ALBERT

Description: Combining ocean models and proxy data via data assimilation is a powerful means to obtain more reliable estimates of past ocean states, but studies using data assimilation for paleo-ocean state estimation are rare. A few studies used the adjoint method, also called 4D-Var, to estimate the state of the ocean during the Last Glacial Maximum (LGM). The adjoint method, however, requires the adjoint of the model code, which is not easily obtained for most models. The method is computationally very demanding and does not readily provide uncertainty estimates. Here, we present a new and computationally very efficient technique to obtain ocean state estimates. We applied a state reduction approach in conjunction with a finite difference sensitivity-iterative Kalman smoother (FDS-IKS) to estimate spatially varying atmospheric forcing fields and to obtain an equilibrium model simulation in consistency with proxy data. We tested the method in synthetic pseudo-proxy data experiments. The method is capable of very efficiently estimating 16 control variables and reconstructing a target ocean circulation from sea surface temperature (SST) and oxygen isotopic composition of seawater data at LGM coverage. The method is advantageous over the adjoint method regarding that it is very easy to implement, it requires substantially less computing time and provides an uncertainty estimate of the estimated control variables. The computing time, however, depends linearly on the size of the control space limiting the number of control variables that can be estimated. We used the method to investigate the constraint of data outside of the Atlantic Ocean on the Atlantic overturning circulation. Our results indicate that while data from the Pacific or Indian Ocean aid in correctly estimating the Atlantic overturning circulation, they are not as crucial as the Atlantic data. We additionally applied the method to estimate the LGM ocean state constrained by a global SST reconstruction and data on the oxygen isotopic composition of calcite from fossil benthic and planktic foraminifera. The LGM estimate shows a large improvement compared to our first guess, but model-data misfits remain after the optimization due to model errors that cannot be corrected by the control variables. The estimate shows a shallower North Atlantic Deep Water and a weaker Atlantic overturning circulation compared to today in consistency with previous studies. The combination of the FDS-IKS and the state reduction approach is a step forward in making ocean state estimation and data assimilation applicable for complex and computationally expensive models and to models where the adjoint is not available.

Description: Images from satellite-based synthetic aperture radar (SAR) instruments contain large amounts of information about the position of flood water during a river flood event. This observational information typically covers a large spatial area, but is only relevant for a short time if water levels are changing rapidly. Data assimilation allows us to combine valuable SAR-derived observed information with continuous predictions from a computational hydrodynamic model and thus to produce a better forecast than using the model alone. In order to use observations in this way a suitable observation operator is required. In this paper we show that different types of observation operator can produce very different corrections to predicted water levels; this impacts on the quality of the forecast produced. We discuss the physical mechanisms by which different observation operators update modelled water levels and introduce a novel observation operator for inundation forecasting. The performance of the new operator is compared in synthetic experiments with that of two more conventional approaches. The conventional approaches both use observations of water levels derived from SAR to correct model predictions. Our new operator is instead designed to use backscatter values from SAR instruments as observations; such an approach has not been used before in an ensemble Kalman filtering framework. Direct use of backscatter observations opens up the possibility of using more information from each SAR image and could potentially speed up the time taken to produce observations needed to update model predictions. We compare the strengths and weaknesses of the three different approaches with reference to the physical mechanisms by which each of the observation operators allow data assimilation to update water levels in synthetic twin experiments in an idealised domain.

Description: The assimilation of satellite-based water level observations (WLOs) into 2-D hydrodynamic models can keep flood forecasts on track or be used for reanalysis to obtain improved assessments of previous flood footprints. In either case, satellites provide spatially dense observation fields, but with spatially correlated errors. To date, assimilation methods in flood forecasting either incorrectly neglect the spatial correlation in the observation errors or, in the best of cases, deal with it by thinning methods. These thinning methods result in a sparse set of observations whose error correlations are assumed to be negligible. Here, with a case study, we show that the assimilation diagnostics that make use of statistical averages of observation-minus-background and observation-minus-analysis residuals are useful to estimate error correlations in WLOs. The average estimated correlation length scale of 7 km is longer than the expected value of 250 m. Furthermore, the correlations do not decrease monotonically; this unexpected behaviour is shown to be the result of assimilating some anomalous observations. Accurate estimates of the observation error statistics can be used to support quality control protocols and provide insight into which observations it is most beneficial to assimilate. Therefore, the understanding gained in this paper will contribute towards the correct assimilation of denser datasets.

Description: Images from satellite-based synthetic aperture radar (SAR) instruments contain large amounts of information about the position of floodwater during a river flood event. This observational information typically covers a large spatial area but is only relevant for a short time if water levels are changing rapidly. Data assimilation allows us to combine valuable SAR-derived observed information with continuous predictions from a computational hydrodynamic model and thus to produce a better forecast than using the model alone. In order to use observations in this way, a suitable observation operator is required. In this paper we show that different types of observation operators can produce very different corrections to predicted water levels; this impacts the quality of the forecast produced. We discuss the physical mechanisms by which different observation operators update modelled water levels and introduce a novel observation operator for inundation forecasting. The performance of the new operator is compared in synthetic experiments with that of two more conventional approaches. The conventional approaches both use observations of water levels derived from SAR to correct model predictions. Our new operator is instead designed to use backscatter values from SAR instruments as observations; such an approach has not been used before in an ensemble Kalman filtering framework. Direct use of backscatter observations opens up the possibility of using more information from each SAR image and could potentially speed up the time taken to produce observations needed to update model predictions. We compare the strengths and weaknesses of the three different approaches with reference to the physical mechanisms with which each of the observation operators allow data assimilation to update water levels in synthetic twin experiments in an idealised domain.

