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See also the full list of methodologies for uncertainty analysis. Follow the tree (strictly, graph) from top to bottom. Rounded nodes are questions; each arc leaving such a node represents a possible answer. Question nodes and answers where these are more complex than "yes" and "no" link to a page of notes. Rectangular nodes represent methods or sets of methods, and link to pages describing those methods. It should be noted that not all of the rectangular nodes appear at the leaves; in some situations indication is made of the need for intermediate processing.
The decision tree is intended to help with the process of choosing a method for quantitative uncertainty analysis. Uncertainty also has qualitative aspects. Both may be important to decision making and should be recorded wherever possible, for example using the NUSAP methodology.
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From unknown Thu May 3 10:06:36 +0100 2007 From: Date: Thu, 03 May 2007 10:06:36 +0100 Subject: Kalman Filter in Uncertainty Analysis? Message-ID: <20070503100636+0100@www.floodrisknet.org.uk>
From unknown Thu May 3 10:17:20 +0100 2007 From: Date: Thu, 03 May 2007 10:17:20 +0100 Subject: It is difficult to understand the concept of "degree of nonlineary?" , Is there any reference? Message-ID: <20070503101720+0100@www.floodrisknet.org.uk>
From dleedal Fri May 25 17:01:53 +0100 2007 From: dleedal Date: Fri, 25 May 2007 17:01:53 +0100 Subject: Kalman Filter in Uncertainty Analysis? Message-ID: <20070525170153+0100@www.floodrisknet.org.uk> In-Reply-To: <20070503100636+0100@www.floodrisknet.org.uk>
A key concept when considering uncertainty in hydrology is the move away from thinking about model output or observations as deterministic and instead see these values as continuous random variables. For example, the model forecast of river level at a particular location in, say, 10 hours time may be 6.5m; however, the observations that went into generating this forecast and the model parameters themselves are all uncertain i.e., the values are best thought of in terms of a mean and some confidence bounds. Given this, the forecast should more correctly be specified as a random variable; for example, 6.5m plus or minus 1m.
The Kalman filter is a method of working with observations and models where it is explicitly assumed that the inputs and outputs are random variables. For the simple Kalman filter, variances are specified for the unobservable noise processes that are assumed present in the model output and the observations. A state covariance matrix is generated by the Kalman filter during the recursive calculation of the state vector. If the process adheres to specific assumptions about the model structure and the nature of the noise, then this matrix provides the variance of the state estimates (otherwise it provides an approximation). Given this information, the model output(s) can be expresses as an expected value with an associated variance i.e., a random variable.
In summary, any hydrological modelling study that makes use of the Kalman filter is considering uncertainty (in the form of process and measurement noise) and generates output in the form of a mean value with an associated variance. This information can be used to attach confidence bounds to the model output thus fulfilling one of the key requirements of an uncertainty study.
From dleedal Fri May 25 17:39:19 +0100 2007 From: dleedal Date: Fri, 25 May 2007 17:39:19 +0100 Subject: It is difficult to understand the concept of "degree of nonlineary?" , Is there any reference? Message-ID: <20070525173919+0100@www.floodrisknet.org.uk> In-Reply-To: <20070503101720+0100@www.floodrisknet.org.uk>
Fundamental properties of a linear model are (1) it obeys the rule of superposition: the sum of the model output for a number of separate inputs is the same as the output that would result if the inputs were summed before applying to the model; (2) the response of a model to an input stimulus should not be dependent on any other external factor (eg. the time that the input was applied).
In hydrology these properties never hold; for example, the sum of the river flow resulting from two separate rainfall events will not be the same as the river flow resulting from an event equal to the sum of the two original rainfalls.
That said, many linear modelling strategies hold up reasonably well for small deviations away from the ideal assumptions of a linear process (often helped by additional mechanisms such as time-varying gain adjustments etc). However, for very nonlinear systems alternate modelling strategies are required. Within the Kalman filtering approach to modelling this includes the extended and ensemble KF.
The degree of nonlinearity referred to here describes the subjective decision made by the modeller as to when to bring in these additional approaches; the benefit being better performance when the process is nonlinear, the drawback being extra numerical complexity. Ultimately, this is a four-way balancing act between choosing a good simple model that works well; a bad simple model that can't explain the data; a bad complex model that is impossible to identify; a good complex model that describes the process well.