Have you ever seen logarithmic burn-in on a Kalman filter before? I'm reviewing a paper that uses a multiplicative factor $\ln i$ where $i$ is the number of steps since the Kalman filter was initialized.  The idea is to slowly build confidence as the filter sees more data.

The authors replace the typical Kalman gain $K$ with one weighted by this multiplicative factor $K_{new}$:


  $K_{new} = ( \ln i) K$


Have you ever seen such an approach used before? I'd like to give the authors a reference to cite to tie them into the literature on this subject if it exists.
The authors cite this work in their response to the review:


  Zhao H, Lu L. Adaptive recursive algorithm with
  logarithmic transformation for nonlinear system identification in
  α-stable noise [J]. Digital Signal Processing, 2015:S1051200415002535.


They take their inspiration from equation (14):


  $J_p(n)=\sum_{i=1}^n \lambda^{n-1} \cdot \log^p (1+|e(i)|)$


It is notable that $e(i)$ is (as one might expect) the error in the signal, and not linearly increasing with the number of steps since the filter was initialized.