The SEEK filter method for data assimilation in oceanography: a synthesis

This paper reviews the developments of the singular evolutive extended Kalman (SEEK) filter method used for data assimilation in oceanography, since the original paper by Pham et al. (J Mar Syst 16:3–4, 323–340 1998a). First, a short review is presented of the context of data assimilation in oceanography and of the variety of numerical ocean codes and configurations in which the SEEK filter has been implemented using different data sets for assimilation. Then, the paper provides an exposition of the different versions of the SEEK filter developed during the past 10 years and discusses their relative merits for scientific or operational implementations. A classification of the algorithmic variants is proposed, and several possible improvements of the generic methodology are mentioned in the perspective of new assimilation challenges.     

Dual state-parameter estimation of root zone soil moisture by optimal parameter estimation and extended Kalman filter data assimilation

With well-determined hydraulic parameters in a hydrologic model, a traditional data assimilation method (such as the Kalman filter and its extensions) can be used to retrieve root zone soil moisture under uncertain initial state variables (e.g., initial soil moisture content) and good simulated results can be achieved. However, when the key soil hydraulic parameters are incorrect, the error is non-Gaussian, as the Kalman filter will produce a persistent bias in its predictions. In this paper, we propose a method coupling optimal parameters and extended Kalman filter data assimilation (OP-EKF) by combining optimal parameter estimation, the extended Kalman filter (EKF) assimilation method, a particle swarm optimization (PSO) algorithm, and Richards’ equation. We examine the accuracy of estimating root zone soil moisture through the optimal parameters and extended Kalman filter data assimilation method by using observed in situ data at the Meiling experimental station, China. Results indicate that merely using EKF for assimilating surface soil moisture content to obtain soil moisture content in the root zone will produce a persistent bias between simulated and observed values. Using the OP-EKF assimilation method, estimates were clearly improved. If the soil profile is heterogeneous, soil moisture retrieval is accurate in the 0-50 cm soil profile and is inaccurate at 100 cm depth. Results indicate that the method is useful for retrieving root zone soil moisture over large areas and long timescales even when available soil moisture data are limited to the surface layer, and soil moisture content are uncertain and soil hydraulic parameters are incorrect.     

A soil moisture assimilation scheme based on the ensemble Kalman filter using microwave brightness temperature

This study presents a soil moisture assimilation scheme, which could assimilate microwave brightness temperature directly, based on the ensemble Kalman filter and the shuffled complex evolution method (SCE-UA). It uses the soil water model of the land surface model CLM3.0 as the forecast operator, and a radiative transfer model (RTM) as the observation operator in the assimilation system. The assimilation scheme is implemented in two phases: the parameter calibration phase and the pure soil moisture assimilation phase. The vegetation optical thickness and surface roughness parameters in the RTM are calibrated by SCE-UA method and the optimal parameters are used as the final model parameters of the observation operator in the assimilation phase. The ideal experiments with synthetic data indicate that this scheme could significantly improve the simulation of soil moisture at the surface layer. Furthermore, the estimation of soil moisture in the deeper layers could also be improved to a certain extent. The real assimilation experiments with AMSR-E brightness temperature at 10.65 GHz (vertical polarization) show that the root mean square error (RMSE) of soil moisture in the top layer (0–10 cm) by assimilation is 0.03355 m³ · m⁻³, which is reduced by 33.6% compared with that by simulation (0.05052 m³ · m⁻³). The mean RMSE by assimilation for the deeper layers (10–50 cm) is also reduced by 20.9%. All these experiments demonstrate the reasonability of the assimilation scheme developed in this study.     

Comparison of reduced-order, sequential and variational data assimilation methods in the tropical Pacific Ocean

This paper presents a comparison of two reduced-order, sequential, and variational data assimilation methods: the singular evolutive extended Kalman filter (SEEK) and the reduced 4D-Var (R-4D-Var). A hybridization of the two, combining the variational framework and the sequential evolution of covariance matrices, is also preliminarily investigated and assessed in the same experimental conditions. The comparison is performed using the twin-experiment approach on a model of the tropical Pacific domain. The assimilated data are simulated temperature profiles at the locations of the TAO/TRITON array moorings. It is shown that, in a quasilinear regime, both methods produce similarly good results. However, the hybrid approach provides slightly better results and thus appears as potentially fruitful. In a more nonlinear regime, when tropical instability waves develop, the global nature of the variational approach helps control model dynamics better than the sequential approach of the SEEK filter. This aspect is probably enhanced by the context of the experiments in that there is a limited amount of assimilated data and no model error.     

Assimilation of SeaWiFS chlorophyll data into a 3D-coupled physical–biogeochemical model applied to a freshwater-influenced coastal zone

In order to predict eutrophication events in coastal areas we tested an assimilation scheme based on sequential data assimilation of SeaWiFS chlorophyll data into a coupled 3D physical–biogeochemical model. The area investigated is a semi-enclosed estuarine system (Gulf of Fos–North-western Mediterranean Sea) closely linked to the Rhone River delta. This system is subjected to episodic eutrophication caused by certain hydrodynamic conditions and intermittent nutrient inputs. The 3D hydrodynamic model Symphonie was coupled to the biogeochemical modelling platform Eco3M. Surface chlorophyll concentrations were derived from SeaWiFS data using the OC5 algorithm and were sequentially assimilated using a singular evolutive extended Kalman filter. Assimilation efficiency was evaluated through an independent in situ data set collected during a field survey that took place in May 2001 (ModelFos cruise). An original approach was used in constructing the state vector and the observation vector. By assimilating pseudo-salinity extracted from the model biogeochemical dynamics in both open sea and plume region were respected. We proved that substantial improvements were made in short-term forecasts by integrating such satellite-estimated chlorophyll maps. We showed that missing freshwater inputs could be corrected to a certain extent by the assimilation process. Simulated concentrations of surface chlorophyll and other basic components of the pelagic ecosystem such as nitrates were improved by assimilating surface chlorophyll maps. Finally we showed the coherent spatial behaviour of the filter over the whole modelled domain.     

Data assimilation with the EnKF in a 1-D numerical model of a North Sea station

A series of numerical experiments for data assimilation with the Ensemble Kalman Filter (EnKF) in a shallow water model are reported. Temperature profiles measured at a North Sea location, 55°30ˊ North and 0°55ˊ East (referred to as the CS station of the NERC North Sea project), are assimilated in 1-D simulations. Comparison of simulations without assimilation to model results obtained when assimilating data with the EnKF allows us to assess the filter performance in reproducing features of the observations not accounted for by the model. The quality of the model error sampling is tested as well as the validity of the Gaussian hypothesis underlying the analysis scheme of the EnKF. The influence of the model error parameters and the frequency of the data assimilation are investigated and discussed. From these experiments, a set of optimal parameters for the model error sampling are deduced and used to test the behavior of the EnKF when propagating surface information into the water column.     

On the role of tides and strong wind events in promoting summer primary production in the Barents Sea

Tides and wind-driven mixing play a major role in promoting post-bloom productivity in subarctic shelf seas. Whether this is also true in the high Arctic remains unknown. This question is particularly relevant in a context of increasing Arctic Ocean stratification in response to global climatic change. We have used a three-dimensional ocean-sea ice-plankton ecosystem model to assess the contribution of tides and strong wind events to summer (June-August 2001) primary production in the Barents Sea. Tides are responsible for 20% (60% locally) of the post-bloom primary production above Svalbard Bank and east of the Kola Peninsula. By contrast, more than 9% of the primary production is due to winds faster than 8 m s⁻¹ in the central Barents Sea. Locally, this contribution reaches 25%. In the marginal ice zone, both tides and wind events have only a limited effect on primary production (<2%). Removing tides or winds faster than 8 m s⁻¹ promotes a regime more sustained by regenerated production with a f-ratio (i.e. the proportion of nitrate-based “new” primary production in the total primary production) that decreases by up to 26% (east of the Kola Peninsula) or 35% (central Barents Sea), respectively. When integrated over all Barents Sea sub-regions, tides and strong wind events account, respectively, for 6.8% (1.55 Tg C; 1 Tg C=10¹² g C) and 4.1% (0.93 Tg C) of the post-bloom primary production (22.6 Tg C). To put this in context, this contribution to summer primary production is equivalent to the spring bloom integrated over the Svalbard area. Tides and winds are significant drivers of summer plankton productivity in the Barents Sea.     

