Inverse Methods for Atmospheric Sounding: Theory and Practice

Transcription

1 Inverse Methods for Atmospheric Sounding: Theory and Practice Clive D. Rodgers July 14, 2016 Page 25, line -3: Remove minus sign. [Thanks to Randy VanValkenburg] Page 26, Figure 2.4: ERRATA The figure labelling is inconsistent. Either the K T S 1 ɛ K should be (K T S 1 ɛ or the other two S s should be S 1 s. [Thanks to Justus Notholt.] Page 27, Equation 2.34: where y a = Kx a, and on page 28, Equation 2.38: K) 1 x = S 1 2 a (x x a ) and ỹ = S 1 2 ɛ (y y a ) (1) x = T T a (x x a ), ỹ = T T ɛ (y y a ) and K = T T ɛ KT T a, (2) [The y a was omitted in both cases.] Page 28, After Equation 2.40: Insert: where y, x and ɛ are now of dimension p, the rank of K. And in line -3, replace I m by I p. [Thanks to Randy VanValkenburg] Page 30, Equation 2.42 Replace both x s by y s. [Thanks to Randy VanValkenburg] 1

2 Page 31, Equation 2.57: Append +m n : d n = tr(s ɛ [KS a K T + S ɛ ] 1 ) = tr([k T S 1 ɛ Page 31, Equation 2.58: K + S 1 a ] 1 S 1 a ) + m n. (3) The term Gy should clearly be G(y Kx a ), from Eq. (2.44). However it seems neater to rewrite the sentence and equation as: For the moment simply note, from Eq. (2.44), that it relates the expected state ˆx to the true state x: [Thanks to Jörn Ungermann] Page 68, after Equation 4.16: ˆx x a = A(x x a ) + Gɛ (4) If the measurement errors are independent we can expand this as P (x y 1, y 2,..., y l ) = P (y 1 x)p (y 2 x)... P (y l x)p (x) (5) P (y 1, y 2,..., y l ) P (x) = P (y i x). (6) P (y 1, y 2,..., y l ) To find the maximum probability solution (i.e. MAP if there is a priori, or ML if there is not) we maximise with respect to x, so the term P (y 1, y 2,..., y l ) can be omitted as it is independent of x. [The point is that the y i are not independent. My thanks to Randy Van Valkenberg for this one.] i Page 69, Equation 4,24: The derivative applies to both terms on the left hand side. brackets is required. [Thanks to Randy VanValkenburg] Another pair of Page 74, Equation 4.52 This equation should read: G = S a K T (KS a K T + γs ɛ ) 1 (7) [Thanks to Mick Christi] 2

3 Page 77, Equation 4.65 Should be r(z) = 12 [z c(z)] 2 A 2 (z, z ) dz / ( A(z, z ) dz ) 2, (8) [The [z c(z)] 2 was [z c(z)] 2. Thanks to Yasjka Meijer.] Page 92, Equation 5.34: [K T K not KK T.] x i+1 = x i + (K T K + γ i I) 1 K T [y F(x i )] (9) Page 97, section Sequential updating [This topic is poorly described, and Equation (5.47) has errors. I have rewritten it.] If the measurement error covariance is diagonal, as it often is, then there is a further economy available in the case of the m-form. Within each iteration cycle, the state estimate can be updated sequentially, one measurement at a time, thus replacing the matrix inverse in Eq. (5.44) by a scalar reciprocal. The process is as follows, where the column vector k j is the weighting function for the j-th channel, i.e. the j-th row of K, and superscript i refers to the iteration cycle. x i+1 0 := x a S 0 := S a for { j := 1 to m do : x i+1 j := x i+1 j 1 + S j 1k j [y j F j (x l ) + k T j (x l x i+1 j 1 )]/(kt j S j 1k j + σj 2) S j := S j 1 S j 1 k j k T j S j 1/(k T j S j 1k j + σj 2) x i+1 := x i+1 m (10) For each iteration cycle, the estimate starts with the a priori, and updates it channel by channel. The choice of linearisation point x l for each of these updates determines the convergence rate. The best linearisation point would of course be the final solution, but this is not known initially. For the first iteration cycle, the best we can do for channel j is the current estimate x 1 j 1. For subsequent cycles, the end point of the previous cycle, x i is likely to be better. This has two potential advantages over updating with a vector of measurements. Some measurements are likely to be more linearly related to the state vector than others; if they are assimilated first in the initial iteration cycle, then the intermediate state will be closer to the final solution when the more non-linear measurements are used, so the linearisation will be more accurate, and fewer iterations may be needed... [N.B. This erratum has been corrected 23 Jan An x a in the fourth line of the display equation has been replaced by x i+1 j 1. My thanks to 3

