In the data assimilation process, the observation operator h is used to interpolate the model forecast field xf to the location in time and space of the observations, y-This enables calculation of the innovations, d = [y - h(xf)]. Once the data assimilation has been performed it is also possible to calculate the equivalent using the analysis field to produce the residuals, r = [y - h(xa)]. The reduction in the errors between the analysis and the forecast can be used as an a posteriori check that the data assimilation process is working as expected, and is fitting the observations to within their error (see for example Cummings 2005).
The increments generated through the data assimilation process also provide an important source of information. The time-average of these increments can indicate areas of significant model bias. However, it is not always obvious how to diagnose the source of these biases.
For validation and verification of the model forecast, it is the innovation statistics that are of most interest, as they provide a pseudo-independent check on its accuracy. The observations being used for this comparison have not previously been assimilated so from that point of view are independent. However, previous observations of the same type will have been assimilated on previous data assimilation cycles so they cannot be viewed as completely independent.
An example of the innovation statistics from a 2-year reanalysis using the global FOAM system (Storkey et al. 2010) is shown in Fig. 22.4. This includes the mean and RMS of the innovations for SSH and for temperature. The mean errors show that the system is able to represent the global average observed SSH and temperature well, although a small positive temperature bias exists below the top 50 m, with a cold bias above this depth for most of the period. The RMS of the innovations provides a measure of the overall accuracy of the system both as a function of time, and of depth (for temperature). The maximum of the global temperature errors is located within the top 200 m with much smaller errors below this depth. These time-series plots also illustrate the stability of the system, with the SSH being relatively stable, whereas the temperature RMS errors appear to have a seasonal cycle with smaller errors in Northern hemisphere winter.
An example of the use of Taylor diagrams for plotting innovation statistics is shown in Fig. 22.5 with results from a hindcast run of the resolution global FOAM system (Storkey et al. 2010). This shows the statistics for a number of different regions for both SST and SSH. The SST statistics are only shown for a comparison with the AATSR data, although other satellite SST data were assimilated. The variability in both these variables is well-reproduced by the model in all regions, but the correlations and RMS differences are clearly regionally dependent with the Mediterranean region having the largest RMS errors and lowest correlation coefficient.
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