The Integrated Brownian Motion for the study of the atomic clock error

1. The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto Elettrotecnico Nazionale “G. Ferraris” Politecnico of Turin Patrizia Tavella Istituto Elettrotecnico Nazionale “G. Ferraris” Turin

2. In the past This work started in 2001 with my graduate thesis developed in collaboration between the University “La Sapienza” (Bruno Bassan) and IEN “Galileo Ferraris” (Patrizia Tavella), one of the Italian metrological institutes. G.Panfilo, B.Bassan, P.Tavella. “The integrated Brownian motion for the study of the atomic clock error”. VI Proceedings of the “Società Italiana di Matematica Applicata e Industriale” (SIMAI). Chia Laguna 27-31 May 2002 Now I have continued this work in my Doctoral study in “Metrology” at Turin Polytechnic and IEN “Galileo Ferraris” also in collaboration with BIPM (Bureau International des Poids et Measures) “The mathematical modelling of the atomic clock error with application to time scales and satellite systems”

3. clock error T(-m,n) n t -m The aim:We are interested in the evaluation of the probability that the clock error exceeds an allowed limit a certain time after synchronization. The atomic clock error can be modelled by stochastic processes T(-m,n) the first passage time of a stochastic process across two fixed constant boundaries Survival probability

4. Survival probability. • Link between the stochastic differential equations (SDE) and the partial differential equations (PDE): infinitesimal generator. Summary • The stochastic model of the atomic clock error obtained by the solution of the stochastic differential equations. • Numerical solution: Monte Carlo method for SDE Finite Differences Method for PDE Finite Elements Method for PDE. • Application: Model of the atomic clock error and Integrated Brownian motion. • Application to rubidium clock used in spatial and industrial applications.

5. The stochastic processes involved in this model are: Brownian Motion (BM) Integrated Brownian Motion (IBM) The atomic clock model The atomic clock model can be expressed by the solution of the following stochastic differential equation: with initial conditions The exact solution is: Observation: The IBM is given by the same system without the term 1W1which represents the contribution of the BM.

6. …and iterative form Innovation clock error 100 50 10 20 30 40 -50 -100 -150 The solution can be expressed in an iterative form useful for exact simulation where

7. The infinitesimal generator The infinitesimal generator A of a homogeneous Markov process Xt , for , is defined by: Ag(x) is interpreted as the mean infinitesimal rate of change of g(Xt) in case Xt=x where: • Tt is an operator defined as: • g is a bounded function • Xt is a realization of a homogeneous stochastic Markov process • is the transition probability density function

8. Stochastic differential equation: Partial differential equation for the transition probability f: (Kolmogorov’s backward equation) Link between the stochastic differential equations and the partial differential equations for diffusions Infinitesimal generator Lt:

9. The survival probability Other functionals verify the same partial differential equation but with different boundary conditions. Example: the survival probabilityp(x,t): where: • 1Dis the indicator function • [0,T]- time domain • D- spatial domain • - boundary of the domain D

10. Integrated Brownian motion Brownian Motion • Numerical Methods applied to PDE: • Finite Differences Method • Finite Elements Method Monte Carlo Method applied to SDE. PDE for the clock survival probability For the complete model (IBM+BM): =0 It is not always possible to derive the analytical solution!!!

11. Example: The Integrated Brownian Motion The Integrated Brownian motion is defined by the following Stochastic Differential Equation: To have the survival probability we have to solve: =0 Numerical Methods: A) Monte Carlo SDE B) Finite Differences PDE C) Finite Elements It doesn’t exist the analytical solution

12. p t Monte Carlo Finite Differences ht=0.05 Finite Differences ht=0.01 The survival probability for IBM It’s not possible to solve analytically the PDE for the survival probability of the IBM process. Appling the Monte Carlo method to SDE and difference finites method to PDE we obtain: hx = 0.04 hy = 0.5 ht = 0.05 ht = 0.01 -m=n = 1 N =105 trajectories τ = 0.01 discretization step The two numerical methods agree to a large extent. Difficulties arises in managing very small discretization steps.

13. p t [days] IBM:Application to atomic clocks Atomic Clock: Rubidium Considering different values for the boundaries m and for the survival probabilities: IBM Experimental data n=-m= 350 ns For example ±10 ns 0.4 days (0.95)

14. Finite Differences Finite Elements Monte Carlo p t p t Complete Model (IBM+BM): Survival Probability By the numerical methods we obtain the survival probability of the complete model: hx = 0.2 hy = 0.5 ht = 0.01 N =105 trajectories τ = 0.01 discretization step -m=n = 1 hx = 0.01 hy = 0.02 ht = 0.003 For the finite elements method The Monte Carlo method and the finite elements method agree for any discretization step. For the difference finites method the difficulties arises in managing very small discretization steps.

15. Atomic Clock: Rubidium IBM Complete Model (IBM+BM) p Experimental data t [days] Complete Model (IBM+BM):Application to atomic clocks Considering different values for the boundaries m and for the survival probabilities: m = 8 ns For example ±10 ns 0.2 days (0.95)

16. Applications In GNSS (GPS, Galileo) the localization accuracy depends on error of the clock carried by the satellite. When the error exceeds a maximum available level, the on board clock must be re-synchronized. Our model estimates that we are confident with probability 0.95 that the atomic clock error is inside the boundaries of 10 ns for0.2 days (about 5 hours) in case of Rubidium clocks. Calibration interval: In industrial measurement process the measuring instrument must be periodically calibrated. Our model estimates how often the calibration is required.

17. We have considered the Ornstein-Uhlembeck process to model the filtered white noise. 30 realizations of the Brownian Motion (red) and Ornstein-Uhlembeck (blue) Perspectives It’s necessary to use other stochastic process to describe the behaviour of different atomic clock error. • Other stochastic processes used to metrological application can be • The Integrated Ornstein-Uhlembeck • The Fractional Brownian Motion

18. clock behavior prediction • Using the atomic clock model Conclusions • Stochastic differential equations helps in modelling the atomic clock errors • By the SDE or related PDE the survival probability of a stochastic process is obtained. • The use of the model of the atomic clock error and the survival probability are very important in many applications like the space and industrial applications. The authors thank Laura Sacerdote and Cristina Zucca from University of Turin for helpful suggestions, support and collaboration.