818 Chapter 18.Integral Equations and Inverse Theory (Don't let this notation mislead you into inverting the full matrix W(z)+AS.You only need to solve for some y the linear system (W(z)+AS).y =R,and then substitute 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 from the 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 of the underlying variable z,so M does not come into play at all.One solves a different N x N set of linear equations for each desired value of z.By contrast,in(18.5.8), one solves an M x M linear set,but only once.In general,the computational burden of repeatedly solving linear systems makes the Backus-Gilbert method unsuitable 81 for other than one-dimensional problems. How does one choose A within the Backus-Gilbert scheme?As already mentioned,you can (in some cases should)make the choice before you see any actual data.For a given trial value of A.and for a sequence of x's,use equation(18.6.12) to calculate q();then use equation(18.6.6)to plot the resolution functions 6(,' as a function of x'.These plots will exhibit the amplitude with which different underlying values z'contribute to the point ()of your estimate.For the same value of A,also plot the function Var()]using equation (18.6.8).(You need an estimate of your measurement covariance matrix for this. As you change A you will see very explicitly the trade-off between resolution and stability.Pick the value that meets your needs.You can even choose A to be a function of A =A(x),in equations (18.6.12)and (18.6.13),should you desire to do so.(This is one benefit of solving a separate set of equations for each z.)For the chosen value or values of A,you now have a quantitative understanding of your 三兰号∽6 inverse solution procedure.This can prove invaluable if-once you are processing OF SCIENTIFIC real data-you need to judge whether a particular feature,a spike or jump for example,is genuine,and/or is actually resolved.The Backus-Gilbert method has found particular success among geophysicists,who use it to obtain information about the structure of the Earth(e.g.,density run with depth)from seismic travel time data 、复公 CITED REFERENCES AND FURTHER READING: Backus,G.E.,and Gilbert,F.1968,Geophysical Journal of the Royal Astronomical Society, vol.16,pp.169-205.I1) Numerica 10621 Backus,G.E..and Gilbert,F.1970,Philosophical Transactions of the Royal Society of London 431 A,vol.266,pp.123-192.2 Recipes Parker,R.L.1977,Annual Review of Earth and Planetary Science,vol.5,pp.35-64.[3] Loredo,T.J.,and Epstein,R.I.1989,Astrophysical Journal,vol.336,pp.896-919.[4] (outside Software. 18.7 Maximum Entropy Image Restoration Above.we commented that the association of certain inversion methodsbreak with Bayesian arguments is more historical accident than intellectual imperative. Maximum entropy methods,so-called,are notorious in this regard;to summarize these methods without some,at least introductory,Bayesian invocations would be to serve a steak without the sizzle,or a sundae without the cherry.We should818 Chapter 18. Integral Equations and Inverse Theory Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). (Don’t let this notation mislead you into inverting the full matrix W(x) + λS. You only need to solve for some y the linear system (W(x) + λS) · y = R, and then substitute 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 from the 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 of the underlying variable x, so M does not come into play at all. One solves a different N × 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 burden of repeatedly solving linear systems makes the Backus-Gilbert method unsuitable for other than one-dimensional problems. How does one choose λ within the Backus-Gilbert scheme? As already mentioned, you can (in some cases should) make the choice before you see any actual 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 functions δ
(x, x
) as a function of x
. These plots will exhibit the amplitude with which different underlying values x
contribute to the point u
(x) of your estimate. For the same value of λ, also plot the function Var[u
(x)] using equation (18.6.8). (You need an estimate of your measurement covariance matrix for this.) As you change λ you will see very explicitly the trade-off between resolution and stability. Pick the value that meets your needs. You can even choose λ to be a function of x, λ = λ(x), in equations (18.6.12) and (18.6.13), should you desire to do so. (This is one benefit of solving a separate set of equations for each x.) For the chosen value or values of λ, you now have a quantitative understanding of your inverse solution procedure. This can prove invaluable if — once you are processing real data — you need to judge whether a particular feature, a spike or jump for example, is genuine, and/or is actually resolved. The Backus-Gilbert method has found particular success among geophysicists, who use it to obtain information about the structure of the Earth (e.g., density run with depth) from seismic travel time data. CITED REFERENCES AND FURTHER READING: Backus, G.E., and Gilbert, F. 1968, Geophysical Journal of the Royal Astronomical Society, vol. 16, pp. 169–205. [1] Backus, G.E., and Gilbert, F. 1970, Philosophical Transactions of the Royal Society of London A, vol. 266, pp. 123–192. [2] Parker, R.L. 1977, Annual Review of Earth and Planetary Science, vol. 5, pp. 35–64. [3] Loredo, T.J., and Epstein, R.I. 1989, Astrophysical Journal, vol. 336, pp. 896–919. [4] 18.7 Maximum Entropy Image Restoration Above, we commented that the association of certain inversion methodsbreak with Bayesian arguments is more historical accident than intellectual imperative. Maximum entropy methods, so-called, are notorious in this regard; to summarize these methods without some, at least introductory, Bayesian invocations would be to serve a steak without the sizzle, or a sundae without the cherry. We should