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21 pages, 3757 KiB  
Article
Runoff Prediction of Tunxi Basin under Projected Climate Changes Based on Lumped Hydrological Models with Various Model Parameter Optimization Strategies
by Bing Yan, Yicheng Gu, En Li, Yi Xu and Lingling Ni
Sustainability 2024, 16(16), 6897; https://doi.org/10.3390/su16166897 - 11 Aug 2024
Cited by 1 | Viewed by 1227
Abstract
Runoff is greatly influenced by changes in climate conditions. Predicting runoff and analyzing its variations under future climates are crucial for ensuring water security, managing water resources effectively, and promoting sustainable development within the catchment area. As the key step in runoff modeling, [...] Read more.
Runoff is greatly influenced by changes in climate conditions. Predicting runoff and analyzing its variations under future climates are crucial for ensuring water security, managing water resources effectively, and promoting sustainable development within the catchment area. As the key step in runoff modeling, the calibration of hydrological model parameters plays an important role in models’ performance. Identifying an efficient and reliable optimization algorithm and objective function continues to be a significant challenge in applying hydrological models. This study selected new algorithms, including the strategic random search (SRS) and sparrow search algorithm (SSA) used in hydrology, gold rush optimizer (GRO) and snow ablation optimizer (SAO) not used in hydrology, and classical algorithms, i.e., shuffling complex evolution (SCE-UA) and particle swarm optimization (PSO), to calibrate the two-parameter monthly water balance model (TWBM), abcd, and HYMOD model under the four objective functions of the Kling–Gupta efficiency (KGE) variant based on knowable moments (KMoments) and considering the high and low flows (HiLo) for monthly runoff simulation and future runoff prediction in Tunxi basin, China. Furthermore, the identified algorithm and objective function scenario with the best performance were applied for runoff prediction under climate change projections. The results show that the abcd model has the best performance, followed by the HYMOD and TWBM models, and the rank of model stability is abcd > TWBM > HYMOD with the change of algorithms, objective functions, and contributing calibration years in the history period. The KMoments based on KGE can play a positive role in the model calibration, while the effect of adding the HiLo is unstable. The SRS algorithm exhibits a faster, more stable, and more efficient search than the others in hydrological model calibration. The runoff obtained from the optimal model showed a decrease in the future monthly runoff compared to the reference period under all SSP scenarios. In addition, the distribution of monthly runoff changed, with the monthly maximum runoff changing from June to May. Decreases in the monthly simulated runoff mainly occurred from February to July (10.9–56.1%). These findings may be helpful for the determination of model parameter calibration strategies, thus improving the accuracy and efficiency of hydrological modeling for runoff prediction. Full article
(This article belongs to the Section Sustainable Water Management)
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Figure 1
<p>Location of the Tunxi catchment with hydrological stations.</p>
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<p>Model structures of the three hydrological models used in this study: (<b>a</b>) TWBM; (<b>b</b>) abcd; and (<b>c</b>) HYMOD models.</p>
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<p>The objective function, (<b>a</b>) 1-KGE, (<b>b</b>) 1-KGE_KMoments, (<b>c</b>) 1-KGE_HiLo, and (<b>d</b>) 1-KGE_KMoments_HiLo values for (<b>A</b>) TWBM, (<b>B</b>) abcd, and (<b>C</b>) HYMOD models during the calibration period.</p>
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<p>Radar charts of the evaluation metrics (<b>a</b>) NSE and (<b>b</b>) RMSE for the TWBM, abcd, and HYMOD models under the four objective functions and six optimization algorithms during the validation period. The pink, black, and orange algorithm names represent the TWBM, abcd, and HYMOD models, respectively.</p>
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<p>Scatter plots depicting simulated and observed runoff under the four objective functions and six optimization algorithms (<b>a</b>) SRS, (<b>b</b>) SSA, (<b>c</b>) GRO, (<b>d</b>) SAO, (<b>e</b>) SCE-UA, and (<b>f</b>) PSO of (<b>A</b>) TWBM, (<b>B</b>) abcd, and (<b>C</b>) HYMOD model calibrations in the validation period.</p>
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<p>Boxplots of performance metrics for (<b>a</b>,<b>b</b>) TWBM, (<b>c</b>,<b>d</b>) abcd, and (<b>e</b>,<b>f</b>) HYMOD models under six optimization algorithms in the validation period. The top line of the box represents the mean with the value label inside the box. The whiskers represent the 95% quantiles.</p>
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<p>Distribution of parameter calibration for 50 iterations under the six algorithm scenarios: (<b>a</b>,<b>b</b>) TWBM, (<b>c</b>–<b>f</b>) abcd, and (<b>g</b>–<b>k</b>) HYMOD models.</p>
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<p>Change process of hydrological elements: (<b>a</b>) <span class="html-italic">P</span>, (<b>b</b>) <span class="html-italic">PET</span>, and (<b>c</b>) runoff under SSP245 and SSP585 scenarios.</p>
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<p>Comparison of runoff under different climate scenarios in the future and reference period: (<b>a</b>) box plot of all observed and simulated monthly runoff; (<b>b</b>) intra-annual distribution of runoff; (<b>c</b>) intra-annual distribution of precipitation. In each box, the black line with circle markers represents the mean, and the whiskers extend from the 5% to 95% quantiles. The bottom and top edges indicate the 25 and 75 percentiles, respectively.</p>
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<p>Change in NSE values of the validation period with an increasing number of contributing calibration years: (<b>a</b>) TWBM, (<b>b</b>) abcd, and (<b>c</b>) HYMOD models.</p>
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<p>Box plots of simulated runoff obtained by the abcd, HYMOD, and TWBM model in 2041–2070 under SSP245 and SSP585 scenarios. In each box, the black line with circle markers represents the mean, and the whiskers extend from the 5% to 95% quantiles. The bottom and top edges indicate the 25 and 75 percentiles, respectively.</p>
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20 pages, 3365 KiB  
Article
Seasonal Streamflow Forecast in the Tocantins River Basin, Brazil: An Evaluation of ECMWF-SEAS5 with Multiple Conceptual Hydrological Models
by Leandro Ávila, Reinaldo Silveira, André Campos, Nathalli Rogiski, Camila Freitas, Cássia Aver and Fernando Fan
Water 2023, 15(9), 1695; https://doi.org/10.3390/w15091695 - 27 Apr 2023
Cited by 7 | Viewed by 1941
Abstract
The assessment of seasonal streamflow forecasting is essential for appropriate water resource management. A suitable seasonal forecasting system requires the evaluation of both numerical weather prediction (NWP) and hydrological models to represent the atmospheric and hydrological processes and conditions in a specific region. [...] Read more.
