Please add!!!

Example one

This example summarises some of the key points of the work published by Young (2002); this is a useful and accessible paper that describes both the practical example - emphasised here - as well as a more philosophical discussion on the concepts of Data Based Mechanistic ([DBM]?) modelling - it's definately worth reading the whole paper.

Problem Description

To generate flood forecasting procedures (prediction of future flow) in real-time using stochastic, online methods that predict not only the most likely flow pprediction but also the level of uncertainty associated with the prediction. The uncertainty information should provide valuable information to aid decision-makers.

Available data

Rainfall - flow data gathered from the 261 km² River Hodder catchment (NW England). Estimation series gathered in January:

• 720 hourly values of rainfall (Thiessen polygon average over catchment).

• corresponding 720 hourly values of flow (in same units as rainfall - divide guaged volumetric flow by catchment area).

Used Software

Captain (TM) toolbox used with Matlab(R) software. The Captain toolbox provides a series of functions for data-based parameter estimation and data analysis. see: http://www.es.lancs.ac.uk/cres/captain/

Methodology

The case study uses (and emphasises some of the advantages of) the Data Based Mechanistic (DBM) approach to modelling environmental systems.

The approach procedes in the follows fashion:

• 1) The training data is used to identify a suitable [transfer function]? model structure, and to estimate the transfer function model parameters. In keeping with DBM philosophy, the model must be both:

  • parsimonious i.e., use as few parameters as possible to explain the observed data.

  • physically meaningful i.e., the model must possess a physical interpretation that makes sense in the context of the modelled process (for example here, a second order model was identified that could be decomposed into two parallel first order models representing quick-flow and slow-flow components.

The parameters of the transfer function model together with the parameters of an effective rainfall function and an Auto Regressive (AR) noise model were estimated using a combination of Matlab (R) and Captain (TM) functions.

• The identification and estimation process generated a model with:

  • two numerator and two denominator parameters to represent the relationship between effective rainfall and flow.

  • a third-order AR model to represent the relationship between white noise and the model residuals.

  • an estimate of the parameters to describe the effective rainfall function (a function designed to describe the non-linearity between observed rainfall and flow using flow as a surrogate for the catchment storage).

The resulting model looks like this: ..........eq(1)

Where y_t is flow (at time-step t), u_t is the effective rainfall input, r_t is the observed rainfall, c is a scalling coefficient, gamma (anyone now how to do greek symbols?) is the power law parameter for the effective rainfall function, and finally xi represents a stochastic noise input (modelled by the 3^rd order AR filter applied to a white noise input).

• 2) The transfer function component of eq(1) was decomposed into fast and slow components i.e., the second order transfer function was decomposed using partial fraction expansion into two first order transfer functions connected in parallel. The two first order transfer functions can be represented as:

...........eq(2)

These where then incorporated into a state space scheme so that eq(1) could be realised using state space methods that permit powerful recursive approaches such as Kalman Filtering. In state space form it is possible to represent eq(1) using the parameters of eq(2). This looks like:

............eq(3)

Here, the noise inputs sigma₁,_t and sigma₂,_t , are assumed to be gaussian.

• 3) Having defined the model in state space form, a recursive form of the Kalman filter can be used for data assimilation. This set of recursive equations looks like:

...............eq(4)

Where the only variables we havn't come across before are: theta, the variance of the (gaussian) observation noise, and Q_r, the noise variance ratio (NVR) matrix. The NVR matrix contains only diagonal entries, with each defined as the standard deviation of the noise applied to each state (x₁ and x₂) divided by the standard deviation of the observation noise (theta). These define the characteristics of the stochastic disturbance applied to each state variable and therefore specify the level of uncertainty in each state relative to the measurement uncertainty.

under construction!!

Results

geography --Fri, 30 Nov 2007 14:51:35 +0000

how does a flood look like?

geography --Fri, 30 Nov 2007 14:51:41 +0000

how does a flood look like?

ho does a flood look like --Wed, 25 Mar 2009 16:54:23 +0000