WAP-1D-VAR v1.0: A One-Dimensional Variational Data Assimilation Model for the West Antarctic Peninsula

Abstract. The West Antarctic Peninsula (WAP) is a rapidly warming region, with substantial ecological and biogeochemical responses to the observed change and variability for the past decades, revealed by multi-decadal observations from the Palmer Antarctica Long-Term Ecological Research (LTER) program. The wealth of these long-term observations provides an important resource for ecosystem modeling, but there has been a lack of focus on the development of numerical models that simulate time-evolving plankton dynamics over the austral growth season along the coastal WAP. Here, we introduce a one-dimensional variational data assimilation planktonic ecosystem model (i.e., the WAP-1D-VAR v1.0 model) equipped with a model parameter optimization scheme. We first demonstrate the modified and newly added model schemes to the pre-existing food web and biogeochemical components of the other ecosystem models that WAP-1D-VAR model was adapted from, including diagnostic sea-ice forcing and trophic interactions specific to the WAP region. We then present the results from model experiments where we assimilate 11 different data types from an example Palmer LTER growth season (October 2002–March 2003) directly related to corresponding model state variables and flows between these variables. The iterative data assimilation procedure reduces the misfits between observations and model results by 58 %, compared to before optimization, via an optimized set of 12 parameters out of a total of 72 free parameters. The optimized model results capture key WAP ecological features, such as blooms during seasonal sea-ice retreat, the lack of macronutrient limitation, and modeled variables and flows comparable to other studies in the WAP region, as well as several important ecosystem metrics. One exception is that the model slightly underestimates particle export flux, for which we discuss potential underlying reasons. The data assimilation scheme of the WAP-1D-VAR model enables the available observational data to constrain previously poorly understood processes, including the partitioning of primary production by different phytoplankton groups, the optimal chlorophyll-to-carbon ratio of the WAP phytoplankton community, and the partitioning of dissolved organic carbon pools with different lability. The WAP-1D-VAR model can be successfully employed to link the snapshots collected by the available data sets together to explain and understand the observed dynamics along the coastal WAP.

Localized depletion of 180 and l3C in a thin subhorizontal marble layer in the Adamello contact aureole, Southern Alps, Italy, resulted from fluid infiltration focused along a crosscutting dike. Values of 0180 and ol3C in calcite from the 1 m long profile decrease systematically from sedimentary values of 0180 = 220/00 (SMOW) and Ol3C= 00/00 (PDB) to 0180 = 12.50/00 and ol3C -70/00near the dike. The presence of clinozoisite and garnet in the 5-15 cm thick marble layers near the granodiorite dike indicates H20-rich fluid conditions (XC02 0.01). The 0 and C isotope profiles were compared with one- and two-dimensional models of advective-dispersive isotope transport. Individually the isotope profiles fit one-dimensional transport models well. However one-dimensional models, using equilibrium fluidrock exchange or a kinetic formulation, do not explain the relative locations or shapes of the two isotope-exchange profiles given the petrologic constraint of XC02 0.01 for the infiltrating fluid. Excellent agreement with the 0180 and ol3C data is obtained using a twodimensional model that specifies (1) a high-permeability zone in marble near the dike that focuses fluid flow parallel to the dike and (2) a lower permeability zone in marble away from the dike where isotope exchange is dominated by molecular diffusion. The combined constraints imposed by phase equilibria and two isotope tracers allow two-dimensional fluid flow to be inferred from one-dimensional data. The results emphasize that isotope distributions resulting from multidimensional flow may fortuitously fit one-dimensional transport models if isotope tracers are considered independently. The use of multiple tracers coupled to fluid-composition constraints is therefore essential to discriminate between various transport models.

Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

Atmospheric chemistry models usually perform badly in forecasting wintertime air pollution because of their uncertainties. Generally, such uncertainties can be decreased effectively by techniques such as data assimilation (DA) and model output statistics (MOS). However, the relative importance and combined effects of the two techniques have not been clarified. Here, a one-month air quality forecast with the Weather Research and Forecasting-Chemistry (WRF-Chem) model was carried out in a virtually operational setup focusing on Hebei Province, China. Meanwhile, three-dimensional variational (3DVar) DA and MOS based on one-dimensional Kalman filtering were implemented separately and simultaneously to investigate their performance in improving the model forecast. Comparison with observations shows that the chemistry forecast with MOS outperforms that with 3DVar DA, which could be seen in all the species tested over the whole 72 forecast hours. Combined use of both techniques does not guarantee a better forecast than MOS only, with the improvements and degradations being small and appearing rather randomly. Results indicate that the implementation of MOS is more suitable than 3DVar DA in improving the operational forecasting ability of WRF-Chem.

