Fire Technology

, Volume 53, Issue 3, pp 1077–1100 | Cite as

A Probabilistic Inferential Algorithm to Determine Fire Source Location Based on Inversion of Multidimensional Fire Parameters



A probabilistic inferential framework is proposed to utilize transient temperature data measured at the ceiling of the compartment to determine location of fire source, as well as size of fire, based on Bayesian inferential theory. This approach treats the problem as one of parameter estimation, expressed as a function of posterior probability distribution based on the qualitative of agreement between predicted temperatures and observed temperatures at the sensor locations. A comparison of the measured temperature from full-scale burn tests with predicted temperatures from in verse problem solution algorithm indicate the error to be less than 5% when fires are small, but the error increases to more than 10% for large size fires. The accuracy of the inverse problem solution algorithm can be improved by utilizing data from sensitivity studies carried out on fire source location errors and heat release rate errors.


Fire source parameters Fire source location Inversion algorithm Bayesian inferential theory Adjoint operator 

List of Symbols

\( \left( {Q,x,y} \right) \)

Three-dimensional parameters of fire source

x, y

Locations of fire source in a bi-dimensional space


Heat relese rate of fire source per volumn

\( q^{\prime} \)

The heat release rate per unit volume from a combustion reaction


Spatial domain of fire source


The smoke propagation domain

\( p\left( {Q,x,y} \right) \)

Prior probability of fire source parameter

\( T\& \left\{ {T_{1} \left( X \right), \ldots ,T_{i} \left( X \right), \ldots ,T_{k} \left( X \right)} \right\} \)

Predicted temperatures

\( M\& \left\{ {M_{1} \left( X \right), \ldots ,M_{i} \left( X \right), \ldots ,M_{k} \left( X \right)} \right\} \)

Measured temperatures

\( R\& \left\{ {R_{1} \left( X \right), \ldots ,R_{i} \left( X \right), \ldots ,R_{k} \left( X \right)} \right\} \)

Theoretical detector temperature


The calculated error

\( \varepsilon_{i} \)

The measured error

\( \sigma_{t,i}^{{}} \), \( \sigma_{m,i}^{{}} \)

Standard deviation

\( p\left( {\left. T \right|\left( {Q,x,y} \right)} \right) \)

Conditional probability

\( l\left( {\left. T \right|\left( {Q,x,y} \right)} \right) \)

Likelihood probability

\( p\left( {\left. {\left( {Q,x,y} \right)} \right|T} \right) \)

Posterior probability



\( h_{s} \)

Special heat capacity

\( \lambda \)

Heat conduct coefficient




Detector response function

\( L^{*} \)

The adjoint operator

\( T^{*} \)

Conjugate temperature field

\( \alpha \)

Acceptance probability

\( \hat{R} \)

The convergence criteria of a Markov chain


Width of pooled interval


Within-chain variance


Between-chain variance

\( \bar{S} \)

Average value of sample parameter in a Markov chain

\( s_{ij} \)

Value of sample parameter


Number of Markov chains


Number of samples in a Markov chain

\( \varepsilon_{loc} \)

Source location error

\( \varepsilon_{Q} \)

Heat release rate error

\( \delta \)

System error



The research, presented in this paper is primarily supported by Sun Yat-sen University and the Michigan State University. And it is also supported by Guangdong Natural Science Foundation (2016A030313347), Guangdong Provincial Key Laboratory of Fire Science and Technology (2014B030301034).


