Quantifying a Threshold of Missing Values for Gap Filling Processes in Daily Precipitation Series

Abstract

Multiple missing levels are explored to quantify a threshold of missing values during gap filling processes in daily precipitation series. An autoregressive model was used to generate rainfall estimates and subsets of data are selected with four sampling windows (whole data, front, middle, and rear section) at different missing levels, including 5, 10, 15, 16, 17, and 18 %. The proposed threshold was found and evaluated based on statistical criteria, including coefficient of determination (R²) and its associated index termed “the R² difference index (RDI).” The result indicates that about 15 % missing level of data is plausible to construct daily precipitation series for further hydrological analysis when the Gamma distribution function (GDF) is used as an estimation method. The threshold determined from this study will contribute to gap filling guidelines, especially for water managers and hydrologists to take advantage of skillful estimates for missing daily precipitation data.

Appendix

The Gamma Distribution Function (GDF):

$$ F\left(x\left|a,\beta \right.\right)={\displaystyle \underset{0}{\overset{x}{\int }}f\left(x\left|a,\beta \right.\right)} $$

(A1)

$$ f\left(x\left|a,\beta \right.\right)=\frac{1}{\beta^2\varGamma (a)}{x}^{a-1}{e}^{-x};x\ge 0 $$

(A2)

$$ \varGamma (a)={\displaystyle \underset{0}{\overset{\infty }{\int }}{x}^{a-1}{e}^{-x}dx} $$

(A3)

$$ a={\left(\frac{\overline{x}}{s}\right)}^2,\kern0.96em \beta =\frac{S^2}{\overline{x}} $$

(A4)

$$ {F}^{-1}\left(x\left|a,\beta \left|a,\beta \right.\right.\right)=x $$

(A5)

Where, F(x) is CDF of gamma distribution, f(x) is the probability density function of the gamma distribution, x is precipitation at the time step, $\overline{x}$ is the mean precipitation of specific month, s is the standard deviation of the specific month, α is the shape parameter, β is the scale parameter, Γ is the gamma function.

Coefficient of determination (R²): R² is given by

$$ {R}^2={\left(\frac{\frac{1}{N}\times {\displaystyle {\sum}_{i=1}^N\left({P}_{Qi}-{\overline{P}}_{Qi}\right)\times \left({P}_{Si}-{\overline{P}}_{Si}\right)}}{\sqrt{\frac{N\times {\displaystyle {\sum}_{i=1}^N{P}_{Qi}^2-{\left({\displaystyle {\sum}_{i=1}^N{P}_{Q1}}\right)}^2}}{N\times \left(N-1\right)}}\times \sqrt{\frac{N\times {\displaystyle {\sum}_{i=1}^N{P}_{Si}^2-{\left({\displaystyle {\sum}_{i=1}^N{P}_{S1}}\right)}^2}}{N\times \left(N-1\right)}}}\right)}^2 $$

(A6)

where, P_Qi is the observed precipitation data at time step i, P_Si is the estimated precipitation data. ${\overline{P}}_{Qi}\; and\;{\overline{P}}_{Si}$ are the mean of the observed and estimated precipitation data, respectively. N is sample size.