View source for Backus–Gilbert method

In mathematics, the '''Backus–Gilbert method''', also known as the '''optimally localized average''' ('''OLA''') '''method''' is named for its discoverers, geophysicists [[George E. Backus]] and [[James Freeman Gilbert]]. It is a [[Regularization (mathematics)|regularization]] method for obtaining meaningful solutions to ill-posed [[inverse problem]]s.  Where other regularization methods, such as the frequently used [[Tikhonov regularization]] method, seek to impose smoothness constraints on the solution, Backus–Gilbert instead seeks to impose stability constraints, so that the solution would vary as little as possible if the input data were resampled multiple times.  In practice, and to the extent that is justified by the data, smoothness results from this.

Given a data array '''X''', the basic Backus-Gilbert inverse is:

:<math>\mathbf{H}_\theta = \frac{\mathbf{C}^{-1}\mathbf{G}_\theta}{\mathbf{G}_\theta^T\mathbf{C}^{-1}\mathbf{G}_\theta}</math>

where '''C''' is the [[covariance matrix]] of the data, and '''G'''<sub>''θ''</sub> is an ''a priori'' constraint representing the source ''θ'' for which a solution is sought. Regularization is implemented by "whitening" the covariance matrix:

:<math>\mathbf{C}' = \mathbf{C} + \lambda \mathbf{I}</math>

with '''C'''<sup>′</sup> replacing '''C''' in the equation for '''H'''<sub>''θ''</sub>. Then,

:<math>\mathbf{H}_\theta^T\mathbf{X}</math>

is an estimate of the activity of the source ''θ''.

==References==
* Backus, G.E., and Gilbert, F. 1968, "The Resolving power of Gross Earth Data", ''[[Geophysical Journal of the Royal Astronomical Society]]'', vol. 16, pp.&nbsp;169–205.
* Backus, G.E., and Gilbert, F. 1970, "Uniqueness in the Inversion of inaccurate Gross Earth Data", ''Philosophical Transactions of the Royal Society of London A'', vol. 266, pp.&nbsp;123–192.
*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=[[Numerical Recipes]] | edition=3rd | publisher=[[Cambridge University Press]] | isbn=978-0-521-88068-8 | chapter=Section 19.6. Backus–Gilbert Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1014 | access-date=2011-08-17 | archive-date=2011-08-11 | archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=1014 | url-status=dead }}

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[[Category:Inverse problems]]
[[Category:Linear algebra]]


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