Table 3. Validation of Eq. (29) to simulate the sampling error of anomaly SST for 8 sub-regions with k3 = 2.0, kσ = 1.0 and γ0 = 1.75 days.
Table 3 gives the quantitative evaluation of the simulation difference. The second row (E1) is areal averaged absolute simulation error for 8 regions, which are about 0.3% of the areal averaged variance for small sampling areas (B, C, E, F and H) and 4% for big sampling error areas (A, D, G). The third row (E3) presents the areal averaged relative simulation difference which are about 3% of the real sampling error for regions B, E and F; 22% for regions A, C, G and H and 93% for equatorial front area D. The absolute averaged relative simulation difference are shown in the fourth row, which are in average about 30% of the real sampling error except for the region D with a simulation error over 100%. Generally speaking, the Eq. (29) provides a sampling error simulation with similar accuracy for anomaly SST in comparison with Eq. (3.14) for high-passed SST. This suggests that the sampling error formula (4.9) can be used for solving SST network design provided that the characteristic length scales of high-passed SST are known.
6 Optimal network design from the formula
By using Eq. (29) with k3 = 2.0, k3 = 1.0 and γ0 = 1.75 days, the optimal network design problem (I") is solved around each local point. In Eq. (29), the sampling error for anomaly SST is only decided by the sampling rates a and b if the high- passed length scales A1 and A2 and variance ratioσ2h/σ2a are given. The optimal zonal and meridional sampling distances are obtained for a prescribed sampling error 4%. Results are shown in Fig. 9a-b. By comparing Fig. 8a with Figs. 9a and b, it is visible that the optimal sampling distance fields closely related to sampling error fields. Large sampling distances (10-20 degrees zonally and 5-10 degrees meridionally) appear in the areas with small sampling error such as central and south-east tropical Pacific while warm water areas and north-east tropical Pacific regions should have smaller sampling distances (about 6 degrees zonally and 4 degrees meridionally).
To Compare the theoretical results with those derived from observed data, we estimate the averaged optimal sampling distances for 8 sub-regions, as shown in the Tab. 4 where Sopt1 = Lopt1 x Mopt1 and Sopt2 = Lopt2 x Mopt2 are optimal sampling box size derived from the formula and the observed anomaly dataset respectively. It is indicated that the spatial distribution is the same as that from observation for zonal optimal sampling distance (Fig. 4a). The optimal zonal sampling separation Lopt2
