# Sensitivity Analysis in Mean Annual Sediment Yield Modeling with Respect to Rainfall Probability Distribution Functions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Applied Methodology

#### 2.2. Rainfall Probability Distribution Functions

^{2}test [50,51]. In general, fitting distributions can be accomplished by the following two methods: (i) the method of moments or (ii) method of maximum likelihood [33]. In this study, parameters are defined analytically based on the method of moments for the functions GBS, GBL and LP3. The SQRT-ET function requires numerical methods for an adjustment with regional parameters [52,53]. To clarify the equations, the parameters of each function are differentiated with an individual index for each case.

#### 2.2.1. Gumbel Type I (Small Sample Size)

_{x}= the typical deviation in the sample (rainfall dataset in the case study); $\overline{x}$ = the arithmetic average of the sample; ${\sigma}_{y}$= the typical deviation in the series of values ${y}_{i}$; ${\mu}_{y}$= the arithmetic average of the series of values ${y}_{i}$. The value of ${y}_{i}$ is calculated as follows:

#### 2.2.2. Gumbel Type I (Large Sample Size)

#### 2.2.3. Log-Pearson Type III

#### 2.2.4. SQRT-Exponential Type Distribution of the Maximum

_{i}and b

_{i}are tabulated parameters used to adjust the dependence relationships with the covariance C

_{v}.

#### 2.3. Probabilistic Combination Model

_{i}

_{+1}− 1/T

_{i}. This model also integrates a rainfall distribution function for each annual yield. The MUSLE model is applied in a semi-distributed way at sub-basin scale (Figure 3b). The use of the HEC-HMS tool allows us to combine the hydrometeorological calculations with MUSLE [64]. The application of MUSLE is reiterated for a series of T.

**.**It should be taken into account that the use of very high values of T (e.g., >100 years) implies the assumption of temporal invariance of the climate, soil, land use and vegetation characteristics for excessively long periods. Notwithstanding the above, it should be taken into account that for T > 100 years, its contribution to the annual yield is very low. In this study, to analyze the formulation, a value of T up to 200 years is considered for safety reasons. This is due to the fact that functions LP3 and SQRT-ET have a relative influence that is slightly more relevant for high T

**,**due to its higher peak discharges (water flow and sediment) regarding the GBS and GBL functions. According to different studies [24], the intervals and probabilities are respectively modifying. The general formulation of the probabilistic combination function is as follows:

_{T}= the mean annual sediment yield for the whole basin (Mg); Y

_{Ti}= the sediment yield that corresponds to a design storm with a return period T

_{i}for the entire basin (Mg); and m, n and n−1 are the first, last and penultimate values of the adopted T series, respectively. For Equation (16) and for the particular case of T intervals between 2, 5, 10, 15, 25, 50, 75, 100, 140 and 200 years (i.e., m = 2, n = 200, and n − 1 = 140 years), by replacing the probabilities for each interval $\left(\mathrm{i}.\mathrm{e}.,1/{T}_{i}-1/{T}_{i+1}\right)$ and replacing the sediment yielded by a design storm with a return period T

_{i}(i.e., Y

_{Ti}), a simplified version can be obtained, which is as follows:

_{Ti}can be used to assess its contribution to the mean annual sediment yield. The value of each Y

_{Ti}in the present work is obtained by applying the MUSLE model in a semi-distributed way using sub-basins. Y

_{Ti}depends on the hydrological connection for each case. By varying the MUSLE factors in each sub-basin, and where is a predominant channel, transport and sedimentation processes are triggered in some reaches. For simple cases of small and almost homogeneous basins, the MUSLE factors are considered to be uniform. The definition of the MUSLE model [15,16] in each sub-basin is as follows:

^{3}); q

_{p}= the peak flow discharge (m

^{3}·s

^{−1}); and K, LS, C and P are the erodibility, slope length and steepness, crop/vegetation and management, respectively, and support the conservation practice factors analogous to the USLE model [8,10,11].

