1,720,980 research outputs found
Uniqueness of an inverse source problem in experimental aeroacoustics
No full textThis paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in R^3 or R^2. As a main result we prove that an unknown, compactly supported source power function is uniquely determined by idealized, noise-free correlation measurements. Our framework further allows for a precise characterization of state-of-the-art source reconstruction methods and their interrelations
Weighted data spaces for correlation-based array imaging in experimental aeroacoustics
No full textThis article discusses aeroacoustic imaging methods based on correlation measurements in the frequencydomain. Standard methods in this field assume that the estimated correlation matrix is superimposed withadditive white noise. In this paper we present a mathematical model for the measurement process coveringarbitrarily correlated noise. The covariance matrix of correlation data is given in terms of fourth ordermoments. The aim of this paper is to explore the use of such additional information on the measurementdata in imaging methods. For this purpose a class of weighted data spaces is introduced, where each dataspace naturally defines an associated beamforming method with a corresponding point spread function. Thisgeneric class of beamformers contains many well-known methods such as Conventional Beamforming, (Ro-bust) Adaptive Beamforming or beamforming with shading. This article examines in particular weightingsthat depend on the noise (co)variances. In a theoretical analysis we prove that the beamformer, weighted bythe full noise covariance matrix, has minimal variance among all beamformers from the described class. Ap-plication of the (co)variance weighted methods on synthetic and experimental data show that the resolutionof the results is improved and noise effects are reduced
The Lepskii Principle for an Inverse Source Problem in Experimental Aeroacoustics
No full textThis conference contribution is concerned with variational inverse methods (e.g. generalized Tikhonov regularization) in experimental aeroacoustics that seek to reconstruct the source power function of an aeroacoustic source from measured correlation data. We investigate a Tikhonov functional composed of a quadratic data misfit functional and a sparsity promoting regularization functional. For an optimal accuracy of the reconstruction, the choice of the regularization parameter is crucial. The Lepski˘ı principle is a parameter choice strategy that is well understood from a mathematical point of view. We will present a procedure, how the Lepski˘ı principle can be applied to the aeroacoustic inverse source problem. The procedure relies on a Monte Carlo
simulation of the propagated data noise error. The source sampling for the Monte Carlo simulation is motivated by the probabilistic setup presented in [HS17]. Furthermore it makes use of the data error statistics that were investigated in [RSHE20]
Beamforming by means of the Mahalanobis distance
Get PDFConventional beamforming is one of the standard methods for array imaging of acoustic sources. It offers a very robust indicator for the sound source distribution. In conventional beamforming the deviation of model and measurement with respect to the standard 2-norm is minimized. This can be interpreted as the Mahalanobis distance of measurement and model, where the covariance matrix of the measurement noise vector is a positive multiple of the Identity matrix. In real life applications this ideal noise model is often violated and hence the components of the noise vector have different variances or are even correlated. In order evaluate the Mahalanobis distance of model and measurement we need to know the covariance matrix of the noise vector. The covariance matrix can be estimated by the measurement samples or alternatively by an explicit formula assuming the noise is Gaussian. If the covariance matrix is assumed to be diagonal, the Mahalanobis distance can be interpreted as a weighted standard 2-norm, where the weights indicate the reliability of each data point. The application of this approach on experimental data of a wind tunnel test show that the resolution of the source maps can be enhanced compared to conventional beamforming.
Conventional beamforming is one of the standard methods for array imaging of acoustic sources. It offers a very robust indicator for the sound source distribution. In conventional beamforming the deviation of model and measurement with respect to the standard 2-norm is minimized. This can be interpreted as the Mahalanobis distance of measurement and model, where the covariance matrix of the measurement noise vector is a positive multiple of the Identity matrix. In real life applications this ideal noise model is often violated and hence the components of the noise vector have different variances or are even correlated. In order evaluate the Mahalanobis distance of model and measurement we need to know the covariance matrix of the noise vector. The covariance matrix can be estimated by the measurement samples or alternatively by an explicit formula assuming the noise is Gaussian. If the covariance matrix is assumed to be diagonal, the Mahalanobis distance can be interpreted as a weighted standard 2-norm, where the weights indicate the reliability of each data point. The application of this approach on experimental data of a wind tunnel test show that the resolution of the source maps can be enhanced compared to conventional beamforming
Numerical Methods for Source Power Reconstruction in Experimental Aeroacoustics
No full textWe consider the inverse source problem of reconstructing the power of a bounded, compactly supported and uncorrelated acoustic source, given the covariance operator of pressure data within a bounded measurement domain. Such problems arise for example from aeroacoustic wind tunnel experiments. It turns out that in this case, the source power function is uniquely determined by the covariance operator. Given noisy correlation measurements, pointwise imaging methods provide a rough estimator of the source power. The resolution of such imaging methods can be increased by considering a weighted inner product in the data space, depending on the statistical properties of the measurement noise. For the reconstruction of the entire source power function, Tikhonov regularization with a sparsity promoting penalty functional is considered
Analysis of an Inverse Source Problem with Correlation Data in Experimental Aeroacoustics
Get PDFIn aeroacoustic testing one seeks to localize and reconstruct the power of an aeroacoustic source, given correlation estimates of pressure fluctuations in the frequency domain. This inverse problem with operator-valued data is the research object of this thesis. We establish a framework on infinite-dimensional spaces which allows for a deeper analysis of the problem. In particular, we prove that the power of a spatially uncorrelated source is uniquely determined by the correlation data. Furthermore, we analyze several reconstruction methods in an infinite-dimensional and discrete setup. In the latter, the concept of weighted norms is introduced together with a theoretical result on on the optimal weighting choice. Finally, we discuss important aspects regarding the regularization and numerical implementation of source power reconstruction methods. These are illustrated with various
numerical experiments
Analyse eines inversen Quellproblems mit Korrelationsdaten in der experimentellen Aeroakustik
No full textIn aeroacoustic testing one seeks to localize and reconstruct the power of an aeroacoustic source, given correlation estimates of pressure fluctuations in the frequency domain.
This inverse problem with operator-valued data is the research object of this thesis.
We establish a framework on infinite-dimensional spaces which allows for a deeper analysis of the problem. In particular, we prove that the power of a spatially uncorrelated
source is uniquely determined by the correlation data.
Furthermore, we analyze several reconstruction methods in an infinite-dimensional and
discrete setup. In the latter, the concept of weighted norms is introduced together with
a theoretical result on on the optimal weighting choice.
Finally, we discuss important aspects regarding the regularization and numerical implementation of source power reconstruction methods. These are illustrated with various
numerical experiments.2021-08-2
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