Motivated by the concept of “location uncertainty”, initially introduced in

Data assimilation is meant to extract information from measurements to improve the state estimate. Kalman-filter-based and particle-filter-based methods are now commonly used for academic studies and operational forecasts. For both methods, the estimate of a state variable and the uncertainty quantification of the estimate of a state variable are repeated at each data assimilation cycle. In the classical Kalman filter, this uncertainty is represented by a covariance matrix. In Monte Carlo-based methods (i.e., the ensemble Kalman filters and particle filters, etc.), it is represented by the spread of the ensemble members or particles. The uncertainty of the state estimate is further part of the input for the next data assimilation cycle. Frequently observed, the uncertainty can be underestimated in nonlinear numerical experiments when there is no model noise

To address the latter issue, “covariance localization” has been developed for both Kalman-filter-based methods and particle filters

In the context of ensemble-/particle-based methods, the uncertainty is usually inflated by artificially perturbing each ensemble member/particle.
We refer the reader to

In the fluid dynamics community, random forcings are not introduced for inflation but to mimic the intermittent backscattering of energy from small scales toward large scales. Among those approaches, we may mention the stochastic Lagrangian models

Several authors proposed schemes specifically to enforce energy conservation or at least a given energy budget

In

From another perspective, the classical optimal transport theory suggests that the difference of two smooth positive density fields (

Motivated by such an optimal transport perspective and the concept of “location uncertainty”, proposed in

Specifically, the idea of covariance inflation can be informally generalized to physical fields that are not always positive, i.e., physical fields other than the density field. Mathematically, a density field

To implement a physically constrained perturbation scheme, the state variable

It will be demonstrated (Sect. 5) that it is indeed sometimes helpful to choose

Correspondingly, at each forecast time step, the covariance inflation should follow four steps.

Step 1: find

Step 2: construct a random diffeomorphism

Step 3: replace

Step 4: calculate the forecast

Associating

In summary, the key perspective of this paper is that the displacement vector field of physical state variables should be determined by the tensor fields associated with the physical fields. The advantage of this perspective is that certain physical quantities can be conserved while applying a displacement vector field to transfer the original physical field. A direct application of this perspective is the physically constrained covariance inflation scheme proposed in this paper. When the tensor fields are positive

This paper is organized as follows. Section 2 is a brief introduction of optimal transport theory. In Sect. 3 we present the perturbation scheme in detail, including the motivation, the specific techniques in derivation, and several examples. In Sect. 4, the resulting perturbation scheme is then compared with the stochastic advection by Lie transport (SALT) equations

The conventions of notation are as follows:

The letter

For Einstein's convention on summation (applies to all indices except

Summation over

Hereafter we briefly summarize some necessary concepts and results in optimal transport theory. Let

Here the probability measures

The convexity of

Consider a compressible flow on a bounded domain

At each time step we construct a small perturbation

Stated in the Introduction, the state variable

This procedure can also be generalized to other types of tensor fields. We refer to

A rigorous mathematical definition and calculation of

Given coordinates

The full derivation of these examples is skipped.
We further express all the terms in coordinates. For instance, we replace

Suppose

As the physical PDE (Eq.

This implies that

To physically interpret this equation, we rewrite

If the original deterministic PDE (

This implies that

Recall, in fluid dynamics, that the Reynolds transport theorem provides an integral conservation equation for the transport of any conserved quantity within a fluid, connected to its corresponding differential equation. The Reynolds transport theorem is central to the LU setting. The present example thus already outlines a closed link between the proposed perturbation approach and the LU formulation.
Accordingly, the SPDE (

For each

This implies

We recognize a diffusion term,

However, the velocity fields appearing in the divergent and advecting terms do not coincide. Indeed, they are even opposite for divergence-free noise (

A major advantage of the proposed perturbation scheme is to possibly prescribe

Define

Then we have

In this section, we demonstrate that both the stochastic advection by Lie transport (SALT) equation

Note that the original LU paper

The original SALT equation

In the second part of Eq. (

The SALT equation regarding the velocity

Compared with

Mentioned above, the Reynolds transport theorem is central to the LU setting, and we already outlined a closed link between the proposed perturbation approach and the LU formulation. This link – related to differential

Dropping the forcing terms,
the LU equation for compressible and incompressible flow reads

To also understand Eq. (

The LU physical justification relies on a stochastic interpretation of fundamental conservation laws, typically conservation of extensive properties (i.e., integrals of functions over a spatial volume) like momentum, mass, matter, and energy

In this section, the proposed approach is applied to derive a stochastic version of thermal shallow water equation. Another stochastic version of thermal shallow water equation can be found in

Stated in

The domain is 2-dimensional. To conserve mass, the only choice for

In sum, we have chosen the following tensor fields:

The starting point of this work was to question how the location of the state variable can be consistently perturbed, motivated by Brenier's theorem

Under this framework, we end up with a stochastic PDE of the state variable

In this paper, we generalize this scheme to a mixed type of tensor fields

Interestingly, similarities and differences can be studied between the proposed perturbation scheme and the existing stochastic physical SALT and LU settings

In order to apply the proposed perturbation scheme to any specific model, the parameters

Suppose that

Given coordinates

Next,

Combining Eqs. (

To simplify Eq. (

No data sets were used in this article.

YZ contributed the main idea and mathematical derivation and wrote the first draft of the paper. VR contributed all the physical interpretations of the derived equations, a much more detailed introduction, and a more detailed comparison of the proposed scheme with the LU equation. BC contributed his insight and enthusiasm to this problem and worked on the writing of the draft. This work was done based on the fruitful discussion of all the three authors since a very early stage.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to express their gratitude towards Wei Pan, Darryl Holm, Dan Crisan, Long Li, and Etienne Mémin for their patient explanation and insightful discussion. The research of Yicun Zhen was supported by the ANR Melody project when he was a postdoc at Ifremer. The research of Valentin Resseguier is supported by the company SCALIAN DS and by France Relance through the MORAANE project. The research of Bertrand Chapron is supported by ERC EU SYNERGY project no. 856408-STUOD and the ANR Melody project.

This research has been supported by the Agence Nationale de la Recherche, Labex Immuno-Oncology (Melody), the European Research Council, H2020 (grant no. STUOD (856408)), the company SCALIAN DS and France Relance through the MORAANE project.

This paper was edited by Jinqiao Duan and reviewed by Daniel Goodair and Baylor Fox-Kemper.