A new approach to select the weights of measurements and constraints to equilibrium codes

1. A new approach to select the weights of measurements and constraints to equilibrium codes M. Gelfusa, A. Murari, E. Peluso, P. Gaudio, D. Mazon, N. Hawkes, G. Point, B. Alper, T. Eich

2. Weightdetermination for the Faraday measurements Question: how to choose the value of the weighting parameter K1=Wfar/Wcoils? Goal of the analysis: Identify a statistically sound methodology to determine the values of the Faraday measurement weights, which give the most accurate equilibrium reconstructions. The approach is based not only on the c2 but also on high order correlations of the residuals, which have been proved to be adequate for nonlinear systems. Statistical method from Billing and Zu (1995)

3. Question: how to choose the value of the weighting parameter K1=Wfar/Wcoils? OUTLINE • Introduction of statistical tools • Details of the application to EFIT • Consistency checks • Validation with independent measurements (IR, SXR) • Summary

4. The correlation tests method Hypothesis: the noise is random and additive Consequence: the residuals of a perfect model should be randomly distributed The model with the distribution of the residuals closer to a random one is preferred Linear correlation function is of the following type Residualcorrelations

5. 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 residual and input and output 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:

6. Implementation of the correlations for equilibrium Implementation of the correlations for the case of EFIT: EFIT Inputs: Pickup coils Faraday measurements Outputs: Pickup coils Faraday angles reconstructed by EFIT u y e Residuals: Differencebetween measurements & EFIT for pickup coils and Faraday In our case the analysis consists of minimising these residual correlations by varying the relative weights of polarimeter and pick-up coils

7. EFIT: Application to EFIT • Input: available magnetic measurements (various coils, Faraday chords) • Least square minimization of residuals Coils Faraday chords • EFIT : Faraday weights equal to 1, coil weights multiplied by a factor K0 varying from 0.1 to 1 to increase the relevance of the polarimeter chords

8. EFIT: EFIT EFIT version: • EFIT-J • Pressure constraint • Polarimeter constraints (ch 3, 5, 7) • P’ and FF’ equal to 0 at the separatrix

9. EFIT: Diagnostics • Main diagnostics are coils and polarimetric measurements Faraday channels (3,5,7) Some magnetic coils

10. Visual inspection of magnetic surfaces pulse #73660 @ t=55.038s -red profiles: no Polarimeter -blue profiles: K0=1 (c), 0.4 (b), 0.1 (a)

11. c Total indicator Indicator c average over coils, Faraday channels and time (one value per shot)

12. Correlations Nonlinear Correlations NACGF agree very well with the c2 results

13. Magnetic part of the indicator Indicatorcurves:

14. Polarimetric part of the indicator c Polarimetric part of the indicator c 73340 74366 73660 75724 73920 K0

15. c Total indicator: less coils Eliminating 30% of the magnetic probes, the best parameter to be used become K0 = 0.3 instead of 0.2.  K0

16. Outer strike point radial position Independent verification: strike point position with IR Good determination of the radial position: Divertor KL9 infrared camera, resolution < 1cm

17. Effect of K0 on the strike point position measured with IR Outer strike point radial position Time traces: K0: 0.1 0.3 1 No Faraday IR meas. pulse #73920

18. Check of the equilibrium quality in the core: SXR Reference used: soft X-rays measurements during sawteeth - the diagnostic: soft X-rays (SXR) camera V: resolution ~ 6 cm with 35 lines of sight

19. Check of the equilibrium quality in the core • q=1 radii at z=zmag from EFIT • Inversion radii • of sawteeth during sawteeth: SXR emission is dropping in the center, incresaes outside but remains constant at (R1,zmag) and (R2,zmag)

20. Equilibrium quality in the core: SXR scalar indicator Scalarindicator: Indicator curves: 73340 74366 73344 75724 73660 K0

21. Summary of the technique to select Faraday weights • Conflict between the reconstruction in the plasma core and at the boundary: the higher the accuracy in one region, the lower in the other • A trade-off can be found for an intermediate value of the weight of the coils K0, more or less between 0.2 and 0.3 depending on the shot (results obtained with two statistically independent methods, one of which is adequate for nonlinear systems) • With a value of the coils weights in this range, the quality of the plasma boundary does not deteriorate significantly and the core region is better reconstructed (as shown by independent measurements and not only by the residual analysis) Details in the paper A.Murari et al ,A Method to determine the Weights of Internal Magnetic Measurements for Optimised Equilibrium Reconstructions , submitted to Nuclear Fusion