Maximum likelihood reconstruction of waveform events

1. Maximum likelihood reconstruction of waveform events Gary C. Hill April 20th 2005

2. The problem….. • We record waveforms in the OMs • These are superpositions of the OM response (time and amplitude smearing) to individual photons that arrive at the OM • They can look quite complicated (several photons close in time that superpose) • … or quite simple (e.g. if a single photon arrives, or several arrive separated by a large spacing in time, or a lot arrive, canceling out fluctuations and leaving a good estimate of the photon density observed)

3. Ideal case – MLE reconstruction where the OM response is a delta function • In this case we know exactly when each photon arrived • The OM time response function is a δ-function • No fluctuations in amplitude either – 1 unit of charge under the δ-function after the OM

4. Observed and expected photon distributions • Call the observed photon distribution f(t) (could be a set of times of individual photons, or a continuous function describing a large number or superposed photons) • The expected distribution of photons μp(t) is given from a simulation (photonics, PTD) or analytically (Pandel function) • p(t) is normalised, μ is the total expectation

5. Probability of f(t) given p(t)? • Suppose we bin the distributions into k bins

6. Probability of {ni } | {μi} ? Product of Poisson probabilities over bins:

7. Probability of f(t) given p(t)?

8. What is this form? Multinomial distribution – what is the probability of arranging exactly N events into k bins Poisson distribution – what is the probability of observing N events given the mean rate μ

9. Take the negative log density terms – “phit-pnohit” constant sum over photons of log p – usual “timing likelihood” – only shape dependent

10. Continuous forms of f(t) and p(t)? • Since we assumed a δ-function response of the OM, we can make our binning infinitesimally small and the log likelihood takes the form again, just sum of log p over all photons!

11. What if the OM response is not a δ-function? • δ-function response, -logP for one photon is

12. Non δ-function OM response? • Define a non δ-function response, Φ(t-t1), which simply spreads the photon in time, i.e. Φ(t-t1) still normalised • What is

13. Try simply replacing δ(t1) with Φ(t-t1) • If log p(t) is slowly varying over the region of t where Φ(t-t1) is non-zero, then and the formulation would still work • This is the case in our OMs… short pulses relative to the scale of the expected distribution of photon times…

14. However….. • The OM has an amplitude response -doesn’t map each photon to same amount of output waveform area… • For large numbers of photons forming a waveform, the amplitude fluctuations would cancel out and f(t) would be a good proxy for the true photon distribution • Need to think a bit more about the cases where only a small number of photons are seen... maybe add a OM amplitude response term into the formulation

15. Conclusion • If you believe that the OM output waveform, f(t), is a good proxy for the true photon distribution, then the log likelhood is basically the integral over the product f(t) log p(t), where p(t) is the normalised expected photon distribution