Recently, researchers have demonstrated that learned index can improve query performance while reducing the storage overhead. It potentially offers an opportunity to address the spatial query processing challenges caused by the surge in location-based services. Although several learned indexes have been proposed to process spatial data, the main idea behind these approaches is to utilize the existing one-dimensional learned models, which requires either converting the spatial data into one-dimensional data or applying the learned model on individual dimensions separately. As a result, these approaches cannot fully leverage or take advantage of the information regarding the spatial distribution of the original spatial data. To this end, in our previous work, we proposed a spatial (multi-dimensional) interpolation function based learned model to develop a spatial learned index and designed efficient range and <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> NN query strategies over it. However, there are some limitations in the proposed learned model, such as the prediction accuracy and index building time. In this paper, we address the limitations of our previous work and propose a new spatial learned model by employing the characteristics of the spatial interpolation functions and a novel dynamic encoding technique. Detailed experiments are conducted with real-world datasets. The results indicate that our new proposed learned model is better than our previous one in terms of building time, prediction accuracy, and storage overhead simultaneously, and the new learned spatial index is better than the existing learned spatial indexes in query execution time and index building time.

Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified. In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution. Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented. Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.

At the National Centers for Environmental Prediction, the global ensemble forecasts from the ensemble Kalman filter scheme in the Global Forecast System are applied in a regional three-dimensional (3D) and a four dimensional (4D) ensemble–variational (EnVar) data assimilation system. The application is a one-way variational method using hybrid static and ensemble error covariances. To enhance impact, three new features have been added to the existing EnVar system in the Gridpoint Statistical Interpolation (GSI). First, the constant coefficients that assign relative weight between the ensemble and static background error are now allowed to vary in the vertical. Second, a new formulation is introduced for the ensemble contribution to the analysis surface pressure. Finally, in order to make use of the information in the ensemble mean that is disregarded in the existing EnVar in GSI, the trajectory correction, a novel approach, is introduced. Relative to the application of a 3D variational data assimilation algorithm, a clear positive impact on 1–3-day forecasts is realized when applying 3DEnVar analyses in the North American Mesoscale Forecast System (NAM). The 3DEnVar DA system was operationally implemented in the NAM Data Assimilation System in August 2014. Application of a 4DEnVar algorithm is shown to further improve forecast accuracy relative to the 3DEnVar. The approach described in this paper effectively combines contributions from both the regional and the global forecast systems to produce the initial conditions for the regional NAM system.

The use of Four-Dimensional variational (4D-Var) data assimilation technology in the context of sea dynamics problems, with a sensitivity analysis of model results to observation errors, is presented. The technology is applied to a numerical model of ocean circulation developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), with the use of the splitting method and complemented by 4D-Var data assimilation with covariance matrices of background and observation errors. The variational data assimilation involves iterative procedures to solve inverse problems so as to correct sea surface heat fluxes for the model under consideration. An algorithm is formulated to study the sensitivity of the model outputs, considered as output functions after assimilation, to the observation errors. The algorithm reveals the regions where the output function gradient is the largest for the average sea surface temperature (SST) in a selected area, obtained by assimilation. In the numerical experiments, a 4D variational problem of SST assimilation for the Baltic Sea area is solved.

Previous descriptions of how localized ensemble covariances can be incorporated into variational (VAR) data assimilation (DA) schemes provide few clues as to how this might be done in an efficient way. This article serves to remedy this hiatus in the literature by deriving a computationally efficient algorithm for using nonadaptively localized four-dimensional (4D) or three-dimensional (3D) ensemble covariances in variational DA. The algorithm provides computational advantages whenever (i) the localization function is a separable product of a function of the horizontal coordinate and a function of the vertical coordinate, (ii) and/or the localization length scale is much larger than the model grid spacing, (iii) and/or there are many variable types associated with each grid point, (iv) and/or 4D ensemble covariances are employed.

Chapter 9 - Data Assimilation and Inverse Modeling

Parameter estimation in the sediment deposition and resuspension process is an important issue in numerical modeling of suspended sediment transport. The sediment settling velocity and resuspension rate are two critical parameters controlling the sediment exchange process between the water column and sediment bed. In this paper a variational inverse data assimilation scheme for estimation of the sediment settling velocity and resuspension rate is developed and tested with a three‐dimensional cohesive sediment transport model. The sediment settling velocity and resuspension rate are treated as poorly known parameters in the model and are estimated by the variational inverse scheme using an adjoint approach. A limited‐memory quasi‐Newtonian conjugate gradient algorithm is used in the minimization process. The variational inverse model is tested in the James River estuary in Virginia by identical twin experiments. Numerical experimental results show that variational inverse data assimilation is a useful tool for retrieving poorly known parameters in a cohesive sediment transport model.

A four-dimensional variational data assimilation (4DVar) system of the LASG/IAP Climate Ocean Model, version 1.0 (LICOM1.0), named LICOM-3DVM, has been developed using the three-dimensional variational data assimilation of mapped observation (3DVM), a 4DVar method newly proposed in the past two years. Two experiments with 12-year model integrations were designed to validate it. One is the assimilation run, called ASSM, which incorporated the analyzed weekly sea surface temperature (SST) fields from Reynolds and Smith (OISST) between 1990 and 2001 once a week by the LICOM-3DVM. The other is the control run without any assimilation, named CTL. ASSM shows that the simulated temperatures of the upper ocean (above 50 meters), especially the SST of equatorial Pacific, coincide with the Tropic Atmosphere Ocean (TAO) mooring data, the World Ocean Atlas 2001 (WOA01) data and the Met Office Hadley Centre’s sea ice and sea surface temperature (HadISST) data. It decreased the cold bias existing in CTL in the eastern Pacific and produced a Nino index that agrees with observation well. The validation results suggest that the LICOM-3DVM is able to effectively adjust the model results of the ocean temperature, although it’s hard to correct the subsurface results and it even makes them worse in some areas due to the incorporation of only surface data. Future development of the LICOM-3DVM is to include subsurface in situ observations and satellite observations to further improve model simulations.

The current four‐dimensional variational (4D‐Var) data assimilation (DA) algorithm, available in the community Weather Research and Forecasting (WRF) model's DA (WRFDA) system, can run only at the same resolution as that of the WRF model forecast, which makes it computationally prohibitive for operational applications at convective scale. The Multi‐Resolution Incremental 4D‐Var (MRI‐4DVar) has been developed in this study in order to speed up WRF 4D‐Var through a three‐stage procedure in each outer loop. One key aspect of WRF MRI‐4DVar is the introduction of the inverse control variable transform within WRFDA that allows the proper resolution change between different outer loops for the control variables projected in the vertical empirical orthogonal function (EOF) space. MRI‐4DVar's computational efficiency and forecast performance are demonstrated by applying it to an afternoon thunderstorm event over northern Taiwan with a 2 km model resolution setting. Comparing to the full‐resolution 4D‐Var experiment, two MRI‐4DVar configurations with a speed‐up of 4.5 and 7.5 times performed similarly well in terms of Fractions Skill Score (FSS) of 6‐hr accumulated rainfall forecasts and hourly variation of total rainfall amount, indicating MRI‐4DVar's potential for operational applications at convective scale.

Kriging-enhanced ensemble variational data assimilation for scalar-source identification in turbulent environments

The effect on weather forecast performance of incorporating ensemble covariances into the initial covariance model of the four-dimensional variational data assimilation (4D-Var) Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) is investigated. This NAVDAS-AR-hybrid scheme linearly combines the static NAVDAS-AR initial background error covariance with a covariance derived from an 80-member flow-dependent ensemble. The ensemble members are generated using the ensemble transform technique with a (three-dimensional variational data assimilation) 3D-Var-based estimate of analysis error variance. The ensemble covariances are localized using an efficient algorithm enabled via a separable formulation of the localization matrix. The authors describe the development and testing of this scheme, which allows for assimilation experiments using differing linear combinations of the static and flow-dependent background error covariances. The tests are performed for two months of summer and two months of winter using operational model resolution and the operational observational dataset, which is dominated by satellite observations. Results show that the hybrid mode data assimilation scheme significantly reduces the forecast error across a wide range of variables and regions. The improvements were particularly pronounced for tropical winds. The verification against radiosondes showed a greater than 0.5% reduction in vector wind RMS differences in areas of statistical significance. The verification against self-analysis showed a greater than 1% reduction from verifying against analyses between 2- and 5-day lead time at all eight vertical levels examined in areas of statistical significance. Using the Navy's summary of verification results, the Navy Operational Global Atmospheric Prediction System (NOGAPS) scorecard, the improvements resulted in a score (+1) that justifies a major system upgrade.