# Estimation of Daily Water Table Level with Bimonthly Measurements in Restored Ombrotrophic Peatland

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}= 0.95, RMSE = 2.48 cm and the lowest AIC), followed by the general linear method (R

^{2}= 0.92, RMSE = 3.10 cm) and support vector machines method (R

^{2}= 0.91, RMSE = 3.24 cm). To estimate daily values, the TSD method, like the six traditional methods, requires daily data from a reference well. However, the TSD method does not require training nor parameter estimation. For the TSD method, changing the measurement frequency to weekly measurements decreases the RMSE by 16% (2.08 cm); monthly measurements increase the RMSE by 13% (2.80 cm).

## 1. Introduction

## 2. An Overview of Daily Water Table Depth Estimation Methods

#### 2.1. Estimation Methods

#### 2.2. General Lineal Model (GLM)

^{2}) of 0.55. The dataset was then used to calculate the optimal range of days for gross ecosystem productivity calculation.

#### 2.3. k-Nearest Neighbours (KNN)

_{i}is made within the range of observed water table depth for the reference well, x

_{min}and x

_{max}, and the estimated value for ŷ

_{i}is the average of the observed values closest to x

_{i}.

#### 2.4. Support Vector Machines (SVM)

^{2}grater than 0.93) for groundwater level forecasting, its interpretability is low.

#### 2.5. Decision-Tree-Based Models: Regression Tree (TREE) and Random Forest (RF)

^{2}grater than 0.9) in hydrology applications (e.g., groundwater pollution and groundwater forecasting) in comparison to the set of data-driven methods (R

^{2}between 0.5 and 0.9), explained before [66,69,71,72].

#### 2.6. Adaptive Boosting (ADABOOST)

#### 2.7. Closing Considerations About Data-Driven Methods

## 3. Time Series Decomposition (TSD) Method: The New Proposed Method

^{e}(t) = λ

^{e}(t) + ρ

^{r}(t),

^{e}(t) = h

^{e}

_{1}·ψ

_{1}(t) + h

^{e}

_{2}·ψ

_{2}(t),

^{e}(t) refers to the trend component for a specific period, h

_{1}and h

_{2}are the observed values of the water table depth at the beginning and the end of the period, respectively, and ψ

_{1}and ψ

_{2}are called the partitions of unity and are functions of t, and they are calculated by Equations (3) and (4):

_{1}(t) = (t

_{2}− t)/(t

_{2}− t

_{1}),

_{2}(t) = (t − t

_{1})/(t

_{2}− t

_{1}),

_{1}and t

_{2}are the time at the beginning and the end of each period.

^{r}) is deducted with Equation (5):

^{r}(t) = h

^{r}(t) − λ

^{r}(t),

^{r}(t) is the observed water table depth in the reference well.

^{e}(t), solid red line) is equal to the sum of its trend component (λ

^{e}(t), dashed red line) and the daily fluctuation component of the reference well (ρ

^{r}(t), gray hatching).

^{e}(t) = λ

^{e}(t) + (1/m) ∑ρ

^{ri}(t),

## 4. Materials and Methods

#### 4.1. Requirements for Testing Methods

#### 4.2. Study Area

#### 4.3. Water Table Depth Monitoring

^{®}Edge—Model 3001 (Solinst Canada Ltd., Georgetown, ON, Canada, accuracy: ±0.1 kPa) during the 2016 and 2017 growing season (20 May to 18 October). Water table depth was measured in six wells per water management systems (basin) and their locations varied according to the type of basin (Figure 3). The data loggers were placed inside the 30 wells of the site to simultaneously record pressure and temperature. The wells were made of 2 in diameter PVC pipe. The wells were installed at a depth of approximately 70 cm using an auger. The wells had nylon stockings on the outer surface to prevent the entry of solids in suspension. All measurements were corrected with the air pressure Barologger Gold—Model 3001 (Solinst Canada Ltd., Georgetown, ON, Canada, accuracy: ±0.1 kPa) with the Solinst Levelogger Series software.

_{max, i}) and the minimum value (h

_{min, i}) recorded during each day of the growing season (Equation (7)).

_{i}= (h

_{max, i}+ h

_{min},

_{i})/2

#### 4.4. Bimonthly Measurements

#### 4.5. Estimation Methods Implementation, Calibration and Validation

#### 4.6. Data Analysis and Method Performance

^{2}), the root-mean-square error (RMSE), the Nash–Sutcliffe coefficient (NS), and the Akaike information criterion (AIC). These coefficients were calculated according to Equations (8)–(11).

^{2}= 1 − [∑ (h

_{i}− ĥ

_{i})

^{2}]/[∑(h

_{i}

^{2}) − (1/N) ∑(ĥ

_{i}

^{2})],

_{i}− ĥ

_{i})

^{2}/ N],

_{i}− ĥ

_{i})

^{2}]/ [∑ (h

_{i}− ħ

_{i})

^{2}],

_{i}− ĥ

_{i})

^{2}/N)] + 2k,

_{i}is the observed water table depth, ĥ

_{i}is the estimated water table depth from each method, ħ

_{i}is the mean of observed water table depth, N is the number of observations and k is the number of estimated parameters. The best fit between simulated and observed data shows the RMSE closer to zero, the AIC lower, the NS and R

^{2}closer to one. In this study, RMSE and NS statistics are used to measure the method performance for forecasting water table depth and AIC is used to compare the performance of methods regarding accuracy and complexity, whereas R

^{2}is used to analyze the linear regression goodness of fit between observed and estimated data. Moreover, for the best fit between simulated and observed data, the intercept and gradient should be close to zero and one respectively to observe over- or under-predictions.

