# A Combined Stochastic–Analytical Method for the Assessment of Climate Change Impact on Spring Discharge

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}emission scenarios. To investigate the temporal distribution of extremities throughout the simulated time periods, the annual numbers of flood and drought days were calculated. Annual predicted flood days show an increasing trend during the first simulation period (2021–2050) and a slightly decreasing trend during the second simulation period (2071–2100), according to the RCP45 climate scenario. The same parameter shows a stagnant trend for the RCP 85 climate scenario. Annual predicted drought days show a decreasing trend both for the RCP 45 and RCP 85 climate scenarios. However, the annual number of drought days shows a large variation over time. There is a periodicity of extremely dry years with a frequency between 5 and 7 years. The number of drought days seems to increase over time during these extreme years. The study confirmed that the applied methodology can successfully be applied for spring-discharge prediction and that it offers a new prospect for its wider application in studying karst aquifers and their behavior under different climate-change scenarios.

## 1. Introduction

## 2. Precedents

^{2}where L is block size [L], T is the hydraulic transmissivity [L

^{2}T

^{−1}], and S is the storativity [-] of the rock matrix. The work of these authors made the estimation of conduit spacing and rock matrix hydraulic properties possible from hydrograph analysis.

## 3. Materials and Methods

#### 3.1. Hydrograph Features

#### 3.2. Hydrograph Analysis

_{t}is the discharge [L

^{3}T

^{−1}] at time t; Q

_{0}is the initial discharge [L

^{3}T

^{−1}] at an earlier time; and α is the recession coefficient [T

^{−1}], which is usually expressed in days. Plotted on a semi-logarithmic graph, this function is represented as a straight line with the slope α. This equation is usually adequate for describing karst systems at low water stages.

#### 3.3. Analytical Solution

^{2}T

^{−1}] is the block transmissivity, S [-] is the block storativity, L

_{x}and L

_{y}[L] are the block size, and β [-] is the asymmetry factor. It follows from Equation (4) that

#### 3.4. Methodology

- While rainfall is distributed in space, rainfall data are punctual or originate from interpolation between discrete data points.
- While rainfall is distributed in time, rainfall data are discretized in time, and each timestep includes aggregated values of rainfall.
- Weather stations are often located outside of the studied catchment.
- Furthermore, the prediction of discharge is based on regional climate model outputs with a spatial resolution of 12.5 km.

- Selection of climate projections assumed to describe future climate conditions. Out of several possible combinations, the selected climate model included the RCP4.5 and RCP8.5 radiative forcing scenarios. The investigated regional climate models are provided in Table 1;
- Selection of hydrograph peaks and creation of peak-discharge dataset;
- Setting up regression models between rainfall and peak discharge;
- Establishment of a Master Recession Curve (MRC);
- Hydrograph decomposition for determining recession coefficients and the baseflow component of spring discharge;
- Simulation of spring discharge for the calibration period, using measured rainfall and comparison between measured and simulated hydrographs;
- Calibration of the combined stochastic–analytical model through the adjustment of initial values of baseflow components;
- Simulation of spring discharge, making use of rainfall time series from RCM projections for the calibration period;
- Determining the bias correction model of RCM-simulated spring hydrograph by regression analysis, using measured hydrograph;
- Simulation of predictive hydrographs based on the calibrated combined stochastic–analytical model and bias correction of simulated discharge time series.
- Analysis and descriptive statistics of predictive model results.

^{2}and 8.5 W/m

^{2}compared to pre-industrial values. In the RCP8.5 scenario, greenhouse gas emissions rise throughout the 21st century, and a global increase of near-surface air temperatures between 2.6 °C and 4.8 °C in 2081–2100 compared to 1986–2005 is likely [65]. This scenario corresponds to the current CO

_{2}emission pattern.

## 4. Test Site

## 5. Simulating Discharge Peaks by Regression Analysis

- Spring discharge is a combination of quickflow (originating from concentrated recharge) and baseflow (originating from diffuse recharge and water release from the rock matrix);
- Quickflow is directly related to rainfall;
- Baseflow is delayed by diffusive processes compared to rainfall.

_{max}is the peak discharge (m

^{3}/s), and p is the daily precipitation (mm).

## 6. Simulating Baseflow by Analytical Model

_{peak}is calculated from rainfall data, using the regression function described in the previous chapters.

## 7. Comparison of Measured vs. Projected Rainfall Data (2006–2010)

- The rainfall statistics indicate that the mean value is similar for each scenario and also for measured data and range between 3.9 and 4.5 mm/day;
- While the maximum values of rainfall range between 87 and 96 mm for model scenarios, they are significantly higher in reality (115 mm/day).

## 8. Comparison of Measured vs. Projected Discharge (2006–2010)

## 9. Bias Correction of Rainfall–Discharge Models

_{measured}= 0.0587 + 2.64 × Q

_{1}− 0.037 × Q

_{1}

^{2}

_{measured}= 0.0583 + 2.37 × Q

_{2}− 0.031× Q

_{2}

^{2}

## 10. Discharge Predictions

#### 10.1. Predictive Simulations for 2021–2050

- Peak discharge largely increased for Scenario 1;
- Peak discharge remained quasi-stagnant for Scenario 2;
- Mid-range discharge increased for Scenario 1;
- Mid-range discharge decreased for Scenario 2;
- Baseflow discharge decreased for both scenarios.

#### 10.2. Predictive Simulations for 2071–2100

- Peak discharges increase for both scenarios;
- Mid-range discharge increases for Scenario 1;
- Mid-range discharge decreases for Scenario 2;
- Baseflow discharge decreases for both scenarios.

## 11. Discussion and Conclusions

- Peak discharge considerably increases during the first half of the 21st century and remains high after that;
- Average discharge increases gradually throughout the 21st century;
- Baseflow discharge decreases during the first half of the 21st century and remains stagnant after that.

- Peak discharge drops during the first half of the 21th century and considerably increases during the second half of the 21st century;
- Average discharge gradually drops throughout the 21st century;
- Baseflow discharge decreases during the first half of the 21st century and remains stagnant after that.

^{3}/s, whereas drought was defined by discharge values below 0.1 m

^{3}/s. Figure 26 indicates the annual flood and drought days for predictive Scenario 1. Figure 27 indicates the annual flood and drought days for predictive Scenario 2.

- Peak discharge is predicted to increase by the end of the 21st century according to both scenarios.
- Baseflow discharge is predicted to drop by the end of the 21st century according to both scenarios.
- While Scenario 1 predicts an increase in average spring discharge, Scenario 2 predicts a decrease in average discharge.
- The annual number of flood days shows little variation and no significant trend over the simulation periods.
- The annual number of drought days shows a decreasing trend over time. At the same time, the annual number of drought days shows a large variation over time.
- There seems to be a periodicity of extremely dry years with a periodicity between 5 and 7 years.
- The number of drought days during extremely dry years seems to increase over the simulated period according to Scenario 2 (no clear trend for Scenario 1).

_{2}emission scenarios. Multi-annual variations seem to play an important role in the length of extreme events.

