# Comparison of cubic, quadratic, and quintic splines for soil erosion modeling

- 357 Downloads
- 1 Citations

## Abstract

Approximate curve is constructed using quadratic, quintic, and cubic splines and examination between these splines. The point of this construction is to predict sediment yield index (SYI) corresponding to curve number. This strategy is outlined with a contextual analysis of Manot watershed of Narmada Basin, India. The relation among calculated SYI and observed SYI esteems is associated with a coefficient of determination (*R*^{2}) of 0.36 and 0.48 for the corresponding quadratic and quintic splines, while the cubic spline showed *R*^{2} of 0.87 (Meshram et al. Arab J Geosci 10:155–168, 2017b; Appl Water Sci 7:1773–1779, 2017c). Numerical results seemed to indicate that the cubic spline method is more accurate than the quadratic/quintic spline method.

## Keywords

Sediment Curve number Quadratic spline Quintic spline## Introduction

Rainfall–runoff–sediment yield modeling, being highly complex, dynamic, and nonlinear, exhibits temporal and spatial variability and comprises several physical processes (Meshram et al. 2017a, b, c). Varying complexity from lumped empirical to physically based space and time-distributed, several models are available in the literature to model runoff and subsequent soil erosion/sediment yield, while physically based models have demonstrated exceptionally helpful as an exploration device, yet they are of restricted use in field, particularly in creating nations like India as they require expansive measure of information.

However, search is still continuing for developing new and simple model. In the present research work, an attempt is therefore made to explore new/modified or improved techniques to model major components of the rainfall–runoff–sediment yield process in a more sound theoretical and mathematical environment. However, in the meantime, simplicity of model structure, practical utility in terms of data requirement as well as easiness in use, and parsimony in data, time, and funds are central to the present study. The present research investigates a few important components of process of the hydrological cycle, specifically the process of rainfall–runoff–sediment yield including evapotranspiration, runoff, soil erosion, and sediment yield.

Watersheds or hydrological units are viewed as more proficient and suitable for important overview and examination for the evaluation of these resources and consequent planning and application of different formative projects like soil and water conservation, order territory advancement, disintegration control in catchment waterways, dryland/rainfed cultivating, recovery of gorge lands, and so forth (Gajbhiye and Sharma 2017).

The definition of appropriate watershed management programs for economic advancement requires a stock of the quantitative soil loss, erosion, and need of watershed. There might be different contemplations for the execution of management programs in the few sub-watersheds. It is constantly better to start management measures from the most fundamental sub-watersheds. Prioritization of watershed is the positioning of various basic sub-watersheds along these lines as indicated by the request in which they should be taken up for the treatment by soil and water management measures (Ambade 2012).

Sediment yield is one of the primary rules for prioritization of watershed to soil erosion. Be that as it may, this model requires constant observation of sediment test at the catchments’ outlet. In India for small watersheds, such information is not really available (Gajbhiye et al. 2014a, b; Gajbhiye et al. 2015; Meshram et al. 2017c; Meshram et al. 2018). In spite of the fact that the sediment yield from large basin can be found from such observations, it is not conceivable to find out the reason for soil loss of small watershed inside a basin. A soil protection program is a costly and bulky procedure, passed out in steps beginning from the most critical area. In this way, there is a requirement to give comparative needs to various areas inside a catchment (Gajbhiye 2015).

The curve number (CN) represents the runoff which is basic for the arrangement of soil erosion control processes. The curve number is at risk to spatial and transient variety (Gajbhiye et al. 2014a, b; Meshram et al. 2017a, b, c). The SCS-CN technique introduced in 1954 processes the surface runoff volume for a given precipitation event from little watersheds (SCS 1956, 1985), which is required for figuring soil erosion.

Spline functions and, all the more by and large, piecewise polynomial functions are the best approximating capacities being used today. A large number of studies have been conducted on the utilization of cubic, quadratic, and quintic splines at territorial and national levels (McAllister and Roulie 1978; Sakai and Usmani 1983; Siddiqi and Akram 2008; Wu and Zhang 2014; Rababah et al. 2017; Herriot and Reinsch 1976; Alayed et al. 2016; Yang and Wang 1994; Holnicki 1996: Tariq and Akram 2016; Meshram et al. 2017a, b, c; Meshram and Powar 2017). Use of spline function is expanding step by step in the areas of different sciences, medicine, farming, and engineering. The cubic spline is the most generally and broadly utilized by spline function; the quadratic spline is also well considered. Thusly, the target of this paper is to apply quadratic and quintic splines to soil erosion modeling and to compare with the cubic spline.

