Proceedings of the 22nd IAHR-APD Congress 2020, Sapporo, JapanTHE ASSIMILATION ANALYSIS OF 2D SHALLOW WATER FLOW MODEL WITH THE
ADJOINT METHOD FOR THE PREDICTION OF DOWNSTREAM WATER LEVELS
Tetsuaki Mikami
Graduate School of Science and Engineering, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan,
e-mail : a19.trp8@g.chuo-u.ac.jp
Shoji Fukuoka
Research and Development Initiative, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan,
e-mail : sfuku@tamacc.chuo-u.ac.jp
Akihide Watanabe
TOKEN C. E. E. Consultant Co., Ltd., Kitaootsuka 1-15-6, Toshima-ku, Tokyo 170-0004, Japan,
e-mail : watanabe-ak@tokencon.co.jp

ABSTRACT
Providing accurate water level forecasts is important to ensure the timely evacuation of residents put at risk by
a levee overflow or breach. The purpose of this study is to clarify the propagation characteristics of flood
flow in Japan’s Tone River and to predict several hours ahead the water level downstream. To this end, wedeveloped a real-time assimilation model with two-dimensional planar flow analysis to determine thetemporal change in the water level observed over a long section of the river. Based on the hydraulic quantities
obtained from the assimilation analysis, the continuity equation is analyzed using the characteristic curve
method, and the propagation time of the flood flow to the downstream prediction section is produced. Results
show that the water level propagation time is 3 to 5 hours during the rising phase and 2.5 hours at peak. Water
levels are then predicted based on the lead time (i.e., propagation time). The predicted water levels are shown
to closely approximate the levels, demonstrating the utility of the method for water level prediction.
Keywords: water surface profiles, water level prediction, propagation characteristics of flood flow, assimilation analysis, adjoint
method1.

INTRODUCTION

The increasing frequency of heavy rainfall due to climate change has increased the risk of levee overflows.
Since overflows can quickly break down earthen levees, residents living near rivers need real-time
information on river flow conditions so that they know when to evacuate. At present, many rivers lack asufficient number of water level observation sites, making it difficult to accurately evaluate and predict
temporal changes in the longitudinal water surface profiles along river banks.
In this study, by using high-water stage data obtained from coarsely arranged water gauges, we effectively
estimate the temporal change in water surface profiles in Japan’s Tone River. The purpose of the study was toclarify the flood flow propagation characteristics and predict water levels in the downstream section from thecurrent time to several hours ahead.
We first developed an assimilation analysis model for the water surface profiles of a 50 km section of theTone River (long section) by applying the adjoint method to a two-dimensional shallow water flow model(e.g., Sanders et al. 2000, Yoshida 2012, Nishiguchi et al. 2017, Watanabe et al. 2017 and Watanabe et al.
2018). When applied to a September 2013 flood event, the assimilation model closely approximated thetemporal changes in water levels observed at 7 sites along the river and simultaneously identified as control
variables the flow rate, roughness coefficients, and the permeability coefficients of vegetation. The data
assimilation method used in this study has adapted the adjoint method, which has high dynamical consistency
and water balance consistency. The inverse estimation of the roughness coefficient has been studied byassimilation analysis using the adjoint method. (e.g., Ding et al. 2000, 2005 and Yoshida et al. 2011).
Next, based on the assimilation analysis results for the long section, the water level for a 17 km downstream
section of the river is predicted by applying two-dimensional flow analysis. The forecasting period for thismethod corresponds to the time of the flood flow propagating to the section targeted for prediction. Using the1cross-sectional averaged quantities of water level, river width, water surface gradient, and roughness
coefficients obtained from the assimilation analysis, characteristics of the longitudinal and temporal changes
in discharge are examined. The propagation time is then estimated by the continuity equation using thecharacteristic curve method. Our intent is to establish a way to predict water levels up to several hours ahead
based on this analysis method using the longitudinal propagation characteristics of the water level. And also,
in order to confirm the practicality of this analysis method as a real-time flood forecasting, we will show thetime required for the analysis and how often the predicted water level information is provided to residents.
2.

