A cheap data assimilation approach for expensive numerical simulations

Available observational climate data sets are the most accurate source of knowledge about the climate system. However they suffer from ill structural conditions: (i) time span of these data sets are usually less than a century, (ii) they change their accuracy in time (environmental changes in station location, usage of different measuring devices, urbanization), (iii) they change their density in time (more stations are available online in recent years), (iv) more numbers of stations over land than over oceans.

Climate proxy archives (tree ring, coral, sediment and glacial) are alternative, indirect climate observations which posses a recording characteristic (Acevedo, 2015; Jones & Mann, 2004). The best resolved features of climate recorded by proxies are of annual and seasonal cycles. However, the data recording process involved in such observations are very complex, encompassing physical, biological and chemical processes (Evans et al., 2013). Differentiating the climate and human impact on proxies is also a challenging work, for example, in terms of historical drought spells. The difficult challenge of inverting proxy records into climate information is traditionally done in the frame work of statistical modeling. However, multivariate linear regression techniques dominate this area (Acevedo, 2015). On the other hand, even using more sophisticated statistical models is problematic in a way that the overlapping time span between the instrumental records (weather station observations) and the proxy records are too short to train the statistical models. Climate model hindcast may serve as a modeling idea to study the long term climate variability. The climate models create dynamically consistent state of the climate using numerical methods. However, their reconstructed states are very sensible to the initial conditions and the forcing used in the model as well as parametrization schemes used for sub-grid scale processes (small processes which are not physically presented by model).

One of the novel and appealing approaches to reconstruct the past climate is Data Assimilation (DA) which blends the proxy records and the climate models (Evensen, 2003; Hughes et al., 2010; Brönnimann, 2011; Bhend et al., 2012; Hakim et al., 2013; Steiger et al., 2014; Matsikaris et al., 2015; Hakim et al., 2016).

For a review of DA techniques applied in climate studies (not the weather forecasting), I refer to the works of Acevedo et al. (2016); Steiger & Hakim (2015). Acevedo et al. (2016); Steiger & Hakim (2015); Steiger et al. (2014) showed that the climate models may not have forecasting skills longer than several years. This fact makes the usage of classical DA methods very limited. The models lose their skill shortly after initialization and will evolve freely till the next step where the observations (in form of proxies) are available.

Considering the implementation ramification of DA methods in climate systems, their extremely computational expenses and the short forecasting skill of the models, an alternative DA approach (“off-line” or “no-cycling”) was applied by several scholars (Acevedo et al., 2016; Steiger & Hakim, 2015; Chen et al., 2015; Steiger et al., 2014). To my knowledge, there exists no study which applies such strategies in high resolution Regional Climate Model (RCM) simulations for long term climate studies.

In this study, I will apply an Off-line DA (ODA) approach in a RCM simulation and test the performance of this method. I will present the problematic features of RCMs in simulating the climate and demonstrate the performance of very cheap statistical methods to correct the model’s bias by means of ODA.

The numerical RCM used in this study is the COSMO-CLM (CCLM) model version cosmo$\_$131108$\_$5.00$\_$clm8 (Asharaf et al., 2012). The horizontal resolution of the simulations is set to 0.44^∘ $\times$ 0.44^∘. The full model set-up file is included in the supplementary materials. “The CCLM model is a non-hydrostatic RCM which uses terrain following height coordinates (Rockel et al., 2008), developed from the COSMO model, the current weather forecast model used by the German weather service (DWD)”(Fallah et al., 2016). In order to investigate the sensitivity of RCM to the initial conditions, two sets of simulations are conducted:

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  Default (NATURE) simulation: 6 years long simulation over Europe driven by global atmospheric reanalysis data (6 hourly) produced by the European Centre for Medium-Range Weather Forecasts (ECMWF), the so-called ERAInterim (Dee et al., 2011) (initial and boundary conditions are taken from ERAInterim). This run will be used as the “Nature” or “True” state of the climate in the investigations.

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  Shifted domain simulation: the same as Default but shifted 4 grid points to the Northwest compared with the Default.

Prior to describing the DA methodology, I give a brief review of the Optimal Interpolation (OI) method (for the full review see Gandin (1966); Barth et al. (2008)). OI or “objective analysis” or “kriging” is one of the most commonly used and fairly simple DA methods applied since 1970s. The unknown state of the climate $X$ is the vector which has to be estimated based on the available observations ($Y$). Given the state vector $X$, the state on observations’ location is obtained by an interpolation method (here nearest neighbor). This operation is noted as matrix $H$ and the state at the observations’ location as $HX$. The ultimate goal is to find the nearest state to the “True” state of the climate ($X^{TRUE}$) the so-called Analysis ($X^{A}$) given the observation ($Y$) and background $X^{B}$ (first guess). The background and observation can be written as :

where $\varepsilon$ and $\eta^{B}$ denote the observation and background errors, respectively.

Models contain systematic errors which may have diverse origins (dynamical core, parametrization, initialization). DA schemes are also based on simplified hypotheses and are imperfect (e.g., here the Gaussian parametrization for $P^{B}$). All these sources of errors may also interact with one another in a way that tracing the source of problem may be impossible. In recent studies (Acevedo, 2015; Acevedo et al., 2016) these error sources are neglected by using a simplified numerical experiment called OSSE.

The OSSE used in this study is described by the schematic shown in Figure 2. First the NATURE model simulation $X^{NATURE}$ (“true” run) is conducted as the final prediction target. Then using the output from NATURE run and adding random draws from a White Noise distribution (with $\mu=0$ and $\sigma=0.5$), the pseudo-observations are created which are interpolated over the observations’ location. And finally, applying the OI scheme on the new forecast (Shifted) run, the observationally constrained run $X^{DA}$ is obtained by assimilating the noisy observations.