#### 4.7. Impact on a Practical Application: Sum of Daily Deficit of Water Table Depth

_{15}) was used to study the error generated by daily estimates from the different methods on the computation of this indicator. This sum is computed for each well via Equation (12).

_{15}= ∑ h

_{i}− 15) for h

_{i}> 15

_{15}values were computed with the data from the 151 days of observation (20 May to 18 October) for both years (2016 and 2017). The SDW

_{15}from estimated and observed water table depths were compared using the same performance criteria as in the previous section.

## 5. Results

#### 5.1. Water Table Observations Statistics

#### 5.2. Methods Performance

^{2}greater than 80%). The TSD method offers the best performance (R = 0.97, R

^{2}= 0.95 and RMSE = 2.48 cm). The GLM and SVM methods show similar performance (for GLM R = 0.96, R

^{2}= 0.92 and RMSE = 3.10 cm; for SVM R = 0.95, R

^{2}= 0.91 and RMSE = 3.24 cm). Finally, KNN, RF, ADABOOST and TREE were the least performing methods.

#### 5.3. Impact on the Computed Daily Indicator

_{15}computed for the six observation wells in each basin. The data show variability within each basin and between basins. Three groups were identified according to the Nemenyi multiple non-parametric comparison test. The first grouping points to where water table depth remained, most of the time, above or close to 15 cm (a small SDW

_{15}value, less than 260 cm·days). This group consists of the wells in basins PC-10 and CC-10. The second group is the basins with a high SDW

_{15}value (greater than 1200 cm·days), which means that the water table depth was repeatedly below 15 cm and/or even reached greater depths. This group consists of the wells located at basins PC-NI and PC-20. Finally, the third group consists of the wells in CC-20, which are somewhere in between the two previous groupings.

_{15}. As expected, estimations of water table depth by the TSD method to compute the SDW

_{15}is the best performing method with the highest R

^{2}and the lowest RMSE. An RMSE of 131 cm·days is quite low in regard of the range of computed SDW

_{15}(Table 5).

#### 5.4. Selection of the Reference Well

- A reference well within the basin: One well was randomly selected per basin as the reference well and, the water table depths for the remaining five wells in the basin were re-estimated. This was done for every basin;
- A reference well within another basin: The same reference wells of the previous case were chosen, but in this case, the estimation of the daily water table depths is made over the wells of all basins. The procedure is repeated for each reference well and is identified as a run in Table 7.

_{i}− ĥ

_{i}) is not influenced by the distance to the reference well belonging to the same basin.

#### 5.5. Measurement Frequency

## 6. Discussion

#### 6.1. TSD Method Performance Explanation

- First, TSD uses an appropriate methodological principle. It estimates the daily water table depth as the result of a local component and a regional component, which is observed in real data (Figure 1). This type of method considers regional sensitivity [80] and uses a physical concept, which is advisable [61]. Moreover, the TSD method keeps the known data (bimonthly observations) for the estimated well. The other methods generate new data, even for the observed data, which is contra-intuitive;
- Second, this method considers the time series properties, which the other methods do not consider. Time series data show auto-correlation from day-to-day data which the TSD method captures by the trend component. The other methods consider daily data as independent. Furthermore, the TSD method also captures the local impact of daily phenomena (precipitation, irrigation) through the daily fluctuation from the trend of the reference well without any additional step. Therefore, the estimated water table hydrograph is more realistic than those obtained by the remaining methods (Figure 5).

#### 6.2. Estimation Performance by the Range of Water Table Depth Variation

^{2}than the wells with greater variation of the water table level (SD value greater than 8 cm, as the example Figure 5, red lines). However, the RMSE is lower for wells with less variation of the water table observations. The range of variation is smaller and for the calculation of R

^{2}(Equation (8)), the denominator [∑(h

_{i}

^{2}) − (1/N) ∑(ĥ

_{i}

^{2})] becomes smaller. This causes the R

^{2}to decrease, even though the RMSE is low.

#### 6.3. Estimation of Daily Indicator

_{15}values (Table 6), TSD is the method that estimates values with the least RMSE value. The performance of the method for estimating SDW

_{15}follows a similar order of the performance for estimating the daily water table depth, which is not surprising. Because SDW

_{15}is a sum, the probable error accumulates according to the square root of the several measurements [90], in this case, 151 measurements. The probable error of the SDW

_{15}based on the estimates can be expressed as Equation (13):

^{2}

_{SDW15}≈ (N − n) RMSE

^{2}+ n E

_{h}

^{2},

#### 6.4. Choice of the Reference Well

## 7. Conclusions

^{2}= 0.95, RMSE = 2.48 cm, NS = 0.95 and the lowest AIC), followed by GLM (R

^{2}= 0.92, RMSE = 3.10 cm, NS = 0.92) and SVM (R

^{2}= 0.91, RMSE = 3.24 cm, NS = 0.91).

_{15}, a way of quantifying daily water stress. This indicator varies according to the location of the well and the basin type with computed values between 0 and 2860 cm·days. The TSD method is the best method computing SWD

_{15}(R

^{2}= 0.98, RMSE = 131 cm·days, NS = 0.98), which is not surprising.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Machine learning architecture for GLM, SVM, KNN and ADABOOST methods. The input layer is composed of the bimonthly water table levels of the estimated (h

^{e}) and the reference well (h

^{r}). The hidden layer depends on the method, as explained in Section 2.

**Figure A2.**Machine learning architecture for TREE and RF. In the case of the TREE method, only one decision tree is established.

## References

- Rydin, H.; Jeglum, J. The Biology of Peatlands, 2nd ed.; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Campbell, D.R.; Rochefort, L. La végétation: Gradients. In Écologie des tourbières du Québec-Labrador; Payette, S., Rochefort, L., Eds.; Presses de l’Université Laval: Québec, QC, Canada, 2001; pp. 129–140. [Google Scholar]
- Andersen, R.; Grasset, L.; Thormann, M.N.; Rochefort, L.; Francez, A.-J. Changes in microbial community structure and function following Sphagnum peatland restoration. Soil Biol. Biochem.
**2009**, 42, 291–301. [Google Scholar] [CrossRef] - Wilson, D.; Blain, D.; Couwenberg, J.; Evans, C.D.; Murdiyarso, D.; Page, S.E.; Renou-Wilson, F.; Rieley, J.O.; Sirin, A.; Strack, M.; et al. Greenhouse gas emission factors associated with rewetting of organic soils. Mires Peat
**2016**, 17, 4. [Google Scholar] [CrossRef] - Taylor, N.; Price, J.; Strack, M. Hydrological controls on productivity of regenerating Sphagnum in a cutover peatland. Ecohydrology
**2016**, 9, 1017–1027. [Google Scholar] [CrossRef] - McCarter, C.P.R.; Price, J.S. The hydrology of the Bois-des-Bel bog peatland restoration: 10 years post-restoration. Ecol. Eng.
**2013**, 55, 73–81. [Google Scholar] [CrossRef] - Ju, W.; Chen, J.M.; Black, T.A.; Barr, A.G.; McCaughey, H.; Roulet, N.T. Hydrological effects on carbon cycles of Canada’s forests and wetlands. Tellus Ser. B Chem. Phys. Meteorol.
**2006**, 58, 16–30. [Google Scholar] [CrossRef] - Van Seters, T.E.; Price, J.S. The impact of peat harvesting nad natural regeneration on the water balance of an abandoned cutover bog, Quebec. Hydrol. Process.
**2001**, 15, 233–248. [Google Scholar] [CrossRef] - Holden, J. Peatland Hydrology and Carbon Release: Why Small-Scale Process Matters. Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci.
**2005**, 363, 2891–2913. [Google Scholar] [CrossRef][Green Version] - Brown, C.M.; Strack, M.; Price, J.S. The effects of water management on the CO2 uptake of Sphagnum moss in a reclaimed peatland. Mires Peat
**2017**, 20, 1–15. [Google Scholar] [CrossRef] - Graf, M.D.; Bérubé, V.; Rochefort, L. Restoration of peatlands after peat extraction: Impacts, restoration goals, and techniques. In Restoration and Reclamation of Boreal Ecosystems; Vitt, D.H., Bhatti, J.S., Eds.; Cambridge University Press: Cambridge, UK, 2012; pp. 259–280. [Google Scholar] [CrossRef]
- Landry, J.; Rochefort, L. The Drainage of Peatlands: Impacts and Rewetting Techniques. Available online: http://www.gret-perg.ulaval.ca/uploads/tx_centrerecherche/Drainage_guide_Web_03.pdf (accessed on 13 June 2020).
- LaRose, S.; Price, J.; Rochefort, L. Rewetting of a cutover peatland: Hydrologic assessment. Wetlands
**1997**, 17, 416–423. [Google Scholar] [CrossRef] - Price, J.S.; Heathwaite, A.L.; Baird, A. Hydrological processes in abandoned and restored peatlands. Wetl. Ecol. Manag.
**2003**, 11, 65–83. [Google Scholar] [CrossRef] - Price, J.S. Hydrology and microclimate of a partly restored cutover bog, Quebec. Hydrol. Process.
**1996**, 10, 1263–1272. [Google Scholar] [CrossRef] - Breeuwer, A.; Robroek, B.J.M.; Limpens, J.; Heijmans, M.M.P.D.; Schouten, G.C.; Berendse, F. Decreased summer water table depth affects peatland vegetation. Basic Appl. Ecol.
**2009**, 10, 330–339. [Google Scholar] [CrossRef] - González, E.; Rochefort, L. Drivers of success in 53 cutover bogs restored by a moss layer transfer technique. Ecol. Eng.
**2014**, 68, 279–290. [Google Scholar] [CrossRef] - Siegel, D.I.; Glaser, P. The hydrology of Peatlands. In Boreal Peatland Ecosystems; Wieder, R.K., Vitt, D.H., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; pp. 289–311. [Google Scholar]
- Paradis, É.; Rochefort, L.; Langlois, M. The lagg ecotone: An integrative part of bog ecosystems in North America. Plant Ecol.
