Discharge estimation for medium-sized river using multi-temporal remote sensing data: a case study in Brazil

Figures & data

Table 1. Hydrological variables inferred from satellite images planimetric properties. ACC: accuracy; RMSE: root mean square error; RRMSE: relative root mean square error (divided by mean of observations); r2: coefficient of determination; $y_i$: predicted value, $x_i$: observed value

Estimated variable	Mission	Results	References
Water surface	SPOT Vegetation	ACC = 95.4%	Haas et al. (2009)
	Landsat	ACC = 97%	Mueller et al. (2016)
	Landsat	ACC = 99.2%	Pekel et al. (2016)
Discharge	ERS 1	r2 = 0.83	Smith et al. (1996)
	Quickbird-2	ACC = 97%	Xu et al. (2004)
	Quickbird-2	ACC = 80%	Zhang et al. (2004)
	MODIS	r2 = 0.81	Smith and Pavelsky (2008)
	Landsat	RRMSE 26–41%	Gleason et al. (2014; 2014)
	MODIS	RMSE = 10%	Elmi et al. (2015)
Propagation speed	MODIS	$y_i$ = 1.01 m/s $x_i$ = 0.97 m/s	Smith and Pavelsky (2008)
	MODIS	-	Sichangi et al. (2018)

Figure 1. Location of the São Francisco River reach used as test area showing the in situ hydrological stations (left) and geographical context in the South American continent (right)

Table 2. Station location

Station location	ANA code	Code	S Latitude/W Longitude	Mean discharge (m^3/s)	Drainage area (km^2)	Altitude (m a.s.l.)
Pirapora	41135000	${\mathrm{S}}_{Q1}$	17°22ʹ9.12ʹ’/44°56ʹ35.2ʹ’	841	62200	480
Cachoeira da Manteiga	42210000	${\mathrm{S}}_{Q2}$	16°39ʹ 25.92ʹ’/45°4ʹ50.9ʹ’	1169	107000	467
São Romão	43200000	${\mathrm{S}}_{Q3}$	16°22ʹ21.00ʹ’/45°4ʹ12.0ʹ’	1569	154000	464
São Francisco	44200000	${\mathrm{S}}_{Q4}$	15°56ʹ57.84ʹ’/44°52ʹ4.1ʹ’	1808	184000	457
Pedras de Maria da Cruz	44290002	${\mathrm{S}}_{Q5}$	15°36ʹ3.96ʹ’/44°23ʹ48.1ʹ’	2017	194000	451
Manga	44500000	${\mathrm{S}}_{Q6}$	14°45ʹ25.92ʹ’/43°55ʹ55.9ʹ’	1953	202000	440
Carinhanha	45298000	${\mathrm{S}}_{Q7}$	14°18ʹ15.84ʹ’/43°45ʹ47.9ʹ’	2192	254000	427

Figure 2. Steps for measuring width. Background: S1 Image; VV; acquisition 08/03/2016

Figure 3. Illustration of the extraction of the river width from a classified image

Figure 4. K-fold cross-validation example for k = 5. Light grey: training data, Dark grey: test data. The method first divides the data set in k subsets, then uses k – 1 subsets for training while keeping the remaining one for validation. k iterations are performed, each one testing a model build from a different fold generating a validation estimate. The mean absolute error (MAE) was used here

Figure 5. The three different methods of virtual (width) stations definitions: (a) individual width ($w$) at-a-station, (b) systematic average width stations ($w_n$) calculated for the whole river reach every 10 km and (c) selected average width stations ($w_s$) defined by statistically selecting river sections with the most width variations

Figure 6. Illustration of the set of 952 river cross-sections widths organized by quartiles (25% = 55.5 m, 50% = 90.62 m, 75% = 228.86 m)

Table 3. Transferred discharge (QWQ) of station SQ3 to a virtual station 64 km downstream that coincides with station SQ4. The PFE gives an idea of the errors generated by the approach

Dates 1–2	Station SQ3 (m^3/s)	Station SQ4 (m^3/s)	QWQ (m^3/s)	PFE QWQ – SQ4 (%)
01–02/03/2016	596.9	658.0	713.2	8.39
08–09/03/2016	909.4	1061.7	1086.5	2.34
03–04/12/2016	685.2	832.6	818.7	–1.67
15–16/12/2016	2169.6	2407.2	2592.3	7.69
13–14/02/2017	1297.0	1661.5	1549.7	–6.73
09–10/03/2017	549.9	630.7	657.1	4.18
13–14/11/2017	559.2	815.0	668.1	–18.02
10–11/12/2017	1751.3	2059.0	2092.5	1.63
28–29/12/2017	426.6	547.5	509.7	–6.90
15–16/01/2018	545.3	678.7	651.5	–4.01
20–21/02/2018	491.2	698.3	586.8	–15.97
16–17/03/2018	1715.0	1934.8	2049.1	5.91
21–22/04/2018	439.2	570.2	524.8	–7.96
07–08/05/2018	294.0	335.1	351.2	4.81
17–18/05/2018	312.3	333.2	373.1	11.96
06–07/06/2018	357.9	383.1	427.6	11.63
21–22/06/2018	338.6	348.3	404.5	16.15
16–17/07/2018	357.9	363.6	427.6	17.60
14–15/08/2018	385.6	387.0	460.7	19.05