4 Jörn Ungermann. Also 13 Jul 2016, when k T i to Nathaniel Livesey.] was replaced by+k T j. Thanks Page 108, Section 6.3 This section is a mess. As well as about seven typos, it is not clearly written. Here is another attempt for the part of this section on page 108: However it is more convenient to use the alternative formulation in terms of singular vectors, mainly because singular vectors are orthogonal. Put C 1 U tλ 2 t V tt, (11) where U t and V t are matrices of left and right singular vectors of C truncated to t columns, and the truncated matrix of singular values has been written as Λ 2 t for reasons which will be apparent later. This gives ˆx = WU tλ 2 t V tt y (12) = WU tλ 2 t V t T Kx + WU t Λ 2 t V tt ɛ. (13) If ɛ has covariance σ 2 ɛ I the error term will have covariance σ 2 ɛ WU tλ 4 t U tt W T, each left singular vector contributing an independent error equal to a column of WU t multiplied by a random normal variable with standard deviation (σ ɛ /λ 2 i ). Rather than retaining a general representation function let us simplify matters by only considering W = K T. In this case C = KK T is symmetric and its singular vectors and eigenvectors are identical, and equal to U, the left singular vectors of K, K = UΛV T. Thus U t = V t = U t. The singular values (and eigenvalues) of C are then the squares of the singular values of K (hence the choice of Λ 2 t above). In this case ˆx = K T U t Λ 2 t U T t y (14) = V t Λ 1 t U T t y (15) = t i=1 λ 1 i v i u T i y (16) because K T U t = V t Λ t. Note that Eq. (15) is equivalent to using a truncation of the pseudo inverse K = VΛ 1 U T based on K = UΛV T. The relation of the retrieval to the state vector is therefore ˆx = V t Λ 1 t U T t Kx + V t Λ 1 t U T t ɛ (17) = V t Vt T x + V t Λ 1 t U T t ɛ (18) showing that the averaging kernel matrix is A = V t Vt T. If ɛ has covariance σɛ 2 I the error term will have covariance σɛ 2 V t Λ 2 t Vt T, the error patterns being given by e i = (σ ɛ /λ i )v i. [Thanks to Lars Hoffmann and Jörn Ungermann] 4

5 Page 124, Equation 7.16: Remove the hats from x t and S t in the last two equations. [Thanks to Bojan Bojkov for pointing this out.] Page 139, Equation 8.23: [Sign error.] Page 144, Equation 9.12: This equation should read: i.e change B i 1 to B i+1. [Thanks to Rachel Hodos] Page 144, Equation 9.16: This equation should read: y = U T (y + K x 0 ) = ΛV T x + ɛ r + ɛ s, (19) L n = (L 0 B 1 )τ 0 + B n 1 n + τ i ( B i B i+1 ) (20) i.e. omit the B i in the last term. [Thanks to Scott Paine] Page 147, Equation = B i + B i 1 B i χ i ( 1 e χ i ) Bi 1 e χi (21) As it stands, p in this equation is the partial pressure of dry air, not the total pressure as implied. However the coefficient 5748 taken from Kaye and Laby appears to be about a factor of 1000 too large in comparison with the formula adopted by the International Association of Geodesy in 1999, based on Ciddor (1996). Replace Equation 9.24 by: ( p T 0 n = 1 + N g p 0 T 11.27e ) 10 6 (22) T where p 0 = mb, T 0 = K, e is the partial pressure of water vapour in mb, and N g is the refractivity of dry air at STP, given by N g = /λ /λ 4 (23) at wavelength λ in micrometers. [Thanks to Susan Sund Kukawik for pointing this out] 5

