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.
Introduction
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
e.g., to find unknown model parameters, such as initial
and/or boundary conditions, right-hand sides in the model equations (forcing
terms), and distributed coefficients, based on minimization of the cost function
related to observations. A necessary optimality condition reduces an optimal
control problem to an optimality system which involves the model equations,
the adjoint problem, and input data functions. The optimal solution depends
on the observation data, and for future forecasts it is very important to
study the sensitivity of the optimal solution with respect to observation
errors .
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 first studies of sensitivity of the response functions
after assimilation with the use of second-order adjoint were done by
for variational data assimilation problems aimed at restoration of
initial condition, where sensitivity with respect to model parameters was
considered. The equations of the forecast sensitivity to observations in a
four-dimensional (4D-Var) data assimilation were derived by . Based
on these results, a practical computational approach was given by
to quantify the effect of observations in 4D-Var data assimilation.
The issue of sensitivity is related to the statistical properties of the
optimal solution see. General
sensitivity analysis in variational data assimilation with respect to
observations for a nonlinear dynamic model was given by to
control the initial-value function. The dynamic formulation of the problem is
important because it shows different implementation options .
This paper is based on the results of and presents the
sensitivity analysis with respect to observations in variational data
assimilation aimed at restoration of unknown parameters of a dynamic model.
We should mention the importance of the parameter estimation problem itself.
A precise determination of the initial condition is very important in view of
forecasting; however, the use of variational data assimilation is not limited
to operational forecasting. In many domains (e.g., hydrology) the uncertainty
in the parameters is more crucial than the uncertainty in the initial
condition e.g.,. In some problems the quantity of interest can
be represented directly by the estimated parameters as controls. For example,
in the sea surface heat flux is estimated in order to
understand its spatial and temporal variability. The problems of parameter
estimation are common inverse problems considered in geophysics and in
engineering applications see. In the last years, interest
in the parameter estimation using 4D-Var is rising .
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 and constitute
a novelty of this paper. Of course, the initial condition function may also be
considered as a parameter; however, in our dynamic formulation we have
two equations for the model: one equation for describing an evolution of the
model operator (involving model parameters such as right-hand sides,
coefficients, boundary conditions, etc.) and another equation is considered
as an initial condition.
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.
Statement of the problem
We consider the mathematical
model of a physical process that is described by the evolution problem
∂φ∂t=F(φ,λ)+f,t∈(0,T)φ|t=0=u,
where the initial state u belongs to a Hilbert space X,
φ=φ(t) is the unknown function belonging to Y=L2(0,T;X) with
the norm ‖φ‖Y=(φ,φ)Y1/2=(∫0T‖φ(t)‖X2dt)1/2, F is a nonlinear operator mapping Y×Yp into Y, Yp is a Hilbert space (space of control parameters, or
control space), and f∈Y. Suppose that for a given u∈X,f∈Y, and
λ∈Yp, there exists a unique solution φ∈Y to
Eq. () with ∂φ∂t∈Y. The
function λ is an unknown model parameter. Let us introduce the cost function
2J(λ)=12(V1(λ-λb),λ-λb)Yp+12(V2(Cφ-φobs),Cφ-φobs)Yobs,
where λb∈Yp is a prior (background) function,
φobs∈Yobs is a prescribed function (observational
data), Yobs is a Hilbert space (observation space), C:Y→Yobs is a linear bounded observation operator, and V1:Yp→Yp and
V2:Yobs→Yobs are symmetric positive definite bounded
operators.
Let us consider the following data assimilation problem with the aim to
estimate the parameter λ: for a given u∈X, and f∈Y, find
λ∈Yp and φ∈Y such that they satisfy Eq. (),
and on the set of solutions to Eq. (), the functional J(λ)
takes the minimum value, i.e.,
∂φ∂t=F(φ,λ)+f,t∈(0,T)φ|t=0=u,J(λ)=infv∈YpJ(v).
We suppose that the solution of Eq. () exists. Let us note that the
solvability of the parameter estimation problems (or identifiability) has
been addressed, e.g., in and . To derive the optimality
system, we assume the solution φ and the operator
F(φ,λ) in Eqs. ()–() are regular enough, and
for v∈Yp find the gradient of the functional J with respect to
λ:
4J′(λ)v=(V1(λ-λb),v)Yp+(V2(Cφ-φobs),Cϕ)Yobs=(V1(λ-λb),v)Yp+(C*V2(Cφ-φobs),ϕ)Y,
where ϕ is the solution to the problem:
∂ϕ∂t=Fφ′(φ,λ)ϕ+Fλ′(φ,λ)v,ϕ|t=0=0.
Here Fφ′(φ,λ):Y→Y,Fλ′(φ,λ):Yp→Y are the Fréchet derivatives of F with
respect to φ and λ, correspondingly, and C* is the adjoint
operator to C defined by
(Cφ,ψ)Yobs=(φ,C*ψ)Y,φ∈Y,ψ∈Yobs.
Let us consider the adjoint operator (Fφ′(φ,λ))*:Y→Y and introduce the adjoint problem:
∂φ*∂t+(Fφ′(φ,λ))*φ*=C*V2(Cφ-φobs),φ*|t=T=0.
Then Eq. () with Eqs. () and () gives
7J′(λ)v=(V1(λ-λb),v)Yp-(φ*,Fλ′(φ,λ)v)Y=(V1(λ-λb),v)Yp-((Fλ′(φ,λ))*φ*,v)Yp,
where (Fλ′(φ,λ))*:Y→Yp is the adjoint operator
to Fλ′(φ,λ). Therefore, the gradient of J is
defined by
J′(λ)=V1(λ-λb)-(Fλ′(φ,λ))*φ*.
From Eqs. ()–() we get the optimality system (the
necessary optimality conditions; ):
∂φ∂t=F(φ,λ)+f,t∈(0,T),φ|t=0=u,∂φ*∂t+(Fφ′(φ,λ))*φ*=C*V2(Cφ-φobs),φ*|t=T=0,V1(λ-λb)-(Fλ′(φ,λ))*φ*=0.
We assume that the system (Eqs. –) has a unique
solution. The system (Eqs. –) may be considered as a
generalized model A(U)=0 with the state variable U=(φ,φ*,λ), and it contains information about observations. In
what follows we study the problem of the sensitivity of functionals of the
optimal solution to the observation data.
If the observation operator C is nonlinear, i.e., Cφ=C(φ),
then the right-hand side of the adjoint equation (Eq. ) contains
(Cφ′)* instead of C* and all the analysis presented below is
similar.
Sensitivity of functionals after assimilation
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 G(φ,λ), which is supposed
to be a real-valued function and can be considered as a functional on
Y×Yp. We are interested in the
sensitivity of G with respect to φobs, with φ and
λ obtained from the optimality system (Eqs. –).
By definition, the sensitivity is defined by the gradient of G with respect
to φobs:
dGdφobs=∂G∂φ∂φ∂φobs+∂G∂λ∂λ∂φobs.
If δφobs is a perturbation on φobs, we
get from the optimality system:
∂δφ∂t=Fφ′(φ,λ)δφ+Fλ′(φ,λ)δλ,δφ|t=0=0,-∂δφ*∂t-(Fφ′(φ,λ))*δφ*-(Fφφ′′(φ,λ)δφ)*φ*=(Fφλ′′(φ,λ)δλ)*φ*-C*V2(Cδφ-δφobs),δφ*|t=T=0,14V1δλ-(Fλφ′′(φ,λ)δφ)*φ*-(Fλλ′′(φ,λ)δλ)*φ*-(Fλ′(φ,λ))*δφ*=0,
and
dGdφobs,δφobsYobs=∂G∂φ,δφY+∂G∂λ,δλYp,
where δφ, δφ*, and δλ are the
Gâteaux derivatives of φ, φ*, and λ in the
direction δφobs (for example,
δφ=∂φ∂φobsδφobs).
To compute the gradient ∇φobsG(φ,λ),
let us introduce three adjoint variables P1∈Y, P2∈Y, and P3∈Yp. By taking the inner product of Eq. () by P1,
Eq. () by P2, and of Eq. () by P3 and adding them,
∂δφ∂t-Fφ′(φ,λ)δφ-Fλ′(φ,λ)δλ,P1Y+-∂δφ*∂t-(Fφ′(φ,λ))*δφ*-(Fφφ′′(φ,λ)δφ)*φ*-(Fφλ′′(φ,λ)δλ)*φ*+C*V2(Cδφ-δφobs),P2Y+V1δλ-(Fλφ′′(φ,λ)δφ)*φ*-(Fλλ′′(φ,λ)δλ)*φ*-(Fλ′(φ,λ))*δφ*,P3Yp=0.
Then, using integration by parts and adjoint operators, we get
δφ,-∂P1∂t-(Fφ′(φ,λ))*P1-(Fφφ′′(φ,λ)P2)*φ*-(Fλφ′′(φ,λ)P3)*φ*+C*V2CP2Y+δφ|t=T,P1|t=TX+δφ*,∂P2∂t-Fφ′(φ,λ)P2-Fλ′(φ,λ)P3Y+δφ*|t=0,P2|t=0X+δλ,V1P3-(Fφλ′′(φ,λ)P2)*φ*-(Fλλ′′(φ,λ)P3)*φ*-(Fλ′(φ,λ))*P1Yp16-δφobs,V2CP2Yobs=0.
Here we put
-∂P1∂t-(Fφ′(φ,λ))*P1-(Fφφ′′(φ,λ)P2)*φ*-(Fλφ′′(φ,λ)P3)*φ*+C*V2CP2=∂G∂φ,
and
V1P3-(Fφλ′′(φ,λ)P2)*φ*-(Fλλ′′(φ,λ)P3)*φ*-(Fλ′(φ,λ))*P1=∂G∂λ,P1|t=,T=0,∂P2∂t-Fφ′(φ,λ)P2-Fλ′(φ,λ)P3=0,P2|t=0=0.
Thus, if P1,P2, and P3 are the solutions of the following system of
equations
-∂P1∂t-(Fφ′(φ,λ))*P1-(Fφφ′′(φ,λ)P2)*φ*=(Fλφ′′(φ,λ)P3)*φ*-C*V2CP2+∂G∂φ,P1|t=T=0,∂P2∂t-Fφ′(φ,λ)P2-Fλ′(φ,λ)P3=0,t∈(0,T)P2|t=0=0,19V1P3-(Fφλ′′(φ,λ)P2)*φ*-(Fλλ′′(φ,λ)P3)*φ*-(Fλ′(φ,λ))*P1=∂G∂λ,
then from Eq. () we get
∂G∂φ,δφY+∂G∂λ,δλYp=δφobs,V2CP2Yobs,
and due to Eq. () the gradient of G is given by
dGdφobs=V2CP2.
We get a coupled system of two differential equations (Eqs.
and ) of the first order with respect to time, and
Eq. (). To study this nonstandard problem
(Eqs. –), we reduce it to a single operator equation
involving the Hessian of the original cost function.
Operator equation via Hessian and response function gradient
Let us denote the auxiliary variable v=P3 and rewrite the nonstandard
problem (Eqs. –) in an equivalent form:
∂P2∂t-Fφ′(φ,λ)P2=Fλ′(φ,λ)v,P2|t=0=0,-∂P1∂t-(Fφ′(φ,λ))*P1-(Fφφ′′(φ,λ)P2)*φ*=(Fλφ′′(φ,λ)v)*φ*-C*V2CP2+∂G∂φ,P1|t=T=0,23V1v-(Fφλ′′(φ,λ)P2)*φ*-(Fλλ′′(φ,λ)v)*φ*-(Fλ′(φ,λ))*P1=∂G∂λ.
Here we have three unknowns: v∈Yp,P1, and P2∈Y. Let us write
Eqs. ()–() in the form of an operator equation for v.
We define the operator H, which acts on w belonging to Yp,
by the successive solution of the following problems:
∂ϕ∂t-Fφ′(φ,λ)ϕ=Fλ′(φ,λ)w,t∈(0,T)ϕ|t=0=0,-∂ϕ*∂t-(Fφ′(φ,λ))*ϕ*-(Fφφ′′(φ,λ)ϕ)*φ*=(Fλφ′′(φ,λ)w)*φ*-C*V2Cϕ,ϕ*|t=T=0,26Hw=V1w-(Fφλ′′(φ,λ)ϕ)*φ*-(Fλλ′′(φ,λ)w)*φ*-(Fλ′(φ,λ))*ϕ*.
Here λ,φ, and φ* are the solutions of the optimality
system (Eqs. –). Then Eqs. ()–()
are equivalent to the following equation in Yp:
Hv=F
with the right-hand side F defined by
F=∂G∂λ+(Fλ′(φ,λ))*ϕ̃*,
where ϕ̃* is the solution to the adjoint problem:
-∂ϕ̃*∂t-(Fφ′(φ,λ))*ϕ̃*=∂G∂φ,t∈(0,T)ϕ̃*|t=T=0.
It is easily seen that the operator H defined by
Eqs. ()–() is the Hessian of the original functional
J considered on the optimal solution λ of the problem
(Eqs. –): J′′(λ)=H. Under the
assumption that H is positive definite, the operator equation
(Eq. ) is correctly and everywhere solvable in Yp,
i.e., for every F there exists a unique solution v∈Yp and
‖v‖Yp≤c‖F‖Yp,c=const>0.
Therefore, under the assumption that J′′(λ) is positive definite on
the optimal solution, the nonstandard problem (Eqs. –)
has a unique solution P1,P2∈Y, and P3∈Yp.
Based on the above consideration, we can formulate the following algorithm to
compute the gradient of the response function G:
For ∂G∂λ∈Yp, and ∂G∂φ∈Y solve the adjoint problem-∂ϕ̃*∂t-(Fφ′(φ,λ))*ϕ̃*=∂G∂φ,ϕ̃*|t=T=0and putF=∂G∂λ+(Fλ′(φ,λ))*ϕ̃*.
Find v by solvingHv=Fwith the Hessian of the original functional J defined
by Eqs. ()–().
Solve the direct problem∂P2∂t-Fφ′(φ,λ)P2=Fλ′(φ,λ)v,t∈(0,T)P2|t=0=0.
Compute the gradient of the response function asdGdφobs=V2CP2.
Equation () allows us to estimate the sensitivity of the
functionals related to the optimal solution after assimilation, with respect
to observation data.
Remark 1. In the above consideration, to show the solvability, we
have assumed that the direct and adjoint tangent linear problems of the form
∂ϕ∂t-Fφ′(φ,λ)ϕ=f,t∈(0,T)ϕ|t=0=0,-∂ϕ*∂t-(Fφ′(φ,λ))*ϕ*=g,t∈(0,T)ϕ*|t=T=0
with f,g∈Y have the unique solutions ϕ,ϕ*∈Y.
Remark 2. The analysis presented above is based on the hypothesis
that the initial state of the system under observation is known, and that it
is only model parameters (boundary conditions, forcing terms, distributed
coefficients, etc.) that are to be determined from the observations. Often, a
more realistic situation would be one where the assimilation is intended to
determine both the initial conditions of the system and, in addition, model
parameters . The sensitivity analysis can be applied as well to
such a situation. To consider the joint state and parameter estimation problem,
we should use the results both of this paper and of the previous one
. In this case we need to introduce an additional term related to
the initial condition into the cost function (Eq. ) to find
simultaneously u and λ. The optimality system
(Eqs. –) will be supplemented by an additional equation
related to the gradient of the cost function with respect to u. The Hessian
in this case is a 2 × 2 operator matrix, acting on the augmented
vector U=(u,λ)T, and all the derivations are made similarly, being,
however, more cumbersome and lengthy.
Below we give a proof-of-concept analytic example to show how the algorithm
(Eqs. –) works. Then, as an application, we consider
a variational data assimilation problem for a sea thermodynamics model.
