Incremental Analysis Updates

Rather than updating the model state directly with the analysis increment, it may be preferable to introduce the increment gradually into the ocean model in order to minimize spurious adjustment processes. This technique is referred to as Incremental Analysis Updates (IAU) [Bloom et al., 1996]. IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI. IAU is used when ln_asmiau is set to true.

With IAU, the model state trajectory ${\bf x}$ in the assimilation window ( $t_{0} \leq t_{i} \leq t_{N}$) is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as additional tendency terms to the prognostic equations:

$${\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\bf x}^{a}$$ (13.1)

where $F_{i}$ is a weighting function for applying the increments $\delta \tilde{\bf x}^{a}$ defined such that $\sum_{i=1}^{N} F_{i}=1$. ${\bf x}^b$ denotes the model initial state and ${\bf x}^a$ is the model state after the increments are applied. To control the adjustment time of the model to the increment, the increment can be applied over an arbitrary sub-window, $t_{m} \leq t_{i} \leq t_{n}$, of the main assimilation window, where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$, Typically the increments are spread evenly over the full window. In addition, two different weighting functions have been implemented. The first function employs constant weights,

$$F^{(1)}_{i} =\left\{ \begin{array}{ll} 0 & {\rm if} \; \; \; t_{i... ... < t_{i} \leq t_{n} \\ 0 & {\rm if} \; \; \; t_{i} > t_{n} \end{array} \right.$$ (13.2)

where $M = m-n$. The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, with the weighting reduced linearly to a small value at the window end-points:

$$F^{(2)}_{i} =\left\{ \begin{array}{ll} 0 & {\rm if} \; \; \; t_{i... ... < t_{i} \leq t_{n} \\ 0 & {\rm if} \; \; \; t_{i} > t_{n} \end{array} \right.$$ (13.3)

where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. The weights described by (13.3) provide a smoother transition of the analysis trajectory from one assimilation cycle to the next than that described by (13.2).