6 Page 163, line 1: subject to z = W x.... Page 179, Equation 11.2 and following text: Ŝ z,ii = trace(s 1 2 a ŜS 1 2 a ) = trace(ŝs 1 a ). (24) i From Eqs. (2.79) and (2.80), we can see that trace(ŝs 1 a ) = trace(i n A) = n d s. (25) so that minimising the trace is the same as maximising the degrees of freedom for signal. Alternatively, the information content of the retrieval, Eq. (2.73), could be optimised by maximising the determinant of ŜS 1 a. The same result should be obtained if the information content of the measurement alone is maximised, provided an information-preserving retrieval is used, e.g. an optimal estimator. [Thanks to Anu Dudhia for this one.] Page 199, Section A.2 This section contains a significant error. In the line after Eq. (A.7) I have assumed without justification that L is normalised. Usually it is not. This could be corrected by replacing this line by Thus L is a matrix of eigenvectors of A T..., when the rest of this section would be correct. However this is unsatisfying because it would be much more sensible to give the relationships for the case where R and L are both normalised. Therefore replace the part after Eq. (A.6) by the following:...then premultiplying and postmultiplying by L = (R T ) 1 A T L = LΛ (A.7) Thus L is a matrix whose columns are eigenvectors of A T, with eigenvalues which are the same as those of the corresponding column of R. However these eigenvectors are not necessarily normalised. Let ν i be the length of l i, the i-th column of L, i.e. ν i = ( l T i l i ) 1/2, and N be a diagonal matrix containing the ν i as its diagonal elements. Then L, the normalised matrix of eigenvectors of A T, is given by L = LN 1 and satisfies: A T L = LΛ (A.8) These are called the left eigenvectors, because they operate on A (rather then A T ) on the left: L T A = ΛL T, while R are the right eigenvectors. By postmultiplying Eq. (A.5) by R 1 = L T = NL T we can express A in terms of its eigenvectors as A = RΛNL T = λ i ν i r i l T i (A.9) i 6

7 which is described as a spectral decomposition of A. In the case of a symmetric matrix S, where S = S T, we must have R = L by symmetry. By premultiplying (A.8) by L T and postmultiplying its transpose by L we see that L T LΛ = ΛL T L, so L T L must be diagonal. As L is normalised we must have L T L = LL T = I or L T = L 1, i.e. the eigenvectors are orthonormal, and N = I. In this case the eigenvalues are all real. If the matrix is positive definite the eigenvalues are all greater than zero, and similarly for a negative definite matrix. The following is a summary of useful relations involving eigenvectors: Asymmetric Matrices AR = RΛ (26) L T A = ΛL T (27) NL T = R 1, NR T = L 1 (28) L T R = R T L = N 1 (29) A = RΛNL T = i λ iν i r i l T i (30) A 1 = RΛ 1 NL T (31) A n = RΛ n NL T (32) L T AR = N 1 Λ (33) L T A n R = N 1 Λ n (34) L T A 1 R = N 1 Λ 1 (35) A = i λ i (36) Symmetric Matrices SL = LΛ (37) L T S = ΛL T (38) L T = L 1 (39) LL T = L T L = I (40) S = LΛL T = i λ il i l T i (41) S 1 = LΛ 1 L T (42) S n = LΛ n L T (43) L T SL = Λ (44) L T S n L = Λ n (45) L T S 1 L = Λ 1 (46) S = i λ i (47) Square roots of matrices The relation A n = RΛ n NL T (where N = I for a symmetric matrix) can be used for arbitrary powers of a matrix, in particular the square root such that A = A 1 2 A 1 2. This square root of a matrix is not unique, because the diagonal elements of Λ in RΛ 2 NL T can have either sign, leading to 2 n possibilities. We only use square roots of covariance matrices in this book. In this case we can see that S 1 2 = LΛ 1 2 L T is symmetric. As well as these roots, symmetric matrices can also have non-symmetric roots satisfying S = (S 1 2 ) T S 1 2, of which the Cholesky decomposition, S = T T T where T is upper triangular, is the most useful, see section and Exercise 5.3. There are an infinite number of nonsymmetric square roots. If S 1 2 is a square root, then clearly so is XS 1 2 where X is any orthonormal matrix. The inverse symmetric square root is S 1 2 = LΛ 1 2 L T, and the inverse Cholesky decomposition is S 1 = T 1 T T. The inverse square root T 1 is triangular, and its numerical effect is implemented efficiently by back substitution. [My thanks to Randy VanValkenburg and Javier Martin-Torres for bringing this to my attention.] 7

