TIME SERIES FORECASTING AND ADAPTIVE FILTERS
2.4 Application of recursive filters to the forecasting of streamftow and spatial rainfall data
2.4.1 Case study - Stream flow forecasting using Kalman filters
Linear reservoirmodel
Inorder to demonstrate a simple use for Kalman filters in the contex t of flood forecasting, a case study is presented. Alinear catchmentmodelisusedto forecast strea m flow on the Lieben bergsvlei catchme nt inSouthAfrica. Similar techniques have been used successfully in the past(Szollosi-Nagyand Mekis, 1987).
Figure 2. 1shows (schematically) the structure of the linear model used. The modelconsists of anarrangement of three interlinked linear reservoirs. The vari- able Si represents the storage in reservoir i while the k, are parameters whic h determin e thereser voirs response to a givenstorage. Eachreservoir has a losspa- rameter associatedwithit and the first reservoir is defined as the one that accepts the rainfa ll input. The model is conveniently represented in state-space form as (Peg ram andSinclair, 2002)
S,
=
ASt-1+
f3Ut- lYt
=
eStwith S, thevector of storagesat time t, while
1- (1-+1-+1-)
1.:1 1.:2 1.:5 0 0A=
1.:111 -(1- +1-)
1.:3 k6 0 and1 1
1-(1- +1-)
k2 k3 k4 k7
25
Precipitation
Loss
Loss
Loss
Fig. 2.1:A linear reservoirmodel for flood forecasting .
The state-space formulation of the model may be directly interpreted in terms of equations2.3 and 2.4. The Kalman filter (equations 2.5 to 2.9) may therefore be readily applied in order to update thestate estimates
S t.
Figure 2.2show schematically how the triple re ervoir model de cribed above has been used in asemi-distributedsense to produce forecast flows based on the application of a Kalman filter for state updates. The catchment has been sub- divided into twelve smaller homogeneousunits. Eachsub-catchment isthen mod- elled in aspatially lumpedsense using average rainfall input and the linear model shown in figure 2.1. The model parameters were fitted using a different flood event ofsimilar magnitude to thosepresentedhere.
Results
Figures 2.3 and 2.4 show forecasts for a number of lead-times of up to twelve hoursahead.The forecasts were produced for floodeventson the Liebenbergsvlei catchment in the Free State province, South Africa. It is clear from these fig- ures that the accuracy of the forecastsreduces with lead-time, as expected. This is inescapabl e as the confidence with which forecastscan be made reduces with lead-time even for a perfect catchment model, if the rainfall input is not known
Pr«1 Itatton
:C :::::J---.
Strtaml10wFig.2.2: An illu tration showing how alinearreservoir model was u ed to model a sub-catchment area . The Liebenbergsvlei was sub-divided into 12 relatively homogeneous areas,each of which ismodelled in aspatially lumped manner.
with certainty in advance . It is pleasingto note that the general character of the events is retained even by the twelvehour ahead forecasts,asthisindicatesa rea- sonable model respon se to the observed rainfall. In fact, it is quite remarkable thatsuch a simple catchment model works so well. It is conjectured that the an- tecedentconditionsof the catchment were such that afairly linear rainfall-runoff relationship existed in both cases. No investigation was made as to whether this is the case when the catchme nt is relativelydry at theon et of arain fall event.
Figures 2.5 and 2.6show the square rootofthe mean sumof squared forecast errors (Root Mean Sum of SquaredErrors -RMSSE)for forecastsmadeusingthe linearreser voirmodel with Kalm an filteringcompared to the RMSSE fromsimple per istenc e forecasts. A persistence forecast is one in which thecurrent value of streamflow isused astheforeca st valueforall lead-tim es.Thevalueis, of course, updated with each change in the forecast origin . This is equivalent to assuming that no change in streamfl ow is expec ted with time. Persistence forecasts take
27
350 300
- Observed data ..... 1hourahead forecasts - 6 hour ahead forecasts 12 houraheadforecasts
100 150 200 250
Timefrom start of event (hours) 50
50 I
j l
250
100 300
350r -- - .--- - .--- - ,--- - ..-;:= ====== ::::=:=::== = :I:j1
:t 150
u:::o
...
...!E.
§. 200
s
EFig. 2.3: Forecasts made using the Linear reservoir model and the Kalman filter (December 1995,Liebenbergsvlei).
800 700
600
... -
§.en
500ell 400
-
Es
0 300u:::
200
100 {
!