#### 2.4. Study Area

#### 2.4.1. Location

#### 2.4.2. Climatology and Rainfall Dataset

#### 2.4.3. Geology, Soils, Forest Cover and MUSLE Parameters

- Algibe sands, which include soils with a sandy-loam texture on the “Algibe” lithological units present in the basin. They are taxonomically described as Eutric Cambisols, Chromic Luvisols and Lithosols with Dystric Cambisols and Rankers. They belong to the USDA Inceptisol Order. Their presence in the areas of greater relief of the basin is clear.
- Clays of the Campo de Gibraltar include clayey soils, with the texture, composition and development characteristics of the corresponding lithological units of the Campo de Gibraltar. They are taxonomically described as Chromic Vertisols and Vertic Cambisols with Calcic Cambisols, Calcareous Regosols and Pellic Vertisols. They belong to the Vertisol Order of the USDA. They are very clearly present in the areas of the basin with lower slopes.

- 1.
- The K factor (erodibility) for each unit of soil is computed using the four parameters included in Table 3. The weighing according to the area of each unit of soil for each sub-basin allows us to obtain the representative value of the K factor for the sub-basins. The values displayed in Table 2 and Table 3 are obtained by textural sampling of the soils [24]. The use of vectorial information from thematical maps of soils and lithological units [40,41] is useful to delimitate the corresponding units and their division by sub-basins.
- 2.
- The LS factor (slope length and steepness) is calculated in various phases. For the purposes of study, a digital elevation model (DEM) is available. The model has a resolution of 5 m. It was obtained by automatic stereo-correlation from the 2013 photogrammetric flight of the Spanish Aerial Orthophotography National Plan. It was created with a resolution of 50 cm/pixel [40]. The phases of the process comprise the following:
- a.
- A slope map is obtained with QGIS [42].
- b.
- Using the algorithm `r.watershed’ provided by GRASS [42] for QGIS, the LS factor is obtained in a distributed manner by the development of a geo-process with this algorithm.
- c.
- Using the raster calculator from QGIS, the average values of the LS for the sub-basins are obtained.

- 3.
- The C factor (crop/vegetation) is calculated in different steps, which are as follows:
- a.
- The use of a vectorial environmental database for soil uses [40]. It allows us to obtain a previous coverage map. The information used for the previous map is vectorial and is from 2018.
- b.
- The use of orthophotos to corroborate the previous information, including orthophotos from the region of Andalusia, with a resolution of 0.5 m. These orthophotos were generated by the Spanish National Geographic Institute in 2016.
- c.
- The use of fieldwork and a general inventory categorized by vegetation layers to determine the parcels [24].
- d.

- 4.
- The P factor (conservation practice) was assigned a value of 1 due to the fact that there are no crops in any of the parcels. There is just one agricultural parcel in sub-basin 6. This crop is not relevant, since it is dedicated to cultivating fodder in a forest context. Due to the physical characteristics and the administrative environmental protection, that restricting viable or permitted land uses [38], it is not currently used for agriculture purposes (nor is such use likely in the future).

#### 2.4.4. Basin Lag Time and Curve Number

_{c}= the time of concentration (h); L = the length of the main channel of the studied basin (km); J = the average slope of the main channel of the studied basin (km·km

^{−1}). In this work, the basin lag time is considered as the time difference between the center of the mass of excess or effective rainfall (net hyetograph) and the peak of the hydrograph (e.g., synthetic unit hydrograph). The lag time of each sub-basin has been estimated according to Equation (19), adopting the empirical relationship observed in Spanish basins [69] (t

_{lag}= 0.35t

_{c}).

_{p}). Figure 4 shows the variation in Q and q

_{p}(for the whole basin) with the return period for the four distribution functions.

## 3. Results and Discussion

#### 3.1. Definition and Adjustment of Probability Distribution Functions

#### 3.2. Sediment Yield for Different Return Periods

_{i}is integrated. The total mean annual sediment yield is doubly coupled to T [24] by the runoff generated by the different design storms, but also by the probability of occurrence associated with these design storms. This will affect the relationship between the mean annual yield and return period, meaning it will be non-linear, as can be observed in Section 3.3.

