Abstract
The prediction of very-long-term time stability is a key issue in various fields, such as time keeping, obviously, but also navigation and spatial applications. This is usually performed by extrapolating the measurement data obtained by estimators such as the Allan variance, modified Allan variance, Hadamard variance, etc. This extrapolation may be assessed from a fit over the variance estimates. However, this fit should be performed on the log-log graph of the estimates, which corresponds to a least-squares minimization of the relative difference between the variance estimates and the fitting curve. However, a bias exists between the average of the log of the estimates and the log of the true value of the estimated variance. This paper presents the theoretical calculation of this log-log bias based on the number of equivalent degrees of freedom of the estimates, shows simulations over a large number of realizations, and provides a reliable method of unbiased logarithmic fit. Extrapolating this fit yields a more confident assessment of the very-long-term time stability.
Highlights
The prediction of very-long-term time stability is a key issue in various fields, such as time keeping, obviously, and navigation and spatial applications
Where η(t) represents the pure random component of the frequency deviation y(t). Such an extrapolation is very sensitive to estimation errors
It is of importance to perform a reliable estimation of the Ci coefficients
Summary
(where h0 is the white FM level) as a parameter; σy2(τ) is a measurement data of σy2(τ = 10 s). Tarantola distinguishes the model space, i.e., the space in which the parameter is given, from the data space, i.e., the space in which the measurement data are given [7]. 1) Model World (Direct Problem): In the direct problem, the task is to determine how the measurement data ξ are distributed when the parameter θ0 is known [see Fig. 3(b)]. The parameter θ0 is precisely the unknown quantity that we want to estimate. This parameter is only known in theory and simulations
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