**2015**, 216, 999–1018. [Google Scholar] [CrossRef] - Pellerin, S.; Lagneau, L.A.; Lavoie, M.; Larocque, M. Environmental factors explaining the vegetation patterns in a temperate peatland. C. R. Biol.
**2009**, 332, 720–731. [Google Scholar] [CrossRef][Green Version] - Jutras, S.; Plamondon, A.P.; Hökkä, H.; Bégin, J. Water table changes following precommercial thinning on post-harvest drained wetlands. For. Ecol. Manag.
**2006**, 235, 252–259. [Google Scholar] [CrossRef] - Jutras, S.; Hökkä, H.; Bégin, J.; Plamondon, A.P. Beneficial influence of plant neighbours on tree growth in drained forested peatlands: A case study. Can. J. For. Res.
**2006**, 36, 2341–2350. [Google Scholar] [CrossRef] - Price, J.S. L’hydrologie. In Écologie des tourbières du Québec-Labrador; Payette, S., Rochefort, L., Eds.; Presses de l’Université Laval: Québec, QC, Canada, 2001; pp. 141–158. [Google Scholar]
- Ireland, A.W.; Booth, R.K. Upland deforestation triggered an ecosystem state-shift in a kettle peatland. J. Ecol.
**2012**, 100, 586–596. [Google Scholar] [CrossRef] - Pinceloup, N.; Poulin, M.; Brice, M.-H.; Pellerin, S. Vegetation changes in temperate ombrotrophic peatlands over a 35 year period. PLoS ONE
**2020**, 15, e0229146. [Google Scholar] [CrossRef] - Holden, J.; Wallage, Z.E.; Lane, S.N.; Mcdonald, A.T. Water table dynamics in undisturbed, drained and restored blanket peat. J. Hydrol.
**2011**, 402, 103–114. [Google Scholar] [CrossRef] - Haapalehto, T.; Kotiaho, J.S.; Matilainen, R.; Tahvanainen, T. The effects of long-term drainage and subsequent restoration on water table level and pore water chemistry in boreal peatlands. J. Hydrol.
**2014**, 519, 1492–1505. [Google Scholar] [CrossRef][Green Version] - Mioduszewski, W.; Kowalewski, Z.; Wierzba, M. Impact of peat excavation on water condition in the adjacent raised bog. J. Water Land Dev.
**2013**, 18, 49–57. [Google Scholar] [CrossRef] - Waddington, J.M.; Strack, M.; Greenwood, M.J. Toward restoring the net carbon sink function of degraded peatlands: Short-term response in CO
_{2}exchange to ecosystem-scale restoration. J. Geophys. Res.**2010**, 115, 1–13. [Google Scholar] [CrossRef][Green Version] - Holden, J.; Chapman, P.J.; Labadz, J.C. Artificial drainage of peatlands: Hydrological and hydrochemical process and wetland restoration. Prog. Phys. Geogr. Earth Environ.
**2004**, 28, 95–123. [Google Scholar] [CrossRef][Green Version] - Taylor, N.; Price, J. Soil water dynamics and hydrophysical properties of regenerating Sphagnum layers in a cutover peatland. Hydrol. Process.
**2015**, 29, 3878–3892. [Google Scholar] [CrossRef] - Price, J.S.; Whitehead, G.S. Developing hydrologic thresholds for sphagnum recolonization on an abandoned cutover bog. Wetlands
**2001**, 21, 32–40. [Google Scholar] [CrossRef] - Guêné-Nanchen, M.; Pouliot, R.; Hugron, S.; Rochefort, L. Effect of repeated mowing to reduce graminoid plant cover on the moss carpet at a sphagnum farm in North America. Mires Peat
**2017**, 20. [Google Scholar] [CrossRef] - Holden, J.; Swindles, G.; Raby, C.; Blundell, A. How well do testate amoebae transfer functions relate to high-resolution water-table records? In Proceedings of the EGU Geophysical Research Abstracts, Vienna, Austria, 27 April–2 May 2014; Volume 16, pp. 2014–2487. [Google Scholar]
- Parry, L.E.; Holden, J.; Chapman, P.J. Restoration of blanket peatlands. J. Environ. Manag.
**2013**, 133, 193–205. [Google Scholar] [CrossRef][Green Version] - Shaffer, P.W.; Cole, C.A.; Kentula, M.E.; Brooks, R.P. Effects of measurement frequency on water-level summary statistics. Wetlands
**2000**, 20, 148–161. [Google Scholar] [CrossRef] - BWSR Minnesota Board of Water & Soil Resources Hydrologic Monitoring of Wetlands MN Board of Water & Soil Resources Supplemental Guidance. Available online: Ttps://bwsr.state.mn.us/sites/default/files/2018-12/WETLANDS_delin_Hydrologic_Monitoring_of_Wetlands_Guidance_BWSR.pdf (accessed on 4 August 2020).
- Pouliot, R.; Hugron, S.; Rochefort, L. Sphagnum farming: A long-term study on producing peat moss biomass sustainably. Ecol. Eng.
**2014**, 74, 135–147. [Google Scholar] [CrossRef] - Hawes, M. The Hydrology of Passive and Active Restoration in Abandoned Vacuum Extracted Peatlands, Southeast Manitoba. Ph.D. Thesis, Brandon University, Brandon, MB, Canada, 2018. [Google Scholar]
- Dimitrov, D.D.; Grant, R.F.; Lafleur, P.M.; Humphreys, E.R. Modeling the Subsurface Hydrology of Mer Bleue Bog. Soil Sci. Soc. Am. J.