#### 3.3. Mean Annual Sediment Yield According to Each Distribution Function

^{−1}·y

^{−1}), and the SQRT-ET function presents the lowest (23,512 Mg·ha

^{−1}·y

^{−1}). Regarding the LP3, which is frequently used in the USA and other countries [35,59], the GBS data function provides a mean annual yield value that is 6.7% higher. With regard to the SQRT-ET function, which is frequently used in Spain [52,62], the differences reach 10%, for which the GBS obtained the highest value. The results must be analyzed cautiously, due to the fact that T values < 2 years and > 200 years have not been considered. Furthermore, the study area imposes its own restrictions (geological, edaphic, climatic and biologic ones). It is not possible to infer a law; thus, it is not possible to state a function. Nevertheless, there is mathematical certitude regarding the case study. In the context of hydrometeorological estimation of sediment yield, the LP3 and SQRT-ET functions do not provide higher values than those obtained with the GBS. Moreover, GBL presents results that are not on the safe or conservative side compared to LP3, or that are not relevant regarding SQRT-ET (Table 5). Some researchers advocate the abandonment of the Gumbel function in statistical hydrology. In this regard, Lettenmaier and Burges [57] propose to avoid the use of the GBL function (i.e., infinite sample assumption) and replace it with the GBS function (i.e., finite sample assumption). This approach is corroborated by the findings of the present research, and we, therefore, cannot justify the abandonment of the Gumbel function per se, but recommend that the sample size should be taken into account in the calculation of the distribution (i.e., as is the case for the GBS function given by Equations (2)–(4)). Indeed, according to our results, for the special case of the hydrometeorological estimation of sediment yield, the use of the GBS function would be preferable compared to the other functions analyzed, if the aim is to obtain predictions on the safe side. Even so, the differences between the predicted sediment yield values are only moderate (maximum value of 10%), which does not allow us to categorically discourage the use of any of the distribution functions analyzed.

## 4. Conclusions

_{y}= 0.5772 and β

_{y}= 1.2825, does not translate into an advantage when calculating the mean annual sediment yield. Nevertheless, due to the minimal differences between the distributions, the usage of a safety coefficient of 1.07 can solve this issue.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Nearing, M.A.; Xie, Y.; Liu, B.; Ye, Y. Natural and anthropogenic rates of soil erosion. Int. Soil Water Conserv. Res.
**2017**, 5, 77–84. [Google Scholar] [CrossRef] - Quine, T.A.; Walling, D.E.; Zhang, X. Tillage erosion, water erosion and soil quality on cultivated terraces near Xifeng in the Loess Plateau, China. Land Degrad. Dev.
**1999**, 10, 251–274. [Google Scholar] [CrossRef] - Mishra, P.K.; Rai, A.; Abdelrahman, K.; Rai, S.C.; Tiwari, A. Land Degradation, Overland Flow, Soil Erosion, and Nutrient Loss in the Eastern Himalayas, India. Land
**2022**, 11, 179. [Google Scholar] [CrossRef] - Butt, M.J.; Waqas, A.; Mahmood, R. The combined effect of vegetation and soil erosion in the water resource management. Water Resour. Manag.
**2010**, 24, 3701–3714. [Google Scholar] [CrossRef] - Gemitzi, A.; Petalas, C.; Tsihrintzis, V.A.; Pisinaras, V. Assessment of groundwater vulnerability to pollution: A combination of GIS, fuzzy logic and decision making techniques. Environ. Geol.
**2006**, 49, 653–673. [Google Scholar] [CrossRef] - Halbac-Cotoara-Zamfir, R.; Smiraglia, D.; Quaranta, G.; Salvia, R.; Salvati, L.; Giménez-Morera, A. Land degradation and mitigation policies in the Mediterranean region: A brief commentary. Sustainability
**2020**, 12, 8313. [Google Scholar] [CrossRef] - European Parlament. Procedure File: 2021/2548(RSP)|Legislative Observatory. 2021. Available online: https://oeil.secure.europarl.europa.eu/oeil/popups/ficheprocedure.do?reference=2021/2548(RSP)&l=en (accessed on 5 November 2022).
- Djoukbala, O.; Hasbaia, M.; Benselama, O.; Mazour, M. Comparison of the erosion prediction models from USLE, MUSLE and RUSLE in a Mediterranean watershed, case of Wadi Gazouana (NW of Algeria). Model. Earth Syst. Environ.
**2019**, 5, 725–743. [Google Scholar] [CrossRef] - Busari, M.A.; Kukal, S.S.; Kaur, A.; Bhatt, R.; Dulazi, A.A. Conservation tillage impacts on soil, crop and the environment. Int. Soil Water Conserv. Res.
**2015**, 3, 119–129. [Google Scholar] [CrossRef] [Green Version] - Wischmeier, W.H.; Smith, D.D. A Universal Soil-Loss Equation to guide conservation farm planning. Trans. 7th Int. Congr. Soil Sci.
**1960**, 1, 418–425. [Google Scholar] - RUSLE2. Conservation Planning, Inventory Erosion Rates and Estimate Sediment Delivery. NRCS, USDA, USA. Available online: https://fargo.nserl.purdue.edu/rusle2_dataweb/About_RUSLE2_Technology.htm (accessed on 11 November 2022).
- Akoko, G.; Le, T.H.; Gomi, T.; Kato, T. A review of SWAT model application in Africa. Water
**2021**, 13, 1313. [Google Scholar] [CrossRef] - Gassman, P.W.; Sadeghi, A.M.; Srinivasan, R. Applications of the SWAT model special section: Overview and insights. J. Environ. Qual.
**2014**, 43, 1–8. [Google Scholar] [CrossRef] [PubMed] - Rivera-Toral, F.; Pérez-Nieto, S.; Ibáñez-Castillo, L.A.; Hernández-Saucedo, F.R. Aplicabilidad del Modelo SWAT para la estimación de la erosión hídrica en las cuencas de México. Agrociencia
**2012**, 46, 101–105. Available online: http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-31952012000200001&lng=es&nrm=iso (accessed on 4 September 2022). - Sadeghi, S.H.R.; Gholami, L.; Khaledi Darvishan, A.; Saeidi, P. A review of the application of the MUSLE model worldwide. Hydrol. Sci. J.
**2014**, 59, 365–375. [Google Scholar] [CrossRef] [Green Version] - Williams, J.R.; Berndt, H.D. Sediment yield prediction based on watershed hydrology. Trans. ASAE
**1977**, 20, 1100–1104. [Google Scholar] [CrossRef] - Arekhi, S.; Shabani, A.; Rostamizad, G. Application of the Modified Universal Soil Loss equation (MUSLE) in prediction of sediment yield (Case study: Kengir Watershed, Iran). Arab. J. Geosci.
**2012**, 5, 1259–1267. [Google Scholar] [CrossRef] - Berteni, F.; Dada, A.; Grossi, G. Application of the MUSLE model and potential effects of climate change in a small alpine catchment in northern Italy. Water
**2021**, 13, 2679. [Google Scholar] [CrossRef] - Pongsai, S.; Schmidt Vogt, D.; Shrestha, R.P.; Clemente, R.S.; Eiumnoh, A. Calibration and validation of the Modified Universal Soil Loss Equation for estimating sediment yield on sloping plots: A case study in Khun Satan catchment of northern Thailand. Can. J. Soil Sci.
**2010**, 90, 585–596. [Google Scholar] [CrossRef] [Green Version] - Odongo, V.O.; Onyando, J.O.; Mutua, B.M.; Becht, R. Sensitivity analysis and calibration of the Modified Universal Soil Loss Equation (MUSLE) for the upper Malewa catchment, Kenya. Int. J. Sediment Res.
**2013**, 28, 368–383. [Google Scholar] [CrossRef] - Rodríguez, A.; García, J.L.; Robredo, J.C.; López, D. Specific sediment yield model for reservoirs with medium-sized basins in Spain: An empirical and statistical approach. Sci. Total Environ.
**2019**, 681, 82–101. [Google Scholar] [CrossRef] - Hrissanthou, V. Estimate of sediment yield in a basin without sediment data. Catena
**2005**, 64, 333–347. [Google Scholar] [CrossRef] - García, C.; Robredo, J.C. Metodología para la evaluación de la emisión interanual de sedimentos por una cuenca vertiente. Rev. Montes
**1996**, 45, 22–24. [Google Scholar] - Rodríguez, C.A.; Rodríguez-Pérez, Á.M.; Mancera, J.; Torres, J.; Carmona, N.; Bahamonde García, M. Applied methodology based on HEC-HMS for reservoir filling estimation due to soil erosion. J. Hydrol. Hydromech.
**2022**, 70, 341–356. [Google Scholar] [CrossRef] - Williams, J.R. Sediment-yield prediction with universal equation using runoff energy factor. In Present and Prospective Technology for Predicting Sediment Yield and Sources; US Department of Agriculture, Agriculture Research Service: Washington, DC, USA, 1975; pp. 244–252. [Google Scholar]
- Lu, J.; Zheng, F.; Li, G.; Bian, F.; An, J. The effects of raindrop impact and runoff detachment on hillslope soil erosion and soil aggregate loss in the Mollisol region of Northeast China. Soil Tillage Res.
**2016**, 161, 79–85. [Google Scholar] [CrossRef] - HEC–HMS (Hydrologic Engineering Center–Hydrologic Modeling System), 4.10 [Computer Software] ed; Army Corps of Engineers: Davis, CA, USA, 2022; Available online: http://www.hec.usace.army.mil/ (accessed on 10 October 2022).
- HEC-HMS (Hydrologic Engineering Center–Hydrologic Modeling System). Technical Reference Manual. CN Tables. Available online: https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/cn-tables (accessed on 14 November 2022).
- Shi, W.; Chen, T.; Yang, J.; Lou, Q.; Liu, M. An improved MUSLE model incorporating the estimated runoff and peak discharge predicted sediment yield at the watershed scale on the Chinese Loess Plateau. J. Hydrol.
**2022**, 614, 128598. [Google Scholar] [CrossRef] - Chimene, C.A.; Campos, J.N.B. The design flood under two approaches: Synthetic storm hyetograph and observed storm hyetograph. J. Appl. Water Eng. Res.
**2020**, 8, 171–182. [Google Scholar] [CrossRef] - Watt, E.; Marsalek, J. Critical review of the evolution of the design storm event concept. Can. J. Civ. Eng.
**2013**, 40, 105–113. [Google Scholar] [CrossRef] - Yen, B.C.; Chow, V.T. Design hyetographs for small drainage structures. J. Hydraul. Div.
**1980**, 106, 1055–1076. [Google Scholar] [CrossRef] - Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill: New York, NY, USA, 1988. [Google Scholar]
- Ilić, A.; Plavšić, J.; Radivojević, D. Rainfall-runoff simulation for design flood estimation in small river catchments. Facta Universitatis. Ser. Archit. Civ. Eng.
**2018**, 16, 029–043. [Google Scholar] [CrossRef] [Green Version] - Maity, R. Probability Distributions and Their Applications. In Civil and Environmental Engineering; Springer: Singapore, 2018; pp. 93–143. [Google Scholar] [CrossRef]
- Haan, C.T. Statistical Methods in Hydrology, 2nd ed.; Iowa State University Press: Ames, IA, USA, 2002. [Google Scholar]
- Stedinger, J.R.; Vogel, R.M.; Foufoula-Georgiou, E. Chapter 18. Frequency analysis of extreme events. In Handbook of Hydrology; Editor in Chief David R. Maidment; McGraw-Hill: New York, NY, USA, 1993. [Google Scholar]
- Ministerio Para la Transición Ecológica y el Reto Demográfico. Gobierno de España. Reserva Natural Fluvial Alto Palmones. Available online: https://www.miteco.gob.es/es/agua/temas/delimitacion-y-restauracion-del-dominio-publico-hidraulico/Catalogo-Nacional-de-Reservas-Hidrologicas/informacion/andalucia/alto-palmones/default.aspx (accessed on 8 November 2022).
- Ministerio Para la Transición Ecológica y el Reto Demográfico. Gobierno de España. Agencia Estatal de Meteorología. Available online: https://www.aemet.es/en/serviciosclimaticos (accessed on 3 September 2021).
- Consejería de Agricultura, Ganadería, Pesca y Desarrollo Sostenible. Junta de Andalucía. Catálogo de la Red de Información Ambiental de Andalucía (REDIAM). Available online: https://www.juntadeandalucia.es/medioambiente/portal/acceso-rediam (accessed on 29 November 2022).
- Ministerio de Ciencia de Innovación. CSIC. Instituto Geológico y Minero de España. Gobierno de España. Información Geocientífica del IGME. Available online: http://info.igme.es/catalogo/default.aspx?lang=spa (accessed on 11 November 2022).
- QGIS. A Free and Open Source Geographic Information System. Available online: https://www.qgis.org/en/site/ (accessed on 8 November 2022).
- Steinmetz, A.A.; Beskow, S.; Terra, F.D.S.; Nunes, M.C.M.; Vargas, M.M.; Horn, J.F.C. Spatial discretization influence on flood modeling using unit hydrograph theory. RBRH
**2019**, 24, 1–12. [Google Scholar] [CrossRef] [Green Version] - Pak, J.H.; Fleming, M.; Scharffenberg, W.; Gibson, S.; Brauer, T. Modeling surface soil erosion and sediment transport processes in the upper North Bosque River Watershed, Texas. J. Hydrol. Eng.
**2015**, 20, 04015034. [Google Scholar] [CrossRef] - Pak, J.H.; Ramos, K.; Fleming, M.; Scharffenberg, W.; Gibson, S. Sensitivity Analysis for Sediment Transport in the Hydrologic Modeling System (HEC-HMS). Proc., Joint Federal Interagency Conf. 2015. Available online: https://acwi.gov/sos/pubs/3rdJFIC/Contents/2A-Pak.pdf (accessed on 8 November 2022).
- Gumbel, E.J. The return period of flood flows. Ann. Math. Stat.
