Principle of maximum entropy in the estimation of suspended sediment concentration

Martins, Patrícia Diniz; Poleto, Cristiano

doi:10.1590/2318-0331.011716058

Patrícia Diniz Martins

Universidade Federal do Triângulo Mineiro, Uberaba, MG, Brazil

Cristiano Poleto

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil

E-mails: patricia_ambiental@hotmail.com (PDM), cristiano.poleto@ufrgs.br (CP)

Patrícia Diniz Martins: Confection and analysis of the article.

Cristiano Poleto: Orientation and review of the article.

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Figure 1
Uniform flow of sediments.

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Figure 2
Profiles of sediment concentration measured by Coleman (1981)COLEMAN, N. L. Velocity profiles with suspended on the open-channel distribution. Water Resources Research, v. 22, n. 10, p. 1377-1384, 1981. http://dx.doi.org/10.1029/WR022i010p01377.
http://dx.doi.org/10.1029/WR022i010p0137... were identified by points and calculated in this work were identified by the solid lines.

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Figure 3
Sediment concentration profiles measured by Einstein and Chien (1955)EINSTEIN, H. A.; CHIEN, N. Effects of heavy sediment concentration near the bed on velocity na sediment distribution. Berkeley: University of California, 1955. (M. R. D. Sediment Series, 8). were identified by points and calculated in this work were identified by the continuous lines.

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Figure 4
Profile of sediment concentration measured by Einstein and Chien (1955)EINSTEIN, H. A.; CHIEN, N. Effects of heavy sediment concentration near the bed on velocity na sediment distribution. Berkeley: University of California, 1955. (M. R. D. Sediment Series, 8). were identified by points, calculated by Cui (2011)CUI, H. Estimation of velocity distribution and suspended sediment discharge in open channels using entropy. 2011. 212 f. Thesis (Doctoral Dissertation) - Office of Graduate Studies, Texas A&M University, Texas, 2011. were dashed and in this work by continuous lines.

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Table 1
Conditions of the Coleman (1981)COLEMAN, N. L. Velocity profiles with suspended on the open-channel distribution. Water Resources Research, v. 22, n. 10, p. 1377-1384, 1981. http://dx.doi.org/10.1029/WR022i010p01377.
http://dx.doi.org/10.1029/WR022i010p0137... experiment.

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Table 2
Conditions of the Einstein and Chien (1955) EINSTEIN, H. A.; CHIEN, N. Effects of heavy sediment concentration near the bed on velocity na sediment distribution. Berkeley: University of California, 1955. (M. R. D. Sediment Series, 8).experiment.

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Table 3
Summary of the results of this work using data from Coleman (1981)COLEMAN, N. L. Velocity profiles with suspended on the open-channel distribution. Water Resources Research, v. 22, n. 10, p. 1377-1384, 1981. http://dx.doi.org/10.1029/WR022i010p01377.
http://dx.doi.org/10.1029/WR022i010p0137... .

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Table 4
Summary of the results of this work using data from Einstein and Chien (1955)EINSTEIN, H. A.; CHIEN, N. Effects of heavy sediment concentration near the bed on velocity na sediment distribution. Berkeley: University of California, 1955. (M. R. D. Sediment Series, 8)..

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Table 5
Comparison of the estimate of this work with other methods.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17a)

(17b)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

Profile	D	Q	U*
Profile	mm	m3/s	m/s
Coleman1	0.105	0.064	0.041
Coleman2	0.105	0.064	0.041
Coleman3	0.105	0.064	0.041
Coleman4	0.105	0.064	0.041
Coleman5	0.105	0.064	0.041
Coleman6	0.105	0.064	0.041
Coleman7	0.105	0.064	0.041
Coleman8	0.105	0.064	0.041
Coleman9	0.105	0.064	0.041
Coleman10	0.105	0.064	0.041
Coleman11	0.105	0.064	0.041
Coleman12	0.105	0.064	0.041
Coleman13	0.105	0.064	0.041
Coleman14	0.105	0.064	0.041
Coleman15	0.105	0.064	0.041
Coleman16	0.105	0.064	0.041
Coleman17	0.105	0.064	0.041
Coleman18	0.105	0.064	0.041
Coleman19	0.105	0.064	0.041
Coleman21	0.210	0.064	0.041
Coleman22	0.210	0.064	0.041
Coleman23	0.210	0.064	0.041
Coleman24	0.210	0.064	0.041
Coleman25	0.210	0.064	0.041
Coleman26	0.210	0.064	0.041
Coleman27	0.210	0.064	0.041
Coleman28	0.210	0.064	0.041
Coleman29	0.210	0.064	0.040
Coleman30	0.210	0.064	0.041
Coleman31	0.210	0.064	0.041
Coleman32	0.420	0.064	0.041
Coleman33	0.420	0.064	0.041
Coleman34	0.420	0.064	0.041
Coleman35	0.420	0.064	0.041
Coleman36	0.420	0.064	0.041
Coleman37	0.420	0.064	0.041
Coleman38	0.420	0.064	0.043
Coleman39	0.420	0.064	0.044
Coleman40	0.420	0.064	0.045

Profile	H	U*	D₅₀
Profile	mm	m/s	mm
RunS1	138	0.115	1.3
RunS2	120	0.129	1.3
RunS3	120	0.133	1.3
RunS4	115	0.144	1.3
RunS5	109	0.144	1.3
RunS6	142	0.118	0.94
RunS7	142	0.118	0.94
RunS8	139	0.115	0.94
RunS9	135	0.118	0.94
RunS10	128	0.125	0.94
RunS11	133	0.0767	0.274
RunS12	132	0.0767	0.274
RunS13	134	0.0767	0.274
RunS14	124	0.0767	0.274
RunS15	124	0.0767	0.274
RunS16	119	0.0767	0.274

