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Appendix A: Frequency correlations for the observables

In this Appendix, we study the correlations in frequency space that result from a finite observation duration T. First, we collect some definitions.

Since observations are discrete, we only consider discrete time points t_j = h_tj, j ∈Z in this paper, to avoid some technical difficulties. As a consequence, the frequency variable ω is 2π/h_t periodic. However, our definitions of the discrete Fourier transform, and its inverse are chosen such that we obtain the time-continuous case in the limit h_t → 0: (A.1)We need the following: an orthogonal projection D_N of L²( [ − π/h_t,π/h_t ]) onto the space Π_N of 2π/h_t-periodic trigonometric polynomials with a degree ≤N and the Dirichlet kernel ; the Fejér smoothing operator F_N: L²( [ − π/h_t,π/h_t ] ) → Π_N with the Fejér kernel ℱ_N; and the projected periodic Hilbert transform H_N: L²( [ − π/h_t,π/h_t ] ) → Π_N with kernel ℋ_N. They are defined by Here, sgn(k): = 1 and sgn(−k): = −1 for k ∈N, and sgn(0): = 0. The transform H_N is related to the standard periodic Hilbert transform H with a convolution kernel ℋ(ω) = cot(ω/ 2) by H_N = HD_N = D_NH. With our convention for the Fourier transform, the Fourier convolution theorem is . In particular (with f(ω) = ℱ_N(h_tω) and , etc.), we have (A.2)To simplify the notations, the cross-covariance (respectively its expectation value) C(x_a,x_b,ω) (respectively ) can be simply written as C_ab(ω) (resp. ), and similarly, the weight functions W(x_a,x_b,ω) can be W_ab(ω). We show the following theorem on the correlation function C_ab(ω₁,ω₂) defined as (A.3)The covariance between the wavefield at two frequencies ω₁ and ω₂ can be expressed as The second term is bounded by (A.7)For a stationary Gaussian time series, the error of the approximate noise model (Eq. (8) in Gizon & Birch 2004) is bounded by (A.8)The proof of the above theorem is given below.

By the definition of φ_T, the covariance betweeen the observations is given by (A.9)where j = k − l, m = −l, and g_ω(j) = ∑ _{|m|,|j − m| ≤ N}e^{iωh_t(j − m)} = ∑ _{|m|,|j + m| ≤ N}e^{− iωh_tm}. For ω₁ = ω₂, we have g₀(j) = 2N + 1 − |j|, so Eq. (A.4) for this case follows from Eq. (A.2). We now consider the case ω₁ ≠ ω₂. For j> 0, we have If t_j< 0, then . Inserting the expression for g_ω in Eq. (A.9), using the identity sin(x − y) = sinxcosy − cosxsiny for x = T(ω₂ − ω₁) / 2 and y = h_t(ω₂ − ω₁) | j | / 2, and finally using Eq. (A.2) leads to To bound II_ab, we may assume that | ω₂ − ω₁ | ≤ π/h_t without loss of generality due to 2π/h_t periodicity. Using the mean value theorem, Eq. (A.2), and the inequality for , we obtain (A.10)This yields Eq. (A.7). It also implies Eq. (A.8) for j ≠ l, since ; that is I_ab(h_ωj,h_ωl) = 0. To show Eq. (A.8) for j = l, we use the bound

Appendix B: Frequency correlations for travel times

In this Appendix we derive the noise covariance matrix for the cross-covariance function C and for the travel time τ when the frequency correlations are taken into account. Appendix A has shown that taking the frequency correlations into account leads to an additional term of order 1 /T in the covariance of the observables at the grid points. As the covariance betweeen two travel times is also of order 1 /T, it is of interest to look if this correction should be taken into consideration. This Appendix proves that the extra term in 1 /T of the observable covariance only leads to an additional term in 1 /T² for the travel times. We also underline the main difficulties that will occur when computing higher order moments of C and τ.

