|
#REDIRECT [[Covariance_matrix#Covariance_mapping]]
{{newpage}}
{{userspace draft}}
In [[statistics]], covariance mapping is an extension of the [[covariance]] concept from [[random variables]] to [[random function|random functions]]. Normal covariance is a scalar (a single number) that measures statistical relation between two random variables. Covariance maps are matrices (arrays of numbers) that show statistical relations between different regions of random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.
{{Rcat shell|
==Application to data analysis==
{{R to related topic}}
Covariance mapping can be applied to any repetitive, fluctuating signal to reveal information hidden in the fluctuations. This technique was first used to analyse [[mass spectra]] of molecules ionised and fragmented by intense laser pulses.<ref name="LJF89">L J Frasinski, K Codling and P A Hatherly "Covariance Mapping: A Correlation Method Applied to Multiphoton Multiple Ionisation" ''Science'' '''246''' 1029–1031 (1989)</ref>
{{R with history}}
}}
Covariance mapping is particularly well suited to [[free-electron laser]] (FEL) research, where the x-ray intensity is so high that the large number of photoelectron and photoions produced at each pulse overwhelms simpler [[Photoelectron photoion coincidence spectroscopy|coincidence techniques]]. Figure 1 shows a typical experiment<ref name="LJF13">L J Frasinski, V Zhaunerchyk, M Mucke, R J Squibb, M Siano, J H D Eland, P Linusson, P v.d. Meulen, P Salén, R D Thomas, M Larsson, L Foucar, J Ullrich, K Motomura, S Mondal, K Ueda, T Osipov, L Fang, B F Murphy, N Berrah, C Bostedt, J D Bozek, S Schorb, M Messerschmidt, J M Glownia, J P Cryan, R Coffee, O Takahashi, S Wada, M N Piancastelli, R Richter, K C Prince, and R Feifel "Dynamics of Hollow Atom Formation in Intense X-ray Pulses Probed by Partial Covariance Mapping" ''Phys. Rev. Lett.'' '''111''' 073002 (2013), [http://hdl.handle.net/10044/1/11746 open access]</ref>. X-ray pulses are focused on neon atoms and ionize them. The kinetic energy spectra of the photoelectrons ejected from neon are recorded at each laser shot using a suitable spectrometer (here a [[Time-of-flight mass spectrometry|time-of-flight spectrometer]]). The single-shot spectra are sent to a computer, which calculates and displays the covariance map.
'''Figure 1: Schematics of a covariance mapping experiment.''' The experiment was performed at the [[LCLS#LCLS|LCLS FEL]] at [[Stanford University]]. <ref name="LJF13"/>
===The need to correlate photoelectrons===
Even in a relatively simple system, such as neon atom, intense x-rays induce a plethora of ionization processes (see Fig. 2). As the kinetic energies of the electrons ejected in different processes largely overlap, it is impossible to identify these processes using simple [[Photoemission spectroscopy|photoelectron spectrometry]]. To do so, one needs to correlate the kinetic energies of the electrons ejected in a given process. Covariance mapping is a method of revealing such correlations.
'''Figure 2: Examples of ionization processes in neon induced by intense x-ray photons of 1062 eV energy.''' When a photon is absorbed, it may eject a photoelectron from the atom core (P) or from its valence shell (P<sub>V</sub>). An Auger process fills any core hole ejecting an Auger electron (A). A core photoelectron on its way out may also kick out an additional valence electron giving double electron ejection (D<sub>KV</sub>) by a single photon. The x-ray intensity so high that several photons can be absorbed by a single atom producing a large variety of ionization sequences.
===The principle===
Consider a random function <math>X_n(E)</math>, where index <math>n</math> labels a particular instance of the function and <math>E</math> is the independent variable. In the context of the FEL experiment, <math>X_n(E)</math> is a digitized electron energy spectrum produced by laser shot <math>n</math>. As the electron energy <math>E</math> takes a range of discrete values <math>E_i</math>, the spectra can be regarded as row vectors of experimental data:
:<math> \mathbf{X}_n = [X_n(E_1), X_n(E_2), X_n(E_3), ... ] </math>.
The simplest way to analyse the data is to average the spectra over <math>N</math> laser shots:
:<math> \langle \mathbf{X} \rangle = \frac{1}{N} \sum^{N}_{n=1} \mathbf{X}_n </math>.
... covariance formula
===How to read the map===
... 1D vs 2D, impurities
... Poisson -> branching ratios
'''Figure 3: A covariance map revealing correlations between electrons emitted from neon (and from some N<sub>2</sub> and water vapour contamination).''' The map is constructed shot by shot from electron energy spectra, which are shown along the x and y axes after averaging over 480 000 FEL shots. Volumes of the features on the map give relative probabilities of various ionization sequences. <ref name="LJF13"/>
===Negative correlations===
... others' research
==Partial covariance mapping==
... N2 pcov stages
<ref name="OK13">O Kornilov, M Eckstein, M Rosenblatt, C P Schulz, K Motomura, A Rouzée, J Klei, L Foucar, M Siano, A Lübcke, F. Schapper, P Johnsson, D M P Holland, T Schlatholter, T Marchenko, S Düsterer, K Ueda, M J J Vrakking and L J Frasinski "Coulomb explosion of diatomic molecules in intense XUV fields mapped by partial covariance" ''J. Phys. B: At. Mol. Opt. Phys.'' '''46''' 164028 (2013), [http://hdl.handle.net/10044/1/12267 open access]</ref>
==Three-dimensional covariance mapping==
... scan N2O pictures
==See also==
* [[Photoelectron photoion coincidence spectroscopy]]
==References==
{{reflist}}
|
