and the variance of space average ΨA is on the similar form:
where S(ν, μ, f) is the space-time spectrum, σ2 the variance at a spatial point and = sin2 (πx)/(πx)2 the rectangular filter, which value becomes smaller than 1.0/π2 if x > 1.0. Clearly, σ2V <σ2,σ2A <σ2 and G(πx)2 < 1. The MSE of sampling problem can be written as follows:
for temporal sampling,
and for spatial sampling,
Here we only focus on the case of spatial sampling. Consider △x = L and △y = M, then we have
In North and Nakamoto (1989) and Nakamoto et al. (1994), the variance of areal average was derived in the following way. At first Eq. (10) is simplified as:
where σ2A1 is the simplified one which is different from the original oneσ2A in Eq. (10). After simple integration for a given spectrum, the variance of areal average can be written in the form:
where k is a constant and α a normalized integration constant for space-time spectrum. With Eq. (16) and Eq. (11), the final sampling error formalism were obtained in North and Nakamoto (1989) and Nakamoto et al. (1994). For example, with the following spectrum formalism for SST:
Nakamoto et al. (1994) derived their sampling error formula as follows:
