How to generate accurate continuous thermal regimes from sparse but regular temperature measurements

Abstract

Abstract In ecology, there is an emerging emphasis on the importance of capturing temperature variation at relevant scales. Temperature fluctuates continuously in nature but is sampled at discrete time points, so how often should ecologists measure temperature to capture its variation? A recent development in thermal ecology is the use of spectral analysis of temperature time series to determine at what frequencies important temperature fluctuations occur. Building on this, I borrow from signal processing theory to show how continuous thermal regimes can be effectively reconstructed from discrete, regular, measurements, and provide a rule of thumb for designing temperature sampling schemes that capture ecologically relevant temporal variation. I introduce sinc interpolation, a method for reconstructing continuous waveforms from discrete samples. Furthermore, I introduce the Nyquist–Shannon sampling theorem, which states that continuous complex waveforms can be perfectly sinc‐interpolated from discrete, regular, samples if sampling intervals are sufficiently short. To demonstrate the power of these concepts in an ecological context, I apply them to several published high‐resolved (15‐min intervals) temperature time series used for ecological predictions of insect development times. First, I use spectral analysis to illuminate the fluctuation frequencies that dominate the temperature data. Second, I employ sinc interpolation over artificially thinned versions of the temperature time series. Third, I compare interpolated temperatures with observed temperatures to demonstrate the Nyquist–Shannon sampling theorem and its relation to spectral analysis. Last, I repeat the ecological predictions using sinc‐interpolated temperatures. Daily, and less frequent, fluctuations dominated the variation in all the temperature time series. Therefore, in accordance with the Nyquist–Shannon sampling theorem, 11‐h measurement intervals consistently retrieved most 15‐min temperature variation. Moreover, previous predictions of insect development times were improved by using sinc‐interpolated, rather than averaged, temperatures. By identifying the highest frequency at which ecologically (or otherwise) relevant temperature fluctuations occur and applying the Nyquist–Shannon sampling theorem, ecologists (or others doing climate‐related research) can use sinc interpolation to produce remarkably accurate continuous thermal regimes from sparse but regular temperature measurements. Surprisingly, these concepts have remained largely unexplored in ecology despite their applicability, not least in thermal ecology.