The assessment of seasonal streamflow forecasting is essential for appropriate water resource management. A suitable seasonal forecasting system requires the evaluation of both numerical weather prediction (NWP) and hydrological models to represent the atmospheric and hydrological processes and conditions in a specific region. In this paper, we evaluated the ECMWF-SEAS5 precipitation product with four hydrological models to represent seasonal streamflow forecasts performed at hydropower plants in the Legal Amazon region. The adopted models included GR4J, HYMOD, HBV, and SMAP, which were calibrated on a daily scale for the period from 2014 to 2019 and validated for the period from 2005 to 2013. The seasonal streamflow forecasts were obtained for the period from 2017 to 2019 by considering a daily scale streamflow simulation comprising an ensemble with 51 members of forecasts, starting on the first day of every month up to 7 months ahead. For each forecast, the corresponding monthly streamflow time series was estimated. A post-processing procedure based on the adjustment of an autoregressive model for the residuals was applied to correct the bias of seasonal streamflow forecasts. Hence, for the calibration and validation period, the results show that the HBV model provides better results to represent the hydrological conditions at each hydropower plant, presenting NSE and NSElog values greater than 0.8 and 0.9, respectively, during the calibration stage. However, the SMAP model achieves a better performance with NSE values of up to 0.5 for the raw forecasts. In addition, the bias correction displayed a significant improvement in the forecasts for all hydrological models, specifically for the representation of streamflow during dry periods, significantly reducing the variability of the residuals. Full article
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<p>Flow chart of the seasonal streamflow forecast procedure.</p>
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<p>Structure of the conceptual hydrological models.</p>
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<p>Location of the Tocantins River Basin and hydrological regimes at six hydropower plants. Black line represents monthly mean streamflow. Red shaded areas represent the 10th and 90th quantiles.</p>
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<p>(<b>a</b>) Comparison of the hindcast and observed daily climatology; (<b>b</b>) Cumulative precipitation between the observed data, the hindcast daily climatology, the raw ensemble mean, and the corrected ensemble mean for the period between 1 March 2017 and 30 September 2017.</p>
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<p>Results of the deterministic metrics in the calibration and validation stages.</p>
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<p>Hydrographs obtained for the seasonal forecasts (corrected and uncorrected) using ECMWF-SEAS5 by all hydrological models for the period 2017–2019 at the Estreito UHE.</p>
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<p>Error bars (with a 95% confidence level) of the PBIAS obtained for each hydrological model as a function of the lead time. The points represent the median.</p>
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<p>Error bars (with a 95% confidence level) of the NSE obtained for each hydrological model as a function of the lead time. The points represent the median.</p>
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<p>Error bars (with a 95% confidence level) of the NSElog obtained for each hydrological model as a function of the lead time. The points represent the median.</p>
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<p>Error bars (with a 95% confidence level) of the DM obtained for each hydrological model as a function of the lead time. The points represent the median.</p>
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<p>Plots of relative error versus forecasted streamflow at three hydropower plants. Red dots represent the error of the uncorrected forecasts, whereas black dots represent the errors of the corrected forecasts.</p>
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<p>Error bars of the continuous ranked probability skill score (CRPSS) as a function of the lead time for each hydrological model. The points and error bars denote the medians and the 95% confidence levels (CLs).</p>
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<p>ROC curves for the Lajeado inflows with different thresholds (Q90, Q75, Q50, Q25, and Q10). The black line represents the random guess prediction.</p>
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<p>Comparison of the AUC for the corrected and uncorrected forecasts for all hydrological models, considering Q90, Q75, Q25, and Q10 as thresholds.</p>
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21 pages, 10654 KiB  
Article
Comparative Evaluation of Five Hydrological Models in a Large-Scale and Tropical River Basin
by Leandro Ávila, Reinaldo Silveira, André Campos, Nathalli Rogiski, José Gonçalves, Arlan Scortegagna, Camila Freita, Cássia Aver and Fernando Fan
Water 2022, 14(19), 3013; https://doi.org/10.3390/w14193013 - 25 Sep 2022
Cited by 22 | Viewed by 5681
Abstract
Hydrological modeling is an important tool for water resources management, providing a feasible solution to represent the main hydrological processes and predict future streamflow regimes. The literature presents a set of hydrological models commonly used to represent the rainfall-runoff process in watersheds with [...] Read more.
Hydrological modeling is an important tool for water resources management, providing a feasible solution to represent the main hydrological processes and predict future streamflow regimes. The literature presents a set of hydrological models commonly used to represent the rainfall-runoff process in watersheds with different meteorological and geomorphological characteristics. The response of such models could differ significantly for a single precipitation event, given the uncertainties associated with the input data, parameters, and model structure. In this way, a correct hydrological representation of a watershed should include the evaluation of different hydrological models. This study explores the use and performance of five hydrological models to represent daily streamflow regimes at six hydropower plants located in the Tocantins river basin (Brazil). The adopted models include the GR4J, HYMOD, HBV, SMAP, and MGB-IPH. The evaluation of each model was elaborated considering the calibration (2014–2019) and validation period (2005–2010) using observed data of precipitation and climatological variables. Deterministic metrics and statistical tests were used to measure the performance of each model. For the calibration stage, results show that all models achieved a satisfactory performance with NSE values greater than 0.6. For the validation stage, only the MGB-IPH model present a good performance with NSE values greater than 0.7. A bias correction procedure were applied to correct the simulated data of conceptual models. However, the statistical tests exposed that only the MGB-IPH model could preserve the main statistical properties of the observed data. Thus, this study discusses and presents some limitations of the lumped model to represent daily streamflows in large-scale river basins (>50,000 km2). Full article
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<p>Flow chart of the hydrological modeling procedure.</p>
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<p>Conceptual diagram of the HYMOD model.</p>
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<p>Conceptual diagram of the GR4J model.</p>
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<p>Conceptual diagram of the SMAP/ONS model.</p>
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<p>Conceptual diagram of the HBV model.</p>
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<p>Schematic diagram of linear reservoirs used by the MGB-IPH model. Adapted from: [<a href="#B55-water-14-03013" class="html-bibr">55</a>].</p>
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<p>Location of the Tocantins river basin and the main hydropower plants.</p>
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<p>Results of deterministric metrics in the calibration stage.</p>
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<p>Boxplots of deterministic metrics for the original and corrected simulated data for the vaildation period.</p>
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<p>Hydrographs for the validation stage comparing the observed and the simulated streamflows by different hydrological models at the Estreito UHE.</p>
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<p>Curves describing the duration of daily streamflow in the Tocantins river basin for the validation period (2005–2010).</p>
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<p>Statistical tests of each hydrological model in the validation stage. The horizontal line indicates the significance level <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>.</p>
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<p>Violin plot describing the low, median, and high flows simulated in the Tocantins river basin by multiple hydrological models. Calibration period.</p>
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<p>Violin plot describing the low, median, and high flows simulated in the Tocantins river basin by multiple hydrological models. Validation period.</p>
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20 pages, 3868 KiB  
Article
A Daily Water Balance Model Based on the Distribution Function Unifying Probability Distributed Model and the SCS Curve Number Method
by Marwan Kheimi and Shokry M. Abdelaziz
Water 2022, 14(2), 143; https://doi.org/10.3390/w14020143 - 6 Jan 2022
Cited by 6 | Viewed by 3814
Abstract
A new daily water balance model is developed and tested in this paper. The new model has a similar model structure to the existing probability distributed rainfall runoff models (PDM), such as HyMOD. However, the model utilizes a new distribution function for soil [...] Read more.
