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

2. The problem….. We record waveforms in the OMs 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 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) 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) … 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 In this case we know exactly when each photon arrived The OM time response function is a δ-function 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 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) 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) 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 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 Suppose we bin the distributions into k bins

6. Probability of {n i } | {μ i } ? 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 sum over photons of log p – usual “timing likelihood” – only shape dependent constant density terms – “phit-pnohit”

10. Where is this form applicable? Our assumption was that we knew the times of individual photons precisely, or could form a good f(t) from a superposition of many photons Our assumption was that we knew the times of individual photons precisely, or could form a good f(t) from a superposition of many photons

11. 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 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!

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

13. Non δ-function OM response? Define a non δ-function response, Φ(t-t 1 ), which simply spreads the photon in time, i.e. Φ(t-t 1 ) still normalised Define a non δ-function response, Φ(t-t 1 ), which simply spreads the photon in time, i.e. Φ(t-t 1 ) still normalised What is What is

14. Try simply replacing δ(t 1 ) with Φ(t-t 1 ) Try simply replacing δ(t 1 ) with Φ(t-t 1 ) If log p(t) is slowly varying over the region of t where Φ(t-t 1 ) is non-zero, then If log p(t) is slowly varying over the region of t where Φ(t-t 1 ) is non-zero, then and the formulation would still work 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… This is the case in our OMs… short pulses relative to the scale of the expected distribution of photon times…

15. However….. The OM has an amplitude response - doesn’t map each photon to same amount of output waveform area… 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 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 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

16. 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 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