where mb is the buoy number available in the box and <> is the averaging in time.
MSE is the mean amplitude of the sampling bias. According to North and Nakamoto (1989), sampling error is defined as the percentage of variance of the areal average taken by MSE, i.e., ε2/σ2A, where
and
ΨA is areal average of the random field. Hereafter we set L = △x and M = △y.
By using anomaly SST, the MSE, variance of areal-average and sampling error were estimated for sampling distances of L varying from 5 to 30 degrees and M from 1 to 7 degrees. Notice that the averaging box also varies with the same parameters L and M in space and averaging time T from 1 to 13 weeks.
Figure 2a-h shows root-mean-square-error (RMSE) distribution varied with sampling distance for the above 8 regions (A-H) with T=4 weeks. It is interesting that all the RMSE distribution on maps on L-M plane shows universal kink along the band between M=1 and M=3. When M is greater than 3 degrees, sampling error increased with M for fixed L. In Far-west and west NEP and north tropical Pacific (Figs. 2a, b, f and g), the RMSE increase rates are slower than in other areas. When M varies from 2 to 3 degrees, the RMSE decreases with increasing meridional sampling distance M in all 8 regions. This feature is also found in the variance of areal average and sampling error (Figures are omitted here). Comparing the RMSE in eight regions with each other, it is noted that the spatial inhomogeneity of RMSE is significant.
Relations between MSE, sampling error and averaging time T were also studied. Figure 3a-h is the relation between sampling error and T for 8 sub-regions. The solid line is for sampling distances of L=15° and M=2° and dashed line for L=11° and M=5°. One feature is that sampling errors for 11° X 5° box are smaller than those of 15° x 2° box, though the later has higher sampling resolution. This suggests that optimal network design study should be of great practical importance.
4.3 Optimal sampling distances from observation
Setting error function to be mean sampling error and the accepted mean sampling error criterion F0 = 4%, optimal network design (I") is solved for each of 8 sub-regions. The solution is shown in Fig. 4a-b. Note that zonal (Lopt) and meridional sampling distances (Mopt) are optimized together.
The spatial inhomogeneity is mainly found in the zonal sampling distances while optimal meridional sampling distances are rather homogeneous, which are about 5