A new daily water balance model is developed and tested in this paper. The new model has a similar model structure to the existing probability distributed rainfall runoff models (PDM), such as HyMOD. However, the model utilizes a new distribution function for soil water storage capacity, which leads to the SCS (Soil Conservation Service) curve number (CN) method when the initial soil water storage is set to zero. Therefore, the developed model is a unification of the PDM and CN methods and is called the PDM–CN model in this paper. Besides runoff modeling, the calculation of daily evaporation in the model is also dependent on the distribution function, since the spatial variability of soil water storage affects the catchment-scale evaporation. The generated runoff is partitioned into direct runoff and groundwater recharge, which are then routed through quick and slow storage tanks, respectively. Total discharge is the summation of quick flow from the quick storage tank and base flow from the slow storage tank. The new model with 5 parameters is applied to 92 catchments for simulating daily streamflow and evaporation and compared with AWMB, SACRAMENTO, and SIMHYD models. The performance of the model is slightly better than HyMOD but is not better compared with the 14-parameter model (SACRAMENTO) in the calibration, and does not perform as well in the validation period as the 7-parameter model (SIMHYD) in some areas, based on the NSE values. The linkage between the PDM–CN model and long-term water balance model is also presented, and a two-parameter mean annual water balance equation is derived from the proposed PDM–CN model. Full article
(This article belongs to the Section Hydrology)
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<p>The structure of the proposed PDM–CN model which unifies the PDM (probability distributed model) and the SCS curve number method.</p>
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<p>The calculation of evaporation for two cases: (<b>a</b>) the entire catchment is saturated; (<b>b</b>) the catchment is partially saturated.</p>
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<p>The AWMB description and structure to simulate runoff.</p>
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<p>The location and boundary of the study catchments where the proposed daily water balance model is applied.</p>
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<p>The exceedance probability of the number of catchments, with respect to NSE, during the calibration (<b>a</b>) and validation (<b>b</b>) periods.</p>
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<p>The frequency distribution of the calibrated parameters for the distribution function for soil water storage capacity: (<b>a</b>) the shape parameter (<span class="html-italic">a</span>); (<b>b</b>) the mean of the distribution <math display="inline"><semantics> <mrow> <mfenced> <mrow> <msub> <mi>S</mi> <mi>b</mi> </msub> </mrow> </mfenced> </mrow> </semantics></math>.</p>
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<p>The estimated cumulative distribution of soil water storage capacity in the Nantahala River, North Carolina (USGS gage #03504000), and sensitivities of the shape parameter (a).</p>
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<p>Simulation results for the Nantahala River, North Carolina (USGS gage #03504000), during the calendar year of 1999: (<b>a</b>) daily streamflow; (<b>b</b>) evaporation.</p>
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<p>Observed and simulated flow duration curves during the validation period (1974–2003) for the Nantahala River in North Carolina.</p>
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<p>The obtained two-parameter, long-term water balance equation from the developed PDM–CN model.</p>
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25 pages, 11769 KiB  
Article
Improving Operational Short- to Medium-Range (SR2MR) Streamflow Forecasts in the Upper Zambezi Basin and Its Sub-Basins Using Variational Ensemble Forecasting
by Rodrigo Valdés-Pineda, Juan B. Valdés, Sungwook Wi, Aleix Serrat-Capdevila and Tirthankar Roy
Hydrology 2021, 8(4), 188; https://doi.org/10.3390/hydrology8040188 - 20 Dec 2021
Cited by 3 | Viewed by 3153
Abstract
The combination of Hydrological Models and high-resolution Satellite Precipitation Products (SPPs) or regional Climatological Models (RCMs), has provided the means to establish baselines for the quantification, propagation, and reduction in hydrological uncertainty when generating streamflow forecasts. This study aimed to improve operational real-time [...] Read more.
The combination of Hydrological Models and high-resolution Satellite Precipitation Products (SPPs) or regional Climatological Models (RCMs), has provided the means to establish baselines for the quantification, propagation, and reduction in hydrological uncertainty when generating streamflow forecasts. This study aimed to improve operational real-time streamflow forecasts for the Upper Zambezi River Basin (UZRB), in Africa, utilizing the novel Variational Ensemble Forecasting (VEF) approach. In this regard, we describe and discuss the main steps required to implement, calibrate, and validate an operational hydrologic forecasting system (HFS) using VEF and Hydrologic Processing Strategies (HPS). The operational HFS was constructed to monitor daily streamflow and forecast them up to eight days in the future. The forecasting process called short- to medium-range (SR2MR) streamflow forecasting was implemented using real-time rainfall data from three Satellite Precipitation Products or SPPs (The real-time TRMM Multisatellite Precipitation Analysis TMPA-RT, the NOAA CPC Morphing Technique CMORPH, and the Precipitation Estimation from Remotely Sensed data using Artificial Neural Networks, PERSIANN) and rainfall forecasts from the Global Forecasting System (GFS). The hydrologic preprocessing (HPR) strategy considered using all raw and bias corrected rainfall estimates to calibrate three distributed hydrological models (HYMOD_DS, HBV_DS, and VIC 4.2.b). The hydrologic processing (HP) strategy considered using all optimal parameter sets estimated during the calibration process to increase the number of ensembles available for operational forecasting. Finally, inference-based approaches were evaluated during the application of a hydrological postprocessing (HPP) strategy. The final evaluation and reduction in uncertainty from multiple sources, i.e., multiple precipitation products, hydrologic models, and optimal parameter sets, was significantly achieved through a fully operational implementation of VEF combined with several HPS. Finally, the main challenges and opportunities associated with operational SR2MR streamflow forecasting using VEF are evaluated and discussed. Full article
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<p>(<b>a</b>) Upper Zambezi River Basin (UZRB) delineated above the Katima Mulilo streamgauge. The green markers represent 9 rain gauges available in the basin. The blue markers represent the streamgauges used in this study; (<b>b</b>) Location of the Zambezi Basin in the African continent. The map also shows the location of major hydropower and water storage projects; (<b>c</b>) Modelling domain selected to implement the real-time HFS (RT-HFS). The modelling domain was set up using grid cells at 0.25° of spatial resolution; (<b>d</b>) Land cover map based on [<a href="#B1-hydrology-08-00188" class="html-bibr">1</a>]. The basin is dominated by broadleaved trees (~66%), herbaceous (16.1%), and shrubs (14.8%), whereas only a little (~0.6%) of the area is managed or represents agricultural; and (<b>e</b>) Digital Elevation Model (DEM) based on Hydrosheds (90 m resolution). The spatial distribution of the vegetation types is consistent with the elevational pattern of the basin, which ranges between approximately 731 and 1671 m above sea level [<a href="#B2-hydrology-08-00188" class="html-bibr">2</a>].</p>
