The hydrological subjects are on very different scales and with different aspects.
The research is within the field of statistics as well as within hydrology and in all of the research projects, empirical measurements are used to estimate unknown parameters.
1.1.1 Paper [A]
Conditional parametric models for storm sewer runoff
In Paper[A], the data originates from a waste water treatment plant in Denmark.
The treatment plant is the outlet of a sewage system with a watershed of 10.89
km2. The sewage system is built in the traditional manner with pipes and node points for pumping stations. Figure 1.1 shows a sketch of a sewage system.
Pumpstation
Pumpstation
Pumpstation Pumpstation
Pumpstation Waste water Treatment plant
Figure 1.1: The sewage system
The input data is precipitation, measured at the waste water treatment plant.
The output data is excess flow data from the last pumping station before the treatment plant.1 The goal is to predict the flow in the last pumping station and use the predictions for on-line automatic control in the waste water treatment plant. Black box models have proven to provide good predictions in hydrological systems e.g., Carstensen et al. (1998) and thus such methods were tested. Linear FIR and linear ARX models were unsatisfactory and thus non-linear methods were used. The non-linear effects are mainly due to two factors; seasonality in the balance and saturation/threshold in the pipe system. Large parts of the measured precipitation do not enter the sewage system but evaporate or infil-trate into the ground. The infiltration rate depends on several factors and the wetness of the root zone plays an important role. Similarly, many factors affect the evaporation and especially the temperature plays a major role. Because of seasonal variations of temperature, plant growth and other physical factors, the variation of infiltration and evaporation varies seasonally and consequently the water balance does too. The other non-linear effect, the saturation/threshold is
1The base flow in the sewage system, also known as dry weather flow, does not originate from rainfall. Consequently, the base flow is subtracted from the flow data and the resulting flow, the excess flow is used in the modelling approach.
1.1 Overview of papers included 3
Figure 1.2: Fnjoskadalur. (Photo Oddur Sigurdsson)
a consequence of limited capacity of the pumps in the sewage system. When a large amount of water enters the system the pump stations in the node points cannot serve all the water. Thus, water accumulates behind the pumping sta-tions waiting to be served. During a very heavy rain storm the water enters the treatment plant with a delay, as compared to a normal rain storm. These two factors were taken into account in a conditional parametric model. Condi-tional parametric models are models where the parameters change as a function (conditioned) of some external variables. In this case the parameters changed as a function of seasonality and as a function of water quantity in the system.
The method of conditional parametric modelling is a significant improvement compared to traditional linear modelling.
1.1.2 Paper [B]
Parameter estimation in a stochastic rainfall-runoff model
The subject in Paper[B] is a classic topic in hydrology, the rainfall-runoff re-lationship. The data originates from a 1132 km2 mountainous watershed in Iceland. Figure 1.2 shows a part of the watershed. It shows the valley Fn-joskadalur and the river Fnjoska. The altitude range is about 1000 meters, stretching from 44 m to 1083 m. More than 50% of the watershed is above 800
−40 −2 0 2 4 6 8 10 0.2
0.4 0.6 0.8 1
Figure 1.3: A sigmoid function with center 4 and scale parameter 1.
meters. The water level gauge is down in the valley, close to the river mouth.
One meteorological observatory is in the watershed and it is located in the valley.
Thus no meteorological observatory is located in the highlands nor close to the watershed in the highlands. The scarcity of meteorological observatories is well known in sparsely populated areas around the world, especially in mountain-ous areas. Because the watershed is large, and with a large altitude range, the weather condition in the watershed can be very different depending on location in the watershed. Furthermore, during winter, snow accumulates, and melts in spring, resulting in large spring floods in the river. Despite of limited data, a rainfall-runoff relationship was required. It was chosen to develop a stochastic conceptual model, and it is found necessary to use a stochastic model since too many effects are unknown and/or not measured.
The system is modelled in a continuous time by using stochastic differential equations. The model structure is kept as simple as possible and with as few parameters as possible in order to be able to use the data to estimate the pa-rameter values. The stochastic differential equations describe a reservoir model with a snow routine. The watershed is not divided into elevation zones, but a smooth threshold function is used in the snow routine both for accumulation and melting, using positive degree day method. The smooth threshold function is the sigmoid function,
φ(T) = 1
1 + exp(b0−b1T) (1.1)
whereT is temperature,b0andb1are constants. The constantb0is the center of the sigmoid function andb1 controls the steepness. Figure 1.3 shows a sigmoid function with centerb0= 4 and scaleb1= 1.