## Study area

## Primilaries of splines

### Quadratic spline

*C*

^{1}piecewise quadratic polynomial. This implies:\(S_{2,n} \left( z \right)\) is piecewise quadratic; that is, among continuous knots \(z_{i}\)

*C*

^{1}; that is,\(S_{2,n} \left( z \right)\) is consistent and has ceaseless first subsidiary wherever in the interim [a, b], specifically, at the knots.

Inside every interim \(\left( {z_{i - 1} z_{i} } \right),\) the comparison of quadratic polynomial is ceaseless and has persistent subordinates of all requests. Accordingly, \(S_{2,n} \left( z \right)\) or one of its subsidiaries can be irregular just at a bunch. Note that the capacity \(S_{2,n} \left( z \right)\) has two quadratic parts occurrence at the inside knot \(z_{i} ;\) to one site of \(z_{i} ,\) it is a quadratic \(p_{i} \left( z \right)\), while to the correct it is a quadratic \(p_{i + 1} \left( z \right)\).

Along these lines of composing, the insertion conditions additionally power \(S_{2,n} \left( z \right)\) to be nonstop at the bunches. Since each of the n quadratic pieces has three obscure coefficients, our portrayal of the capacity \(S_{2,n} \left( z \right)\) includes 3*n* obscure coefficients. Guaranteeing coherence of the principal subsidiary forces (*n* − 1) direct imperatives on its coefficients, and interjection forces an extra 2*n* straight limitations. In this way, there is an aggregate of 3*n* − 1 straight requirements on the 3*n* obscure coefficients. All together, we have an indistinguishable number of conditions from questions and we require one progressive (straight) limitation.

### Quintic spline

*L*] into the subintervals \(\left[ {z_{i - 1} z_{i} } \right]\). Let \(u\left( z \right)\) be an adequately differentiable capacity characterized on [0,

*L*] and \(S\) be a quin \(\left( z \right)\) tic polynomial spline to \(u\left( z \right)\). Consider that every quintic polynomial spline section \(P_{i} \left( z \right)\) has the accompanying structure:

*S*can be composed as far as \(S_{i} s\) and any three derivatives at the limits of each subinterval. To characterize spline as far as \(S_{i} s\) and \(F_{i} s\), the coefficients presented in Eq. (5) are figured as:

*ρ*= 1 and 3, and the accompanying helpful relations are gotten as

## Construction of splines

Manot dataset utilized in the present study for quadratic/quintic spline

Sub-watershed | SYI | CN |
---|---|---|

MN 1 | 991.61 | 73.87 |

MN 2 | 987.42 | 77.73 |

MN 3 | 986.82 | 72.03 |

MN 4 | 1032.24 | 84.95 |

MN 5 | 1173.33 | 79.51 |

MN 6 | 1088.42 | 78.23 |

MN 7 | 1031.08 | 64.75 |

MN 8 | 1099.92 | 59.36 |

MN 9 | 940.97 | 76.71 |

MN 10 | 1030.19 | 66.42 |

MN 11 | 1347.6 | 76.71 |

MN 12 | 1238.36 | 67.84 |

MN 13 | 1007.85 | 73.76 |

MN 14 | 925.38 | 74.15 |

We built the quadratic and quintic splines as discussed in “Quadratic spline” section for the Manot watershed as:

Given information at the knots \(\delta \left( {n_{1} } \right), \delta \left( {n_{2} } \right), \delta \left( {n_{3} } \right), \delta \left( {n_{4} } \right), \delta \left( {n_{5} } \right), \delta \left( {n_{6} } \right),\, \delta \left( {n_{7} } \right)\), we built a piecewise quadratic polynomial.

### Quadratic spline

### Quintic splines

## Sediment yield index (SYI) approximation validation

*C*

_{1}–

*C*

_{7}) (Tables 2, 3) were utilized in Eqs. 16–27 for the calculation of SYI. This figured SYI named as computed sediment yield index (SYI

_{C}), and it was compared with the conventionally derived sediment yield index named as observed sediment yield index (SYIo). The observed SYI values are plotted against the computed SYI (Figs. 3, 4) and compared through a line of perfect fit. The coefficient of determination (

*R*

^{2}) is appeared in Table 4. It is observed that the quadratic and quintic splines have not predicted well the SYI from CN.