**1941**, 12, 163–190. Available online: https://www.jstor.org/stable/2235766 (accessed on 9 November 2022). [CrossRef] - Wanielista, M.; Robert, K.; Ron, E. Hydrology: Water Quantity and Quality Control; John Wiley and Sons: Hoboken, NJ, USA, 1997. [Google Scholar]
- Molin, P.; Abdi, H. New Table and Numerical Approximations for Kolmogorov-Smirnov/Lilliefors/van Soest Normality Test; University of Bourgogne: Dijon, France, 1998; Available online: https://personal.utdallas.edu/~herve/MolinAbdi1998-LillieforsTechReport.pdf (accessed on 24 October 2022).
- Lilliefors, H.W. On the Kolmogorov-Smirnov test for normality with mean and variance unknown. JASA
**1967**, 62, 399–402. [Google Scholar] [CrossRef] - Huang, Y.P.; Lee, C.H.; Ting, C.S. Improved estimation of hydrologic data using the chi-square goodness-of-fit test. J. Chin. Instig. Eng.
**2008**, 31, 515–521. [Google Scholar] [CrossRef] - Coronado-Hernández, Ó.E.; Merlano-Sabalza, E.; Díaz-Vergara, Z.; Coronado-Hernández, J.R. Selection of hydrological probability distributions for extreme rainfall events in the regions of Colombia. Water
**2020**, 12, 1397. [Google Scholar] [CrossRef] - Zorraquino, C. La función SQRT-ET max. Revista de Obras Públicas
**2004**, 3447, 33–37. [Google Scholar] - Ferrer, F.J. El Modelo de Función de Distribución SQRT et MAX en el Análisis Regional de Máximos Hidrológicos. Aplicación a Lluvias Diarias. Ph.D. Thesis, Universidad Politécnica de Madrid, Madrid, Spain, 1996. [Google Scholar]
- Carter, D.J.T.; Challenor, P.G. Methods of fitting the Fisher-Tippett type 1 extreme value distribution. Ocean Eng.
**1983**, 10, 191–199. [Google Scholar] [CrossRef] - Gumbel, E.J. Les valeurs extrêmes des distributions statistiques. Annales de l’institut Henri Poincaré
**1935**, 5, 115–158. Available online: http://www.numdam.org/item/AIHP_1935__5_2_115_0.pdf (accessed on 9 November 2022). - Lehmer, D.H. Euler constants for arithmetical progressions. Acta Arith.
**1975**, 27, 25–142. [Google Scholar] [CrossRef] [Green Version] - Lettenmaier, D.P.; Burges, S.J. Gumbel’s extreme value I distribution: A new look. J. Hydraul. Eng.
**1982**, 108, 502–514. [Google Scholar] [CrossRef] - Pearson, K. Contributions to the mathematical theory of evolution. Philos. Trans. R. Soc.
**1894**, 185, 71–110. [Google Scholar] [CrossRef] [Green Version] - Huynh, N.P.; Thambirajah, J.A. Applications of the log Pearson type-3 distribution in hydrology. J. Hydrol.
**1984**, 73, 359–372. [Google Scholar] [CrossRef] - Bobee, B.B.; Robitaille, R. The use of the Pearson type 3 and log Pearson type 3 distributions revisited. Water Resour. Res.
**1977**, 13, 427–443. [Google Scholar] [CrossRef] - Etoh, T.; Murota, A.; Nakanishi, M. SQRT-exponential type distribution of maximum. In Hydrologic Frequency Modeling; Springer: Dordrecht, The Netherlands, 1987; pp. 253–264. [Google Scholar]
- Ferrer, F.J. Recomendaciones Para el Cálculo Hidrometeorológico de Avenidas. Ed. CEDEX, Spain. 2000. Available online: https://hispagua.cedex.es/node/92786 (accessed on 24 October 2022).
- Ministerio de Fomento. Dirección General de Carreteras. Gobierno de España. Máximas Lluvias Diarias en la España Peninsular. Ed. Secretaría de Estado de Infraestructuras y Transportes, Spain. 1999. Available online: https://www.mitma.gob.es/recursos_mfom/0610300.pdf (accessed on 24 October 2022).
- Kaffas, K.; Hrissanthou, V. Annual sediment yield prediction by means of three soil erosion models at the basin scale. In Proceedings of the 10th World Congress of EWRA “Panta Rhei”, Athens, Greece, 5–9 July 2017; pp. 5–9. Available online: https://www.ewra.net/ew/pdf/EW_2017_58_46.pdf (accessed on 10 November 2022).
- Gómez-Zotano, J.; Alcántara-Manzanares, J.; Olmedo-Cobo, J.A.; Martínez-Ibarra, E. La sistematización del clima mediterráneo: Identificación, clasificación y caracterización climática de Andalucía (España). Rev. Geogr. Norte Gd.
**2015**, 61, 161–180. [Google Scholar] [CrossRef] [Green Version] - Trinh, T.; Kavvas, M.L.; Ishida, K.; Ercan, A.; Chen, Z.Q.; Anderson, M.L.; Nguyen, T. Integrating global land-cover and soil datasets to update saturated hydraulic conductivity parameterization in hydrologic modeling. Sci. Total Environ.
**2018**, 631, 279–288. [Google Scholar] [CrossRef] [PubMed] - Allue Andrade, J.L. Atlas Fitoclimático de España. Taxonomías; Ministerio de Agricultura Pesca y Alimentación: Madrid, Spain, 1990. [Google Scholar]
- Dysarz, T.; Wicher-Dysarz, J. Application of Hydrodynamic Simulation and Frequency Analysis for Assessment of Sediment Deposition and Vegetation Impacts on Floodplain Inundation. Pol. J. Environ. Stud.
**2011**, 20, 1441–1451. Available online: http://www.pjoes.com/Application-of-Hydrodynamic-Simulation-r-nand-Frequency-Analysis-for-Assessment-r,88695,0,2.html (accessed on 11 November 2022). - Témez, J.R. Facetas del cálculo hidrometeorológico y estadístico de máximos caudales. Rev. Obras Públicas
**2003**, 3430, 47–51. [Google Scholar] - Lai, C.-D.; Murthy, D.N.; Xie, M. Weibull distributions and their applications. In Springer Handbooks; Springer: Berlin/Heidelberg, Germany, 2006; pp. 63–78. [Google Scholar] [CrossRef]
- Razali, N.M.; Wah, Y.B. Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. JOSMA
**2011**, 2, 21–33. [Google Scholar]

