Sensitivity of functionals in variational data assimilation for a sea thermodynamics model

The sensitivity of functionals of the optimal solution to a variational data assimilation problem for the sea thermodynamics model is studied. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary condition. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is determined by the gradient of the response function and reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. The sensitivity analysis of the response function with respect to errors of observation data is carried out. Numerical examples are presented for the Black Sea thermodynamics model.


Introduction
The methods of data assimilation are designed to combine mathematical models, observational data and a priori information. The data assimilation is aimed at constructing or refining the initial and/or boundary conditions (and other model parameters) to improve the accuracy of a prediction model. Mathematically, such problems may be formulated as optimal control problems [1][2][3][4][5][6]. Together with the development and justification of algorithms for the numerical solution to variational data assimilation problems [7], the key role belongs to the properties of the optimal solution and its sensitivity (see [8][9][10][11][12][13][14][15]).
As an extension of the approach in [14,15], this paper considers the problem of sensitivity of functionals of the optimal solution of the variational data assimilation of the sea surface temperature for the sea thermodynamics model. The statement of the problem with the aim to restore the initial state and the boundary heat flux is given. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. The algorithm to compute the gradient of the response function, based on the Hessian of the original cost functional is presented. To illustrate the theory, numerical examples for the Black Sea thermodynamics model are presented.

Statement of the variational data assimilation problem
The sea thermodynamics problem is considered in the form [16], [17]: where = (|Ū n | −Ū n )/2, andŪ n is the normal component ofŪ . We represent (1) in the form of an operator equation in (W 1 2 (D)) * : being understood in the weak sense that Here The functions a T , Q T , f T , Q are supposed to be regular enough such that equality (3) makes sense. The operator L was studied in [16]. Due to (3), equation (2) is considered in Y * = L 2 (t 0 , t 1 ; (W 1 2 (D)) * ) because the operator B : L 2 (Ω × (t 0 , t 1 )) → Y * maps the function Q ∈ L 2 (Ω × (t 0 , t 1 )) to BQ ∈ Y * so that (BQ, T ) = Ω Q T | z=0 dΩ ∀ T ∈ W 1 2 (D) and, therefore, BQ is a linear bounded functional on Y = L 2 (t 0 , t 1 ; W 1 2 (D)). We consider the following data assimilation problem [18]: find T 0 ∈ L 2 (D) and Q ∈ L 2 (Ω × (t 0 , t 1 )) such that where and ) are the given functions, α, β = const > 0. The function T obs (x, y, t) is the sea surface temperature given from observations on some subset of Ω × (t 0 , t 1 ) with the characteristic function m 0 . It is easy to see that for α, β > 0, problem (4) has a unique solution. The solution T of (1) depends continuously on the initial condition T 0 and the flux Q, and a priori estimates are valid in the corresponding functional spaces. The functional J is weakly lower semicontinuous, and the space of admissible controls L 2 (D) × L 2 (Ω × (t 0 , t 1 )) is weakly compact, which gives the existence of the optimal solution to (4).
The necessary optimality condition gradJ = 0 of the variational data assimilation problem under consideration is reduced to the optimality system of the form: where L * and B * are the operators adjoint to L and B, respectively. We will study the sensitivity of the response functions as functionals of the optimal solution T 0 , Q with respect to the observational data T obs .

The sensitivity of functionals of the optimal solution
The response functions as functionals of the optimal solution play an important role in applications. They are related to the outputs of the model after assimilation and it is often necessary to estimate their sensitivity with respect to observation errors.
Let us consider a response function G(T ), which is supposed to be a real-valued function and can be considered as a functional on X = L 2 (D × (t 0 , t 1 )). We would like to study the sensitivity of the response function G with respect to the observation data T obs , with T, Q, T 0 obtained after the data assimilation from (6)- (9).
Assuming that from (15), (17) we obtain dG dT obs , δT obs = − m 0 P 2 | z=0 , δT obs L 2 (Ω×(t 0 ,t 1 )) . Therefore, the following statement is valid: Theorem 2.1. Let P 1 , P 2 ∈ Y , P 3 ∈ L 2 (Ω × (t 0 , t 1 )), P 4 ∈ L 2 (D) be the solutions of the following system of equations: Then the gradient of G with respect to T obs is given by The functions P 1 , P 2 ∈ Y are the solutions of a non-standard problem, being a coupled system of two differential equations (18) and (19) of the first order with respect to time with the mutually dependent initial and boundary values.
From (18)-(27) we obtain the algorithm to find the gradient of the response function G(T ): 1) Find the solution of the adjoint problem and set (ϕ * | t=0 , ϕ * | z=0 ) T .

2) Solve the Hessian equation
3) Find the solution of the problem 4) Find the gradient of the response function by Using this algorithm, one can estimate the sensitivity of the response functions after the data assimilation with respect to observation errors.

Numerical experiments for the Black Sea dynamics model
The numerical experiments have been performed using the three-dimensional numerical model of the Black and the Azov seas hydrothermodynamics developed at the Marchuk Institute of Numerical Mathematics, the Russian Academy of Sciences. This model is based on the splitting method [19]. The data assimilation problem [18] was considered for the sea surface temperature T obs with the aim to find the initial state T 0 and the heat flux Q.
The parameters of the domain are the following: σ-grid is 306×200×27 (in latitude, longitude, and depth, respectively). The first point of the "grid C" [20] has the coordinates 26.65 • E and 40.15 • N. The mesh sizes in x and y are equal to 0.05 and 0.036 degrees, respectively. The time step is ∆t = 2.5 minutes.
The sea surface temperature (SST) observational data were provided by the "See the Sea" satellite service being a part of the CKP "IKI-Monitoring", which collects and processes various data on the state of the Earth's surface and focuses on the satellite observations [21]. In this 7 experiment, the SST data for October 1, 2019 from the VIIRS spectrometer on the SNPP satellite were selected (at certain temporal points) as T obs , the data were recalculated on the model grid used for the numerical calculations. For Q (0) , we took the mean climatic flux obtained from the NCEP (National Centers for Environmental Prediction) reanalysis.
With the use of the model considered, calculations for the Black and the Azov area were carried out, with the sea surface assimilation algorithm working at certain time moments t 0 , with t 1 = t 0 + ∆t. The sensitivity of functionals of the optimal solution Q with respect to observation errors over the interval (t 0 , t 1 ) was numerically studied.
For G(T ), the following functional was considered: where F * (x, y, t) is the weight function related to the temperature on the sea surface z = 0. If the mean temperature of a specific region ω of the sea for z = 0 on the intervalt − τ ≤ t ≤t is considered, then F * is defined by with mes ω denoting the area of the region ω. In this case, (31) gives Formula (31) may be represented in the form The water area considered in the numerical experiments is shown in figure 1. Figure 2 presents the SST observation data used in the numerical experiment at a certain time moment.
The gradient of the response function G(T ) defined by (33) with respect to the observation data T obs on the sea surface is presented in figure 3, according to algorithm (28)-(30). Here ω = Ω, τ = ∆t,t = t 1 , α = β = 10 −5 . The sub-areas (in red) are revealed near the regions with a small depth, where the response function G(T ) is most sensitive to errors in the observation data during variational assimilation. The direct computation of the response function G(T ) by (33), with perturbations of the observation data T obs , has confirmed this result.

Conclusion
We have considered the problem of sensitivity of functionals of the optimal solution of the variational data assimilation of the sea surface temperature for the sea thermodynamics model. The variational data assimilation problem was formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is defined by the