With our convention Eq. (A.1), the Fourier transform is unitary up to the factor . It follows from definition (3)that Therefore, (B.1)The first difficulty is to evaluate the quantity, Cov [ C₁₂(ω₁),C₃₄(ω₂) ]. For a higher order moment, we also need to evaluate Cov [ C₁₂(ω₁)C₃₄(ω₂),C₅₆(ω₃) ] and Cov [ C₁₂(ω₁)C₃₄(ω₂),C₅₆(ω₃)C₇₈(ω₄) ]. The way to deal with these terms is presented in Appendix C, where it is shown that (B.2)It leads to (B.3)The second difficulty comes from the evaluation of these integrals; that is the evalution of linear functionals of the expectation value of the cross-covariance C given by the weight functions W. Similarly, we need to be able to evaluate, for higher order moments, The method to compute these terms is presented in Appendix D. Applying the result for the second order moment presented in Appendix D.1 leads to the following result:

The travel-time covariance for finite T is given by the travel-time covariance for infinite observation time (Eq. (13)) plus a correction that decreases as 1 /T²(B.6)where m corresponds to the regularity (the number of derivatives) of the functions and W_ab, and (B.7)

Remark concerning the setting of Gizon & Birch (2004)

In Gizon & Birch (2004), it was assumed that (B.8)so the covariance of C is (B.9)We note that Eq. (B.9) is exact. It differs slightly from Eq. (C8) in Gizon & Birch (2004), which incorrectly contained an additional term. It leads to the covariance between travel times (B.10)We note that Eq. (B.10) is identical to Eq. (28) in Gizon & Birch (2004), as the extra term in the covariance of C was actually neglected by the authors. Taking the frequency correlations into account, Eq. (B.8) is no longer valid, and correction terms have to be added to Eqs. (B.9), (B.10). These correction terms are given in the previous result.

Appendix C: Noise covariance matrix for high order cross-covariances

In this section, we present the way to compute the noise covariance matrices for the cross-covariance function C: As the cross-covariance function can be written as a function of the observables (C.4)Equations (C.1)–(C.3) imply that the moments of 4, 6, and 8 of the observables have to be computed. In the next section we present a formula to compute high order moment of Gaussian variables. Then, we apply this formula to compute Eqs. (C.1)–(C.3).

Appendix C.1: Expectation value of high-order products of Gaussian random variables

We have seen that the moments of order 4, 6 and 8 of the observables have to be computed to find the noise covariance matrix for cross-covariances and products of cross-covariances. A formula to compute the (2J)th-order moment of a multivariate complex normal distribution with zero-mean can be found in Isserlis (1918): (C.5)where μ and ν have distinct values in ˜1,2J¨ and the set ℳ^J is defined by (C.6)Here, we used the notation ˜1,2J¨ for the set of all integers between 1 and 2J. To better understand Eq. (C.5), let us explain it for the case J = 2. In this case, Eq. (C.5) can be written as (C.7)where the indices i,j,k,l must satisfy i<j, i<k and k<l according to Eq. (C.6). This forces that i = 1. Then, we can have k = 2 or k = 3. If k = 3, then l = 4 and j = 2. If k = 2, then we have again two possibilities: l = 3 and j = 4 or l = 4 and j = 3. So three combinations are possible: (1,2,3,4), (1,4,2,3), and (1,3,2,4). This leads to (C.8)In particular, we have (C.9)which is the formula required to compute the moment of order 4 in Eq. (C.1). For J = 3, Eq. (C.5) becomes (C.10)where the indices i,j,k,l,m,n must satisfy i<k<m (since the sequence (μ_i) must increase) and i<j, k<l and m<n (since μ_i<ν_i), according to Eq. (C.6). Hence, we obtain (C.11)A problem is that the cardinality of the set ℳ^J is (4J) ! / [ (2J) ! 4^J ] (Isserlis 1918) increases exponentially. The sum in Eq. (C.5) contains three terms for J = 2 and 15 for J = 3, as shown above. Unfortunately, it leads to 105 terms for J = 4 so it is not convenient to write them down explicitly, and we just list the main guidelines in Sect. C.4.

Appendix C.2: Second order moment of C

In the original paper, the fourth order moment of the observables was guessed after looking at all the possible cases in the Fourier domain. Using Eq. (C.9) and the definitions of C_ab and C_ab (Eqs. (C.4), (A.3)) and recalling that as φ_j(t) is real-valued, the covariance matrix between two cross-covariances is readily computed as follows: (C.12)

Appendix C.3: Third order moment of C

In this section, we compute the sixth order moment of the observables defined by Eq. (C.2). After writing the cross-correlations as a function of the observables, we need to compute the moment of order 6 of the observables. This can be done using Eq. (C.11) with z₁ = φ₁(ω₁), z₂ = φ₂(ω₁), z₃ = φ₃(ω₂), z₄ = φ₄(ω₂), z₅ = φ₅(ω₃), and z₆ = φ₆(ω₃). After integration against weight functions, it will turn out that the order of the different terms in 1 /T depends on their degree of separability. Therefore, we denote by the sum of the terms, which can be written as product of at most N functions of disjoint subsets of the set of variables { ω₁,ω₂,ω₃ }. Then (C.13)where (C.14)and (C.15)

