# Spreadsheet-based modelling of hysteresis-affected curves

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## Abstract

Design, operation and management of water resource projects are influenced by the amount of discharge passing through the stream. Discharge at the gauging site is generally estimated by developing single-valued simple rating curves. However, in case of unsteady flows, hysteresis affect is introduced in the stage discharge relationship and as such single-valued rating curves are no longer valid for such situations. The present paper presents a simple spreadsheet-based optimization approach for modelling the hysteresis-affected discharge rating curves. Generalized reduced gradient (GRG) technique has been reported as a reliable tool for handling optimization problems; therefore, in the present paper, it has been applied to estimate discharge for two sites with hysteresis affect based on Jones formula. Comparison of results shows that discharge estimated by GRG technique is as efficient as genetic algorithm and the goodness-of-fit criteria shows that the rating curves obtained by using Jones formula fit the observed data better than single-valued simple rating curves for both the sites considered in the present study. Application of spreadsheet-based GRG optimization technique could prove very helpful to the hydrometric offices.

## Keywords

GRG Hysteresis Optimization Rating curves Spreadsheet## Introduction

Design and operation of hydraulic structures mainly rely on the amount of discharge reaching these structures. Quantification of discharge passing through a river section is of utmost importance in river engineering practices, water management, water distribution systems, design of hydraulic structures and water quality monitoring. Therefore, accurate estimation of streamflow is of utmost importance in water resource engineering. Streamflow estimation is generally accomplished through developing stage–discharge relationship at the gauging site so that measured stage may be converted into discharge as direct measurement of discharge in a river is very costly and time-consuming process. Generally, a single-valued relationship is fitted to the observed data of stage and discharge. The single-valued stage–discharge relationship usually performs well in estimation of discharge in case of steady flow but may lead to inaccurate estimation of discharge if significant unsteadiness is introduced in the flow as in case of flood waves (Herschy 1995; Zakwan et al. 2017a).

When a flood wave propagates through a river corresponding to same stage higher discharges are observed during rising stage than in falling stages resulting in looped rating curves. This affect is popularly known as hysteresis in stage–discharge relationship. Several methods have been proposed by various researchers to account for hysteresis affect in discharge rating curves. However, Jones (1916) formula is the most acceptable approach to account hysteresis affect in rating curves. Fread (1975) Faye and Cherry (1980) also proposed models which could mimic the looped rating curves efficiently, but their scope was limited as application of these models required information on channel slope, Manning roughness coefficient and cross-sectional geometry. Many researchers have commented on the logic behind Jones formula but the logical aspect behind the Jones formula was justified by Perumal and Ranga Raju (1999) by suggesting that the Jones formula is based on approximate convection–diffusion (ACD) equation. Several modifications in the Jones formula has been proposed (Perumal et al. 2004), but the extensive requirement of hydraulic parameters could limit their application (Petersen-Øverleir 2006). Petersen-Øverleir (2006) proposed a simple approach based on Jones formula and nonlinear regression which requires only stage–discharge data and time of measurement, to model the hysteresis-affected rating curves. Petersen-Øverleir (2006) successfully applied this approach for various gauging sites affected by hysteresis in USA.

Tawfik et al. (1997) applied a three-layer back propagation artificial neural network (ANN) to model the hysteresis-affected rating curves of two sites Melut and Malakal of river White Nile. Based on the comparative analysis of Boyer’s approach, falling and rising approach and ANN modelling, they concluded ANN-based modelling as the most accurate among the three. Analysing the stage–discharge data of 1993 flood for four stations in middle Mississippi Westphal et al. (1999) reported that the rating curves at these sites were affected by hysteresis. They developed single-valued stage–discharge rating curves using power law and second-degree polynomial and found that these approaches could not satisfactorily mimic the discharge rating curves in case of floods. Jain and Chalisgaonkar (2000) used back propagation feed forward ANN for modelling hypothetical looped rating curves. However, using the same data set as by Jain and Chalisgaonkar (2000), Sudheer and Jain (2003) demonstrated that radial basis function (RBF) neural network are much superior to back propagation neural network in modelling the hysteresis-affected rating curves. Goel and Pal (2011) used support vector machine (SVM) to model the hypothetical looped rating curve by dividing the data set into rising and falling stages. However, training complex ANN periodically is a cumbersome task. Therefore, the present paper proposes a very simple spreadsheet-based optimization approach to model the hysteresis-affected rating curves.

*Q*

_{n}= steady state discharge in stream;

*h*= stage height;

*a*= a constant representing the gauge reading corresponding to zero discharge;

*K*and

*n*are the rating curve constants.

*S*= bottom slope and

*C*= flood wave celerity.

Equation (3) is much similar to the nonlinear regression equation used by Petersen-Øverleir (2006) to model the hysteresis rating curves. In the present paper, Eq. (3) has been used to model the hysteresis-affected rating curves. The four parameters in Eq. (3) were determined by using nonlinear optimization approaches generalized reduced gradient (GRG) and genetic algorithm (GA). Brief description of these approaches is as follows.

