We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves).We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region.1.We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minnaert resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of these resonances in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way.2.In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.

中文翻译：

---

在几乎共振的频率下使用气泡作为造影剂的声学成像模态的数学分析

我们使用气泡作为造影剂，对声学成像模态进行数学分析。这些气泡是通过质量密度和体积模量进行对比的，它们具有相反的比例。这些对比的标度允许它们在某些入射频率上产生共振。我们考虑两种类型的对比。在第一个中，气泡是轻的，与背景的气泡相比具有较小的体积模量，因此它们会产生Minnaert共振（对应于局部表面波）。在第二个中，气泡具有中等的质量密度，但体积模量仍然较小，因此它们会产生一系列共振（与局部体波相对应）。我们建议将注入气泡前后的远场用作测量值，设置在目标域中的给定位置，单向散射方向。然后，我们用这种气泡扫描目标域并收集相应的远场。我们的目标是重建目标区域背景的可变，质量密度和体积模量。1我们表明，对于每个固定使用的气泡，在远近，入射，近频处，对比的远场均达到最大值。到Minnaert共振（或体波共振，这取决于我们使用的气泡类型）。因此，我们可以从我们的数据中重建这种共振。这些共振对背景的背景质量密度的显着依赖性使我们能够以一种简单的方式恢复它，即质量密度。2。此外，该实测的对比远场使我们能够恢复整个场。在气泡的位置点（即在没有气泡的情况下的总场）。