Towards the assimilation of tree-ring-width records using ensemble Kalman filtering techniques

Clim Dyn (2016) 46:1909–1920 DOI 10.1007/s00382-015-2683-1 Towards the assimilation of tree‑ring‑width records using ensemble Kalman filtering techniques Walter Acevedo1 · Sebastian Reich2 · Ulrich Cubasch1 Received: 1 July 2014 / Accepted: 22 May 2015 / Published online: 10 June 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for treering-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. This research work was produced within the Helmholtz graduate research school GeoSim. * Walter Acevedo ‑berlin.de 1 Institut für Meteorologie, Freie Universität Berlin, Carl‑Heinrich‑Becker‑Weg 6‑10, 12165 Berlin, Germany 2 Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes. Keywords Proxy forward modeling · Data assimilation · Fuzzy logic · Ensemble Kalman filter · Paleoclimate reconstruction 1 Introduction In recent years the assimilation of proxy data into climate models has emerged as a very promising way to bring physical consistency to paleoclimate reconstructions, as well as to reduce their uncertainty levels. “High resolution” proxy records (Hughes and Ammann 2009) offer information about inter-annual and slower internal climate variabilities which in principle could be constrained by a suitable DA scheme. As a result, several research groups have been actively working with the purpose of paving the way towards an eventual paleo-reanalysis (Hughes et al. 2010; Guiot et al. 2009; Brönnimann 2011). To date, several very diverse DA schemes have been tested on climate models—such as pattern nudging (von Storch et al. 2000), forcing singular vectors (Barkmeijer et al. 2003), adjoint method (Kurahashi-Nakamura et al. 2014), particle filter (Dubinkina and Goosse 2013; Mathiot et al. 2013) and ensemble Kalman filter (Bhend et al. 2012; Pendergrass et al. 2012)—providing very encouraging results regarding the potential constraint of the inter-annual internal variability. As for the observations employed in these experiments, the dominant approach has been to use either pseudo-proxies or statistically reconstructed climate 13 1910 time series, both considering a univariate linear relationship between climate state and climate-driven proxy signals. Nowadays, the many developments in the booming field of high-resolution paleoclimatology (Hughes and Ammann 2009) have made evident the complexity of climate proxy systems. The formation of a proxy record might comprise biological, physical, and chemical mechanisms, each of these able to introduce non-linear processes. Accordingly, there is a growing interest in adopting more realistic approaches regarding the relation between proxy data and climate forcing (Evans et al. 2013). One of the most promising approaches in this direction is “Proxy Forward Modeling” (Hughes et al. 2010; Evans et al. 2013), which takes climate forcing as input data and generate artificial proxy records that can be directly compared to actual proxy records. This strategy is diametrically opposed to the traditional inverse approach where climate conditions are directly inverted from proxy data. Proxy forward models can be used for different purposes such as model-paleodata comparison in the proxy space (Evans et al. 2013) and prediction of future evolution of proxy archives (Vaganov et al. 2006). On the other hand, despite its forth direction, forward models can also be used for reconstruction purposes by resorting to probabilistic inversion strategies, such as Bayesian hierarchical modeling (Tolwinski-Ward et al. 2014), Markov Chain Monte Carlo method (Boucher et al. 2014) and DA (Hughes et al. 2010). Contributing to the development of the latter approach is the objective of the present paper. Different proxy data kinds react in distinctive ways to specific sets of climate variables. Accordingly, the assimilation of a given proxy type should be pursued on its own merits, using a forward model able to simulate the climate recording processes specific to that particular kind of proxy record. Nowadays, there exist forward models for many different proxy types, most notably TRW (Vaganov et al. 2006; Evans et al. 2006; Tolwinski-Ward et al. 2011), treering isotopes (Roden et al. 2000; Danis et al. 2012; Evans 2007), coral isotopes (Thompson et al. 2011), ocean sediments (Heinze 2001; Schmidt 1999) and stalagmite isotopes (Baker et al. 2012). There are also initiatives to model the transport of isotopes within the atmosphere (Sturm et al. 2010). Depending on the particular application and the availability of data, proxy forward models assume different complexity levels which go from the minimalistic linear pseudo-proxies to comprehensive models that simulate the proxy generation process as realistically as possible. Following this train of thought, we will investigate the role of the observation operator in the assimilation of the most traditional climate proxy: TRW. This proxy record type is particularly suitable for DA purposes given that it exhibits the largest spatial coverage, albeit strongly biased towards the Northern hemisphere, as well as one of the 13 W. Acevedo et al. highest and most stable temporal resolutions, among the different paleoclimate proxies currently available. Additionally, tree-ring growth forward models have reached a rather mature state of development. In particular within climate applications, the (...truncated)