NWPSAF 1D-Var User Manual

Here the observations, y, have error covariances R; B is the error covariance matrix of the a priori (background) profile  x₀; and y(x) is the observed radiance that would result for a given atmospheric state x.

Rodgers (1976) gives three variations of iterative solution to the minimisation of J(x), two of which are:

and

where  x_n is the n^th estimate of the atmospheric profile (the background profile being the 0^th estimate) and .

Equation (2a) is the more efficient version when there are more profile elements than channels (i.e., R is smaller than B). However, when there are more channels than profile elements the form in Equation (2b) should be used.

In Rodgers's paper, Eqn. 2a above is Eqn 101, while Eqn. 2b is Eqn. 100. Therefore in this code, when the minimisation is to be done using the former method NWPSAF_Minimise_101 is called while NWPSAF_Minimise_100 is called in the latter case.

Equation (2b) also differs from (2a) in that it can also be extended to allow for extra terms in the cost function (e.g., to penalise solutions with supersaturated levels or with superadiabatic lapse rates) and can be modified to become the Marquardt-Levenberg descent algorithm (Rodgers, 2000). This case is dealt with in the NWPSAF_Minimise_100ML subroutine

The Marquardt-Levenberg version of Eqn. (2b) with the additional cost function terms is then:

Here J' is the first differential of the additional cost function with respect to x, evaluated at  x_n, and J'' is the second differential. Thus, J' is a vector with the same length as x_n and J'' is a matrix with the same size as B.

In the Marquardt-Levenberg minimisation method, the value of   is varied depending on the non-linearity of the problem and the proximity to the desired solution. When , the equation reduces to the Newtonian inverse Hessian method of Eqn (2b). In non-linear cases, it is possible that the step taken by using the Newtonian method will result in an increased cost function. In this case the value of    is increased (thereby reducing the step size) until a point is reached where the cost function decreases (as   , Eqn. (3) becomes the method of steepest descent). After an improved solution (i.e., lower cost function) has been found, the value of    may once again be increased to allow larger steps in the next iteration.

The NWPSAF_Minimise subroutine allows for either the Newtonian or Marquardt-Levenberg algorithms to be used and for the inclusion of additional cost function terms.