Appendix C.4: Fourth order moment of C

This section is devoted to the computation of the eigth order moment of the observables defined by Eq. (C.3). Writing the cross-correlations as a function of the observables leads to (C.16)In Eq. (C.16) and in the following we omit the argument ω_j of the observables φ_{2j − 1} = φ_{2j − 1}(ω_j) and φ_2j = φ_2j(ω_j). As for the moments of order 4 and 6, we can calculate this expression. As explained in Sect. C.1, the moment of order 8 contains 105 terms, so we do not write explicitely all the terms. As for the moments of order 6, we arrange the terms as (C.17)where is the the sum of all terms, which can be written as a product of at most N functions of disjoint subsets of the set of variables { ω₁,ω₂,ω₃,ω₄ }. The three terms will be computed below.

Expression for

These terms are the ones from the subset given by Eq. (C.6) from which the observables use the same frequencies in two expectation values, for example . It leads to the following formula: (C.18)Calculating all the expectation values implies which can be written in terms of the covariance between two cross-covariance functions,

Expression for

Two kinds of products in Eq. (C.5) will lead to terms with only two frequency integrals:

• In two expectation values, the constraints on ω are the same; for example, (they lead to the first two terms in Eq. (C.19));

• In one expectation value, the observables use the same frequencies; for example, .

Computing all the terms, one can show that (C.19)The terms Cov [ C,C ] and Cov [ CC,C ] that appear in this expression can be computed using Eqs. (C.12), (C.13).

Expression for

All other terms lead to terms that contain only one frequency integral in the covariance of the product of travel times. After reorganizing all the terms, one can show that can be written as

Appendix D: Evaluation of separable linear functionals of nonseparable products of C_ab’s

In this section, we derive asymptotic expansions of the terms, in 1 /T as T → ∞ and explicit formulae for the leading order terms. Recall that C defined in Eq. (A.3) depends on T, although this is suppressed in our notation.

Appendix D.1: Functionals of nonseparable products of two C_ab functions

In this subsection, we show that (D.4)where is defined by Eq. (B.7) if and have m derivatives and W₁₂ and W₃₄ have m − 1 derivatives.

Plugging Eq. (A.4) into the left hand side of Eq. (D.4), we arrive at a sum (2π)²(X + 2Y + Z) involving the following three terms: We repeatedly use the following transformation of variables formula for functions f(ω₁,ω₂), which are 2π/h_t periodic in both variables: (D.8)Even though the Jacobian of this transformation of variables is 1 / 2, no factor appears, since we integrate over a domain that can be reassembled to two periodicity cells on the right hand side.

Using Eq. (D.8) and noting that , the first term can be written as We want to interpret the inner product as a convolution with ℱ_2N evaluated at 0. First, we note that by a change of variables: . Let f(ω₁,ω₂) be 2π/h_t periodic in both arguments and . Then and hence (D.9)where F_2N always acts on the second argument. As , it follows that Since , we get an additional if we omit the orthogonal projections D_2N in the last equation.

To bound Y (Eq. (D.6)), we again apply the change of variables in Eq. (D.8) to obtain where f has uniformly bounded derivatives of order m − 1. When T tends to infinity, this corresponds to a high order Fourier coefficient and thus can be made as small as desired. In particular, by repeated partial integration, (D.10)The term Z (Eq. (D.7)) can be transformed, in the same way, and after using that , we find that where the higher order term comes from which is similar to Eq. (D.10). As lim_{n → ∞}H_2Nf = Hf and all the terms in the integrals are bounded it follows that X is of order 1 /T². Gathering the expressions for the three terms X, Y, and Z leads to Eq. (D.4).

Appendix D.2: Functionals of nonseparable products of three C_ab functions

Let C be defined by Eq. (A.3), and W_i represent some functions of ω. Then, we have the following expension: (D.11)Using Eq. (A.4) in the left hand side of Eq. (D.11), four different types of terms have to be studied

where the expressions I and II are given by Eqs. (A.5), (A.6) respectively.

