big sampling error areas (a < 1) but the simulation difference in small sampling error areas (a > 1) is up to about 4%. Though γ0 = 90 days produces an acceptable result in small sampling error areas by formula N, the simulated differences in big sampling error areas are too large (about 13.3 % for E1) to be used. In opposite, formula S with both of γ0 = 1.75 days and γ0 = O.31γe creates acceptable results in large sampling error areas, which averaged simulation differences are -0.64% and 1.34%.
The absolute averaged relative simulation difference is more strict index than E1 to validate the formula. It is indicated in Tab. 1b that formula N has a 39% error for small sampling areas and 95% for big sampling error areas with γ0 = 90 days while the new one S has 37% for small sampling areas and 26% for big sampling error areas with γ0 = 1.75 days respectively.
Table 1c. Averaged simulation difference in 8 sub-regions with γ0 = 0.31γe (%)
In Section 4, we divided the whole NETP ocean into 8 sub-regions. Table 1c shows the averaged simulation difference (E1) and the averaged relative simulation difference (E3) with γ0 = 0.3lγe for these 8 sub-regions. The error value in Tab. 1c can be regarded as a practical validation of the new sampling error formula for the purpose of optimal network design. It is visible that the simulation difference decreased in comparison with Tab. 1b, especially in NEP areas. In terms of sub-regional averaging, the simulation difference is just 2% of the area-averaged variance and 15% of the real sampling error in the NEP areas. The simulation difference in tropical areas (regions F, G and H) is larger, about 25% of the real sampling error.
The above is the calibration of the new formula for high-passed SST. How about for low-passed SST? Considering the spatial pattern of low-passed MSE is not similar to that of spacescales of low-passed SST, the new formula may not simulate the low-passed sampling error well. This point is confirmed by the comparison between simulated results and observed results for low-passed SST using formula both N and S (Figures are omitted here). In fact, the simulation from the two formula is too poor to be applicable. In the next section, another method will be developed to cope with this difficulty.
5.5 Sampling error formula for anomaly SST
As was pointed out in the last section, the improved sampling error formula can only be applied to the high-passed SST but not anomaly SST directly. To apply the new sampling error formula to the optimal design problem (I") for anomaly SST, we have to introduce some approximations based on basic SST statistics.
The first one is to assume that the anomaly MSE ε2a can be described in the form:
