Locating structural changes in a multiple scattering domain with an irregular shape

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Locating structural changes in a multiple scattering domain with an irregular shape

Qi Xue, Éric Larose, Ludovic Moreau

To cite this version:

Qi Xue, Éric Larose, Ludovic Moreau. Locating structural changes in a multiple scattering domain

with an irregular shape. Journal of the Acoustical Society of America, Acoustical Society of America,

2019. �hal-02170497�

Locating structural changes in a multiple scattering domain with an irregular shape

June 26, 2019

Qi XUE

, Eric LAROSE

, Ludovic MOREAU

1 - ISTerre, CNRS & Univ. Grenoble Alpes, CS 40700 38058 GRENOBLE Cedex 9 France

1

Locadiff is a method for imaging local structural changes in a random, heterogeneous medium.

2

It relies on the combination of a forward model to calculate the sensitivity kernel of the source-

3

receiver pairs, with an inversion method to determine the position of the changes. So far, the

4

sensitivity kernel has been evaluated based on an analytical solution of the diffusion equation, which

5

lacks the flexibility to handle problems where the domain has boundaries with an irregular shape.

6

Moreover, the accuracy of the previous inversion method, based on linear algebra tools, was very

7

sensitive to the values of the inversion parameters. This paper introduces a more generic approach

8

to solve both these issues. The first problem is tackled by the implementation of numerical

9

method as an alternative for solving the diffusion equation. The second problem is tackled by the

10

introduction of enhanced optimization algorithms to improve the stability of the inversion. This

11

improved version of Locadiff is validated via both numerical examples and experimental data from

12

an actual civil engineering problem.

13

I Introduction

14

The imaging of multiple scattering media is of interest in a variety of research domains, ranging from

15

the micrometer scale in optical tomography to the kilometer scale in seismology [1]. For example, mon-

16

itoring changes associated with earthquakes or volcanic activity is a major research topic in seismology.

17

In ultrasonic nondestructive testing (NDT), which is the main focus of this paper, monitoring defects

18

in civil engineering structures is a key aspect. The difficulty for imaging such media is the presence of

19

heterogeneities with dimensions of the order of the wavelength (some millimeters to some centimeters).

20

For example, the monitoring of structures made of concrete is a challenging problem, because concrete

21

is a highly heterogeneous medium composed of sand, gravel, pore etc. [2]. Temperature or pressure

22

changes and the human or industrial activity on the structure result in the apparition of small cracks in

23

the concrete [3, 4]. Early detection and characterization of such defects is very important to preserve

24

the integrity of the structure. However, at the early stage of their formation, cracks have a scattering

25

cross section that is generally less than that of the heterogeneities in the concrete, which makes them

26

particularly difficult to detect.

27

The multiple reflection and scattering of waves propagating in a heterogeneous medium yield a long

28

lasting waveform known as the coda wave which exhibits strong sensitivity to small perturbation of the

29

medium, because it provides broad sampling of the medium, by interacting with the heterogeneities a

30

number of times that is proportional to the propagation time. This property has been used for almost

31

20 years in optics with diffuse wave spectroscopy [5, 6], in seismology with coda wave interferometry

32

applied to the Earth’s crust [7, 8] and volcanoes [9, 10], and in concrete damage assessment [11, 12,

33

13, 14, 15].

34

Apart from overall monitoring of small changes inside a heterogeneous medium, Locadiff was pro-

35

posed to locate and characterize the changes [16, 17], which consists of solving an inverse problem

36

where the position and scattering cross-section of structural changes are recovered via analytical for-

37

ward modeling of the diffuse wave and a linear inversion scheme. It has been applied both at the

38

ultrasound scale [18, 19] and at the seismic scale [20, 21]. It was successfully applied to characterize

39

several local scatterers in a multiple scattering medium [22], and also showed promising results to

40

image extended millimeter cracks in civil engineering structures [23, 24]. However, several technical

41

limitations are listed in the following.

42

1. Only infinite domains or finite domains with a regular squared shape (cuboids) can be processed.

43

This is because Locadiff requires the solution of the diffusion or the radiative transfer equation.

44

The diffusion equation has an explicit solution in infinitely homogeneous domain. In the previous

45

implementation of Locadiff, boundaries were handled using the image source method: to take

46

into consideration of the reflection from a plane boundary, a virtual source is added symmet-

47

rically on the other side of the plane. The final solution is obtained through superposition of

48

all these sources. Therefore, so far Locadiff could not handle more complex boundaries. The

49

first improvement presented in this paper consists of replacing the analytical model with a nu-

50

merical model for the diffusion equation based on a finite element solver. This enables the use

51

of Locadiff for civil engineering applications, where the concrete structures are not simple, e.g.,

52

bridges, buildings and dams. Other possible methods, including solving the radiative transfer

53

equation [25], using the partial or photon method [26], or computing the waveforms [27], can

54

also be applied to compute the sensitivity kernel.

55

2. The previous version of Locadiff is based on a regularized least squares problem with L-curve

56

choosing the regularization parameter [22]. Negative components of the solution are decreased

57

via a projected iterative step where negative components were forced to zeros at the beginning

58

of each iteration. The solution should be positive because of its physical meaning, the scattering

59

cross section. The drawbacks of this method are that it is time consuming and it cannot ensure

60

convergence, i.e., the final solution still has negative components. In this paper, we demonstrate

61

the importance of regularization through the singular value decomposition (SVD) [28] of the

62

linear equation system generated by the inverse problem, where the solution is expanded into a

63

sum of a series of vectors corresponding to each singular value. The technique of regularization

64

cuts off component vectors of the solution corresponding to small singular values which are highly

65

contaminated by the noise. Meanwhile the importance of nonnegativity is demonstrated via the

66

Picard condition [29] (Definition IV.1), which tells us that to obtain a reasonable solution, large

67

part of the component vectors should not be polluted by the noise. We find out that our problem

68

doesn’t satisfy the Picard condition. That is to say, it is difficult to get a reasonable solution

69

with only the regularization of the original problem. Combining the regularization of the linear

70

system and the nonnegativity of the solution, we arrive at a regularized least squares problem

71

with nonnegative constraints, which belongs to the category of convex optimization [30]. We

72

suggest to solve this problem with the interior point method [31], which ensures the convergence

73

and the computational efficiency. We consider two kinds of regularizations: the first one controls

74

the magnitude (L

norm) of the solution and the second one controls its smoothness (norm of

75

its Laplacian).

76

The manuscript is organized as follows. Section II is a brief description of the fundamental aspects

77

of Locadiff. Section III introduces the numerical forward model used to overcome the limitations of

78

the analytical model so far used in Locadiff. In Section IV, we carefully review the properties of the

79

linear system generated by Locadiff and introduce more generic solvers. In section V we validate the

80

improved Locadiff methodology on two examples.

81

II Review of Locadiff

82

Let us consider the basic problem of locating an isolated change inside a multiple scattering medium.

83

For simplicity we assume that the background velocity keeps unchanged and the scatterers are isotropic.

84

At time t

, waves are generated at position s and propagated into the medium. The receiver at position

85

r records the waveform h(s, r, t). We now assume a structural change such as the apparition of a new

86

scatterer. The same waves are generated at the same position to obtain the waveform h

(s, r, t). The

87

decorrelation coefficient(DC) between h(s, r, t) and h

(s, r, t) defined by

88

DC

(s, r, t) = 1 −

R

t−T

h(s, r, τ )h

(s, r, τ )dτ q R

t−T

h(s, r, τ )

dτ R

t−T

h

(s, r, τ )

dτ

(1)

is a sensitive and stable indicator of a localized change of the medium. Here the superscript E means

89

that the DC is computed from experimental data. It was proved in ref. [18] that, for an isolated new

90

scatterer located at x, the theoretical DC is

91

DC

(s, r, x, t) = cσ(x)

2 K(s, r, x, t), (2)

K(s, r, x, t) = R

0

P(s, x, τ )P (x, r, t − τ)dτ

P(s, r, t) , (3)

where c is the wave speed and σ the total cross section of the new scatterer. The function P(s, r, t) is

92

the intensity of the wave, and can be approximated by the solution of the diffusion equation in highly

93

heterogeneous medium [32, 33]:

94

∂

P (s, r, t) − D∆

P (s, r, t) = δ(r − s)δ(t).

In practice, we employ several sources s

, s

, . . . , s

and receivers r

, r

, . . . , r

. The notation DC

(t)

95

means the DC corresponding to the source s

and the receiver r

.

