Ill-Posedness and Accuracy in Connection with the Recovery of a Single Parameter from a Single Measurement

Abstract

A particular one-parameter problem, that of the reflection and refraction of a plane wave at a planar boundary between two media, which can be solved in closed form in both the direct and inverse contexts, is chosen to illustrate the ill-posed nature of a class of more general inverse problems and to show how the quality of the reconstruction of the parameter varies with the accuracy of the data on the one hand, and the accuracy of the interaction model employed in the inversion scheme on the other hand.

Recovery of the location, size, orientation and shape of a rigid cylindrical body from simulated and experimental scattered acoustic field data

“…This problem is at the very heart of our contribution, and we show that the observed non-uniqueness is due to data and/or estimator error. A6 Other contributions [10,11] dealing with simpler inverse problems also place stress on this particular aspect of boundary recovery via scattered waves. In these works, it was shown that the key to success in resolving the problem is that the error in the data be identical to that in the estimator (substantiated by R5).…”

“…However, this situation, which is equivalent to that of the inverse crime, is impossible to attain in a real-life setting appealing to real data, because there is no way to know the error of the data, nor to tailor the error of the estimator to the error of the data (if this was known). In [10] it was stated that the best way to solve the problem is to use data that is as accurate as possible, and a model for the estimator which is as exact as possible. Herein, we took no heed of this recommendation, in that we relied on an approximate model and made use of (experimental) data that was obtained carefully, but not overly so (probably less so than in the experiments giving rise to the data employed in the studies of [16]).…”

“…The solutions furnished by SYSNOISE are generally not identical to those provided by the ICBA, except when the body is a circular cylinder with center at O (for which the ICBA furnishes the exact solution), so that the estimator usually provides a solution which is different from the data for the same scattering body, meaning that the inverse crime (situation in which the data is identical to the estimation thereof) is avoided [6,10,11].…”

Apparent Attenuation and Dispersion Arising in Seismic Body-Wave Velocity Retrieval

“…2 and 3 show that the cost function exhibits multiple relative minima [this is an illustration of the non-uniqueness of inverse problems (WIRGIN 2002)] for all numbers of sensors, but a single, deep feature emerges when the number of sensors is greater or equal to three. Moreover, the position of the global minimum stabilizes as soon as four sensors are placed within and on the faces of the layer.…”

“…Note that in all these graphs, the intersection of the vertical and horizontal white lines indicates the retrieved value of G 0 and G 00 , i.e.,G 0 ;G 00 , whose position is that of the global minimum of K 2 ðG 0 ; G 00 Þ. Figures 4, 5 and 6 show that the cost function exhibits multiple relative minima [this is another illustration of the non-uniqueness of inverse problems (WIRGIN 2002)] for all numbers of sensors, but a single, deep (black) feature, indicative of a welldefined global minimum of the cost function, 8 and 9, which are obtained for the four-sensor configuration, 121 angular frequencies in x 2 ½0:05 Hz; 2:05 Hz, and 121 trial values of G 0 2 ½2; 6, depict the spectra, true and reconstructed signals at the four sensor positions, for the case in which I retrieve G 0 only. What is changing from one figure to the next is the uncertainty of B (which is nil when B ¼ b).…”