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Aim
Karlsruhe
Strasbourg (France)
Stuttgart
Freiburg
Basel (Switzerland)
N
100 km
Fig. 1.1 Map of the study area
Aim
Q90 = ?
Wiese
Site of the proposed hydropower plant
How can we gain information on the expected relationship between the Q90 and the catchment descriptors?
Fig. 1.2 Wiese catchment (No. 532)
Aim
AimAvailable Catchment Descriptors
Soil
Percentage of soils with high infiltration capacity 0.19%
Percentage of soils with medium infiltration capacity 0%
Percentage of soils with low infiltration capacity 99.81%
Percentage of soils with very low infiltration capacity 0%
Mean hydraulic conductivity of the soils 201.69 cm/d
Percentage of soils with low hydraulic conductivity 0%
Percentage of soils with high waterholding capacity in the effective root zone 0%
Mean waterholding capacity in the effective root zone 109 mm
Q90 = ?
Morphometry
Catchment area 206.28 km2
Drainage density 1.31 km/ km2
Highest elevation 1485.5 m a.m.s.l.
Lowest elevation 423.6 m a.m.s.l.
Average elevation 898.74 m a.m.s.l.
Maximum slope 45.54%
Minimal slope 0 %
Average slope 18.08%
Climate
Annual precipitation 1891 mm
Land Use
Percentage of urbanisation 2%
Percentage of forested area 63%
Hydrogeology
Percentage of rock formations with a very low hydraulic permeability 0%
Weighted mean of hydraulic conductivity 9.62 * 10–4 m/s
back
Aim
Information on the relationship between the Q90 and a certain set of catchment descriptors can be gained by looking at other catchments in the same region for which both flow data and catchment descriptors are available.
Q90 = ?
By means of multiple regression analysis among the other qualifying catchments in this region, we may be able to find a common
regional pattern which describes this relationship between catchment descriptors and the Q90. This equation is called the regional transfer function.
Wiese
Site of the proposed hydropower plant
Assuming that the same relationship is true for the Wiese catchment, we can use the regional transfer function and estimate the desired Q90 value at our ungauged site based on the respective Wiese catchment descriptors.
Fig. 1.2 Wiese catchment (No. 532)
Aim
Aim
Karlsruhe
Stuttgart
Strasbourg (France)
Freiburg
Basel (Switzerland)
N
100 km
Fig. 1.1 Map of the study area
Study Area
The Oberrheinische Tiefebene(Rhine Rift Valley) is a 300 km long and 2030 km wide tectonic rift, which is filled with fluvioglacial deposits. The river Rhine flows through the valley from South to North. It interacts with the sediments to form terraces, alluvial fans, gravel bars, etc.. It is here that the lowest elevations of the study area, 85 m to 250 m a.m.s.l., are found.
The region is among the warmest in Central Europe, with mean air temperatures around 10°C, and it receives 600 to 900 mm rainfall per year. (BORCHERDT 1991). The favourable climate and fertile soil on extensive loess deposits are the basis for the high agricultural productivity of this region, where wine and fruit are grown (MOHR 1992).
Karlsruhe
Stuttgart
Strasbourg (France)
Freiburg
Basel (Switzerland)
N
100 km
back toOverview
Fig. 1.1 Map of the study area
Study Area
The Schwarzwald (Black Forest) is a mountain range, characterized by steep valleys on the West side toward the river Rhine and more gentle slopes on the Eastern side towards the Danube.
The Northern and Eastern Schwarzwald has an average elevation of 600 to 800 m a.m.s.l. (highest peak: Hornisgrinde 1164 m a.m.s.l.) and is dominated by New Red Sandstone. Due to relatively permeable bedrock, the drainage network is not particularly dense.
The Southern and Western part of the Schwarzwald is the most elevated part of the study area with mean elevations of 1000 m a.m.s.l.. Feldberg is the highest elevation in the study area with 1493 m a.m.s.l. and an average air temperature of 3.2 °C (BORCHERDT 1991). The area also receives the most precipitation in the study area; up to 2100 mm/year. Since the top bedrock is composed of granite and gneiss with relatively low permeability, a significant amount of water is drained on the surface and a dense drainage network with a mean drainage density of 1.94 km/km2 and a maximum of 5.0 km/km2 (WUNDT 1953) has developed.
Karlsruhe
Stuttgart
Strasbourg (France)
Freiburg
Basel (Switzerland)
N
100 km
back toOverview
Fig. 1.1 Map of the study area
Study Area
The Südwestdeutsches Schichtstufenland(literally: “Southwest German steplayered land”) is characterized by a relatively level to rolling topography, which is slightly tilted towards the South East. Its elevation ranges from 700 to 1000 m a.m.s.l.. Mean annual air temperatures range from 6 to 9°C. The region receives between 650 and 900 mm rainfall per year.
The bedrock is composed of layers of sedimentary rocks, such as New Red Sandstone, Coquina, Keuper, and Jurassic, which exhibit karstic phenomena, such as dolines and sinkholes. Dry valleys are relics from periods of colder climate when the ground was frozen so that more water drained on the surface. Drainage density today may be as low as 0.03 km/km2 (WUNDT 1953).
In some areas, limestone is covered with loess, which causes an increase in drainage density. New Red Sandstone is also found in this region in alternating layers with Marl. Due to differential erosion and an inclination of these layers a sequence of steps has been formed in the landscape.
Karlsruhe
Stuttgart
Strasbourg (France)
Freiburg
Basel (Switzerland)
N
100 km
back toOverview
Fig. 1.1 Map of the study area
Study Area
The Alpenvorland (PreAlps) is an area where unconsolidated sediments have been rearranged by glaciers.
The area around Bodensee (Lake Constance) has been affected by the most recent (Würm) ice age, and has a quite pronounced relief with drumlins, lakes, and bogs of glacial origin. For the most part the area drains to Lake Constance, which is part of the river Rhine system. The lake is the result of glacial scouring. With a surface area of 538 km2 and a maximum depth of 254 m it is the largest German lake supplying Stuttgart and several other cities with drinking water.
The Northern part of this landscape is a relic of the preceding (Riss) ice age and is therefore more levelled. Along the Danube, gravel with loess deposits can be found. The region lies at a mean elevation of 600 m a.m.s.l.. It receives 750 to 1400 mm precipitation and its mean annual air temperature is between 6 and 7°C.
Karlsruhe
Stuttgart
Strasbourg (France)
Freiburg
Basel (Switzerland)
N
100 km
back toOverview
Fig. 1.1 Map of the study area
Study Area
N
Fig. 2.2 Mean annual precipitation (196190) [mm]
more
Study Area
N
Fig. 2.2 Mean annual precipitation (196190) [mm]
back toOverview
Study Area
Rhine
Danube
Fig. 2.3 Catchments in Southwest Germany
more
Study Area
Fig. 2.4 Drainage density in Southwest Germany
more
Study Area
1
Study AreaRunoff RegimesFigure 2.5 shows the Pardé coefficients (mean annual monthly flow divided by mean flow) for two catchments in our region, ranging between 0.5 in late summer and 2.0 in the spring.
Breg
(at Hammereisenbach)
Elz
(at Mosbach)
2
2
Fig. 2.5 Pardé coefficients
back toOverview
Study Area
Model Design (Step 2)
Data Acquisition (Step 1)
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Set (56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Calibration (Step 3)
BASE = b0 + bj * Xij + ei MAM(10) = b0 + bj * Xij + ei Q90 = b0 + bj * Xij + ei
Validation Data Set (27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Evaluation (Step 4)
Model Validation (Step 5)
Check for agreement between observed and estimated values
Model Application
You may chose a specific step of the procedure or click on the arrows (bottom right) to proceed in sequence
Procedure
Model Design (Step 2)
Data Acquisition (Step 1)
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Set (56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Calibration (Step 3)
BASE = b0 + bj * Xij + ei MAM(10) = b0 + bj * Xij + ei Q90 = b0 + bj * Xij + ei
Validation Data Set (27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Evaluation (Step 4)
Model Validation (Step 5)
Check for agreement between observed and estimated values
Model Application
You may chose a specific step of the procedure or click on the arrows (bottom right) to proceed in sequence
Procedure
1
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Sets
(56 Stations)
Validation Data Sets
(27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Procedure
1
back
Procedure
2
N
0 80 160 km
Fig. 3.1 Spatial distribution of selected catchments
Procedure
3
spatial representation with respect to climate, land use, morphometry, soil, and hydrogeology
experience in using these independent variables in other studies
availability for the study area
relatively easycalculations
possible interpretation as areal means
Tab. 1 Selected catchment descriptors
Soil
Percentage of soils with high infiltration capacity [%]
Percentage of soils with medium infiltration capacity[%]
Percentage of soils with low infiltration capacity[%]
Percentage of soils with very low infiltration capacity [%]
Mean hydraulic conductivity of the soils [cm/d]
Percentage of soils with low hydraulic conductivity [%]
Percentage of soils with high waterholding capacity in the effective root zone [%]
Mean waterholding capacity in the effective root zone [mm]
Hydrogeology
Percentage of rock formations with a very low hydraulic permeability [%]
Weighted mean of hydraulic conductivity [m/s]
Climate
Average annual precipitation [mm]
Land Use
Percentage of urbanisation [%]
Percentage of forest [%]
Morphometry
Catchment area [km2]
Drainage density [km/ km2]
Highest elevation [m a.m.s.l.]