Profile	c measured (g/L)	Parameters		Calculated sediment concentration profile
Profile	c measured (g/L)	λ₁	a	c estimated (g/L)	D %		Error	NSE		R²		PBIAS			RSR				RMSE
2	3.3	0.020	1.03 a 8.48	8.9	396.6%	Sup	61.8	-4.24	I	0.99	S	-169%	I		1.54		I		5.65
3	5.7	0.050	1.02 a 4.9	6.5	126.6%	Sup	9.5	0.84	Exc	0.88	S	-14%	S		0.43		S		1.98
4	8.9	0.072	1.03 a 7.63	9.6	74.5%	Sup	8.4	0.97	Exc	0.98	S	-8%	S		0.18		S		1.48
5	12.1	0.085	1.03 a 8.05	12.2	41.7%	Sup	0.3	0.97	Exc	0.98	S	0%	S		0.16		S		1.82
6	15.0	0.100	1.03 a 9.42	14.2	39.3%	Sup	-9.6	0.35	S	0.41	S	5%	S		0.83		I		11.66
7	18.7	0.105	1.03 a 8.94	18.5	20.5%	S	-3.0	0.95	Exc	0.95	S	1%	S		0.22		S		4.04
8	22.3	0.115	1.03 a 8.48	20.8	10.4%	B	-18.1	0.96	Exc	0.96	S	7%	S		0.20		S		4.50
9	25.7	0.122	1.03 a 8.71	24.0	9.8%	MB	-20.7	0.96	Exc	0.97	S	7%	S		0.19		S		5.04
10	29.7	0.130	1.03 a 8.48	26.3	4.7%	MB	-41.7	0.95	Exc	0.96	S	12%	S		0.23		S		7.25
11	32.8	0.138	1.03 a 8.48	29.6	8.3%	MB	-38.9	0.95	Exc	0.97	S	10%	S		0.21		S		7.94
12	34.9	0.139	1.03 a 8.48	31.4	-0.5%	MB	-41.3	0.96	Exc	0.97	S	10%	S		0.20		S		7.70
13	38.4	0.142	1.03 a 8.48	33.2	-5.2%	MB	-62.6	0.94	Exc	0.96	S	14%	S		0.23		S		9.47
14	41.3	0.145	1.03 a 8.48	35.9	-5.4%	MB	-64.3	0.94	Exc	0.96	S	13%	S		0.23		S		10.45
15	43.9	0.150	1.03 a 8.48	38.3	-5.9%	MB	-67.2	0.95	Exc	0.96	S	13%	S		0.22		S		11.22
16	46.3	0.152	1.03 a 8.71	40.8	-4.4%	MB	-66.4	0.95	Exc	0.96	S	12%	S		0.21		S		11.50
17	46.2	0.155	1.03 a 8.48	42.2	-2.0%	MB	-49.0	0.96	Exc	0.97	S	9%	S		0.19		S		10.80
18	47.0	0.156	1.03 a 8.48	42.7	-5.7%	MB	-51.9	0.95	Exc	0.97	S	9%	S		0.20		S		11.40
19	50.3	0.158	1.03 a 8.48	45.7	-1.8%	MB	-55.3	0.96	Exc	0.97	S	9%	S		0.19		S		11.41
20	52.6	0.164	0	45.7	-5.1%	MB	-82.5	0.96	Exc	0.97	S	13%	S		0.19		S		12.87
22	2.8	0.040	1 a 3.32	7.1	416.2%	Sup	47	-4.21	I	0.97	S	-141%		I		1.11		I		4.39
23	5.5	0.058	1 a 3.32	8.1	157.8%	Sup	31.0	0.76	MB	0.96	S	-47%	S		0.48		S		2.83
24	8.5	0.080	1 a 3.04	9.1	78.4%	Sup	7.3	0.97	Exc	0.98	S	-7%	S		0.18		S		1.63
25	11.5	0.100	1 a 3.32	13.2	57.9%	Sup	20.3	0.94	Exc	0.96	S	-15%	S		0.24		S		3.37
26	14.2	0.100	1 a 3.32	13.9	29.2%	Sup	-3.1	0.97	Exc	0.97	S	2%	S		0.18		S		2.78
27	16.9	0.114	1 a 3.32	16.7	26.6%	Sup	-3.0	0.96	Exc	0.97	S	1%	S		0.18		S		3.60
28	19.9	0.122	1 a 3.32	19.6	18.4%	S	-2.7	0.97	Exc	0.97	S	1%	S		0.17		S		3.96
29	22.6	0.130	1 a 3.32	21.9	14.4%	B	-8	0.95	Exc	0.96	S	3%	S		0.20		S		5.74
30	25.2	0.136	1 a 3.32	24.5	10.6%	B	-8.6	0.95	Exc	0.96	S	3%	S		0.21		S		6.86
31	26.9	0.135	1 a 3.52	27.7	14.1%	MB	9.2	0.91	Exc	0.92	S	-3%	S		0.29		S		10.10
33	0.7	0.030	1.02 a 1.59	5.8	1658.7%	Sup	62.4	-57.67	I	0.90	S	-799%	I		1.71		I		5.47
34	1.1	0.030	1.02 a 1.68	5.9	1085.9%	Sup	57.4	-12.19	I	0.92	S	-421%	I		1.56		I		4.92
35	1.9	0.030	1.02 a 1.68	6.0	689.5%	Sup	49.4	-1.83	I	0.95	S	-218%	I		1.28		I		4.16
36	3.2	0.050	1.02 a 1.68	5.3	328.9%	Sup	24.9	0.72	MB	0.95	S	-65%	I		0.55		S		2.41
37	3.8	0.056	1.02 a 1.68	5.4	214.1%	Sup	19.6	0.83	Exc	0.94	S	-43%	S		0.42		S		2.09
38	4.7	0.062	1.02 a 1.68	5.6	150.4%	Sup	10.9	0.86	Exc	0.89	S	-19%	S		0.38		S		2.22
39	5.3	0.072	1.02 a 1.68	6.1	116.7%	Sup	9.3	0.92	Exc	0.93	S	-15%	S		0.29		S		2.04
40	5.2	0.070	1.02 a 1.68	6.0	110.1%	Sup	9.6	0.91	Exc	0.92	S	-15%	S		0.30		S		2.053

