1. DirectFit reconstruction of the Aya’s two HE cascade events Dmitry Chirkin, UW Madison Method of the fit: exhaustive search simulate cascade events with various x,y,z, ,E,t 0, and compare them to the data event. the simulated event that looks most like the data event is the result Advantages: simple and robust (is not stuck in local minima) most precise description of ice can be used in reconstruction (SPICE Lea: including tilt and anisotropy) Drawbacks: can be very slow (~1 day/Aya’s event)

2. Comparing a simulation event with a data event Use the same likelihood function as in the SPICE Lea fit: includes Poisson fluctuations in data, simulation, and a 20% allowance for non-Poisson errors (in description of ice and others). All feature-extracted waveforms (charge histogram vs. time in a DOM) are binned in 25 ns bins, and then processed with a Bayesian blocks procedure, which combines low-count or nearly-same-charge bins. This is the same procedure as was used in the SPICE Lea fit. So, we are using exactly the same comparison procedure as was used in the SPICE Lea model fit.

3. Search algorithm 1.Start with x,y,z of COG,  =0,  =0, E=10 5 GeV, t 0 =0 2.Propose 25 sets of cascade parameters x,y,z, ,  from a gaussian distribution with rms=10 m in x,y,z and rms=30 degrees in , . Keep the values of E and t 0. 3.For each proposed simulated event find the best E (by scaling the simulated event) and t 0 (by time-shifting the hits in the simulated event); calculate the likelihood L. 4.Out of these 25 event select the one with the best value of L and update the 7 cascade parameters; remember the best value of L: L *. 5.Repeat steps 2-4 40 times. Use 20 events resulting from step 4 with the best values of L * to update the rms in x,y,z, and rms in , , and to establish correlation between these (important since the brightest point of the cascade is some distance away from the starting point along the cascade direction; also the Cherenkov light is emitted predominantly forward). 6.Repeat steps 2-5 10 times; The final result is calculated by averaging simulated events with the best 160 values of L *. The rms of x,y,z, and the rms in ,  are also computed to provide a measure of uncertainties.

4. Other search algorithms and uncertainties The algorithm described on the previous slide is an optimized variant of Localized random search. Other methods that I tried are: Simultaneous perturbation stochastic approximation with and without the estimate of the second derivative (Newton-like method). Markov chain with transitional probability defined by condition L i+1 <L i. Although the rms values in cascade parameters obtained in the localized random search and Markov chain methods are probably related to the uncertainties of the measurement, the well-defined values of the uncertainties should probably be calculated by applying the reconstruction to a few (dozen?) cascade events simulated with the same parameters.

5. Results: llh vs. step number Event 118545Event 119316

6. Results: z vs. llh Event 118545Event 119316

7. Results: llh Event 118545Event 119316 1…200 all 201…400

8. Results: z Event 118545Event 119316 1…200 all 201…400

9. Results: t 0 vs. E Event 118545Event 119316

10. Results: E Event 118545Event 119316 1…200 all 201…400

11. Results: t 0 Event 118545Event 119316 1…200 all 201…400

12. Results: y vs. x Event 118545Event 119316

13. Results: x Event 118545Event 119316 1…200 all 201…400

14. Results: y Event 118545Event 119316 1…200 all 201…400

15. Results:  vs.  Event 118545Event 119316

16. Results:  Event 118545Event 119316 1…200 all 201…400

17. Results:  Event 118545Event 119316 1…200 all 201…400

18. Summary Event 118545Event 119316 llh = 2866.0+-19.6 x = -201.3 m y = 96.73 m z = 121.1 m  = 76.0 o  = -44.3 o E = 1.115e6 +- 1.5%  r = 2.7 m  = 10.4 o dist. to pivot = -16.0 m llh = 3487.5+-23.7 x = -77.6 m y = 261.0 m z = 25.1 m  = 28.8 o  = -80.7 o E = 1.336e6 +- 3.8%  r = 3.1 m  = 8.5 o dist. to pivot = -18.3 m SPICE Lea

19. Summary Event 118545Event 119316 llh = 2866.0 +- 19.6 E = 1.115e6 +- 1.5% llh = 3487.5 +- 23.7 E = 1.336e6 +- 3.8% SPICE Lea llh = 2940.0 +- 20.8 E = 1.168e6 +- 1.7% llh = 3557.6 +- 23.2 E = 1.398e6 +- 3.0% SPICE Mie llh = 3151.7 +- 20.3 E = 931966 +- 1.5% llh = 3713.4 +- 23.8 E = 1.127e6 +- 3.1% SPICE 1 llh = 3467.8 +- 22.5 E = 1.247e6 +- 1.9% llh = 4161.6 +- 21.0 E = 1.565e6 +- 2.4% WHAM!

20. Simulation at best point vs. data with SPICE Lea

21. Event 118545Event 119316

22. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

23. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

24. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

25. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

26. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

27. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

28. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

29. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

30. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

31. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

32. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

33. Simulation at best point vs. data with SPICE Lea Event 118545Event 119316

34. Comparison SPICE Lea vs. SP1 for event 118545

35. SPICE LeaSPICE 1

36. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

37. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

38. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

39. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

40. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

41. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

42. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

43. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

44. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

45. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

46. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

47. Comparison SPICE Lea vs. SP1 for event 118545 SPICE LeaSPICE 1

48. More plots at http://icecube.wisc.edu/~dima/work/IceCube-ftp/mcmc/.http://icecube.wisc.edu/~dima/work/IceCube-ftp/mcmc/