- Modeling was undertaken based on the data that were available at the time that this work commenced. Continuous discharge time series applied for model calibration (i.e., establishment of regression functions) were available only for three years, while model predictions had to be provided for 80 years. This resulted in the uncertainty of model predictions.
- Predicted discharge time series reflect the results of downscaled numerical climate models. Discharge prediction thus inherits the uncertainties and assumptions involved in the regional climate projections.
- The applied method is based on regression between rainfall and discharge. The other components of the water budget, such as runoff and evapotranspiration, are only implicitly represented in the model, which entails the uncertainty of the results.
- The methodology introduced in this paper proved to be efficient and is assumed to be applicable to other sites. However, further testing is required on sites with different karst characteristics and longer calibration and validation data series available.
- The goal of the study was to develop and test the utilization of a combined stochastic–analytical method, rather than investigating the climate sensitivity of a geographical area. For this reason, only two scenarios of one regional climate model projection were applied. Additional model projections would provide a more detailed picture of the spring behavior of the investigated site.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Stevanović, Z. Karst waters in potable water supply: A global scale overview. Environ. Earth Sci.
**2019**, 78, 662. [Google Scholar] [CrossRef] - Chen, Z.; Auler, A.S.; Bakalowicz, M.; Drew, D.; Griger, F.; Hartmann, J.; Jiang, G.; Moosdorf, N.; Richts, A.; Stevanović, Z.; et al. The World Karst Aquifer Mapping Project—Concept, mapping procedure and map of Europe. Hydrogeol. J.
**2017**, 25, 771–785. [Google Scholar] [CrossRef] - Nerantzaki, S.D.; Nikolaidis, N.P. The response of three Mediterranean karst springs to drought and the impact of climate change. J. Hydrol.
**2020**, 591, 125296. [Google Scholar] [CrossRef] - Hengeveld, H.G. A Discussion of Recent Simulations with CGCM. Climate Change Digest; Environment Canada Special Edition CCD 00-01; Environment Canada, 2000; 32p. [Google Scholar]
- Mearns, L.O.; Hulme, M.; Carter, T.R.; Leemans, R.; Lal, M.; Whetton, P. Climate scenario development. In Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change; Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K.B., Tignor, M., Miller, H.L., Eds.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2007; pp. 739–768. [Google Scholar]
- Le Treut, H.; Somerville, R.; Cubasch, U.; Ding, Y.; Mauritzen, C.; Mokssit, A.; Peterson, T.; Prather, M. Historical Overview of Climate Change. In Climate Change 2007: The Physical Science Basis Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change; Solomon, S.D.Q., Manning, M., Chen, Z., Marquis, M., Averyt, K.B., Tignor, M., Miller, H.L., Eds.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2007. [Google Scholar]
- Seneviratne, S.I.N.; Nicholls, D.; Easterling, C.M.; Goodess, S.; Kanae, J.; Kossin, Y.; Luo, J.; Marengo, K.; McInnes, M.; Rahimi, M.; et al. 2012: Changes in climate extremes and their impacts on the natural physical environment. In Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation; A Special Report of Working Groups I and II of the Intergovernmental Panel on Climate Change (IPCC); Field, C.B.V., Barros, T.F., Stocker, D., Qin, D.J., Dokken, K.L., Ebi, M.D., Mastrandrea, K.J., Mach, G.-K., Plattner, S.K., Allen, M., et al., Eds.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2012; pp. 109–230. [Google Scholar]
- Christensen, J.H.; Carter, T.R.; Rummukainen, M.; Amanatidis, G. Evaluating the performance and utility of regional climate models: The PRUDENCE project. Clim. Change
**2007**, 81, 1–6. [Google Scholar] [CrossRef] - van der Linden, P.; Mitchell, J.F.B. (Eds.) ENSEMBLES: Climate Change and Its Impacts: Summary of Research and Results from the ENSEMBLES Project; Met Office Hadley Centre: Exeter, UK, 2009; 160p. [Google Scholar]
- Giorgi, F.; Jones, C.; Asrar, G.R. Addressing climate information needs at the regional level: The CORDEX framework. World Meteorol. Organ. (WMO) Bull.
