Data assimilation has been adapted in paleoclimatology to reconstruct past climate states. A key component of some assimilation systems is the background-error covariance matrix, which controls how the information from observations spreads into the model space. In ensemble-based approaches, the background-error covariance matrix can be estimated from the ensemble. Due to the usually limited ensemble size, the background-error covariance matrix is subject to the so-called sampling error. We test different methods to reduce the effect of sampling error in a published paleoclimate data assimilation setup. For this purpose, we conduct a set of experiments, where we assimilate early instrumental data and proxy records stored in trees, to investigate the effect of (1) the applied localization function and localization length scale; (2) multiplicative and additive inflation techniques; (3) temporal localization of monthly data, which applies if several time steps are estimated together in the same assimilation window. We find that the estimation of the background-error covariance matrix can be improved by additive inflation where the background-error covariance matrix is not only calculated from the sample covariance but blended with a climatological covariance matrix. Implementing a temporal localization for monthly resolved data also led to a better reconstruction.

Estimating the state of the atmosphere in the past is important to enhance our understanding of the natural climate variability, the underlying mechanisms of past climate changes and their impacts. To infer past climate states, two basic sources of information are available: observations and numerical models. Climate models constrained with realistic, time-dependent external forcings provide fields that are consistent with these forcings and the model physics. Observations can be instrumental meteorological measurements, which are mainly available from the mid-19th century. Prior to this time, information from proxies stored in natural archives (like trees, speleothems, marine sediments, ice cores) or documentary data can be exploited. Observations provide important local information; however, their spatial and temporal coverage is sparse.

In recent years, a novel technique, the data assimilation (DA) approach, has been adapted for paleoclimatological research. DA creates a framework to combine information from different sources. If information from observations is optimally blended with climate model simulations, the result is the best estimate of the climatic state, given the observations, given the external forcings and given the known climate physics. The field of paleoclimate data assimilation (PDA) has undergone profound developments, and many DA techniques have been implemented to reconstruct past climate states, such as forcing singular vectors and pattern nudging

One popular DA method is the Kalman filter

An essential component of the KF is the uncertainty of the background state. In ensemble-based approaches, an ensemble of the background state provides estimation of the truth, represented by the ensemble mean, and the perturbations from the mean are used to estimate the uncertainty, represented by the background-error covariance matrix. Ensemble-based KFs are approximations of the KF, because the true state is usually sampled with a few tens to a few hundreds of ensemble members. The limited ensemble size leads to errors in the estimation of the background-error covariance matrix. This effect is known as the sampling error.

Two methods are commonly used in online ensemble-based KF approaches to reduce the negative effect of sampling error: inflation

In stationary offline PDA studies, the time-dependent background-error covariance matrix is replaced by a constant covariance matrix

Covariance inflation and localization techniques are used and under improvement in weather forecasting

The first possibility involves using a two-dimensional multivariate Gaussian function as a horizontal localization function to test the hypothesis of longer correlation length scales in zonal than meridional direction.

The second method is by applying covariance inflation techniques. In the multiplicative inflation technique, a constant factor is used to inflate the deviations from the ensemble mean. In the additive method, the background-error covariance matrix is calculated as the sum of the sample covariance matrix plus a climatological background matrix, where the climatological background is based on all ensemble members of multiple years. This larger sample size decreases the chances of spurious correlations.

The third possibility is adding temporal localization to the background-error covariance matrix. Multiple time steps are combined in one assimilation window to efficiently assimilate seasonal paleoclimate data. In the case of monthly observations, covariances between the months have been used to update all 6 months

This paper is structured as follows: an overview of our PDA approach, introducing the model, the observational network and the offline DA technique is given in Sect. 2. Section 3 describes the experimental framework. In Sect. 4, the results are presented and each experiment followed directly by a discussion. We summarize our experiments in Sect. 5.

We start from an existing DA system, which is described in

In this study, we use the same observational network of tree-ring proxies, documentary data and early instrumental measurements as described in

The observational network in 1904, before the quality check.

In our paleoclimate reconstruction, we combine the CCC400 model simulation with the observations as described above by implementing a modified version of the ensemble square root filter

Defined localization length scale parameters.

The use of DA in an offline manner is typical in paleoclimate reconstructions

As

The main steps of the blending experiment in one assimilation window. The blended covariance matrix

Summary of the experiments carried out in this study. The names of the experiments indicate which settings were used in the assimilation. Localization refers to the shape of the localization function applied on

In most of the studies, the localization function is implemented in an isotropic manner. In the original setup, the same horizontally isotropic localization function was used with different localization parameters. However, such spatial symmetries may not be realistic. In the real atmosphere, correlation lengths might be longer in the zonal than in the meridional direction, due to the prevailing winds and the weaker large-scale temperature gradients in this direction. On multi-annual to multi-decadal timescales, multiple processes act in the meridional direction, e.g., a widening/shrinking of the Hadley cell, shifts of the Intertropical Convergence Zone or changes in atmospheric modes like the Atlantic Multi-Decadal Oscillation or the North Atlantic Oscillation. These can shift the zonal circulation northward or southward, but the zonal coherence will be less effected. Hence, instead of using a circular Gaussian function, we conducted an experiment with a spatially anisotropic localization function

Covariance inflation techniques are another possible method to compensate for errors in the DA system

The other methodology that we adapt shows similarities with the additive inflation technique

Figure

Since observations are assimilated serially, we also update

Localizing observations in time is a special feature of the EKF due to its 6-month assimilation window. Having the state vector in half-year format, every month within the October–March or April–September time window is updated by each single observation. To test whether the covariances between a single observation and the multivariate climate fields are correctly captured, we ran an instrumental-only experiment with temporal localization. We set covariances between different months to zero.

