Chapman 1991

Original Form

Chapman, T.G., 1991. Comment on the evaluation of automated techniques for base flow and recession analyses, by R.J. Nathan and T.A. McMahon. Water Resource Research 27(7): 1783-1784

\[f_k = \frac{3\alpha-1}{3-\alpha}f_{k-1} + \frac{2}{3-\alpha}\left(y_k - \alpha y_{k-1}\right)\]

set

\[y_k = f_k + b_k\]

into

\[b_k = y_k - \frac{3\alpha-1}{3-\alpha}f_{k-1} - \frac{2}{3-\alpha}\left(y_k - \alpha y_{k-1}\right)\]

set $f_{k-1}=y_{k-1}-b_{k-1}$ and re-arrange

\[b_k = \frac{3\alpha-1}{3-\alpha} b_{k-1} + \left(1- \frac{2}{3-\alpha}\right) y_k + \left(\frac{2\alpha}{3-\alpha} - \frac{3\alpha-1}{3-\alpha}\right)y_{k-1}\] \[b_k = \frac{3\alpha-1}{3-\alpha} b_{k-1} + \left(\frac{1-\alpha}{3-\alpha}\right) y_k + \left(\frac{1-\alpha}{3-\alpha}\right)y_{k-1}\]

where $\alpha=k$, the recession coeficient.

General form

\[b_t = \alpha b_{t-1} + \beta q_t + \gamma q_{t-1}\] \[\alpha = \frac{3k-1}{3-k} \qquad \beta = \frac{1-k}{3-k} \qquad \gamma=\frac{1-k}{3-k}\]