Technical Note: Correcting for signal attenuation from noisy proxy data in climate reconstructions

Regression-based climate reconstructions scale one or more noisy proxy records against a (generally) short instrumental data series. Based on that relationship, the indirect information is then used to estimate that particular measure of climate back in time. A well-calibrated proxy record(s), if stationary in its relationship to the target, should faithfully preserve the mean amplitude of the climatic variable. However, it is well established in the statistical literature that traditional regression parameter estimation can lead to substantial amplitude attenuation if the predictors carry significant amounts of noise. This issue is known as “Measurement Error” (Fuller, 1987; Carroll et al., 2006). Climate proxies derived from tree-rings, ice cores, lake sediments, etc., are inherently noisy and thus all regression-based reconstructions could suffer from this problem. Some recent applications attempt to ward off amplitude attenuation, but implementations are often complex (Lee et al., 2008) or require additional information, e.g. from climate models (Hegerl et al., 2006, 2007). Here we explain the cause of the problem and propose an easy, generally applicable, data-driven strategy to effectively correct for attenuation (Fuller, 1987; Carroll et al., 2006), even at annual resolution. The impact is illustrated in the context of a Northern Hemisphere mean temperature reconstruction. An inescapable trade-off for achieving an unbiased reconstruction is an increase in variance, but for many climate applications the change in mean is a core interest. Correspondence to: C. M. Ammann ([email protected]) 1 The problem of noisy predictors Random noise in any linear system will affect the estimation process of regression coefficients that tie explanatory variable(s) X to the response Y . Uncertainty in estimation of Y can be quantified through the variance of the error from an ordinary least squares (OLS) fit, which by definition, in this case, provides unbiased parameter estimates (thus it is known as “BLUE”: best linear unbiased estimator). Errors in the predictor(s) X, however, cause the regression slope to get attenuated towards zero and the resulting signal in the prediction or reconstruction period will invariably be biased (Fuller, 1987). Figure 1a illustrates this effect for a simple 1:1-linear process where the response Y is only observed (available for calibration) over the interval 0.9 to 1 while X is available over the full range of 0 to 1. Increasing the noise contained in X attenuates the OLS-derived slope parameter away from the true linear relationship. Why does noise in the predictors cause attenuation of the true signal? Consider a simple linear regression model Y=β0+β1X+ε for which we have instrumental observations Y and the noisy proxy record W =X+U , where X is the desired climate signal and U is the contaminating noise. An OLS regression of instrumental data Y is therefore not directly on X but actually on W , and thus the result is not a consistent estimate of the desired regression coefficient β1 (Fuller, 1987; Carroll et al., 2006). Rather, the regression slope is, in fact, σ 2 X/(σ 2 X+σ 2 U ) ·β1, where σ 2 X and σ 2 U denote the variance of X and U , respectively. Therefore, the larger the noise U , the stronger the attenuation of the regression slope will be. Published by Copernicus Publications on behalf of the European Geosciences Union. 274 C. M. Ammann et al.: Correcting for signal attenuation from noisy proxy data in climate reconstructions 0.0 0.2 0.4 0.6 0.8 1.0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0