1. Understanding the 3DVAR in a Practical Way
Lecture One:
Understanding the 3DVAR in a Practical Way
Jidong Gao Center for Analysis and Prediction of Storms University of Oklahoma

2. A simple example
A variational data assimilation algorithm can involve:
Variational calculus;
inverse problem theory;
estimation theory;
optimal control theory; and
various computational theories
Can we present a relatively easy but comprehensive outline of variational data assimilation without being annoyed by all those complicated theories ?
Let’s begin with a very simple example, that involves all major aspects of the variational data assimilation.

3. Consider the temperature in our class room
Consider the temperature in our class room. This room is air-conditioned. According to room characteristics, power of AC, etc., we estimate (forecast) that the room temperature should be Tb=180C. We call it background T. This cannot be perfect, since there are various random disturbances, for example, the door of the room can be opened randomly, the air conditioner does not work perfectly according to spec., etc. Let the standard deviation of background σb=10 c.
On the other hand, suppose there is a thermometer installed in this room. The reading of the thermometer is, say, To=20o C. We call it observation. The reading is not perfect either, Let the standard deviation of the measured temperature from the thermometer be σo=0.50 c.
The question is, what should be the best estimate of the room temperature ?

4. The simple way is to make a weighted average,
We know that the above weights are optimal weights determined by the minimum variance estimation theory, as
you have learnt from Drs. Carr and Xue’s lectures.

5. Based on Bayesian estimation, or maximum likelihood estimate theory, we can derive that the best solution can be obtained by minimizing the following cost function,
The minimization of J is the solution of the equation
It should be easy to find that this is the maximum likelihood estimate. The answer is the same as the weighted average derived.

6. Why the analysis is more close to the observation?
Posterior error
You may ask how good this estimate is? This is actually a crucial question. In the world of data assimilation, the estimate of the accuracy of result is of the same importance as the result itself !! By using error variance of the estimate by defining,
Go through some algebraic manipulations, yield,
Obviously,
and
Why the analysis is more close to the observation?
(Bkgrd: 18oC, Obs: 20oC and Analysis: 19.6oC)

7. Comments:
This simple example shows how to solve the data assimilation problem in the variational way. We need to minimize the cost function J. In order to do this, we have to calculate the gradient of cost function with respect to analysis variable T, dJ/dT, and set to zero.
For real 3DVAR/4DVAR, a scalar T is replaced by a vector, the two scalar error variances are replaced by error covariance matrices. The procedures to solve them are quite similar.

8. The 3DVAR Formulation A general cost function is defined as
Goal: Find the analysis state x that minimizes J.
Where,
x is the NWP model state;
xb is the background state;
B is the background error covariance matrix;
R is the observation error covariance matrix;
yo is the observation vector;
y = H(x) is the operator that brings the model state x to the
observational state variables.
For 4DVAR, H includes the full prediction model
Jc represents the dynamic constraints.

9. What should we know before we can solve the 3DVAR problem ?
Cost function J measures the fit of analysis x to background xb and observation yo, and dynamic constraints (Jc) are also satisfied in some way.
What really is J? a vector, a numerical number, or a matrix ?
What we should know before minimization ?
B: unknown, but can be estimated. It is a priori, 3DVAR is thus also called a priori estimate. B is vitally important!! It decides how observations spread to nearby grid points. However, B is also most difficult one to get. Its dimension is huge and its inverse is impossible. Simplification is necessary…And this is a very active research area for data assimilation in the past 30 years. Among the data assimilation community, there are two basic methods: 1) assume B to be diagonal. This can be done only in spectral space (Parrish and Derber 1992). However, this approximation is not acceptable for grid point models. 2) B is modeled by parameterized formulism. This reduces the dimension and the inversion of B can be avoided through judiciary choices of control variables (Huang 2000, Purser et al. 2003a, b).
R: observation error covariance matrix, also includes the representative error, usually diagonal, can be decided “off-line” based on each type of observation used.
xb: background state usually comes from previous forecast.
yo: obs. Every new type of observation may have positive, or negative impact to the whole 3DVAR system. Active research area: OU, radar data; Wisconsin, satellite data.
Jc: One, or more equation constraints. Also can be a good research topic.
y=H(x): forward observational operator (including interpolation operator). Also a lot of research in this area.
With all of the above being readily taken care of and coded, we can begin to think about the minimization.