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<p>Structure and major modules required to implement an operational Realtime Hydrologic Forecasting System (RT-HFS) at (<b>a</b>) short-range timescales RT-HFS<sub>SR</sub> and at (<b>b</b>) medium-range timescales RT-HFS<sub>MR</sub>.</p>
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<p>Simplified hydrologic modelling paradigm (HMP) used in the operational HFS implementation for the UZRB and its sub basins. Each of the hydrologic processing strategies (HPS) can help quantifying the propagation of total hydrological uncertainty (THU) from different sources, i.e., input data, model structures and parameters, modelling assumptions, and initial conditions of the models, etc.</p>
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<p>Modelling structures used in the operational HFS implementation for the UZRB. (<b>a</b>) HBV_DS, (<b>b</b>) HYMOD_DS, and (<b>c</b>) VIC 4.2.b. Details about model states, fluxes, and parameters are provided in the <a href="#app1-hydrology-08-00188" class="html-app">Appendix A</a>.</p>
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<p>Daily streamflow records for Katima Mulilo, Senanga, Lukulu, Kabompo, Kalabo, Chivata, and Kalene sub-catchments. Daily average rainfall data from CHIRPS are also included. The numbers with “x” next to the names represent an amplification along the <span class="html-italic">y</span>-axis for a better visualization of the hydrographs.</p>
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<p>(<b>Top</b>) UZRB and sub-catchments utilized during the daily calibration process. (<b>Bottom</b>) Calibration performances for Katima Mulilo Streamgauge (2002–2015).</p>
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<p>Multi-Input, Multimodel, and Multiparameter Variational Ensemble Forecasting (VEF). Each Hydrologic Processing Strategy (HPS) is shown in the context of VEF.</p>
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<p>Modelling paradigm implemented in an operational HFS context in the UZRB. The hydrological processing hypotheses that can be implemented to improve streamflow forecasts are displayed as sources of uncertainty (<span class="html-italic">h</span>) propagated from the input data (<span class="html-italic">y<sub>i</sub></span>), and/or from the applied hydrological preprocessing and/or processing techniques. The output data (<span class="html-italic">ŷ<sub>i</sub></span>) can also propagate uncertainty, which can be minimized through the implementation of hydrological postprocessing techniques.</p>
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<p>Scatter plots for daily average rainfall in the UZRB. The observed daily rainfall records from CHIRPS (with drizzle effect removed for rainfall ≤ 0.1 mm) are compared to raw (<b>a</b>–<b>c</b>) and corrected (<b>d</b>–<b>f</b>) satellite-based rainfall estimates from CMORPH, TMPA-RT, and PERSIANN for the period 2001–2017. Three error measures are included for comparison: the Root Mean Squared Error (RMSE), the Nash-Sutcliffe Efficiency (NSE), and the Correlation Coefficient (R).</p>
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<p>(<b>a<sub>1</sub></b>–<b>a<sub>7</sub></b>) All 72 possible SR2MR streamflow forecasts simulated for the UZRB and its sub-basins using an operational VEF approach. (<b>b<sub>1</sub></b>–<b>b<sub>7</sub></b>) Ranking of Total Skill (R<sup>2</sup>) propagated from SR2MR streamflow forecasts. (<b>c<sub>1</sub></b>–<b>c<sub>7</sub></b>) Best 10 VEF simulations ranked by R<sup>2</sup>. (<b>d<sub>1</sub></b>–<b>d<sub>7</sub></b>) Ranking of Root Mean Squared Error (RMSE) propagated from SR2MR forecasts. (<b>e<sub>1</sub></b>–<b>e<sub>7</sub></b>) Best 10 VEF simulations ranked by RMSE. Best 10 VEF simulations. The basins and sub-basins are organized from larger to smaller catchment area (<b>left</b> to <b>right</b>).</p>
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<p>(<b>a</b>) Peak streamflow hydrograph (January to June of 2017) for the UZRB at Katima Mulilo, and (<b>b</b>) Ranked Predictive Skill (R<sup>2</sup>) for the best 20 simulations obtained from the operational VEF approach. The acronyms represent the Hydrological Models (HYM for Hymod; HBV for HBV; and VIC for VIC); the Satellite Precipitation Products or Climatology (CH for CHIRPS and CH2 for drizzle removed effect; CM for CMORPH and CM2 for its bias corrected version; TM for TMPA and TM2 its bias corrected version; PE for PERSIANN and PE2 its bias corrected version); and the utilized parameter set (CH is CHIRPS parameter set; CM is CMORPH parameter set; TM is TMPA parameter set; and PE is PERSIANN parameter set).</p>
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<p>(top) Catchment-average satellite-based precipitation for the UZRB and its sub-basins. SR2MR streamflow forecasts for Katima Mulilo, and its sub-basins Senanga, Lukulu, Kabompo, Kalabo, Chivata, and Kalene (organized from left to right according to their size). The <span class="html-italic">y</span>-axis represents the daily streamflow forecasts, and the <span class="html-italic">x</span>-axis represents the validation and testing periods (2002–2004). The initial streamflow forecasts with their respective uncertainty bands (RMSE in mm) are shown in red. The hydrologically postprocessed (HPP) forecast is shown in light green. BSP is the best streamflow forecast; AVSP is the average streamflow forecast; IVW-SP is the Inverse-Variance Weighting streamflow forecast; IVP-SP is the Inverse-Probability Weighting streamflow forecast. The letter “c” at the end of each acronym represents the regularization applied by combining Multivariate Combinatorial Linear Regression (MCLR) and Inference-Based methods for SR2MR daily streamflow forecasting.</p>
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<p>Hydrologiska Byrans Vattenbalansavdelning (HBV) Model Structure (states, fluxes, and parameters).</p>
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<p>HyMod Hydrologic Model Structure (states, fluxes, and parameters).</p>
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<p>Variable Infiltration Capacity (VIC) Model Structure (states, fluxes, and parameters).</p>
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<p>Effect of short and long-term memory (moving window) on the performance of daily streamflow forecasts for the UZRB and its sub-basins. From left to right each plot represents a memory window ranging between 5 and 180 days. The following windows were used: 5, 8, 15, 30, 45, 60, 90, 120, 150, and 180 days. Reddish colors represent aggregated streamflow forecasts and blueish colors represent weighted streamflow forecasts.</p>
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23 pages, 5536 KiB  
Article
A Prior Estimation of the Spatial Distribution Parameter of Soil Moisture Storage Capacity Using Satellite-Based Root-Zone Soil Moisture Data
by Yifei Tian, Lihua Xiong, Bin Xiong and Ruodan Zhuang
Remote Sens. 2019, 11(21), 2580; https://doi.org/10.3390/rs11212580 - 3 Nov 2019
Cited by 6 | Viewed by 3437
Abstract
Integration of satellite-based data with hydrological modelling was generally conducted via data assimilation or model calibration, and both approaches can enhance streamflow predictions. In this study, we assessed the feasibility of another approach that uses satellite-based soil moisture data to directly estimate the [...] Read more.
