Sampling Statistics

To estimate the required variances simple averages are taken over the required quantities. That is:

$$V_0 = \frac{1}{N} \sum_{\ensuremath{\mathbf{x}}} \frac{1}{N_T} \sum_{t=1}^{N_T} \left( S_t(\ensuremath{\mathbf{x}}) \right)^2$$ (50)

$$V_1 = \frac{1}{N} \sum_{\ensuremath{\mathbf{x}}} \frac{1}{N_T} \sum_{t=... ...{\mathbf{x}}+ \ensuremath{\mathbf{d}}) - S_t(\ensuremath{\mathbf{x}}) \right)^2$$ (51)

$$V_2 = \frac{1}{N} \sum_{\ensuremath{\mathbf{x}}} \frac{1}{N_T} \sum_{t=... ...math{\mathbf{x}}+ \ensuremath{\mathbf{d}}) S_t(\ensuremath{\mathbf{x}}) \right)$$ (52)

where $S_t(\ensuremath{\mathbf{x}})$ is the observed smooth field at time $t$.

The required smoothing parameters are then calculated using equations 48 and 49. That is:

$${\sigma_x}^2 = \frac{- d^{\, 2}}{4 \ln\left( 1 - \frac{V_1}{2 V_0} \right)}.$$ (53)

or

$${\sigma_x}^2 = \frac{- d^{\, 2}}{4 \ln\left( \frac{V_2}{V_0} \right)}$$ (54)

Note that, because ratios are used, any constant scaling factors relating $S(\ensuremath{\mathbf{x}})$ to $F_S(\ensuremath{\mathbf{x}})$ will cancel.