Proof-of-concept analytic example
Let us consider a simple evolution problem for the ordinary differential
equation
dφdt+aφ=λg,t∈(0,T)φ|t=0=u,
where u∈R; a,λ∈R; g=g(t)≥0. Here, in
the notations of Sect. 2, we have X=R, Y=L2(0,T), and
F(φ,λ)=-aφ+λg. Let us formulate the data
assimilation problem to find the parameter λ if we have observation
data for φ at the end of the time interval t=T. We need to minimize
the cost function
J(λ)=infv∈RJ(v),
where J(v)=12|φ̃|t=T-φobs|2, and φ̃ is the
solution to Eq. () with λ=v. Thus, here we have Yp=R, V1=0, V2=1, and Cφ=φ|t=T. In this case, the optimality system (Eqs. –) has the
form:
dφdt+aφ=λg,t∈(0,T)φ|t=0=u,dφ*dt-aφ*=0,t∈(0,T)φ*|t=T=φ|t=T-φobs,(g,φ*)=∫0Tg(t)φ*(t)=0.
It is easy to see that the problem of data assimilation
(Eqs. –) has a unique solution
λ=λopt=φobs-φ0φ1,
where φ0=u0e-aT, φ1=∫0Te-a(T-t′)g(t′)dt′.
Indeed, if λ has the form (Eq. ), the solution of the
problem (Eq. ) satisfies φ|t=T=φobs, and
the functional J from Eq. () attains its minimal value J=0. In
this case φ*=0, and the optimality system
(Eqs. –) is satisfied. Let us consider the response function in the form
Gφ,λ=∫0Tφ(t)dt.
Let a≠0. After assimilation, taking into account the solution of the
problem (Eq. ), we have
G(φ,λ)=ua1-e-aT+λopta∫0Tg(t)dt-φ1,
where λopt is given by Eq. (). Then, by differentiation
of G with respect to φobs we have the gradient
dGdφobs=1aφ1∫0Tg(t)dt-φ1.
Let us now apply the algorithm (Eqs. –) to compute
the gradient of the function G. Since ∂G∂φ=1, and (Fφ′(φ,λ))*=-a, then on the first step of the
algorithm, we solve the problem (Eq. ) and get the solution
ϕ̃*(t)=1a1-e-a(T-t).
Taking into account that ∂G/∂λ=0 and
(Fφ′(φ,λ))*ϕ̃*=(g,ϕ̃*), we get
F=(g,ϕ̃*), i.e.,
F=∫0Tgϕ*̃dt=1a∫0Tg(t)dt-φ1.
On the second step of the algorithm, one needs to solve the equation
Hv=F with the Hessian H defined by
Eqs. ()–(). Since all the second-order derivatives of
F(φ,λ) equal zero, then it is easily seen that H
in this case is the operator of multiplication by the scalar
H=∫0Tg(t)ψ|t=Te-a(T-t)dt=(ψ|t=T)2,
where ψ(t) is the solution of the problem (Eq. ) with u=0,
and λ=1. Then, after the second step of the algorithm we get
v=H-1F=(ψ|t=T)-2F.
On the third step of the algorithm, we need to solve the problem
(Eq. ). Since Fλ′(φ,λ)=g, the solution
of this problem has the form P2(t)=vψ(t). Finally, using
Eq. (), we get the gradient of G with respect to
φobs:
dGdφobs=P2|t=T=vψ(T)=ψ(T)Fψ2(T)=Fψ(T).
Moreover, since φ1=ψ(T), then from Eqs. () and
() we have
dGdφobs=1aφ1∫0Tg(t)dt-φ1.
Thus, the gradient obtained by the algorithm (Eqs. –)
exactly coincides with the value of the gradient obtained in Eq. ()
by direct differentiation, which is the expected result.
