## 1. Introduction

[2] Proper smoothing of climate time series, particularly those exhibiting non-stationary behavior (e.g., substantial trends late in the series) is essential for placing recent trends in the context of past variability. The smoothing of a time series can be posed as an inverse problem with non-unique boundary constraints [*Park*, 1992], for which additional objective considerations must be made to determine the behavior near the boundaries. Various different boundary constraints have recently been employed, for example, in the smoothing of the Northern Hemisphere mean temperature series [*Folland et al.*, 2001; *Mann and Jones*, 2003; *Soon et al.*, 2004]. The approach used by *Mann and Jones* [2003], as noted therein, employed a smoothing boundary constraint optimized to resolve the non-stationary late behavior of the time series in comparison with previously employed constraints involving e.g., the padding of the series with mean values after the boundary [*Folland et al.*, 2001; *Mann*, 2002; *Mann et al.*, 2003]. The approach used by *Mann and Jones* [2003] (which is incorrectly assumed by *Soon et al.* [2004] to be a ‘wavelet’ approach), is described in more detail in this study.

[3] The three lowest order boundary constraints that can be applied to a smooth [see, e.g., *Park*, 1992] involve the minimization near the boundaries of either: (1) the zeroth derivative of the smooth (yielding the ‘smallest’ or ‘minimum norm’ solution), (2) the 1st derivative of the smooth (yielding the ‘minimum slope’ constraint), and (3) the 2nd derivative of the smooth (yielding the smoothest or ‘minimum roughness’ solution). Application of constraint (1) favors the tendency of the smooth to approach the mean value (i.e., ‘climatology’) near the boundaries. Application of (2) favors the tendency of the smooth to approach a constant local value near the boundary. Application of (3) favors the tendency of the smooth to approach the boundary with a constant *slope*. The first two approaches will underestimate the behavior of the time series near the boundaries in the presence of a long-term trend, but the 3rd approach may lead to an extrapolation error in the presence of leverage by outliers near the boundaries. Without additional considerations, none of these three constraints can be favored on *a priori* grounds. An objective choice, nonetheless, can be motivated as that particular constraint of the three which minimizes some measure of misfit of the smooth with respect to the original time series.

[4] In this paper, we describe both time-domain and frequency-domain approaches to implementing each of these three alternative boundary conditions, and employ an objective measure of the quality of fit of the various candidate smooths.

[5] We note that while our focus here is on smoothing of time series, similar considerations can be applied to alternative statistical time series modeling such as change point analysis [*Tomé and Miranda*, 2004]. We provide applications to two relevant instrumental climate time series, the Northern Hemisphere (NH) annual mean series from 1856–2003 of *Jones et al.* [1999], and the cold-season North Atlantic Oscillation (NAO) times series of *Jones et al.* [1997] from 1825/26 to 1999/2000.