Correlated errors in geodetic time series: Implications for time‐dependent deformation

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Abstract
Analysis of frequent trilateration observations from the two‐color electronic distance measuring networks in California demonstrate that the noise power spectra are dominated by white noise at higher frequencies and power law behavior at lower frequencies. In contrast, Earth scientists typically have assumed that only white noise is present in a geodetic time series, since a combination of infrequent measurements and low precision usually preclude identifying the time‐correlated signature in such data. After removing a linear trend from the two‐color data, it becomes evident that there are primarily two recognizable types of time‐correlated noise present in the residuals. The first type is a seasonal variation in displacement which is probably a result of measuring to shallow surface monuments installed in clayey soil which responds to seasonally occurring rainfall; this noise is significant only for a small fraction of the sites analyzed. The second type of correlated noise becomes evident only after spectral analysis of line length changes and shows a functional relation at long periods between power and frequency of 1/ƒα, where ƒ is frequency and α≈2. With α=2, this type of correlated noise is termed random‐walk noise, and its source is mainly thought to be small random motions of geodetic monuments with respect to the Earth's crust, though other sources are possible. Because the line length changes in the two‐color networks are measured at irregular intervals, power spectral techniques cannot reliably estimate the level of 1/ƒα noise. Rather, we also use here a maximum likelihood estimation technique which assumes that there are only two sources of noise in the residual time series (white noise and random‐walk noise) and estimates the amount of each. From this analysis we find that the random‐walk noise level averages about 1.3 mm/√yr and that our estimates of the white noise component confirm theoretical limitations of the measurement technique. In addition, the seasonal noise can be as large as 3 mm in amplitude but typically is less than 0.5 mm. Because of the presence of random‐walk noise in these time series, modeling and interpretation of the geodetic data must account for this source of error. By way of example we show that estimating the time‐varying strain tensor (a form of spatial averaging) from geodetic data having both random‐walk and white noise error components results in seemingly significant variations in the rate of strain accumulation; spatial averaging does reduce the size of both noise components but not their relative influence on the resulting strain accumulation model.