A nonlinear pendulum plus 10 % white noise has been used as an example y’’ + g∙y’ + a∙sin(y) = b∙sin(p∙t) Accurate model  Good correlations (inside the 95% confidence interval)

6. Correlation analysis Example: pendulum. In the case of an error in the nonlinear term the correlations exceed the confidence interval y’’ + g∙y’ + a∙sin(y) = b∙sin(p∙t) Error added  Poor correlations (outside the 95% confidence interval) See paper by A.Murari, M. Gelfusa et al,Residual Analysis for the Assessment of the Equilibrium Reconstruction Quality on JET, Nuclear Fusion, 51, 053012 (2011).

7. Polarimeter: calibration curves Phase shift calibration curve. Comparison between the experimental calibration curve (red asterisks) and the estimate of the new calibration model (black circles). The x axis Theta is the rotation angle of the half-wave plate. Comparison between the experimental calibration curve (black dotted line) and the estimate of the new calibration model (red line). Shot: 79692 chord #3.

8. Application to Polarimetry: calibration global tests For the calibration model, the input is the half-wave plate signal and the outputs are the Faraday angle and the Cotton-Mouton effect, so we have: Input - Residual (Calibration). Shot: 77650 Residual - Output (Calibration). Shot: 77650 Left: Non linear correlation between the model outputs (times the residuals) and the sum of the squared residuals. Dashed lines indicate a 95% confidence interval which correspond to where N is the number of sampled points of the trajectory. Right: Non linear correlation between the model inputs and the sum of the squared residuals.

9. Propagation code To confirm the validity of the polarimeter optical model, developed to interpret the calibration data, a numerical code, based on the Stokes formalism, has been implemented, which calculates the evolution of the state of polarisation of a laser beam crossing JET plasmas. Starting from the Lidar Thomson scattering density profiles and the equilibrium magnetic field as input, this code gives as output the Faraday rotation angle and the Cotton Mouton effect. As it is well known, the radiation propagating in a magnetized plasma in the absence of dissipation is described by the vector equation: Details in F. P. Orsitto , A. Boboc, C Mazzotta, E. Giovannozzi, L. Zabeo, Plasma Phys Contr Fusion 50 115009, (2008)

10. Propagation code: global tests For the propagation code the previous quantities are defined as follows: Residual - Output (Plasma). Shot: 77650 Input - Residual (Plasma). Shot: 77650

11. Nonlinear Local tests The results showed before refer to the system as a whole and therefore are called “non linear global tests”. These “non linear global tests” show clearly that the propagation model is not adequate to describe the experimental measurements. Non linear “local tests” can be performed to determine which “submodels” are incorrect in the full wave propagation code. These local tests are basically the same high order correlations but now with individual parts of the model, individual inputs in our case.

12. Nonlinear local tests: LIDAR Given the results obtained with the non linear global tests, which show clearly that the model is not adequate to describe the experimental measurements, the non linear local tests have been performed to determine which “submodels” are incorrect in the full wave propagation code. Non Linear correlation: Cotton_Mouton-Lidar (Shot: 77650) Non Linear correlation: Faraday-Lidar (Shot: 77650) Left: Non linear local tests. Output: Faraday angle, input: LIDAR density Right: Non linear local test. Output: Cotton-Mouton, input: LIDAR density.

13. Nonlinear local tests: magnetic field Non Linear correlation: Faraday- bt (Shot: 77650) Non Linear correlation: Faraday-br (Shot: 77650) Non linear local test. Output: Faraday angle, input: z-component magnetic field Non linear local tests. Output: Faraday angle, input: toroidal magnetic field Non Linear correlation: Faraday-bz (Shot: 77650) Non linear local test. Output: Faraday angle, input: radial component of the magnetic field.

14. Nonlinear local tests: magnetic field Non Linear correlation: CM-br (Shot: 77650) Non Linear correlation: CM-bt (Shot: 77650) Non linear local test. Output: Cotton-Mouton, input: z-component magnetic field Non linear local test. Output: Cotton-Mouton, input: toroidal magnetic field Non linear local test. Output: Cotton-Mouton, input: radial component magnetic field Non Linear correlation: CM-bz (Shot: 77650)

15. Conclusions A systematic validation, based on statistical correlations of the residuals, has been performed to assess the quality of the model for the calibration of JET polarimeter and of the propagation code, developed to interpret the results of the diagnostic. In the case of the calibration, whose input is the position of the half wave plate, both the linear and the non linear analysis confirm the null hypothesis that the model is fully adequate to interpret the measurements within the chosen 95% confidence interval. In the case of the propagation code, the residuals present both linear and nonlinear correlations well outside the 95% confidence intervals. Therefore the null hypothesis of an adequate propagation model has to be rejected.

16. Conclusions The local tests, aimed at identifying the source of the problems, indicate quite clearly that the LIDAR input is adequate whereas there are significant issues with the components of the magnetic field. Since the components of the magnetic fields along the laser beam path are not direct measurements but are provided as outputs by the equilibrium code EFIT, the situation is not surprising. The same methodology has already been applied to the qualification of equilibrium code. See paper by A.Murari, M. Gelfusa et al,Residual Analysis for the Assessment of the Equilibrium Reconstruction Quality on JET, Nuclear Fusion, 51, 053012 (2011).