Average elevation [m a.m.s.l.]
Lowest elevation [m a.m.s.l.]
Maximum slope [%]
Average slope [%]
Minimal slope [%]
Procedure
3
Climate
Morphology and Morphometry
Soil
Land Use
Hydrogeology
Fig 3.2 Catchment Descriptors (PLATE 1992)
Procedure
3
HMIN – Lowest elevation [m a.m.s.l.]
HMAX – Highest elevation [m a.m.s.l.]
HMEAN – Average elevation [m a.m.s.l.]
The elevation data are based on a digital elevation model (50 m by 50 m cells), provided by the Water and Soil Atlas of the State of BadenWürttemberg (WaBoA) and the RIPSPool.
SLOPEMIN  Minimal slope [%]
SLOPEMAX  Maximum slope [%]
SLOPEMEAN  Mean slope [%]
Minimum, maximum and mean slopes were deduced using a digital elevation model.
back to Catchment Descriptors  Overview
Procedure
3
GEOHCMEAN –Weighted mean of hydraulic
conductivity [m/s]
GEOVLHP – Percentage of rock formations with a
very low hydraulic permeability [%]
From a 1:350,000 scale map produced by the Regional Authority for Geology, Commodities, and Mining of BadenWürttemberg (LGRB), 98 geological classes were reduced to 54 hydrogeological classes and aggregated to eight groups.
Each group was associated with a mean hydraulic conductivity of the upper hydrogeological unit. From these values, a weighted mean was produced for each catchment. From the same data, the proportion of rock formations with a mean hydraulic conductivity of less than 105 m/s was derived.
back to Catchment Descriptors  Overview
Procedure
3
SOILM – Percentage of soils with medium
infiltration capacity [%]
Examples of soils which feature a medium infiltration capacity are loamy soils and loess of medium depth.
SOILL – Percentage of soils with low
infiltration capacity [%]
The low infiltration capacity of these soils is due to their fine texture and/or the impermeability of one or more layers, as found in shallow sandy and loamy soils.
SOILVL – Percentage of soils with very low
infiltration capacity [%]
The infiltration capacity in these soils is very low because they are shallow, composed of hardly permeable material (such as clay) or have a high ground water level.
more
back to Catchment Descriptors  Overview
Procedure
3
The data for this descriptor is based on a map produced by the Regional Authority for Geology, Commodities, and Mining of BadenWürttemberg (LGRB), which shows the distribution of waterholding capacity for a theoretical soil depth of 100 cm.
Waterholding capacity is defined as “water in the soil available to plants. It is normally taken as the water in the soil between wilting point and field capacity. In this context waterholding capacity is used and is identical to the available water” (IHP/OHP 1998).
Based on the information of soil type, land use, root depth, and water logging conditions the waterholding capacity values were adjusted to the estimated effective root zone. These values were then used to compute the areal mean. A threshold mean waterholding capacity was set at 200 mm. Above this threshold, all classes were aggregated to “soils with high waterstorage capacity in the effective root zone” and its proportion was calculated.
back to Catchment Descriptors  Overview
Procedure
3
N
Fig. 2.2 Mean annual precipitation (196190) [mm]
back to Catchment Descriptors  Overview
Procedure
4
You may use the arrow buttons to view the lowflow indices in sequence or proceed to the next section.
Procedure
5
Streamflow [m3/s]
Rank
Fig. 3.3 Monthly minimum runoff values, ranked in ascending order (Elsenz at Meckesheim, No. 460, 196690)
Procedure
6
critical value
5% value
50% value
Streamflow [m3/s]
BASE
Rank
Fig. 3.4 Monthly minimum runoff values, ranked in ascending order (Elsenz at Meckesheim, No. 460, 196690)
Procedure
7
Type I Type II
Streamflow[m3/s]
Streamflow[m3/s]
Rank
Rank
Fig. 3.5 Type I and Type II curves
Procedure
8
Streamflow [m3/s]
Year
Fig. 3.6 Annual 10day minimum values of discharge and their arithmetic mean (Elsenz at Meckesheim, No. 460, 196690)
Procedure
9
Streamflow [m3/s]
Percentiles
Fig. 3.7 Flow Duration curve and deduction of the 90 percentile (Elsenz at Meckesheim, No. 460, 196690)
Procedure
10
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Data Splitting
Validation Data Set
(27 Stations)
Calibration Data Set
(56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices dependent variables)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Procedure
Model Design (Step 2)
Data Acquisition (Step 1
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Set (56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Calibration (Step 3)
BASE = b0 + bj * Xij + ei MAM(10) = b0 + bj * Xij + ei Q90 = b0 + bj * Xij + ei
Validation Data Set (27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Evaluation (Step 4)
Model Validation (Step 5)
Check for agreement between observed and estimated values
Model Application
You may chose a specific step of the procedure or click on the arrows (bottom right) to proceed in sequence
Procedure
1
By applying the regression approach we assume that the relationship between a lowflow index Y and its catchment descriptors X can be expressed as follows:
Yi = b0 + bj * Xij + ei
with i = 1, ..., N
and j = 1, ..., P
where Yi is the dependent variable and b0 and bj are constants or coefficients respectively.
Xij signifies the catchment descriptor j of the catchment i. N is the total number of data sets (samples) and P is the total number of independent variables; finally, ei is the error term (DEMUTH 1993).
P
j = 1
Procedure
2
1. The model is free of specification error
2. The data set is free of measurementerror
3. Homoscedasticity: The variance of theerror term is constant for all values ofthe independent variables
4. The error term is neither autocorrelatednor correlated with the independentvariables
5. The error term follows normaldistribution
6. The model is free of multicolinearity
Procedure
3
2. The data set is free of measurement error
The model relies on the quality of the data. We must be confident that the variables Xi and Yi have been measured accurately.
The fulfilment of this condition is problematic, particularly since low flows are usually associated with an error in the magnitude of 10 to 30% (GLOS & LAUTERBACH 1972).
back toAssumptions  Overview
Procedure
4
StandardisedResiduals
0
Predicted Values of Y
Fig. 3.8 Check for Homoscedasticity
more
Procedure
5
transformation of Y or
inclusion of polynomial terms of X in the model
may prove as a remedy (HOLDER 1985).
StandardisedResiduals
0
Predicted Values of Y
Fig. 3.9 Check for Homoscedasticity
more
Procedure
6
StandardisedResiduals
0
Predicted Values of Y
Fig. 3.10 Check for Homoscedasticity
back toAssumptions  Overview
Procedure
7
a
StandardisedResiduals
0
Time
b
StandardisedResiduals
0
Time
Fig. 3.11 Assessment of the effect of time on the error term
more
Procedure
8
a
StandardisedResiduals
0
Time
b
StandardisedResiduals
0
Time
Fig. 3.11 Assessment of the effect of time on the error term
more
Procedure
9
StandardisedResiduals
0
Values of X
Fig. 3.12 Assessment of correlation between the error term and an independent variable
back toAssumptions  Overview
Procedure
10
Frequency
0
Residuals
Fig. 3.13 Frequency distribution of residuals
Predicted cumulative probability of residuals
Observed cumulative
probability of residuals
back toAssumptions  Overview
Fig. 3.14 Probability plot of residuals
Procedure
11
Frequency
0
Residuals
Fig. 3.13 Frequency distribution of residuals
Predicted cumulative probability of residuals
Observed cumulative
probability of residuals
back toAssumptions  Overview
Fig. 3.14 Probability plot of residuals
Procedure
12
The problem of multicolinearity can be addressed by enlarging the sample size or by combining the problematic variables to forma single indicator (e.g. through principle component analysis).
The third option, excluding the problematic variable, introduces specification error to the model (see assumption 1)! Comparing the new (reduced) model with the original model can help in the assessment of the significance of this error (LEWISBECK 1986).
The seriousness of violations of the above assumptions is argued controversially in scientific literature. What can be said is that there are different degrees of robustness among the above conditions. For example, while the assumption of normality (5) is relatively robust for large samples, specification errors (1) generally cause grave problems (LEWISBECK 1986).
back to Assumptions – Overview
Procedure
Model Design (Step 2)
Data Acquisition (Step 1)
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Set (56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Calibration (Step 3)
BASE = b0 + bj * Xij + ei MAM(10) = b0 + bj * Xij + ei Q90 = b0 + bj * Xij + ei
Validation Data Set (27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Evaluation (Step 4)
Model Validation (Step 5)
Check for agreement between observed and estimated values
Model Application
You may chose a specific step of the procedure or click on the arrows (bottom right) to proceed in sequence
Procedure
1
However, it must be kept in mind that adding further variables to the model increases the risk of introducing variables whose correlation with the target value is coincidental.
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Regional Transfer Function
Pool of independent variables
Procedure
2
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Regional Transfer Function
Pool of independent variables
Procedure
3
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Regional Transfer Function
Pool of independent variables
Procedure
4
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Regional Transfer Function
Pool of independent variables
Procedure
5
Table 2 Results of the regression analysis for BASE
Click here to review the definition of BASE.
Independent variables corrected R2
AREA 0.81
AREA, SOILVL 0.82
AREA, SOILVL; SLOPEMEAN 0.84
AREA, SOILVL; SLOPEMEAN, ROOTSMEAN 0.87
AREA Catchment area [km2]
SOILVL Percentage of soils with very low infiltration capacity [%]
SLOPEMEAN Mean slope [%]
ROOTSMEAN Mean waterholding capacity in the
effective root zone [mm]
BASE = AREA*7.3*103  SOILVL*0.416 + SLOPEMEAN*3.9*102  ROOTSMEAN*2.5*103
R2 = 0.87
s.e. = 0.33
Click here to learn about the definition of R2 and s.e.
