hydrological model parameters based ... 1Technische Universität Darmstadt,
Germany, ... physically oriented hydrological models and the simulation time step
.
Identification of time time-step step dependent hydrological model parameters based on data from the Poellau experimental basin Ostrowski, M.1, Gamerith, V.2, Bach, M.1, DeSimone,S.1 1Technische Universität Darmstadt, Germany, 2Graz University of Technology, Austria
INTRODUCTION
The estimation of parameters in deterministic hydrological models is highly disturbed by multiple, partly unkown error impacts. While some parameter variations are stochastic in nature, others are deterministic, of which some are induced by the application of different time steps. The data resolution is critical both for f climatic model forcing as well as for measurements used for parameter estimation. i i
H Hypothesis th i & Methodology M th d l Hypotheses: H th There are functional relationships between sensitive parameters of physically oriented hydrological models and the simulation time step Identification of optimum physically based model structures must not rely on daily values Methodology: A physically oriented model is applied to two neighbouring experimental catchments with varying time steps steps. Optimum parameter sets are estimated with evolutionary search algorithms using a two criteria objective function (Nash & Sutcliffe 1- NSC, Mass Balance Error MBE) Optimum empirical relationships are identified Proposals for solutions are presented
E Empirical i i l relationships l ti hi 35
tc=time of concentration (h) tstep = recomm. time step (h) potential function tstep = f (A) potential function tc = f(A)
30
time e (h)
25
tc = 0.51A
0.6
20
15
tstep p = 0.2A
10
0.54
5
0 0
100
200
300
400
500
600
700
800
catchment area A (km2)
This is a very old requirement (see Maniak 1997)
900
1000
Th modelling The d lli problem bl Unmeasured input
Measured output
Reality
Σgoodness of fit
Measured input
y
Model
x Computed output
Local or global?
Parameter estimation
OF
p From Bruce Beck, 1982
Th modelling The d lli problem bl
D t E Data Errors Syst. Syst Errors
Random Errors
Parameter E ti ti Estimation Errors Meth. Meth Errors
M d lli Errors Modelling E
Appl. Appl Errors
Struct. Struct Errors
Appl. Appl Errors
+- Superposition of single g errors and uncertainties with little chance of identification of single impacts F From Ostrowski&Wolf, O t ki&W lf 1984
Th parameter The t estimation ti ti problem bl
Model drivers (precipitation, temperature etc) temperature,
Hydro-meteorological time series with changing temporal resolution l ti Δt
Data for parameter estimation (runoff)
Determination of functional relationships between time step Δt and parameter vector p Foreward propagation by model drivers, backward propagation during model inversion Remark: I try to avoid calibration, which would pretend non existing accuracy
C t h Catchments t and d climate li t time ti series i Gauge R Runoff ff Saifenbach S if b h Runoff Oelmuehle Runoff Praetisbach Runoff Duerre Saifen Streams Catchment Rainfall stations
Th model The d l sett up (BlueM) (Bl M) Sub catchment River reach Reservoir
WHR
DS
The p problem: The step p from differential towards difference equation SN = precipitation mass curve tan α = momentaneous intensity, tan β intensity for discrete Δt = t – t0
=Δt
If Δt is too large, you cannot describe physical processes
Mean rainfall intensities: decreasing g intensities with increasing time step (model driver)
Types yp of flow discretisation-mean flow ((true in volume) versus linear (true at moment)
Movement of p peak values with time step p reduction 5 60
360 720 1440
Effect of the use of constant p parameters for different time steps
Volume error direct runoff due to use of constant parameter (60 min)
accumula ated effect. precipitation n [mm]
16
Optimum 60 minutes Optimum 15 minutes
14
Optimum 360 minutes 15 minutes 60 minutes parameters
12
Over estimation
360 minutes 60 minutes parameters 10 8 6
