where a and b are sampling rates as defined in the last section;
c2 = 0.93 and c3(a) is a convergent integral function of parameter a > 0, which can be calculated numerically.
Substituting the spectrum (17), H function of spatial sampling (13) and variance (23) into Eq. (11), we can obtain the new sampling error formula which is written as follows:
5.4 Validation of new sampling error formula
In the new sampling error formula, the ε2/σ2A-T relation is still the same as in Nakamoto et al. (1994). However, the modulation of spacescales on sampling error changes. When sampling separations L and M become larger than length scale λ1 andλ2, the denominator in Eq. (24) becomes large. This effectively restrains the extraordinary increase of sampling error as in the old formula. Furthermore, γ0 can be selected as the same order as the e-folding timescale.
With the sampling parameters of L = 15°, M = 2° and T = 4 weeks, the new sampling error formula (24) is validated by comparing with observed sampling error for high-passed SST (Fig. 7a). The theoretical sampling error field obtained from Eq. (18) with γ0 = 90 days is shown in Fig. 7b and Fig. 7c-d shows the sampling error fields derived from the new sampling error formula Eq. (24) with γ0 = 1.75 days and γ0 = 0.31γe(x, y), where γe(x, y) is e-folding timescale as shown in Fig. 6a. Both Fig. 7c and Fig. 7d illustrate the similar spatial patterns with observation results (Fig. 7a). The sampling error formula still predicts larger value in equatorial front area than that in observation but the value is acceptable. The quantitative validation of previous (Eq. (18), hereafter referred as N) and new (Eq. (24), referred as 5) sampling error formula is shown in Tab. 1a (average simulation difference), Tab. 1b (absolute average of relative difference) and Tab. 1c (for sub-region average). The average simulation difference (E1) is defined in the form:
