# Comparison of Three Imputation Methods for Groundwater Level Timeseries

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## Abstract

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## 1. Introduction

- Spline interpolation (mathematical approach), which smoothly interpolates the groundwater trends;
- The autoregressive linear model (statistical approach), based on random processes where imputed values can be identified by linearly combining values previously observed;
- Patched kriging (geostatistical approach), the sequential application of the ordinary kriging, preserving the available spatio-temporal information [34].

## 2. Materials and Methods

#### 2.1. Overview of the Study Area

#### 2.2. Dataset Pre-Processing

#### 2.3. Imputation Methods

- The minimum spline value must be greater than the recorded minimum less one quarter of the recorded range;
- The maximum spline value must be smaller than the recorded maximum plus one quarter of the recorded range.

#### 2.4. Comparing Procedures and Applied Statistical Analyses

- Using the formula of [56]:$$\lambda =\sqrt{-\frac{f\left(0\right)}{{f}^{\u2033}\left(0\right)}}$$
- Evaluating the intersection with the abscissa axis of the parabola passing by point $(0,1)$ and leaned against the first autocorrelation values;
- Using Equation (6) but evaluating it starting from the parabola previously defined. Giving the parabola general equation $y=a{x}^{2}+bx+c$, the formula becomes$$\lambda =\sqrt{-\frac{fc}{2a}}$$

## 3. Results and Discussion

#### 3.1. Comparison on Three Complete Long Timeseries

#### 3.2. Comparison on the Whole Dataset of the Recorded Timeseries

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of the analysis applied to compare the results of the imputation methods: pre-processing phase, imputation phase applying three different methods and their average for the interpolation of missing values, the comparison phase comprising two parallel procedures working on different datasets (three complete long timeseries vs. the whole dataset), and both comparing the results of three statistical analyses on the imputed timeseries. Blue squares and arrows define the main phases of the analysis.

**Figure 2.**Map of the sensors within the study area: the 79 sensors included in the pre-processed dataset (blue) and the three identified sensors for the first comparative procedure on complete long timeseries (red).

**Figure 3.**Cubic Spline interpolation applied to the three timeseries (sensors 31, 37, and 57). For each sensor, a plot shows the comparison of the original complete recorded timeseries (in gray) with the interpolated timeseries (in red) of the fragmented timeseries (in black). On the right, the map shows the location of the three sample timeseries within the domain.

**Figure 4.**The autoregressive linear model applied to the three long timeseries (sensors 31, 37, and 57). For each sensor, a plot shows the comparison of the original complete recorded timeseries (in gray) with the interpolated timeseries (in red) of the fragmented timeseries (in black). On the right, the map shows the locations of the three sample timeseries within the domain.

**Figure 5.**(

**a**) Representation in time of the maps obtained with the ordinary kriging. (

**b**) Example of the extraction of a timeseries in a certain location. Source: [34].

**Figure 6.**The patched kriging applied to the three long timeseries (sensors 31, 37, and 57). For each sensor, a plot shows the comparison of the original complete recorded timeseries (in gray) with the interpolated timeseries of the fragmented timeseries (in black). The plots show three different patched kriging interpolations differing on the applied range adjustment (RA): no RA (in red), series RA (in yellow), and interval RA (in violet). On the right, the map shows the locations of the three sample timeseries within the domain.

**Figure 7.**Statistical results comparison for the interpolation of sensor 31: (

**a**) Records in time domain; (

**b**) Phase Spectra; (

**c**) Power Spectra, and a table of the three most relevant periodicities (P1, P2, P3) identified in the original and in each interpolated timeseries; (

**d**) Autocorrelation functions; (

**e**) Ranges of evaluated microscales—plotted as horizontal green bars. Within plots ‘e’, continuous lines are the autocorrelation functions, while the dashed lines are the fitting parabolas. In all plots, black lines/bars represent the original complete observed record (named ‘Rec’) where missing data have been artificially located and then interpolated. The tested imputation methods are cubic spline (CS in pink), autoregressive linear model (AR in light blue), patched kriging (PK in red), and their average (Av in blue)—the legend is shown only in plots ‘a’, but it is valid for all plots. Note that the Power Spectra (plots ‘c’) has two y-axes due to the different magnitudes: Power of the CS on the left, and Power of the other three imputation methods and the original record on the right.

