Experimental measurement of the acoustic sensitivity kernel

Sensitivity kernels (SKs) describe the changes in the acoustic field that propagate in a medium, with respect to a local perturbation. The perturbation can be a change in density (Marandet et al., 2011), sound speed (Skarsoulis and Cornuelle, 2004), or interface conditions (Sarkar et al., 2012). In practice, SKs are calculated as the field perturbations associated with a local heterogeneity between a source–receiver pair. Naturally, the position of a perturbation and its physical nature influences the acoustic field at the receiver. As well as from its straightforward use in the forward problem, the SK formulation can be used to invert for environment fluctuations between a source–receiver pair (Woodward, 1992; Dahlen and Baig, 2002; Roux et al., 2011).

One approach used to model SKs is the perturbation method (Skarsoulis and Cornuelle, 2004). This method deals with small fluctuations of the acoustic field in the Born approximation that are due to a local change, the size of which is much smaller than the acoustic wavelength. In the frequency domain, the SK is proportional to the product of two Green's functions: the Green's function from the source to the local change, and the Green's function from the local change to the receiver. When the propagation medium is complex, with scattering, reflection or refraction, the SK can then include more complicated features than the diffraction-based oscillations classically observed around the Fresnel zone associated to one source–receiver pair in free space.

The aim of this study is to show the experimental measurement of the SK between a source and a receiver that compares favorably to numerical calculations. The SKs associated to both the direct path and a reflected path will be imaged at the ultrasonic scale. To experimentally measure the SK between a source and a receiver, a small perturbation has to be placed at every point in the propagation medium. In this study, the perturbation is a metallic sphere, the size of which is on the order of the acoustic wavelength. During the SK experiment, the sphere is shifted in the propagation medium to infer the spatially dependent acoustic fluctuations between a fixed source and a fixed receiver. The perturbation is not important and should remain small, so that the SK is calculated in the first-order scattering Born approximation. For each position of the target, the sensitivity kernel is calculated at the time of the unperturbed field as the difference between the source–receiver pressure field with and without the perturbation.

This paper is organized as follows. Section II deals with the experimental configuration and the results for the direct-path and reflected-path SK are presented. In Sec. III, the theoretical formulation of the SK is introduced, and a comparison with the experimental results is discussed.

The experimental configuration is illustrated in Fig. 1. Two identical source and receiver piezo-transducers face each other in a water tank at a distance apart of L. Each element is centered at F ∼ 1.1 MHz with a 50% bandwidth. The element size is 0.75 mm along the z-dimension, and 12 mm long along the y-dimension. The purpose of this shape is to optimize the power transmitted by each element, through the collimating of the acoustic beam in the x-z plane (Fig. 1). The emitted signal is a broadband pulse of T ∼ 1 μs at the central frequency of the transducer.

The probe used to perturb the acoustic field is a metallic sphere of diameter 2a = 3.5 mm. The sphere is mounted on a thin string of diameter 0.06 mm, and is suspended from a remotely controlled step-motor that enables two-dimensional displacement in the x-z or y-z plane.

Two different experiments are performed. First, a section of the SK is measured at the time of the direct path in the y-z plane, as shown in Fig. 1(a). In this configuration, the acoustic field is perturbed in a perpendicular plane (with respect to the direct path) at the middle of the source–receiver distance L = 1 m. The course of the sphere starts at the top-left corner and goes down along the z axis, and then it moves one step along the y axis for the same z axis displacement. The step is approximately 1.5 mm with a 90-mm scan in both directions, for a total of 61 × 61 = 3721 acquisitions of the source-to-receiver signal. Second, the SK is measured successively at the time of the direct and surface-reflected paths in the x-z plane, as shown in Fig. 1(b). In this case, the sphere motion covers the source–receiver range L = 0.5 m with a 2-mm step along the x axis and an 80-mm scan down from the surface, with a 0.5-mm step along the z axis, for a total of 251 × 161 = 40 411 acquisitions of the source-to-receiver signal. Note that, due to motor limitations, the range L is different between the two experimental configurations.

To measure the SK, a reference field without the target was necessary. In Fig. 1(b), the SK was determined separately for the direct and surface-reflected paths by measuring the relative amplitude change, ΔA/A, between the maximum pressure field without and with the target for each position of the lead sphere [Figs. 2(a), 3(a), and 4(a)]. Using the 3.5-mm metallic sphere, the maximum value of the SK is around 5%. The SKs in Figs. 2 and 3 are slices across and along the vertical propagation plane between the source and the receiver for the direct path (Skarsoulis and Cornuelle, 2004; Iturbe et al., 2009; Marandet et al., 2011).