A trophic model for the Danshuei River Estuary, a hypoxic estuary in northern Taiwan

The estuary of the Danshuei River, a hypoxic subtropical estuary, receives a high rate of untreated sewage effluent. The Ecopath with Ecosim software system was used to construct a mass-balanced trophic model for the estuary, and network analysis was used to characterize the structure and matter flow in the food web. The estuary model was comprised of 16 compartments, and the trophic levels varied from 1.0 for primary producers and detritus to 3.0 for carnivorous and piscivorous fishes. The large organic nutrient loading from the upper reaches has resulted in detritivory being more important than herbivory in the food web. The food-chain length of the estuary was relatively short when compared with other tropical/subtropical coastal systems. The shortness of food-chain length in the estuary could be attributed to the low biomass of the top predators. Consequently, the trophic efficiencies declined sharply for higher trophic levels due to low fractions of flows to the top predators and then high fractions to detritus. The low biomass of the top predators in the estuary was likely subject to over-exploitation and/or hypoxic water. Summation of individual rate measurements for primary production and respiration yielded an estimate of −1791 g WW m⁻² year⁻¹, or −95 g C m⁻² year⁻¹, suggesting a heterotrophic ecosystem, which implies that more organic matter was consumed than was produced in the estuary.     

太湖叶绿素a同化系统敏感性分析

太湖叶绿素a同化系统对于不同参数的敏感性将直接影响到该系统能否精确的估算太湖叶绿素a的浓度分布.利用2009年4月21日环境一号卫星(HJ-1B CCD2)影像数据反演太湖叶绿素a浓度场信息.以此作为背景场信息,结合基于集合均方根滤波的太湖叶绿素a同化系统,分析和评价了样本数目、同化时长、背景场误差、观测误差和模型误差对于同化系统性能的影响.结果表明:从计算成本、系统运行时间和同化效果等方面分析,当集合样本数目达到30~40左右时同化系统取得了较好的结果;同化系统对于背景场误差的估计变化不是很敏感,即初始场的估计是否准确对于同化系统的性能影响不是很大;同化系统对于模型误差和观测误差的变化较为敏感,不同的测试点位由于水体动力学性质不一,其敏感性的表现形式有所差异;利用数据同化方法可以有效地估算太湖叶绿素a浓度.     

Data assimilation methods for estimating a heterogeneous conductivity field by assimilating transient solute transport data via ensemble Kalman filter

An ensemble Kalman filter (EnKF) is developed to identify a hydraulic conductivity distribution in a heterogeneous medium by assimilating solute concentration measurements of solute transport in the field with a steady‐state flow. A synthetic case with the mixed Neumann/Dirichlet boundary conditions is designed to investigate the capacity of the data assimilation methods to identify a conductivity distribution. The developed method is demonstrated in 2‐D transient solute transport with two different initial instant solute injection areas. The influences of the observation error and model error on the updated results are considered in this study. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating solute concentration measurements. The larger area of the initial distribution and the more observed data obtained, the better the calculation results. When the standard deviation of the observation error varies from 1% to 30% of the solute concentration measurements, the simulated results by the data assimilation method do not change much, which indicates that assimilation results are not very sensitive to the standard deviation of the observation error in this study. When the inflation factor is more than 1.0 to enlarge the model error by increasing the forecast error covariance matrix, the updated results of the hydraulic conductivity by the data assimilation method are not good at all. Copyright © 2012 John Wiley & Sons, Ltd.     

