On the uncertainty of the uncertainty of long-term trends derived from geophysical and climate time series

Quantifying the uncertainty associated with parameter estimates is crucial for a wide range of geophysical and climate applications. This is particularly important for interpreting the long-term trends of quantities of interest, such as ground displacement, sea level, and water storage (among others), estimated from geophysical time series. Unfortunately, our imperfect understanding of measurement error sources and of the intrinsic stochastic behavior of the quantities of interest often makes it difficult to realistically assess the uncertainty of long-term trend estimates. 

One pragmatic approach to obtaining realistic trend uncertainties is to model all the stochastic variations observed in the time series (that is, the “noise”) by stochastic processes, and then derive the trend uncertainty using the variance propagation law. In practice, such noise models often include unknown stochastic parameters controlling, e.g., the amplitudes or time correlations of the stochastic processes, which need to be estimated from the observations. Estimated stochastic parameters, however, come with uncertainty, just like any estimated quantity. And an uncertainty on the parameters of the noise model implies an uncertainty on the long-term trend uncertainty based on that noise model. In view of trend analysis from geophysical and climate time series data, the importance of considering such “uncertainty on the uncertainty” remains so far to be investigated.

In this study, we address this issue by assessing, using numerical simulation, how the uncertainty of stochastic models derived from sparse geophysical time series (a few hundred data points) translates into the uncertainty of long-term trend uncertainty estimates. We demonstrate that uncertainty in the time-correlation structure can result in significant uncertainty on trend uncertainty estimates. We then discuss the impact of such “uncertainty on the uncertainty” on the assessment of long-term trend significance from geodetic time series and provide recommendations on how to deal with the issue in practice.