Values of the coefficient and constructed quadratic spline

CN | Coefficient | Splines | ||
---|---|---|---|---|

| | | ||

59.36–64.75 | 0.0000 | − 12.7718 | 1858.0540 | \(p_{1} \left( x \right) = 1858.0540 - 12.7718x + 0 x^{2}\) |

64.75–67.84 | 25.8423 | − 3359.3496 | 110,203.5103 | \(p_{2} \left( x \right) = 110203.5103 - 3359.3496x + 25.8423x^{2}\) |

67.84–73.76 | − 31.3971 | 4406.8986 | − 153,227.6271 | \(p_{3} \left( x \right) = - 153227.6271 + 4406.8986x - 31.3971x^{2}\) |

73.76–76.71 | 68.5212 | − 10,333.0503 | 390,381.6870 | \(p_{4} \left( x \right) = 390381.6870 - 10333.0503x + 68.5212x^{2}\) |

76.71–79.51 | − 34.4573 | 5465.9098 | − 215,587.4299 | \(p_{5} \left( x \right) - 215587.4299 + 5465.9098x - 34.4573x^{2}\) |

79.51–84.95 | − 2.2869 | 350.1714 | − 12,211.2477 | \(p_{6} \left( x \right) = - 12211.2477 + 350.1714x - 2.2869x^{2}\) |

Values of the coefficient and constructed quintic spline

CN | Coefficient | Splines | |||||
---|---|---|---|---|---|---|---|

| | | | | | ||

59.36–64.75 | − 0.0002 | 0.0554 | − 4.5962 | 168.2128 | − 2292.8540 | − 177.4029 | \(p_{1} \left( x \right) = - 177.4029 - 2292.8540x + 168.2128 x^{2} - 4.5962 x^{3} + 0.0554 x^{4} - 0.0002 x^{5}\) |

64.75–67.84 | − 0.0002 | 0.0302 | − 1.2753 | − 12.7640 | 1238.0507 | 120.8112 | \(p_{2} \left( x \right) = 120.8112 + 1238.0507x - 12.7640x^{2} - 1.2753x^{3} + 0.0302x^{4} - 0.0002 x^{5}\) |

67.84–73.76 | − 0.0002 | 0.0302 | − 1.2739 | − 12.8494 | 1240.5799 | 91.5073 | \(p_{3} \left( x \right) = 91.5073 + 1240.5799x - 12.8494x^{2} - 1.2739x^{3} + 0.0302 x^{4} - 0.0002 x^{5}\) |

73.76–76.71 | − 0.0001 | 0.0239 | − 1.6269 | 35.6846 | 20.4722 | 1.2750 | \(p_{4} \left( x \right) = 1.2750 + 20.4722x + 35.6846 x^{2} - 1.6269x^{3} + 0.0239x^{4} - 0.0001x^{5}\) |

76.71–79.51 | − 0.0001 | 0.0241 | − 1.6488 | 36.8027 | − 0.9618 | − 0.1627 | \(p_{5} \left( x \right) = - 0.1627 - 0.9618x + 36.8027x^{2} - 1.6488x^{3} + 0.0241x^{4} - 0.0001 x^{5}\) |

79.51–84.95 | − 0.0001 | 0.0242 | − 1.6611 | 37.4543 | − 13.9050 | − 0.9337 | \(p_{6} \left( x \right) = - 0.9337 - 13.9050x + 37.4543x^{2} - 1.6611x^{3} + 0.0242x^{4} - 0.0001 x^{5}\) |

Performance evaluation of splines

Spline | |
---|---|

Quadratic spline | 0.36 |

Quintic spline | 0.48 |

Cubic spline | 0.87 |

## Inter-comparison of models

In order to compare the applied spline (quadratic/quintic) with each other and another spline, i.e., cubic spline (Meshram et al. 2017a, b, c), Table 4 shows the *R*^{2} criteria of the selected splines and cubic spline. According to Table 4, it is evident that, as it is expected, the cubic spline was the best model based on coefficient of determination (i.e., *R*^{2}). This confirms that the cubic spline can be a powerful tool for sediment estimation at daily scales. The quintic spline (with a value of *R*^{2} = 0.48) and the quadratic spline (with a value of *R*^{2} = 0.36) ranked as the second and the third best models, respectively.

## Conclusion

We anticipate the rough estimation of SYI of a given set of CN ahead of time without doing any examination. What’s more, we have settled the issue of labor and time limitation up to some degree. In this way, the SYI (observed) information of Manot watershed shows a solid relationship with SYI inferred utilizing cubic spline (*R*^{2} = 0.87), and some connection with SYI determined utilizing the quadratic spline (*R*^{2} = 0.36) contrasted with quintic spline (*R*^{2} = 0.48). Numerical outcomes appeared to demonstrate that the cubic spline technique is more exact than the quadratic/quintic spline strategy.