**2009**, 58, 175. [Google Scholar] - Harris, G.R.; Collins, M.; Sexton, D.M.H.; Murphy, J.M.; Booth, B.B.B. Probabilistic projections for 21st century European climate. Nat. Hazards Earth Syst. Sci.
**2010**, 10, 2009–2020. [Google Scholar] [CrossRef] - Lionello, P.; Scarascia, L. The relation between climate change in the Mediterranean region and global warming. Reg. Environ. Change
**2018**, 18, 1481–1493. [Google Scholar] [CrossRef] - Hiscock, K.; Sparkles, R.; Hodgson, A. Evaluation of future climate change impacts on Europe groundwater resources. In Climate Change Effects on Groundwater Resources: A Global Synthesis of Findings and Recommendations; Treidel, H., Martin-Bordes, J.L., Gurdak, J., Eds.; CRC: Boca Raton, FL, USA, 2012; pp. 351–366. [Google Scholar]
- Hartmann, A.; Goldscheider, N.; Wagener, T.; Lange, J.; Weiler, M. Karst water resources in a changing world: Review of hydrological modeling approaches. Rev. Geophys.
**2014**, 52, 218–242. [Google Scholar] [CrossRef] - Stevanović, Z.; Blagojević, M. (Eds.) Hydrogeology and Climate Changes Impact on Aquifer Systems of Drina River Basin; Ministry of Agriculture, Forestry and Water Management of Montenegro: Podgorica, Montenegro, 2021; p. 315. [Google Scholar]
- Jenkins, G.M.; Watts, D.G. Spectral Analysis and Its Applications; Holden Days: San Francisco, CA, USA, 1968. [Google Scholar]
- Mangin, A. Etude des débits classés d’exutoires karstiques portant sur un cycle hydrologique. Ann. Spéléologie
**1971**, 28, 21–40. [Google Scholar] - Mangin, A. Contribution a l‘Étude Hydrodynamique des Aquifères Karstiques. Ph.D. Thesis, Institut des Sciences de la Terre de l‘Université de Dijon, Dijon, France, 1975. [Google Scholar]
- Mangin, A. Utilisation des analyses correlatoire et spectrale dans l’approche des systèmes hydrologiques. Comptes Rendus De L’académie Des Sci.
**1981**, 293, 401–404. [Google Scholar] - Mangin, A. Pour une meilleure connaissance des systèmes hydrologiques à partir des analyses corrélatoire et spectrale. J. Hydrol.
**1984**, 67, 25–43. [Google Scholar] [CrossRef] - Padilla, A.; Pulido-Bosch, A. Study of hydrographs of karstic aquifers by means of correlation and cross-spectral analysis. J. Hydrol.
**1995**, 168, 73–89. [Google Scholar] [CrossRef] - Larocque, M.; Mangin, A.; Razack, M.; Banton, O. Characterization of the La Rochefoucauld karst aquifer (Charente, France) using correlation and spectral analysis. Bull. d’Hydrogéologie l’Université Neuchâtel
**1998**, 16, 49–57. [Google Scholar] - Grasso, D.A. Interprétation des Réponses Hydrauliques et Chimiques des Sources Karstiques. Ph.D. Thesis, Centre d’Hydrogéologie, Université de Neuchâtel, Neuchâtel, Switzerland, 1998. [Google Scholar]
- Grasso, D.A.; Jeannin, P.-Y. Etude critique des méthodes d’analyse de la réponse globale des systèmes karstiques. Application au site de Bure (JU, Suisse). Bull. d’Hydrogéologie l’Université Neuchâtel
**1994**, 13, 87–113. [Google Scholar] - Grasso, D.A.; Jeannin, P.-Y. Statistical approach to the impact of climatic variations on karst spring chemical response. Bull. d’Hydrogéologie l’Université Neuchâtel
**1998**, 16, 59–74. [Google Scholar] - Fiorillo, F.; Esposito, L.; Guadagno, F.M. Analyses and forecast of water resources in an ultra-centenarian spring discharge series from Serino (Southern Italy). J. Hydrol.
**2007**, 336, 125–138. [Google Scholar] [CrossRef] - Fiorillo, F.; Doglio, A. The Relation between Karst Spring Discharge and Rainfall by the Cross-Correlation Analysis. Hydrogeol. J.
**2010**, 18, 1881–1895. [Google Scholar] [CrossRef] - Kiraly, L. Karstification and groundwater flow. In Speleogenesis and Evolution of Karst Aquifers; Karst Research Institute: Postojna, Slovenia, 2003; Volume 1, p. 26. [Google Scholar]
- Gabrovsek, F. (Ed.) Carsologica; Zalozba ZRC: Ljubljana, Slovenia, 2002; pp. 155–190. [Google Scholar]
- Mero, F. Application of the groundwater depletion curves in analyzing and forecasting spring discharges influenced by well fields. In Proceedings of the Symposium on Surface Waters, General Assembly of Berkeley of IUGG, Berkeley, CA, USA, 19–31 August 1963; IAHS Publication: Wallingford, UK, 1963; Volume 63, pp. 107–117. [Google Scholar]
- Mero, F. An approach to daily hydrometeorological water balance computations for surface and groundwater basins. In Proceedings of the ITC-UNESCO, Seminar for Integrated River Basin Development, Delft, The Netherlands, 1 May 1969. [Google Scholar]
- Mero, F.; Gilboa, Y. A methodology for the rapid evaluation of groundwater resources, Sao Paulo State, Brazil. Bull. Sci. Hydrogéologiques
**1974**, 19, 347–35821. [Google Scholar] [CrossRef] - Guilbot, A. Modélisation des Écoulement d‘un Aquifère Karstique (Liaisons Pluie-Debit), Application Aux Bassins de Saugras et du Lez. Ph.D. Thesis, Université des Sciences et Techniques du Languedoc, Montpellier, France, 1975. [Google Scholar]
- Bezes, C. Contribution a la Modélisation des Systèmes Aquifères Karstiques. Ph.D. Thesis, Université des Sciences et Techniques du Languedoc, Montpellier, France, 1976. [Google Scholar]