Profile		Parameters		Sediment concentration profile calculated
Profile	c measured (g/L)	λ₁	a	c estimated (g/L)	D %		Error	NSE		R²		PBIAS		RSR		RMSE
RunS1	34.83	0.29	1.22 a 2.93	30.49	6.4%	MB	-2.92	1.00	Exc	0.983	S	2%	S	0.027	S	0.60
RunS2	59.92778	0.32	1.36 a 3.6	56.27	19.6%	S	19.65	1.00	Exc	0.988	S	-3%	S	0.034	S	1.40
RunS3	77.91111	0.19	1.3 a 2.93	71.86	13.7%	B	12.39	1.00	Exc	0.976	S	-1%	S	0.021	S	1.11
RunS4	106.9867	0.09	1.35 a 2.93	95.77	0.6%	MB	-13.91	1.00	Exc	0.965	S	2%	S	0.016	S	1.18
RunS5	172.2778	0.03	1.03 a 1.59	162.02	12.8%	B	55.53	1.00	Exc	0.998	S	-3%	S	0.022	S	2.36
RunS6_1	11.00333	12	1.48 a 2.37	10.43	15.4%	S	6.54	0.95	MB	0.999	S	-8%	S	0.285	S	1.28
RunS6_2	17.9625	0.2	0.91 a 0.66	16.89	15.3%	S	11.23	0.98	Exc	0.941	S	4%	S	0.170	S	1.50
RunS7	34.49	0.47	1.33 a 2.86	30.17	22.6%	S	2.38	1.00	Exc	0.993	S	0%	S	0.028	S	0.58
RunS8	42.4775	0.45	1.22 a 2.93	37.46	23.8%	S	-3.53	1.00	Exc	0.981	S	2%	S	0.020	S	0.63
RunS9	75.22143	0.11	1.17 a 1.88	67.53	15.4%	S	10.23	1.00	Exc	0.996	S	-1%	S	0.022	S	1.13
RunS10	106.4688	0.09	1.2 a 2.13	98.89	18.7%	S	32.60	1.00	Exc	0.988	S	-3%	S	0.025	S	1.90
RunS11	21.09429	13	1.27 a 2.6	21.25	-1.1%	MB	2.29	1.00	Exc	0.977	S	1%	S	0.061	S	0.53
RunS12	85.95125	1.09	1.17 a 3.91	81.22	21.6%	S	35.62	1.00	Exc	0.985	S	-4%	S	0.029	S	1.99
RunS13	151.4222	0.54	1.19 a 5.55	133.74	7.2%	MB	-29.96	1.00	Exc	0.973	S	3%	S	0.009	S	1.17
RunS14	157.72	0.59	1.19 a 6.27	141.84	22.9%	S	-52.48	1.00	Exc	0.969	S	43%	S	0.015	S	2.01
RunS15	269.6642	0.352	1.2 a 6.39	243.83	23.8%	S	-72.02	1.00	Exc	0.947	S	16%	S	0.011	S	2.35
RunS16	286.1833	0.45	1.18 a 5.44	267.27	18.3%	S	23.86	1.00	Exc	0.962	S	26%	S	0.006	S	1.35

Profile	Tsallis			Rouse Equation			In this work
Profile	Square Error	NSE	R²	Square Error	NSE	R²	Square Error	NSE	R²
RunS1	84.17	0.98	0.98				65.5	0.98	0.98
RunS11	21.4	0.97	0.98	61.33	0.91	0.97	16.43	0.97	0.98
RunS13	47447	0.59	0.79	22910	0.85	0.94	4269	0.97	0.97

H (c) = \frac{1}{m - 1} {1 - \int_{C h}^{C m} {[f (c)]}^{m} d c} = \frac{1}{m - 1} \int_{C h}^{C m} f (c) {1 - - {[f (c)]}^{m - 1}} d c

L = \int_{c_{h}}^{c_{m}} \frac{f (c)}{m - 1} {1 - {[f (c)]}^{m - 1}} d c - λ_{0} [\int_{c_{h}}^{c_{m}} f (c) d c - 1] - - λ_{1} [\int_{c_{h}}^{c_{m}} c f (c) d c - c_{D}]

\frac{1}{λ_{1}} {(\frac{m - 1}{m})}^{\frac{m}{m - 1}} [{(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{m})}^{\frac{m}{m - 1}} - {(\frac{1}{m - 1} - λ_{0} - - λ_{1} c_{h})}^{\frac{m}{m - 1}}] = 1

\begin{array}{l} - λ_{1} c_{D} {(\frac{m}{m - 1})}^{\frac{m}{m - 1}} = c_{m} {(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{m})}^{\frac{m}{m - 1}} - \\ - c_{h} {(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{h})}^{\frac{m}{m - 1}} + \frac{m - 1}{2 m - 1} \frac{1}{λ_{1}} \\ [{(\frac{1}{m - 1} - λ_{0} - - λ_{1} c_{m})}^{\frac{2 m - 1}{m - 1}} - {(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{h})}^{\frac{2 m - 1}{m - 1}}] \end{array}

F (c) = {(\frac{m - 1}{m})}^{\frac{m}{m - 1}} \frac{1}{λ_{1}} [{(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{h})}^{\frac{m}{m - 1}} - {(\frac{1}{m - 1} - λ_{0} - - λ_{1} c)}^{\frac{m}{m - 1}}]

F (c) = {(\frac{m - 1}{m})}^{\frac{m}{m - 1}} \frac{1}{λ_{1}} [{(\frac{1}{m - 1} - λ_{0})}^{\frac{m}{m - 1}} - {(\frac{1}{m - 1} - λ_{0} - - λ_{1} c)}^{\frac{m}{m - 1}}]

H (c) = \frac{1}{m - 1} {\begin{cases} c_{m} - c_{h} + {(\frac{m - 1}{m})}^{\frac{m}{m - 1}} + + \frac{1}{(2 m - 1) λ_{1}} \\ [{(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{m})}^{\frac{2 m - 1}{m - 1}} - {(\frac{1}{m - 1} - λ_{0} - - λ_{1} c_{h})}^{\frac{2 m - 1}{m - 1}}] \end{cases}}

F (c) = {(\frac{m - 1}{m})}^{\frac{m}{m - 1}} \frac{1}{λ_{1}} [{(\frac{1}{m - 1} - λ_{0} - λ_{1} c_{h})}^{\frac{m}{m - 1}} - {(\frac{1}{m - 1} - λ_{0} - - λ_{1} c)}^{\frac{m}{m - 1}}] = 1 - \frac{y}{h_{0}}

c = \frac{λ_{*}}{λ_{1}} - \frac{1}{λ_{1}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} (1 - \frac{y}{h_{0}}) + {[\frac{m - 1}{m} (λ_{*} - - λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}

c = \frac{λ_{*}}{λ_{1}} - \frac{1}{λ_{1}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} (1 - \frac{y}{h_{0}}) + {(\frac{m - 1}{m} λ_{*})}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}

\frac{c}{c_{m}} = \frac{λ_{*}}{λ_{1} c_{m}} - \frac{1}{λ_{1} c_{m}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} (1 - \frac{y}{h_{0}}) + {(\frac{m - 1}{m} λ_{*})}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}