We use the change of variables: (D.16)where the Jacobian 1 / 3 does not appear for the same reason as in Eq. (D.8). Applying this to X, we obtain where ω_i can be replaced by the corresponding value in . The role of the Fejér kernel is played by the function: where we have used the change of variables m = j − l, n = k + l, and o = l. If denotes the corresponding convolution operator and is the two-dimensional orthogonal projection, we can use the inequality max( | m | , | n | , | m − n | ) ≤ | m | + | n | to obtain (D.17)If f(ω₁,ω₂,ω₃) is 2π/h_t periodic in all of its arguments and , we find in analogy to Sect. D.1 that and hence . With Eq. (D.17), we obtain (D.18)The terms Y₁ is of very high order using the same method than in Sect. D.1. The term Y₂ can be treated in the same way as it also contains a cosine that oscillates with T. Finally, Z is of order 1 /T³ using a similar demonstration than in Sect. D.1.

Appendix D.3: Functionals of nonseparable products of four C_ab functions

Let C be defined by Eq. (A.3), and W_i represent some functions of ω. Then, we have the following expension: (D.19)As in the previous proof, different terms have to be treated. The terms with combinations of the expressions I and II can be bounded by the same methods as in Sect. D.2, and the term involving only expressions II can be bounded as in Sect. D.1. The only different term is (D.20)Here, small adaptions of the argument in Section D.2 with the change of variables, (D.21)lead to the formula (D.22)

Appendix E: Noise covariance matrix for products of travel times

Appendix E.1: Third order moment of the travel times

Using the definition of the travel times, we obtain that the covariance for the product of travel times is given by: Using Eq. (C.13) and the two results presented in Sects. D.1 and D.2, we can express the covariance for three travel-times as (E.1)where is the expectation value of τ_j and the covariance involving two travel times can be computed with Eq. (13).

Appendix E.2: Analytic formula for the covariance matrix for products of travel times

In this section, we derive the main result of this paper. It gives an analytic expression for the covariance matrix between a product of travel times. Using the definition of the travel times, one can show that the covariance of the product of travel times is given by (E.2)In Appendix D, we have shown that not all the terms lead to the same number of frequency integrals. This implies that the covariance given by Eq. (E.2) has terms of different order with respect to the observation time T. The terms containing three integrals in ω are of order T^-1, while the other ones are of order T^-2 and T^-3. We write the covariance as the sum between three terms for the different orders: (E.3)The terms of order 1 /T⁴ come from the correlation between the frequencies in the frequency domain as detailed in Sect. B for the covariance between travel times. The other terms are detailed below.

Term Z₁ of order T^-1

Looking at Eq. (E.2), one can see that this term is composed of

• all the terms involving Cov [ C,C ],

• the terms with two integrals in ω for the terms with Cov [ CC,C ] (term ),

• the terms with three integrals in ω for the terms with Cov [ CC,CC ] (term ),

where C is a generic cross-covariance. Reorganizing terms leads to the formula Eq. (16) for Z₁.

Term Z₂ of order T^-2

Looking at Eq. (E.2) one can see that this term is composed of

• the terms with one integral in ω for the terms with Cov [ CC,C ] (term ),

• the terms with two integrals in ω for the terms with Cov [ CC,CC ] (term ).

Reorganizing terms leads to the formula Eq. (18) for Z₂.

Term Z₃ of order T^-3

The terms of order T^-3 come from the terms with only one integral in ω in Cov [ CC,CC ] (term ). This yields Eq. (20) for Z₃.

Appendix F: Far-field approximation for

In this section, we give approximate expressions for the different terms that compose Eq. (15) for in the far field (Δ → ∞). We start with the definitions of Z₁, Z₂, and Z₃:

In the far field, we have . If we suppose that , then the global behavior of the four terms is (F.1)We can thus see that the global behavior of the terms is

As we can conclude that the first term in Z₂ and the one in Z₃ are dominant in this case. We can go further to see for which observation time T_c these two last terms intersect in the case of difference travel times. If the window function f(t) in the definition of W_diff, as defined by Eq. (4), is a Heavyside function, then we have (Gizon & Birch 2004) (F.2)For a p-mode ridge κ_r = κ_r(ω), the function can be written in the far field as Gizon & Birch (2004): (F.3)where κ_i is the imaginary part of the wavenumber at resonance and represents attenuation of the waves. The sums in Eq. (F.1) can be approximated because the cosine in Eq. (F.3) oscillates many times within the frequency width ξ of the envelope of : Using the numerical value of ξ/ 2π = 1mHz, the observation time T_c at which the two terms are equal is (F.4)For T>T_c, Z₂/T² is the dominant term. As the observation time is traditionally of at least eight hours in helioseismology, the term of order 1 /T³ can be neglected.

© ESO, 2014