96

To locate the position of the new scatterer, we need to find x and σ that minimize the cost function

97

e(x) = X

m,n

DC

(t) − DC

(x, t)

.

A maximum likelihood method was presented in ref. [18] to do the minimization. If the interaction of

98

newly appearing scatterers is neglected, the total effect is the linear superposition of each individual

99

one, i.e.,

100

DC

(s, r, t) =

Z cσ(x)

2 K(s, r, x, t)dx. (4)

The Locadiff method is extended to locate all these new scatterers by solving a linear least squares

101

problem in ref. [22]. Let us separate the whole medium into Q equal voxels with volume δV . Then

102

the discrete formula for the theoretical DC (4) is

103

DC

(s, r, t) = cδV 2

Q

X

q=1

K(s, r, x

, t)σ(x

). (5)

Let us choose L discrete time t

, l = 1, . . . , L. We can summarize all these measurements into a linear

104

system

105

Gm = d (6)

with d the DC, G the sensitivity kernel and m the scattering cross section distribution. Each line of

106

the matrix G corresponds to one of the acquisitions at source s

, receiver r

and time t

.

107

III Sensitivity kernels for irregular shapes

108

In this section we will use a disk as an example to demonstrate our modification of Locadiff. Since

109

the boundary of a disk is a curve, previous version of Locadiff kernel do not apply. To deal with

110

irregular shapes, a numerical solver for the diffusion equation is designed. Let us take a disk denoted

111

by Ω (radius 0.4 meters) as an example. We impose reflecting boundary conditions for the diffusion

112

equation, that is,

113







∂

P (s, r, t) − D∆

P(s, r, t) = δ(r − s)δ(t), r ∈ Ω,

∂

P (s, r, t) = 0, r ∈ ∂Ω, where n is the unit outer normal direction of ∂Ω.

114

Figure 1: Triangular mesh of a disk.

We use the finite element method [34] to solve this diffusion equation. The whole medium Ω is cut

115

off into small triangles as shown in Figure 1. The space derivative is approximated with those nodes

116

and the time derivative is approximated using the Crank-Nicolson method. Let me clarify that the

117

discretization strategy used here is different from the one in equation (5). In equation (5), the mesh

118

is generated from the discrete approximation of the integral equation (4), while here the mesh is used

119

to solve the diffusion equation.

120

Let us recall the formula of the kernel (3). We need to compute P(s

, x

, t), P(x

, r

, t) and

121

P(s

, r

, t) for all m = 1, 2, . . . , M , n = 1, 2, . . . , N and q = 1, 2, . . . , Q. In practice, M and N are of

122

the order of 10, i.e., we use tens of sources and tens of receivers. The value Q is the discretization size

123

of the medium, which is chosen by the user. Let us note that Q does not depend on the finite element

124

method mentioned previously. A large Q means a fine meshing of the medium, which results in more

125

unknowns in the linear system (6). The linear system becomes more under-determined, and therefore

126

more difficult and time-consuming to solve.

127

We show a series of images demonstrating the sensitivity kernel. In this experiment we choose the

128

diffusivity D = 125m

/s. The disk is represented by about 5000 equally spaced points (Q ≈ 5000).

129

We put 16 equally spaced sources on the boundary. In the middle of each adjacent source pair we put

130

a receiver. We plot the sensitivity kernel of source 1 and receiver 2 at three different times: 0.2ms,

131

0.4ms and 0.6ms in Figure 2. With increasing time, the probed region becomes larger. Meanwhile,

132

the impacting strength becomes stronger.

133

0 1 2 3 4 5

(a) Sensitivity kernel at 0.2ms

0 1 2 3 4 5

(b) Sensitivity kernel at 0.4ms

0 1 2 3 4 5

(c) Sensitivity kernel at 0.6ms

Figure 2: Sensitivity kernels of source 1 and receiver 2 at different times. Red circles show positions of sources and green circles are positions of receivers.

IV Ill-posedness, regularization and nonnegativity

134

In this section we focus on demonstrating the difficulty of solving the linear system (6). We simulate

135

to obtain the data based on the setup in Section III. The basic tool to study the property of a linear

136

system is the singular value decomposition (SVD) [28]. For the matrix G, we have the following

137

decomposition

138

G = U ΛV

,

where U and V are orthogonal matrices (U

U = I, V

V = I), and V

means the transpose of V . The

139

matrix Λ is a diagonal matrix with non-negative, real, decreasing diagonal elements σ

, i = 1, 2, . . . , L.

140

The blue line in Figure 3 plots the singular values of the matrix G. The largest singular value has a

141

magnitude of 10

and the smallest one has a magnitude of 10

. Besides, the singular values decrease

142

smoothly without any gap, i.e., the problem is ill-posed.

143

With the help of the SVD of G, the solution to the linear system (6) is

144

m = X

i

u

d σ

v

,

with u

the i-th column of the matrix U and v

the i-th column of the matrix V . The vector d usually

145

contains a certain quantity of noise, i.e., d = ˜ d + e, where e is the noise and ˜ d is the noise-free part

146

which is not accessible in practice. In this section, ˜ d is simulated by adding four new scatterers which

147

are shown by white points in Figure 5, and therefore we know ˜ d. Correspondingly, m is composed of

148

two parts

149

m = X

i

u

d ˜ σ

v

+ X

i

u

e σ

v

, (7)

the noise-free components and the noisy components. We assume that the noise is white and the noise

150

level is η, that is, |u

e| ≈ η.

151

Regularization is necessary to obtain a reasonable solution to an ill-posed problem [35]. The linear

152

0 500 1000 1500 2000 2500 10^-20

10^-15 10^-10 10^-5 10⁰ 10⁵

Amplitude

Picard plot

Figure 3: Singular values, decomposition of the solution and Picard condition.

system (6) is equivalent to the minimization problem

153

min

m

kGm − dk

. (8)

Let us consider a general kind of regularization, Tikhonov regularization. Instead of minimizing (8),

154

an additional term is appended,

155

min

kGm − dk

+ λ

kBmk

, (9) where λ is called regularization parameter and B is a matrix or an operator chosen by the user. A

156

general choice of B could be B = I, the identity matrix, which controls the magnitude of the solution,

157

or B = ∆, the Laplacian operator, which controls the smoothness of the solution. The idea of the

158

regularization is to filter out the noisy components from the final solution. For a very small singular

159

value (large i), the noisy component

i

≈

i

becomes extremely large. The solution is spoiled by

160

these large noisy components. Meanwhile, the corresponding singular vector v

is highly oscillating.

161

A regularized solution is able to provide a much more reasonable solution by diminishing or removing

162

those noisy oscillating components.

163

A potential precondition for the regularization to be effective is that enough components of the

164

solution (7) should not be contaminated by the noise. Usually speaking, components corresponding

165

to small singular values are more susceptible to be polluted by the noise. We hope that the true

166

solution has more proportion on large singular values than small ones, that is, we need to have enough

167

components |u

d| ˜ larger than the noise level η. This assumption, the discrete Picard condition, is

168

precisely proposed by ref. [29].

169

Definition IV.1 (The discrete Picard condition) Let ε denote the level at which the computed

170

singular values σ

level off due to rounding errors. The discrete Picard condition is satisfied if for all

171

0 50 100 150 200 250 300 10^-4

10^-2 10⁰ 10² 10⁴

Amplitude

Figure 4: The decomposition of the true solution and the noise.

singular values larger than ε, the corresponding coefficients |u

d|, on average, decay faster than the ˜

172

σ

.

173

The discrete Picard condition indicates a decreasing tendency of the figure |u

d|/σ ˜

, while the white

174

noise |u

e|/σ usually has an increasing tendency. In the ideal case, we cut off the components after

175

the crossing of these two lines. Unfortunately, our linear system does not satisfy the discrete Picard

176

condition. The figure |u

d|/σ ˜

does not have a decreasing tendency and stays at a certain level. Let us

177

create a 15% Gaussian noise by e = ˜ d ∗ ξ ∗ 15%, where ξ is the normalized Gaussian distribution. We

178

see in Figure 4 that the noise begins to dominate after the first 30 components, i.e., most components

179

of the solution are highly polluted, which indicates that the regularized least squares problem (9) is

180

not a proper model. It is shown in ref. [22] that the nonnegative projection of the solution results in a

181

better imaging of new scatterers. We propose the following model, a regularized least squares problem

182

with the nonnegative constraint,

183

m≥0

min

kGm − dk

+ λ

kBmk

. (10) In ref. [22], authors proposed to do projections to eliminate negative components, that is, the least

184

squares problem (9) is solved and negative components are forced to be zero, and then this approxi-

185

mated solution is used as the initial guess to start the next iteration. The nonnegative projection has

186

no guarantee of convergence. In fact, the problem (10) belongs to “convex quadratic optimizations”

187

[30]. This kind of optimization problem has a unique solution. Efficient methods, such as interior

188

point [31], are able to solve this problem fast and stably.