Figure 7. Transferred discharge data (SQ QWQ) of station S_Q3 to a point 64 km downstream that coincides with station S_Q4.

Figure 8. Model performance evaluation for virtual stations generated from the three methods

Figure 9. Evaluation metrics used to compare the results from the three discharge estimation methods: (a) individual width, $w$; (b) systematic average, $w_n$ and (c) selected average, $w_s$

Figure 10. The best curves for the $w-Q$ relationship by method and the respective analysis sections in the river

Table 4. Best fit power functions achieved by each method

Method	Equation	NSE	NRMSE	Pbias
$w$	$Q={\left(\frac{W}{34.70}\right)}^{\frac{1}{0.39}}$	0.92	27.42	5.1
$w_n$	$Q={\left(\frac{We}{59.06}\right)}^{\frac{1}{0.29}}$	0.89	31.5	6.0
$w_s$	$Q={\left(\frac{We}{100.41}\right)}^{\frac{1}{0.24}}$	0.87	34.82	2.8

Table 5. Models $w_s$ method. Bold values indicate best model results

Virtual station	a	b	NSE	NRMSE	Pbias
$w$s1	4.968	0.662	0.760	47.2	2.8
$w$s2	8.358	0.635	0.686	54.0	3,9
$w$s3	70.446	0.296	0.856	36.6	–3.1
$w$s4	100.417	0.243	0.869	34.8	–3.0
$w$s5	56.224	0.313	0.835	39.1	9.5
$w$s6	3.839	0.709	0.706	52.2	5.9
$w$s7	87.266	0.237	0.746	48.6	–1.6
$w$s8	129.009	0.236	0.795	43.6	2.7
$w$s9	80.714	0.297	0.605	60.2	–9.3
$w$s10	10.870	0.582	0.849	37.2	–4.8
$w$s11	0.284	1.083	–0.044	94.6	17.6
$w$s12	263.828	0.109	–0.222	102.3	–19.0
$w$s13	40.654	0.376	0.652	53.8	8.0

Figure 11. Location of the 13 virtual $w_s$ stations

Figure 12. Daily in situ discharge (Q_obs) data (line) compared with estimated discharge calculated from the $w_s$ method for the period March 2016–August 2018

Figure 13. Cross-sections at six locations along this river reach $w_{s4}$

Figure 14. Relationships between (a) exponent $b$ and width amplitude, and (b) NRMSE and width amplitude

Figure A1. Top Sentinel-1B SAR image of the Pampulha Reservoirs complex. Bottom: difference maps between the SAR image classification results and the precise delineation. Grey pixels represent false negative errors (omitted pixels) and black pixels represent false positive errors (falsely included pixels). Reproduced with permission from the authors

Figure A2. Left: PlanetScope false colour image. Right: water surfaces extracted from Lansat-8, Sentinel-1 and Sentinel-2 images. The background cyan colour shows the reference surface from the Planet Scope image used for determination of accuracy. Reproduced with authorization from the authors

Table A1. List of images acquired with some of their specification and the associated water stage level

Sensor	Image date	Pixel size (m)	Stage (cm)*	Product level
Sentinel-1 (S-1)	15/01/18	10	223.5	Level-1GRD
Sentinel-2 (S-2)	17/01/18	10	217	1 C
PlanetScope	17/01/18	3	217	3A
Landsat/OLI (L-8)	18/01/18	30	213	L1

Table A2. Water surface classification accuracy by sensor, spectral bands and polarizations

Rank	Bands L8	Accuracy L8	Bands S-2	Accuracy S-2	Pol band S-1	Accuracy S-1
1	6	86.43	8,12	94.70	VV	93.79
2	3,6	86.22	3,8	94.48	VH	84.97
3	3,7	86.09	8A,11	94.41	VV+VH	82.00
4	6,7	85.92	8,11	94.32
5	3,6,7	85.88	6,8A,11	94.28
6	3,4,6	85.73	8,11,12	94.25
7	7	85.59	11,12	94.24
8	3,4,6,7	85.53	8A,11,12	94.21
9	3,4,7	85.46	6,8A,11,12	94.16
10	4,6,7	85.28	6,11,12	94.13

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