Procedure
Model Calibration Review: BASE
Streamflow [m3/s]
Rank
Fig. 3.3 Monthly minimum runoff values, ranked in ascending order (Elsenz at Meckesheim, No. 460, 196690)
more
more
Procedure
Model Calibration Review: BASE
critical value
5% value
50% value
Streamflow [m3/s]
BASE
Rank
Fig. 3.4 Monthly minimum runoff values, ranked in ascending order (Elsenz at Meckesheim, No. 460, 196690)
back
back
Procedure
Y
unexplained deviation
explained deviation
Y
X
Fig. 3.15 Components of deviation from Y
Sum of squared explained deviations
R2 =
Sum of squared total (explained
and unexplained) deviations
more
Procedure
The standard error of estimate of Y is defined as follows (LEWISBECK 1986):
where Yi, obs is the observed value of the dependent variable Y and Yi, pred is the predicted value of the dependent variable Y.The difference between Yi ,obs and Yi, pred is also called prediction error.
(Yi, obs – Yi, pred)2
s.e. =
n2
back
Procedure
Model Calibration Regional Transfer Function
6
Click here to review the definition of MAM(10).
Table 3 Results of the regression analysis for MAM(10)
independent variables corrected R2
AREA 0.84
AREA, SOILVL 0.87
AREA, SOILVL, SLOPEMEAN 0.88
AREA, SOILVL, SLOPEMEAN, DD 0.89
AREA Catchment area [km2]
SOILVL Percentage of soils with very low
infiltration capacity [%]
SLOPEMEAN Mean slope [%]
DD Drainage density [km/km2]
MAM(10) = AREA*4.5*103  SOILVL*0.4 + SLOPEMEAN*2.1*102  DD*0.1
R2 = 0.89
s.e. = 0.19
Click here to learn about the definition of R2 and s.e.
Procedure
Y
unexplained deviation
explained deviation
Y
X
Fig. 3.15 Components of deviation from Y
Sum of squared explained deviations
R2 =
Sum of squared total (explained
and unexplained) deviations
more
more
Procedure
The standard error of estimate of Y is defined as follows (LEWISBECK 1986):
where Yi, obs is the observed value of the dependent variable Y and Yi, pred is the predicted value of the dependent variable Y.The difference between Yi, obs and Yi, pred is also called prediction error.
(Yi, obs – Yi, pred)2
s.e. =
n2
back
Procedure
Model Calibration Review: MAM(10)
Streamflow [m3/s]
Year
Fig. 3.6 Annual 10day minimum values of flow and their arithmetic mean (Elsenz at Meckesheim, No. 460, 196690)
back
Procedure
7
Click here to review the definition of Q90.
Table 4 Results of the regression analysis for Q90
independent variables corrected R2
AREA 0.81
AREA, SOILVL 0.83
AREA, SOILVL; SLOPEMEAN 0.86
AREA, SOILVL; SLOPEMEAN, ROOTSMEAN 0.88
AREA Catchment area [km2]
SOILVL Percentage of soils with very low infiltration capacity [%]
SLOPEMEAN Mean slope [%]
ROOTSMEAN Mean waterholding capacity in the effective root zone [mm]
Q90 = AREA*4.9*103  SOILVL*0.5 + SLOPEMEAN*2.5*102  ROOTSMEAN*1.5*103
R2 = 0.88
s.e. = 0.21
Click here to learn about the definition of R2and s.e.
Procedure
Y
unexplained deviation
explained deviation
Y
X
Fig. 3.15 Components of deviation from Y
Sum of squared explained deviations
R2 =
Sum of squared total (explained
and unexplained) deviations
more
more
Procedure
The standard error of estimate of Y is defined as follows (LEWISBECK 1986):
where Yi, obs is the observed value of the dependent variable Y and Yi, pred is the predicted value of the dependent variable Y.The difference between Yi, obs and Yi, pred is also called prediction error.
(Yi, obs – Yi, pred)2
s.e. =
n2
back
Procedure
Model Calibration Review: Q90
Streamflow [m3/s]
Percentiles
Fig. 3.7 Flow Duration curve and deduction of the 90 percentile (Elsenz at Meckesheim, No. 460, 196690)
back
Procedure
Model Design (Step 2)
Data Acquisition (Step 1)
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Set (56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Calibration (Step 3)
BASE = b0 + bj * Xij + ei MAM(10) = b0 + bj * Xij + ei Q90 = b0 + bj * Xij + ei
Validation Data Set (27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Evaluation (Step 4)
Model Validation (Step 5)
Check for agreement between observed and estimated values
Model Application
You may chose a specific step of the procedure or click on the arrows (bottom right) to proceed in sequence
Procedure
?
