A Wavelet Approach to Reconstructing Near-Surface Temperature Time Series From Observations of Subsurface Temperatures at Several Meters Depth

Here we demonstrate a method that uses wavelet analysis to reconstruct a near-surface temperature time series from subsurface temperature observations taken at depth. Our approach is as follows: (1) we apply a wavelet transform to a 3-m subsurface temperature time series and construct a wavelet-filtered time series over a continuous set of frequency bands ranging from 0.5 to 65 years; (2) we upward continue each wavelet-filtered time series to a near-surface depth by shifting the spectral component in each band backward in time and amplifying the amplitude using the steady-state solution of the one-dimensional thermal diffusion equation; (3) we reconstruct the near-surface temperature time series by summing the upward continued time series associated with each band; and (4) we optimize the reconstruction by repeating steps 1 through 3 over a range of thermal diffusivities to minimize the root mean square error between observation and reconstruction at the near-surface depth. The method is applied to data observed at Fargo, North Dakota, comprising 6938 daily time steps between 1 September 1981 and 31 August 1999. We reconstruct several near-surface depths and show the method to accurately reconstruct observed temperatures, therefore demonstrating the wavelet approach as a promising means of characterizing subsurface thermodynamics.