Data assimilation (DA) is widely used in geosciences to combine observations with information from a dynamical model (the background) in a statistically optimal way . In its standard formulation, DA aims to estimate the full system state on a spatial grid.

In this study, we do not focus on the full gridded state, but rather on the estimation of specific geophysical structures (GSs), such as ocean eddies or cyclones, viewed as coherent objects characterized by a small set of parameters. When classic, linear-Gaussian DA methods, such as optimal interpolation or the Kalman filter, are applied directly to a gridded field without prior structural adjustment, the presence of a coherent structure can cause the linear combination of background and observation to distort the reconstructed structure: spatially displaced or differently-shaped realizations are averaged rather than reconciled, which smears the structure whenever displacement or shape, rather than amplitude, uncertainty dominates .

Several strategies have been proposed to introduce structural information explicitly into the assimilation problem, spanning a spectrum from field-based to more object-oriented representations. A first family augments the state vector with structure parameters directly: added vortex position to the ensemble Kalman filter (EnKF) state, allowing the filter to separate displacement from amplitude errors. This was extended to tropical cyclones by including position, translation speed, and axisymmetric wind structure , and further to operationally relevant track, wind-radius, and intensity parameters . These approaches define state variables that are directly interpretable as object properties, though they remain tied to a predefined parametric model, in a spirit close to the parametric representation adopted here. A second family instead treats structural errors through geometric transformations or alternative measures of distance in the field representation. Alignment and morphing approaches modify the field geometry to reconcile displaced or differently shaped structures , while displacement data assimilation explicitly estimates the displacement field required for such alignment while enforcing area preservation . Alternatively, optimal-transport-based approaches modify the metric used to compare background and observations, using the Wasserstein distance to better account for positional errors of coherent structures . A third family operates on adaptive or moving computational meshes that track coherent structures rather than resolving the full domain at uniform resolution: developed ensemble DA for adaptive, non-conservative moving-mesh models, and mesh-node positions have since been included directly within the assimilation state itself , though in this case the assimilated geometry reflects the numerical discretization rather than the structure's own properties. Feature-informed and Lagrangian-based approaches extract collective descriptors of coherent structures directly from observations, rather than assimilating raw trajectories or fields . Non-Gaussian methods such as particle filters can, in principle, handle the nonlinearity associated with displacement and shape uncertainty; part of the difficulty in relating a gridded and a reduced representation, however, lies not solely in this nonlinearity, but in correctly marginalizing, rather than merely interpolating, uncertainty between the two spaces, a distinction examined further in Sect. .

Correctly linking uncertainty across representation spaces is itself motivated by a broader principle: Rao-Blackwellization formalizes the idea of analytically marginalizing a conditionally-linear-Gaussian subset of a state while treating the remaining nonlinear component separately, seeking a representation in which a problem becomes, and remains, Gaussian-tractable. This principle underlies Rao-Blackwellized particle filters , including applications to environmental data assimilation that combine Kalman-type updates with particle-based treatment of low-dimensional non-Gaussian parameters . The parametric transformation explored in this study follow this approach: rather than operating directly on the full gridded field, it seeks a lower-dimensional, physically interpretable representation in which relating background and observation uncertainty is more tractable, provided this relationship is established by propagating the full uncertainty through the mapping between spaces, rather than by a local linear approximation of it.

The goal of this study is to investigate, in a controlled and idealized framework, how the choice of representation space influences the behavior of a standard Gaussian linear DA update when applied to coherent structures with uncertain position and shape. In particular, we compare assimilation performed in the classic gridded space with assimilation performed directly in a reduced parametric space describing the structure.

The idea of working in a low-dimensional parameter space rather than on a full-state vector has a long history in the control and stochastic filtering literature . However, in these frameworks, the parameters to be estimated are typically internal coefficients of a dynamical model, rather than geometric descriptors of coherent structures. The present work applies a related principle to an explicitly geometric and physically interpretable representation of a coherent structure, defined here by amplitude, position and radius.

This parametric perspective is also related to reduced-order and subspace approaches, where the system state is represented in a lower-dimensional space, such as empirical orthogonal functions or learned latent variables . In contrast to these approaches, which define reduced coordinates in an abstract basis, the present work considers an explicit physically interpretable parameterization of the structure. This allows a direct interpretation of assimilation updates in terms of physically meaningful geometric properties.

We emphasize that this framework does not aim to replace existing Lagrangian, feature-based, or non-Gaussian DA methods, but rather to provide a simplified setting to analyze how standard Gaussian updates behave when applied either in gridded space or in a low-dimensional parameter space representing coherent structures. To the best of our knowledge, this specific comparison of the assimilation behavior between gridded space and an explicitly defined geometric parameter space of a coherent structure has not been analyzed in the literature. In this sense, the study is primarily methodological and diagnostic, focusing on the mechanism of error representation rather than on operational forecasting performance.

In the proposed framework, the state of the system is fully described by the parameters of a single idealized eddy, which allows a direct analysis of how uncertainties in amplitude, position, and radius propagate through the assimilation step in both representations. The computational implications of this lower-dimensional representation are discussed in Sect. .

