has been defined in a number of papers see for example Young (1974), Young and Beven (1994) or, for an online source: Young (2001)

The assumption being that, considered at a catchment scale, the realationship between rainfall (r_t) and flow (y_t) can be decomposed into two components:

• a non-linear relationship between rainfall and effective rainfall (u_t). In this relationship effective rainfall is a function of rainfall, flow, catchment storage (which is unobservable), temperature T_t, evaporation E_t, and possibly various other factors.

• a flow routing component describing the relationship between effective rainfall and flow. This relationship can often be closely approximated with a linear equation; for example, a discrete time transfer function.

This process is illustrated in Figure 1.

........Fig(1)

The inclusion of y_t in the non-linear function is important. Numerous studies have shown that flow data alone often contains enough information to justify its use as a surrogate for the other variables in the function. This is useful as the other variables are often either impossible or expensive to observe.

If applying a Data Based Mechanistic [(DBM)]? approach to modelling, it would be seen as advantagious to produce a simple model with parameters identified from observational data as far as possible. To this end several studies (see Young and Beven (1994), Young and Tomlin (2000)) have adopted the simple form:

............Eq(1)

In Eq(1) c is a scaling coefficient chosen to ensure the volume of effective rainfall is equal to the flow volume over the estimation period, and gamma (the power law constant) is estimated from the data. The estimation process usually takes place in two stages:

1. Firstly the rainfall flow data is used with a State Dependent Parameter algorithm to produce a look-up table (x-axis = y_t, y-axis = f(y_t)) describing the shape of the non-linearity.

2. The shape of the look-up table is approximated by a parametric function of y_t; in this case, a power law.

The aim of this approach is to provide a fast and often effective method for removing the main non-linearity within the rainfall flow relationship. This speed and simplicity is useful for applications such as real-time flood forecasting where data must be processed online. Once this non-linearity has been removed, the process of flow estimation can be greatly simplified with recourse to linear modelling and forecasting theory such as the use of Kalman filtering.

As well as Eq(1), various other transforms are used to convert rainfall to effective rainfall, this is also an ongoing area of research (please add examples...).

References

• Young, P.C. 1974 Recursive approaches to time-series analysis. Bull. of Inst. Maths and its Applications, 10, 209-224.

• Young, P.C. & Beven, K.J. 1994 Data-based mechanistic modelling and the rainfall-flow nonlinearity, Environmetrics, 5, 335-363.

• Young, P. C. & Tomlin, C. M. 2000 Data-based mechanistic modelling and adaptive flow forecasting. In Flood forecasting: what does current research offer the practitioner? (M.J. Lees & P. Walsh eds.), pp 26-40. BHS Occasional paper No 12, produced by the Centre for Ecology and Hydrology on behalf of the British Hydrological Society.

• Young, P. C. 2001 Advances in real-time forecasting. Centre for Research on Environmental Systems and Statistics, Lancaster university, Report no. TR/176.