Under estimation
4 2 0 0
20
40
60
80
100
120
140
160
precipitation p [[mm]] accumulated p
Result: Use of parameter function leads to error compensation
Maximum rainfall intensity y as a function of time step 120 00 120.00 rain gage 2 Rainfall intens sity (mm/h)
100.00
rain gage 3 rain gage 5 rain gage 6
80.00
rain gage 7 0 74 2 rain i gage 3 imax = 403 tstep-0.74 ,R = 0.99 -0.63 2 rain gage 7 imax = 178 tstep ,R = 0.99
60.00
rain gage 5 imax = 211 tstep-0.64,R2 = 0.99 2 rain gage 6 imax = 296 tstep-0.71,R R = 0.96 0 96
40.00
rain gage 2 imax = 215 tstep-0.65,R2 = 1.00 20.00
0.00 0
200
400
600
800
1000
1200
Time step p ((min))
The parameter is partly determined by the maximum rainfall intensity
1400
M i Maximum Runoff R ff as function f ti off time ti step t 7.00
mea an maximum m flow [m3/s s]
6.00 Runoff Peaks DS Runoff Peaks WHR logarithmic DS logarithmic WHR
5.00 4.00 3.00
2
2 00 2.00
R =0 0.95 95
1.00
R = 0.97
2
0.00 0
200
400
600 800 time step [min]
1000
The parameter is partly determined by the maximum runoff
1200
1400
Th model The d l structure t t (BlueM) (Bl M)
Hydrological response unit scale soil parameters Infmax Rd
kf, Θs, Θfc, Θwp
I(t)
Subbasin scale retention parameters ß, Ro,f o f,Ro, o
Infiltration zone
s
Root zone Transient zone Groundwater
Ri
Rb
S il moisture Soil i t module d l – The Th roott zone
Ep(t)
I(t) Ieff(t) Infp(Θ) Infa(t)
Ea(t)
Infmax Pv(Θ)
Θ(t) Pl(t)
kf
Ea(Θ) Pl(Θ)
Pv(t)
0
Θr
Θwp
Θfc
dΘ = Inf a (t ) − Pv (t ) − Pl (t ) − E a (t ) dt
Θs
Solution: Modified analytic y Euler integrator g with most efficient computational performance
V(t) I1(t) Q1(t) I2(t)
Q3(t)
V(t)
Q2(t)
Q1(t)
Q2(t) I3((t)) Q3(t) Qj(t) developed by Ostrowski, 1992
Sub surface retention constants versus time step 240 1000 h
Inter flow retention DS Inter flow retention WHR Base flow retention DS Base flow retention WHR Linear (Base flow retention DS) Linear (Base flow retention WHR)
900
retention con nstants [h]
800 700
2
Rb (DS) = 0.38 tstep + 380, R = 0.70
600 500 2
Rb (WHR) = 0.07 tstep + 419,R = 0.77
400
Ri (DS)=constant= 320 h
300 Ri (WHR) =constant=240 h
200 100 0 0
200
400
600
800
Time step [h]
Result: real constants or linear functions
1000
1200
1400
Fast and slow direct flow retention constants versus time step 18
3
2
Rd,s WHR= -0.21Ln(tstep) + 15.2 ,R = 0.79
14
2.5
2
Rd,s (DS)= -0.63Ln(tstep) + 16.6 ,R = 0.74
12
2
2
Rd,f (WHR) = 4E-05(tstep)+ 1.68 ,R = 0.01
10 1.5
2
Rd,f (DS) = -0.28Ln(tstep) + 3.08 ,R = 0.79
8
Direct runoff slow DS Direct runoff slow WHR Direct runoff fast DS Direct runoff fast WHR logarithmic direct runoff slow WHR logarithmic slow direct runoff DS logarithmic g fast direct runoff DS linear fast direct runoff WHR
6 4 2 0 0
200
400
600 800 time step [min]
Result: Close to linear functions or constants
1 0.5 0 1000
1200
1400
fast diirect retentio on constant [h]
slow d direct retention constant[h]
16
S il storage Soil t parameters t versus time ti step t 600
Wilting point Field capacity Total pore volume potential wilting p gp point potential total pore volume
Soil storage [mm/m]
500
400
y = 502,91x-0,0272,R2 = 0,6845
300
200 0,0882
y = 87,74x
2
,R = 0,738
100
0 0
200
400
600
800
1000
Timestep [min]
Result: Close to linear except for very small time steps
1200
1400
H d Hydraulic li conductivity d ti it versus time ti step t
8
sat. h hydr. conducttivity [mm/h]
7
hydr cond. hydr. cond DS
6
hydr. cond. WHR
5
logarithmic DS logarithmic WHR
2
4
kf(WHR) = -1,23Ln(tstep) + 9,36 ,R = 0,96
3 2 2 1 kf(DS) = -0,47Ln(x) + 4,53,R = 0,90
0 0
200
400
600
800
1000
Time step [min]
Result: Close to functional non linear relationships
1200
1400
M Max. infiltration i filt ti versus time ti step t 3000 maximum infiltration DS
max ximum infiltra ation [mm/h]
2500
maximum i iinfiltration filt ti WHR DS > 6h
2000
WHR > 6h logarithmic Duerre Saifen
2
infm (DS, (DS < 6h) = -58Ln(tstep) 58Ln(tstep) + 3400 3400,R R =0 0,9 9
1500
logarithmic Wildholzrechen logarithmic DS > 6h logarithmic WHR > 6h
1000 2
infm(WHR,