We present a set of idealized numerical experiments designed to compare a single analysis step performed in the classic gridded space and in this reduced parametric space. The eddy representation and assimilation methodology are detailed in Sect. , the experimental setup in Sect. , and the results are presented in Sect. . Conclusions and perspectives are discussed in Sect. .

The objective of this paper is to compare two methods of blending two sources of information, a background and an observation, using a basic DA algorithm. We focus exclusively on the analysis step and do not perform repeated assimilation cycles (forecast-analysis-reanalysis). The stability of the system over multiple cycles is beyond the scope of this study and will be investigated in future work.

We consider a simplified one-dimensional cross-shore section of an ocean eddy, described by three parameters: peak amplitude a, peak position p, and maximal velocity radius r, such that xparam=(a,p,r). In practice, these parameters can be estimated using standard eddy detection and fitting algorithms, such as local maxima identification combined with Gaussian profile fitting e.g..

For our idealized experiments, however, the parameters are prescribed manually rather than obtained from a detection algorithm, in order to control uncertainties and systematically study the assimilation behavior. The physical profile along the cross-section is given by the mapping 1xgrid=h(lon;xparam)=aexp⁡-12lon-pr2, where lon is the vector of longitudinal positions along the cross-section. Here, h serves as a mapping function from the parametric space xparam to the physical gridded space xgrid. It is neither an observation operator nor a numerical dynamical model, but simply defines the physical profile corresponding to a given set of parameters. We note that the Gaussian shape assumed here for the eddy profile is a modeling choice, distinct from the Gaussian statistical assumptions made elsewhere in this study (e.g. for the background and observation error distributions); every detected or analyzed eddy is consequently constrained to this Gaussian shape by construction. The framework could in principle be generalized to other parametric profile shapes, which would only change the form of h and its Jacobian in Eq. () and Sect. .

Two spaces are considered for the eddy representation: the parametric space (param), of dimension n=3 corresponding to (a,p,r), and the gridded space (grid), of dimension m equal to the length of the longitude vector. The notation used throughout this manuscript is summarized in Table .

The background state in each space is modeled as a perturbation around a reference value: 2x=xb+ηb, where xb is the reference background state and ηb∼N(0,B) is the background error, assumed Gaussian with covariance matrix B. The observation is related to the true state by: 3y=H(x)+ε, where H is the observation operator and ε∼N(0,R) is the observation error with covariance R. Equations () and () are written generically and apply separately in each space with the appropriate superscript (param or grid) on all variables (see Table ).

In our case, since the state vector is fully observed in both spaces, the operator H is the identity matrix, leading to a simple linear observation equation.

The data assimilation is performed in each space with the appropriate superscript (param or grid) on all variables using the following cost function J: J(x)=12x-xbTB-1x-xb4+12(y-x)TR-1(y-x). Under the assumptions of linearity and Gaussian errors, the minimizer of J is given analytically by the Best Linear Unbiased Estimator (BLUE). No iterative minimization algorithm is therefore required; the solution is computed directly for each configuration. Let Xb=[x1b,…,xNb] and Y=[y1,…,yN] denote the background and observation ensemble matrices, each of size n×N (or m×N in grid space), where N is the ensemble size. The covariance matrices are estimated from these ensembles as: 5B=Cov(Xb,Xb), and R=Cov(Y,Y), following the Ensemble Optimal Interpolation (EnOI) framework. The BLUE update of each ensemble member i reads: 6xia=xib+Kyi-xib,i=1,…,N, from which the analysis ensemble mean x‾a=1N∑ixia is derived, and the analysis error covariance Pa is given by: 7Pa=(I-K)B, where K=B(B+R)-1 is the Kalman gain. Equations ()–() apply separately in each space with the appropriate superscript (param or grid) on all variables.

Two experimental frameworks are considered, differing in how the uncertainty is introduced and how the covariance matrices are specified, as described in the following subsections. In the tutorial framework (Sect. ), the uncertainty originates in parameter space and is propagated to grid space via h. In the realistic framework (Sect. ), the uncertainty originates in grid space and must be inverted back to parameter space through a fitting procedure, providing a more balanced end-to-end comparison.

Four experiments are presented, organized into two sets corresponding to the two frameworks described in Sect. . Experiments #1, #2, and #3 follow the tutorial framework (Sect. ) and are designed to isolate the effect of a single uncertain parameter on the assimilation outcome. Experiment #4 follows the realistic framework (Sect. ) and provides a more balanced end-to-end evaluation by starting from noisy gridded observations. The parameter values for all four experiments are given in Table .

This study investigates data assimilation (DA) performed in a reduced parametric space as well as classic linear and Gaussian DA in gridded space, focusing on a simplified one-dimensional representation of an ocean eddy and on a single analysis step.