The EKF method is tested with different localization functions and with a set of mixed background-error covariance matrices as described above. We have performed the experiments by assimilating either only proxy records (proxy-only experiment) or only instrumental data (instrumental-only experiment). The proxy-only experiments were carried out between 1902 and 1959, because many proxy records already end in the 1960s, while the instrumental-only experiments were tested over the 1902–2002 period. We separated the different observation types to see whether different settings perform better depending on the type of data being assimilated.
We do not compare proxy-only results with instrumental-only results; hence, the difference in time periods used does not matter; we simply use the longest possible time period.
To evaluate the reconstructions, we examined two verification measures: correlation coefficient and reduction of error (RE) skill score

To test which experiments have significantly different skill compared with the original skill, we carried out a permutation test following the method described in

In the next section, we will focus on analyzing the result of the experiments mainly over the extratropical Northern Hemisphere (ENH), because most of the data are located in this region. The skill scores refer to seasonal averages of the ensemble mean.

We compared the original setup applying an isotropic localization function and the experiment in which an anisotropic localization function was used, to test whether we can obtain a more skillful reconstruction by implementing anisotropic localization method.
As an example of the spatial reconstruction skill, we show the RE values of temperature (Fig.

Spatial skill of temperature reconstruction presented by RE values, assimilating only instrumental data

Difference between the aniso experiment and the original setup in terms of skill scores over the ENH region. Distributions of correlation values and of RE values are on the left and right figures, respectively. Distribution of temperature

In a previous ozone reconstruction study, a seasonally and latitudinally varying localization method was tested which mostly positively affected the analysis

The main problem of ensemble-based DA techniques is the computationally affordable limited ensemble size. Due to the finite ensemble size, the estimation of

Distribution of correlation coefficients differences between the mixed background-error covariance matrix experiments and the original setup over the ENH region. Panels

Using the multiplicative inflation method, the deviations from the ensemble mean are multiplied with a small factor (

In the other set of experiments, we used

Distribution of RE value differences between the mixed background-error covariance matrix experiments and the original setup over the ENH region; otherwise, it is the same as in Fig.

We expect that estimating the covariances from a bigger ensemble size (

For the ENH region, we present how the verification measures changed by replacing

Figures

Spatial reconstruction skill of precipitation in terms of RE values, assimilating only instrumental data. Panels

We also investigated the effect of the ensemble size in the estimation of

Furthermore, we conducted two experiments in which only

Distribution of skill scores over the ENH region. The skill of the original setup is compared with experiment 75c_PbL_constPc2L, 75c_PbL_Pc2L, 100c_constPcL and 100c_PcL. Distribution of correlation coefficients in the winter (left column) and in the summer (right column) seasons. Distribution of RE values in the winter (left column) and in the summer (right column) seasons.

We have tested a number of configurations of the mixed covariance matrix

Difference of the RE skill between the temporally localized experiment and the original setup, when only instrumental data are assimilated. Temperature

In our implementation,

Since 6-monthly time steps were combined in one state vector (one assimilation window), covariances between different months also need to be considered. An additional experiment was conducted in which the (localized)

The higher skill scores with temporal localization (Fig.

In this study, a transient offline data assimilation approach was used to test the effect of the estimation of the background-error covariance matrix in a climate reconstruction. Several experiments were evaluated with different validation measures to see which background-error covariance matrix estimation techniques improve the skill of the reconstruction. The evaluation of the presented techniques suggests the following: (1) applying an anisotropic localization function on the sample covariance matrix did not improve the reconstruction; (2) most of the settings, which make use of covariance estimates from a larger climatological sample, result in significantly improved skills compared to an estimation from the 30-member ensemble; (3) assimilating early instrumental data with temporal localization leads to a better analysis. To which extent the different techniques helped in the estimation of the background-error covariance matrix varies geographically and also depends on the climate variable being reconstructed. The cross-variable covariances of the background-error covariance matrix can provide information from unobserved climate variables.
Including climatological information in the estimation of precipitation has led to a better reconstruction, especially in Europe. Estimating sea-level pressure with the blended

The EKF400 reanalysis is available at the World Data Center for Climate at Deutsches Klimarechenzentrum (DKRZ) in Hamburg, Germany (

All authors were involved designing the study and contributed to writing the paper. VV conducted the experiments and performed most of the analyses. JF developed the original code and helped with the analyses.

The authors declare that they have no conflict of interest.

The CCC400 simulation was performed at the Swiss National Supercomputing Centre CSCS. The comments of the two anonymous reviewers are gratefully acknowledged.

This research has been supported by the Swiss National Science Foundation (grant no. 162668) and the European Commission – Horizon 2020 (grant no. 787574).

This paper was edited by Bjørg Risebrobakken and reviewed by two anonymous referees.