Integration of satellite-based data with hydrological modelling was generally conducted via data assimilation or model calibration, and both approaches can enhance streamflow predictions. In this study, we assessed the feasibility of another approach that uses satellite-based soil moisture data to directly estimate the parameter β to represent the degree of the spatial distribution of soil moisture storage capacity in the semi-distributed Hymod model. The impact of using historical root-zone soil moisture data from the Soil Moisture Active Passive (SMAP) mission on the prior estimation of the parameter β was explored. Two different ways to incorporate the root-zone soil moisture data to estimate the parameter β are proposed, i.e., one is to derive a priori distribution of β , and the other is to derive a fixed value for β . The simulations of the Hymod models employing the two ways to estimate β are compared with the results produced by the original model, i.e., the one without employing satellite-based data to estimate the parameter β , at three study catchments (the Upper Hanjiang River catchment, the Xiangjiang River catchment, and the Ganjiang River catchment). The results illustrate that the two ways to incorporate the SMAP root-zone soil moisture data in order to predetermine the parameter β of the semi-distributed Hymod model both perform well in simulating streamflow during the calibration period, and a slight improvement was found during the validation period. Notably, deriving a fixed β value from satellite soil moisture data can provide better performance for ungauged catchments despite reducing the model freedom degrees due to fixing the β value. It is concluded that the robustness of the Hymod model in predicting the streamflow can be improved when the spatial information of satellite-based soil moisture data is utilized to estimate the parameter β . Full article
(This article belongs to the Special Issue Microwave Remote Sensing for Hydrology)
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Graphical abstract

Graphical abstract
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<p>(<b>a</b>) Location, topography, and river systems of three study catchments and the hydrological stations selected for sub-catchments (SC). Sub-catchments are indicated by red lines. (<b>b</b>) The upper Hanjiang River catchment, (<b>c</b>) the Xiangjiang River catchment, and (<b>d</b>) the Ganjiang River catchment.</p>
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<p>Temporal average values of SMAP root-zone soil wetness (dimensionless) over (<b>a</b>) the Hanjiang River catchment, (<b>b</b>) the Xiangjiang River catchment, and (<b>c</b>) the Ganjiang River catchment.</p>
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<p>(<b>a</b>) Schematic illustration of the Hymod model. (<b>b</b>) Distributions of the soil moisture storage capacity of a given catchment, while the parameter <span class="html-italic">β</span> changes from 0.1 to 1.0 based on Equation (1).</p>
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<p>Distributions of soil moisture storage capacity of three study catchments for different values of <span class="html-italic">β</span> based on two different methods (i.e., estimation using SMAP data and model calibration).</p>
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<p>Empirical distributions (sampled values) for the <span class="html-italic">β</span> parameter at (<b>a</b>) the Hanjiang River catchment, (<b>b</b>) the Xiangjiang River catchment, and (<b>c</b>) the Ganjiang River catchment.</p>
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<p>Boxplots of NSE/KGE values obtained by the three model setups during the validation period. For the Hanjiang River catchment (<b>a</b>,<b>b</b>), the Xiangjiang River catchment (<b>c</b>,<b>d</b>), and the Ganjiang River Catchment (<b>e</b>,<b>f</b>), the optimized parameter sets obtained from DREAM were used to perform all validation tests. The boxplots show the 0.1, 0.25, 0.5, 0.75, and 0.9 percentiles.</p>
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<p>Posterior distributions of <span class="html-italic">β</span> using Hd_0 (left) and Hd_S1 (right) at the Hanjiang River catchment (<b>a</b>,<b>b</b>), the Xiangjiang River catchment (<b>c</b>,<b>d</b>), and the Ganjiang River Catchment (<b>e</b>,<b>f</b>).</p>
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<p>Flow duration curves for observed and modeled streamflow at (<b>a</b>) the Hanjiang River catchment, (<b>b</b>) the Xiangjiang River catchment, and (<b>c</b>) the Ganjiang River catchment during the calibration period (left) and validation period (right).</p>
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<p>Flow duration curves for observed and modeled streamflow at (<b>a</b>) the Hanjiang River catchment, (<b>b</b>) the Xiangjiang River catchment, and (<b>c</b>) the Ganjiang River catchment during the calibration period (left) and validation period (right).</p>
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<p>Performance for Xiangtan station (<b>a</b>,<b>b</b>) and Hengyang station (<b>c</b>,<b>d</b>) at the Xiangjiang River catchment in terms of NSE and KGE using parameter sets from the donor catchment. The boxplots represent the variation due to 2000 simulations.</p>
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<p>Performance for Waizhou station (<b>a</b>,<b>b</b>) and Xiajiang station (<b>c</b>,<b>d</b>) at the Ganjiang River catchment in terms of NSE and KGE using parameter sets from the donor catchment. The boxplots represent the variation due to 2000 simulations.</p>
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26 pages, 4497 KiB  
Article
Practical Experience of Sensitivity Analysis: Comparing Six Methods, on Three Hydrological Models, with Three Performance Criteria
by Anqi Wang and Dimitri P. Solomatine
Water 2019, 11(5), 1062; https://doi.org/10.3390/w11051062 - 22 May 2019
Cited by 29 | Viewed by 5863
Abstract
Currently, practically no modeling study is expected to be carried out without some form of Sensitivity Analysis (SA). At the same time, there is a large number of various methods and it is not always easy for practitioners to choose one. The aim [...] Read more.