  1. 1.
    May A, Mitchell V, Piper J (2014) A user centred design evaluation of the potential benefits of advanced wireless sensor networks for re-in-tunnel emergency response. Fire Saf J 63:79–88CrossRefGoogle Scholar
  2. 2.
    Ko BC, Cheong K, Nam J (2009) Fire detection based on vision sensor and support vector machines. Fire Saf J 44:322–329CrossRefGoogle Scholar
  3. 3.
    Heskestad G, Newman JS (1992) Fire detection using cross-correlations of sensor signals. Fire Saf J 18:355–374CrossRefGoogle Scholar
  4. 4.
    Milke JA (1999) Using multiple sensors for discriminating fire detection. In: The fire suppression and detection research application symposium//proceeding, Feb 24–26, 1999, Orlando, pp 150–164.Google Scholar
  5. 5.
    Jahn W, Rein G, Torero J (2011) Forecasting fire growth using an inverse zone modelling approach. Fire Saf J 46(3):81–88CrossRefGoogle Scholar
  6. 6.
    Jahn W, Rein G, Torero J (2011) Forecasting fire growth using an inverse CFD modelling approach in a real-scale fire test. Fire Saf Sci 10:1349–1358.CrossRefGoogle Scholar
  7. 7.
    Jahn W, Rein G, Torero J (2012) Forecasting fire dynamics using inverse computational fluid dynamics and tangent linearisation. Adv Eng Softw 47:114–126CrossRefGoogle Scholar
  8. 8.
    Koo S, Fraser-Mitchel J, Welch S (2010) Sensor-steered fire simulation. Fire Saf J 45(3):193–205.CrossRefGoogle Scholar
  9. 9.
    Overholt KJ, Ezekoye O (2012) Characterizing heat release rates using an inverse fire modeling technique. Fire Technol 48:893–909.CrossRefGoogle Scholar
  10. 10.
    Beji T, Verstockt S, Van de Walle R, Merci B (2014) On the use of real-time video to forecast fire growth in enclosures. Fire Technol 50:1021–1040Google Scholar
  11. 11.
    Price M, Marshall A, Trouve A (2015) A multi-observable approach to the III-posed nature of inverse modeling problems fire. Fire Technol 19, Dec (published online)Google Scholar
  12. 12.
    Wang Y, Yu C, Tu R, Zhang Y (2011) Fire detection model in Tibet based on grey-fuzzy neural network algorithm. Expert Syst Appl 38:9580–9586CrossRefGoogle Scholar
  13. 13.
    Ko BC, Cheong K, Nam J (2014) Fire detection and 3D surface reconstruction based on stereoscopic pictures and probabilistic fuzzy logic. Fire Saf J 68:61–70CrossRefGoogle Scholar
  14. 14.
    Richards RF, Munk BN, Plumb OA (1997) Fire detection, location and heat release rate through inverse problem solution. Part I: theory. Fire Saf J 28:323–350CrossRefGoogle Scholar
  15. 15.
    Richards RF, Munk BN, Plumb OA (1997) Fire detection, location and heat release rate through inverse problem solution Part II: experiment. Fire Saf J 28:323–350CrossRefGoogle Scholar
  16. 16.
    Ko BC, Cheong K, Nam J (2010) Early fire detection algorithm based on irregular patterns of flames and hierarchical bayesian networks. Fire Saf J 45:262–270CrossRefGoogle Scholar
  17. 17.
    Chen J, Fu J (2012) Fire alarm system based on multi-sensor Bayes network. In: 2012 international workshop on information and electronics engineering (IWIEE)//procedia engineering 2012, vol 29, pp 2551–2555Google Scholar
  18. 18.
    Overholt KJ, Ezekoye O (2014) Quantitative testing of fire scenario hypotheses: a Bayesian inference approach. Fire Technol 51:335–367.CrossRefGoogle Scholar
  19. 19.
    Yee E (2008) Theory for reconstruction of an unknown number of contaminant sources using probabilistic inference. Boundary-Layer Meteorol 127:359–394.CrossRefGoogle Scholar
  20. 20.
    Keats A, Yee E, Lien F (2007) Bayesian inference for source determination with applications to a complex urban environment. Atmos Environ 41:465–479CrossRefGoogle Scholar
  21. 21.
    Guo SD, Yang R, et al (2009) Source identification for unsteady atmospheric dispersion of hazardous materials using Marcov Chain Monte Carlo method. Int J Heat Mass Transf 52:3955–3962CrossRefMATHGoogle Scholar
  22. 22.
    Metropolis N, Rosenbluth AW et al (1953) Equation of state calculations by fast computing machine. J Chem Phys 21:1087–1092.CrossRefGoogle Scholar
  23. 23.
    Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gregory PC (2008) Bayesian logical data analysis for the physical sciences: a comparative approach with mathematica R support. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  25. 25.
    Gelman A, Carlin J et al (2003) Bayesian Data Analysis. Chapman & Hall/CRC, LondonGoogle Scholar
  26. 26.
    Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov Chain Monte Carlo in practice. Chapman & Hall/CRC, LondonMATHGoogle Scholar
  27. 27.
    Zhao W, Zong R, Wei T et al (2015)The physical model and validation study of ceiling-jet flow in near-field of corridor fires. Int J Heat Mass Transf 88:91–100CrossRefGoogle Scholar
  28. 28.
    Dunkley J, Bucher M, Ferreira PG, Moodley K, Skordis C (2005) Fast and reliable MCMC for cosmological parameter estimation. Mon Not R Astron Soc 356:925–936CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of EngineeringSun Yat-sen UniversityGuangzhouChina
  2. 2.Guangdong Provincial Key Laboratory of Fire Science and TechnologyGuangzhouChina
  3. 3.Civil and Environmental EngineeringMichigan State UniversityEast LansingUSA

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