- Thiéry, D. Logiciel GARDÉNIA, Version 8.2. Guide d’utilisation. BRGM/RP-62797-FR. 2014. Available online: https://www.brgm.fr/sites/default/files/documents/2022-01/logiciel-gardenia-v8-2-rp-62797-fr-notice.pdf (accessed on 1 September 2022).
- Boussinesq, J. Recherches théoriques sur l’écoulement des nappes d’eau infiltrées dans le sol et sur le débit des sources. J. Mathématiques Pures Appliquées
**1904**, 10, 5–78. [Google Scholar] - Berkaloff, E. Limite de validité des formules courantes de tarissement de débit. Chronique d’Hydrogéologie
**1976**, 10, 31–41. [Google Scholar] - Kovács, A. Geometry and Hydraulic Parameters of Karst Aquifers: A hydrodynamic Modelling Approach. Ph.D. Thesis, CHYN, University of Neuchatel, Neuchâtel, Switzerland, 2003; 131p. Available online: http://doc.rero.ch/search.py?recid=2603&ln=fr (accessed on 1 May 2019).
- Kovács, A.; Perrochet, P.; Király, L.; Jeannin, P.-Y. A quantitative method for the characterization of karst aquifers based on spring hydrograph analysis. J. Hydrol.
**2005**, 303, 152–164. [Google Scholar] [CrossRef] - Hornik, K.; Stinchcombe, M.; White, H. Multilayer Feedforward networks are universal approximator. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - Maier, H.R.; Dandy, G.C. Neural networks for the prediction and forecasting of water resources variables: A review of modelling issues and applications. Environ. Model. Softw.
**2000**, 15, 101–124. [Google Scholar] [CrossRef] - Corzo, G.; Solomatine, D. Knowledge-based modularization and global optimization of artificial neural network models in hydrological forecasting. Neural Netw.
**2007**, 20, 528–536. [Google Scholar] [CrossRef] - Filho, A.J.; dos Santos, C.C. Modeling a densely urbanized watershed with an artificial neural network, weather radar and telemetric data. J. Hydrol.
**2006**, 317, 34–48. [Google Scholar] - Toukourou, M.; Johannet, A.; Dreyfus, G.; Ayral, P.-A. Rainfall-runoff modeling of flash floods in the absence of rainfall forecasts: The case of “Cévenol flash floods”. J. Appl. Intell.
**2011**, 35, 1078–1189. [Google Scholar] [CrossRef] - Artigue, G.; Johannet, A.; Borrell, V.; Pistre, S. Flash flood forecasting in poorly gauged basins using neural networks: Case study of the Gardon de Mialet basin (southern France). Nat. Hazards Earth Syst. Sci.
**2012**, 12, 3307–3324. [Google Scholar] [CrossRef] - Kong-A-Siou, L.; Johannet, A.; Borrell, V.; Pistre, S. Complexity selection of a neural network model for karst flood forecasting: The case of the Lez basin (southern France). J. Hydrol.
**2011**, 403, 367–380. [Google Scholar] [CrossRef] - Kurtulus, B.; Razack, M. Evaluation of the ability of an artificial neural network model to simulate the input-output responses of a large karstic aquifer: The La Rochefoucauld aquifer (Charente, France). Hydrogeol. J.
**2007**, 15, 241–254. [Google Scholar] [CrossRef] - Kong-A-Siou, L.; Cros, K.; Johannet, A.; Borrell-Estupina, V.; Pistre, S. KnoX method or Knowledge eXtraction from neural network model. Case study on the Lez karst Aquifer (southern France). J. Hydrol.
**2013**, 507, 19–32. [Google Scholar] [CrossRef] - Kong-A-Siou, L.; Fleury, P.; Johannet, A.; Borrell-Estupina, V.; Pistre, S.; Dörflinger, N. Performance and complementarity of two systemic models (reservoir and neural networks) used to simulate spring discharge and piezometryfor a karst aquifer. J. Hydrol.
**2014**, 519, 3178–3192. [Google Scholar] [CrossRef] - Maillet, E. Essais d‘Hydraulique Souterraine et Fluviale; Hermann: Paris, France, 1905. [Google Scholar]
- Forkasiewicz, J.; Paloc, H. Le régime de tarissement de la Foux-de-la-Vis. Etude préliminaire. Chronique d‘Hydrogéologie BRGM
**1967**, 3, 61–73. [Google Scholar] - Schoeller, H. Hydrodynamique Dans le Karst, Hydrologie des Roches Fissurées; Coedition IAHS/UNESCO: Dubrovnik, Croatia, 1965. [Google Scholar]
- Kovács, A.; Perrochet, P. A quantitative approach to spring hydrograph decomposition. J. Hydrol.
**2008**, 352, 16–29. [Google Scholar] [CrossRef] - Kovács, A.; Perrochet, P. Well Hydrograph Analysis for the Estimation of Hydraulic and Geometric Parameters of Karst Aquifers. In Environmental Earth Sciences, H2Karst Research in Limestone Hydrogeology; Springer International Publishing: Cham, Switzerland, 2014; pp. 97–114. ISBN 978-3-319-06138-2. [Google Scholar]
- Kovács, A.; Perrochet, P.; Darabos, E.; Lénárt, L.; Szűcs, P. Well hydrograph analysis for the characterisation of flow dynamics and conduit network geometry in a karstic aquifer, Bükk Mountains, Hungary. J. Hydrol.