8 Page 206, Equation B.8 and line before: The other expression follows from using this with Eq. (2.45), giving d n = tr(i m KS a K T [KS a K T + S ɛ ] 1 ) = m tr(kg) = m tr(k[k T S 1 ɛ K + S 1 a ] 1 K T S 1 = m n + tr(i n [K T S 1 ɛ = tr([k T S 1 ɛ K + S 1 a ɛ ) Minor clarifications and typographic errors Page 1, line 15: insert is after it. Page 4, line 12: insert of after kind. Page 9, line 2: replace b by w. Page 10, line -9: replace G j (ζ) by G i (ζ). Page 13, line -10: separate vector and x. Page 13, line -8: insert than after rather. K + S 1 a ] 1 K T S 1 ɛ K) ] 1 S 1 a ) + m n (48) Page 17, first line of second paragraph of 2.2.1: italicise coordinate system (RVV) Page 17, line 13: replace k j by k i for consistency with page 16. Page 17, middle of first paragraph of 2.2.1: Even when m > n,..., the rank cannot be greater than n,... (RVV) Page 17, line 19: change to called the row space of K or range of K T. Page 18, lines 13, 21 and 23: replace k j by k i for consistency with page 16. Page 20, line after Eqn. 2.8, replace and that by where. Page 21, line 1: delete is used. Page 21, Eqn. 2.15, replace S 1 by S 1 y. (DW) Page 22, line -16: insert of after pdf. Page 23, lines 5 and 7: replace P(y,x) by P(x,y). Page 23, line -3: replace This is what we want by This equation is what we need in order. (RVV) Page 25, line 3: replace constant by independent of x. (RVV) 8

9 Page 25, line after Eqn. 2.28: delete is used. (DW) Page 28, line 14: replace T T by T T ɛ. Page 28, line -2: replace Element by Elements. Page 29, line 13: replace ln p by ln(p/p 0 ). (RVV) Page 29, line 19: replace.. by.. Page 30, line -12: insert be after therefore. Page 33, section 2.5.2, line 2: replace it by its. (DW) Page 33, section 2.5.2, para. 2, line 1: delete one about. (DW) Page 33, section 2.5.2, para. 3, line 1: can be defined. (DW) Page 35, Eqn. 2.71, line 1: e should be e. Page 35, line -16: replace first the by a. Page 37, last para.,line 2: delete therefore. (DW) Page 39, Figure 2.5 caption: Solid: eigenvalues of a non-diagonal model covariance in decreasing order of size; dotted: the same for a diagonal covariance matrix with the same total variance. Page 40, line 8: replace on by one. Page 44, line -2: replace an by a. (DW) Page 45, Equations (3.3), (3.5), and (3.7): replace R(...) by R[...]. Page 46, section 3.1.4, last line:... total error in the measured signal relative.... (DW) Page 46, section 3.1.5, line 4:... need to know.... (DW) Page 47, line 13: replace first a i by a T i. Page 48, line 4: insert the before bias. Page 48, lines 8 and 9: replace b by ˆb. Page 49, middle paragraph, line 4: delete that. (DW) Page 50, section 3.2.3, line 3: delete second not. (DW) Page 50, line -5: insert that after terms. Page 51, section 3.2.6, line 4: hyper-ellipsoid. (DW) Page 54, line 6: replace it by its. 9

10 Page 54, line -11: delete second should. (DW) Page 55, section 3.3, line -2: delete give. Page 55, line -10: replace matrices by matrix. (DW) Page 56, line 7 of text, replace that by they. (DW) Page 60, Figure 3.8 caption: replace eight by ten. Page 61, line 1: delete first to. (DW) Page 61, line 3: replace S ɛ + K T b S bk b by S ɛ + K b S b K T b Page 68, line before Equation 4.19: replace maximising by minimising. (RVV) Page 62, Figure 3.11 caption: at end insert Original sinusoid: dashed. Response: solid.. Page 66, line 4: replace x 0 by x 0. Page 73, lines -14, -7: replace measurement error by retrieval noise. Page 72, lines -6, -5: replace maximum likelihood by MAP. Page 73, section 4.3, line -4: delete the. (DW) Page 76, line -3: replace (3.23) by (4.59). Page 77, line 3: insert the before standard. Page 77, line 8: delete the first of. Page 78, line -10, replace large by small! (DW) Page 82, line 1: insert treatment of after exhaustive. Page 83, line 12:... example has... Page 84, line after Eqn. 5.3: delete second is. (DW) Page 84, Equation (5.4) : delete 2 on LHS of Equation (5.4). Page 85, line 3 before Eqn. 5.8: literature. (DW) Page 85, line 3 after Eqn. 5.7: replace Jacobean by Jacobian. (RJW) Page 86, lines 2, -5: replace moderately linear by moderately non-linear. Page 90, line 3: delete so after this. Page 90, line -11: replace then by than. 10

