Multiscale combination of climate model simulations and proxy records over the last millennium

Abstract

To highlight the compatibility of climate model simulation and proxy reconstruction at different timescales, a timescale separation merging method combining proxy records and climate model simulations is presented. Annual mean surface temperature anomalies for the last millennium (851–2005 AD) at various scales over the land of the Northern Hemisphere were reconstructed with 2° × 2° spatial resolution, using an optimal interpolation (OI) algorithm. All target series were decomposed using an ensemble empirical mode decomposition method followed by power spectral analysis. Four typical components were obtained at inter-annual, decadal, multidecadal, and centennial timescales. A total of 323 temperature-sensitive proxy chronologies were incorporated after screening for each component. By scaling the proxy components using variance matching and applying a localized OI algorithm to all four components point by point, we obtained merged surface temperatures. Independent validation indicates that the most significant improvement was for components at the inter-annual scale, but this became less evident with increasing timescales. In mid-latitude land areas, 10–30% of grids were significantly corrected at the inter-annual scale. By assimilating the proxy records, the merged results reduced the gap in response to volcanic forcing between a pure reconstruction and simulation. Difficulty remained in verifying the centennial information and quantifying corresponding uncertainties, so additional effort should be devoted to this aspect in future research.

Appendices

Appendix 1

A Monte Carlo simulation test process was applied to quantify the significant levels of correlation coefficient for the time series with reduced freedom (Mann et al. 2007; Mann et al. 2009; Xing et al. 2016). The Monte Carlo simulated series was generated with the same auto-correlations as the original series by a first-order auto-regression (AR-1) model plus red noise. There were 3000 simulated series produced for target component series, i.e., the four typical components of instrument series (CRU series). This Monte Carlo test was applied to (1) screen the proxy records candidates and (2) test the significance of correlation between the segments (the components at different timescale) of merged results and instrument series.

Appendix 2

The localized OI algorithm used in this paper at four typical timescales used a dynamically updated B matrix within the same frame as follows in Eq. 2.

$$ {T}_i^a={T}_i^b+\sum_k^M{G}_{i k}\left[{T}_k^o- H\left({T}^b\right)\right] $$

(2)

where ${T}_i^a$ is the analyzed (merged) component at a certain timescale. ${T}_i^a$ is the ensemble mean of background. M is the number of available proxy records within the range with diagnostic radius. H is the observation operator, in this study functions to transfer the nearest value on regular grid to the proxy sites. G is the gain matrix value for the ith grid and kth proxy component. The G matrix is the solution of the set of linear equations (3):

$$ {BH}^T= G\left( R+{HBH}^T\right) $$

(3)

where the R matrix is “observation error covariance matrix.” In this study, we use the diagonal form of R, which is estimated by the covariance of scaled proxy components at certain timescales. B is the background error covariance matrix. B was not calculated directly and it was stored as BH ^T (orHB) implicitly.

When deriving G, as the H operator transferred the nearest grid value to the scattered proxy site, it is likely that BH ^T might become ill-conditioned; in that case, the linear equations would be solved by the technique of singular value decomposition (SVD), and the truncated results would be given.