Currently, practically no modeling study is expected to be carried out without some form of Sensitivity Analysis (SA). At the same time, there is a large number of various methods and it is not always easy for practitioners to choose one. The aim of this paper is to briefly review main classes of SA methods, and to present the results of the practical comparative analysis of applying them. Six different global SA methods: Sobol, eFAST (extended Fourier Amplitude Sensitivity Test), Morris, LH-OAT, RSA (Regionalized Sensitivity Analysis), and PAWN are tested on three conceptual rainfall-runoff models with varying complexity: (GR4J, Hymod, and HBV) applied to the case study of Bagmati basin (Nepal). The methods are compared with respect to effectiveness, efficiency, and convergence. A practical framework of selecting and using the SA methods is presented. The result shows that, first of all, all the six SA methods are effective. Morris and LH-OAT methods are the most efficient methods in computing SI and ranking. eFAST performs better than Sobol, and thus it can be seen as its viable alternative for Sobol. PAWN and RSA methods have issues of instability, which we think are due to the ways Cumulative Distribution Functions (CDFs) are built, and using Kolmogorov–Smirnov statistics to compute Sensitivity Indices. All the methods require sufficient number of runs to reach convergence. Difference in efficiency of different methods is an inevitable consequence of the differences in the underlying principles. For SA of hydrological models, it is recommended to apply the presented practical framework assuming the use of several methods, and to explicitly take into account the constraints of effectiveness, efficiency (including convergence), ease of use, and availability of software. Full article
(This article belongs to the Section Hydrology)
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<p>Graphical expression of Sensitivity Analysis.</p>
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<p>Location of the Bagmati catchment. Triangles denote the rainfall stations and circles denote the discharge gauging stations.</p>
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<p>Hydrograph of the Bagmati Catchment from 1 March 1991 to 31 December 1995.</p>
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<p>Diagram of GR4J model.</p>
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<p>Diagram of Hymod model.</p>
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<p>Diagram of Hidrologiska Bryåns Vattenbalansaldevning (HBV) model [<a href="#B50-water-11-01062" class="html-bibr">50</a>].</p>
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<p>Scatter plot of Root Mean Square Error (RMSE) against parameter values with 10,000 runs for GR4J model.</p>
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<p>Scatter plot of RMSE against parameter values with 10,000 runs for Hymod model.</p>
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<p>Scatter plot of RMSE against parameter values with 10,000 runs for HBV model.</p>
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<p>Sensitivity Indices (normalized) of six Sensitivity Analysis (SA) methods with benchmark run for GR4J (<b>a</b>), Hymod (<b>b</b>) and HBV (<b>c</b>) model, the number in the grid indicates the rank of the parameter within each SA method.</p>
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<p>Sensitivity Indices of different Sensitivity Analysis (SA) methods with different number of runs for GR4J (<b>left column</b>), Hymod (<b>mid column</b>), and HBV (<b>right column</b>) model, the horizontal axis is in log scale.</p>
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<p>Estimate of mean and 95% Confidence Interval (CI) of different Sensitivity Analysis (SA) methods with different number of runs for GR4J (<b>left column</b>), Hymod (<b>mid column</b>) and HBV (<b>right column</b>) model, the horizontal axis is in log scale.</p>
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<p>Framework for Sensitivity Analysis and Uncertainty Analysis of hydrological model.</p>
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20 pages, 6209 KiB  
Article
Evaluating the Effect of Numerical Schemes on Hydrological Simulations: HYMOD as A Case Study
by Shiyan Zhang and Khalid Al-Asadi
Water 2019, 11(2), 329; https://doi.org/10.3390/w11020329 - 14 Feb 2019
Cited by 10 | Viewed by 3046
Abstract
The importance of numerical schemes in hydrological models has been increasingly recognized in the hydrological community. However, the relationship between model performance and the properties of numerical schemes remains unclear. In this study, we employed two types of numerical schemes (i.e., explicit Runge-Kutta [...] Read more.
The importance of numerical schemes in hydrological models has been increasingly recognized in the hydrological community. However, the relationship between model performance and the properties of numerical schemes remains unclear. In this study, we employed two types of numerical schemes (i.e., explicit Runge-Kutta schemes with different orders of accuracy and partially implicit Euler schemes with different implicit factors) in the hydrological model (HYMOD) to simulate the flow hydrograph of the Leaf River basin from 1948 to 1988. Results computed by different numerical schemes were compared and the relationships between model performance and two scheme properties (i.e., the order of accuracy and the implicit factor) were discussed. Results showed that the more explicit schemes generally lead to the overestimation of flow hydrographs, whereas the more implicit schemes lead to underestimation. In addition, the numerical error tended to decrease with increasing orders of accuracy. As a result, the optimal parameter sets found by low-order schemes significantly deviated from those found by the analytical solution. The findings of this study can provide useful implications for designing suitable numerical schemes for hydrological models. Full article
(This article belongs to the Section Hydrology)
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<p>Configuration of HYMOD.</p>
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<p>Flow hydrographs simulated with analytical solutions and the explicit Runge-Kutta schemes for: January 1970–January 1972 (<b>top</b>) and 20 February 1971–7 March 1971 (<b>bottom</b>).</p>
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<p>Flow hydrographs simulated with analytical solutions and the partially implicit Euler schemes for: January 1970–January 1972 (<b>top</b>) and 20 February 1971–7 March 1971 (<b>bottom</b>).</p>
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<p>Comparison of the mean square error (MSE) computed by the analytical solutions and those by the (<b>a</b>) 1st order Runge-Kutta scheme (RK1); (<b>b</b>) 2nd order Runge-Kutta scheme (RK2); (<b>c</b>) 3rd order Runge-Kutta scheme (RK3); (<b>d</b>) 4th order Runge-Kutta scheme (RK4); (<b>e</b>) explicit Euler scheme (EXP, equivalent to RK1); (<b>f</b>) partially implicit Euler scheme (PIM, <span class="html-italic">θ</span> = 1/3); (<b>g</b>) partially implicit Euler scheme (PIM, <span class="html-italic">θ</span> = 2/3); and (<b>h</b>) implicit Euler scheme (IMP, <span class="html-italic">θ</span> = 1). ANA represents the analytical solution and <span class="html-italic">θ</span> represents the implicit factor.</p>
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<p>One-dimensional sensitivity analysis (1DSA) of the mean square error (MSE) with different numerical schemes for the six parameters in HYMOD.</p>
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<p>Two-dimensional sensitivity analysis (2DSA) with different numerical schemes for <span class="html-italic">α</span> and <span class="html-italic">K<sub>q</sub></span>. First row: mean square error (MSE) response surfaces for RK1 to RK4; second row: MSE difference between the analytical and numerical solutions for RK1 to RK4; third row: MSE response surfaces for the Euler schemes; fourth row: MSE difference between the analytical and numerical solutions for the Euler schemes.</p>
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<p>The optimal parameter values and the variations of the mean square error (MSE) values computed by the eight numerical schemes. First row: Normalized optimal parameter values (NPV) for RK1 to RK4; second row: normalized MSE (NMSE) variations for RK1 to RK4; third row: NPV for the Euler schemes; fourth row: normalized MSE (NMSE) variations for the Euler schemes.</p>
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<p>Comparison between the analytical and numerical solutions with: (<b>a</b>) 1st order Runge-Kutta scheme (RK1); (<b>b</b>) 2nd order Runge-Kutta scheme (RK2); (<b>c</b>) 3rd order Runge-Kutta scheme (RK3); (<b>d</b>) 4th order Runge-Kutta scheme (RK4); (<b>e</b>) explicit Euler scheme (EXP, equivalent to RK1); (<b>f</b>) partially implicit Euler scheme (PIM, <span class="html-italic">θ</span> = 1/3); (<b>g</b>) partially implicit Euler scheme (PIM, <span class="html-italic">θ</span> = 2/3); and (<b>h</b>) implicit Euler scheme (IMP, <span class="html-italic">θ</span> = 1). ANA represents the analytical solution, <span class="html-italic">θ</span> represents the implicit factor and n is the number of linear reservoirs in the Nash cascade process. Curves from left to right in each plot correspond to n increasing from one to ten.</p>
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22 pages, 3299 KiB  
Article
Comparing the Hydrological Responses of Conceptual and Process-Based Models with Varying Rain Gauge Density and Distribution
by Zhaokai Yin, Weihong Liao, Xiaohui Lei, Hao Wang and Ruojia Wang
Sustainability 2018, 10(9), 3209; https://doi.org/10.3390/su10093209 - 7 Sep 2018
Cited by 15 | Viewed by 3134
Abstract
Precipitation provides the most crucial input for hydrological modeling. However, rain gauge networks, the most common precipitation measurement mechanisms, are sometimes sparse and inadequately distributed in practice, resulting in an imperfect representation of rainfall spatial variability. The objective of this study is to [...] Read more.
