Projection in Hilbert space for flood forecasting
Extended Abstract: A proper forecast of floods based on forecasted or occurring rainfalls is of great necessity for communities, government agencies and local authorities for the proper warning and management of the flood event before and during its occurrence. This abstract presents a new approach for flood forecasting of the Leith and Pomahaka Rivers in Otago, New Zealand. This methodology is based on the projection of hourly flows of a river at a desired site on the span of hourly rainfall data and/or previous flows of the same flow site or other flow sites. The projection theorem in Hilbert Space guarantees that the estimated parameters will produce a model with the least mean square error. The streamflow at time t “Qt” (noted here as Q) is considered an element of a Hilbert space H, and other hydrologic variables, on which Q to be projected, are considered to constitute a closed sub-space S of H. These “other” hydrologic variables can be lagged flows Qi,t-L and rainfalls Ri,t-L (noted here as Xi) at sites i, i=1 to N, and lagged L hours before time t, where L can range from zero to any integer Lmax. The projection of Q in the sub-space S is denoted by Q ̂ , where Q ̂ ϵ S. In Hilbert space, < Q - Q ̂ , Xi > should equal zero for all Xi ϵ S, i=1 to N to produce the predictor Q ̂ with the minimum least squared error. .This is a consequence of Q - Q ̂ being orthogonal to Xi By the linearity of Hilbert space, this results in the system of N equations <Q,Xi> = <Q ̂,Xi>, i= 1 to N.. Q ̂ can be a non-linear function of the “other” hydrologic variables X, but in this application, the linear projection of Q on the span of S (Xi, i=1 to N) is used (Q ̂=∑_(i=1)^N▒〖∝_i X_i)〗. For this case, a system of N linear equations are produced which are solved simultaneously to obtain the corresponding coefficients for the “other” hydrologic variables Xi, i = 1 to N. Note that in Hilbert space <X,Y> = E[XY]. The Leith River goes through Dunedin city in New Zealand, with a small catchment of about 45 km2 with two rainfall sites within its catchment and one flow site in the city near the river’s outlet to the harbor. To the contrary, the Pomahaka catchment is much bigger, with rural aspects and mainly agricultural activities. The Pomahaka catchment is about 1871 km2, and has several flow and rainfall sites. The model for the Pomahaka is to forecast flows at Burkes ford site, which is the closest to its outlet to the Clutha River. Only 4 rainfall sites have been used for the Pomahaka catchment due to data availability. Hourly flows and rainfall data for the two catchments have been investigated and 25 high events of the Leith River at the University Foot Bridge in Dunedin, and 24 high flow events of the Pomahaka River at Burkes Ford, have been identified for the modeling process. Cross correlation analyses have been carried out between the river flows and other variables (rainfalls and/or previous flows) on which these flows will be projected. These cross correlations were utilized in estimating the lags between these variables and the modeled flows in the projection process. Results of the modeling process for producing a 10 hr forecast model for the Pomahaka River, achieved an overall value of 0.8 for R squared and 0.9 for Filliben correlation coefficient, while they were 0.8 and 0.89, respectively, for a 12 hr forecast model. This is not a significant difference, and of course it is preferred to have a longer forecast. The model underestimated several events, especially some big events, as its accuracy for simulating the maximum recorded flow was 80%. However, this might be acceptable for a forecast, as variability is usually high and it is hard to forecast precisely. The model performed well for many other high events. Non linear relations can be included in the modeling process between the projected flows and the variables on which they are projected. Future implementation of non linear relationship could result in an overall improvement of the model’s performance. The produced model is easy to apply for real forecast, doesn’t require a lot of information on the hydrological aspects/characteristic of the catchment areas, and can be used for real time forecast during an ongoing event or long term forecast based on forecasted rainfall over the following few days.... [Show full abstract]
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