00
- Observed data ... 1 hour aheadforecasts - 6 hour ahead forecasts 12 hourahead forecasts
50 100 150 200
Time from start of event (hours)
Fig. 2.4: Forecast made using the Linear reservoir model and the Kalman filter (February 1996,Liebenb ergsvlei).
"*
Persistence forecast~ Model forecast 50
Cil 40
-
M
-
EW 30
tJ) tJ)
~ 20 et::
10
0 2 4 6 8
Forecast lead time (hours)
10 12
Fig. 2.5: Comparisonof the root mean sum of squared forecasterrors (December 1995, Liebenbergsvlei).
no cognisance of the streamflow history and figures 2.5 and 2.6 show that the Kalman filtered model foreca ts outperform the persistence forecast (despite a relatively simple rainfall-runoff model), due to their incorporation of the observed characteristics of the recent flows via the Kalman filter.
Figure 2.7shows a set of scatter plots comparing observedand forecast stream- flows for the 1995 event of figure 2.3. The lefthand column of the plotsshow the comparison for forecasts made using the linear reservoir model and Kalman filter while the right hand column shows the comparisons for persistence forecasts - note the larger R2 values in the plots in the left hand column. Figure 2.8 shows the ame comparisons for the 1996 event of figure 2.4. These figures provide an alternative way of showing the model forecasts to be an improvement over the persistence forecasts.
29
150.---.---,---,----,---r---~
"'*
Persistenceforecast~ Model forecast
~ 100
M
-
l1JEen en
~
50o 2 4 6 8
Forecast lead time (hours) 10 12 Fig.2.6: Comparison of the root mean sum of squared forecasterrors (February
1996, Liebenbergsvlei).
325
. .
,
.
o
u g : ; , ; ; - - - 'o
325325 Y=0.86137x+10.40, R2
=
0.74203 ,,/ 325 Y=0.99497x+0.363,0R2=0.98964 ,/
325
325
o
K....- ~o
o
~---'o
325, . . . . - - - -__----:?I
Y
=
1.0726x+2.2326/R2=0.8483 '... ,/
!.
..,,,,',.
~"-.~,
325 Y
=
1.0275x _0.3584~'
R2=0.99379 /
-
rnn:so.E e
"C n:s(1)
.cn:s
'-:J
.c
o
CD
325
.. ,,~...l
o"'---''--'-'-'---- ---' o
325.--- ::-:---:--:--::__:_::-'1
y
=
0.6274x+28.521,6'R2
=
0.39584.
'.
. " ,/ /: " ,...t
"
,
"
Observed flow (m3/s)
Fig.2.7: Observed and forecast scatter-plots (December 1995, Liebenbergsvlei).
Left-hand column model and Kalman filter, right-hand column persistence . Both axes of the plots have units of m3/s.
31
770
770
. . . ..
y=0.86235x+31.21!6 R2=0 74285
.
,/.. -
"/ )
,/
,;;.
\
. . .
:770..---,,---=----:-~
770. - - --:-:-::--:-:---=-=-=-=""71 y
=
0.61618x+92.76,1 R2=0.38897 ,/770 y=0.99547x+0.996)' R2
=
0.99075 , /770 770
••
. .
. . .
0'4"-- - - - - - ----'
o
o • o
o
~ " " , - - - - _ - - - - J 0"'---- - - -=-==----o
770 0 770Observed flow (m3/s) 770 Y
=
0.99343x+8.936}'R2
=
0 82128.
• ,/770. -----=-=----.r::_=_:::":'"""'I
y
=
0.81387x+5.4.55)'R2=0.50462 i ,/
,/-\WI.'",/
'1ft" ~..
.
,," ..
"...
.
. "
lW
770 Y=1.0155x -1.706 / R2
=
0.99441 , /-
Ineu(,).s e
"C
euQ)
.ceu
...
:::J
o
.c-
CDIneuo.s e
"C
euQ)
.ceu
...
:::J
o
.cN
~
Fig. 2.8: Observed and forecas t scatter-plots (February 1996, Liebenbergsvlei).
Left-hand column model and Kalm an filter, right-hand column persistence. Both axesoftheplots haveunitsof m3/s.
Fig. 2.9: A time series plot of WAR and SMF. The data has been sampled at 5 minute intervals over the duration of a rainfall event lasting 42 hours during the 24th and 25th of February 1996. The observations of the event were made with the SAWS MRL5 radar in Bethlehem,South Africa.