Trade-off between number of constraints and primary-statement robustness in entropy models: the case of the open-channel velocity field

SILVA FILHO, ANTONIO VIANA DA; ARAÚJO, JOSÉ CARLOS DE; RAABE, ARMIN

doi:10.1590/0001-3756202020200594

ANTONIO VIANA DA SILVA FILHO

Federal Rural University of Pernambuco (UFRPE), Av. Gregório Ferraz Nogueira, Caixa Postal 063, 56909-535 Serra Talhada, PE, Brazil

https://orcid.org/0000-0002-7926-3609

JOSÉ CARLOS DE ARAÚJO

Federal University of Ceará, Department of Agricultural Engineering, Campus do Pici, Bloco 804, Pici, 60455-760 Fortaleza, CE, Brazil

https://orcid.org/0000-0002-2649-151X

ARMIN RAABE

Leipzig University, Leipzig Institute of Meteorology, Stephanstr. 3, 04103, Leipzig, Saxonia, Germany

https://orcid.org/0000-0002-8819-3294

Correspondence to: Antonio Viana da Silva Filho E-mail: viana.filho.ufrpe@hotmail.com

AUTHOR CONTRIBUTIONS

Antonio Viana da Silva Filho codified the models; developed the model with three restrictions and curvilinear coordinates (U3ξ); parameterized and simulated the models; co-wrote; and formatted the text. José Carlos de Araújo designed the research project; reviewed the models, their parameterization and their results; and co-wrote the text. Armin Raabe reviewed the models and their results; and co-wrote the text.

Figures (5)
Tables (2)

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Figure 1
Curvilinear coordinate system based on Chiu and Chiou (1986).

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Figure 2
Correlation between measured (experiments SRP1, SRP2, and SRP3: Steffler et al. 1985STEFFLER PM, RAJARATNAM N & PETERSON AW. 1985. LDA measurements in open channel. J Hydraul Eng 111(1): 119-130.) and simulated velocities using the weak statement (Cartesian coordinates): (a) model U1y; (b) model U2y; and (c) model U3y. The black dots represent points located further than 3% of the channel width from the sidewalls and from the channel bed, whereas the plus (+) signs refer to the points located elsewhere (near wall and/or channel bed).

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Figure 3
Correlation between measured (experiments SRP1, SRP2, and SRP3: Steffler et al., 1985) and simulated velocities using the strong statement (curvilinear coordinates): (a) model U1ξ; (b) model U2ξ; and (c) model U3ξ. The black dots represent points located further than 3% of the channel width from the sidewalls and from the channel bed, whereas the plus (+) signs refer to the points located elsewhere (near wall and/or channel bed).

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Figure 4
Performance of the entropy models according to the number of constraints. (a) Nash-Sutcliffe coefficient (NSE). (b) Root mean square error (RMSE). U1 encompasses the joint results for both U1y and U1ξ models. The same applies to U2 and U3.

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Figure 5
Performance of the entropy models as a function of the distance to the vertical wall. Each point refers to a measured vertical profile. The variable D_W is the horizontal distance to the nearest wall and B is channel width. Vertical grey lines indicate the threshold of the model applicability (D_W/B > 0.03). (a) Nash-Sutcliffe coefficient (NSE). (b) Root mean square error (RMSE).

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Table I
Parameters of the experiments: run 1 (SRP1), run 2 (SRP2), and run 3 (SRP3).

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Table II
Model performance of the velocity-field simulations for the 42 vertical profiles of three experiments: SRP1, SRP2, and SRP3.

Experiment	n	n.vert	Q (m³/s)	D (m)	B (m)	B/D (-)	A (m²)	u_av (m/s)	U_max (m/s)	ξ (m)
SRP1	526	10	0.126	0.146	1.143	7.83	0.167	0.755	0.844	0.00
SRP2	663	16	0.126	0.225	1.143	5.08	0.257	0.490	0.530	0.00
SRP3	541	16	0.032	0.093	1.143	12.29	0.106	0.301	0.372	0.00
All	1730	42	(-)	(-)	(-)	(-)	(-)	(-)	(-)	(-)

Experiment (n.vert)	Nash-Sutcliffe coefficient (NSE)				Root mean square error (RMSE)
Experiment (n.vert)	Min	Average	Median	Max	Min	Average	Median	Max
Model U1y: one constraint and Cartesian coordinates
SRP1 (10)	-14.96	-9.59	-8.68	-6.85	25%	35%	36%	46%
SRP2 (16)	-9.68	-6.43	-6.28	-2.84	17%	25%	26%	31%
SRP3 (16)	-7.50	-4.13	-3.80	-2.34	10%	15%	14%	19%
All (42)	-14.96	-6.31	-5.99	-2.34	10%	23%	22%	46%
Model U2y: two constraints and Cartesian coordinates
SRP1 (10)	-18.37	-3.76	-0.17	0.98	2%	14%	13%	32%
SRP2 (16)	-17.37	-1.13	0.82	0.96	2%	7%	4%	22%
SRP3 (16)	-24.16	-2.57	0.48	0.97	1%	7%	5%	18%
All (42)	-24.16	-2.31	0.51	0.98	1%	9%	6%	32%
Model U3y: three constraints and Cartesian coordinates
SRP1 (10)	-21.72	-4.88	-0.73	0.90	4%	17%	15%	34%
SRP2 (16)	-29.12	-3.20	0.05	0.68	6%	12%	10%	28%
SRP3 (16)	-16.76	-1.29	0.79	0.94	2%	6%	3%	15%
All (42)	-29.12	-2.87	0.32	0.94	4%	11%	9%	34%
Model U1ξ: one constraint and curvilinear coordinates
SRP1 (10)	-29.14	-15.93	-15.80	-6.99	34%	43%	45%	51%
SRP2 (16)	-14.62	-8.41	-8.59	-3.53	15%	27%	28%	32%
SRP3 (16)	-13.45	-6.43	-6.46	-2.79	11%	17%	18%	21%
All (42)	-29.14	-9.45	-8.41	-2.79	11%	27%	26%	51%
Model U2 ξ: two constraints and curvilinear coordinates
SRP1 (10)	-0.86	0.48	0.85	0.95	3%	6%	5%	10%
SRP2 (16)	-0.34	0.68	0.83	0.97	2%	4%	4%	6%
SRP3 (16)	0.48	0.75	0.76	0.90	2%	3%	3%	5%
All (42)	-0.86	0.66	0.77	0.97	2%	4%	4%	10%
Model U3ξ: three constraints and curvilinear coordinates
SRP1 (10)	-1.70	0.14	0.55	0.76	6%	9%	9%	12%
SRP2 (16)	-11.51	-1.02	0.18	0.44	7%	10%	9%	18%
SRP3 (16)	-0.01	0.65	0.79	0.94	2%	3%	3%	4%
All (42)	-11.51	-0.11	0.42	0.94	2%	7%	8%	18%

[1] Correspondence to: Antonio Viana da Silva Filho E-mail: viana.filho.ufrpe@hotmail.com

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Trade-off between number of constraints and primary-statement robustness in entropy models: the case of the open-channel velocity field

Abstract