**2010**, 74, 680–694. [Google Scholar] [CrossRef] - St-Hilaire, F.; Wu, J.; Roulet, N.T.; Frolking, S.; Lafleur, P.M.; Humphreys, E.R.; Arora, V. McGill wetland model: Evaluation of a peatland carbon simulator developed for global assessments. Biogeosciences
**2010**, 7, 3517–3530. [Google Scholar] [CrossRef][Green Version] - Frolking, S.; Roulet, N.T.; Moore, T.R.; Lafleur, P.M.; Bubier, J.L.; Crill, P.M. Modeling seasonal to annual carbon balance of Mer Bleue Bog, Ontario, Canada. Global Biogeochem. Cycles
**2002**, 16, 1030. [Google Scholar] [CrossRef] - Daliakopoulos, I.N.; Coulibaly, P.; Tsanis, I.K. Groundwater level forecasting using artificial neural networks. J. Hydrol.
**2005**, 309, 229–240. [Google Scholar] [CrossRef] - Coppola, E.; Poulton, M.; Charles, E.; Dustman, J.; Szidarovszky, F. Application of artificial neural networks to complex groundwater management problems. Nat. Resour. Res.
**2003**, 12, 303–320. [Google Scholar] [CrossRef] - Taormina, R.; Chau, K.-W.; Sethi, R. Artificial neural network simulation of hourly groundwater levels in a coastal aquifer system of the Venice lagoon. Eng. Appl. Artif. Intell.
**2012**, 25, 1670–1676. [Google Scholar] [CrossRef][Green Version] - Mohanty, S.; Jha, M.K.; Kumar, A.; Sudheer, K.P. Artificial neural network modeling for groundwater level forecasting in a river island of eastern India. Water Resour. Manag.
**2010**, 24, 1845–1865. [Google Scholar] [CrossRef] - Coulibaly, P.; Anctil, F.; Aravena, R.; Bobée, B. Artificial neural network modeling of water table depth fluctuations. Water Resour. Res.
**2001**, 37, 885–896. [Google Scholar] [CrossRef] - Nayak, P.C.; Satyaji Rao, Y.R.; Sudheer, K.P. Groundwater level forecasting in a shallow aquifer using artificial neural network approach. Water Resour. Manag.
**2006**, 20, 77–90. [Google Scholar] [CrossRef] - Freedman, D.A. Chapter 2. Regression Line. In Statistical Models: Theory and Practice; Freedman, D.A., Ed.; Cambridge University Press: Berkeley, CA, USA, 2009; pp. 18–28. [Google Scholar]
- Lu, W.X.; Zhao, Y.; Chu, H.B.; Yang, L.L. The analysis of groundwater levels influenced by dual factors in western Jilin Province by using time series analysis method. Appl. Water Sci.
**2014**, 4, 251–260. [Google Scholar] [CrossRef][Green Version] - Choubin, B.; Malekian, A. Combined gamma and M-test-based ANN and ARIMA models for groundwater fluctuation forecasting in semiarid regions. Environ. Earth Sci.
**2017**, 76, 1–10. [Google Scholar] [CrossRef] - Wang, W. Stochasticity, Nonlinearity and Forecasting of Streamflow Processes; IOS Press: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Cover, T.M.; Hart, P.E. Nearest Neighbor Pattern Classification. IEEE Trans. Inf. Theory
**1967**, 13, 21–27. [Google Scholar] [CrossRef] - Lall, U.; Sharma, A. A Nearest Neighbor Bootstrap For Resampling Hydrologic Time Series. Water Resour. Res.
**1996**, 32, 679–693. [Google Scholar] [CrossRef] - Raschka, S.; Vahid, M. Python Machine Learning. Machine Learning Deep Learning with Python, Scikit-Learn, and TensorFlow, 2nd ed.; Packt Publishing Ltd.: Brimingham, UK, 2017. [Google Scholar]
- Modaresi, F.; Araghinejad, S. A comparative assessment of support vector machines, probabilistic neural networks, and K-nearest neighbor algorithms for water quality classification. Water Resour. Manag.
**2014**, 28, 4095–4111. [Google Scholar] [CrossRef] - Sakizadeh, M.; Mirzaei, R. A comparative study of performance of K-nearest neighbors and support vector machines for classification of groundwater. J. Min. Environ.
**2016**, 7, 149–164. [Google Scholar] [CrossRef] - Vapnik, V.N.; Chervonenkis, A.Y. ОБ ОДНОМ КЛАССЕ АЛГОРИТМОВ ОБУЧЕНИЯ РАСПОЗНАВАНИЮ ОБРАЗОВ (A Class of Algorithms of Learning). Available online: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=at&paperid=11678&option_lang=rus (accessed on 16 June 2020).
- Cortes, C.; Vapnik, V. Support-Vector Networks. Mach. Learn.
**1995**, 20, 273–297. [Google Scholar] [CrossRef] - Witten, I.H.; Frank, E. Data Mining: Practical Machine Learning Tools and Techniques, 2nd ed.; Diane, C., Ed.; Morgan Kaufmann Publishers: Burlington, MA, USA, 2005. [Google Scholar] [CrossRef][Green Version]
- Zhao, T.; Zhu, Y.; Ye, M.; Mao, W.; Zhang, X.; Yang, J.; Wu, J. Machine-Learning Methods for Water Table Depth Prediction in Seasonal Freezing-Thawing Areas. Groundwater
**2020**, 58, 419–431. [Google Scholar] [CrossRef] [PubMed] - Rahman, A.T.M.S.; Hosono, T.; Quilty, J.M.; Das, J.; Basak, A. Multiscale groundwater level forecasting: Coupling new machine learning approaches with wavelet transforms. Adv. Water Resour.
**2020**, 141, 103595. [Google Scholar] [CrossRef] - Breiman, L.; Friedman, J.H.; Olshen, R.A.; Stone, C.J. Classification and Regression Trees; CRC Press: Belmont, CA, USA, 1984; ISBN 9781351460491. [Google Scholar] [CrossRef]
- Quinlan, J.R. Induction of decision trees. Mach. Learn.
**1986**, 1, 81–106. [Google Scholar] [CrossRef][Green Version] - Rokach, L.; Maimon, O. Data Mining with Decision Trees, 2nd ed.; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2015. [Google Scholar]
- Rodriguez-Galiano, V.; Mendes, M.P.; Garcia-Soldado, M.J.; Chica-Olmo, M.; Ribeiro, L. Predictive modeling of groundwater nitrate pollution using Random Forest and multisource variables related to intrinsic and specific vulnerability: A case study in an agricultural setting (Southern Spain). Sci. Total Environ.
**2014**, 476–477, 189–206. [Google Scholar] [CrossRef] - Ho, T.K. Random decision forests. In ICDAR ’95, Proceedings of the Third International Conference on Document Analysis and Recognition, Montreal, QC, Canada, 14–15 August 1995; IEEE Computer Society: Washington, DC, USA, 1995; Volume 1, pp. 278–282. [Google Scholar] [CrossRef]
- Biau, G.; Scornet, E. A random forest guided tour. Test
**2016**, 25, 197–227. [Google Scholar] [CrossRef][Green Version] - Tyralis, H.; Papacharalampous, G. Variable Selection in Time Series Forecasting Using Random Forests. Algorithms
**2017**, 10, 114. [Google Scholar] [CrossRef][Green Version] - Amaranto, A.; Munoz-Arriola, F.; Corzo, G.; Solomatine, D.P.; Meyer, G. Semi-seasonal groundwater forecast using multiple data-driven models in an irrigated cropland. J. Hydroinform.
**2018**, 20, 1227–1246. [Google Scholar] [CrossRef] - Koch, J.; Berger, H.; Henriksen, H.J.; Sonnenborg, T.O. Modelling of the shallow water table at high spatial resolution using random forests. Hydrol. Earth Syst. Sci.
**2019**, 23, 4603–4619. [Google Scholar] [CrossRef][Green Version] - Brédy, J.; Gallichand, J.; Celicourt, P.; Gumiere, S.J. Water table depth forecasting in cranberry fields using two decision-tree-modeling approaches. Agric. Water Manag.
**2020**, 233, 106090. [Google Scholar] [CrossRef] - Schapire, R.E. The Strength of Weak Learnability. Mach. Learn.
**1990**, 5, 197–227. [Google Scholar] [CrossRef][Green Version] - Freund, Y.; Schapire, R.E. Experiments with a new boosting algorithm. In Machine Learning, Proceedings of the Thurteenth International Conference, Bari, Italy, 3–6 July 1996; Morgan Kaufmann: Burlington, MA, USA, 1996; pp. 149–156. [Google Scholar]
- Kégl, B. The return of AdaBoost.MH: Multi-class Hamming trees. In Proceedings of the International Conference on Learning Representations, Banff, AB, Canada, 14–16 April 2014. [Google Scholar]
- Xiao, T.; Zhu, J.; Liu, T. Bagging and Boosting statistical machine translation systems. Artif. Intell.
**2013**, 195, 496–527. [Google Scholar] [CrossRef][Green Version] - Albon, C. Adaboost Classifier. Available online: https://chrisalbon.com/machine_learning/trees_and_forests/adaboost_classifier/ (accessed on 17 June 2020).
- Zanotti, C.; Rotiroti, M.; Sterlacchini, S.; Cappellini, G.; Fumagalli, L.; Stefania, G.A.; Nannucci, M.S.; Leoni, B.; Bonomi, T. Choosing between linear and nonlinear models and avoiding overfitting for short and long term groundwater level forecasting in a linear system. J. Hydrol.
**2019**, 578, 124015. [Google Scholar] [CrossRef] - Amaranto, A.; Pianosi, F.; Solomatine, D.; Corzo, G.; Muñoz-Arriola, F. Sensitivity analysis of data-driven groundwater forecasts to hydroclimatic controls in irrigated croplands. J. Hydrol.
**2020**, 587, 124957. [Google Scholar] [CrossRef] - Corzo, G.; Solomatine, D. Baseflow separation techniques for modular artificial neural network modelling in flow forecasting. Hydrol. Sci. J.
**2007**, 52, 491–507. [Google Scholar] [CrossRef] - Hyndman, R.J.; Athanasopoulos, G. Forecasting: Principles and Practice, 2nd ed.; OTexts: Melbourne, Australia, 2018; Available online: https://otexts.com/fpp2/ (accessed on 21 June 2020).
- Ginzburg, I.F. Discrete and continuous description of physical phenomena. J. Phys. Conf. Ser.
**2017**, 873, 012046. [Google Scholar] [CrossRef] - Rochefort, L.; Quinty, F.; Campeau, S.; Johnson, K.; Malterer, T. North American approach to the restoration of wetlands. Wetl. Ecol. Manag.
**2003**, 11, 3–20. [Google Scholar] [CrossRef] - Van Rossum, G.; Drake, F.L. Python 3 Reference Manual; CreateSpace: Scotts Valley, CA, USA, 2009; ISBN 10.5555/1593511. [Google Scholar]
- Oliphant, T.E. Guide to NumPy; Massachusetts Institute of Technology: Cambridge, MA, USA, 2006. [Google Scholar]
- McKinney, W. Data structures for statistical computing in Python. In Proceedings of the 9th Python in Science Conference (SciPy 2010), Austin, TX, USA, 28 June–3 July 2010; van der Walt, S., Millman, J., Eds.; pp. 56–61. [Google Scholar] [CrossRef][Green Version]
- Seabold, S.; Perktold, J. Statsmodels: Econometric and statistical modeling with Python. In Proceedings of the 9th Python in Science Conference, Austin, TX, USA, 28 June–3 July 2010; pp. 92–96. [Google Scholar] [CrossRef][Green Version]
- Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef][Green Version] - Lagacé, R. Chapitre 12. Calcul de l’erreur. Available online: http://www.grr.ulaval.ca/gaa_7003/index.html (accessed on 20 June 2020).