Precipitation provides the most crucial input for hydrological modeling. However, rain gauge networks, the most common precipitation measurement mechanisms, are sometimes sparse and inadequately distributed in practice, resulting in an imperfect representation of rainfall spatial variability. The objective of this study is to analyze the sensitivity of different model structures to the different density and distribution of rain gauges and evaluate their reliability and robustness. Based on a rain gauge network of 20 gauges in the Jinjiang River Basin, south-eastern China, this study compared the performance of two conceptual models (the hydrologic model (HYMOD) and Xinanjiang) and one process-based distributed model (the water and energy transfer between soil, plants and atmosphere model (WetSpa)) with different rain gauge distributions. The results show that the average accuracy for the three models is generally stable as the number of rain gauges decreases but is sensitive to changes in the network distribution. HYMOD has the highest calibration uncertainty, followed by Xinanjiang and WetSpa. Differing model responses are consistent with changes in network distribution, while calibration uncertainties are more related to model structures. Full article
(This article belongs to the Special Issue Impacts of Climate Change on Hydrology, Water Quality and Ecology)
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<p>Map of the study area showing the locations of streams, stream gauges, and rainfall gauges overlain on a digital elevation model (DEM) of the Jinjiang River Basin.</p>
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<p>NSE (<b>a</b>–<b>c</b>), CC (<b>d</b>–<b>f</b>), and BIAS (<b>g</b>–<b>i</b>) distributions for HYMOD (<b>a</b>,<b>d</b>,<b>g</b>), XAJ (<b>b</b>,<b>e</b>,<b>h</b>), and WetSpa (<b>e</b>,<b>f</b>,<b>i</b>) for 16 scenarios illustrating performance during the entire validation period. Colors of the boxes indicate the locations of concentrated rain gauges: central (black), south-west (red), south-east (green), north-west (blue), and north-east (gray). Boxplots show the 0.10, 0.25, 0.50, 0.75, and 0.90 percentiles.</p>
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<p>Monthly average runoff and precipitation in the Jinjiang River Basin in 2012.</p>
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<p>Heatmaps of monthly performance for HYMOD (<b>a</b>,<b>d</b>), XAJ (<b>b</b>,<b>e</b>), and WetSpa (<b>c</b>,<b>f</b>) based on NSE (<b>a</b>–<b>c</b>) and BIAS (<b>d</b>–<b>f</b>). For each heatmap, <span class="html-italic">X</span> axes refer to the months in 2012 and <span class="html-italic">Y</span> axes refer to the different rain gauge distribution scenarios. The colors represent the average NSE or BIAS values for all trials in the current scenario and current month.</p>
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<p>Interpolated mean rainfall distribution in June 2012 over the Jinjiang River Basin using 20 rain gauges.</p>
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<p>Comparison of observed and simulated runoff from scenarios 20_C (<b>a</b>–<b>c</b>), 10_SE (<b>d</b>–<b>f</b>), and 5_NE (<b>g</b>–<b>i</b>) in HYMOD (<b>a</b>,<b>d</b>,<b>g</b>), XAJ (<b>b</b>,<b>e</b>,<b>h</b>), and WetSpa (<b>c</b>,<b>f</b>,<b>i</b>) in June 2012. Error bars represent the standard deviation of precipitation data measured by rain gauges in the indicated scenario.</p>
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4249 KiB  
Article
Multiobjective Automatic Parameter Calibration of a Hydrological Model
by Donghwi Jung, Young Hwan Choi and Joong Hoon Kim
Water 2017, 9(3), 187; https://doi.org/10.3390/w9030187 - 6 Mar 2017
Cited by 21 | Viewed by 6526
Abstract
This study proposes variable balancing approaches for the exploration (diversification) and exploitation (intensification) of the non-dominated sorting genetic algorithm-II (NSGA-II) with simulated binary crossover (SBX) and polynomial mutation (PM) in the multiobjective automatic parameter calibration of a lumped hydrological model, the HYMOD model. [...] Read more.