## GRG technique

Lasdon et al. (1978) developed GRG optimization code to solve complex nonlinear programming problems. GRG technique is basically a nonlinear extension of simplex method of linear programming that determines the search direction and performs a line search to solve system of nonlinear equations at each step. GRG technique has been found to be one of the most reliable approach to solve highly complex nonlinear programming problems (Lasdon and Smith 1992). Depending on the available storage, GRG technique involves either of the two techniques viz. Quasi-Newton method or Conjugate Gradient method to determine the search direction. However, the Quasi-Newton method is the default choice which maintains an approximation to the Hessian matrix and requires more storage space.

Strong graphical interface and ease in use has attracted the researchers towards the application of spreadsheet to solve engineering problems over the years. Analysis of hydraulic design projects was carried out on spreadsheet by Weiss and Gulliver (2001). GRG technique was applied by Bhattacharjya (2011) to obtain the optimal solution of groundwater flow inverse problem. Che et al. (2014) found GRG solver as reliable as general algebraic modelling system (GAMS) to determine optimal unit hydrographs of watersheds. Muzzammil et al. (2015) applied GRG technique to model the discharge rating curve. Zakwan and Muzzammil (2016) applied GRG technique to model the nonlinear form of Muskingum flood routing equation demonstrating that nonlinear form of Muskingum flood routing equation estimates the outflow more accurately. Zakwan et al. (2016) used GRG technique to estimate the parameters of various infiltration models. Recently, Zakwan et al. (2017b) compared the performance of GRG solver and GA for establishing stage discharge curve in case of steady flow and reported the same result for either optimization technique.

## Genetic algorithm

Genetic algorithm was developed by Holland (1975); however, it became popular after Goldberg (1989). Genetic algorithm is a form of biologically inspired optimization technique and comes under the stochastic class of optimization technique. GA is capable of handling wide variety optimization problem with continuous or discontinuous objective functions. Darwin’s rule of survival of the fittest was the source of inspiration behind the development of this algorithm (Haupt and Haupt 2004; Pandey et al. 2018). In this algorithm, random population of solutions is generated to start the search and the population evolves through operators based on natural genetic variation and natural selection. In GA at each step, parents are selected from current population based on the selection rules; these parents then combine to generate elite children, crossover children and mutation children. GA differs from other heuristic search algorithms as in this algorithm search is conducted based on population information that consist of a subset of solutions. The population of chromosomes is updated till convergence, or until a specified number of updates are completed.

## Data

Statistical characteristics of flood data

Data | Station | Quantity | Max. | Min. | Mean (µ) |
---|---|---|---|---|---|

Calibration | Littles Ferry Bridge | Discharge (ft | 7670 | 830 | 3664.56 |

Stage (ft) | 902.26 | 893.56 | 898.16 | ||

Georgia Highway 141 | Discharge (ft | 6550 | 1490 | 4140.96 | |

Stage (ft) | 885.96 | 881.01 | 883.993 | ||

Validation | Littles Ferry Bridge | Discharge (ft | 7500 | 1090 | 3836.67 |

Stage (ft) | 902.26 | 894.56 | 898.40 | ||

Georgia Highway 141 | Discharge (ft | 6520 | 1490 | 3978.39 | |

Stage (ft) | 885.95 | 881.01 | 883.25 |

## Analysis, results and discussion

*X*is the observed discharge, and

*Y*is the estimated discharge.

Hysteresis rating curve parameters for two sites by different method

Station | Method | | | | |
---|---|---|---|---|---|

Littles Ferry Bridge | SRC | 3.55 | 886.14 | 2.74 | – |

Jones (GRG) | 2.07 | 885.46 | 2.90 | 816.45 | |

Jones (GA) | 2.06 | 885.46 | 2.90 | 816.46 | |

Georgia Highway 141 | SRC | 24.91 | 875.29 | 2.34 | – |

Jones (GRG) | 98.20 | 876.98 | 1.90 | 628.47 | |

Jones (GA) | 98.20 | 876.97 | 1.90 | 628.46 |

*X*is the observed discharge,

*Y*is the estimated discharge, and \(\overline{X}\) is the average discharge.

Performance indices for two sites during calibration

Station | Method | RMSE | IA | Correlation |
---|---|---|---|---|

Littles Ferry Bridge | SRC | 648.95 | 0.86 | 0.96 |

Jones (GRG) | 144.25 | 0.97 | 0.99 | |

Jones (GA) | 144.26 | 0.97 | 0.99 | |

Georgia Highway 141 | SRC | 297.29 | 0.91 | 0.98 |

Jones (GRG) | 98.70 | 0.97 | 0.99 | |

Jones (GA) | 98.70 | 0.97 | 0.99 |

Performance indices for two sites during validation

Station | Method | RMSE | IA | Correlation |
---|---|---|---|---|

Littles Ferry Bridge | SRC | 684.01 | 0.88 | 0.95 |

Jones (GRG) | 161.09 | 0.97 | 0.99 | |

Jones (GA) | 161.09 | 0.97 | 0.99 | |

Georgia Highway 141 | SRC | 310.01 | 0.90 | 0.97 |

Jones (GRG) | 93.09 | 0.97 | 0.99 | |

Jones (GA) | 93.09 | 0.97 | 0.99 |

## Conclusion

In this paper, discharge rating curves were established for two sites that were affected by hysteresis. The rating curve was modelled by single-valued simple rating approach and the Jones formula. The discharge estimated by Jones formula was found to be more accurate than the simple rating curve approach. Further, the Jones formula was modelled using GRG technique and GA. It has been found that the spreadsheet-based optimization GRG technique results in as efficient estimation of discharge as genetic algorithm. However, GRG technique is a very simple spreadsheet-based optimization technique which does not require complex parameter tuning and could prove beneficial to the hydrometric departments as most of the hydrometric departments assemble their data on spreadsheets.