DESCRIPTION OF THE ADJOINT MODEL

The cost function for this adjoint method can be expressed as
J T Qbound , n, K  J H  J Q  J n _ init  J K _ initT

      x  xmH , y  y mH , t  t k 
Ta A mHTQc x, t   Qo x, t 222 Qo

Ta A mQ k
num _ annid   ninit id 2

id 1

2 n2_ init id 



2
2 Ho

k

    x  xmQ , t  t k H c x, y, t   H o x, y, t 2



(1)

dAdt

num _ akK id   K init id 2

id 1

2 K2 _ init id 



dAdt

Here, JT is the total cost to be minimized and is a function of three uncertain control variables. Variable Qboundis the discharge at the upstream end, n is Manning’s roughness coefficient for beds, and K is the permeability
constant for tree communities.
Terms JH and JQ on the right-hand side of the equation are the total misfit for the water level and the discharge,
respectively. Jn_init and JK_init are the sum of the squared differences between the values obtained from theassimilation analysis and the initially assumed (standard) values of n and K, respectively. The values for n
and K were constrained in some ranges to avoid any locally abnormal values. Among the four terms, the mostinfluential is the first regarding errors in water level. The meanings of the other parameters are as follows: Ta
and T are the initial and last time of calculation, respectively; (xmH, ymH) are the coordinates of the location of
the water gauge; tk is the observation time; xmQ is the location of the discharge observation station; H0 and Q0
are the observed water level and discharge, respectively; Hc and Qc are calculated water level and discharge,
respectively; H0 and Q0 are the standard deviation of the error of the observed water level and the discharge,
respectively; A is the area of the horizontal plane; num_an and num_ak are numbers representing theclassifications of Manning’s bed surface roughness and the permeability coefficient, respectively; ninit and Kinit
are the initial values of n and K, respectively, in the assimilation analysis; and n_init and K_init are the standard
deviations of n and K, respectively.
The continuity equation is written as Fh (U, V, h) = 0, where U and V are longitudinal and transverse velocity,respectively, and h is water depth. Then, the longitudinal and transverse momentum equations are written as
FU (U, V, h) = 0 and FV (U, V, h) = 0, respectively. Finally, the functional to be minimized in the Lagrange
multiplier method under the constraint of the shallow water equations is written as follows:
𝐽

𝐽

𝜆 ∙𝐹

𝜆 ∙𝐹

𝜆 ∙ 𝐹 𝑑𝐴𝑑𝑡

(2)

The adjoint matrix that the Lagrange multipliers satisfy was obtained by taking the variation of U, V, and haccording to Eq. (2). The gradient vectors for the control variables obtained by differentiating Eq. (2) weresolved using a quasi-Newton method. The iteration process is summarized in Figure 1. The steps in theprocess are described in more detail below.
(a) The unsteady shallow water equations are solved with initial values of Qbound, n, and K to obtain U(x, y;t+Δt), V(x, y; t+Δt), and h(x, y; t+Δt) over the entire calculation area for the next calculation time step.
(b) Misfits are evaluated using Eq. (1).
(c) Lagrange multipliers are calculated from the adjoint matrix.
(d) The functional L is calculated from Eq. (2) and gradient vectors of L relative to Qbound, n, and K are
estimated. If convergence is achieved, the process ends.

2(e) The variables Qbound, n, and K are updated using a quasi-Newton method if L did not converge. Then, theprocess returns to step (a) for the next iteration.
Target section for prediction(150 km - 133 km)

Long section(182 km - 133 km)

Figure 1. Procedure for assimilation analysis

3.

Figure 2. Location of study field in the Tone River watershed

ASSIMILATION ANALYSIS FOR THE WATER SURFACE PROFILES DURING THE
SEPTEMBER 2013 FLOOD IN THE TONE RIVER

3.1 The study field in the Tone river
Figure 2 shows the location of the study field. The Tone River has its source in the central mountains ofHonshu Island, Japan, and flows across the alluvial plain on the north side of Tokyo, eventually discharging
into the Pacific Ocean. The model was applied to the flood event observed in September 2013 over a 50 km
reach from 182 km to 133 km (long section), where water gauges are installed at 7 points. Based on theassimilation analysis results for the long section, the water level for the 17 km downstream section extending
from 150 km to 133 km (target section) was predicted using two-dimensional flow analysis．
Figure 3 is an enlarged view of the channel showing the locations of the water gauges (red triangles). The
river channel has a composite cross section with a high-water channel width of approximately 800 m. The
low-water channel has an average width and depth of 400 m and 6 m, respectively. Most of the floodplain iscovered by grass, but some reed colonies and tree communities remain along the banks of the low-water
channel, as indicated in the figure. Channel bed elevations were taken from aerial laser profiling dataobtained in 2013; the channel cross-sectional measurement data were obtained in 2012. Vegetation was
identified from river environmental information maps prepared in 2011.
The reasons why planar two-dimensional analysis method adopted to the flood flow analysis is as follows.
The Tone River is a meandering channel with compound cross-sections (Figure 2 and 3). The flow velocity
distribution and fluid resistance is influenced by bed material, ground cover conditions and planar
topographical shape of the river. The planar two-dimensional flow analysis can generally consider the above
effects except for the flow around the river structure.
Manning’s roughness n for each ground cover classification were determined based on report (2003). For thepermeability constants of the tree communities K, we adopted the values recommended by Fukuoka et al.
(2006 and 2007). These parameter values are listed in Table 1. The time increment for unsteady flow
calculation was set to 0.5 seconds in real time.
3.2 Results of the assimilation analysis
Figure 4 (a) shows that the ratio of the cost function to its initial value in the assimilation analysis for thewater level observed over two days during the flood period converges as the iteration steps proceed. The ratio
decreased sharply through the first 5 iteration steps and became almost constant after the 10th iteration step.
Figure 4 (b) shows the convergence process of the values of n and K, which indicate the resistancecharacteristics of each ground cover classification. All values became stable by the tenth iteration step.
Figure 5 compares the observed discharge and the discharge hydrographs identified by the assimilation
analysis. As described below, since the water level in the assimilation analysis agrees with the time change ofthe observed water levels at stations 1 to 7, the identified discharge hydrographs are smaller than the observed
discharge.

3

station 1

station 2
station 5
station 3
station 4

station 1

station 6

station 2

station 7
station 5
station 3
station 4
station 6
station 7
Figure 3. Topography and ground cover of study field
Table 1. Standard values of Manning’s roughness and permeability coefficients
Pre-setvalue

Minimumvalue

Maximum
valueStandard
deviation

Low-water channel (182km – 169.5 km)

0.032

0.030

0.035

0.0032

Low-water channel (169.5 km – 162 km)

0.030

0.025

0.032

0.0030

Low-water channel (162 km – 154 km)

0.027

0.020

0.030

0.0027

0.027

0.020

0.030

0.0027

Classification

Unit

Low-water channel (154 km – 143 km)
Low-water channel (143 km – 133 km)

m-1/3s

0.025

0.018

0.030

0.0025

Low grass

0.035

0.025

0.050

0.0035High grass

0.050

0.030

0.080

0.0050

Plain ground or levee

0.025

Ground surface of wooded area
Tree communities (permeability constant)

Fixed Value

0.035
m/s

40

Fixed Value
20

80

8

Figure 6 shows a comparison of the computed and observed water level hydrographs. The curves in Figure 7
are the cross-sectional averaged longitudinal water surface profiles of the assimilation analysis during therising (black), falling (blue), and peak at Station 4 (red). The triangles indicate the observed values at thewater gauge stations corresponding to the three lines. As shown, the assimilated water levels are in good
agreement with the observed values throughout the flood period.
We consider the reason why the discharge by the assimilation analysis was evaluated smaller than theobserved discharge around the peak time in Figure 5. Roughness coefficient changes during the flood due togeneration of sand waves and flattening of vegetation, but in this analysis, the roughness coefficient isassumed to be constant through the flood period. The number of water level observation points in the target
flood is not enough to examine the time variation of the roughness coefficients. Therefore, the temporalchange in the roughness coefficients with the temporal change in the water level is not evaluated in thisassimilation analysis, but only the temporal change in the flow rate is estimated. In addition, the preset
values of the Manning’s roughness coefficients and vegetation permeability coefficients shown in Table 3 areconfirmed to be past observations, but the appropriate weights for each term (i.e. JH , JQ , Jn_init , JK_init) in thecost function (1) due to the variance used are unknown. In this analysis we give 1.0 for all values of weights.
From the above, it can be seen that the water levels obtained from the assimilation analysis closely
approximate the observed temporal water levels at 7 points. Furthermore, the flow rate, roughness coefficients,
and permeability coefficients of the trees are simultaneously estimated.

4

0.6
0.4
0.2
0.0
4

6

50

0.05

40

0.04

30

0.03

20
10

0.02

8 10 12 14 16

0

2

4

number of iterations(a) Relative values of cost function
Figure 4. Convergence process in the assimilation analysis

Station 1(obs.)

Starion 2(obs.)
Starion 6(obs.)
Starion 5(calc)

Station 1(calc)

7,000

5,000
4,000
3,000
2,000
1,000

Starion 3(obs.)
Starion 2(calc)
Starion 6(calc)

Date and timeat 3:00 on 17th(obs.)
at 3:00 on 17th(calc,receding)35

Station 517540

at 22:00 on 16th(obs.)
at 22:00 on 16th(calc,peak)

Station 4

180

45Starion 5(obs.)
Starion 4(calc)

Figure 6. Comparison of the computed and observed
water level hydrographs

Station 3

at 17:00 on 16th(obs.)
at 17:00 on 16th(calc,rising)
H.W.L

Station 2

Station 1

Water Level (m)50Starion 4(obs.)
Starion 3(calc)

Date and time

Figure 5. Comparison of the computed and observed
discharge hydrographs
55

16

31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12

9/16 0:00
9/16 3:00
9/16 6:00
9/16 9:00
9/16 12:00
9/16 15:00
9/16 18:00
9/16 21:00
9/17 0:00
9/17 3:00
9/17 6:00
9/17 9:00
9/17 12:00
9/17 15:00
9/17 18:00
9/17 21:00
9/18 0:00

014

9/16 0:00
9/16 3:00
9/16 6:00
9/16 9:00
9/16 12:00
9/16 15:00
9/16 18:00
9/16 21:00
9/17 0:00
9/17 3:00
9/17 6:00
9/17 9:00
9/17 12:00
9/17 15:00
9/17 18:00
9/17 21:00
9/18 0:00

Water Level (m)

6,000

12

(b) Values of n and K

8,000

Flow rate (m3/s)

6
8
10
number of iterations

Station 7

2

0.06

Station 6

0

Low‐water channel (169.5 km – 162 km)
Low‐water channel (154 km – 143 km)
Low grassTree community permeability

K : Permeability Constant (m/s)

0.8

n : Roughness Coefficient

Jn / J0

Low‐water channel (182km – 169.5 km)
Low‐water channel (162 km – 154 km)
Low‐water channel (143 km – 133 km)
High grass

1.0

30
25
Tone Weir

20
15
10
185

170

165
160
155
150
Distrance from river mouth（km）

145

140

135

130

Figure 7. Longitudinal water surface profiles in rising and receding phases and highest water level profile

5

4.

PROPAGATION CHARACTERISTICS AND WATER LEVEL PREDICTION

Based on the results of the assimilation analysis, the propagation characteristics of the flood flow in the Tone
River were established, after which the propagation time to the upstream edge of the predicted section for the
September 2013 flood was determined. Water level predictions up to the lead time were then calculated. The
validity of the approach is confirmed by comparing the predicted water level to the observed level.
4.1 Propagation characteristics of flood flow
Figure 8 shows the cross-sectional average hydraulic quantities obtained from the assimilation analysis up to
the upstream edge of the prediction section. These include water level (H), water surface gradient (Iw),discharge (Q) and conveyance (Co). Here, conveyance (Co) is defined as
C = A R2⁄3 (e.g., Chow 1959 and Motonaga et al. 2012)
1n

Q=Co Iw 1⁄2
where Ac is the cross-sectional area and R is the hydraulic radius.
In this study, the amount obtained from the water level is redefined as the conveyance (CH).
Q

function Ac (H), R(H), Iw (H), α

αAc R2⁄3 Iw 1⁄2

αCH

(3)

31

24

46

30

23

45
44
43

180k

42
41

Water Level(m)47Water Level(m)Water Level(m)From Figure 8 (a), it can be seen that the relationship between water level and discharge loops downstream(150 km) since the riverbed gradient is smaller there than upstream (180 km). Figure 8 (b) shows that thewater surface gradient differs in the rising and receding phases of the flood and that the change in the water
surface gradient during the flood period is larger downstream than upstream. Figure 8 (c) shows that therelationship between the flow rate and the water flow capacity is nearly linear.
29
28
27

165k

26

1,000 2,000 3,000 4,000 5,000 6,000 7,000
Discharge(m3/s)

20

150k

19
17

24

0

21

18

25

40

22

0

0

1,000 2,000 3,000 4,000 5,000 6,000 7,000
Discharge(m3/s)

1,000 2,000 3,000 4,000 5,000 6,000 7,000
Discharge(m3/s)

300
250

1.5E‐03

CH=AR2/3Iw1/2

Water Surface Gradient (Iw)(a) Relationship between discharge and water level
2.0E‐03

1.0E‐03
180k
165k
150k

5.0E‐04

200
150
180k
165k
150k

100
50
0

0.0E+00
001,000 2,000 3,000 4,000 5,000 6,000 7,000
Discharge(m3/s)

1,000 2,000 3,000 4,000 5,000 6,000 7,000
Discharge(m3/s)(b) Relationship between discharge and(c) Relationship between discharge and conveyancewater surface gradientFigure 8. Characteristics of flood flow obtained from assimilation analysis

Based on the above results, we can examine the propagation characteristics of the water level. For thecontinuity equation, it is assumed that the longitudinal change in the discharge can be expressed by thelongitudinal change in water level, river width, water surface gradient, and total roughness. This is expressed
as follows:
5 ∂A
2 Q ∂B
1 Q ∂Iw
Q ∂α∂A ∂Q ∂A
+
= + V
+ +
+
=0
3
∂x
3 B ∂x2 Iw ∂xα ∂x∂t ∂x ∂t∂A∂A
+C =-D∂x∂t
5
2 Q ∂B
1 Q ∂Iw
Q ∂α) +( )
C= V, D=(- ) +(
3

3B

∂x

2 Iw

∂x

α

(4)

∂x

The propagation distance and time of the water surface profile can be obtained by tracking equation (4) using
the characteristic curve method. In this case, C (t, x) and D (t, x) are given as the average values in the crosssection obtained from the assimilation analysis results at each time and each place.
As shown in Figure 9 (a), the assimilated water level and the water level given by the characteristic curve
method closely agree. Figure 9 (b) shows the path diagram of the characteristic curve. Since equation (4) is
6not applicable to the Tone Weir, the calculation is performed separately for the section from the upstreamedge to the weir and the section downstream from the weir. From this result, it can be seen that the
propagation speed is large during the flood’s rising phase (until 17:00 on Sept. 16) but becomes graduallysmaller in the receding phase. It can also be seen that the propagation speed changes around the Tone Weir.

characteristic curve(from Upstream edge)
characteristic curve(from Tone Weir)

Station 2(calc)
analysis of the characteristic curve method
Date and time

9/17 18:00

9/17 12:00
9/17 9:00

9/17 3:00
9/17 0:00
9/16 21:00
9/16 18:00

9/18 0:00

9/17 21:00

9/17 18:00

9/17 15:00

9/17 9:00

9/17 12:00

9/17 6:00

9/17 3:00

9/17 0:00

9/16 21:00

9/16 18:00

9/16 15:00

9/16 9:00

9/16 12:00

9/16 15:00

Date and time(a) Comparison of the characteristic method andassimilation analysis for water level

9/16 12:00
Tone Weir

9/16 9:00
185

175
165
155
145
Distrance from river mouth（km）

135(b) Path diagram of water level propagation

6
Propagation time (hour)

9/17 15:00

9/17 6:00

9/16 6:00

9/16 3:00

Station 1(calc)
Station 3(calc)
49
47
45
43
41
39
37
35
33
31
29
27
25
23

9/16 0:00

Water Level (m)Figure 9 (c) shows the propagation time required to reach the upstream edge (150 km: Station 4) of the waterlevel prediction section. The propagation time is 3 to 5 hours during the rising phase and about 2.5 hours at
the peak. These propagation times represent predictable lead times that do not require a rainfall intensity
distribution, water level, or discharge hydrograph several hours ahead to predict the water level in the section
downstream of 150 km.

Acquisition of the observed water levels at 7 points

5

Delivery delay time4

Numerical calculation for data assimilation using theadjoint method32from upstream edge to Tone Weir

1from upstream edge to 150.3k
9/17 12:00

9/17 9:00

9/17 6:00

9/17 3:00

9/17 0:00

9/16 21:00

9/16 18:00

9/16 15:00

9/16 12:00

9/16 9:00

0Date and time(c) Propagation time of water level
Figure 9. Propagation characteristics of flood flow obtained fromassimilation analysis

・The temporal change in the water levels observed
over the preceding 9 hours is assimilated withconvergence calculation

Planar two-dimensional flow analysis for forecastingwater surface profiles up to the lead time (Table 2)
Figure 10. Procedure of analysis method to theongoing flood

4.2 Water level prediction in the downstream section based on propagation characteristics
Establishing the propagation time of the flood flow up to 150 km (as described above) enables us to conduct
our water level prediction analysis for the downstream section. Table 2 shows the forecast start time and thelead time for the analysis. For each of the times indicated in the table, the temporal change in the water level
observed over the preceding 9 hours at 7 points in the long section of the river (182 km – 133 km) is
assimilated using the adjoint method. The temporal change in the longitudinal water surface profile is thenpredicted up to the lead time for the target section (150 km – 133 km) using planar two-dimensional flow
analysis.
Figure 11 shows a comparison of the predicted and observed water levels at stations 4 and 5. As can be seenin the figure, the predicted water level agrees well with the observed water level until the peak time, although
it is slightly higher than the observed water level in the receding phase of the flood. Since the water level atpeak is what matters for practical use, the results here affirm the efficacy of the approach.
The analysis method to the ongoing flood evacuation warning for residents is shown in Figure 10 and theapplicability of the method is confirmed as follows. The acquisition time of the observed water level (delaytime) and the time required for numerical calculation for data assimilation are about 3 minutes and about 1
hour, respectively. The calculation time to estimate the water surface profiles along the embankment from 3
to 5 hours ahead is about 3 minutes. As a result, the total time is within 1.5 hours. These calculations are
executed with 6 MPI processes on IBM Power8 (18core, 3.45GHz, 2CPUs).
7

Time to start
forecast(September 2013)

Lead time(hour)

at 9:00 on 16th

5.2

at 15:00 on 16th

3.0

at 18:00 on 16th

2.8

at 21:00 on 16th

3.1

at 0:00 on 17th

3.6

5.

Water Level (m)Table 2 Time to start forecast and lead time
for the water level prediction analysis

24
23
22
21
20
19
18
17
16
15
14

Station 4(obs.)
Station 5(obs.)
6:00

9:00

12:00

15:00

16th

18:00

21:00

Station 4(predict)
Station 5(predict)
0:00

Date and time
Figure 11. Comparison of the predicted and observed water level

3:00

6:00

17th

CONCLUSION

The study produced several important findings that show the significance and usefulness of the method:
1. The assimilation analysis model accurately reproduced the temporal changes in the water surface profile
observed for a 50 km reach of the Tone River during the September 2013 flood.
2. The propagation characteristics of the flood flow and the lead time (propagation time) that are required to
predict the water level in the downstream section were quantitatively determined.
3. The predicted water level up to the lead time approximately agrees with the observed water level.
Moreover, if this analysis method is used as a flood forecasting system, it is possible to provide theresidents with predicted water level information 3-5 hours ahead every 1.5 hours. Therefore, theapplicability of water level prediction method for disaster prevention is considered to be high.

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Hydr. Eng., ASCE,Vol.130, No.6, pp.501-510.
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equation, Int. J. Comput. Fluid Dynam., Vo.19, No.1, pp.3-13.
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