**2015**, 530, 484–499. [Google Scholar] [CrossRef] - Kovács, A. Quantitative classification of carbonate aquifers based on hydrodynamic behaviour. Hydrogeol. J.
**2021**, 29, 33–52. [Google Scholar] [CrossRef] - Allen, M.D. Climate change impacts on valley-bottom aquifers in mountain regions: Case studies from British Columbia, Canada. In Climate Change Effects on Groundwater Resources: A Global Synthesis of Findings and Recommendations; Treidel, H., Martin-Bordes, J.L., Gurdak, J., Eds.; CRC: Boca Raton, FL, USA, 2012; pp. 249–264. [Google Scholar]
- Jacob, D.; Petersen, J.; Eggert, B.; Alias, A.; Christensen, O.B.; Bouwer, L.M.; Braun, A.; Colette, A.; Déqué, M.; Georgievski, G.; et al. EURO-CORDEX: New high-resolution climate change projections for European impact research. Reg. Environ. Change
**2014**, 14, 563–578. [Google Scholar] [CrossRef] - Jacob, D.; Teichmann, C.; Sobolowski, S.; Katragkou, E.; Anders, I.; Belda, M.; Benestad, R.; Boberg, F.; Buonomo, E.; Cardoso, R.M.; et al. Regional climate downscaling over Europe: Perspectives from the EURO-CORDEX community. Reg. Environ. Change
**2020**, 20, 51. [Google Scholar] [CrossRef] - Dosio, A.; Paruolo, P. Bias correction of the ENSEMBLES high-resolution climate change projections for use by impact models: Evaluation on the present climate. J. Geophys. Res.
**2011**, 116, 1–22. [Google Scholar] [CrossRef] - Piani, C.; Haerter, J.O.; Coppola, E. Statistical bias correctionfor daily precipitation in regional climate models over Europe. Theor. Appl. Climatol.
**2010**, 99, 187–192. [Google Scholar] [CrossRef] - Piani, C.; Weedon, G.P.; Best, M.; Gomes, S.M.; Viterbo, P.; Hagemann, S.; Haerter, J.O. Statistical bias correction of global simulateddaily precipitation and temperature for the application of hydrologicalmodels. J. Hydrol.
**2010**, 395, 199–215. [Google Scholar] [CrossRef] - Yang, W.; Andréasson, J.; Graham, L.P.; Olsson, J.; Rosberg, J.; Wetterhall, F. Distribution-based scaling to improve usability of regional climate model projections for hydrological climate change impacts studies. Hydrol. Res.
**2010**, 41, 211–229. [Google Scholar] [CrossRef] - Landelius, T.; Dahlgren, P.; Gollvik, S.; Jansson, A.; Olsson, E. A high-resolution regional reanalysis for Europe. Part 2: 2D analysis of surface temperature, precipitation and wind. Q. J. R. Meteorol. Soc.
**2016**, 142, 2132–2142. [Google Scholar] [CrossRef] - Meinshausen, M.; Smith, S.J.; Calvin, K.; Daniel, J.S.; Kainuma, M.; Lamarque, J.; Matsumoto, K.; Montzka, S.; Raper, S.; Riahi, K.; et al. The RCP greenhouse gas concentrations and their extensions from 1765 to 2300. Clim. Change
**2011**, 109, 213–241. [Google Scholar] [CrossRef] - IPCC. Climate Change 2013. The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change; Stocker, T.F., Qin, D., Plattner, G.-K., Tignor, M., Allen, S.K., Boschung, J., Nauels, A., Xia, Y., Bex, V., Midgley, P.M., Eds.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2013; 1535p. [Google Scholar] [CrossRef]
- Voldoire, A.; Sanchez-Gomez, E.; Salas y Mélia, D.; Decharme, B.; Cassou, C.; Sénési, S.; Valcke, S.; Beau, I.; Alias, A.; Chevallier, M.; et al. Chauvin (2011): The CNRM-CM5.1 global climate model: Description and basic evaluation. Clim. Dyn.
**2013**, 40, 2091–2121. [Google Scholar] [CrossRef] - CLMcom. CLMcom CORDEX Data for Europe (EUR-11) Based on CCLM4-8-17 Model Simulations. World Data Center for Climate (WDCC) at DKRZ. 2016. Available online: http://cera-www.dkrz.de/WDCC/ui/Compact.jsp?acronym=CXEU11CLCL (accessed on 1 May 2019).
- Radulović, M. Karst hydrogeology of Montenegro. In Special Issue of Geological Bulletin Vol XVIII; Geol. Survey of Montenegro: Podgorica, Montenegro, 2000. [Google Scholar]
- Gregor, M.; Malík, P. Construction of master recession curve using genetic algorithms. J. Hydrol. Hydromech.
**2012**, 60, 3. [Google Scholar] [CrossRef] - Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Gupta, H.V.; Kling, H.; Yilmaz, K.K.; Martinez, G.F. Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. J. Hydrol.
**2009**, 377, 80–91. [Google Scholar] [CrossRef] - Kling, H.; Fuchs, M.; Paulin, M. Runoff conditions in the upper Danube basin under an ensemble of climate change scenarios. J. Hydrol.
**2012**, 424–425, 264–277. [Google Scholar] [CrossRef] - Knoben, W.J.M.; Freer, J.E.; Woods, R.A. Technical note: Inherent benchmark or not? Comparing NashSutcliffe and Kling-Gupta efficiency scores. Hydrol. Earth Syst. Sci.
**2019**, 23, 4323–4331. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Hydrograph decomposition according to the concept of Forkasiewitz and Paloc [51].

