Covariance of the Smoothed Field

Consider the covariance of the smoothed field:

$$E \{ F_S(\ensuremath{\mathbf{x_1}}) F_S(\ensuremath{\mathbf{x_2}}) \} = E \left\{ \int \int G(\ensuremath{\mathbf{x_1}}- \ensuremath{\mat... ...athbf{p_2}}) \, d\ensuremath{\mathbf{p_1}}\, d\ensuremath{\mathbf{p_2}}\right\}$$ (5)

$$= \int \int G(\ensuremath{\mathbf{x_1}}- \ensuremath{\mathbf{p_1}})... ...emath{\mathbf{p_2}}) \, d\ensuremath{\mathbf{p_1}}\, d\ensuremath{\mathbf{p_2}}$$ (6)

$$= \int G(\ensuremath{\mathbf{x_1}}- \ensuremath{\mathbf{x_0}}) G(\ensuremath{\mathbf{x_2}}- \ensuremath{\mathbf{x_0}}) \, d\ensuremath{\mathbf{x_0}}$$ (7)

$$= \left( \int G_{\sigma_x}(x_1 - x_0) G_{\sigma_x}(x_2 - x_0) \, dx... ...t G_{\sigma_y}(y_1 - y_0) G_{\sigma_y}(y_2 - y_0) \, dy_0 \right) \times \cdots$$

$$\qquad \qquad \left( \int G_{\sigma_z}(z_1 - z_0) G_{\sigma_z}(z_2 - z_0) \, dz_0 \right)$$ (8)

Each 1D integral is of the form:

$$I_1 = \int G_{\sigma}(x_1 - x_0) G_{\sigma}(x_2 - x_0) \, dx_0$$ (9)

$$= \frac{1}{2\pi \sigma^2} \int \exp\left( \frac{-1}{2 \sigma^2} \left[ (x_1 - x_0)^2 + (x_2 - x_0)^2 \right] \right) \, dx_0$$ (10)

$$= \frac{1}{2\pi \sigma^2} \int \exp\left( \frac{-1}{2 \sigma^2} \fr... ... \frac{-1}{\sigma^2} \left[ x_0 - \frac{x_1 + x_2}{2} \right]^2 \right) \, dx_0$$ (11)

$$= \frac{1}{2\pi \sigma^2} \exp\left( \frac{- (x_1 - x_2)^2}{4 \sigma^2} \right) \sqrt{ \pi \sigma^2}$$ (12)

$$= \frac{1}{\sqrt{4 \pi \sigma^2}} \exp\left( \frac{- (x_1 - x_2)^2}{4 \sigma^2} \right)$$ (13)

Therefore:

$$E \{ F_S(\ensuremath{\mathbf{x_1}}) F_S(\ensuremath{\mathbf{... ...ght) \exp\left( \frac{- (z_1 - z_2)^2}{4 {\sigma_z}^2} \right)$$ (14)