## Notes

## References

- Bhattacharjya RK (2011) Solving groundwater flow inverse problem using spreadsheet solver. J Hydrol Eng 16(5):472–477CrossRefGoogle Scholar
- Che D, Nangare M, Mays LW (2014) Determination of optimal unit hydrographs and Green–Ampt parameters for watersheds. J Hydrol Eng 19(2):375–383CrossRefGoogle Scholar
- Faye RE, Cherry RN (1980) Channel and dynamic flow characteristics of the Chattahoochee River, Buford Dam to Georgia Highway 141. US Geological Survey Water Supply Paper-2063Google Scholar
- Fread DL (1975) Computation of stage–discharge relationships affected by unsteady flow. Water Resour Bull 11:213–218CrossRefGoogle Scholar
- Goel A, Pal M (2011) Stage-discharge modelling using support vector machine. IJE Trans A Basics 25(1):1–9Google Scholar
- Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, BostonGoogle Scholar
- Haupt RL, Haupt SE (2004) Practical genetic algorithms, 2nd edn. Wiley, HobokenGoogle Scholar
- Herschy RW (1995) Streamflow measurement, 2nd edn. E & FN Spon, LondonGoogle Scholar
- Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann ArborGoogle Scholar
- Jain SK, Chalisgaonkar D (2000) Setting up stage-discharge relations using ANN. J Hydrol Eng 5(4):428–433CrossRefGoogle Scholar
- Jones BE (1916) A method of correcting river discharge for a changing stage. US Geological Survey Water Supply Paper 375-EGoogle Scholar
- Lasdon LS, Smith S (1992) Solving sparse nonlinear programs using GRG. ORSA J Comput 4(1):2–15CrossRefGoogle Scholar
- Lasdon LS, Waren AD, Jain A, Ratner M (1978) Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Trans Math Softw 4(1):34–50CrossRefGoogle Scholar
- Muzzammil M, Alam J, Zakwan M (2015) An optimization technique for estimation of rating curve parameters. In: National symposium on hydrology, New Delhi, India, pp 234–240Google Scholar
- Pandey M, Zakwan M, Sharma PK, Ahmad Z (2018) Multiple linear regression and genetic algorithm approaches to predict temporal scour depth near circular pier in non-cohesive sediment ISH. J Hydraul Eng. https://doi.org/10.1080/09715010.2018.1457455 Google Scholar
- Perumal M, Ranga Raju KG (1999) Approximate convection–diffusion equations. J Hydrol Eng 4:160–164CrossRefGoogle Scholar
- Perumal M, Shrestha KB, Chaube UC (2004) Reproducing hysteresis in rating curves. J Hydraul Eng 130:870–878CrossRefGoogle Scholar
- Petersen-Øverleir A (2006) Modelling stage–discharge relationships affected by hysteresis using the Jones formula and nonlinear regression. Hydrol Sci 51(3):365–388CrossRefGoogle Scholar
- Sudheer KP, Jain SK (2003) Radial basis function neural network for modelling rating curves. J Hydrol Eng 8(3):161–164CrossRefGoogle Scholar
- Tawfik M, Ibrahim A, Fahmy H (1997) Hysteresis sensitive neural network for modelling rating curves. J Comput Civ Eng 11(3):206–211CrossRefGoogle Scholar
- Weiss PT, Gulliver JS (2001) What do students need in hydraulic design projects? J Hydraul Eng. https://doi.org/10.1061/(ASCE)0733-9429(2001)127:12(984) Google Scholar
- Westphal J, Thompson D, Stevens G Jr., Strauser C (1999) Stage–discharge relations on the middle Mississippi River. Water Resour Plan Manag 125:48–53CrossRefGoogle Scholar
- Zakwan M, Muzzammil M (2016) Optimization approach for hydrologic channel routing. Water Energy Int 59(3):66–69Google Scholar
- Zakwan M, Muzzammil M, Alam J (2016) Application of spreadsheet to estimate infiltration parameters. Perspect Sci. https://doi.org/10.1016/j.pisc.2016.06.064 Google Scholar
- Zakwan M, Muzzammil M, Alam J (2017a) Application of data driven techniques in discharge rating curve—an overview. Aquademia Water Environ Technol 1(1):02Google Scholar
- Zakwan M, Muzzammil M, Alam J (2017b) Developing stage–discharge relations using optimization techniques. Aquademia Water Environ Technol 1(2):05CrossRefGoogle Scholar

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