**Figure 3.**Conceptual model of karst systems. T

_{m}[L

^{2}T

^{−1}] is matrix transmissivity, S

_{m}[-] is matrix storativity, K

_{c}[L

^{3}T

^{−1}] is conduit conductance, S

_{c}[L] is conduit storativity, A [L

^{2}] is catchment area, L

_{x}and L

_{y}[L] are block dimensions, x and y are distance of observation well from block center, and α

_{H}and α

_{Q}are well and spring hydrograph recession coefficients. Based on Kovács and Perrochet [53].

**Figure 6.**Regional hydrogeological map of the Drina River Basin, Montenegro. Black square indicates Bukovica spring location. Based on Stevanović and Blagojević [15].

**Figure 8.**Measured spring hydrograph and discharge-peak identification, Bukovica spring. Q

_{max(2 days}

_{)}indicates that two days’ monotonously rising and then falling discharge was used as peak selection criterion.

**Figure 16.**Forecasted spring hydrographs for the period 2006–2010. Discharge was calculated from RCM projections, using the combined stochastic–analytical model. Red line indicates measurements available for the 2008–2010 time period.

**Figure 17.**Comparison of flow duration curves of the measured and predicted spring hydrographs for the 2008–2010 period.

**Figure 19.**Raw and bias-corrected predicted spring hydrographs calculated from RCM projections for the period 2006–2010.

**Figure 20.**Flow duration curves for the calibration period before and after bias correction. Spring discharges calculated from RCM projections, using the combined stochastic–analytical method.

**Figure 21.**Spring hydrograph forecasts for the 2021–2050 (

**a**) and 2071–2100 (

**b**) periods based on RCMs. Bias-corrected combined stochastic–analytical model was applied.

**Figure 22.**Flow duration curves of spring-discharge forecasts for the 2021–2050 (

**a**) and the 2071–2100 (

**b**) periods based on RCMs. Bias-corrected combined stochastic–analytical model was applied.

**Figure 23.**Box-and-whisker plots of predicted bias-corrected spring discharge for the 2021–2050 (

**a**) and 2071–2100 (

**b**) periods based on RCMs. Measured data from 2008 to 2010 are included for comparison.

**Figure 24.**Flow duration curves of spring-discharge forecasts for the 2006–2010, 2021–2100, and 2071–2100 periods based on Scenario 1’s RCM projection. A bias-corrected combined stochastic–analytical model was applied.

**Figure 25.**Flow duration curves of spring-discharge forecasts for the 2006–2010, 2021–2100, and 2071–2100 periods based on Scenario 2’s RCM projection. A bias-corrected combined stochastic–analytical model was applied.

GCM | RCM | Scenario | Scenario Number |
---|---|---|---|

CNRM-CERFACS-CNRM-CM5 | CLMcom-CCLM4-8-17 | RCP45 | 1 |

CNRM-CERFACS-CNRM-CM5 | CLMcom-CCLM4-8-17 | RCP85 | 2 |

Statistical Measure | Value |
---|---|

Correlation coefficient | 0.82 |

R-squared | 0.66 |

Standard error of the estimate | 4.86 |

Mean absolute error | 3.82 |

Durbin–Watson statistic | 1.01 |

Lag 1 residual autocorrelation | 0.49 |

Correlation coefficient | 0.66 |

Q_measured_(m^{3}/s) | Q_simulated_(m^{3}/s) | |
---|---|---|

Count | 1096 | 1095 |

Average | 2.39 | 1.93 |

Standard deviation | 3.27 | 3.11 |

Variance | 10.72 | 9.65 |

Coeff. Of variation | 137% | 161% |

Minimum | 0.06 | 0.06 |

Maximum | 36.45 | 37.60 |

Range | 36.38 | 37.54 |

Stnd. Skewness | 57.40 | 65.75 |

Stnd. Kurtosis | 192.13 | 229.96 |

Median | 1.36 | 1.05 |

Count | Average | Standard Deviation | Coeff. of Variation | Minimum | Maximum | Range | Stnd. Skewness | |
---|---|---|---|---|---|---|---|---|

Scenario 1 | 1725 | 3.93 | 7.88 | 200% | 0 | 87.30 | 87.30 | 66.59 |

Scenario 2 | 1725 | 4.56 | 8.51 | 187% | 0 | 95.80 | 95.80 | 59.77 |

p_meas | 1725 | 4.24 | 11.91 | 281% | 0 | 115.00 | 115.00 | 74.12 |

Count | Mean | Stnd. Error | Lower Limit | Upper Limit | |
---|---|---|---|---|---|

Sce1 | 1725 | 3.93 | 0.23 | 3.61 | 4.25 |

Sce2 | 1725 | 4.56 | 0.23 | 4.24 | 4.88 |

p_meas | 1725 | 4.24 | 0.23 | 3.92 | 4.56 |

**Table 6.**Summary statistics of multiple sample comparison amongst predicted and measured spring hydrographs.

Count | Average | Standard Deviation | Coeff. of Variation | Minimum | Maximum | Range | |
---|---|---|---|---|---|---|---|

Q_meas | 1096 | 2.39 | 3.27 | 137% | 0.06 | 36.45 | 36.38 |

Q1 | 1825 | 0.90 | 1.28 | 141% | 0.0046 | 21.67 | 21.66 |

Q2 | 1825 | 1.04 | 1.45 | 139% | 0.0051 | 26.09 | 26.09 |

**Table 7.**Summary statistics of bias-corrected predicted and measured spring hydrographs for the 2021–2050 simulation period.

