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Youonly need to solve for some y the linear system (W(x) + λS) · y = R, and thensubstitute y into both the numerators and denominators of 18.6.12 or 18.6.13.)Equations (18.6.12) and (18.6.13) have a completely different character fromthe linearly regularized solutions to (18.5.7) and (18.5.8). The vectors and matrices in(18.6.12) all have size N , the number of measurements. There is no discretization ofthe underlying variable x, so M does not come into play at all. One solves a differentN × N set of linear equations for each desired value of x. By contrast, in (18.5.8),one solves an M × M linear set, but only once.

In general, the computational burdenof repeatedly solving linear systems makes the Backus-Gilbert method unsuitablefor other than one-dimensional problems.How does one choose λ within the Backus-Gilbert scheme? As alreadymentioned, you can (in some cases should) make the choice before you see anyactual data. For a given trial value of λ, and for a sequence of x’s, use equation(18.6.12) to calculate q(x); then use equation (18.6.6) to plot the resolution functionsb x0 ) as a function of x0 . These plots will exhibit the amplitude with whichδ(x,different underlying values x0 contributepto the point ub(x) of your estimate.

For theu(x)] using equation (18.6.8). (Yousame value of λ, also plot the function Var[bneed an estimate of your measurement covariance matrix for this.)As you change λ you will see very explicitly the trade-off between resolutionand stability. Pick the value that meets your needs.

You can even choose λ to be afunction of x, λ = λ(x), in equations (18.6.12) and (18.6.13), should you desire todo so. (This is one benefit of solving a separate set of equations for each x.) Forthe chosen value or values of λ, you now have a quantitative understanding of yourinverse solution procedure. This can prove invaluable if — once you are processingreal data — you need to judge whether a particular feature, a spike or jump forexample, is genuine, and/or is actually resolved.

The Backus-Gilbert method hasfound particular success among geophysicists, who use it to obtain information aboutthe structure of the Earth (e.g., density run with depth) from seismic travel time data..

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