Figure 3.8: Laser Doppler velocity meter
(Figure is from http://www.efunda.com/designstandards/sensors /laser doppler/laser doppler effect theory.cfm).
Q
h0 h
Figure 3.9: Water level data, flow data and a rating curve.
enough so this difference is negligible. But when that is not the case, enough data is usually not available to determine the complete loop.
3.3 River ice
In a colder climate icing in rivers disturbs water level measurements. The fol-lowing section is based on; (Bengtson 1988), (Chow 1964), and (Rist 1962).
3.3.1 Construction of river ice
When the temperature of the surface water has dropped to the freezing point, net heat loss due to the atmosphere will cause ice production. Ice formation frequently starts with the development of many tiny disk formed crystals called frazil as shown in Figure 3.10, left. Frazil ice particles, frequently collected by
Figure 3.10: To left frazil ice and to right is ice pan.
(http://www.clarkson.edu/ htshen)
adhesion, form larger masses which move along with the current. As the ice content of the water increases, the water becomes oily or milky in appearance and the viscosity of the water increases.
Ice rapidly forms on the surface of most flowing rivers. At first this consists of agglomeration of broken surface crystals and frazil ice which unite to form round pans. Figure 3.10, right shows an ice pan. These pans grow by accretion and the open water between them becomes smaller. When the ice cover on the surface is complete, the river regime changes from open water flow to flow beneath the ice cover. However, the same flow is to be carried and as a consequence the water level is increased.
Ice can also form on underwater objects. This is called anchor ice, shown in Figure 3.11. The coating of anchor ice may be several inches thick and may then grow more rapidly on sharp corners. This anchor ice may eventually dam up the stream. Thus, it is possible to develop a staircase of a series of small ice dams with some still water trapped behind them. The increased viscosity due to the ice content, the damping of turbulent eddies, and the rise of the river bed due to the formation of anchor ice also cause an increase in the river stage.
3.3 River ice 41
Figure 3.11: Anchor ice. (http://www.clarkson.edu/ htshen)
3.3.2 Melting of river ice
The melting of ice in an icecovered river causes the river stage to rise in the spring. At one point the ice cover begins to break and to move with the stream.
This initiates a chain reaction so that a complete ice cover can be removed in a matter of hours.
If the water rises fast the ice cover can be fragmented and forced to move while it still has almost its full strength. At particular locations depending on river morphology ice jams are formed. Very high water levels are caused by ice jams present during break-ups.
3.3.3 Ice reduction
The term ice reduction is used for the complete process of correcting the observed water levels so that these reach the levels that would have been observed if there were no ice present in the entire river system. The difference between the water level with ice present and the water level during summer conditions at the same discharge, is referred to as backwater from ice formation.
In countries with a predominantly continental climate, the river state can be catagorized into following phases:
1. Freezing-over period
2. Ice cover period from early to late winter and 3. Breaking-up period in early spring
4. Ice free summer season
In some countries this cycle is so stable that the beginning and the end of each of its phases or periods may be predicted with an error of a few days only. In such circumstance a wintertime rating curve can be made and used during the winter.
In Iceland, this regularity is more or less absent. Freezing-over may start during a cold period in early winter, but before the rivers are frozen the temperature rises, the ice is broken up without any intervening ice cover period and an ice free period may then follow. Each winter this may be repeated several times.
The winter period in Iceland is rather long and the weather is changing. It is sometimes very hard to detect when an icing begins and if there is still ice in the river after a thawing period.
The Hydrological Service (HS) has no automatic procedures for detecting sus-picious periods. All series are treated manually, and it is up to the operator to detect the period. At most gauging stations in Iceland, the Hydrological Service (HS) does not employ a local observer to follow and report on the building-up of ice jams in the river. Consequently, the usual situation is that no information from local sources on potential ice problems is available. In this case the ex-pertise of the hydrologist working on ice correction is the only thing to rely on.
The usual procedure is to use weather data from a nearby meteorological station showing both temperature and precipitation. The water level data is compared with the meteorological data and usually backwater effects are identified as an
3.3 River ice 43
abrupt increase in water level during periods where this would not be expected due to cold and dry weather. Sometimes this could be very difficult procedure especially if the backwater effects are not very remarkable and only last for short periods of time. Also, the effects of frazil ice and in particular anchor ice can be very difficult to detect, but ice problems due to these types of ice formation are quite common in Icelandic rivers. Discharge measurements during winter conditions are performed once every winter.
Finally, Figure 3.12 shows a graph of ice corrupted water level data in the River Fnjoska in northern Iceland in late winter 1996. Furthermore, the temperature and precipitation at Akureyri, also in northern Iceland, is shown. The temper-ature in January is cold and at the beginning of February the water level rises drastically and stays high until a long warm period in March sets in. Then the water level falls.
The problem of river ice is, indeed, one of the reasons for applying stochastic models in hydrology.
Chapter 4
The stochastic dynamic modelling
This PhD thesis consists of stochastic modelling of hydrological systems. Four projects have been studied and the results are illustrated in Papers [A], [B] [C]
and [D].
Journal papers are often written in a compact form as only a limited number of pages are allowed. In this section the modelling approaches will be further described.
4.1 Model categorization
Modelling approach may be grouped into three categories; Black box models, white box models and grey box models.
Black box models are purely data based model. They approach the system in terms of input and output with the internals hidden in a black box.
White box models are the opposite of the black box ones. The internal of the system is fully known. In order the develop a white box the modeller must know
the system in details and occasionally the model becomes over-specified.
Grey box models are placed in between black- and white- box modelling ap-proaches. A model is viewed as grey box if physical knowledge about the sys-tem is used along with the data. Thus the model is not completely described by physical equations but the equations and the parameters are physically in-terpretable.
In this project both grey box models and black box models have been used.
The choice of model depends on the aim and on the available knowledge in each situation.
Models can be developed in continuous time and discrete time. As the time is continuous all the physical systems studied are defined in continuous time.
However, the data are sampled in discrete time and occasionally it is more convenient to describe the model in discrete time. The dynamical data series studied are called the time series.
The systems, or the models, can be grouped into linear models and non-linear models, black box systems can be both linear and non-linear.