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# decorrelation

by Laurent Roblou — last modified Apr 23, 2012 01:55 PM

Observations influence in the model error subspace can be decreased using decorrelation functions. The data assimilation code provide users with a large range of isotropic functions for this purpose.

The following lines describes the shape functions collection.

• linear function: $\mbox{ w(r) = 1 - r }$

• step function: $w(r) = 1 \thickspace \mbox{for} \thickspace r \leq 1, \thickspace \mbox{0 otherwise}$

• spherical function: $w(r) = 1 - \frac{3}{2}r + \frac{1}{2} r^{3}$

• sinusoidal function: $w(r) = \frac{1 + \cos(r)}{2}$

• exponential function class
• $w(r) = e^{-r}$
• $w(r) = e^{-r} \thickspace \mbox{for} \thickspace r \leq 1, \thickspace \mbox{0 otherwise}$
• $w(r) = (1 + r + \frac{1}{6} r^{2} - \frac{1}{6}r ^{3} ) e^{-r}$

• Gaussian functions class:
• $w(r) = e^{\frac{-r ^ {2}}{2 \sigma ^ {2} }}, \thickspace \sigma = \frac{1}{\sqrt{2 \ln(2)}}$
• $w(r) = e^{\frac{-r ^ {2}}{2 \sigma ^ {2} }}, \thickspace \sigma = \frac{1}{\sqrt{2 \ln(2)}} \thickspace \mbox{for} \thickspace r \leq 2 \sigma, \thickspace \mbox{0 otherwise}$
• $w(r) = (1 - \frac{1}{6} r) e^{\frac{-r ^ {2}}{2 \sigma ^ {2} }} , \thickspace \sigma = \frac{1}{\sqrt{2 \ln(2)}} \thickspace \mbox{for r \leq 1}, \thickspace \mbox{0 otherwise}$

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