1
AREA*4.9*103  SOILVL*0.5 + SLOPEMEAN*2.5*102  ROOTSMEAN*1.5*103
R2 = 0.88
s.e. = 0.21
Q90 =
Procedure
?
2
AREA*4.9*103  SOILVL*0.5 + SLOPEMEAN*2.5*102  ROOTSMEAN*1.5*103
R2 = 0.88
s.e. = 0.21
Q90 =
Procedure
?
3
AREA*4.9*103 SOILVL*0.5 + SLOPEMEAN*2.5*102  ROOTSMEAN*1.5*103
R2 = 0.88
s.e. = 0.21
Q90 =
Procedure
?
4
AREA*4.9*103  SOILVL*0.5 + SLOPEMEAN*2.5*102  ROOTSMEAN*1.5*103
R2 = 0.88
s.e. = 0.21
Q90 =
Procedure
?
5
AREA*4.9*103  SOILVL*0.5 + SLOPEMEAN*2.5*102 ROOTSMEAN*1.5*103
R2 = 0.88
s.e. = 0.21
Q90 =
Procedure
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6
Predicted values of Q90
Observed values of Q90
Fig. 3.16 Predicted vs. observed values of Q90
Procedure
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Predicted values of Q90
Observed values of Q90
Fig. 3.16 Predicted vs. observed values of Q90
Procedure
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1. The model is free of specification error
2. The data set is free of measurementerror
3. Homoscedasticity: The variance of theerror term is constant for all values ofthe independent variables
4. The error term is neither autocorrelatednor correlated with the independentvariables
5. The error term follows normaldistribution
6. The model is free of multicolinearity
You may click on any of the six assumptions, use the arrow buttons to view them in sequence, or click here to proceed.
Procedure
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(2) The data set is free of measurement error
In Step 1 we checked the data for inconsistencies and excluded those catchments for which the data did not comply with our requirements.
However, we must be aware, that even consistent data are subject to error since our lowflow measurements could only be obtained with limited precision.
Furthermore, it must be noted that the independent variables include error as well, all of which is by nature of the regression analysis attributed to the error associated with Y.
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Procedure
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Residuals of Y
Values of Y
Fig. 3.17 Residuals of Y as a function of values of Y (for the Q90 regression)
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Procedure
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N
0 80 160 km
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Fig. 3.18 Catchments that produced „outliers“
Procedure
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N
0 80 160 km
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Fig. 3.18 Catchments that produced „outliers“
Procedure
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Procedure
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Residuals
Catchment area [km2]
Fig. 3.19 Residuals as a function of catchment area (for the Q90 regression)
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Procedure
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1
Predicted cumulative probability
Frequency
0
Residuals
0
0
1
Observed cumulative probability
Fig. 3.21 Probability plot of standardized residuals (Q90 regression)
Fig. 3.20 Frequency distribution of residuals (Q90 regression)
Procedure
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Table 5 Multicolinearity of catchment descriptors
Variable combinationcorrected R2
AREA = f (SOILVL, SLOPEMEAN, ROOTSMEAN) 0.04
SOILVL = f (AREA, SLOPEMEAN, ROOTSMEAN) 0.08
SLOPEMEAN = f (AREA, SOILVL, ROOTSMEAN) 0.29
ROOTSMEAN = f (AREA, SOILVL, SLOPEMEAN) 0.25
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Procedure
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Step 3Model Calibration
Step 4Model Evaluation
Step 5Model Validation
Procedure
Model Design (Step 2)
Data Acquisition (Step 1)
Multiple Linear Regression ModelYi = b0 + bj * Xij + ei
Catchment Selection
Calculation of LowFlow Indices
Data Splitting
Calibration Data Set (56 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Calibration (Step 3)
BASE = b0 + bj * Xij + ei MAM(10) = b0 + bj * Xij + ei Q90 = b0 + bj * Xij + ei
Validation Data Set (27 Stations)
Catchment Descriptors (independent variables)
LowFlow Indices (dependent variables)
Model Evaluation (Step 4)
Model Validation (Step 5)
Check for agreement between observed and estimated values
Model Application
You may chose a specific step of the procedure or click on the arrows (bottom right) to proceed in sequence
Procedure
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Model ValidationObserved vs. Predicted Q901
Predicted Q90
Observed Q90
Fig. 3.23 Validation: predicted Q90 vs. observed Q90
Procedure
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Model ValidationModel Bias2
Deviation of the predicted
from the observed Q90 [%]
Q90 predicted – Q90 observed
* 100%
Q90 observed

Observed Q90 [m3/s]
Fig. 3.24 Validation: Relative Deviation of the predicted from the observed Q90
Procedure
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Model ValidationRelative Deviation3
Deviation of the predicted
from the observed Q90 [%]

Observed Q90 [m3/s]
Fig. 3.24 Validation: Relative Deviation of the predicted from the observed Q90
Procedure
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Model ValidationSummary4
Deviation of the predicted
from the observed Q90 [%]
Table 6 Summary of the validation results
Evaluation Deviation in % % of Data sets
Q90 MAM10 BASE
Very good <10 15 11 7
Good 1030 15 26 15
Satisfactory 3050 15 11 11
Unsatisfactory >50 56 52 67
Observed Q90 [m3/s]
Fig. 3.25 Validation: Relative deviation of the predicted from the observed Q90 (+/ 50% range shown)
Procedure
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Model ValidationConclusion5
Model Validation (Step 5)
Model Application
Procedure
Q90 = ?
