We study the inverse problem for the Helmholtz equation using partial data from one-side illumination. In order to reduce the ill-posedness of the problem, the model to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors, following regularization by discretization approach. The eigenvectors result from a diffusion equation and we compare several choices of weighting coefficient from image processing theory. We first investigate their efficiency for image decomposition (accuracy of the representation with a small number of variables, denoising). Depending on the model geometry, we also highlight potential difficulties in the choice of basis and underlying parameters. Then, we implement the method in the context of iterative reconstruction procedure, following a seismic setup. Here, the basis is defined from an initial model where none of the actual structures are known, thus complicating the process. We note that the method is more appropriate in case of salt dome media (which remains a very challenging situation in seismic), where it can compensate for lack of low frequency information. We carry out two and three-dimensional experiments of reconstruction to illustrate the influence of the basis selection, and give some guidelines for applications.

中文翻译：

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求解亥姆霍兹方程地震反问题的特征向量模型

我们使用来自一侧照明的部分数据研究亥姆霍兹方程的逆问题。为了减少问题的不适定性，在通过离散化方法进行正则化之后，使用与特征向量基相关联的有限数量的系数来表示要恢复的模型。特征向量来自扩散方程，我们比较了图像处理理论中加权系数的几种选择。我们首先研究它们的图像分解效率（具有少量变量的表示准确性，去噪）。根据模型几何形状，我们还强调了选择基础和基础参数方面的潜在困难。然后，我们在地震设置之后在迭代重建程序的上下文中实施该方法。这里，基础是根据初始模型定义的，其中没有任何实际结构是已知的，因此使过程复杂化。我们注意到该方法更适用于盐丘介质（这在地震中仍然是一个非常具有挑战性的情况），它可以补偿低频信息的缺乏。我们进行了二维和三维重构实验来说明基选择的影响，并给出一些应用指导。