Water Research 37 (2003) 429–435 Technical note Forward and backward uncertainty propagation: an oxidation ditch modelling example A. Abusam*, K.J. Keesman, G. van Straten Systems and Control Group, Wageningen University, Bomenweg 4, 6703 HD Wageningen, The Netherlands Received 16 March 2001; received in revised form 1 June 2002; accepted 27 June 2002 Abstract In the field of water technology, forward uncertainty propagation is frequently used, whereas backward uncertainty propagation is rarely used. In forward uncertainty analysis, one moves from a given (or assumed) parameter subspace towards the corresponding distribution of the output or objective function. However, in the backward uncertainty propagation, one moves in the reverse direction, from the distribution function towards the parameter subspace. Backward uncertainty propagation, which is a generalisation of parameter estimation error analysis, gives information essential for designing experimental or monitoring programmes, and for tighter bounding of parameter uncertainty intervals. The procedure of carrying out backward uncertainty propagation is illustrated in this technical note by working example for an oxidation ditch wastewater treatment plant. Results obtained have demonstrated that essential information can be achieved by carrying out backward uncertainty propagation analysis. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Wastewater; Activated sludge; Oxidation ditch; Modeling; Uncertainty analysis 1. Introduction Uncertainty analysis is a very important step in the model building process. It contributes directly to the reliability and applicability of the developed mathema- tical model. It is mainly concerned with the effect that various sources of uncertainty have on the model output. Model sources of uncertainty can be in: (i) model inputs, (ii) model parameter values, (iii) initial state conditions, and in (iv) model structure. The method illustrated in this paper is particularly applicable to types (i)–(iii) of sources of uncertainty. In the field of water technology, uncertainty analysis or error propagation—if it is carried out at all—is usually executed in one direction: forward direction. That is, starting the uncertainty analysis from a given (or assumed) parameter subspace, defined in terms of ranges or distributions, and moving towards the corresponding distribution of the output or objective function. However, the backward uncertainty propaga- tion is rarely performed. Clearly, the backward un- certainty propagation is the reverse of the forward uncertainty propagation, and it can be seen as a generalisation of parameter estimation error quantifica- tion from given experimental data. It can be used for obtaining, in a systematic way, essential information about which part of the parameter space, or which parameter combinations, contributed mostly to the part of the distribution function that will be analysed. For instance, which inputs, parameters or initial conditions lead to extreme or off-normal process conditions? Such information will be the important ingredients that help the researcher or the plant manager in designing and carrying out a monitoring programme. Through a monitoring programme, one usually wants to find out accurate values for model parameters or initial condi- tions that are suspected of causing or contributing to a certain part of interest in the distribution function found *Corresponding author. E-mail address: abdallah.abusam@user.aenf.wau.nl (A. Abusam). 0043-1354/02/$-see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII:S0043-1354(02)00288-9