The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a nonstandard problem involving the coupled system of direct and adjoint equations. The nonstandard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is presented. A numerical example is given for the variational data assimilation problem related to sea surface temperature for the Baltic Sea thermodynamics model.

The methods of data assimilation have become an important tool for
analysis of complex physical phenomena in various fields of science and
technology. These methods allow us to combine mathematical models, data
resulting from observations, and a priori information. The problems of
variational data assimilation can be formulated as optimal control problems

The necessary optimality condition is related to the gradient of the original
cost function; thus, to study the sensitivity of the optimal solution, one
should differentiate the optimality system with respect to observations. In
this case, we come to the so-called second-order adjoint problem

The issue of sensitivity is related to the statistical properties of the
optimal solution

This paper is based on the results of

We consider a dynamic formulation of the variational data assimilation problem
for parameter estimation in a continuous form, but the presented sensitivity
analysis formulas with respect to observations do not follow from our
previous results for the initial condition problem

This paper is organized as follows. In Sect. 2, we give the statement of the variational data assimilation problem for a nonlinear evolution model to estimate the model parameters. In Sect. 3, sensitivity of the response function after assimilation with respect to observations is studied, and its gradient is related to the solution of a nonstandard problem. In Sect. 4 we derive an operator equation involving the Hessian to study the solvability of the nonstandard problem, and give an algorithm to compute the gradient of the response function. A proof-of-concept analytic example with a simple model is given in Sect. 5 to demonstrate how the sensitivity analysis algorithm works. Section 6 presents an application of the theory to the data assimilation problem for a sea thermodynamics model. Numerical examples are given in Sect. 7 for the Baltic Sea dynamics model. The main results are discussed in Sect. 8.

We consider the mathematical
model of a physical process that is described by the evolution problem

Let us consider the following data assimilation problem with the aim to
estimate the parameter

We suppose that the solution of Eq. (

Let us consider the adjoint operator

From Eqs. (

We assume that the system (Eqs.

If the observation operator

In geophysical applications the observation data cannot be measured precisely; therefore, it is important to be able to estimate the impact of uncertainties in observations on the outputs of the model after assimilation.

Let us introduce a response function

If

To compute the gradient

Here we put

We get a coupled system of two differential equations (Eqs.

Let us denote the auxiliary variable

It is easily seen that the operator

Based on the above consideration, we can formulate the following algorithm to
compute the gradient of the response function

For

Find

Solve the direct problem

Compute the gradient of the response function as

Equation (

Below we give a proof-of-concept analytic example to show how the algorithm
(Eqs.

Let us consider a simple evolution problem for the ordinary differential
equation

Indeed, if

Let us now apply the algorithm (Eqs.

On the second step of the algorithm, one needs to solve the equation

On the third step of the algorithm, we need to solve the problem
(Eq.

Consider the sea thermodynamics problem in the form

Equation (

Due to Eq. (

Consider the data assimilation problem for the sea surface temperature

Consider the data assimilation problem for the surface temperature in the
following form: find

For

For

The optimality system determining the solution of the formulated variational
data assimilation problem according to the necessary condition

Here the boundary-value function

To illustrate the above-presented theory, we consider the problem of
sensitivity of functionals of the optimal solution

Equation (

In our notations, Eq. (

We are interested in the sensitivity of the functional

By definition, the sensitivity is given by the gradient of

Since

For

Find

Solve the direct problem

Compute the gradient of the response function as

Equation (

The numerical experiments have been performed using the three-dimensional
numerical model of the Baltic Sea hydro-thermodynamics developed at the
Institute of Numerical Mathematics, Russian Academy of
Sciences on the base of the splitting method

The object of simulation is the Baltic Sea water area. The parameters of the
considered domain and its geographic coordinates can be described in the
following way: the

The Baltic Sea daily-averaged nighttime surface-temperature data were used
for

The mean climatic flux obtained from the NCEP (National Centers for
Environmental Prediction) reanalysis was taken for

Using the hydro-thermodynamics model mentioned above, which is supplied with
the assimilation procedure for the surface temperature

The aim of the experiment was the numerical study of the sensitivity of
functionals of the optimal solution

Implementing the assimilation procedure, we considered a system of the form
in Eqs. (

We use here the SI units, namely K (kelvin) is used for temperature,
m s

Let us present some results of numerical experiments. The calculation results for

The gradient of the functional

We can see the sub-areas (in red) in which the functional

Baltic Sea topography (m).

The above studies allow us to solve the problem of the definition of sea sub-areas in which the functional of the optimal solution is most sensitive to errors in the observations during variational data assimilation, when the error values are not a priori known.

In this paper we have considered numerical algorithms to study the sensitivity of functionals of the optimal solution of the variational data assimilation problem aimed at the reconstruction of unknown parameters of the model. The optimal solution obtained as a result of assimilation depends on the observations that may contain uncertainties. Computing the gradient of the functionals with respect to observations reduces to the solution of a nonstandard problem which is a coupled system involving direct and adjoint equations with mutually dependent variables. Solvability of the nonstandard problem is related to the properties of the Hessian of the original cost function. An algorithm developed to compute the gradient of the response function is based on the second-order adjoint techniques. A numerical example for the variational data assimilation problem related to sea surface temperature for the Baltic Sea thermodynamics model demonstrates the result of the gradient computation of the response function associated with the mean surface temperature. The presented algorithm may be used to determine the sea sub-areas in which the functionals of the optimal solution are most sensitive to errors in the observations during variational data assimilation.

The Baltic Sea daily-averaged surface-temperature data
(Copernicus product ID ST_BAL_SST_L4_REP_OBSERVATIONS_010_016) can be
found at the Copernicus Marine Environment Monitoring Service website
(

The authors declare that they have no conflict of interest.

This article is part of the special issue “Numerical modeling, predictability and data assimilation in weather, ocean and climate: A special issue honoring the legacy of Anna Trevisan (1946–2016)”. It is a result of A Symposium Honoring the Legacy of Anna Trevisan – Bologna, Italy, 17–20 October 2017.

The authors are greatly thankful to Olivier Talagrand and the reviewers for providing helpful comments that resulted in substantial improvements for the paper. This work was carried out within the Russian Science Foundation project 17-77-30001 (studies in Sects. 1–4), the AIRSEA project (INRIA Grenoble Rhône-Alpes), and the project 18-01-00267 of the Russian Foundation for the Basic Research. Edited by: Olivier Talagrand Reviewed by: Olivier Talagrand and one anonymous referee