Phytoplankton biomass and primary production responses to physico-chemical forcing across the northeastern New Zealand continental shelf

Phytoplankton biomass and primary production were monitored in the Hauraki Gulf and on the northeastern continental shelf, New Zealand - using ship surveys, moored instruments and satellite observations (1998-2001) - capturing variability across a range of space and time scales. A depth-integrated primary production model (DIM) was used to predict integrated productivity from surface parameters, enabling regional-specific estimates from satellite data. The shelf site was dominated by pico-phytoplankton, with low chlorophyll-a (<1 mg m⁻³) and annual production (136 g C m⁻² yr⁻¹). In contrast, the gulf contained a micro/nano-phytoplankton-dominated community, with relatively high chlorophyll-a (>1 mg m⁻³) and annual production (178 g C m⁻² yr⁻¹). Biomass and productivity responded to physico-chemical factors; a combination of light, critical mixing depths and/or nutrient limitation—particularly new nitrate-N. Relatively low biomass and production was observed during 1999. This coincided with inter-annual variability in the timing and extent of upwelling- and downwelling-favourable along-shelf wind-stress, influencing the fluxes of new nitrate-N to the shelf and gulf. Relationships with the Southern Oscillation Index are also discussed. Our multi-scaled sampling highlighted details associated with stratification and de-stratification events, and deep sub-surface chlorophyll-a not visible to satellite sensors. This study demonstrates the importance of multi-scaled sampling in gaining estimates of regional production and its responses to physico-chemical forcing.     

The growth dynamics of Karenia brevis within discrete blooms on the West Florida Shelf

As part of the ECOHAB: Florida Program, we studied three large blooms of the harmful bloom forming dinoflagellate Karenia brevis. These blooms formed on the West Florida Shelf during Fall of 2000 off Panama City, and during Fall 2001 and Fall 2002 off the coastline between Tampa Bay and Charlotte Harbor. We suggest that these blooms represent two different stages of development, with the 2000 and 2001 blooms in an active growth or maintenance phase and the 2002 bloom in the early bloom initiation phase. Each bloom was highly productive with vertically integrated primary production values of 0.47–0.61, 0.39–1.33 and 0.65 g C m⁻² d⁻¹ for the 2000, 2001 and 2002 K. brevis blooms, respectively. Carbon specific growth rates were low during each of these blooms with values remaining fairly uniform with depth corresponding to generation times of 3–5 days. Nitrogen assimilation by K. brevis was highest during 2001 with values ranging from 0.15 to 2.14 μmol N L⁻¹ d⁻¹ and lower generally for 2000 and 2002 (0.01–0.64 and 0.66–0.76 μmol N L⁻¹ d⁻¹ for 2000 and 2002, respectively). The highest K. brevis cell densities occurred during the 2001 bloom and ranged from 400 to 800 cells mL⁻¹. Cell densities were lower for each of the 2000 and 2002 blooms relative to those for 2001 with densities ranging from 100 to 500 cells mL⁻¹. The 2000 and 2001 blooms were dominated by K. brevis in terms of its contribution to the total chlorophyll a (chl a) pool with K. brevis accounting generally for >70% of the observed chl a. For those populations that were dominated by K. brevis (e.g. 2000 and 2001), phytoplankton C biomass (C_p,0) constituted <30% of the total particulate organic carbon (POC). However, in 2002 when diatoms and K. brevis each contributed about the same to the total chl a, C_p,0 was >72% of the POC. The fraction of the total chl a that could be attributed to K. brevis was most highly correlated with POC, chl a and salinity. Nitrogen assimilation rate and primary production were highly correlated with a greater correlation coefficient than all other comparisons.     

Towards the estimation root-zone soil moisture via the simultaneous assimilation of thermal and microwave soil moisture retrievals