Wiese
Site of the proposed hydropower plant
Fig. 1.2 Wiese catchment (No. 532)
Application
Q90 = ?
Wiese
Site of the proposed hydropower plant
Fig. 1.2 Wiese catchment
Application
Soil
Percentage of soils with high infiltration capacity 0.19%
Percentage of soils with medium infiltration capacity 0%
Percentage of soils with low infiltration capacity 99.81%
Percentage of soils with very low infiltration capacity 0%
Mean hydraulic conductivity of the soils 201.69 cm/d
Percentage of soils with low hydraulic conductivity 0%
Percentage of soils with high waterholding capacity in the effective root zone 0%
Mean waterholding capacity in the effective root zone 109 mm
Q90 = ?
Morphometry
Catchment area 206.28 km2
Drainage density 1.31 km/ km2
Highest elevation 1485.5 m a.m.s.l.
Lowest elevation 423.6 m a.m.s.l.
Average elevation 898.74 m a.m.s.l.
Maximum slope 45.54%
Minimal slope 0 %
Average slope 18.08%
Climate
Annual precipitation 1891 mm
Land Use
Percentage of urbanisation 2%
Percentage of forested area 63%
Hydrogeology
Percentage of rock formations with a very low hydraulic permeability 0%
Weighted mean of hydraulic conductivity 0.000962 m/s
Application
Soil
Percentage of soils with high infiltration capacity 0.19%
Percentage of soils with medium infiltration capacity 0%
Percentage of soils with low infiltration capacity 99.81%
Percentage of soils with very low infiltration capacity 0%
Mean hydraulic conductivity of the soils 201.69 cm/d
Percentage of soils with low hydraulic conductivity 0%
Percentage of soils with high waterholding capacity in the effective root zone 0%
Mean waterholding capacity in the
effective root zone 109 mm
AREA*4.879*103
 SOILVL*0.457
+ SLOPEMEAN* 2.506*102
 ROOTSMEAN*1.540*103
Q90 =
Morphometry
Catchment area 206.28 km2
Drainage density 1.31 km/ km2
Highest elevation 1485.5 m a.m.s.l.
Lowest elevation 423.6 m a.m.s.l.
Average elevation 898.74 m a.m.s.l.
Maximum slope 45.54%
Minimal slope 0 %
Average slope 18.08%
Climate
Annual precipitation 1891 mm
Land Use
Percentage of urbanisation 2%
Percentage of forested area 63%
Q90 = 1.29 m3/s
Hydrogeology
Percentage of rock formations with a very low hydraulic permeability 0%
Weighted mean of hydraulic conductivity 0.000962 m/s
Application
In this selfguided tour you have learned in which hydrological context you may use a multiple linear regression procedure to estimate lowflow indices at the ungauged site.
The learningbyscreening method (step by step) gave you the opportunity not only to learn at your own pace but also to simultaneously apply the method to your own data. You have seen an example of how to develop a conceptual model, translate a conceptual model into a statistical model, calibrate, evaluate, and validate the model.
You should note that the selfguided tour is a practical introduction towards the design and application of multiple regression models. A detailed discussion about the theoretical background is found in the appropriate literature. For your own exercise or review we have included the data sets used in this selfguided tour, both lowflow indices and catchment descriptors.
Application
The documents can be opened directly by clicking on the respective name
Data
Catchment Descriptors
Acronyms, means of deduction, units
Data Sources
Data pools, projects, and organisations
References
Background and previous research
Acknowledgements
Thanks to the numerous contributors
Contact Information
We appreciate your feedback
Appendices
3
Climate
Morphology and Morphometry
Soil
Land Use
Hydrogeology
Fig. 3.2 Catchment Descriptors (PLATE 1992)
Appendices
3
HMIN – Lowest elevation [m a.m.s.l.]
HMAX – Highest elevation [m a.m.s.l.]
HMEAN – Average elevation [m a.m.s.l.]
