Atmospheric and Oceanic Optics

, Volume 27, Issue 1, pp 75–87 | Cite as

Measurement technique of turbulence characteristics from jitter of astronomical images onboard an aircraft: Part 1. Main ergodic theorems

  • V. V. Nosov
  • V. P. Lukin
Optical Instrumentation


Aspects of constructing statistical characteristics of random functions for discrete-continuous averaging that corresponds to the end-time response of a measuring device are investigated. This averaging is usually implemented in practice, and any discrete sequence of empirical values of a random function is partially averaged over some argument range. The rate of convergence of the variance of deviation of the time-average from the ensemble average (generalizations of the Taylor ergodic theorem) is estimated. These estimates provide the probability of convergence. It is shown that the rate of convergence depends on the integral scales of random function correlation. The scales are determined by the type of averaging; they differ for continuous, discrete, and discrete-continuous averaging. Relationships between the scales are found. An equation that connects the correlation functions of nonaveraged and partially averaged random processes is found. It is ascertained that the correlation function of a nonaveraged process can be satisfactorily retrieved from partially averaged data, even at very long ranges of the partial averaging.


Correlation Function Random Process Random Function Ensemble Average Oceanic Optic 
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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • V. V. Nosov
    • 1
  • V. P. Lukin
    • 1
  1. 1.V.E. Zuev Institute of Atmospheric OpticsTomskRussia

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