For travel-time tomography in oceanography and seismology, the travel-time SK associated with a sound-speed change shows the well-known banana/doughnut shape that explained the diffraction-limited resolution of tomography inversion in the late 1990s (Marquering et al., 1999). For a target with a density mismatch (stronger than the velocity mismatch), the banana/doughnut shape is observed on the SK amplitude, with zero sensitivity along the source–receiver ray path [which crosses the two-dimensional observation plane at position z = y = 0 in Fig. 2(a)].

The patterns observed in Figs. 2(a) and 3(a) correspond to the frequency-dependent oscillations around the Fresnel zone that are typically observed in diffraction physics for a point-like target. For a spherical target and point-to-point SKs, these oscillations are expected to be axisymmetric. However, the piezoelectric transducer dimension along the y axis (12 mm) slightly breaks this symmetry, which results in the stronger damping of the diffraction-based oscillations along the y axis in Fig. 2.

These experimental results confirm the spatial extension of the source–receiver sensitivity far outside the ray path that connects the two points [Figs. 1(a), 1(b), dotted lines]. This justifies the use of the SK approach as an optimal tool for inversion in the case of small perturbations, when the high-frequency, large bandwidth approximation is no longer valid (Roux et al., 2011; Marandet et al., 2011).

Figure 4 presents the SK for the surface-reflected path for the same experimental configuration as in Fig. 3 (L = 0.5 m). The surface reflection leads to an interference pattern close to the surface that strongly modifies the SK pattern. Two points must be highlighted. First, according to the image theorem, the reflected field at the receiver can be interpreted as the signal obtained in free space from the image of the source with respect to the interface (with a minus sign that takes into account the reflection coefficient at the surface). The dominant part of the SK pattern appears then as the folded version of the free-space SK [similar to Fig. 3(a)] between the source image and the receiver. Note that, as will be shown in Sec. IV, the minus sign of the reflection coefficient does not appear in the SK before and after the surface bounce, contrary to the Green's function.

Second, the bottom oscillations observed in Fig. 4(a) correspond to the residual of the direct-path SK on the surface-reflected SK. Indeed, the reflected SK is calculated as the amplitude difference of the field with and without the target at the arrival time of the surface-reflected echo. However, this travel-time also corresponds to the scattered field associated to the direct field for a target located far away from the straight path. A physical interpretation of this contribution to the reflected SK is given in Sec. III.

In this section, we present a theoretical formulation of the SK that is compared to the experimental results. The goal is not to change the classical expression of the SK, but to adjust the physical parameters of the perturbation to obtain the optimal match with the data.

The SK expression is derived from perturbation theory in the first-order scattering Born approximation. The perturbed Green's function, ΔG (at frequency ω), due to a local sound-speed perturbation in r′ was formulated between a source in r_s and a receiver in r_r, as (Skarsoulis and Cornuelle, 2004)

where k is the wavenumber and $Δc/c$ characterizes the sound-speed change of volume δV in r′. The SK is traditionally defined as

for a local change in a unit volume δV = 1 m³. The SK arises from the product of the Green's functions from the source/receiver to the perturbation modulated by a scattering coefficient that is associated to the local sound-speed perturbation.

In the case of a density and sound-speed change (e.g., in the case of a hard sphere), the SK can be generalized as (Marandet et al., 2011)

where $f∞$ is the scattering form function of the local perturbation that depends on the frequency ω and the scattering angles φ_s and φ_r [defined as in Fig. 5(a)].

When taking the Fourier transform of the source spectrum, P_s(ω),in the time domain, the pressure-field perturbation becomes

From Eq. (4), the SK formulation that is compared to the experimental measurements in Figs. 2–4 is written as follows:

where t_i is the travel-time at the maximum of the unperturbed source–receiver field.

The utility of the SK is that the field change due to a general perturbation can be computed by breaking this perturbation into a weighted sum of elementary perturbations, so that the field change is a sum of weighted SKs. In the present study, the SK deals with fluctuations due to a hard target of size comparable to the wavelength (Fig. 1). The target is characterized by size, mass density, and longitudinal and transversal wave speed. The non-linear nature of this perturbation problem becomes evident when looking for scattering due to a target having different parameters. This means that the SK formulation in Eq. (3) is only valid with small targets in the Rayleigh scattering regime (⁠$ka≪1$⁠, with a defined as the target characteristic size).

