Blue arrow: basketball basket.
Red arrow: top of a so-called wall, a skatebaoarding obstacle.
Photo by Günter Auzinger.
For modelling the water flow, we sketch a cross section of the river and make some simplifying assumptions.
Due to the reduced flow speed in the area subject to flooding, we neglect the increased width of the Danube but assume its regular width of 252m (source) and a rectangular shape. The highest water level on June 4, 2013 was 9.30m (source).
The most difficult part in modelling is the flow speed. The flow speed of the Danube for regular water levels (which means around 4 metres in Linz) is between 1.8 and 2m/second. For high water levels, flow speeds of 3.5 to 4.5 m/s are observed. Michael Fürst reports flow speeds of 4m/s during the high water wave of August 2002 in the East of Upper Austria (in his diploma thesis: Überflutungsraumanalysen anhand von Beispielen an der österreichischen Donau – Beitrag zur Verbesserung einer Evaluierungsmethodik, 2011. Institute of Water Management, Hydrology and Hydraulic Engineering, University of Natural Resources and Life Sciences, Vienna.)
Of course, the flow speed is lower close to the bottom due to the roughness of the river bed. Therefore, if we use width=252m, depth =9.3m, and a velocity=4m/s, we should get an upper bound for the flow rate. Multiplying and rounding gives 9400 cubic metres (or 9.4 million litres) per second. Can this be true?
Result validation
It is not that easy to obtain Danube flow rates for Austria. At least the following (semi-)governmental organisations are candidates: The water management departments of Upper and of Lower Austria, the traffic ministry (responsible for shipping regulations) and Verbund Hydro Power, which operates electric power plants and is therefore responsible for controlling the watergates.
Mauthausen (downstream of Linz and also downstream of the influent river Traun) reports a flow rate of 10.000 cubic metres/second on June 4 (source ).
A telephone call at the Upper Austrian governmental department yields between 8900 and 9000 cubic metres/second for the highest flow rate in Linz in June.
Relevance for model calibration
Why do I bother you with these details? At least for two reasons.
First, to show that simple models can deliver quite reasonable results. In finance, the preferred mathematical model for a specific task should be as simple as possible and as complicated as necessary.
Second: The parameters of your mathematical model are often not easily accessible. You might have data of good quality for simpler situations (regular water levels in the example, vanilla instruments in finance) and might have measurements only for proxies (Mauthausen instead of Linz, market price of one structured instrument instead of a similar one.)
Especially for the extreme cases, model validation is an essential task in modelling and simulation.
Risk modelling continues on next mathematics Wednesday