**Figure 3.**Thematic maps: (

**a**) slope map, (

**b**) land-cover map, (

**c**) geological and soil map; (

**d**) MUSLE factors.

**Figure 4.**(

**a**) Peak flow discharge (q

_{p}), and (

**b**) runoff volume (Q) during return period for the probability distribution functions and for the whole basin.

**Figure 5.**Cumulative probability against precipitation depth for the probability distribution functions.

**Figure 7.**Differences between theoretical and sample distributions (absolute value) against precipitation depth.

**Figure 8.**Differences between theoretical distributions with respect to Gumbel type I with a small sample size (small s.s.), as a reference, against precipitation depth.

**Figure 9.**Differences between theoretical distributions with respect to Gumbel type I with a small sample size (small s.s.), as a reference, against precipitation depth. Log scale is used for horizontal axis.

**Figure 10.**Differences between theoretical distributions with respect to Gumbel type I with a small sample size (small s.s.), as a reference, against return period.

**Figure 11.**Design hyetographs for T = 2, 25, 50 and 100 years. The blocks that correspond to GBS are highlighted in green.

Sample Size n | Mean (mm) | Standard Deviation (mm) | Covariance | Skewness Coefficient |
---|---|---|---|---|

50 | 98.62 | 45.15 | 0.46 | 0.24 |

Sandy Algibe | Campo de Gibraltar Clays | |
---|---|---|

Diameter (mm) | Percentage Lower Than | |

0.0005 | 0 | 0 |

0.0009 | 6 | 15 |

0.002 | 10 | 40 |

0.005 | 15 | 45 |

0.01 | 20 | 50 |

0.063 | 42 | 56 |

0.1 | 70 | 63 |

0.25 | 90 | 75 |

0.5 | 95 | 85 |

1 | 100 | 100 |

Properties | Type of Soil | |
---|---|---|

Sandy Algibe | Clays of the Campo de Gibraltar | |

Soil texture | Sandy loam | Clayey |

Soil structure | Simple grain, weak and thin | Crumbly and lumpy on the surface. Arranged in blocks below the surface. |

