| In modeling the temporal and spatial behavior of geophysical time series, we must address the inter-dependence of these time series and discriminate between possible trends and long range dependencies. Long memory processes have long been suggested as well suited tools to discriminate between trends and long range dependencies (Künsch 1986). We propose models based on a type of such processes, i.e, Fractional Gaussian noise processes (FGn). We first demonstrate the quality of a proposed approximation to the likelihood of a FGn process. This approximation accelerates the evaluation of the likelihood and allows us to model the spatial dependencies through the dual, frequency domain, representation of the time series. We show that even in the most extreme case, the approximation remains conservative. For the analysis of a single time series, we propose a model with two noise components. The first component is considered as part of the “signal” (and contains long range dependencies) while the second represents local measurement errors. We present a fully Bayesian estimating procedure relying on a Markov chain Monte Carlo. Special care is taken to accelerate the convergence of the proposed MCMC. Finally, to address the spatial dependence of the frequency domain representation of the time series, we use complex Gaussian random variables. For the signal part we use 2-D Fourier transforms and a hypothesis of a long memory process behavior of its spectrum. The remaining “noise” is assumed to be white noise. The result of this modeling is a method for smoothing an image in the frequency domain, which may also be used to clean an,estimated climate signal present in the original time domain data. |
