# Analysis-Forecast Cycle Observation Impacts

Figure: A schematic showing the typical configuration of an analysis-forecast cycle. Analysis cycle spans the interval ${\displaystyle [t_{0}^{j},t_{0}^{j}+\tau ]}$, and is associated with 4D-Var analysis ${\displaystyle x_{a}^{j}}$ and the forecast ${\displaystyle x_{f}^{j}}$. At the start of each cycle there are three available circulation estimates: two forecasts initialized from the previous two adjacent analysis cycles, and the analysis for the current time. These are illustrated at time ${\displaystyle t_{0}^{j+2}+\tau }$ at the start of cycle ${\displaystyle j+3}$.
In the figure above, each analysis cycle is assumed to be of length and analysis cycle spans the interval${\displaystyle [t_{0}^{j},t_{0}^{j}+\tau ]}$. The circulation estimate at time ${\displaystyle t_{0}^{j}+\tau }$ (i.e. the end of analysis cycle ) is denoted as ${\displaystyle x_{a}^{j}}$ and is the initial condition for the forecast spanning the next analysis interval ${\displaystyle [t_{0}^{j+1},t_{0}^{j+1}+\tau ]}$. In sequel it is assumed that the forecast duration is an integer multiple of , but this is not a necessary constraint. The figure shows the analyses and forecasts that result from three adjacent analysis cycles, namely cycles , ${\displaystyle j+1}$ and cycle ${\displaystyle j+2}$. The analysis ${\displaystyle x_{a}^{j}}$ at the end of cycle is used as the initial condition for the forecast ${\displaystyle x_{f}^{j}}$ of duration ${\displaystyle 2\tau }$ that terminates at time ${\displaystyle t_{0}^{j+2}+\tau }$, the end of analysis cycle ${\displaystyle j+2}$. Similarly, the analysis ${\displaystyle x_{a}^{j+1}}$ at the end of cycle ${\displaystyle j+1}$ is used as the initial condition for the forecast ${\displaystyle x_{f}^{j+1}}$ of duration and also terminates at time ${\displaystyle t_{0}^{j+2}+\tau }$, the end of analysis cycle ${\displaystyle j+2}$. After sufficient time has elapsed, a new analysis ${\displaystyle x_{a}^{j+2}}$ will be computed at this time. Since ${\displaystyle x_{a}^{j+2}}$ represents our best estimate of the ocean circulation at time ${\displaystyle t_{0}^{j+2}+\tau }$ it can be used to quantify the veracity of the forecasts ${\displaystyle x_{f}^{j}}$ and ${\displaystyle x_{f}^{j+1}}$. For this reason, ${\displaystyle x_{a}^{j+2}}$ is usually referred to as the “verifying analysis.” However, as discussed, later other sources of information can be used to verify the forecasts, such as new or independent observations.
It should be clear from the figure that the forecast ${\displaystyle x_{f}^{j+1}}$ benefits from the observations assimilated into the model during analysis cycle ${\displaystyle j+1}$ (i.e. during the interval ${\displaystyle [t_{0}^{j+1},t_{0}^{j+1}+\tau ]}$). Therefore, providing that ${\displaystyle x_{f}^{j}}$ and ${\displaystyle x_{f}^{j+1}}$ are subject to identical surface forcing and open boundary conditions during the interval ${\displaystyle [t_{0}^{j+2},t_{0}^{j+2}+\tau ]}$, any differences in forecast error must be associated with the observations assimilated into the model during the interval ${\displaystyle [t_{0}^{j+1},t_{0}^{j+1}+\tau ]}$. The impact of each observation on the forecast error can be quantified as described next which is based on the work of Langland and Baker (2004), Errico (2007) and Gelaro et al. (2007).