This study proposes variable balancing approaches for the exploration (diversification) and exploitation (intensification) of the non-dominated sorting genetic algorithm-II (NSGA-II) with simulated binary crossover (SBX) and polynomial mutation (PM) in the multiobjective automatic parameter calibration of a lumped hydrological model, the HYMOD model. Two objectives—minimizing the percent bias and minimizing three peak flow differences—are considered in the calibration of the six parameters of the model. The proposed balancing approaches, which migrate the focus between exploration and exploitation over generations by varying the crossover and mutation distribution indices of SBX and PM, respectively, are compared with traditional static balancing approaches (the two dices value is fixed during optimization) in a benchmark hydrological calibration problem for the Leaf River (1950 km2) near Collins, Mississippi. Three performance metrics—solution quality, spacing, and convergence—are used to quantify and compare the quality of the Pareto solutions obtained by the two different balancing approaches. The variable balancing approaches that migrate the focus of exploration and exploitation differently for SBX and PM outperformed other methods. Full article
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<p>Probability distributions of simulated binary crossover (SBX) with three crossover distribution index (C<sub>DI</sub>) values: C<sub>DI</sub> = 0.5 (solid gray line), C<sub>DI</sub> = 2.0 (dotted black line), and C<sub>DI</sub> = 20 (dashed black line).</p>
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<p>Changes in crossover distribution index (C<sub>DI</sub>) using the two parameter-setting-free (PSF) methods over generations when the number of generations (NGEN) = 500.</p>
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<p>Schematic description of the modified hydrological model (HYMOD).</p>
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<p>Measurements in the Leaf River basin: (<b>a</b>) daily hyetograph; and (<b>b</b>) hydrograph.</p>
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<p>Flow chart of the application results section.</p>
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<p>Three representative plots showing 1000 randomly generated points in the two-objective space: (<b>a</b>) Nash–Sutcliffe Measure (NSF) versus Total Mean Squared Error (TMSE); (<b>b</b>) TMSE versus Three Peak Flow Difference (TPFD); (<b>c</b>) Percent Bias (PB) versus TPFD; and (<b>d</b>) NSF versus PB.</p>
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<p>Three representative plots showing 1000 randomly generated points in the two-objective space: (<b>a</b>) Nash–Sutcliffe Measure (NSF) versus Total Mean Squared Error (TMSE); (<b>b</b>) TMSE versus Three Peak Flow Difference (TPFD); (<b>c</b>) Percent Bias (PB) versus TPFD; and (<b>d</b>) NSF versus PB.</p>
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<p>The 511 global Pareto solutions (black diamonds) identified from the non-dominated sorting of 5400 Pareto solutions (magenta squares) obtained from the cases of combinations of NGEN = 100, 500, and 1000 and NPOP = 30, 50, and 100.</p>
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<p>Comparison of the Pareto optimal solutions obtained by using static and variable approaches: (<b>a</b>) Pareto solutions identified by the two approaches, each of which was run independently; (<b>b</b>) comparison of the two global Pareto fronts, where each global Pareto front was extracted independently; and (<b>c</b>) global Pareto fronts identified by the two best cases: Crate = 0.6 and Approach 3, represented by the x symbol, and Crate = 0.9 and Approach 4, indicated by the cross symbol.</p>
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<p>Normalized parameter ranges of the global Pareto solutions obtained by the variable balancing approaches.</p>
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<p>Global Pareto front region covered by Groups 1 and 2 with three selected solutions for the hydrograph: Group 1 has two quick flow tanks (<span class="html-italic">n</span> = 2, normalized <span class="html-italic">n</span> = 0.5), and Group 2 has a single quick flow tank (<span class="html-italic">n</span> = 1).</p>
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<p>Observed and simulated hydrographs obtained by using three representative global Pareto solutions. The first has a percent bias (PB) of 3.04 × 10<sup>−8</sup>% and three peak flow difference (TPFD) of 9.07 mm/day (simulated streamflow 1), the second has a PB of 29.71% and TPFD of 3.96 mm/day (simulated streamflow 2), and the third has a PB of 1.87% and TPFD of 8.35 mm/day (simulated streamflow 3).</p>
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<p>Closer view of the hydrograph of the three peak flows in <a href="#water-09-00187-f011" class="html-fig">Figure 11</a>: (<b>a</b>) Peak 1; (<b>b</b>) Peak 2; and (<b>c</b>) Peak 3.</p>
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1114 KiB  
Article
Hydrological Model Calibration by Sequential Replacement of Weak Parameter Sets Using Depth Function
by Shailesh Kumar Singh and András Bárdossy
Hydrology 2015, 2(2), 69-92; https://doi.org/10.3390/hydrology2020069 - 23 Apr 2015
Cited by 8 | Viewed by 6004
Abstract
It is always a dream of hydrologists to model the mystery of complex hydrological processes in a precise way. If parameterized correctly, a simple hydrological model can represent nature very accurately. In this study, a simple and effective optimization algorithm, sequential replacement of [...] Read more.
It is always a dream of hydrologists to model the mystery of complex hydrological processes in a precise way. If parameterized correctly, a simple hydrological model can represent nature very accurately. In this study, a simple and effective optimization algorithm, sequential replacement of weak parameters (SRWP), is introduced for automatic calibration of hydrological models. In the SRWP algorithm, a weak parameter set is sequentially replaced with another deeper and good parameter set. The SRWP algorithm is tested on several theoretical test functions, as well as with a hydrological model. The SRWP algorithm result is compared with the shuffled complex evolution-University of Arizona (SCE-UA) algorithm and the robust parameter estimation (ROPE) algorithm. The result shows that the SRWP algorithm easily overcomes the local minima and converges to the optimal parameter space. The SRWP algorithm does not converge to a single optima; instead, it gives a convex hull of an optimal space. An ensemble of results can be generated from the optimal space for prediction purpose. The ensemble spread will account for the parameter estimation uncertainty. The methodology was demonstrated using the hydrological model (HYMOD) conceptual model on upper Neckar catchments of southwest Germany. The results show that the parameters estimated by this stepwise calibration are robust and comparable to commonly-used optimization algorithms. SRWP can be an alternative to other optimization algorithms for model calibration. Full article
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<p>Example of convex hull, high and low depth in the two-dimensional case (P1 and P2 are two parameters of a model. The red star (A) is a point with depth 0 and red square (B) is a point with depth 3).</p>
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<p>Schematical explanation of the sequential replacement of weak parameters algorithm.</p>
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<p>Rastrigin function.</p>
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<p>Contour map for the Rastrigin function and the optimal space obtained by the SRWP algorithm.</p>
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<p>Six-hump camel back function.</p>
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<p>Contour map for the six-hump camel back function and the optimal space obtained by the SRWP algorithm.</p>
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<p>Study area: Upper Neckar catchment in southwest Germany.</p>
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<p>Schematic representation of the hydrological model (HYMOD) model.</p>
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<p>Improvement of the model performance curve in terms of the Nash–Sutcliffe coefficient (NS) during the calibration by the SRWP algorithm for Rottweil catchment (Diff. is difference; std.is the standard deviation). ROPE, robust parameter estimation; SCE-UA, shuffled complex evolution-University of Arizona.</p>
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<p>Example of the observed and model hydrographs during the calibration period from the ROPE, SRWP and SCE-UA algorithms for Rottweil catchment.</p>
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<p>Example of the observed and model hydrographs during the validation period (1971–1980) from the ROPE, SRWP and SCE-UA algorithms for Rottweil catchment.</p>
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<p>Model parameters at different stages of the SRWP algorithm (the y-axis is the normalized parameter value, and the x-axis shows all of the parameters of the HYMOD model).</p>
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<p>Plot matrix of the model parameter at the initial iteration of the SRWP algorithm.</p>
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<p>Plot matrix of the parameter at the final iteration of the SRWP algorithm.</p>
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