Proof of validity of first-order travel estimates

Proof of v alidity of first-order tra vel estimates Len Bos Dipartimento di Informatica, Uni versità di V erona, Italy Michael A. Slawinski Department of Earth Sciences, Memorial Uni versity of Ne wfoundand, St. John’ s, Ne wfoundland, Canada March 12, 2022 Abstract In the seminal paper by Dahlen et al [1] the authors formulate an important expression as a first-order estimate of trav eltime delay . The authors left out a term which would at first glance seem nontrivial, on the basis that their intention was to derive the Fréchet deriv ativ e linking the observed delay to the model perturbation (Nolet 2009, pers. comm.). Here we show that the deri vation by [1] results in a first-order estimate e ven without anticipating a Fréchet deri vati ve, b ut instead remaining deductively in their T aylor -series formulation. Although a mathematical technicality , this strengthens the result of [1] by showing that it is intrinsically v alid, requiring no e xternal justification. W e sho w also that ignoring the aforementioned term is not valid in general and needs to be supported by careful ar gument. 1 Intr oduction The purpose of this paper is to clarify and strengthen the deriv ation of formula (65) of the seminal paper by [1], in which the use of crosscorrelation to estimate tra veltime delays is discussed. Formula (65) is a first-order estimate of δ τ , the shift in the peak position of the crosscorrelation of observed and synthetic pulses. At a certain point in the deriv ation, which we explain in detail belo w , the T aylor series used by [1] is of one order less than would be normally expected, and for a general equation it would not be possible to make the claim that the estimate is indeed of first order . Howe ver , we show that in this special context the claim is mathematically true for rather subtle reasons, without the need of justifying it by a subsequent geophysical application. Even though, certain subtleties might be of no interest to geophysicists using the results of [1], we offer this proof in the context of geomathematics and in a spirit of respect for the contributions of the late T ony Dahlen. Let us describe the deriv ation in question. The main entity studied is the crosscorrelation Γ( τ ) = t 2 Z t 1 s ( t − τ ) s obs ( t )d t, (1) where s is the synthetic signal and s obs is the observ ed signal; this is equation (58) of [1]. Subsequently , they assume that s obs can be written as s obs ( t ) = s ( t ) + δ s ( t ) , where δ s ( t ) is a small perturbation; this is equation (59). This permits them to write equation (60): Γ( τ ) = γ ( τ ) + δ γ ( τ ) , (2) 1 where γ ( τ ) = t 2 Z t 1 s ( t − τ ) s ( t )d t is the crosscorrelation of s with itself, and δ γ ( τ ) = t 2 Z t 1 s ( t − τ ) δ s ( t )d t (3) is the crosscorrelation of the synthetic signal and the perturbation. The maximum of the autocorrelation, γ ( τ ) , is at τ = 0 ; the goal of [1] is to give a first-order estimate of the delay giv en by the maximum of Γ( τ ) , which they call δ τ . They obtain this estimate by taking the T aylor series of Γ( τ ) about τ = 0 , to the second order, differentiating this second-order series, and taking δ τ to be giv en by the zero of this deriv ativ e. Herein is a technical subtlety . The second-order T aylor series should be Γ( τ ) = γ (0) + ∂ τ γ (0) τ + 1 2 ∂ τ τ γ (0) τ 2 + δ γ (0) + ∂ τ δ γ (0) τ + 1 2 ∂ τ τ δ γ (0) τ 2 + · · · , (4) so that ∂ τ Γ( τ ) = ∂ τ γ (0) + ∂ τ τ γ (0) τ + ∂ τ δ γ (0) + ∂ τ τ δ γ (0) τ + · · · , and the critical point of Γ( τ ) , using the fact that ∂ τ γ (0) = 0 , would be δ τ ≈ − ∂ τ δ γ (0) ∂ τ τ γ (0) + ∂ τ τ δ γ (0) . (5) Howe ver , instead of the second-order series (4), [1] use Γ( τ ) = γ (0) + ∂ τ γ (0) τ + 1 2 ∂ τ τ γ (0) τ 2 + δ γ (0) + ∂ τ δ γ (0) τ + · · · , (6) which is their equation (63); in other words, they ne