Data assimilation problem for a sea thermodynamics model
Consider the sea thermodynamics problem in the form :
Tt+(U‾,Grad)T-Div(a^T⋅GradT)=fTinD×(t0,t1),T=T0fort=t0inD,-νT∂T∂z=QonΓS×(t0,t1),∂T∂n=0onΓw,c×(t0,t1),U‾n(-)T+∂T∂n=QTonΓw,op×(t0,t1),∂T∂n=0onΓH×(t0,t1),
where T=T(x,y,z,t) is an unknown temperature function, t∈(t0,t1),
(x,y,z)∈D=Ω×(0,H), Ω⊂R2, H=H(x,y) is the
function of the bottom relief, Q=Q(x,y,t) is the total heat flux,
U‾=(u,v,w), a^T=diag((aT)ii),
(aT)11=(aT)22=μT, (aT)33=νT, and fT=fT(x,y,z,t)
are given functions. The boundary of the domain Γ≡∂D is
represented as a union of four disjoint parts ΓS,
Γw,op, Γw,c, and ΓH, where
ΓS=Ω (the unperturbed sea surface),
Γw,op is the liquid (open) part of vertical lateral
boundary, Γw,c is the solid part of the vertical lateral
boundary, ΓH is the sea bottom,
U‾n(-)=(|U‾n|-U‾n)/2, and
U‾n is the normal component of U‾. The other
notations and a detailed description of the problem statement can be found in
.
Equation () can be written in the form of an operator equation:
Tt+LT=F+BQ,t∈(t0,t1),T=T0,t=t0,
where the equality is understood in the weak sense, namely,
(Tt,T^)+(LT,T^)=F(T^)+(BQ,T^)∀T^∈W21(D),
in this case L, F, and B are defined by the following relations:
(LT,T^)≡∫D(-TDiv(U‾T^))dD+∫Γw,opU‾n(+)TT^dΓ+∫Da^TGrad(T)⋅Grad(T^)dD,F(T^)=∫Γw,opQTT^dΓ+∫DfTT^dD,(Tt,T^)=∫DTtT^dD,(BQ,T^)=∫ΩQT^|z=0dΩ,
and the functions a^T, QT, fT, and Q are such that
Eq. () makes sense. The properties of the operator L were studied
by .
Due to Eq. (), Eq. () is considered in Y*=L2(t0,t1;(W21(D))*), and the operator B:L2(Ω×(t0,t1))→Y*
maps the function Q∈L2(Ω×(t0,t1)) into the function BQ∈Y* such that (BQ,T^)=∫ΩQT^|z=0dΩ, ∀T^∈W21(D).
Therefore, BQ is a linear and bounded functional on L2(0,T;W21(D)).
Consider the data assimilation problem for the sea surface temperature
see. Suppose that the function Q∈L2(Ω×(t0,t1)) is unknown in Eq. (). Let Tobs(x,y,t) also be the
function on Ω‾≡Ω∪∂Ω obtained for
t∈(t0,t1) by processing the observation data, and this function in
its physical sense is an approximation to the surface temperature function on
Ω, i.e., to T|z=0. We suppose that Tobs∈L2(Ω×(t0,t1)), but the function Tobs may not possess
greater smoothness and hence it cannot be used for the boundary condition on
ΓS. We admit the case when Tobs is defined only on
some subset of Ω×(t0,t1) and denote the indicator
(characteristic) function of this set by m0. For definiteness sake, we
assume that Tobs is zero outside this subset.
Consider the data assimilation problem for the surface temperature in the
following form: find T and Q such that
Tt+LT=F+BQinD×(t0,t1),T=T0,t=t0J(Q)=infvJ(v),
where
J(Q)=α2∫t0t1∫Ω|Q-Q(0)|2dΩdt52+12∫t0t1∫Ωm0|T|z=0-Tobs|2dΩdt,
and Q(0)=Q(0)(x,y,t) is a given function, and α=const>0.
For α>0 this variational data assimilation problem has a unique
solution. The existence of the optimal solution follows from the classic
results of the theory of optimal control problems , because it is
easy to show that the solution to Eq. () continuously depends on
the flux Q (a priori estimates are valid in the corresponding functional
spaces), the functional J is weakly lower semi-continuous, and the space of
admissible controls L2(Ω×(t0,t1)) is weakly compact.