**Figure 8.**Statistical results comparison for the interpolation of sensor 37: (

**a**) Records in time domain; (

**b**) Phase Spectra; (

**c**) Power Spectra, and a table of the three most relevant periodicities (P1, P2, P3) identified in the original and in each interpolated timeseries; (

**d**) Autocorrelation functions; (

**e**) Ranges of evaluated microscales—plotted as horizontal green bars. Within plots ‘e’, continuous lines are the autocorrelation functions, while dashed lines are the fitting parabolas. In all plots, black lines/bars represent the original complete observed record (named ‘Rec’), where missing data have been artificially located and then interpolated. Tested imputation methods are cubic spline (CS in pink), autoregressive linear model (AR in light blue), patched kriging (PK in red), and their average (Av in blue)—the legend is shown only in plots ‘a’, but it is valid for all plots. Note that the Power Spectra (plots ‘c’) has two y-axes due to the different magnitudes: Power of the CS on the left, and Power of the other three imputation methods and the original record on the right.

**Figure 9.**Statistical results comparison for the interpolation of sensor 57: (

**a**) Records in time domain; (

**b**) Phase Spectra; (

**c**) Power Spectra, and a table of the three most relevant periodicities (P1, P2, P3) identified in the original and in each interpolated timeseries; (

**d**) Autocorrelation functions; (

**e**) Ranges of evaluated microscales—plotted as horizontal green bars. Within plots ‘e’, continuous lines are the autocorrelation functions, while dashed lines are the fitting parabolas. In all plots, black lines/bars represent the original complete observed record (named ‘Rec’) where missing data have been artificially located and then interpolated. Tested imputation methods are cubic spline (CS in pink), autoregressive linear model (AR in light blue), patched kriging (PK in red), and their average (Av in blue)—the legend is shown only in plots ‘a’, but it is valid for all plots. Note that the Power Spectra (plots ‘c’) has two y-axes due to the different magnitudes: Power of the CS on the left, and Power of the other three imputation methods, and the original record on the right.

**Figure 10.**Statistical results comparison for the interpolation of sensor 92: (

**a**) Records in time domain, (

**b**) Phase Spectra, (

**c**) Power Spectra, (

**d**) Autocorrelation functions, (

**e**) Ranges of evaluated microscales—plotted as horizontal green bars. Within plots ‘e’, continuous lines are the autocorrelation functions, while dashed lines are the fitting parabolas. Tested imputation methods are cubic spline (CS in pink), autoregressive linear model (AR in light blue), patched kriging (PK in red), and their average (Av in blue)—the legend is shown only in plots ‘a’, but it is valid for all plots. In red are the percentages of missing values in the recorded timeseries.

**Figure 11.**Relationships between variability indicators and the percentage of missing data. Estimated statistical indicators are (

**a**) the variance ${\sigma}^{2}$ of five identified main periodicities for the Fourier analysis, (

**b**) the average maximum distance between evaluated autocorrelation functions (Equation (9)), the coefficient of variation $CV$, considering (

**c**) maximum, (

**d**) minimum, (

**e**) mean microscale values, and (

**f**) their ranges (Equation (10)). Tested imputation methods are Cubic Splice CS, Autoregressive linear model AR, Patched Kriging PK, and their average.

**Figure 12.**(

**a**) Map of the variability indicator for the Fourier Analysis and the periodicities variance. (

**b**) Map of the prefered imputation method for each sensor (Cubic Spline CS, Autoregressive linear model AR, Patched Kriging PK, and their average). In both maps, the symbol dimension is related to the percentage of missing values in the recorded timeseries, and the yellow crosses represent the 26 timeseries without missing data.

**Table 1.**Percentage daily errors of patched kriging interpolations differing on the applied range adjustment (RA): no RA, series RA, and interval RA for sensors 31, 37, and 57.

No RA | Series RA | Interval RA | |
---|---|---|---|

Sensor 31 | 5.00 | 4.76 | 11.71 |

Sensor 37 | 165.08 | 11.12 | 19.52 |

Sensor 57 | 13.76 | 12.24 | 16.34 |

**Table 2.**Estimated R2 between the imputed timeseries, using the imputation methods (Cubic Spline CS, Autoregressive linear model AR, Patched Kriging PK, and their average) and the original recorded timeseries for each analyzed long timeseries. R2 estimated only for the interpolated intervals, roughly $35\%$ of the complete timeseries.

