6.3.4: 3-dimensional variational assimilation (3D-Var) and its relationship to OI
We saw in section 6.2 that there is an important equivalence between the formulation of the optimal analysis of a scalar by minimizing the analysis error variance (finding the optimal weights through a least squares approach), and solving the same problem through a variational approach (finding the analysis that minimizes a cost function measuring its distance to the background and to the observations). The same is true when the analysis involves a full 3-dimensional field, as in section 6.3.1. In the derivation of Optimal Interpolations, we found the optimal weight matrix W that minimized the analysis error covariance (a matrix). Lorenc (1986) showed that this is equivalent to a variational assimilation problem: Find the optimal analysis xafield that minimizes a (scalar) cost function. The cost function must be defined as the distance between x and the background xb, weighted by the inverse of the background error covariance, plus the distance to the observations yo:
(1)
The minimum of J(x) is attained for x=xa,i.e., the analysis is given by the solution of
(2)
As we did in 6.3.1, we can expand the second term of equation (1), the observational differences, assuming that the analysis is a close approximation to the truth and therefore to the observations, and linearizing H around the background value:
(3)
Replacing (3) into (1) we get
(4)
Or, expanding the products, using the rules for transpose of matrix products:
(5)
The cost function is a quadratic function of the analysis increments and therefore we can use the remark d) of the previous section. Given a quadratic function , where A is a symmetric matrix, d is a vector and c a scalar, the gradient is given by . The gradient of the cost function J wrt is then
(6)
We now set to ensure that J is a minimum, and obtain an equation for
(7)
or
(8)
Formally, this is the solution of the variational (3D-Var) analysis problem, but in practice the solution is obtained through minimization algorithms for J(x) using iterative methods for minimization such as conjugate gradient or quasi-Newton.
Note that the control variable for the minimization (i.e., the variable with respect to which we are minimizing the cost function J) is now the analysis, not the weights. The equivalence between the minimization of the analysis error variance (finding the optimal weights through a least squares approach), and the variational cost function approach (finding the optimal analysis that minimizes the distance to the observations weighted by the inverse of the error variance) is an important property.
We now demonstrate the equivalence of this solution to the OI analysis solution obtained in the previous section (Lorenc (1986). We have to show that the weight matrix that multiplies the innovation in (8) is the same as the weight matrix obtained with OI.
The 3D-Var weight matrix in equation (8) can be transformed, using the rules for inverse and transpose of matrix products, in the following way:
(9)
The last equality is the optimal weight matrix for Optimal Interpolation, demonstrating the formal equivalence of 3D-Var and OI.
Although 3D-Var and Optimal Interpolation have been shown to formally solve the same problem, there are important differences in the method of solution. As indicated before, OI in practice requires introducing a number of approximations, and solving the analysis locally, grid point by grid point, or small volume by small volume (Lorenc, 1981). This requires using a "radius of influence" and selecting only the stations closest to the grid point or volume being analyzed. The background error covariance matrix has to be also locally approximated.
3D-Var has several important advantages with respect to OI, because the cost function (1) is minimized using global minimization algorithms, and as a result it avoids making the approximations required by OI (Parrish and Derber, 1992, Derber et al, 1991, Courtier et al, 1998, Rabier et al, 1998, Andersson et al, 1998):
a)There is no data selection, all available data is used simultaneously, and thus avoiding the jumpiness in the boundaries between regions which have selected different observations.
b)The background error covariance matrix for 3D-Var, although simplified, can also be more general and "globally" defined, rather than a local approximation. In particular, in recent years most centers have adopted the "NMC method" for estimating the forecast error covariance:
(10)
The forecast or background error covariance is estimated as the average over many cases of the difference between two short-range model forecasts. The actual magnitude is then appropriately scaled. In this approximation, rather than estimating the structure of the forecast error covariance from differences with rawinsondes (Thiebaux and Pedder, 1987, Hollingsworth and Lonnberg, 1986), the model-forecast differences themselves provide a multivariate global forecast difference covariance. This forecast covariance, although strictly speaking is the covariance of the forecast differences and not necessarily the forecast errors, has been shown to produce better results than previous estimates. One of the reasons is that the rawinsonde network does not have enough density to allow a proper estimate of the global structures (Parrish and Derber, 1992, Rabier et al, 1998).
c)It is possible to add constraints to the cost function without increasing the cost of the minimization. For example, Parrish and Derber (1992) included a "penalty" term in the cost function (1) forcing simultaneously the analysis increments to approximately satisfy the global quasi-geostrophic balance equation. These replaced the previous approach of performing first an OI analysis and then follow it with a nonlinear normal mode initialization. As a result, the NCEP global model spin up (indicated for example by the change of precipitation over the first 12 hours of integration) was reduced by at least an order of magnitude.
d)It is possible to incorporate important nonlinear relationships between observed variables and model variables in the H operator in the minimization of the cost function (1) by performing "inner" iterations with the linearized H observation operator kept constant and "outer" iterations in which it is updated. This is harder to do in the OI approach.
e)The introduction of 3D Var has also allowed performing 3D variational assimilation of radiances (Derber and Wu, 1998). In this approach, there is no attempt to perform retrievals, and instead, each satellite sensor is taken as an independent observation. As a result, for each satellite observation spot, even if some channel measurements are rejected because of cloud contamination, others may still be used. In addition, because all the data is assimilated simultaneously, information from one channel at a certain location can influence the use of satellite data at a different geographical location.