A Heuristic Gap Filling Method for Daily Precipitation Series

Abstract

The gap filling is common practice to complete hydrological data series without missing values for environmental simulations and water resources modeling in a changing climate. However, gap filling processes are often cumbersome because physical constraints, such as complex terrain and density of weather stations, often limit the ability to improve the performance. Although several studies of gap filling methods have been developed and improved by researchers, it is still challenging to find the best gap filling method for broad applications. This research explores a gap filling method to improve climate data estimates (e.g., daily precipitation) using gamma distribution function with statistical correlation (GSC) in conjunction with cluster analysis (CA). The daily dataset at the source stations (SSs) is utilized to estimate missing values at the target stations (TSs) in the study area. Three standard gap filling methods, including Inverse Distance Weight (IDW), Ordinary Kriging (OK), and Gauge Mean Estimator (GME) are evaluated along with cluster analysis based on statistical measures (RMSE, MAE, R) and skill scores (HSS, PSS, CSI). The result indicates that cluster analysis can improve estimation performances regardless of the gap filling methods used. However, the GSC method associated with cluster analysis, in particular, outperformed other methods when the performance comparison task was conducted under rain and no-rain conditions in the study area. The proposed method, GSC, therefore, will be used as a case toward advancing gap filling methods in the field.

Appendix

1.1 Gauge Mean Estimator (GME)

The Gauge Mean Estimator (GME) is to use an arithmetic average of all SSs. It is a special case of Inverse Distance Weighting (IDW) and it is a similar method being used for the average precipitation estimation method presented by McCuen (1998). The estimation of missing precipitation is given by

$$ {X}_m = \frac{{\displaystyle {\sum}_{i=1}^{i=n}}\frac{N_m}{N_i}\ {X}_i}{n} $$

(11)

Where, X_m = estimated value at the TS_m, n = the number of stations, X_i = the observed value at SS_i, N_m and N_i = the average monthly precipitation at TS_m and SS_i, respectively.

1.2 Inverse Distance Weighting (IDW) Method

As proposed by Simanton and Osborn (1980), the Inverse Distance Weighting (IDW) is used to estimate the missing precipitation values and it is implemented by Watson and Philip (1985). IDW has been commonly used to fill the missing values. It is also widely used many water resources applications (ASCE 1996). For the precipitation estimation at TSs, IDW provides weighting values inversely depending on the distance between TS and SS. The missing values at TSs can be computed by Eq. (12).

$$ {X}_m = \frac{{\displaystyle {\sum}_{i=1}^n}{X}_i{d}_{mi}^{-k}}{{\displaystyle {\sum}_{i=1}^n}{d}_{mi}^{-k}} $$

(12)

Where, m = the selected TS, X_m = the estimated value at the TS m; n is the number of stations, X_i = the observed value at SS I, d_mi = the distance from the station i to station m, and k = referred to as friction distance (Vieux 2001) that ranges from 1.0 to 6.0. In this study, k = 2 is used (Teegavarapu et al. 2009).

1.3 Ordinary Kriging (OK) Method

The Ordinary Kriging (OK) is the standard approach for surface interpolation method based on scalar measurement at different locations (Journel and Huijbregts 1978; Isaake and Srivastava 1989). OK is spatially dependent variance (Vieux 2001). The degree of spatial dependence in OK method can be determined using a semivariogram. The weights of OK are based on the distance between SS and TS as well as. The equations for OK method to estimate missing values include Eqs. (13) – (15).

$$ {X}_m = {\displaystyle \sum_{i=1}^n}\delta\ {X}_i $$

(13)

$$ \updelta = {\tau}^{-1}\gamma $$

(14)

$$ \upgamma \left(\mathrm{d}\right) = \frac{1}{2\ n\ (d)}\ {\displaystyle \sum_{dij}}{\left({X}_i - {X}_j\right)}^2 $$

(15)

where, δ = the weight obtained from the fitted simivariogram, τ = the gamm matrix, which is the model semivariance for all sampled pairs, γ(d) = the semivariance which is defined over observations X _i and X _j lagged successively by distance d, n(d) = the number of distinct pairs in n(d), X _i and X _j = data values at spatial location I and j, respectively.

1.4 RMSE, MAE, and R

$$ RMSE = \sqrt{\frac{{\displaystyle {\sum}_{i = 1}^N}{\left({P}_{Si} - {P}_{Qi}\right)}^2}{N}} $$

(16)

$$ MAE = \frac{1}{N}\ {\displaystyle \sum_{i = 1}^N}\left|\ {P}_{Si} - {P}_{Qi}\right| $$

(17)

$$ R = \frac{\frac{1}{N}\times {\displaystyle {\sum}_{i=1}^N}\left({P}_{Qi}-{\overline{P}}_{Qi}\right)\times \left({P}_{Si}-{\overline{P}}_{Si}\right)}{\sqrt{\frac{N\times {\displaystyle {\sum}_{i=1}^N}{P}_{Qi}^2 - {\left({\displaystyle {\sum}_{i=1}^N}{P}_{Q1}\right)}^2}{N\times \left(N-1\right)}} \times \sqrt{\frac{N\times {\displaystyle {\sum}_{i=1}^N}{P}_{Si}^2 - {\left({\displaystyle {\sum}_{i=1}^N}{P}_{S1}\right)}^2}{N\times \left(N-1\right)}}\kern0.75em } $$

(18)

where, P_Qi = the observed precipitation data at time step I, P_Si = the estimated precipitation data, ${\overline{P}}_{Qi}\kern0.5em and\ {\overline{P}}_{Si}$ = the mean of the observed and estimated precipitation data, respectively; N = the number of sample sizes.