\frac{c}{c_{m}} = 1 - \frac{1}{μ} {1 - {[{(\frac{m}{m - 1})}^{\frac{m}{m - 1}} \frac{μ}{c_{m}} (- \frac{y}{h_{0}}) + 1]}^{\frac{m - 1}{m}}}

\frac{c}{c_{m}} = 1 - \frac{1}{μ} (1 - {{(1 - μ)}^{\frac{m}{m - 1}} + [1 - {(1 - μ)}^{\frac{m}{m - 1}}] (1 - - \frac{y}{h_{0}})}^{\frac{m - 1}{m}})

c = \frac{λ_{*}}{λ_{1}} + \frac{1}{λ_{1}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} {(1 - \frac{y}{h_{0}})}^{a} + {[\frac{m - 1}{m} (λ_{*} + + λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}

c = \frac{λ_{*}}{λ_{1}} + \frac{1}{λ_{1}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} {(1 - \frac{y}{h_{0}})}^{a} + {[\frac{m - 1}{m} (λ_{*} + + λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}

c_{m} = \frac{λ_{*}}{λ_{1}} + \frac{1}{λ_{1}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} {(1)}^{a} + {[\frac{m - 1}{m} (λ_{*} + λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}

\frac{1}{c_{m}} = \frac{1}{c} (\frac{λ_{*}}{λ_{1} c_{m}} - \frac{1}{λ_{1} c_{h}} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} {(1 - \frac{y}{h_{0}})}^{a} + {[\frac{m - 1}{m} (λ_{*} - - λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}})

\frac{1}{c_{m}} = \frac{λ_{1}}{1 + λ_{1}} {- λ_{1} \frac{m - 1}{m} {(1)}^{a} + {[\frac{m - 1}{m} (λ_{*} - λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{1 - m}{m}}

\begin{array}{l} \frac{1}{c} (\frac{λ_{*}}{λ_{1} C m} - \frac{1}{λ_{1} C m} \frac{m}{m - 1} {- λ_{1} \frac{m - 1}{m} {(1 - \frac{y}{h_{0}})}^{a} + {[\frac{m - 1}{m} (λ_{*} - - λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{m - 1}{m}}) = \\ \frac{λ_{1} (m - 1)}{λ_{1} (m - 1) - m} {- λ_{1} \frac{m - 1}{m} {(1)}^{a} + {[\frac{m - 1}{m} (λ_{*} - - λ_{1} c_{h})]}^{\frac{m}{m - 1}}}^{\frac{1 - m}{m}} \end{array}

N S E = 1 - \frac{\sum_{i = 1}^{N} {(c_{o b s i} - c_{c a l c i})}^{2}}{\sum_{i = 1}^{N} {(c_{o b s i} - \bar{c_{o b s}})}^{2}}

R^{2} = \frac{\sum_{i = 1}^{N} (c_{o b s i} - \bar{c_{o b s}}) * (c_{c a l c i} - \bar{c_{c a l c}})}{{[\sum_{i = 1}^{N} {(c_{o b s i} - \bar{c_{o b s}})}^{2}]}^{0.5} * {[\sum_{i = 1}^{N} {(c_{c a l c i} - \bar{c_{c a l c}})}^{2}]}^{0.5}}

R S R = \frac{R M S E}{S T D E V_{o b s}} = \frac{\sqrt{\sum_{i = 1}^{N} {(c_{c a l c i} - c_{o b s i})}^{2}}}{\sqrt{\sum_{i = 1}^{N} {(c_{o b s i} - c_{m é d i o})}^{2}}}

[1] E-mails: patricia_ambiental@hotmail.com (PDM), cristiano.poleto@ufrgs.br (CP)

Brasil

Brasil

Principle of maximum entropy in the estimation of suspended sediment concentration

Princípio da entropia máxima na estimativa da concentração de sedimentos em suspensão

ABSTRACT