For α=0 the problem does not always have a solution, but, as was shown
by , there is unique and dense solvability, and it allows one
to construct a sequence of regularized solutions minimizing the functional,
which is related to a sequence of coefficients αn, with αn→0 when n→∞.
The optimality system determining the solution of the formulated variational
data assimilation problem according to the necessary condition
gradJ=0 has the form:
53Tt+LT=F+BQinD×(t0,t1),T=T0,t=t0,54-(T*)t+L*T*=Bm0(Tobs-T)inD×(t0,t1),T*=0,t=t1,55α(Q-Q(0))-T*=0onΩ×(t0,t1),
where L* is the operator adjoint to L.
Here the boundary-value function Q plays the role of λ from
Sect. 2, φ=T, the operator F has the form F(T,Q)=-LT+BQ, and
FT′=-L,FQ′=B. Since the operator F(T,Q) is bilinear in this case, the
Hessian H acting on some ψ∈L2(Ω×(t0,t1)) is
defined by the successive solution of the following problems:
∂ϕ∂t+Lϕ=Bψ,t∈(t0,t1)ϕ|t=t0=0,-∂ϕ*∂t+L*ϕ*=-Bm0ϕ,t∈(t0,t1)ϕ*|t=t1=0,Hψ=αψ-B*ϕ*.
To illustrate the above-presented theory, we consider the problem of
sensitivity of functionals of the optimal solution Q to the observations
Tobs. Let us introduce the following functional (response
function):
G(T)=∫t0t1dt∫Ωk(x,y,t)T(x,y,0,t)dΩ,
where k(x,y,t) is a weight function related to the temperature field on
the sea surface z=0. For example, if we are interested in the mean
temperature of a specific region of the sea ω for z=0 in the
interval t‾-τ≤t≤t‾, then as k we take the
function
k(x,y,t)=1/(τmes ω)if(x,y)∈ω,t‾-τ≤t≤t‾0else,
where mesω denotes the area of the region ω. Thus,
Eq. () is written in the form:
G(T)=1τ∫t‾-τt‾dt1mes ω∫ωT(x,y,0,t)dΩ.
Equation () represents the mean temperature averaged over the time
interval t‾-τ≤t≤t‾ for a given region
ω. The functionals of this type are of most interest in the theory of
climate change .
In our notations, Eq. () may be written as
G(T)=∫t0t1(Bk,T)dt=(Bk,T)Y,Y=L2(D×(t0,t1)).
We are interested in the sensitivity of the functional G(T), obtained for
T after data assimilation, with respect to the observation function
Tobs.
By definition, the sensitivity is given by the gradient of G with respect
to Tobs:
dGdTobs=∂G∂T∂T∂Tobs.
Since ∂G∂T=Bk, then, according to the theory
presented in Sect. 4, to compute the gradient (Eq. ) we need to
perform the following steps:
For k defined by Eq. () solve the adjoint problem
-∂ϕ̃*∂t+L*ϕ̃*=Bk,t∈(t0,t1)ϕ̃*|t=t1=0
and put Φ=B*ϕ̃*.
Find χ by solving Hχ=Φ with the Hessian
defined by Eqs. ()–().
Solve the direct problem
∂P2∂t+LP2=Bχ,t∈(t0,t1)P2|t=t0=0.
Compute the gradient of the response function as
dGdTobs=m0P2|z=0.
Equation () allows us to estimate the sensitivity of the
functionals related to the mean temperature after data assimilation, with
respect to the observations on the sea surface.
Numerical example for the Baltic Sea dynamics model
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
and supplied with the assimilation procedure for
the surface temperature Tobs with the aim to reconstruct the heat
fluxes Q.
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 σ-grid is 336×394×25 (latitude,
longitude, and depth, respectively). The first point of the “grid C”
has the coordinates 9.406∘ E and 53.64∘ N.
The mesh sizes in x and y are constant and equal to 0.0625 and 0.03125
degrees, respectively. The time step is Δt=5 min. The initial condition for the
whole model, including T0, was chosen in the following way: the model was
started running with zero initial conditions and ran with atmospheric forcing
obtained from reanalysis of about 20 years, and after that the result of
calculation was taken as an initial condition for further running of the
model. The assimilation procedure worked only during some time windows. To
start the assimilation procedure for the heat flux estimation, the initial
condition was taken as a model forecast for the previous time interval.