This being said, we limit our goal here to the match between the spatial pattern of the field fluctuations in the presence of the sphere (Figs. 2–4) and a theoretical prediction. For the specific case of an elastic sphere with finite ka, the form function $f∞$ was calculated as an infinite series of Legendre polynomials (Faran, 1951):

where $ηn$ includes the density, sound speed (longitudinal and transversal), and size of the sphere. Note, however, that Eq. (6) requires the incoming wave to be locally planar at the target location, since $f∞$ is calculated in the plane-wave approximation.

Figures 2–4 show good agreement between the experimental and the synthetic SKs. The effects of the transducer width along the y axis were taken into account in the theoretical computations [Fig. 2(b)]. Note that the far-field approximation assumed in Eq. (6) was fulfilled in Fig. 2, as the source–target and target–receiver distance was ∼330 λ. This approximation is no longer valid close to the source and receiver as can be seen from the field perturbation amplitude ΔA/A in Figs. 3(b) and 4(b).

The Fresnel zone oscillations in Figs. 2–4 carry the footprint of the target properties, as well as the characteristics of the acoustic wave (e.g., frequency, bandwidth). They also show the spatial features of the propagation medium, as shown for the surface-reflected SK in Fig. 4. Indeed, in the presence of one or more reflectors, the Green's function is a sum of several contributions. As the SK is the product of two Green's functions [Eq. (3)], it contains all of the cross-terms associated to the different eigenrays between the source–receiver pair. For example, in the case of one single reflection, as in Fig. 1(b), we have

where G₀ corresponds to the direct and G₁ to the surface-reflected contributions of the point-to-point Green's function. Then, the SK can be written as [Eq. (3)]

where the scattering angles φ_s0 and φ_r0 associated to the direct path are the same as in Fig. 5(a), and the set of angles (φ_s1,φ_r1), (φ′_s1,φ′_r1) and (φ_s2,φ_r2) associated to the reflected paths can be equivalently defined from the image theorem [see Fig. 6(b) in Marandet et al. (2011)].

As shown in Fig. 5(b), the G₀G₀ term in Eq. (8) corresponds to the direct SK, while the two middle terms (G₁G₀ and G₀G₁) contribute to the surface-reflected SK. The G₁G₁ term is associated to two reflections at the interface.

Remember now that the SK represents the (weak) wavefield fluctuations computed at the travel-time of the unperturbed source–receiver signal. Given the short duration of the emitted pulse, only the G₀G₀ product contributes to the direct SK at the time of the direct arrival. For the surface-reflected arrival, the SK is dominated by the G₁G₀ and G₀G₁ products, but there also remain the two contributions of (1) the G₀G₀ term, when the perturbation in r′ is far away from the direct path; and (2) the G₁G₁ term, when the perturbation in r′ gets close to the surface around the surface-reflected path. The first contribution is clearly seen in the bottom oscillations in Fig. 4, while the second one contributes to the interference pattern observed just below the air-water interface in Fig. 4.

Finally, we note that despite the presence of interferences, the main lobe of the Fresnel oscillations in Figs. 3 and 4 corresponds to a negative $ΔA/A$ that is independent of the sign change between the direct and surface-reflected Green's functions G₀ and G₁, respectively. From Eq. (5), which was evaluated at the time of the direct path, the negative sign of the SK is due to the −G₀G₀ at the numerator and the G₀ contribution at the denominator. Similarly, at the time of the surface-reflected path, the positive sign of the $−(G1G0+G0G1)$ terms at the numerator is balanced by the negative sign of G₁ at the denominator. In both cases, the SKs should be mainly negative outside the areas where all of the terms in Eq. (8) interfere.

In conclusion, we have presented experimental results at the ultrasonic scale that illustrate the spatial pattern of the sensitivity kernel physics in the finite-frequency approach. The amplitude change of the pressure field was measured for a metallic sphere, and compares favorably to the theoretical expectation of the SK.

C.M. was sponsored by Direction Générale de l'Armement (DGA) during this study. This work was also funded by the Office of Naval Research Global (ONR-G).