Use of reduced-rank covariance estimates for objective analyses of historical data sets

Alexey Kaplan
Lamont-Doherty Earth Observatory of Columbia University

The only ultimate source of our knowledge of the climate system is observed data. Yet the existing data records are stunningly incomplete, erratic or of short time coverage. Spatial or spatiotemporal analyses of historical climate data sets (?reconstructions?) are steadily gaining popularity over raw observations. In most cases the reconstructions are produced by objective analysis techniques, e.g. optimal interpolation, and obtained as least-squares estimates with an assumed spatial or spatiotemporal covariance structure. It can be shown that with realistic assumptions on statistical properties of climate signal and data error, the objective solution is always smoother than the true signal. In other words, its eigenvalue spectrum for the covariance in analyzed dimensions is systematically redder than that of the signal. This gives a theoretical basis to the effectiveness of the ?reduced space? approach to objective analyses, i.e. a simplification which seeks for a solution among linear combinations of a small set of predetermined patterns. Unfortunately, the difference in spectral redness increases with increase in data error (and decrease in data coverage). Therefore statistical studies which use an objective analysis as input data may misinterpret the analysis? artifact, reflecting an improvement in data quality from the past to present with the change in spectral properties of climate signal. Actual reconstructions of climate fields from ship records illustrate this problem. Statistical studies need to use not a single objective solution, but an ensemble of possible realizations, consistent with the observed data and reflecting the uncertainty in the solution.

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