Lognormal error distribution instead of the Gaussian

[BenjMy]

Example with Ks
See Botto

Example with LAI

Nicola Paciolla1, Chiara Corbari1, ...

• Manuscript Number: PRAG-D-25-00299
• Full Title: Salinity stress and water availability inferred from satellite leaf area index assimilated into a water-energy-crop model

On the other hand, perturbations on LAI are done using a lognormal error distribution instead of the Gaussian one to account for the constrain of non-negative values (Kondrashov et al., 2011). The observation error is equal to the standard deviation over the years.
Thus, in the DA scheme, the LAI variable is assimilated through its logarithm (Λ): 𝐿𝐴𝐼~𝐿𝑛(𝜇𝐿𝐴𝐼, 𝜎𝐿𝐴𝐼) → 𝛬 = log(𝐿𝐴𝐼)~𝑁(𝜇𝐿𝑁(𝐿𝐴𝐼), 𝜎𝐿𝑁(𝐿𝐴𝐼)) (3)
Where the LAI variable has a logarithmic distribution with average μLAI and standard deviation σLAI, meaning that its logarithmic transform (Λ) is normally distributed, with mean μLN(LAI) and standard deviation σLN(LAI).
Whenever new observations are available, the model state update will be performed on the logarithmic transform, between the logarithms of modelled and measured LAI. Finally, the updated logarithmic transform variable (𝛬(𝑢𝑝𝑑..)) will be reverted to the original modelled one (𝐿𝐴𝐼(𝑢𝑝𝑑.)) to be reintegrated into the model.