The results obtained in this idealized framework indicate that DA in gridded space provides satisfactory reconstructions when uncertainty mainly affects linear parameters such as amplitude, as illustrated by Experiment #1, where both approaches yield identical results (RMSE = 0.00 m, zero parameter error). However, when uncertainty involves nonlinear geometric parameters such as position and radius, the gridded-space update leads to a deformation of the reconstructed structure. In Experiment #2, the gridded analysis underestimates the amplitude by 23 % (error of 0.24 m) and overestimates the radius by 29 % (error of 2.15 km), while the RMSE relative to the true profile reaches 0.10 m, compared with 0.00 m in parametric space. In Experiment #3, the analyzed profile is no longer Gaussian, and the RMSE in gridded space (0.03 m) exceeds that in parametric space (0.00 m). These structural distortions affect derived physical quantities: the total EKE is reduced by 67 % in Experiment #2 and by 11 % in Experiment #3 relative to the parametric analysis. These EKE figures are reported as an illustration of the physical consequences of the profile deformation, rather than as validation metrics.

The tutorial experiments (#1–3) are useful to isolate and illustrate this mechanism, but they do not constitute a balanced end-to-end comparison, since the uncertainty originates in parameter space and is propagated to grid space. Experiment #4, which starts from noisy gridded observations, provides a more symmetric test. The parametric approach retains its qualitative advantage: the RMSE of the gridded analysis (0.09 m) remains higher than that of the parametric analysis (0.02 m), and the gridded analysis underestimates the amplitude by 0.21 m and overestimates the radius by 1.94 km relative to the truth, compared to errors of 0.02 m and 0.24 km for the parametric analysis. However, the calibration of the parametric error covariances is reduced compared to the tutorial experiments, due to the additional uncertainty introduced by the nonlinear fitting step. Conclusions drawn from this study should therefore be understood as pertaining to these specific idealized configurations and not as general statements about the relative performance of gridded and parametric DA approaches.

In contrast, the parametric formulation provides a direct update of the structure-defining parameters in this controlled setting. This leads to a more consistent representation of amplitude, position and radius within the assumptions of the experiments. In this idealized setting, the parametric formulation also operates in a lower-dimensional space than the gridded approach (n=3 vs. m=101).

In the above experiments the assimilation in the parametric space presents several advantages. The structure is analyzed directly through a small set of physically interpretable parameters, and structural consistency is preserved in this space: the analyzed profile remains Gaussian by construction (Experiment #3), with the associated EKE diagnostic (Table ) deviating from the truth by at most 1 %, compared with reductions of up to 55 % in gridded space. The parametric formulation also operates in a markedly lower-dimensional space (n=3 vs. m=101), which reduces the cost of the analysis step. This reduction comes at the expense of the additional parameter-extraction step required for each ensemble member. As discussed in Sect. , the overall computational benefit therefore depends on the relative cost of parameter extraction and analysis, and hence on the grid resolution and ensemble size.

The Rao-Blackwellization principle introduced in Introduction to motivate the parametric transformation used in this study suggests a concrete direction for extending it. As noted in Sect. , this transformation only partially achieves Gaussian tractability, the radius parameter, in particular, remains significantly non-Gaussian after fitting, unlike the exact conditional Gaussianity that Rao-Blackwellization guarantees by construction; identifying a transformation that more fully achieves this property, or a marginalized/Rao-Blackwellized particle filter that treats the residual non-Gaussianity explicitly, is a natural target for future work. We emphasize that the comparison in Sect.  concerns the propagation of uncertainty through an explicitly known analytic mapping h, for which the failure of local linearization could be directly diagnosed against a Monte Carlo reference. In realistic applications, however, the correspondence between gridded and parametric (or feature-based) representations of a coherent structure is generally not available in closed form. In that setting, more general nonlinear, non-Gaussian approaches would be needed to relate uncertainty between representation spaces directly from data, rather than through explicit parametric fitting as done here. Diffusion maps, which construct data-driven geometric embeddings relating different function spaces and have recently been combined with generative particle systems to produce non-Gaussian samples directly from data , or transport-map-based methods that characterize non-Gaussian joint distributions across representations via measure transport , remain candidate tools for this purpose. Exploring either avenue lies beyond the scope of the present idealized study and is identified as a direction for future work.

This work is intentionally limited to a controlled and idealized setting and does not include sequential assimilation cycles or prognostic model dynamics. Future work will investigate the extension of this framework to realistic, time-evolving situations, where eddy parameters must be inferred from observations through detection and fitting procedures, and updated sequentially. In such a context, more general dynamical models for parameter evolution will be required, which may include stochastic parameter models, reduced-order models, tracking-based models, or hybrid approaches, rather than being restricted to any specific method. In particular, the dynamics of eddy parameters in a sequential context are expected to be strongly nonlinear, which may necessitate ensemble-based or nonlinear assimilation methods. The stability of the system over multiple cycles will also be investigated in future work, when this parametric approach is applied in an operational, sequential assimilation context.

Finally, the parametric and gridded formulations provide two complementary perspectives: the former emphasizes the evolution of physically interpretable structure parameters, while the latter operates directly on spatial fields. Depending on the application, analyses can be projected back into gridded space for diagnostic purposes or retained in parameter space for object-oriented applications that directly exploit coherent structure properties.