Estimating the Instabilities ofNClocks by Measuring Differences of their Readings

Abstract

The estimation of individual instabilities of N clocks, when only differences of clock readings are available, is examined without any a priori assumption about the lack of correlation between clocks. The instability of the N clocks is described by the N x N covariance matrix R but only the ( N -1)x ( N -1) covariance matrix S of the differences referred to the N-th clock may be estimated directly from measurements. The estimation of R from S is not unique but the solution domain is limited by the positive definiteness of R. The features of this domain are analysed and the conditions for the validity of the uncorrelation hypothesis are established. The modifications of the solution domain, when the number N of compared clocks increases, is examined showing that each added clock can reduce the domain and therefore the arbitrariness in estimating R. Examples with some experimental data illustrate the capabilities of this approach.

Uncertainties of the Gravity Recovery and Climate Experiment time-variable gravity-field solutions based on three-cornered hat method

“…43 However, Premoli and Tavella 41 have pointed out that this condition is not sufficient to determine a unique solution for R. They have proposed an optimal choice criteria for the free parameters based on the minimization of the global correlation among the noises of the involved time series, maintaining the positive definiteness of R. Galindo and Palacio 44 have suggested the constrained minimization problem aiming to reach a unique solution (minimum of the quadratic mean of covariances) using the Kuhn-Tucker theorem, where the objective function F is given by 44 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 ; 1 1 6 ; 6 5 1…”

Recent progress in the performances of ultrastable quartz resonators and oscillators

“…We note that the accuracy of the Three-Cornered Hat method [25][26][27] is discussed when not simultaneously applied to the determination of the contribution on short term frequency stability performed on ultra-stable quartz oscillator. It led to a validation of the results [28].…”

Estimating the stability of N clocks with correlations

“…We assume here, of course, that the process of averaged fractional frequencies is stationary and ergodic so that the definition (3) is well-defined. Most authors use the notation σ 2 i,N (τ ) to represent the quantity in (3), but we use the notation set forth in references [4] and [5]. For what follows we need to define the Allan covariance of clocks i and j referenced to clock N to be:…”

“…In a later paper [5] Tavella and Premoli refined their analysis. They show that, when the number of clocks in a comparison increases, the amount of arbitrariness in determining absolute frequency stability decreases.…”

“…The goal of this paper is to generalize the methods of [4] and [5] and to estimate absolute frequency stability for N clocks where N > 3. We derive optimal clock combinations in the presence of correlations.…”