where a = △x/λ1, b = △y/λ2, c = T/γ0. λ1 andλ2 are zonal and meridional characteristic scales.
5.3 The improvement of existing formula
In the Eq. (16), the variance of spatial average has no relation with characteristic length scale A1 and A2. Is this consistent with the observation? We calculate the ratio of σ2A/σ2 for high-passed SST which is shown in Fig. 5. The e-folding timescale and zonal scale for high-passed SST are shown in Fig. 6a-b. It is visible that e-folding timescale is rather homogeneous in the whole tropical Pacific, which is consistent with the former conclusion by Smith and Meyers(1997). From Eq. (16), this suggests that the ratio σ2A/σ2 should also be spatial homogeneous. However, the spatial inhomogeneity in Fig. 5 is obvious and its spatial pattern of σ2A/σ2 is very similar to that of e-folding zonal scale (Fig. 6b). This indicates that Eq. (16) conflicts with our observation results.
The origin of this conflict can be found by qualitative analysis. Since for any given frequency f0 and "red" type spectrum S(ν, μ, f_) > 0, we have S(0, 0, f0) 〜 Max[S(ν, μ, f0)]. It is concluded that σ2A1/σ2A. This means sampling error formula by Nakamoto et al. (1994) underestimates the sampling error owing to undue simplification in the integration in Eq. (10). This is why we have to use a very large r0 to get a sampling error comparable with observation. The description of σ2A should be modified.
By using
and substituting spectrum (17) into Eq. (10), then we have:
