This document discusses forecasting daily streamflow in two Kenyan watersheds, Yala and Sondu, using the Kalman filter. Concurrent daily rainfall and streamflow data from 1972-1981 were used to identify transfer function models relating rainfall to streamflow. The Kalman filter was applied to dynamically update the model parameters based on new observations. Results showed the transfer function models adequately described the rainfall-runoff relationship and recursive simulations using the Kalman filter reproduced the temporal variations and mass balance of observed streamflows. However, the models tended to underestimate flows during extreme events.

1. Forecating Daily Streamflow Using the Kalman
Filter for Dynamic Stochastic
Hydrologic System
Phillip M. Mutulu1, J.A.Rod Blais2 and Daniel Mutua3
1Dept. of Geomatics Engineering,
2Pacific institute for the Mathematical Sciences
University of Calgary
3DANCARE Engineering and Telecoms Inc., Calgary
CGU Presentation 2005 – Banff Alberta

2. Presentation Topics
•Data and study area
•Modeling Strategy
•Results
•Concluding remarks

3. Data
•Concurrent daily rainfall-runoff data for period
1972-1981
•Watersheds: Yala, Area 2388 km2
Sondu, Area 3287 km2
•Hydrometeorological network:
-Yala: 16 rainfall gauges 1 automatic streamflow gage
station.
-Sondu 15 rainfall gauges, 1 automatic streamflow
station
•Rainfall input into the watershed system is areal
average using isopercentile method found suitable
for watersheds in East Africa.

4. Location of Study Watersheds

5. Potential Application – Real Time Forecasting
Sondu Hydro-Power Project - Kenya

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−−= −−−−−
q
r
qtttrtttt UUUYYYY
ω
ω
ω
δ
δ
δ
.
......
1
0
2
1
121
+εt
General Model Structure
Code Built in Fortran Language
OR, Transfer Function representation, TF(r,q)
Where r,q are the orders of the TF model- weighting for
memory and forcing functions
tbt
s
st
r
r UBBYBB εωωωδδ ++++=+++ −)...()...1( 1
101

7. 0
50
100
150
200
250
300
Depth(mm)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Norma Annual Rainfall (mean over study areas)
A B
•Every year is divided into two blocks representing wet
seasons in the watersheds: First season (A) runs from
APR to May, second season (B) runs from Aug to
Oct
Modeling Approach

8. Model Identification
• (Analysis of autocorrelation, partial autocorrelation,
spectral density function
• Design of (AR) output filter from input rainfall
• Filtering output discharge
cross-correlation and cross-spectral density
function [phase, coherence,cospectrum and
quadracture spectrum])
•Based on classical time series (time and frequency
domain) analysis and modeling by Box and Jenkins technique
(1970), Jenkins and Watts (1968)
Modeling Approach
Model Estimation: Least squares estimation of model
parameters to initialize state vector
•Prediction: Model verification

9. Formulation of the Kalman filter
Watershed dynamic linear system model (DLM)
Observation model
System Model
,teθXY ttt += ),0(~ tt Ne V
,tn+= −1tt Gθθ ),0(~ tt Nn W
where Yt is output discharge at time t, Xt is input
rainfall, and θt is the parameter (state) vector of the
system, Gt is state weighting matrix. Vt and Wt are
noise variance matrices of the observation and state
models.
Different formulations exist. A simple approach
equivalent to Bayesian forecasting is followed
,teθXY ttt += ),0(~ tt Ne V
,tn+= −1tt Gθθ
,teθXY ttt += ),0(~ tt Ne V
,tn+= −1tt Gθθ
,teθXY ttt += ),0(~ tt Ne V
),0(~ tt Nn W

10. State prediction: tGVGθθ 1t1t/t += −−
ˆ
State covariance extrapolation: tWGGCC T
1t1t/t += −−
Forecast dependent variable : tθXY 1tt
ˆˆ
−=
Variance of Y : t
T
ttt VXCXD +=t
State update : ( )t1t/ttt1t/tt WθXKθθ −−= −−
ˆˆˆ
State Cov Update : 1t/t
T
tt1t/tt CXKCC −− −=
Kalman Gain matrix :
1−
−− += ]XCX[VXCK t1t/t
T
tt1t/tt
The Kalman Filter Algorithm
1t/t
T
tt1t/tt CXKCC −− −=

11. The Kalman Filter Algorithm
G zI
F
Γ
Kt
Xt
F
zIG
Yt
Yt
et
ηt
XtVt
et
θt
θt
SYSTEM
K-FILTER

12. Initialization of state vector and state
covariance matrix
• Block Least Squares Estimation applied to data
averaged over the calibration period 1972-1976
•The final model parameters are estimated from daily data
averaged for block A and B over the calibration period.
These parameters are then used to initialize the state
vectors of the DLM models of the watersheds
•Parameter standard errors are used to initialize the state
covariance matrix

13. Results

14. Model Identification
Correlogram YALA72(B)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Time lag (days)
Arcoefficient
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5 0.6
Frequency
Powerspectrum
Rainfall-Dicharge Cross-Correlation:
Yala 1972 (B)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30
Lag (days)
CCFcoefficient
-200
-150
-100
-50
0
50
100
150
200
0 0.1 0.2 0.3 0.4 0.5 0.6
Phase(degrees)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Coherence
Phase
Coh

15. Best Models Fitted to Block Data
YALA
PERIOD MODEL 1972 1973 1 974 1975 1976 1977
A TF(2,2 ) 21.71 24·12 17·41 21.69 10.90 14.12
A TF(3,2) 23.74 26.74 9·47 21.29 8.76 14.48
A TF(4,2, 20.83 20.78 7.47 21·52 7.66 13.46
B TF(2,2 ) 9.93 21.68 15.03 20.10 21.87 7.73
B TF(3,2) 11.05 20·30 13.58 19.71 20.08 8.43
B TF(4,2 ) 10.37 16.97 14.08 19.18 18.48 7.57
SONDU
PERIOD Model 1972 1973 1974 1975 1976 1977
A TF(2,2) - 15·21 15.13 12.18 18.14 25.33
A TF(3,2) - 18.20 17.24 15·29 17.56 24.32
A TF(4,2) - 13·22 12.92 12.24 17.84 22.91
B TF(2,2 ) 31.08 12.64 19·74 - - 25.01
B TF(3,2 ) 27.67 12.69 19·52 - - 24.12
B TF(4,2 ) 28.24 10.60 18.77 - - 22.74
Table for Porte Manteau lack of fit test
Max Lag 20
Chi Sq
TF(2,2) TF(3,2) TF(4,2)
Chi Sq 26.3 25 23.7
DF 16 15 14

16. Average Parameter Values
YALA
Period α1 α2 ω0 ω1 ω2 R2
se
A 1.029 -0.117 -0.041 0.16
6
0.016 0.92 0.29
B 0.971 -0.044 0.044 0.06
6
0.116 0.96 0.21
SONDU
Period α1 α2 ω0 ω1 ω2 R2
se
A 1.000 -0.063 0.009 0.005 0.096 0.92 0.29
B 1.145 -0.168 0.047 0.066 0.000 0.96 0.21
In subsequent analyses, parameter ω0 has been
omitted to enable practical application of the models.
For application in data assimilation problems it is
impractical to use this parameter.

17. State Vector Tracking and Recursive Flow Simulation
State vector evolution; Yala 1978 (A)
-2
-1
0
1
2
3
0 20 40 60 80 100
time (days)
parameter
d1 d2 w1 w2
0
20
40
60
80
100
120
0 20 40 60 80 100
time (days)
flowm
3
/s
Observed
Simulated

18. Mass Curve Analysis
Mass curves: Yala 1978(A)
y = 1.2836x
R
2
= 0.9544
-5
0
5
10
15
20
25
30
35
40
-5 0 5 10 15 20 25 30 35
cumulative simulated flow
cumulativeobservedflow
Ideal fit Model curve

19. Flow Simulations (cont’d)
Recursive Yala river flow simulation based on TF(2,2) model
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100
Time (days) [Aug-Sept,1979]
Flow(m3
/s)
Observed
simulated

20. Concluding Remarks
•It has been shown that dynamic linear stochastic model can
adequately describe the daily rainfall-runoff data for study
watershed.
•The parsimoneous transfer function model with 4
parameters is found to be statistically adequate
•The effectiveness of incorporation of Bayesian forecasting
using the Kalman gain has been demonstrated
•The simulated flows reproduce the temporal
variabilities of the observed hydrographs but the
model underestimates the flows especially during
times of extreme events
• Results are sensitive to initial conditions and there
need further for further studies