189

As a conclusion of this section, we present the result of the example problem from Section (III).

190

First we demonstrate the effect of different regularizations and the nonnegative constraint. Figure 5

191

shows the results. In all these experiments, we fix the regularization parameter λ = 100. We see that

192

the nonnegative constraint makes the solution more contracted near positions of new scatterers. The

193

nonnegative constraint also stabilizes the result. New scatterers near source-receiver pairs are easier

194

to be detected than those far away due to the impact region demonstrated in Figure 2. Compared to

195

B = I, the regularization B = ∆ seems to be more proper for this problem. It has the ability to detect

196

the new scatterer inside the domain and the result is more compact.

197

(a)

=

negative constraint.

(b)

= ∆, without non- negative constraint.

(c)

=

, with nonnega- tive constraint.

(d)

= ∆, with nonneg- ative constraint.

Figure 5: Comparison of results of the imaging problem with four new scatterers in a multiple scattering medium, with different regularizations and with/without the nonnegative constraint. The four white points inside the images are positions of new scatterers.

In Figure 6, we compare results with different regularization parameters. If the regularization

198

parameter is too large, the problem is over regularized. The potential region of new scatterers is large

199

(low resolution) as shown in the first picture, that is, we are not sure where are the scatterers exactly.

200

On the other hand, if the regularization parameter is too small, the result is too sensitive to the noise.

201

We have the result shown in the last picture, where the cross section of a scatterer is split into two.

202

For this problem, a regularization parameter λ = 10 provides a good balance between stability and

203

resolution.

204

V Validation examples

205

In this section, we apply the Locadiff method with numerical kernel and new inversion technique to

206

more realistic problems. We revisit previous simulations and real field experiments to test the validity

207

of our new scheme. The first experiment is similar to the one from ref. [22]. In this experiment we

208

simulate the propagation of acoustic waves in a multiple scattering medium using the finite difference

209

time domain (FDTD) method. Then, three small groups of new scatterers are added and acoustic

210

waves are simulated with the same sources and receivers. The second experiment is from ref. [24] in

211

which sensors are glued on the surface of a concrete wall to generate and receive ultrasonic waves at

212

two different states. Between these two states, the concrete wall changes a little (cracks open) because

213

(a)

= 1000. (b)

= 100. (c)

= 10. (d)

= 1.

Figure 6: Comparison of results with different regularization parameters. In this experiment, we choose B = ∆. The four white points inside the images are positions of new scatterers.

of the pressure change inside the building.

214

A Numerical simulation of coda waves

215

In this experiment, we simulate the propagation of acoustic waves inside a 75cm × 75cm multiple-

216

scattering medium using the FDTD method with a point source of central wavelength λ

= 0.375cm.

217

Positions of sources and receivers are denoted by red crosses and green squares in Figure 7a, respec-

218

tively. The same simulation is repeated after adding three new scatterers which have radii of 3, 5 and

219

7 spatial pitches, respectively. The theoretical values of the cross section of these three new scatterers

220

are 1.05λ

, 1.67λ

and 2.19λ

, respectively. The other parameters are set up the same with those in

221

ref. [22] where the scattering mean free path is smaller than the size of the medium to ensure the

222

multiple scattering regime to occur.

223

The DC is computed from formula (1). Then 15% Gaussian noise is added. We tried different

224

kinds of regularization and regularization parameters, and chose the one with B = ∆ and λ = 400.

225

The result is demonstrated in Figure 7b. The reconstructed cross section values of three new scatterers

226

can be obtained by a localized summation which are 0.97λ

, 1.54λ

and 1.99λ

, respectively. They are

227

quite similar to the theoretical values. Compared to the results from ref. [22], these new results are

228

more compact and stable.