Percent of organic matter | 1.72 | 1.93 |

Soil permeability | Moderate | Very slow |

K factor | 0.30 | 0.41 |

Gumbel Type I with a Small Sample Size | Gumbel Type I with a Large Sample Size | Log-Pearson Type III | SQRT-ET Max | |||||
---|---|---|---|---|---|---|---|---|

α _{(}_{n}_{)} | β _{(}_{n}_{)} | α _{(}_{∞}_{)} | β _{(}_{∞}_{)} | β _{(LP3)} | γ _{(LP3)} | X_{0 (LP3)} | α _{(SQRT)} | k _{(SQRT)} |

0.02597 | 77.4991 | 0.0284 | 78.3037 | 0.0521 | 67.5027 | 0.9818 | 0.4993 | 69.3027 |

Interval (Years) | Probability of Occurrence ^{a} | Average Yield ^{b} (Mg) | Sediment Yield Per Interval ^{c} (Mg·y^{−1}) | ||||||
---|---|---|---|---|---|---|---|---|---|

GBS | GBL | LP3 | SQRT-ET | GBS | GBL | LP3 | SQRT-ET | ||

2–5 | 0.300 | 35,707 | 34,208 | 32,584 | 31,711 | 10,712 | 10,262 | 9775 | 9513 |

5–10 | 0.100 | 58,835 | 54,911 | 53,844 | 51,755 | 5884 | 5491 | 5384 | 5175 |

10–15 | 0.033 | 75,658 | 69,869 | 70,608 | 67,806 | 2522 | 2329 | 2354 | 2260 |

15–25 | 0.027 | 90,637 | 83,158 | 86,639 | 83,413 | 2417 | 2218 | 2310 | 2224 |

25–50 | 0.020 | 111,345 | 101,598 | 110,285 | 106,459 | 2227 | 2032 | 2206 | 2129 |

50–75 | 0.007 | 131,540 | 119,114 | 135,093 | 130,102 | 877 | 794 | 901 | 867 |

75–100 | 0.003 | 144,622 | 130,639 | 151,789 | 146,387 | 482 | 435 | 506 | 488 |

100–140 | 0.003 | 157,266 | 141,779 | 167,217 | 161,991 | 449 | 405 | 478 | 463 |

140–200 | 0.002 | 171,394 | 154,354 | 186,507 | 181,300 | 367 | 331 | 400 | 388 |

Total sum of mean annual sediment yield (Mg) | 25,937 | 24,297 | 24,313 | 23,509 |

^{a}Calculated as $\left(\frac{1}{{T}_{i}}-\frac{1}{{T}_{i+1}}\right);$

^{b}Calculated as $\frac{1}{2}\left({Y}_{Ti}+{Y}_{Ti+1}\right)$;

^{c}Calculated as $\left(\frac{1}{{T}_{i}}-\frac{1}{{T}_{i+1}}\right)\frac{1}{2}\left({Y}_{Ti}+{Y}_{Ti+1}\right)$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rodríguez González, C.A.; Rodríguez-Pérez, Á.M.; López, R.; Hernández-Torres, J.A.; Caparrós-Mancera, J.J.
Sensitivity Analysis in Mean Annual Sediment Yield Modeling with Respect to Rainfall Probability Distribution Functions. *Land* **2023**, *12*, 35.
https://doi.org/10.3390/land12010035

**AMA Style**

Rodríguez González CA, Rodríguez-Pérez ÁM, López R, Hernández-Torres JA, Caparrós-Mancera JJ.
Sensitivity Analysis in Mean Annual Sediment Yield Modeling with Respect to Rainfall Probability Distribution Functions. *Land*. 2023; 12(1):35.
https://doi.org/10.3390/land12010035

**Chicago/Turabian Style**

Rodríguez González, César Antonio, Ángel Mariano Rodríguez-Pérez, Raúl López, José Antonio Hernández-Torres, and Julio José Caparrós-Mancera.
2023. "Sensitivity Analysis in Mean Annual Sediment Yield Modeling with Respect to Rainfall Probability Distribution Functions" *Land* 12, no. 1: 35.
https://doi.org/10.3390/land12010035