Parameter Estimation Using the Inverse Problem Method for Simulating Lateral Inflow and Runoff Depth in a small catchment of Amazon

FALCÓN, CINDY T.; BLANCO, CLAUDIO JOSÉ C.; ESTUMANO, DIEGO C.

doi:10.1590/0001-3765202420230570

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ENGINEERING SCIENCES • An. Acad. Bras. Ciênc. 96 (3) • 2024 • https://doi.org/10.1590/0001-3765202420230570 copy

Parameter Estimation Using the Inverse Problem Method for Simulating Lateral Inflow and Runoff Depth in a small catchment of Amazon

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The inverse problem method can be applied to determine the properties of hydrological phenomena and estimate the parameters, which cannot be measured directly. This type of inverse focus can facilitate the implementation of the kinematic wave model (direct model-DM), to fill gaps for lateral inflow rate and runoff depth in watersheds. Thus, the goal of the study was the application of the inverse problem method (IP). The lateral inflow rate was generally obtained as a Fourier transform to represent any watersheds. The study was developed using a small catchment in the Amazon where intense rainfall events occur, producing runoff and sediments, which affect rural populations. Lateral inflow rate and runoff depth were derived using precipitation data and parameters estimated through the KINEROS2 (K2)/direct model (DM) model and the ensuing solution methods with MCMC (Markov chains Monte Carlo)/Fourier transform. The developed method was applied to four rainfall-runoff events, leading to a good fit between the observed and predicted data (Nash-Sutcliffe coefficients between 0.76 and 0.85 and RMSE values between 1.80 mm and 6.72 mm).

Key words
Kinematic waves; KINEROS2; Inverse problem; Parameter estimation

Parameter	Symbol	Units	Values used in the study	Values suggested	Estimation method/ Reference
Length	L	m	3000	-	Field measurements
Width	W	m	20	-	Field measurements
Slope	S	-	0,025	0-1	Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001...
Density	Ss	-	0.80-1.50	0-2.80	An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Thickness	Thick	mm	300	0-500	An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Soil porosity	ɸ	-	0.38	0-45.77	Al-Qurashi et al. 2008AL-QURASHI A, MCINTRYE N, WHEATER H & UNKRICH C. 2008. Application of the KINEROS2 Rainfall-Runoff Model to an Arid Catchment in Oman. J Hydrol 355: 91-105. https://doi.org/10.1016/j.jhydrol.2008.03.022. https://doi.org/10.1016/j.jhydrol.2008.0... , An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Saturation initial	Si	-	0,2	0-1	Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001... , An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Manning	n	-	0,3	0.01-0.8	Memariam et al. 2012MEMARIAM H, BALASUNDRAM SK, TALIB J, SUNG CTB, SOOD AM, ABBASPOUR KC & HAGHIZADEH A. 2012. Hydrologic Analysis of a Tropical Watershed using KINEROS2. Environment Asia 5(1): 84-93., An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Saturated hydraulic conductivity	Ks	mm.hr¯¹	0.60	0-50	Al-Qurashi et al. 2008AL-QURASHI A, MCINTRYE N, WHEATER H & UNKRICH C. 2008. Application of the KINEROS2 Rainfall-Runoff Model to an Arid Catchment in Oman. J Hydrol 355: 91-105. https://doi.org/10.1016/j.jhydrol.2008.03.022. https://doi.org/10.1016/j.jhydrol.2008.0... , Memariam et al. 2012MEMARIAM H, BALASUNDRAM SK, TALIB J, SUNG CTB, SOOD AM, ABBASPOUR KC & HAGHIZADEH A. 2012. Hydrologic Analysis of a Tropical Watershed using KINEROS2. Environment Asia 5(1): 84-93.
Coefficient of variation of Ks	Cv	-	0.1	0.01-50	Memariam et al. 2012MEMARIAM H, BALASUNDRAM SK, TALIB J, SUNG CTB, SOOD AM, ABBASPOUR KC & HAGHIZADEH A. 2012. Hydrologic Analysis of a Tropical Watershed using KINEROS2. Environment Asia 5(1): 84-93., Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001...
Mean capillary drive	G	mm	12.70	0-500	Memariam et al. 2012MEMARIAM H, BALASUNDRAM SK, TALIB J, SUNG CTB, SOOD AM, ABBASPOUR KC & HAGHIZADEH A. 2012. Hydrologic Analysis of a Tropical Watershed using KINEROS2. Environment Asia 5(1): 84-93., An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Pore size distribution index	λ	mm	0.45	0.03-1.43	Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001... , An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Rainsplash coefficient	Cf	-	0	0-625, 2500, 5625	An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Soil cohesion coefficient	Co	-	0.5	0.001-0.1	An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Micro topographic Relief	re	mm	2	0-400	Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001...
Micro topographic Spacing	rs	m	0.3	0.001-10	Al-Qurashi et al. 2008AL-QURASHI A, MCINTRYE N, WHEATER H & UNKRICH C. 2008. Application of the KINEROS2 Rainfall-Runoff Model to an Arid Catchment in Oman. J Hydrol 355: 91-105. https://doi.org/10.1016/j.jhydrol.2008.03.022. https://doi.org/10.1016/j.jhydrol.2008.0... , Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001...
Rock cover	r	%	0-1	0-1	An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...
Interception depth	In	mm	1.0	0.01-300	Kim et al. 2014KIM K, WHELAN G, PURUCKER T, BOHRMANN TF, CYTERSKI MJ, MOLINA M, GU Y, PACHEPSKY Y, GUBER A & FRANKLIN DR. 2014. Rainfall-runoff model parameter estimation and uncertainty evaluation on small plots. Hydrol Proc 28: 5220-5235. https://doi.org/10.1002/hyp.10001. https://doi.org/10.1002/hyp.10001...
Plant cover	p	%	0-1	0-1	An et al. 2019AN M, HAN Y, XU L, XIURU W, AO C & PANG D. 2019. KINEROS2-based simulation of total nitrogen loss on slopes under rainfall events. CATENA 177: 13-21. https://doi.org/10.1016/j.catena.2019.01.039. https://doi.org/10.1016/j.catena.2019.01...