## Notes

### Acknowledgements

The authors are thankful to anonymous reviewers for their valuable suggestions and critical comments to improve the quality of this paper. The first author is thankful to UGC, New Delhi, for providing financial support under the scheme of Dr. D.S. Kothari Postdoctoral Fellowship (DSKPDF).

## References

- Alayed O, Ying TY, Saaban A (2016) Quintic spline method for solving linear and nonlinear boundary value problems. Sains Malays 45(6):1007–1012Google Scholar
- Ambade B (2012) Physico-chemical assessment of rain, fog and runoff water. Lap-Lambert Academic Publishing, Germany. ISBN 978-3-659-30271-8Google Scholar
- Gajbhiye S (2015) Estimation of surface runoff using remote sensing and geographical information system. Int Jf u-and e-Serv Sci Technol 8(4):118–122Google Scholar
- Gajbhiye S, Sharma SK (2017) Prioritization of watershed through morphometric parameters: a PCA based approach. Appl Water Sci 7:1505–1519CrossRefGoogle Scholar
- Gajbhiye S, Mishra SK, Pandey A (2014a) Relationship between SCS-CN and Sediment Yield. Appl Water Sci 4(4):363–370CrossRefGoogle Scholar
- Gajbhiye S, Sharma SK, Meshram C (2014b) Prioritization of watershed through sediment yield index using RS and GIS approach. Int Jf u-and e-Serv Sci Technol 7(6):47–60CrossRefGoogle Scholar
- Gajbhiye S, Mishra SK, Pandey A (2015) Simplified sediment yield index model incorporating parameter CN. Arab J Geosci 8(4):1993–2004CrossRefGoogle Scholar
- Herriot JG, Reinsch CH (1976) Procedures for quintic interpolation. ACM Trans Math Softw 2(3):281–289CrossRefGoogle Scholar
- Holnicki P (1996) A piecewise-quintic interpolation scheme. J Comput Phys 127:316–329CrossRefGoogle Scholar
- McAllister DF, Roulie JA (1978) Interpolation by convex quadratic splines. Math Comput 32(144):1154–1162CrossRefGoogle Scholar
- Meshram SG, Powar PL (2017) Piecewise regression using cubic spline—a case study. Int J Hybrid Inf Technol 10(1):75–84CrossRefGoogle Scholar
- Meshram SG, Powar PL, Singh VP, Meshram CS (2018) Application of cubic spline in soil erosion modelling from Narmada Watersheds, India. Arab J Geosci. https://doi.org/10.1007/s12517-018-3699-8 CrossRefGoogle Scholar
- Meshram SG, Powar PL, Singh VP (2017b) Modelling soil erosion from a watershed using cubic splines. Arab J Geosci 10:155–168CrossRefGoogle Scholar
- Meshram SG, Sharma SK, Tignath S (2017c) Application of remote sensing and geographical information system for generation of runoff curve number. Appl Water Sci 7:1773–1779CrossRefGoogle Scholar
- Rababah A, Khalid MB, Shakhaterh MK (2017) Quadratic spline interpolation with minimized error. AIP Conf Proc 1863:060007. https://doi.org/10.1063/1.4992221 CrossRefGoogle Scholar
- Sakai M, Usmani R (1983) Quadratic spline and two-point boundary value problem. Publ RIMS Kyoto Univ 19:7–13CrossRefGoogle Scholar
- SCS (1956) Hydrology, national engineering handbook, supplement A, section 4, chapter 10, soil conservation service. USDA, WashingtonGoogle Scholar
- Siddiqi SS, Akram G (2008) Quintic spline solutions of fourth order boundary-value problems. Int J Numer Anal Model 5(1):101–111Google Scholar
- Tariq H, Akram G (2016) Quintic spline technique for time fractional fourth-order partial differential equation. Numer Methods Partial Differ Equ. https://doi.org/10.1002/num CrossRefGoogle Scholar
- Wu J, Zhang X (2014) Integro quadratic spline interpolation. Appl Math Model. https://doi.org/10.1016/j.apm.2014.11.015 CrossRefGoogle Scholar
- Yang DCH, Wang FC (1994) A quintic spline interpolator for motion command generation of computer-controlled machines. Trans ASME 116:226–231CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.