Sensor 31 | Sensor 37 | Sensor 57 | |
---|---|---|---|

CS | 0.34 | 0.18 | 0.00 |

AR | 0.69 | 0.40 | 0.08 |

PK | 0.84 | 0.39 | 0.01 |

Average | 0.80 | 0.48 | 0.07 |

**Table 3.**Euclidean distances between the autocorrelation functions estimated from the timeseries interpolated by the imputation methods (Cubic Spline CS, Autoregressive linear model AR, Patched Kriging PK, and their average) and the autocorrelation function estimated from the original recorded timeseries.

Sensor 31 | Sensor 37 | Sensor 57 | |
---|---|---|---|

CS | 0.0680 | 0.1420 | 0.1243 |

AR | 0.0543 | 0.0783 | 0.0855 |

PK | 0.0373 | 0.0359 | 0.0738 |

Average | 0.0455 | 0.0503 | 0.0703 |

**Table 4.**Five main periodicities (years), identified using each imputation method, observing the estimated power spectrum-sensor 92.

1st Peak | 2nd Peak | 3rd Peak | 4th Peak | 5th Peak | |
---|---|---|---|---|---|

CS | 0.42 | 7.17 | 0.80 | 1.19 | 1.79 |

AR | 1.79 | 2.39 | 7.17 | 0.42 | 0.51 |

PK | 1.79 | 2.39 | 7.17 | 0.42 | 0.51 |

Average | 1.79 | 0.42 | 7.17 | 2.39 | 1.19 |

**Table 5.**Five main standardized periodicities identified by each imputation method, observing the estimated power spectrum-sensor 92. For each power peak, the variances ${\sigma}^{2}$ of the standardized periodicities of the four elaborated power spectra are reported in blue.

1st Peak | 2nd Peak | 3rd Peak | 4th Peak | 5th Peak | |
---|---|---|---|---|---|

CS | 0.06 | 1.00 | 0.11 | 0.17 | 0.25 |

AR | 0.25 | 0.33 | 1.00 | 0.06 | 0.07 |

PK | 0.25 | 0.33 | 1.00 | 0.06 | 0.07 |

Average | 0.25 | 0.06 | 1.00 | 0.33 | 0.17 |

${\sigma}_{92}^{2}\times {10}^{-2}$ | 0.9 | 16.1 | 19.7 | 1.68 | 0.74 |

**Table 6.**Calculations for the four microscale variability indicators referring to the maximum, minimum, mean, and overall range of the estimated microscale-sensor 92. For each indicator, the standard deviation $\sigma $, the mean $\mu $, and the coefficient of variation $CV$ are reported in blue.

Max | Min | Mean | Range | |
---|---|---|---|---|

CS | 11.31 | 6.67 | 8.65 | 4.64 |

AR | 10.80 | 7.55 | 9.14 | 3.24 |

PK | 11.02 | 7.55 | 9.26 | 3.46 |

Average | 10.74 | 6.92 | 8.65 | 3.82 |

${\sigma}_{92}$ | 0.26 | 0.45 | 0.32 | 0.61 |

${\mu}_{92}$ | 10.97 | 7.17 | 8.92 | 3.79 |

$C{V}_{92}\times {10}^{-2}$ | 2.38 | 6.26 | 3.59 | 16.2 |

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**MDPI and ACS Style**

Meggiorin, M.; Passadore, G.; Bertoldo, S.; Sottani, A.; Rinaldo, A.
Comparison of Three Imputation Methods for Groundwater Level Timeseries. *Water* **2023**, *15*, 801.
https://doi.org/10.3390/w15040801

**AMA Style**

Meggiorin M, Passadore G, Bertoldo S, Sottani A, Rinaldo A.
Comparison of Three Imputation Methods for Groundwater Level Timeseries. *Water*. 2023; 15(4):801.
https://doi.org/10.3390/w15040801

**Chicago/Turabian Style**

Meggiorin, Mara, Giulia Passadore, Silvia Bertoldo, Andrea Sottani, and Andrea Rinaldo.
2023. "Comparison of Three Imputation Methods for Groundwater Level Timeseries" *Water* 15, no. 4: 801.
https://doi.org/10.3390/w15040801