## Spatiotemporal Statistics

Time Uncertainty and Climate Spectra

One of the largest sources of error in paleoclimate (and geologic) timeseries data is the timing of samples. Because many analytical techniques were developed for contexts in which there is near-perfect knowledge of sample times (or equivalently locations), the statistical implications of time uncertainty are not well-understood. For example, what happens when you compute a power spectrum using the wrong timescale? If the signal has much narrowband energy, the result can be disastrous:

One of the largest sources of error in paleoclimate (and geologic) timeseries data is the timing of samples. Because many analytical techniques were developed for contexts in which there is near-perfect knowledge of sample times (or equivalently locations), the statistical implications of time uncertainty are not well-understood. For example, what happens when you compute a power spectrum using the wrong timescale? If the signal has much narrowband energy, the result can be disastrous:

*Narrowband effects of time-uncertainty upon spectral estimates. A narrowband spectrum (from a time-limited sine wave) is shown in blue. The signal is distorted by compression in the first half and expansion in the second half (grey spectrum) to represent a systematic bias. However, timing errors will be closer to cumulative noise. Cumulative timing errors are applied 1,000 times to the original signal, and the mean resulting spectrum is shown in black. For narrowband signals, time-uncertainty scatters energy over a wide frequency range.*But if the spectrum is better characterized as a continuum (as they are for most geophysical signals), there is hardly any error at all:

In RH2010, we lay some groundwork in exploring these effects, explaining the distinction between time-uncertain narrowband and broadband spectra. We also discuss one of the time-error models we developed, which permits Monte-Carlo simulation of timescale errors in layer-counted stratigraphic records such as ice cores from Greenland. Similar methods apply to tree rings, ocean sediment cores, and other continuously-deposited records. The narrowband effects have implications for significance estimation, particularly in the problem of identifying Milankovitch forcing in paleoclimate records. The broadband effects are important when studying the connections between short-term variability, climatic timescales (~5 - 1000 yr), orbital timescales (10 kyr - 500 kyr), and geologic timescales (Myr - Gyr).