10. a scalar Iteration loop
The procedure for minimization of J by iteration
First Guess
x=(u,v,p,t…)
Minimization algorithm
Find the new control
Variable x=(u,v,p,t…)
Calculate cost function J
Iteration loop
a scalar No
Calculate gradient of
cost function, dJ/dx
Convergence criterion
YES
Output x=(u,v,p,t…)
Model forecast

11. From the flow chart, there are three important tasks for the minimization procedure:
Calculate the cost function.
Calculate the gradient of the cost function.
Select a minimization algorithm.
The first task was already discussed; the second task usually requires the use of the adjoint technique; and the third one is also crucial! To develop an efficient minimization algorithm is an active research topic in applied mathematics for the past 40 years...
For us we just need to pick up a good one
and know how to use it. You may find one from book “Numerical Recipe” on-line:

12. Simple example of how to use minimization algorithm
To solve this in a variational way, define a cost function first, let it to be,

13. Then, we need the gradient of the cost function, it’s a vector of 3 variables,
Subroutine FCN(N, X, F)
integer::N; real::X(N), F
F=(x(1)+x(2)+x(3)-6)**2+(2*x(1)-x(2)-x(3)-3)**2 &
+(x(1)+2*x(2)+x(3)-8)**2
return; end
Subroutine GRAD(N, X, G)
integer::N; real::X(N), G(N)
G(1)=6*x(1)+x(2)-20;
G(2)=x(1)+6*x(2)+4*x(3)-19
G(3)=4*x(2)+3*x(3)-11

14. Program main
Integer::n
parameter(n=3)
Integer:: I, maxfn
Real:: Fvalue, X(n), G(n), Xguess(n)
Real:: dfred, gradtl
External:: fcn, grad
Do i=1,n; Xguess(i) = 0.0; end do ! Provide the first guess
dfred= ! Accuracy criterion about cost function
Gradtl=1.0E ! Accuracy criterion about the norm of
! the gradient
Maxfn=50
Call umcgg(fcn, grad, n, xguess, gradtl, maxfn, dfred, x, g, fvalue) !algorithm
print*, ‘x=‘,x(1), x(2), x(3)
end
The minimization algorithm, like this umcgg, one of the conjugate gradient methods, requires you to provide subroutines, FCN and GRAD, for calculating J and its gradient. We can quickly get the answer: (x, y, z)= (3, 2, 1) after only 2, or 3 iterations. Because the problem is simple, you do not need many iterations.

15. Comments:
All variational data assimilation algorithms work in similar ways; you define a cost function, get its gradient, and feed them into a minimization algorithm along with a first guess of the solution.
But, large, real problems are not that easy ti implement. One of the outstanding problem is how to calculate the gradient of cost function efficiently. This brings out the adjoint technique, which allows us to efficiently calculate transposes of large matrices found in the gradient calculation.
What is the adjoint, and how to use it?

16. It’s only a mathematical tool to help you to get the gradient of the cost function.
In R. M. Errico paper What is an adjoint model?, It is said “…the adjoint is used as a tool for efficiently determining the optimal solutions. Without this tool, the optimization problem (including minimization and maximization) could not be solved in a reasonable time for application to real-time forecasting”. This is a good statement. Then what does it mean exactly ?

17. A simple maximization example
Suppose we have a fence with 200m, and want to use it to make a rectangular yard with a maximum possible area. How can we do it ? Let x, and y are the long and wide of the yard respectively, then we can define a cost function, like,
and the gradient of the cost function,
The ‘multiplier’ is equivalent to the ‘adjoint variable’, the role of this parameter is help to calculate the gradient of cost function!