The elevation data are based on a digital elevation model (50 m by 50 m cells), provided by the Water and Soil Atlas of the State of BadenWürttemberg (WaBoA) and the RIPSPool.
SLOPEMIN  Minimal slope [%]
SLOPEMAX  Maximum slope [%]
SLOPEMEAN  Mean slope [%]
Minimum, maximum and mean slopes were deduced using a digital elevation model.
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Appendices
Appendices  Catchment Descriptors Land Use and Hydrogeology
1
Data AcquisitionLand Use and Hydrogeology3
GEOHCMEAN –Weighted mean of hydraulic
conductivity [m/s]
GEOVLHP – Percentage of rock formations with a
very low hydraulic permeability [%]
From a 1:350,000 scale map produced by the Regional Authority for Geology, Commodities, and Mining of BadenWürttemberg (LGRB), 98 geological classes were reduced to 54 hydrogeological classes and aggregated to eight groups.
Each group was associated with a mean hydraulic conductivity of the upper hydrogeological unit. From these values, a weighted mean was produced for each catchment. From the same data, the proportion of rock formations with a mean hydraulic conductivity of less than 105 m/s was derived.
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Appendices
Appendices  Catchment Descriptors Soil (1)
1
3
SOILM – Percentage of soils with medium
infiltration capacity [%]
Examples of soils which feature a medium infiltration capacity are loamy soils and loess of medium depth.
SOILL – Percentage of soils with low
infiltration capacity [%]
The low infiltration capacity of these soils is due to their fine texture and/or the impermeability of one or more layers, as found in shallow sandy and loamy soils.
SOILVL – Percentage of soils with very low
infiltration capacity [%]
The infiltration capacity in these soils is very low because they are shallow, composed of hardly permeable material (such as clay) or have a high ground water level.
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Appendices
The data for this descriptor is based on a map produced by the Regional Authority for Geology, Commodities, and Mining of BadenWürttemberg (LGRB), which shows the distribution of waterholding capacity for a theoretical soil depth of 100 cm.
Waterholding capacity is defined as “water in the soil available to plants. It is normally taken as the water in the soil between wilting point and field capacity. In this context waterholding capacity is used and is identical to the available water” (IHP/OHP 1998).
Based on the information of soil type, land use, root depth, and water logging conditions the waterholding capacity values were adjusted to the estimated effective root zone. These values were then used to compute the areal mean. A threshold mean waterholding capacity was set at 200 mm. Above this threshold, all classes were aggregated to “soils with high waterstorage capacity in the effective root zone” and its proportion was calculated.
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Appendices
Fig. 2.2 Mean annual precipitation (1961 90) [mm]
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Appendices
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Appendices
GLOS, E. & LAUTERBACH, D. (1972): Regionale Verallgemeinerung von Niedrigwasserdurchflüssen mit Wahrscheinlichkeitsaussage. Mitteilungen des Institutes für Wasserwirtschaft. H. 37. VEB Verlag für Bauwesen, Berlin.
HAAS, M. (2000): Regionalisierung des Quotienten Basisabfluss/Gesamtabfluss (Qbas/Qges) für Einzugsgebiete BadenWürttemberg. Freiburg, Germany.
HOLDER, R. L. (1985): Multiple Regression in Hydrology. Institute of Hydrology, Wallingford, Great Britain.
HUTTENLOCHER, F. (1972): BadenWürttemberg. Kleine geographische Landeskunde. Schriftenreihe der Kommission für geschichtliche Landeskunde, H. 2. Karlsruhe.
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Appendices
PLATE, E. J. (1992): Regionalisierung in der Hydrologie. Deutsche Forschungsgemeinschaft. Mitteilung XI der Senatskommission für Wasserforschung. Hrsg. KLEEBERG, H.B. Cambridge, NY.
SCHREIBER, P. (1996): Regionalisierung des Niedrigwassers mit statistischen Verfahren. Freiburger Schriften zur Hydrologie, Band 4. Freiburg, Germany.
VILLINGER, E. (1982): Hydrogeologische Aspekte zur geothermischen Anomalie im Gebiet UrauchBoll am Nordrand der Schwäbischen Alb (Südwestdeutschland). Geologisches Jahrbuch, H. 32, 342.
WUNDT, W. (1953): Gewässerkunde. Berlin, Göttingen, Heidelberg .
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Appendices
Thanks is due to individuals for their support in producing the selfguided tour:
Volker Abraham, for providing the cartographic skills, maps and the layout of the front page of the selfguided tour.
Kerstin Stahl, for calculating the flow regimes.
Helmut Straub, Environmental Agency, Regional Office, State of BadenWürttemberg, Germany, for providing the flow data of BadenWürttemberg and the permission to use the data on this CDROM.
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Appendices
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Appendices