Linear Inverse Calculation

Without loss of generality, let us consider the fine tuning of heat fluxes. Assuming no heat exchange with the continents, (43) can be integrated to produce an estimate of the vertically and zonally integrated meridional heat transport

where cartesian coordinates have been adopted for notational convenience. In (45), is a northern latitude where is specified as a boundary condition. In addition to latitude, the meridional heat transport also depends on the particular choice of parameters entering the bulk formulas. We explicitly denote this dependence by

Let us denote by the current choice of parameters in the bulk formulas. The goal is to find small adjustments so that agrees in some sense with oceanographic measurements , . For the cases considered here, , leading to an underdetermined problem. We make the following assumptions:

1. The bulk formula parameter errors are normally distributed and uncorrelated, with an error covariance matrix denoted by .

2. The measurement errors are normally distributed and uncorrelated, with an error covariance matrix denoted by .

3. Bulk formula parameter errors and measurement errors are uncorrelated.

For small adjustments the meridional heat transport can be linearized about at the measurement locations:

where are the elements of the sensitivity matrix ,

A solution of this underdetermined problem is found by minimizing the functional

where denotes transpose, and contains the imbalances between the measurements and the meridional heat transport with the adjusted parameters , viz.

In a least square sense, the first term in the RHS of (49) penalizes large changes in from the reference value , while the second term penalizes large imbalances between measured and computed meridional heat transport. Therefore, the measurements enter (50) as a weak constraint. Both terms of the functional are weighted by the inverse of the error covariance matrices, which in principle need not be diagonal as assumed here. Under the assumption of normality stated above, the functional (49) follows from a maximum likelihood principle (Menke 1984).

Using (47), a linearized version of (49) can be found

where , , denotes the imbalance between the measurements and the meridional heat transport computed with the current set of parameters . Setting one finds that the desired minimum satisfies

or equivalently,

Using the following matrix identity (e.g., Jazwinski 1970, p. 261)

one arrives at the final result

(Notice that this equation differs slightly from eq. (13'') in Isemer et al. (1989) where the solution is shown as . This typographic error has been acknowledged by H.-J. Isemer [personal communication]. )

In the particular case of a single measurement (), eq. (55) reduces to

where (the index has been omitted).

A plausibility condition to be checked a posteriori is whether the adjustments are small enough compared to the assumed errors . Following Isemer et al. (1989), we accept a solution if

As discussed by Isemer et al. (1989), the standard -test, used to assess the goodness-of-fit of the parameter estimates, is problematic given that one has very little confidence on the error variances .