Chemical Engineering Journal 84 (2001) 309–314 Reconciliation of censored measurements in chemical processes: an alternative approach V.G. Dov` ı ∗ , C. Solisio DICheP, “G.B. Bonino” Genoa University, Via Opera Pia 15, Genova 16145, Italy Received 28 April 1998; accepted 13 November 2000 Abstract The importance of considering the censoring of measured data in the reconciliation of process flow rates has been shown in a previous paper [Chem. Eng. Sci. 52 (17) (1997) 3047]. The purpose of the present paper is to introduce a new technique for carrying out the actual reconciliation procedure and compare its significance and performance with those of previous methods. A numerical example shows how nontrivial differences are to be expected. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Data reconciliation; Detection limits; Censored data 1. Introduction The measurement of concentrations and flowrates close to, or even below, the detection limits of many industrial on-line samplers is becoming more and more frequent. This is due to more and more stringent constraints on the emis- sion of pollutants in industrial effluents and to the require- ment of high purity products, which results in both toxic substances and impurities being measured at extremely low concentrations. Traditional techniques, such as those developed by Vaclavek [2], Vaclavek and Loucka [3], Mah et al. [4], Ro- magnoli and Stephanopoulos [5], Crowe et al. [6], Crowe [7] and recently by Sanchez and Romagnoli [8] are not suit- able for the reconciliation of process measurements when some data may be below the detection limits, because they do not consider the presence of constraints. On the other hand, the introduction of positivity bounds on the rectified data, as proposed by Narasimhan and Harikumar [9], would overlook the functional form of the distribution function of the measurements close to the detection limits. Similarly the maximum entropy approach recently pro- posed by Crowe [10] would neglect the statistical informa- tion available. Measurements subject to detection limits are said to be censored. The following error distribution function has been previously proposed for them [1]: ∗ Corresponding author. Tel./fax: +39-010-3532921. E-mail address: dovi@istic.unige.it (V.G. Dov` ı). p(ε) = p(ξ ′ - ˆ ξ) =      if ξ ′ = 0  1/T ξ 0 ≤ ε ≤ T ξ 0 ε ≥ T ξ if ξ ′ ≥ T ξ N(0,σ 2 ξ ) (1) where ξ ′ is the experimental observation, ˆ ξ the unknown exact value, T ξ the detection limit and σ 2 ξ the variance of the error distribution when the measurement is above T ξ . In other words, a uniform distribution between zero and T ξ was assumed if a zero concentration value had been observed and a normal distribution otherwise. In this paper we propose an alternative distribution func- tion based on a slightly modified assumption, i.e. p(ε) = p(ξ ′ - ˆ ξ) =        1 √ 2πσ ξ T ξ  T ξ 0 e (η- ˆ ξ) 2 /2σ 2 ξ dη if ξ ′ = 0 N(0,σ 2 ξ ) if ξ ′ ≥ T ξ (2) Apparently, the error distribution function assumed in (2) is physically more correct than that in (1), because it is the measurement ξ ′ that is subject to censoredness, not the unknown true value ˆ ξ , which is not necessarily below the threshold T ξ if ξ ′ = 0, as implied by the error distribution function (1). On the other hand, assuming a Gaussian distribution for the experimental error between the censored measurement and the unknown exact value (and integrating over all the 1385-8947/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII:S1385-8947(00)00380-6