Identifying potential systemic changes in a regulated river: a diagnostic methodology based on whiteness property of innovation sequences

Figures & data

Figure 1. Schematic representation of systematic change detection in a routing model structure.

Table 1. True parameter formulation for both pre- and post-regulation periods.

Period	LMM		NMM
	$X^L$	$K^L$	$X^N$	$K^N$	$m$
Pre-change	$\left\{\begin{array}{c}-0.05sin\left(\frac{\pi t^{†}}{125}\right)+0.35forquickflo{w}^{‡}\\ 0.35forbaseflow\end{array}\right.$	$\left\{\begin{array}{c}-0.1sin\left(\frac{\pi t}{125}\right)+1.1forquickflow\\ 1.1forbaseflow\end{array}\right.$	0.2	1.1	1.3
Post-change	$\left\{\begin{array}{c}-0.05sin\left(\frac{\pi t}{125}\right)+0.3forquickflow\\ 0.3forbaseflow\end{array}\right.$	$\left\{\begin{array}{c}-0.5sin\left(\frac{\pi t}{125}\right)+1.6forquickflow\\ 1.6forbaseflow\end{array}\right.$	0.2	1.1	1.3

LMM: linear Muskingum model. NMM: non-linear Muskingum model. “t” is the time counter when the value of quick flow is non-zero. It ranges from 1 to 120. Quick flow is the portion of a streamflow hydrograph that occurs from overland flow.

Table 2. Detail of scaling factor implementation to capture the future structural changes before and after the river regulation for four plausible scenarios – coefficients of linear (${\varphi }_t^L$) and non-linear (${\varphi }_t^N$) models.

	Transformation coefficients	Graphical description
Scenario 1^†	${\varphi }_t^L=1,fort=1:T$
	${\varphi }_t^N=0,fort=1:T$
Scenario 2	${\varphi }_t^L=\left\{\begin{array}{c}1,fort=1:t_1\\ 1-0.75sin\left(\frac{\pi x}{t_2-t_1}\right),for{t}_1<t<t_{2}andx=1:t_2-t_1\\ 1,fort=t_2:T\end{array}\right.$
	${\varphi }_t^N=\left\{\begin{array}{c}0,fort=1:t_1\\ 0.75sin\left(\frac{\pi x}{t_2-t_1}\right),for{t}_1<t<t_{2}andx=1:t_2-t_1\\ 0,fort=t_2:T\end{array}\right.$
Scenario 3	${\varphi }_t^L=\left\{\begin{array}{c}1,fort=1:t_1\\ 1-tanh\left(\frac{x}{3000}\right),fort>t_{1}andx=1:T-t_1\end{array}\right.$
	${\varphi }_t^N=\left\{\begin{array}{c}0,fort=1:t_1\\ tanh\left(\frac{x}{3000}\right),fort>t_{1}andx=1:T-t_1\end{array}\right.$
Scenario 4	${\varphi }_t^L=\left\{\begin{array}{c}1,fort=1:t_1\\ 1-tanh\left(\frac{x}{6600}\right),for{>}_{1}andx=1:T-t_1\end{array}\right.$ ${\varphi }_t^N=\left\{\begin{array}{c}0,fort=1:t_1\\ tanh\left(\frac{x}{6600}\right),fort>t_{1}andx=1:T-t_1\end{array}\right.$

^†See Table 3 for more details on the background of the developed scenarios.

Table 3. More detailed description of concepts used in formulating four plausible scenarios.

Scenario	Underlying concept
#1	A relatively stable landscape to resist controlling factors and forcing processes.
#2	A typical landscape to recover rapidly following disturbance events.
#3	A typical landscape to respond rapidly at the early stage of recovery pathway and at a steadily declining rate thereafter.
#4	A typical landscape to respond more gradually compared to the recovery pathway of scenario #3.

Figure 2. Normalized innovation sequences for: (a) scenario #1, (b) scenario #2, (c) scenario #3, and (d) scenario #4.

Table 4. Root mean square error under the four devised scenarios.

				Streamflow RMSE^† $\left(\frac{m^3}{s}\right)$
		Pre-regulation (Year–Year)		Post-regulation (Year–Year)							Whole dataset
Scenario	Simulation model	1928 – 1932	1932 – 1937	1937 – 1942	1942 – 1947	1947 – 1952	1952 – 1957	1957 – 1962	1962 – 1967	1967 – 1972	1928 – 1972
1	LMM^⁎	15.83	18.04	22.24	29.96	33.41	37.38	27.02	31.44	23.82	26.89
	LMM+Dual KF^⁑	1.64	1.96	2.86	3.57	4.58	5.31	4.52	5.13	3.81	3.85
	RMSE Ratio^⁂	9.62	9.19	7.77	8.38	7.29	7.04	5.97	6.12	6.25	6.99
2	LMM	16.90	15.69	22.49	46.29	34.45	38.12	28.17	31.46	23.49	29.34
	LMM+Dual KF	1.77	1.81	2.93	3.88	4.75	5.46	4.66	5.18	3.85	3.94
	RMSE Ratio	9.56	8.67	7.68	11.92	7.26	6.98	6.05	6.08	6.10	7.44
3	LMM	16.74	16.66	22.08	54.08	91.64	96.63	99.10	105.50	81.22	75.36
	LMM+Dual KF	1.75	1.81	2.76	3.89	5.74	6.50	6.41	6.58	5.14	4.89
	RMSE Ratio	9.57	9.20	8.01	13.89	15.96	14.86	15.46	16.05	15.79	15.42
4	LMM	15.88	17.29	21.77	28.91	49.19	61.95	75.44	87.72	72.94	57.82
	LMM+Dual KF	1.68	1.82	2.75	3.27	4.75	5.68	5.62	6.15	4.87	4.41
	RMSE Ratio	9.47	9.50	7.91	8.84	10.36	10.91	13.42	14.27	14.98	13.12

^†RMSE = $\sqrt{\frac{1}{L}\sum _{t=1}^L{\left[Q_{sim}\left(t\right)-Q_{obs}\left(t\right)\right]}^2}$.

^⁎LMM: Simulated streamflow without kernel updating.

^⁑LMM+Dual KF: Simulated streamflow obtained by dual state–parameter updating.

^⁂RMSE ratio: $\frac{RMS{E}_{LMM}}{RMS{E}_{DualKF}}$.

Figure 3. Visual representation of model performance for the two routing models, the LMM coupled with dual state–parameter updating kernel and LMM without coupling any kernel updating, under scenario #1 (left column) and Scenario #3 (right column) over time.

Figure 4. $J_K$ versus $K$ under the four devised scenarios and for the two change criteria: (1) change in the mean (left column) and (2) change in the variance (right column). The points on the convex hull (blue line) are shown with blue squares and the others are shown with red triangles.

Table 5. Variations of normalized contrast value and the second-order derivative against the number of segments for both the mean and variance measures.

$K$	Scenario #1				Scenario #2				Scenario #3				Scenario #4
	Changing in				Changing in				Changing in				Changing in
	Mean		Variance		Mean		Variance		Mean		Variance		Mean		Variance
	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$	${\tilde{J}}_K$	$D_K$
1	1	DNE^†	51	DNE	10	DNE	51	DNE	10	DNE	51	DNE	10	DNE	51	DNE
2	1	0	44.72	0.04	6.73	0	43.93	0.64	7.4	0.74	37.08	11.17	7.41	0.81	29.02	19.58
3	1	0	38.48	1.13	3.47	1.93	37.5	1.27	5.54	0.14	34.34	0.77	5.64	0	26.62	0.82
4	1	0	33.37	3.3	2.13	0.78	32.33	2.38	3.81	0	32.36	0.1	3.86	0.11	25.03	0.34
5	1	0	31.56	0.06	1.56	0	29.56	0.4	2.09	1.08	30.49	0.33	2.19	0.91	23.79	0
6	1	0	29.8	0.13	1	0.56	27.17	1.1	1.44	0.22	28.95	0	1.43	0.54	22.55	0
7	1	0	28.18	0.47	1	0	25.89	0	1.02	0.4	27.41	0	1.22	0	21.31	0.21
8	1	0	27.03	0	1	0	24.61	0.07	1	0.02	25.87	0.26	1	0.22	20.28	0
9	1	0	25.88	0.18	1	0	23.39	0	1	0	24.59	0	1	0	19.25	0.22
10	1	-	24.9	0	1	-	22.18	0.07	1	-	23.31	0.17	1	-	18.44	0

^†Does not exist.

Table 6. Number and location of detected change points based on change in the mean and variance for four devised scenarios.

Scenario	Number of change points			Location of change points (day)
	Mean		Variance	Mean		Variance
#1	0		3	-		3478
						6795
						15 994
#2	3		5	5748		2879
				6095		3134
				6485		5770
						6060
						15 994
#3	4		2	6805		3510
				8737		6790
				11 227
				13 111
#4	4		2	7190		3519
				11 227		6796
				13 843
				15 963

Figure 5. The pathway of structural evolution for the four devised scenarios along with the change points detected based on the two change criteria: (1) change in the mean (left column) and (2) change in the variance (right column).

Figure 6. Temporal variability of environmental flow components along with the locations of change points in the variance for scenario #1.