The Baltic Sea daily-averaged nighttime surface-temperature data were used
for Tobs. These are the data of the Danish Meteorological Institute
based on measurements of radiometers (AVHRR, AATSR, and AMSRE) and
spectroradiometers (SEVIRI and MODIS) . Data interpolation
algorithms were used to convert observations on computational
grid of the numerical model of the Baltic Sea thermodynamics. For each time
step the heat flux was determined at each surface point; therefore, the
number of scalar parameters to be determined were equal to the number of
scalar observations.
The mean climatic flux obtained from the NCEP (National Centers for
Environmental Prediction) reanalysis was taken for Q(0). We need to
mention that Q(0) has a physical meaning here, it is not only an initial
guess but also a parameter calculated from atmospheric data and taken in the
model for temperature boundary condition on the sea surface when the model
runs without the assimilation procedure.
Using the hydro-thermodynamics model mentioned above, which is supplied with
the assimilation procedure for the surface temperature Tobs, we
have performed calculations for the Baltic Sea area where the assimilation
algorithm worked only at certain time moments t0; in this case
t1=t0+Δt.
The aim of the experiment was the numerical study of the sensitivity of
functionals of the optimal solution Q to observation errors in the interval
(t0,t1).
Implementing the assimilation procedure, we considered a system of the form
in Eqs. ()–(), where Eqs. ()–()
mean the finite-dimensional analogues of the corresponding problems
. For the statement of a data assimilation problem we introduce
the cost function (Eq. ) with a regularization parameter
α, which weights the squared difference |Q-Q(0)|2. Since in all
numerical experiments α was chosen very small, the impact of the first
term in the functional was also small, and therefore Q was different from
Q(0).
We use here the SI units, namely K (kelvin) is used for temperature,
m s-1 for velocities, and mK s-1 for the heat flux Q. The parameter
α is defined as s2 m-2 to give both terms in
Eq. () the same dimension. It is easily seen that in this case
the units of the gradient dGdTobs from
Eq. () are defined as m-2 s-1.
Let us present some results of numerical experiments. The calculation results for t0=50 h (600 time steps for the model) are presented in Fig. 1 showing the
gradient of the functional G(T) defined by Eq. () and related to
the mean temperature after data assimilation, with respect to the
observations on the sea surface, according to
Eqs. ()–(). Here ω=Ω, τ=Δt, t‾=t1, and α=10-5 s2 m-2.
The gradient of the functional G(T) (m-2 s-1).
We can see the sub-areas (in red) in which the functional G(T) is most
sensitive to errors in the observations during assimilation. The largest
values of the gradient of G(T) correspond to the points x,y lying near
the regions with a small depth (cf. sea topography, Fig. 2). One explanation
of this phenomenon may be the fact that in the areas with depths of about
50 m, rapid convection occurs in the upper mixed layer. With the
assimilation of the surface temperature, information is transmitted faster to
shallower depths, which in turn contributes to a higher sensitivity to data
in these places, in contrast to deeper regions.
Baltic Sea topography (m).
Remark 3. We use the discretize-then-optimize approach, and for numerical experiments
all the presented equations are understood in a discrete form, as
finite-dimensional analogues of the corresponding problems, obtained after
approximation. This allows us to consider model equations as a perfect model,
with no approximation errors. Therefore, the accuracy of the sensitivity
estimates given by Eqs. ()–() are determined
by the accuracy of solving the Hessian equation Hχ=Φ (step 2
of the algorithm). Due to Eqs. ()– (), this equation
is equivalent to a linear data assimilation problem, and an approximate
solution to the minimization problem is obtained by an iterative procedure.
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.
Conclusions
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.
Data availability
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
(http://marine.copernicus.eu/services-portfolio/access-to-products/).
Competing interests
The authors declare that they have no conflict of
interest.
Special issue statement
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.
Acknowledgements
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
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