**Figure 1.**Example of recorded water table depth decomposition. (

**a**) Recorded water table depths in two nearby wells at a restored site in Eastern Canada in 2017 and decomposition into two components: the trend (

**b**) and the difference from the trend (

**c**).

**Figure 3.**Schematic representation of the site with the location of the basins and typical basin configuration.

**Figure 4.**Taylor diagram with different statistics (correlation coefficient—R, standard deviation, and root mean squared error—RMSE) of the estimated daily water table depth by the seven estimation methods.

**Figure 5.**Comparison between observed (continuous line) and estimated (dotted line) water table depth for two cases: a well in the middle of the PC-NI basin (red lines), and a well located 1 m from the channel in PC-10 basin (blue lines). Data from 2017.

**Figure 6.**Computed values of the error between observed (h

_{i}) and estimated (ĥ

_{i}) water table depths. The estimated water table depths are based on the reference wells belonging to the same basin.

**Table 1.**Set up description for the five experimental basins for Sphagnum farming in Eastern Canada.

Basin ID | Chanel Configuration | Target Water Table Depth (cm) |
---|---|---|

PC-NI | Peripheral, non-irrigated | None |

CC-20 | Central | 20 |

CC-10 | Central | 10 |

PC-20 | Peripheral | 20 |

PC-10 | Peripheral | 10 |

Method | Hyperparameters of the Method | Estimated Parameters for Regression | k |
---|---|---|---|

TSD | No training | 0 | 0 |

GLM | No special consideration | 2 | a_{1}, b_{1} |

SVM | degree of the polynomial function = 1 linear kernel gamma coefficient automatic | 2 | w_{2}, b_{2} |

RF | random state equals zero n_estimators = 2 max depth of the tree = 2 | 2 | f_{1}, e |

KNN | n_neighbours = 1 weights based in distance | 2 | f_{1}, e |

ADABOOST | random state equals zero n_estimators = 1 | 2 | f_{1}, e |

TREE | split criteria set by default random state equals zero max depth = 2 min_samples_leaf = 0.3 | 2 | f_{1}, e |

**Table 3.**Summary of water table depth in the five experimental basins. Number of observations per well (N obs), number of wells per basin (N wells) and descriptive statistics of water table depth: mean, standard deviation (SD), minimum (min) and maximum (max).

