Efficient Model-free Deconvolution of Measured Ultrafast Kinetic Data Ernő Keszei and Péter Pataki Department of Physical Chemistry, and Reaction Kinetics Laboratory, Eötvös University (ELTE) H-1518 Budapest, P.O. Box 32, Hungary; Purpose of this work A direct model-free deconvolution method has been implemented, using a genetic algorithm. The method has been thoroughly tested on femtosecond pump-probe transient absorption and fluorescence upconversion data. What is deconvolution? To get the undistorted kinetic function o(t) from the detected (convolved) signal i(t), the spread function s(t) should be known and the integral equation (Eq. C) should be solved. The procedure yielding an estimate of the original o(t) function is called deconvolution. Pros and contras of direct deconvolution Advantages: 2 Does not require any prior knowledge of the kinetic mechanism After deconvolving the detected signal, it is much easier to find the appropriate mechanism Instrumental response parameters can be determined without correlation with kinetic and photochemical parameters Difficulties of „classical” signal processing methods: 2,3,4 Possible appearence of mathematically acceptable, but physically nonsense solutions as artefacts Unavoidable low-frequency wavy oscillations and high frequency noise Artefacts can largegely be reduced using a genetic algorithm for deconvolution, due to the highly flexible genetic operators 1 E. Keszei: J. Chemomet., (sent for publication) 2 Á. Bányász, E. Keszei, J. Phys Chem. A 110, 6192 (2006) Convolution in ultrafast laser chemistry Experiment: femtosecond pump-probe transient absorption femtosecond fluorescence upconversion Limitation: due to uncertainty relation: 100 fs ≤ pulse width Problem:characteristic times of the studied reactions and the temporal width of the laser pulse are comparable Result of the measurement: a distorted curve (image, i ); the convolution of the kinetic response function (object, o ) and the instrumental distortion function (spread, s ) (Eq. C) Mostly used method to evaluate ultrafast kinetic data: R econvolution = least-squares fitting of a suitable model function o(t) convolved with the distortion function s(t) Problem: reconvolution requires a particular kinetic model, which is usually not known prior to kinetic inference. Model-free deconvolution enables to get undistorted kinetic data without presupposing any particular kinetic model. Additional advantage: instrumental distortion parameters can be determined without any correlation with kinetic and photochemical parameters, as there is no need for an additional adjustable „zero time” parameter. Relevance of model-free deconvolution References Conclusion and Perspectives Thomas Gustavsson and Ákos Bányász for detailed experimental data Balaton exchange project 11038YM OTKA project T Acknowledgement Model-free deconvolution using a genetic algorithm Tests on simulated kinetic data 1 Kinetic mechanism used to test transient absorption: consecutive two steps reaction: τ 1 = 200 fs, τ 2 = 500 fs; transient absorption with residual bleaching:  A = 5,  B = 30,  C = – 10 Spread: 255 fs fwhm Gaussian Experimental error: random noise with a normal distribution of 2 % variance of the maximum amplitude. Test on real-life experimental data 1 Tests were also performed on experimental fluorescence decay data of adenosine monophosphate in aqueous solution obtained by femtosecond fluorescence upconversion (excited at 267 nm, observed at 310 nm; T. Gustavsson and Á. Bányász, unpublished data). Contraryly to model-free deconvolution via time-domain iterative methods 2 and inverse filtering in the frequency domain 2-4, use of genetic algorithms results in a distortion-free deconvolved kinetic signal that does not have low-frequency wavy behaviour correctly reproduces sudden steplike features of kinetic functions efficiently damps experimental error without signal distortion fully recovers the whole frequency spectrum of the undistorted kinetic function Experience shows that there is less systematic distortion if nonparametric (model-free) deconvolution is applied, even in the case if an established photophysical and kinetic model is known and used to perform statistical inference. The procedure can efficiently be applied to both synthetic and real-life experimental data. Further work concentrates on improving the quality of deconvolution by applying a genetic algorithm to create the initial population, deconvolving more real-life experimental data, and developing a user-friendly graphical interface to perform the deconvolution. time domain results 3 Á. Bányász, E. Mátyus, E. Keszei, Rad. Phys. Chem. 72, 235 (2005) 4 Á. Bányász, G. Dancs, E. Keszei, Rad. Phys. Chem. 74, 139 (2005) frequency domain results temporally widened signal lowered amplitude shallower rise and descent smoothenedsteplike jumps  = convolution results in: To get the undistorted signal, these effects should be inversed This is achieved by: 1) creation of an initial population (whose members represent fairly good solutions) 2) fine-tuning of the population members (to best reproduce the detected signal when convolved with the instrumental response) rise and decay steepening ”cutoff” of the first few data amplitude increase temporal compression creation operators to generate one individual of the initial population randomly generated initial population evolution of the population by crossover random mutation natural selection genetic operators best deconvolved (”winner”) 1) 2) Model function used to test transient fluorescence: biexponential decay with τ 1 = 100 fs, τ 2 = 500 fs spread: 310 fs fwhm Gaussian - - signal processing frequency domain resultstime domain results Implementation: a package of user defined Matlab functions and scripts Input: a project descriptor text file with parameters of the creation of initial population and evolution operators + files containing measured data Output: all input parameters in the same format as the project descriptor, measured input data and all results in a matrix format as a text file + a four-panel graphical window signal processing results frequency domain resultstime domain results Code of the deconvolution procedure via genetic algorithm 1 Initial conditions: [A] = 1, [B] = [C] = 0 at t = 0. Kinetic response function (F): Code available at