This paper deals with the estimation of the noise covariance matrices of systems described by state-space models. Stress is laid on the systematic survey and classification of both the recursive and batch processing methods proposed in the literature with a special focus on the correlation methods. Besides the correlation methods, representatives of other groups are introduced also with respect to their basic idea, estimate properties, assumptions and possible extensions, and user-defined parameters. Common and dual properties of the methods are highlighted, and a simulation comparison using exemplary MATLAB implementations of the methods is provided. KEYWORDSadaptive systems, noise covariance matrix, state estimation, state-space model, system identification INTRODUCTIONKnowledge of a system model is a key prerequisite for many state estimation, signal processing, fault detection, and optimal control problems. The model is often designed to be consistent with random behaviour of the system quantities and properties of the measurements. While the deterministic part of the model often arises from mathematical modelling on the basis of physical, chemical, or biological laws governing the behaviour of the system, the statistics of the stochastic part are often difficult to find by the modelling and have to be identified using the measured data. Incorrect description of the noise statistics may result in significant worsening of estimation, signal processing, detection, or control quality or even in a failure of the underlying algorithms.In the last 5 decades, therefore, a significant research interest has been focused on a design of the methods for the estimation of the properties of the stochastic part of the model. The attention has been devoted to both the input-output models 1-5 and the state-space (SS) models 6-14 and both recursive and batch processing methods. This paper focuses on the methods estimating the properties of the stochastic part of the system described by an SS model discrete in time. In particular, the methods estimating the covariance matrices (CMs) † of noises in the state and measurement equation from a sequence of measured data are of interest. The methods are further denoted as the noise CM estimation methods.

Abstract. Cave monitoring studies clarify the climatic, surface vegetation, and karst processes affecting the cave system and lay the foundation to interpreting geochemical stalagmite records. Here we report monitoring of cave air, bedrock chemistry, and drip water δ13CDIC, δ18O and δD as well as 16 trace elements covering a full annual cycle spanning 16 months between November 2019 and March 2021 in La Vallina cave in the Northwest Iberian Peninsula. While decreased rainfall and increased evapotranspiration in summer months lead to a strong reduction in drip rates, there is little seasonal variation of δ18O and δD in a given drip, likely reflecting discrete moderately- to well-mixed karst water reservoirs. Small differences in δ18O and δD between drip sites are attributed to variable evaporation intensity and/or transit times. The dissolved inorganic carbon composition of drip water (δ13CDIC) is likely driven by seasonal changes in temperature controlling biological processes (vegetation and microbial soil activity) resulting in minimum δ13CDIC in summer and autumn months. Increased bedrock dissolution due to higher soil pCO2 in summer and autumn results in increased trace element concentrations of congruently dissolved elements. Cave air measurements indicate seasonal ventilation (winter) and stagnation (summer) of cave air. The opposite effects of reduced cave air pCO2, seasonally variable biological activity and increased drip rate limit the extent of seasonal variation of degassing and prior calcite precipitation (PCP) supported by trace elements (Sr/Ca-index). Estimated stalagmite growth rates using monitoring data suggest calcite precipitation is restricted to certain seasons (summer and winter) at certain locations within the cave, which has important implications on proxy interpretation of stalagmite records.

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The rate and consequences of future high latitude ice sheet retreat remain a major concern given ongoing anthropogenic warming. Here, new precisely dated stalagmite data from NW Iberia provide the first direct, high-resolution records of periods of rapid melting of Northern Hemisphere ice sheets during the penultimate deglaciation. These records reveal the penultimate deglaciation initiated with rapid century-scale meltwater pulses which subsequently trigger abrupt coolings of air temperature in NW Iberia consistent with freshwater-induced AMOC slowdowns. The first of these AMOC slowdowns, 600-year duration, was shorter than Heinrich 1 of the last deglaciation. Although similar insolation forcing initiated the last two deglaciations, the more rapid and sustained rate of freshening in the eastern North Atlantic penultimate deglaciation likely reflects a larger volume of ice stored in the marine-based Eurasian Ice sheet during the penultimate glacial in contrast to the land-based ice sheet on North America as during the last glacial.