In Figure 1.4 the modelling principle is illustrated. Precipitation enters the system and is divided into snow and rain, depending on the temperature. It can be rain only, snow only and partly snow and rain. The rain enters the first
1.1 Overview of papers included 5
Snow
Reservoir Precipiation
Rain Snow
k2 φ
repr. base flow
Σ Runoff
Melt
k1, time contant Reservoir
φ: The sigmoid function
φ
infiltration
Constant
Figure 1.4: The modelling principles.
reservoir which delivers water partly directly into the river and partly into the second reservoir that finally delivers the water into the river. The snow, how-ever, enters the snow container and stays in the snow container until it melts, and is then delivered into the first container. As mentioned earlier, the same smooth threshold function is used for precipitation division and snow melting.
Thus, at a same time a precipitation can be divided into partly rain and partly snow while some ratio of the snow is melting. This modelling method computes precipitation division and melting on an average basis. This works well, partic-ulary since no meteorological observatory is located in higher altitudes so that temperature lapse rate and precipitation lapse rate can be estimated and used as a basis for elevation division.
During the winter the snow container, because of its nature, swallows the snow and accumulates it until the temperature rises and the snow begins to melt.
During the melting, the snow container delivers water into the system until the snow container is emptied. During summer, the snow container is inactive.
Consequently, the snow routine causes the system to be both non-linear and stiff and, therefore, difficult to cope with numerically.
The parameters are estimated by using the program CTSM (Continuous Time
Data:
Input/Output
updating with
Optimize Parameters A
Model
Output=prediction
Model Data:
Optimize updating Input
without Paramters B Output=simulation
Figure 1.5: The input and output for different optimization principles.
Stochastic Modelling) (Kristensen et al. 2003). The estimation method is the maximum likelihood method and the principle of extended Kalman filter is used. Three different filtering methods or ODE solvers are implemented and it depends on the system’s stiffness which one is the ”optimal” to use. Fur-thermore, it is possible to choose to optimize the parameters with or without the traditional Kalman filter updating. Figure 1.5 illustrates the optimization principles. The optimized parameter values will not be the same we call them A and B. Optimization with Kalman filter updating results in a parameter Set A and those parameters are optimal for making model prediction. Contrarily, optimization without Kalman filter updating results in parameter Set B which is optimal for making model simulations if the true model exists.
1.1 Overview of papers included 7
1.1.3 Paper [C]
Assessment of serious water shortage in the Ice-landic water resource system.
The topic in Paper[C] is a risk assessment of a water shortage in a hydropower plant. The data originates from the river Tungna´a in southern Iceland, measured at Mariufossar. The watershed is 3470 km2, of which 555 km2 is glacier. The data consist, of daily values of discharge over a period of 50 years. Figure 1.6 shows a hydropower plant and the corresponding reservoir. The water in the reservoir is led to the hydropower plant in pipes, located in the mountain, and if necessary bypass flow is led into the canyon which is on left side of the reservoir.
Figure 1.6: The hydropower plant Burfell and its reservoir. (Photo: Oddur Sigurdsson)
When a hydropower plant is designed, two major quantities are taken into con-sideration. One is the regulated flow,Qreg, which is the flow of water delivered into the hydropower plant for electricity production. The other quantity is the size of the reservoir, V. For a given regulated flow Qreg and for a given dis-charge series, a volume V exists, which is the smallest volume that can secure regulated flowQreg. The largest possibleQregwhich can be served without any water shortage is the mean value of the discharge. The relationship (Qreg,V) is known as the regulation curve. Figure 1.7 shows the regulation curve for the discharge series at Mariufossar.
40 50 60 70 80 0
0.5 1 1.5 2 2.5x 109
Regulated flow m3/s
Volume m3
Regulation curve
Figure 1.7: The regulation curve.
All available data are used to construct the regulation curve, and using a differ-ent discharge series will lead to another regulation curve. Thus, for a given point (Qreg, V) on the regulation curve, the risk of a water shortage is zero using the data series which was used to construct the regulation curve. However, when the hydropower plant has been designed by choosing (Qreg, V), the future discharge series will not be exactly the same as the past discharge series and, thus, water shortage might occur. Consequently, a stochastic model must be developed in order to construct a simulated discharge series to be used for risk assessment.
It is very important to have an estimation of the risk of water shortage in the lifetime of the hydropower plants, about 30-60 years.
A stochastic periodic model in the spirit of Yevjevich, (Yevjevich 1976) was developed and the available data were used to estimate the parameters in the stochastic model. The stochastic model is then used to simulate flow series in order to estimate the water shortage probabilities. The goal is to estimate prob-abilities of rare events and it turned out that it was necessary to simulate the daily flow for 50000 years in order to achieve a stable estimate of the risk of water shortage. Using the simulated data it was concluded that the water short-age probabilities can be described by the Weibull distribution. However, even though the distribution of the water shortage probabilities is known, simulations are required in order to estimate the parameters in the distribution.