. zyxw t c * zyxwvutsrqp c zyxwvutsrqponmlkjih i * zyxwvutsrqponm VARIANCE ANALYSIS OF UNEVENLY SPACED TIME SERIES zyx DATA Christine Hackman and Thomas E. Parker National Institute of Standards and Technology Time and Frequency Division Boulder, Coolorado 80303 Abstract zyxwv We have investigafed the efiecf of uneven dufa spacing on the computation of zyxw uz(r). Evenly spaced simulated data sets were generated for noise processes ranging from white PM to random walk FM. uz(r) was then &&fed for each noise type. zyxwv Dcrtcr were subsequently removed from each simulcrted data set using fypical TWSTFT data patterns to create two unevenly spaced sets with average zyxwvuts intervals of 2.8 and 3.6 days. cz(r) was then calculated for euch sparse data set using two dinerent approaches. First, the missing data points were replaced by linear interpolation and u,(T) &lafed from this now fuU data set. The second appraach ignored the fad that the data were unevenly spaced and calculated uz(r) as if the data were equally spaced with average spacing of 2.8 or 3.6 days. B d h approaches have advantages and disadvanhges, and techniques are presented for correcting errors caused by uneven data spacing in typical TWSTFT data sets. INTRODUCTION Data points obtained from an experiment are often not evenly spaced. In this paper, we examine the application of u,(T) = 3-'/*~(mo&,(~))[1] to the unevenly spaced time-series data obtained from two-way satellite time and frequency transfer (TWSTF"). We do so by using u,(T) with both evenly and unevenly spaced simulated data of known power-law noise type and magnitude. The noise types examined are white phase modulation (WHPM), flicker phase modulation (FLPM), white frequency modulation (WHFM), flicker frequency modulation (FLFM), and random walk frequency modulation (RWFM)[zl. Vemotte et aZ.[3] studied the analysis of noise and drift in unevenly spaced pulsar data. However, the data obtained from pulsar studies are much more sparse in time, with only about 2% of the possible data available. In TWSTR, the task is less daunting: time transfers are typically measured on Monday, Wednesday, and Friday, so, in a perfect world, we would have a data density of 3 data points present out of a possible 7. This paper is not intended to be a rigorous treatment of how to calculate o,(T) in all possible cases of unevenly spaced data. Rather, our purpose is to suggest methods and corrections which may be applied to data such as those produced by TWSTFT in order to obtain a more accurate assessment of the underlying time stability and noise type. 323