The upcoming deployment of satellite-based microwave sensors designed specifically to retrieve surface soil moisture represents an important milestone in efforts to develop hydrologic applications for remote sensing observations. However, typical measurement depths of microwave-based soil moisture retrievals are generally considered too shallow (top 2–5 cm of the soil column) for many important water cycle and agricultural applications. Recent work has demonstrated that thermal remote sensing estimates of surface radiometric temperature provide a complementary source of land surface information that can be used to define a robust proxy for root-zone (top 1 m of the soil column) soil moisture availability. In this analysis, we examine the potential benefits of simultaneously assimilating both microwave-based surface soil moisture retrievals and thermal infrared-based root-zone soil moisture estimates into a soil water balance model using a series of synthetic twin data assimilation experiments conducted at the USDA Optimizing Production Inputs for Economic and Environmental Enhancements (OPE³) site. Results from these experiments illustrate that, relative to a baseline case of assimilating only surface soil moisture retrievals, the assimilation of both root- and surface-zone soil moisture estimates reduces the root-mean-square difference between estimated and true root-zone soil moisture by 50% to 35% (assuming instantaneous root-zone soil moisture retrievals are obtained at an accuracy of between 0.020 and 0.030 m³ m⁻³). Most significantly, improvements in root-zone soil moisture accuracy are seen even for cases in which root-zone soil moisture retrievals are assumed to be relatively inaccurate (i.e. retrievals errors of up to 0.070 m³ m⁻³) or limited to only very sparse sampling (i.e. one instantaneous measurement every eight days). Preliminary real data results demonstrate a clear increase in the R² correlation coefficient with ground-based root-zone observations (from 0.51 to 0.73) upon assimilation of actual surface soil moisture and tower-based thermal infrared temperature observations made at the OPE³ study site.     

Radiance data assimilation for operational snow and streamflow forecasting

Estimation of seasonal snowpack, in mountainous regions, is crucial for accurate streamflow prediction. This paper examines the ability of data assimilation (DA) of remotely sensed microwave radiance data to improve snow water equivalent prediction, and ultimately operational streamflow forecasts. Operational streamflow forecasts in the National Weather Service River Forecast Center (NWSRFC) are produced with a coupled SNOW17 (snow model) and SACramento Soil Moisture Accounting (SAC-SMA) model. A comparison of two assimilation techniques, the ensemble Kalman filter (EnKF) and the particle filter (PF), is made using a coupled SNOW17 and the microwave emission model for layered snow pack (MEMLS) model to assimilate microwave radiance data. Microwave radiance data, in the form of brightness temperature (TB), is gathered from the advanced microwave scanning radiometer-earth observing system (AMSR-E) at the 36.5 GHz channel. SWE prediction is validated in a synthetic experiment. The distribution of snowmelt from an experiment with real data is then used to run the SAC-SMA model. Several scenarios on state or joint state-parameter updating with TB data assimilation to SNOW-17 and SAC-SMA models were analyzed, and the results show potential benefit for operational streamflow forecasting.     

Calibration framework for a Kalman filter applied to a groundwater model

The paper presents a novel approach to the setup of a Kalman filter by using an automatic calibration framework for estimation of the covariance matrices. The calibration consists of two sequential steps: (1) Automatic calibration of a set of covariance parameters to optimize the performance of the system and (2) adjustment of the model and observation variance to provide an uncertainty analysis relying on the data instead of ad-hoc covariance values. The method is applied to a twin-test experiment with a groundwater model and a colored noise Kalman filter. The filter is implemented in an ensemble framework. It is demonstrated that lattice sampling is preferable to the usual Monte Carlo simulation because its ability to preserve the theoretical mean reduces the size of the ensemble needed. The resulting Kalman filter proves to be efficient in correcting dynamic error and bias over the whole domain studied. The uncertainty analysis provides a reliable estimate of the error in the neighborhood of assimilation points but the simplicity of the covariance models leads to underestimation of the errors far from assimilation points.     

Sediment accumulation in the western Gulf of Lions,France: The role of Cap de Creus Canyon in linking shelf and slope sediment dispersal systems

Previous work in the Gulf of Lions (western Mediterranean Sea) has suggested that significant amounts of sediment escape through the western part of this tectonically passive margin, despite it being far removed from the primary sediment source (the Rhone River, ∼160 km to the NE). The primary mechanism behind this export is hypothesized to be the interaction of a regional, southwestward sediment-transport path with a canyon deeply incising the southwestern part of the shelf, Cap de Creus Canyon.     