Q_meas. (m^{3}/s) | Q1_biascorr. (m^{3}/s) | Q2_biascorr. (m^{3}/s) | |
---|---|---|---|

Count | 1096 | 10,957 | 10,957 |

Average | 2.39 | 2.62 | 2.16 |

Standard deviation | 3.27 | 3.96 | 2.83 |

Coeff. of variation | 137% | 151% | 130% |

Minimum | 0.063 | 0.059 | 0.058 |

Maximum | 36.45 | 47.13 | 35.87 |

Range | 36.38 | 47.07 | 35.81 |

Stnd. skewness | 57.40 | 213.11 | 162.42 |

Stnd. kurtosis | 192.13 | 799.15 | 533.03 |

**Table 8.**Summary statistics of bias-corrected predicted and measured spring hydrographs for the 2071–2100 simulation period.

Q_meas. (m^{3}/s) | Q1_biascorr. (m^{3}/s) | Q2_biascorr. (m^{3}/s) | |
---|---|---|---|

Count | 1096 | 10,957 | 10,957 |

Average | 2.39 | 2.93 | 1.92 |

Standard deviation | 3.27 | 4.02 | 3.24 |

Coeff. of variation | 137% | 137% | 168% |

Minimum | 0.06 | 0.06 | 0.06 |

Maximum | 36.45 | 47.06 | 45.33 |

Range | 36.38 | 47.00 | 45.27 |

Stnd. skewness | 57.40 | 194.24 | 296.69 |

Stnd. kurtosis | 192.13 | 672.02 | 1430.04 |

**Table 9.**Summary statistics of Scenario 1’s bias-corrected predicted and measured spring hydrographs for the 2008–2010, 2021–2050, and 2071–2100 simulation periods.

Q1 | 2008–2010 | 2021–2050 | 2071–2100 |
---|---|---|---|

Count | 1825 | 10,957 | 10,957 |

Average | 2.36 | 2.62 | 2.93 |

Standard deviation | 2.97 | 3.96 | 4.02 |

Coeff. of variation | 126% | 151% | 137% |

Minimum (m^{3}/s) | 0.07 | 0.06 | 0.06 |

Maximum (m^{3}/s) | 39.89 | 47.13 | 47.06 |

Range (m^{3}/s) | 39.82 | 47.07 | 47.00 |

Stnd. skewness | 71.82 | 213.11 | 194.24 |

Stnd. kurtosis | 260.29 | 799.15 | 672.02 |

**Table 10.**Summary statistics of Scenario 2’s bias-corrected predicted spring hydrographs for the 2008–2010, 2021–2050, and 2071–2100 simulation periods.

Q2 | 2008–2010 | 2021–2050 | 2071–2100 |
---|---|---|---|

Count | 1825 | 10,957 | 10,957 |

Average | 2.42 | 2.16 | 1.92 |

Standard deviation | 2.97 | 2.82 | 3.24 |

Coeff. of variation | 122% | 130% | 168% |

Minimum (m^{3}/s) | 0.07 | 0.06 | 0.06 |

Maximum (m^{3}/s) | 40.79 | 35.87 | 45.33 |

Range (m^{3}/s) | 40.72 | 35.81 | 45.27 |

Stnd. skewness | 69.80 | 162.42 | 296.69 |

Stnd. kurtosis | 260.59 | 533.03 | 1430.04 |

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## Share and Cite

**MDPI and ACS Style**

Kovács, A.; Stevanović, Z.
A Combined Stochastic–Analytical Method for the Assessment of Climate Change Impact on Spring Discharge. *Water* **2023**, *15*, 629.
https://doi.org/10.3390/w15040629

**AMA Style**

Kovács A, Stevanović Z.
A Combined Stochastic–Analytical Method for the Assessment of Climate Change Impact on Spring Discharge. *Water*. 2023; 15(4):629.
https://doi.org/10.3390/w15040629

**Chicago/Turabian Style**

Kovács, Attila, and Zoran Stevanović.
2023. "A Combined Stochastic–Analytical Method for the Assessment of Climate Change Impact on Spring Discharge" *Water* 15, no. 4: 629.
https://doi.org/10.3390/w15040629