229

B Experiment using ultrasound in concrete

230

Here we re-investigate data from an aeronautical wind tunnel made of concrete at the French ONERA

231

Toulouse center [24]. Due to changes of the pressure inside the tunnel, cracks in the concrete close or

232

open. Our aim is to use the Locadiff method to image those cracks.

233

The experiment is done on a 2m × 2m area of the 35cm-thick concrete wall. 16 receivers and

234

16 sources are glued on the surface of the wall. The sources are triggered alternatively, emitting an

235

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

x(m)

y(m)

(a) Setup of the acoustic simulation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x(m)

y(m)

σ ( λ

)

0.005 0.01 0.015 0.02 0.025

(b) Inversion result of the acoustic experiment.

Figure 7: Red crosses in the left figure are positions of sources and green squares are positions of receivers. The background cyan dots denote positions of scatterers. Three new scatterers (blue dots) are added in the second simulation. The top-left one is a disk with a 3-spatial-pitch radius. The bottom one is a disk with a 5-spatial-pitch radius. The top-right one is a disk with a 7-spatial-pitch radius. The result is shown in the right figure. In this experiment we choose B = ∆ and λ = 400.

ultrasonic sound with frequency ranging from 80KHz to 100KHz. The experiment is repeated after

236

the increasing of the pressure inside the tunnel. The whole concrete wall is much larger than our

237

experimental domain. It is not necessary to compute the sensitivity kernel on the whole concrete wall,

238

because the diffusion solution decreases exponentially with respect to the distance. On the other hand,

239

in order to eliminate reflections from the truncated boundary in the model, the computational domain

240

is chosen to be a 4m × 4m area surrounding the inspected area.

241

We choose B = I and λ = 0.2 to perform the inversion. The reconstructed cross section is compared

242

with the one from ref. [24] in Figure 8. Since the original data set of ref. [24] is no longer available,

243

we work on a different one, which results in the magnitude difference. Nevertheless, we still find the

244

position of the most prominent crack (three black segments in Figure 8), which validates our new

245

kernel and inversion methodology.

246

VI Conclusion and future work

247

In this paper we discussed some limitations of the Locadiff method to locate small changes in heteroge-

248

neous multiple scattering media. We propose to replace the explicit diffusion solution by a numerical

249

Depth 3.5cm

0 1 2 3 4

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0.2 Depth 17.5cm

0 1 2 3 4

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0.2 Depth 31.5cm

0 1 2 3 4

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Figure 8: Reconstructed changes in cross section of the ONERA experiment. The first line is the result from ref. [24]. The second line is our result. The three black segments denote positions of the crack.

The magnitude difference is due to different data sets and different implementations of Locadiff.

solver, which makes it possible to use Locadiff for civil applications without simple structure, e.g.,

250

bridges, buildings and dams.

251

Second we have a detailed study of the sensitivity kernel and emphasize the importance of the

252

nonnegative property of the cross section function (the unknown m in (10)). We propose to use the

253

interior-point method to solve the inversion problem. The comparison with existing results validates

254

the modification of Locadiff. There are still several problems remaining unsolved:

255

• The choice of the regularization and regularization parameter. The regularization depends on

256

the noise. In practice we choose the regularization B = ∆ if new scatterers are highly localized,

257

and B = I if new scatterers are extended. There seems to be no general answer to the problem of

258

how to choose regularization parameters. Usually we start from a large enough λ and gradually

259

decrease the scale until a reasonable result.

260

• The computation of the sensitivity kernel. From our experiments, the sensitivity kernel is the

261

most time-consuming part because of the fine mesh everywhere. Replacing the equally spaced

262

mesh by an adaptive mesh, which is fine near sources and receivers and coarse other places, is

263

able to reduce the size of the problem dramatically. Since the solving of the diffusion equation

264

with sources locating at different positions are totally independent, the computation can be

265

parallelized easily. In practice, if we fix source, receiver and time positions, the sensitivity kernel

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does not change. We only need to compute it once and save it for later experiments.

267

Acknowledgements

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The first author acknowledges a grant from IDEX UGA. The authors also acknowledge supports from

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ANR ENDE, the UGA VOR Program, and the Labex OSUG@2020.

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