Variations of phytoplankton productivity and biomass over an annual cycle in Saanich Inlet,a British Columbia fjord

Saanich Inlet is a highly productive temperate fjord with the capability to record inter-annual patterns of water-column primary production in undisturbed laminated sediments. We investigated spatial and temporal variations in primary productivity, total and size-fractionated phytoplankton chl a, dissolved nutrients, temperature and salinity at the head and mouth of Saanich Inlet from May 2005 to November 2006. New primary productivity was also measured from May to October 2006. During the growing season (spring, summer and fall), primary productivity was 1.5 times higher at the mouth than at the head of Saanich Inlet and, averaged across stations, total productivity was 460 g C m⁻² y⁻¹. Average new productivity was 53% and 58% of total primary productivity at the head and mouth of the inlet, respectively, and during the growing season micro-phytoplankton (>20 μm; mainly diatoms) was the most abundant size-class of phytoplankton. These rates of primary production are as high as or higher than those measured in other fjords, possibly because of a tidally-driven fortnightly gravity exchange that supplies nutrients to surface waters that enhance biological production when nutrients would otherwise be limiting. This exchange delivers nutrients at least as far inland as the head station, while nutrients associated with an eddy near the mouth may be the cause of even higher productivity there. We discuss the impact of these nutrient sources to Saanich Inlet on the records of paleoproduction generated from two Ocean Drilling Program cores extracted from this fjord, and suggest that the fortnightly exchange buffers variations in nutrient supply occurring on sub-decadal or decadal scales.     

Limestone assimilation and the origin of CO2 emissions at the Alban Hills (Central Italy): Constraints from experimental petrology

The Alban Hills volcanic region (20 km south of Rome, in the Roman Province) emitted a large volume of potassic magmas (> 280 km³) during the Quaternary. Chemical interactions between ascending magmas and the ∼ 7000–8000-m-thick sedimentary carbonate basement are documented by abundant high temperature skarn xenoliths in the eruptive products and have been frequently corroborated by geochemical surveys. In this paper we characterize the effect of carbonate assimilation on phase relationships at 200 MPa and 1150–1050 °C by experimental petrology. Calcite and dolomite addition promotes the crystallization of Ca-rich pyroxene and Mg-rich olivine respectively, and addition of both carbonates results in the desilication of the melt. Furthermore, carbonate assimilation liberates a large quantity of CO₂-rich fluid. A comparison of experimental versus natural mineral, glass and bulk rock compositions suggests large variations in the degree of carbonate assimilation for the different Alban Hills eruptions. A maximum of 15 wt.% assimilation is suggested by some melt inclusion and clinopyroxene compositions; however, most of the natural data indicate assimilation of between 3 and 12 wt.% carbonate. Current high CO₂ emissions in this area most likely indicate that such an assimilation process still occurs at depth. We calculate that a magma intruding into the carbonate basement with a rate of ∼ 1 – 2 · 10⁶ m³/year, estimated by geophysical studies, and assimilating 3–12 wt.% of host rocks would release an amount of CO₂ matching the current yearly emissions at the Alban Hills. Our results strongly suggest that current CO₂ emissions in this region are the shallow manifestation of hot mafic magma intrusion in the carbonate-hosted reservoir at 5–6 km depth, with important consequences for the present-day volcanic hazard evaluation in this densely populated and historical area.     

Evaluating the utility of the ensemble transform Kalman filter for adaptive sampling when updating a hydrodynamic model

This paper compares two Monte Carlo sequential data assimilation methods based on the Kalman filter, for estimating the effect of measurements on simulations of state error variance made by a one-dimensional hydrodynamic model. The first method used an ensemble Kalman filter (EnKF) to update state estimates, which were then used as initial conditions for further simulations. The second method used an ensemble transform Kalman filter (ETKF) to quickly estimate the effect of measurement error covariance on forecast error covariance without the need to re-run the simulation model. The ETKF gave an unbiased estimate of EnKF analysed error variance, although differences in the treatment of measurement errors meant the results were not identical. Estimates of forecast error variance could also be made, but their accuracy deteriorated as the time from measurements increased due in part to model non-linearity and the decreasing signal variance. The motivation behind the study was to assess the ability of the ETKF to target possible measurements, as part of an adaptive sampling framework, before they are assimilated by an EnKF-based forecasting model on the River Crouch, Essex, UK. The ETKF was found to be a useful tool for quickly estimating the error covariance expected after assimilating measurements into the hydrodynamic model. It, thus, provided a means of quantifying the ‘usefulness’ (in terms of error variance) of possible sampling schemes.