Description: The assimilation of satellite-based water level observations (WLOs) into 2D hydrodynamic models can keep flood forecasts on track or be used for reanalysis to obtain improved assessments of previous flood footprints. In either case, satellites provide spatially dense observation fields, but with spatially correlated errors. To date, assimilation methods in flood forecasting either incorrectly neglect the spatial correlation in the observation errors or, in the best of cases, deal with it by thinning methods. These thinning methods result in a sparse set of observations whose error correlations are assumed to be negligible. Here, with a case study, we show that the assimilation diagnostics that make use of statistical averages of observation-minus-background and observation-minus-analysis residuals are useful to estimate error correlations in WLOs. The estimated correlations do not behave as expected; however, analysis shows that the diagnostic can also be used to highlight anomalous observation datasets. Accurate estimates of the observation error statistics can be used to support quality control protocols and provide insight into which observations it is most beneficial to assimilate. Furthermore, the understanding gained in this paper will contribute towards the correct assimilation of denser datasets.

Description: In climate reanalyses for multi-decadal or longer scales with coupled atmosphere-ocean General Circulation models (CGCMs) it can be assumed that the growth of prediction errors arises chiefly from imprecisely known model parameters, which have a nonlinear relationship with the climate observations (paleoclimate proxies). Also, high-resolution CGCMs for climate analysis are extremely expensive to run, which constrains the applicability of assimilation schemes. In a model framework where we assume that model dynamic parameters account for (nearly) all forecast errors at observation times, we compare two computationally efficient iterative schemes for approximate nonlinear model parameter estimation and joint flux estimation (taking the specific shape of freshwater from melting in the Greenland ice sheet), and its physically consistent state. First, a trivial adaptation of the strong constraint incremental 4D-Var formulation leads to what we refer to as the parameter space iterative extended Kalman smoother (pIKS); a Gauss-Newton scheme. Second, a so-called parameter space fractional Kalman smoother (pFKS) is an alternative controlled-step line search, which can potentially be a more stable approach. While these iterative schemes have been used in data assimilation, we revisit them together within the context of parameter estimation in climate reanalysis, as compared to the more general 4D-Var formulation. Then, the two schemes are evaluated in numerical experiments with a simple 1D energy balance model (Ebm1D) and with a fully-coupled Community Earth System Model (CESM v1.2). Firstly, with Ebm1D the pFKS obtains a cost function similar to the adjoint method with highly reduced computational cost, while an ensemble transform Kalman filter with an m = 60 ensemble size (ETKF60) behaves slightly worse. The pIKS behaves worse than the ETKF60, but an ETKF10 (m = 10) is even worst. Accordingly, with CESM we evaluate the pKFS and the ETKF60 along with an ETKF with Gaussian Anamorphosis (ETKF-GA60). From all the options, the pFKS has the lowest cost function and seems the favored overall option under heavy computational restrictions, but the ETKF obtains better estimates of the flux term.

Description: Paleoclimate reconstruction based on assimilation of proxy observations requires specification of the control variables and their background statistics. As opposed to numerical weather prediction (NWP), which is mostly an initial condition problem, the main source of error growth in deterministic Earth system models (ESMs) regarding the model low-frequency response comes from errors in other inputs: parameters for the small-scale physics, as well as forcing and boundary conditions. Also, comprehensive ESMs are non-linear and only a few ensemble members can be run in current high-performance computers. Under these conditions we evaluate two assimilation schemes, which (a) count on iterations to deal with non-linearity and (b) are based on low-dimensional control vectors to reduce the computational need. The practical implementation would assume that the ESM has been previously globally tuned with current observations and that for a given situation there is previous knowledge of the most sensitive inputs (given corresponding uncertainties), which should be selected as control variables. The low dimension of the control vector allows for using full-rank covariances and resorting to finite-difference sensitivities (FDSs). The schemes are then an FDS implementation of the iterative Kalman smoother (FDS-IKS, a Gauss–Newton scheme) and a so-called FDS-multistep Kalman smoother (FDS-MKS, based on repeated assimilation of the observations). We describe the schemes and evaluate the analysis step for a data assimilation window in two numerical experiments: (a) a simple 1-D energy balance model (Ebm1D; which has an adjoint code) with present-day surface air temperature from the NCEP/NCAR reanalysis data as a target and (b) a multi-decadal synthetic case with the Community Earth System Model (CESM v1.2, with no adjoint). In the Ebm1D experiment, the FDS-IKS converges to the same parameters and cost function values as a 4D-Var scheme. For similar iterations to the FDS-IKS, the FDS-MKS results in slightly higher cost function values, which are still substantially lower than those of an ensemble transform Kalman filter (ETKF). In the CESM experiment, we include an ETKF with Gaussian anamorphosis (ETKF-GA) implementation as a potential non-linear assimilation alternative. For three iterations, both FDS schemes obtain cost functions values that are close between them and (with about half the computational cost) lower than those of the ETKF and ETKF-GA (with similar cost function values). Overall, the FDS-IKS seems more adequate for the problem, with the FDS-MKS potentially more useful to damp increments in early iterations of the FDS-IKS.