18.6 Backus-Gilbert Method 817 Substituting equation(18.6.4)into equation(18.6.3),and comparing with equation (18.6.2),we see that x,x)=∑g(ar(x) (18.6.6) We can require this averaging kernel to have unit area at every z,giving 1=,r=∑9a回r(dr=∑oR三qR(I86.7) where g(z)and R are each vectors of length N,the number of measurements. Standard propagation of errors,and equation (18.6.1),give B=Var(z=∑∑9(o)Sg(a)=q(a)·s·q(a) (18.6.8) ij where Si;is the covariance matrix (equation 18.4.6).If one can neglect off-diagonal server 2 covariances (as when the errors on the ci's are independent),then Sj=j? (Nort University is diagonal. Press. THE We now need to define a measure of the width or spread of 6(,')at each value ofr.While many choices are possible,Backus and Gilbert choose the second moment of its square.This measure becomes the functional A, Program A三w(x)= (d'-x)26,x']2dr OF SCIENTIFIC( (18.6.9) =∑∑9()Ww(r()三q()w(·q) 6 where we have here used equation(18.6.6)and defined the spread matrix W()by W()三(x'-x)2rn(rr(x'）dr (18.6.10) Numerical 10621 The functions gi(z)are now determined by the minimization principle idge.org 431 Recipes minimize:A+λB=q(x)·W(x)+λS·q(x) (18.6.11) (outside subject to the constraint (18.6.7)that q(x).R =1. North The solution of equation (18.6.11)is [W(z)+λS-1.R q)=R·w国)+S-1R (18.6.12) (Reference [4]gives an accessible proof.)For any particular data set c (set of measurements ci),the solution (z)is thus aa-图+SR (18.6.13)18.6 Backus-Gilbert Method 817 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). Substituting equation (18.6.4) into equation (18.6.3), and comparing with equation (18.6.2), we see that δ
(x, x
) = i qi(x)ri(x
) (18.6.6) We can require this averaging kernel to have unit area at every x, giving 1 = δ
(x, x
)dx
= i qi(x) ri(x
)dx
= i qi(x)Ri ≡ q(x) · R (18.6.7) where q(x) and R are each vectors of length N, the number of measurements. Standard propagation of errors, and equation (18.6.1), give B = Var[u
(x)] = i j qi(x)Sij qj (x) = q(x) · S · q(x) (18.6.8) where Sij is the covariance matrix (equation 18.4.6). If one can neglect off-diagonal covariances (as when the errors on the ci’s are independent), then Sij = δijσ2 i is diagonal. We now need to define a measure of the width or spread of δ
(x, x
) at each value of x. While many choices are possible, Backus and Gilbert choose the second moment of its square. This measure becomes the functional A, A ≡ w(x) = (x
− x) 2[δ
(x, x
)]2dx
= i j qi(x)Wij (x)qj (x) ≡ q(x) · W(x) · q(x) (18.6.9) where we have here used equation (18.6.6) and defined the spread matrix W(x) by Wij (x) ≡ (x
− x) 2ri(x
)rj (x
)dx
(18.6.10) The functions qi(x) are now determined by the minimization principle minimize: A + λB = q(x) ·  W(x) + λS  · q(x) (18.6.11) subject to the constraint (18.6.7) that q(x) · R = 1. The solution of equation (18.6.11) is q(x) = [W(x) + λS] −1 · R R · [W(x) + λS] −1 · R (18.6.12) (Reference [4] gives an accessible proof.) For any particular data set c (set of measurements ci), the solution u
(x) is thus u
(x) = c · [W(x) + λS] −1 · R R · [W(x) + λS] −1 · R (18.6.13)