Parameter		Optimal values with IP per event
Parameter		Event A	Event B	Event C	Event D
Upstream width	W_u	13.49	17.935	14.142	16.104
Width	W	10.826	9.469	9.378	9.711
Upstream slope and roughness	α_u	112.672	147.632	158.668	149.114
Slope and roughness	α	7.394	6.179	6.278	6.671
Fourier transform coefficients	Ω	0.009	0.009	0.009	0.009
	A0	0.456	0.438	0.462	0.458
	A1	41.189	39.702	39.609	41.182
	B1	26.673	16.947	15.872	17.186
	A2	15.28	14.866	14.189	14.404
	B2	2.12E-04	2.33E-04	2.13E-04	2.32E-04

Events		Parameters K2/IP
Events		W	W _u	α	α _u	Ω	A0	A1	B1	A2	B2
Range		0-50	0-50	0-200	0-200	0-0.01	0-0.5	0-50	0-50	0-50	0-0.01
A	Average	13.49	10.83	112.67	7.39	0.009	0.46	41.19	26.67	15.28	2.12E-04
	CI	13.4-13.5	10.8-10.9	112.6-112.7	7.3-7.4	0.008-0.009	0.45-0.46	41.18-41.19	26.6-26.7	15.2-15.3	2.12E-04-2.13E-04
	95%	13.4-13.5	10.8-10.9	112.6-112.7	7.3-7.4	0.008-0.009	0.45-0.46	41.18-41.19	26.6-26.7	15.2-15.3	2.12E-04-2.13E-04
B	Average	17.93	9.47	147.63	6.18	0.009	0.44	39.7	16.95	14.87	2.33E-04
	CI	17.7-18.1	9.32-9.60	145.5-148.9	6.10-6.25	0.008-0.009	0.43-0.44	38.11-40.62	15.8-18.3	14.6-15.1	2.31E-04-2.36E-04
	95%	17.7-18.1	9.32-9.60	145.5-148.9	6.10-6.25	0.008-0.009	0.43-0.44	38.11-40.62	15.8-18.3	14.6-15.1	2.31E-04-2.36E-04
C	Average	17.14	9.38	158.67	6.28	0.009	0.46	39.61	15.87	14.19	2.13E-04
	CI	16.9-17.4	9.28-9.51	153.5-164.1	6.21-6.34	0.008-0.009	0.45-0.47	38.35-41.18	14.8-17.0	13.9-14.6	2.11E-04-2.15E-04
	95%	16.9-17.4	9.28-9.51	153.5-164.1	6.21-6.34	0.008-0.009	0.45-0.47	38.35-41.18	14.8-17.0	13.9-14.6	2.11E-04-2.15E-04
D	Average	16.1	9.71	149.11	6.67	0.009	0.46	41.18	17.19	14.4	2.32E-04
	CI	16.0-16.2	9.60-9.84	143.7-156.7	6.61-6.72	0.009	0.45-0.46	39.28-43.02	16.0-18.5	14.3-14.5	2.30E-04-2.34E-04
	95%	16.0-16.2	9.60-9.84	143.7-156.7	6.61-6.72	0.009	0.45-0.46	39.28-43.02	16.0-18.5	14.3-14.5	2.30E-04-2.34E-04

Events	r	R²	NSE	RMSE	PBIAS
A	0.88	0.77	0.76	6.72	-3.35
B	0.93	0.86	0.83	1.95	-5.32
C	0.93	0.87	0.84	1.80	-5.38
D	0.93	0.86	0.85	2.25	-3.88

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[1] Correspondence to: Claudio José Cavalcante Blanco E-mail: blanco@ufpa.br