Issue |
A&A
Volume 567, July 2014
|
|
---|---|---|
Article Number | A137 | |
Number of page(s) | 20 | |
Section | The Sun | |
DOI | https://doi.org/10.1051/0004-6361/201423580 | |
Published online | 30 July 2014 |
Online material
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 tj =
htj,
j ∈Z in
this paper, to avoid some technical difficulties. As a consequence, the frequency
variable ω
is 2π/ht
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 ht →
0: (A.1)We need the
following: an orthogonal projection DN of L2( [ −
π/ht,π/ht
]) onto the space ΠN of 2π/ht-periodic
trigonometric polynomials with a degree ≤N and the Dirichlet kernel
;
the Fejér smoothing operator FN: L2( [
−
π/ht,π/ht
] ) → ΠN with the Fejér kernel
ℱN; and the projected periodic Hilbert
transform HN: L2(
[ −
π/ht,π/ht
] ) → ΠN with kernel ℋN. They are
defined by
Here,
sgn(k): =
1 and sgn(−k): = −1 for
k ∈N, and
sgn(0): =
0. The transform HN is related to the
standard periodic Hilbert transform H with a convolution kernel ℋ(ω) =
cot(ω/ 2) by HN =
HDN =
DNH.
With our convention for the Fourier transform, the Fourier convolution theorem is
.
In particular (with f(ω) =
ℱN(htω)
and
,
etc.), we have
(A.2)To simplify the
notations, the cross-covariance (respectively its expectation value) C(xa,xb,ω)
(respectively
) can
be simply written as Cab(ω)
(resp.
), and similarly, the weight
functions W(xa,xb,ω)
can be Wab(ω).
We show the following theorem on the correlation function Cab(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 φT, the covariance
betweeen the observations is given by (A.9)where
j = k −
l, m = −l, and gω(j) = ∑
|m|,|j −
m| ≤
Neiωht(j
− m) = ∑
|m|,|j +
m| ≤ Ne−
iωhtm.
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
tj<
0, then
.
Inserting the expression for gω in Eq. (A.9), using the identity sin(x − y) =
sinxcosy −
cosxsiny for x =
T(ω2 − ω1)
/ 2 and y =
ht(ω2 −
ω1) | j | /
2, and finally using Eq. (A.2) leads to
To
bound IIab,
we may assume that |
ω2 − ω1 | ≤
π/ht
without loss of generality due to 2π/ht
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 Iab(hωj,hωl)
= 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
Wab, 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
ℳJ 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 (μi) must
increase) and i<j,
k<l
and m<n
(since μi<νi),
according to Eq. (C.6). Hence, we
obtain
(C.11)A
problem is that the cardinality of the set ℳJ is (4J) ! / [
(2J) ! 4J ] (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
Cab and
Cab (Eqs. (C.4), (A.3)) and recalling that as
φj(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 ωj of the
observables φ2j − 1 =
φ2j −
1(ωj) and
φ2j =
φ2j(ωj).
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 Cab’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 Cab 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/ht
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 ℱ2N
evaluated at 0. First, we
note that by a change of variables:
.
Let f(ω1,ω2)
be 2π/ht
periodic in both arguments and
.
Then
and
hence
(D.9)where F2N always acts on
the second argument. As
, it follows
that
Since
, we get an additional
if
we omit the orthogonal projections D2N 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 limn →
∞H2Nf =
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 Cab functions
Let C be defined by Eq.
(A.3), and Wi 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
ωi 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/ht
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 Cab functions
Let C be defined by Eq.
(A.3), and Wi 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 τj 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
Tc these two last
terms intersect in the case of difference travel times. If the window function
f(t) in the definition of
Wdiff, as defined by Eq. (4), is a Heavyside function, then we have
(Gizon & Birch 2004)
(F.2)For a p-mode ridge
κr =
κr(ω),
the function
can be written in the far field
as Gizon & Birch (2004):
(F.3)where κi 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 Tc at which the two terms are equal is
(F.4)For T>Tc,
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
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