glect 1 2 ∂ τ τ δ γ (0) τ 2 , (7) which results in δ τ ≈ − ∂ τ δ γ (0) ∂ τ τ γ (0) , (8) which is estimate (65) of [1]. The absence of the term (7) is not tri vial mathematically , even if one might argue for it heuristically in the context of geophysical approximations. W e have examined the documents of the late T ony Dahlen, which he sketched in preparation for [1], and which are a vailable at Princeton Univ ersity . Not ha ving found a mathematical justification therein, we give one in Sections 3 and 4, below . Before doing so, ho wev er , we give an example that shows that care must be taken when using such a method to estimate critical points. 2 2 Illustrativ e Example W e gi ve an example sho wing that small perturbations may cause critical points to mov e great distances. Consider f ( t ) = 1 + at 2 1 + t 2 , where a := 1 −  4 (1 +  2 ) 2 , with 1 >  > 0 , which implies that 0 < a < 1 . Thus, we can write f ( t ) = (1 + t 2 ) + ( a − 1) t 2 1 + t 2 = 1 − (1 − a ) t 2 1 + t 2 , and conclude that a ≤ f ( t ) ≤ 1 , t ∈ R , and that f ( t ) has a unique maximum at t = 0 : f (0) = 1 . Furthermore, f 0 ( t ) = 2( a − 1) t (1 + t 2 ) 2 , which attains its maximum absolute value at t = ± √ 3 / 3 :    f 0 ( ± √ 3 / 3)    = 3 √ 3 8 (1 − a ) = 3 √ 3 8  4 (1 +  2 ) 2 ; in other words, | f 0 ( t ) | ≤ 3 √ 3 8  4 (1 +  2 ) 2 , t ∈ R . (9) Now let g ( t ) =  1 + ( t − b ) 2 where b := 1 / , be a perturbation to f ( t ) . Since | g ( t ) | ≤ , t ∈ R , it is a small perturbation. Also, g 0 ( t ) = 2  b − t (1 + ( t − b ) 2 ) 2 , and we see that g 0 ( t ) > 0 for t ≤ 0 . Hence, h ( t ) := f ( t ) + g ( t ) is such that h 0 ( t ) > 0 for t < 0 , and the perturbed function, h , has no negati ve critical points. Since for 0 ≤ t ≤ b/ 2 , | g 0 ( t ) | ≥ b (1 + b 2 ) 2 = 1 (1 + 1 / 2 ) 2 =  4 (1 +  2 ) 2 = 1 − a, 3 it follows that | g 0 ( t ) | ≥ 1 − a > 3 √ 3 8 (1 − a ) ≥ | f 0 ( t ) | , and, consequently , h 0 ( t ) 6 = 0 , 0 ≤ t ≤ b/ 2 . In other words, any critical points of the perturbed function, h ( t ) , must be greater than b/ 2 = 1 / (2  ) , which can be arbitrarily large and distant from the original critical point of f ( t ) , which is at t = 0 . 3 First-order estimates As illustrated in the above example, it is important to understand what is happening if one estimates a critical point by using T aylor series in this way . A critical point, c, of f ( t ) is a root of its deriv ative, f 0 ( t ) . In the expectation that this critical point is near t = 0 , it is reasonable to estimate f ( t ) by its T aylor series about t = 0 , f ( t ) = f (0) + f 0 (0) t + 1 2 f 00 (0) t 2 + · · · . Differentiating, we get f 0 ( t ) = f 0 (0) + f 00 (0) t + · · · , (10) and the estimate of the critical point is obtained by setting expression (10) to zero and solving for t to get c = − f 0 (0) f 00 (0) , which is analogous to expression (65) of [1]. Notice that this is one iteration, c = t 1 , of Newton’ s method for finding a root of f 0 ( t ) with t 0 = 0 as starting point. Hence the relev ant error estimates are those for Ne wton’ s method. Theorem 3.1 ([2], p. 56) Suppose that Newton’ s method is applied to the function F ( t ) with starting point t 0 to obtain the estimate t 1 of the true r oot t = z . Then t 1 − z = − F 00 ( η ) 2( F 0 ( η )) 3 ( F ( t 0 )) 2 , for some η between z and t 0 . If | t 0 − z | is “small” then we may refine this error bound as follo ws. In this case η ≈ z so that − F 00 ( η ) 2( F 0 ( η )) 3 ≈ − F 00 ( z ) 2( F 0 ( z )) 3 , which we may regard as approximately constant, pro vided F 0 ( z ) 6 = 0 . Furthermore, we may expand F ( t 0 ) = F ( z ) + F 0 ( z )( t 0 − z ) + · · · = 0 + F 0 ( z )( t 0 − z ) + · · · so that | F ( t 0 ) | ≈ | F 0 ( z ) | | t 0 − z | . Hence we arriv e at, under suitable technical assumptions, that | t 1 − z | ≤ C | t 0 − z | 2 (11) 4 t t 0 t 1 Figure 1: An example of the Newton