11 Page 91, line 11: replace 11 by 12. Page 91, line 14: replace moderately linear by moderately non-linear. Page 91, line -10: may be needed... (DW) Page 93, line -6: evaluation of its Jacobian... (DW) Page 94, line -18: delete be after been. Page 95, line -11: by the pivot (B (DW) Page 95, line before Eqn. 5.40: replace involved by involves. Page 95, line -3: insert it after if. Page 99, 5/6 lines after Equation 5.49: replace corresponding by correspond. Page 103, Equation (6.8): insert minus on RHS of equation. Page 103, Equation (6.9): insert minus on LHS of equation. Page 105, Equation (6.12): change i =... m, to i = 1,... m,. Page 107, line 12: replace Not by Note. Page 113, line -9: insert a after Thus. Page 116, Figure 6.3a: Abcissa should be labelled eigenvector. Page 119, line 8: replace give by gives. Page 120, line 11: replace give by gives. Page 124, Equation (7.16): Remove the hats from x t+1 and S t+1 in the first two equations. They are superfluous as the double prime already indicates a backwards estimate. Also, remove the commas in the subscripts on the left hand side of these equations. Page 145, Equation (9.19): insert minus before κ k. Page 146, line 6: replace is by a. Page 146, line 7: delete first the. Page 146, Equation (9.23): Encase expression on RHS after in square brackets. Page : replace all occurrences of (a), (b) and (c) by (i), (ii) and (iii) respectively. Page 159, line -4: replace and by an. (DW) 11

12 Page 160, line 10:... has been chosen for... (DW) Page 160, line -5:... is to be... (DW) Page 161, line 8:... or a coarse representation... Page 162, section , line 3: replace reduces by reduced. Page 164, Subtitle : replace a priori by a posteriori. Page 167, line 9: replace measure by measured. Page 168, line 10: insert this after Unfortunately. Page 168, line -11: insert a comma after strictly Page 171, Equation (10.38): replace + ɛ r + ɛ s by + Gɛ r + Gɛ s. Page 172, line -7: replace moderately linear by moderately non-linear. Page 173, Equation (10.50): insert W T between terms in brackets (i.e. change...w] 1 [Ŝ 1... to...w] 1 W T [Ŝ 1... ). Page 177, line 4:... stage be regarded... (DW) Page 186, line -2: delete second priori. Page 187, line 13: delete is being used. (DW) Page 193, Equation (12.19): change S T ɛ 2 to S ɛ2. Page 194, line 11: replace our by or. Page 194, line 11: replace compared by compare. Page 195, line 12: insert in after Substituting. Page 207, line 4: replace.. by.. Page 229, Bretherton reference: J. Climate. (DW) My thanks to Daniela Wurl for finding these. My thanks to Mick Christi for his amazing diligence in finding all these! (RVV) My thanks to Randy VanValkenburg for finding these. (RJW) My thanks to Robert Watson for finding this. The following are corrected in the published version, but may exist in some late pre-publication drafts: 12

13 Page 53, Figure caption 3.2: The most significant ten error patterns of the a priori covariance matrix S ij = σ 2 a exp( i j δz/h), for σ a 3 K and h = 1 Page 162, Equation 10.8: P195, Eqn W = (W T S 1 e W) 1 W T S 1 e. (49) (K 1 S c K T 1 + S ɛ1 ) 1 2 K1 S c K T 2 (K 2 S c K T 2 + S ɛ2 ) 1 2 (50) P195, line 6 from bottom:... more or less... 13