SIO 224. m(r) =(ρ(r),k s (r),µ(r))

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1 SIO A bref look at resoluton analyss Here s some background for the Masters and Gubbns resoluton paper. Global Earth models are usually found teratvely by assumng a startng model and fndng small perturbatons to t. Usually, the smallest or smoothest perturbaton s found that allows the data to be ft to some tolerance. Ths means that the fnal model can end up wth features of the startng model whch are not requred by the data. These notes descrbe one way of assessng the true resoluton of the data. For concreteness, we consder the free oscllaton nverse problem. Suppose we let m(r) be the true Earth and m (r) be a startng model where m s usually taken to be the trplet of functons: then let m(r) =(ρ(r),k s (r),µ(r)) δm = m m then we obtan for the free-oscllaton problem δω = G (m ),δm + O δm 2 (1) where, for the th mode The braket notaton s shorthand for: G (m ),δm = δω = ω obs ω model (G ρ δρ(r)+g K δk s (r)+g µ δµ(r)) dr and the G s can be computed from the egenfunctons of the th mode for the startng model and so are mplctly a functon of m. Equaton 1 s solved under the assumpton that δm s small enough so that terms of order δm 2 can be neglected. Equaton 1 s a lnearzaton of the problem and so, n prncple, t s possble to fnd many dfferent m whch satsfy the data and whch may not be lnearly close to one another. Ths occurs f the G are rapdly changng functons of m. For the case of free-oscllatons, the lnearzaton s probably vald snce G,δm correctly predcts the perturbaton n frequency, δω, for perturbatons up to several percent. (Ths sn t true for all modes but s true for the vast majorty). A few percent s the typcal uncertanty n the models of the sphercally averaged Earth. Suppose we have found a model whch gves a satsfactory ft to the data,.e. the resduals, δω, are normally dstrbuted wth zero mean. How do we fnd those features of the model whch are truly resolved by the data? Ths queston s addressed by the paper of Backus and Glbert (197). For smplcty, consder a one-dmensonal model (the model conssts of only one functon of radus): 1

2 δω ± σ = G (r)δm(r) dr for =1to N. Suppose we take a lnear combnaton of data A(r)δm(r) dr where A(r) = a G (r) Suppose we choose the multplers, a so that A(r) approxmates a δ-functon peaked at a partcular radus, r. If we acheved ths perfectly, we would have δ(r r )δm(r) dr = δm(r ) Wth a fnte amount of data we cannot make A(r) a perfect δ-functon but we can try and make t as δ-lke as possble. We then have = A(r)δm(r) dr A(r)(m(r) m (r)) dr = where we have assumed that the model, m (r) s lnearly close to the real Earth, m(r) and that the model fts the data so that the expected value of the resduals s zero. We thus obtan A(r)m(r) dr = A(r)m (r) dr = m(r ) say where m(r ) s an average of the real Earth (averaged wth our approxmaton to a δ-functon) and s dentcal to the same average of our model. We force the average to be unbased by makng A(r) unmodular,.e. A(r) dr =1 The data also have errors (σ ) and we suppose that the uncertantes n the data are characterzed by a covarance matrx, E j. We usually don t know what the covarances between our data are so we assume that the data are ndependent n whch case E j s dagonal wth elements along the dagonal whch are the varances of the data: σ 2. The varance of out estmate, m s then gven by 2

3 σ 2 = j a a j E j We would lke ths to be as small as possble. We want our multplers to make A(r) as δ-lke as possble at a radus r to localze nformaton about m(r) around r and at the same tme we want the localzed nformaton to be precse. Backus and Glbert show that these ams are mutually exclusve. How do we choose the a s to make A(r) peaked? Consder mnmzng the form S = f(r)a 2 (r) dr If f(r) s dpped near r then we would expect A(r) to be peaked at r. Backus and Glbert suggest usng a parabola: f(r) = 12(r r ) 2 The factor of 12 s ntroduced to make S a measure of the peak wdth of A whch we shall call the spread. (If A(r) s a boxcar of unt area centered at r then S s exactly the wdth of the boxcar). We now have S = j a a j S j where If we defne S j =12 (r r ) 2 G (r)g j (r) dr the unmodularty constrant reads u = G (r) dr a u =1 Snce σ 2 and S cannot be mnmzed smultaneously, we consder the followng combnaton: and mnmse M where M j = S j cos θ + we j sn θ θ π/2 M = a a j M j subject to a u =1 θ s called a tradeoff parameter. When θ =, we choose the a to mnmze the spread. When θ = π/2 we choose the a to mnmze σ 2. At ntermedate values we compute a compromse between 3

4 spread and error. w s a weghtng factor to make w E and S to be about the same the tradeoff calculaton wll then be centered about θ 45. Wrtten n vector form, our problem s mnmze a M a wth a u =1 The s a problem n calculus of varatons and s solved by ntroducng a Lagrange multpler, λ: mnmze a M a + λ(a u 1) Dfferentatng wth respect to a and settng equal to zero gves Thus 2M a + λu = a = λ 2 M 1 u We can evaluate the Lagrange multpler by dottng the above equaton wth u whch gves So ellmnatng λ/2 gves a u =1= λ 2 u M 1 u a = M 1 u u M 1 u Note that a must be recalculated for each value of r and θ and the calulaton s made much more effcent f M s dagonal. Ths can be acheved (Glbert 1971) but we don t consder numercal ncetes any further here. Once a s computed, the spread: a S a; the varance: a E a, and the resolvng kernel: a G(r) can all be computed. Some results of applyng ths technque to the mode problem are gven n the accompanyng PEPI artcle. In the example, we smultaneously try and peak nformaton about densty at some target radus whle removng senstvty to the elastc modul. Rather than computng a tradeoff curve, we choose a specfc error level (.5% say) then adjust θ untl the a that gves ths error level s found. N.B. The form chosen for S j above s relatvely arbtrary. We mght decde (as s done n the PEPI paper) that we want our resolvng kernel to be a boxcar between rad r 1 and r 2 wth unt area between these lmts: A(r) dr =1 r 1 If B(r) s the desred boxcar then we would mnmze r 2 (A(r) B(r)) 2 dr Substtutng n A = a G and expandng the square gves 4

5 a S a 2a u + B(r) 2 dr where we have redfned S and u: S j = G (r)g j (r) dr and u = r 2 G (r) dr r 1 Snce a u s forced to be one, the only part of the above equaton that depends on a s a S a. We now form M = S cos θ + we sn θ and fnd that we get the same answer for a as before: a = M 1 u u M 1 u but wth the redefned M and u. Ths form s computatonally effcent snce M no longer depends on the target depth range for the boxcar. Only u has to be recomputed for new r 1,r 2. 5