estimate being f ar from a root where C is a constant that depends on F , z and t 0 . Notice that this is a restatement of the fact that Ne wton’ s method is quadratically con ver gent. But we must be careful; estimate (11) is local in nature — it depends strongly on the assumptions that t 0 is already close to z and that F 0 ( z ) is not close to 0 . If this is not the case, then there is nothing reasonable to say about | t 1 − z | . Figure 1 illustrates this problem. Returning to our problem, we use F ( τ ) := ∂ τ { γ ( τ ) + δ γ ( τ ) } with τ 0 = 0 , so that τ 1 = − ∂ τ δ γ (0) ∂ τ τ γ (0) + ∂ τ τ δ γ (0) ; τ 1 ≡ δ τ of expression (5). Hence we conclude that indeed estimate (5) does provide a first-order approximation to the delay , provided that the true critical point is close to zero and that ∂ τ τ γ (0) + ∂ τ τ δ γ (0) is not close to zero. 4 Conclusion: Why estimate (8) is of first order W e consider the dif ference between expression (5), which we kno w to be first-order, and estimate (8) of [1], which we wish to show is also first-order . In fact we claim that the difference between these two expressions is of second order, from which our claim follows. Specifically , we hav e     ∂ τ δ γ (0) ∂ τ τ γ (0) + ∂ τ τ δ γ (0) − ∂ τ δ γ (0) ∂ τ τ γ (0)     = | ∂ τ δ γ (0) |     1 ∂ τ τ γ (0) + ∂ τ τ δ γ (0) − 1 ∂ τ τ γ (0)     = | ∂ τ δ γ (0) |     ∂ τ τ δ γ (0) ∂ τ τ γ (0)( ∂ τ τ γ (0) + ∂ τ τ δ γ (0))     . (12) 5 Follo wing expression (3), we write | δ γ (0) | =       t 2 Z t 1 s ( t ) δ s ( t )d t       ≤ max t 1 ≤ t ≤ t 2 | δ s ( t ) | t 2 Z t 1 | s ( t ) | d t, which we rewrite symbolically as | δ γ (0) | ≤ C 1 | δ s | . Herein and below , C 1 , C 2 , etc. stand for generic constants. T aking the deri vati ve, we ha ve | ∂ τ δ γ (0) | =       t 2 Z t 1 ∂ t s ( t ) δ s ( t )d t       ≤ max t 1 ≤ t ≤ t 2 | δ s ( t ) | t 2 Z t 1 | ∂ t s ( t ) | d t, which we rewrite symbolically as | ∂ τ δ γ (0) | ≤ C 2 | δ s | . (13) Differentiating ag ain, we have | ∂ τ τ δ γ (0) | ≤ C 3 | δ s | . (14) Using estimates (13) and (14) in expression (12), we ha ve, for suf ficiently small | δ s | and ∂ τ τ γ (0) 6 = 0 ,     ∂ τ δ γ (0) ∂ τ τ γ (0) + ∂ τ τ δ γ (0) − ∂ τ δ γ (0) ∂ τ τ γ (0)     ≤ C 4 | δ s | 2 , which means that the difference between the first-order estimate (5) and estimate (8) of [1] is second-order in δ s . Hence, we conclude that estimate (8) is indeed a first-order estimate. T o demonstrate that estimate (8) is of the first order , we remained in the context of steps (58) to (65) of [1], which are expressions (1) and (8), herein. There is no need to anticipate Fréchet deriv atives that are to result from this deri vation; in other words, there is no need to let δ γ tend to zero, as suggested by Nolet (2009, pers. comm.). Hence, the formulation herein, albeit technical, strengthens the first-order validity of estimate (8) of [1]. Acknowledgments The authors are grateful for discussions with Qinya Liu, T arje Nissen-Meyer , Daniel Peter , Michael Rochester and Jeroen T romp, for email comments of Adam Baig, Shu-Huei Hung and Guust Nolet, for editorial help of David Dalton, for help of Frederik Simons in making T ony Dahlen’ s notes av ailable. MS’ s research was supported by the Natural Sciences and Engineering Research Council of Canada, and a part of his work was done at Princeton Uni versity during a sabbatical year . The authors are grateful for the encouraging atmosphere at Casaleggio, where the ideas presented came to fruition. Refer ences [1] F . A. Dahlen and S.-H. Hung and G. Nolet (2000) Fréchet kernels for finite-frequency traveltimes—I. Theory , Geo- phys. J. Int, 141, 157-174. [2] A. Ostrowski (1973) Solution of Equations in Euclidean and Banach Spaces, Academic Press. [3] G. Nolet (2008) A bre viary of seismic tomography: Imaging the interior of the Earth and Sun, Cambridge Univ ersity Press. 6