Basin ID | N Obs | N Wells | Mean ^{1} | SD | Min | Max ^{2} |
---|---|---|---|---|---|---|

2016 | ||||||

PC-NI | 151 | 6 | 25.33 ^{d} | 11.06 | 54.25 | 0.1 |

CC-20 | 151 | 6 | 21.14 ^{c} | 7.6 | 40.05 | −0.6 |

CC-10 | 151 | 6 | 12.52 ^{b} | 6.94 | 34.3 | −1.85 |

PC-20 | 151 | 6 | 24.44 ^{d} | 8.66 | 44.55 | 2.25 |

PC-10 | 151 | 6 | 9.34 ^{a} | 4.75 | 25.3 | −3.5 |

2017 | ||||||

PC-NI | 151 | 6 | 26.9 ^{d} | 12.48 | 51.6 | −1.4 |

CC-20 | 151 | 6 | 21.39 ^{c} | 7.45 | 44.4 | −0.45 |

CC-10 | 151 | 6 | 11.73 ^{b} | 6.27 | 35.4 | −0.5 |

PC-20 | 151 | 6 | 24.99 ^{d} | 8.24 | 43.05 | −1.4 |

PC-10 | 151 | 6 | 5.43 ^{a} | 3.3 | 21.25 | −1.4 |

^{1}Means followed by different letters indicate differences, according to Nemenyi (non-parametric test).

^{2}Negative values represent levels above ground level.

**Table 4.**Performance criteria for daily water table depth (cm) estimations by the different methods.

Performance Criteria | Methods | ||||||
---|---|---|---|---|---|---|---|

TSD | GLM | SVM | RF | KNN | ADABOOST | TREE | |

R^{2} | 0.95 | 0.92 | 0.91 | 0.88 | 0.88 | 0.86 | 0.82 |

RMSE (cm) | 2.48 | 3.10 | 3.24 | 3.84 | 3.87 | 4.19 | 4.68 |

NS | 0.95 | 0.92 | 0.91 | 0.88 | 0.87 | 0.85 | 0.82 |

AIC | 7628 | 10,241 | 10,659 | 12,202 | 12,257 | 12,983 | 13,989 |

**Table 5.**Mean SDW

_{15}values (cm·days) computed with the water table level observed in the six wells of each basin. Values between parentheses represent the 95% confidence interval.

Basin ID | 2016 | 2017 |
---|---|---|

PC-NI | 1729 ^{b} (1299–2158) | 1991 ^{b} (1552–2430) |

CC-20 | 1065 ^{ab} (803–1326) | 1107 ^{ab} (644–1573) |

CC-10 | 261 ^{a} (0–531) | 216 ^{a} (0–467) |

PC-20 | 1495 ^{b} (1008–1983) | 1582 ^{b} (1142–2021) |

PC-10 | 23 ^{a} (0–51) | 1 ^{a} (0–3) |

Performance Criteria | Methods | ||||||
---|---|---|---|---|---|---|---|

TSD | GLM | SVM | RF | KNN | ADABOOST | TREE | |

R^{2} | 0.98 | 0.95 | 0.94 | 0.95 | 0.96 | 0.89 | 0.95 |

RMSE (cm·days) | 131 | 200 | 215 | 198 | 182 | 377 | 201 |

NS | 0.98 | 0.95 | 0.94 | 0.95 | 0.96 | 0.87 | 0.95 |

Basin ID | Nearest Well | Within the Basin | In Another Basin | ||||
---|---|---|---|---|---|---|---|

Run 1 | Run 2 | Run 3 | Run 4 | Run 5 | |||

PC-NI ^{1} | CC-20 ^{1} | CC-10 ^{1} | PC-20 ^{1} | PC-10 ^{1} | |||

PC-NI | 3.38 | 3.29 | - | 3.89 | 3.41 | 2.99 | 3.47 |

CC-20 | 2.71 | 3.97 | 3.76 | - | 2.68 | 2.92 | 2.94 |

CC-10 | 1.78 | 2.29 | 4.47 | 4.50 | - | 3.26 | 1.83 |

PC-20 | 2.85 | 2.36 | 3.34 | 3.40 | 3.01 | - | 2.98 |

PC-10 | 1.10 | 0.97 | 4.52 | 4.45 | 2.52 | 3.21 | - |

Aggregated | 2.48 | 2.77 | 3.45 |

^{1}basin of the reference well.

**Table 8.**Performance criteria for daily estimations of water table depth by the TSD method using different measurement frequencies for infrequent data.

Measurement Frequency | Performance Criteria | ||
---|---|---|---|

R^{2} | RMSE (cm) | NS | |

Weekly | 0.96 | 2.08 | 0.96 |

Bimonthly | 0.95 | 2.48 | 0.95 |

Monthly | 0.94 | 2.80 | 0.93 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gutierrez Pacheco, S.; Lagacé, R.; Hugron, S.; Godbout, S.; Rochefort, L. Estimation of Daily Water Table Level with Bimonthly Measurements in Restored Ombrotrophic Peatland. *Sustainability* **2021**, *13*, 5474.
https://doi.org/10.3390/su13105474

**AMA Style**

Gutierrez Pacheco S, Lagacé R, Hugron S, Godbout S, Rochefort L. Estimation of Daily Water Table Level with Bimonthly Measurements in Restored Ombrotrophic Peatland. *Sustainability*. 2021; 13(10):5474.
https://doi.org/10.3390/su13105474

**Chicago/Turabian Style**

Gutierrez Pacheco, Sebastian, Robert Lagacé, Sandrine Hugron, Stéphane Godbout, and Line Rochefort. 2021. "Estimation of Daily Water Table Level with Bimonthly Measurements in Restored Ombrotrophic Peatland" *Sustainability* 13, no. 10: 5474.
https://doi.org/10.3390/su13105474