Comparison of some inverse methods for wave propagation in layered media

The one-dimensional wave equation in a layered medium is considered. The inverse problem consists of computing the acoustic impedance of the layered medium from the reflection response measured at the surface. For a discrete medium consisting of homogeneous layers of equal traveltime the Levinson algorithm is used to compute the reflection coefficients at the interfaces between the layers. For a medium with continuously varying parameters, an iterative frequency-domain method based on the Riccati equation is used. When these methods are applied to band-limited synthetic seismic data, the result is a filtered version of the acoustic impedance. When the noise level is increased, both methods diverge. For a medium consisting of homogeneous layers of unknown thickness, the reflection coefficients and the traveltimes are estimated simultaneously by using a detection scheme combined with a numerical solution of the wave equation. The performance of three different methods were compared on synthetic data. The first method is based on downward continuation of the upgoing and downgoing wavefield. The second method is based on the computation of the wavefield at the surface, and progressively removing the effect of the layers once they have been identified. The third method is based on layer removal in the frequency domain. In all these cases, the seismic pulse was assumed to be known, and the same detection scheme was used. Numerical simulations indicate that, with the detection scheme used, the method based on surface calculations gave slightly better results than the method using downward continuation. Both these methods gave improved results compared to the layer removal scheme when applied to data with medium noise level. All three detection methods proved to have superior performance compared to the classical method using the Levinson algorithm or the iterative frequency-domain method. For small noise levels, the detection methods all gave a very good reconstruction of the acoustic impedance. For medium and high noise levels, the detection methods remained stable, although a number of false reflectors were found.

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