Random and bias errors in simple regression-based calculations of sea-level acceleration

We examine the random and bias errors associated with three simple regression-based methods used to calculate the acceleration of sea-level elevation (SL). These methods are: (1) using ordinary least-squares regression (OLSR) to fit a single second-order (in time) equation to an entire elevation time series; (2) using a sliding regression window with OLRS 2nd order fits to provide time and window length dependent estimates; and (3) using a sliding regression window with OLSR 1st order fits to provide time and window length dependent estimates of sea level rate differences (SLRD). A Monte Carlo analysis using synthetic elevation time series with 9 different noise formulations (red, AR(1), and white noise at 3 variance levels) is used to examine the error structure associated with the three analysis methods. We show that, as expected, the single-fit method (1), while providing statistically unbiased estimates of the mean acceleration over an interval, by statistical design does not provide estimates of time-varying acceleration. This technique cannot be expected to detect recent changes in SL acceleration, such as those predicted by some climate models. The two sliding window techniques show similar qualitative results for the test time series, but differ dramatically in their statistical significance. Estimates of acceleration based on the 2nd order fits (2) are numerically smaller than the rate differences (3), and in the presence of near-equal residual noise, are more difficult to detect with statistical significance. We show, using the SLRD estimates from tide gauge data, how statistically significant changes in sea level accelerations can be detected at different temporal and spatial scales.