Backus-Gilbert

Backus and Gilbert (1967) have developed a solution of the linear inverse problem that estimates the resolution of the model parameters. In this section, we will outline the basic ideas of this approach and will show how the regularization can be introduced in the Backus-Gilbert method. 

The Backus-Gilbert Method from which we have... 

Thus, wc have shown that the ideas of the regularizat ion tliciory can be naturally implemented in the Backus-Gilbert, method, providing it with o])t.iinal stability... 

A retrieval formulation developed by Backus Gilbert (1970) and used in geophysical inverse problems of the solid Earth has also found application in remote sensing of atmospheres. Although this formulation is developed from a somewhat different point of view, it is formally related to the constrained linear inversion as well as other techniques. We include a brief description here because of the additional physical insight it can provide into inversion theory. 

Thus, from the Backus-Gilbert point of view Eq. (8.2.24) is that retrieval for which the averaging kernel lies closest to a Dirac-delta function in the least squares sense, subject to the constraint imposed by a limited set of measurements in the presence of measurement error. 

Fig. 8.2.3 Examples of trade-off curves for measurements within the 15 /rm CO2 band in the Earth s atmosphere as defined in the Backus-Gilbert theory. The spread is a measure of the vertical resolution for the retrieved profiles, while aj/

Backus, G. Gilbert, F. (1967). Numerical applications of a formalism for geophysical inverse problems. Geophys. J. R. Astron. Soc., 13, 247-76. 

Other modifications are possible to the same basic approach of seeking a filter that is optimum in the sense of least mean-square error. Backus and Gilbert (1970), for example, derive a linear filter by minimizing a sum of terms in which noise and resolution criteria are separately weighted. Frieden (1975) discusses variations of this technique. 

Backus, G. Gilbert, F. (1970). Uniqueness in the inversion of inaccurate gross earth data. Philosophical Transactions Royal Society London, A266, 123-92. 

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