Daily streamflow forecasting using a wavelet transform and artificial neural network hybrid models

Figures & data

Fig. 1 Examples of (a) simple filtering process and (b) filtering process with downsampling decomposition.

Fig. 2 Decomposition tree at three levels.

Fig. 3 Daubechies wavelets (from order 1 to 10).

Fig. 4 MLP network with one intermediate (hidden) layer.

Fig. 5 Location of Sobradinho Dam and reservoir within the São Francisco River basin.

Table 1 Descriptive statistics for the observed daily streamflow into Sobradinho Dam (1931–1982).

Statistic	Value	Units
Arithmetic mean	2 811	m^3/s
Mode	1 381	m^3/s
Median	1 992	m^3/s
Harmonic mean	1 802	m^3/s
Geometric mean	2 217	m^3/s
Range	17 923	m^3/s
First quartile	1 267	m^3/s
Third quartile	3 960	m^3/s
Interquartile range	2 693	m^3/s
Median absolute deviation	1 643	m^3/s
Variance	4 374 353	(m^3/s)^2
Standard deviation	2 091	m^3/s
Coefficient of variation	0.74	—
Skewness	1.71	—
Kurtosis	4.61	—
Excess kurtosis	1.61	—

Fig. 6 General structure of the proposed wavelet–ANN model.

Fig. 7 Reconstructed approximations (A) and details (D) for 10 levels of decomposition.

Table 2 The four best tested WA hybrid models and combinations of approximations used.

Model	Approximations
WA1	A = A1
WA3	A = A2
WA5	A = A3 + A4
WA7	A = A3 + A4 + A5

Table 3 Assessed models (n, m, k).

Model	Days ahead	Architecture (n, m, k)	Input variables	Inputs	Output	Training			Testing
						RMSE	MARE	CE	RMSE	MARE	CE
ANN1	1	(7, 8, 1)	Q	Qt, Qt–1,…, Qt–6	Qt+1	47.1811	0.0111	0.9994	33.8431	0.0059	0.9997
ANN3	3	(7, 8, 1)	Q	Qt, Qt–1,…, Qt–6	Qt+3	166.7856	0.0385	0.9935	135.3477	0.0267	0.9957
ANN5	5	(6, 10, 1)	Q	Qt, Qt–1,…, Qt–5	Qt+5	309.1333	0.0713	0.9779	276.6977	0.0546	0.9820
ANN7	7	(7, 10, 1)	Q	Qt, Qt–1,…, Qt–6	Qt+7	457.1007	0.1097	0.9518	437.3646	0.0882	0.9537
WA1	1	(7, 8, 1)	A = A1	At, At–1,…, At–6	Qt+1	24.9232	0.0060	0.9998	17.9712	0.0035	0.9999
WA3	3	(7, 12, 1)	A = A3	At, At–1,…, At–6	Qt+3	111.9636	0.0257	0.9971	83.8876	0.0165	0.9984
WA5	5	(6, 15, 1)	A = A3 + A4	At, At–1,…, At–5	Qt+5	139.6978	0.0327	0.9955	113.2199	0.0211	0.9971
WA7	7	(7, 15, 1)	A = A3 + A4 + A5	At, At–1,…, At–6	Qt+7	178.7082	0.0428	0.9926	152.5620	0.0302	0.9947

Fig. 8 Residual errors for 1-, 3-, 5- and 7-day ahead forecasts with ANN and WA hybrid models.

Fig. 9 Forecasting 1 day ahead with the ANN₁ model and WA₁ hybrid model compared with observed streamflow (Q_o).

Fig. 10 Forecasting 3 days ahead with the ANN₃ model and WA₃ hybrid model compared with observed streamflow (Q_o).

Fig. 11 Forecasting 5 days ahead with the ANN₅ model and WA₅ hybrid model compared with the observed streamflow (Q_o).

Fig. 12 Forecasting 7 days ahead with the ANN₇ model and WA₇ hybrid model compared with the observed streamflow (Q_o).