Equivalent Source Method Research Articles

Optimization of the Equivalent Source Configuration for the Equivalent Source Method

The equivalent source method is widely applied to study structural acoustic radiation in an underwater environment. However, there is still uncertainty in arranging the equivalent source, and the current mainstream configuration method needs a large number of equivalent sources, limiting its practical applicability. In this paper, an equivalent source configuration method that is simple, effective, and easy to implement, and which based on a tradeoff between the ill condition of the transfer matrix and the adequacy of the simulated structure’s radiated sound field, is proposed. The optimization method can derive the appropriate positions and quantity of monopole equivalent sources simultaneously. The method does not yield an optimal solution in a strict mathematical sense but provides satisfactory results compared with those obtained by uniformly distributed equivalent sources. Numerical simulation results showed that the optimization method derives accurate sound field calculation results with a relatively small number of equivalent sources, significantly reducing the number of subsequent calculations needed. Finally, the experiments conducted with a cylindrical shell structure verified the validity and practicality of the proposed method.

Stable downward continuation of the total-field magnetic anomaly derived by using equivalent sources

The stable downward continuation is an important and difficult problem in magnetic data processing and interpretation. The downward continuation of the magnetic field is an unstable process, wherein high-frequency signals will be rapidly amplified and overwhelm useful signals. Although the suppression of high-frequency signals can allow for the acquisition of smooth downward continuation results, this will lead to a loss of amplitude in the data. Therefore, it remains challenging to reproduce the amplitude and shape of a magnetic anomaly well synchronously. Based on the good performance of the dual-layer equivalent source method when processing magnetic data, a stable and high precision downward continuation approach is proposed. The synthetic data tests show that the proposed approach can obtain results with a high accuracy. Notably, this new technique provides a stable scheme for the downward continuation of the magnetic anomaly field. Application of the approach to aeromagnetic data helps to verify the stability of the proposed method.

Construction and selection of a directivity wave function improving ill-condition problems in equivalent source method based near-field acoustic holography

Sound-field reconstruction via the equivalent source method presents an ill-conditioned problem. This paper proposes a method in which traditional monopole equivalent source spherical wave functions are replaced with strong directional wave functions such that an ill-conditioned problem can be improved upon when building the integral equations of a near-field acoustic holography model. Using directional wave functions as an integral kernel function can cause the transfer matrix to be main-diagonally dominant. The accuracy of the proposed method for sound-field reconstruction is verified using numerical simulations, and reconstruction stability in the presence of measurement noise is considered. Furthermore, a method for selecting directional wave functions that are appropriate for various model parameters is introduced. The results show that the proposed method can greatly reduce the condition number of the transfer matrix and improve reconstruction stability.

Identification of panel acoustic contribution based on patch nearfield acoustic holography

To identify panel acoustic contribution to the interior sound field, a new method using a modified sound field separation technique based on patch nearfield acoustic holography is presented. An improved sound field separation technique with single semi-closed surface covering the panel of interest is proposed to simplify the measurement process. The outward and inward pressures relative to the hologram surface can be separated directly by establishing the relationship between the measured acoustic pressures and calculated values based on the equivalent source method. The acoustic quantities on the vibrating panel surface can be reconstructed from the free radiated field by getting rid of the inward and the scattered field on the surface. The acoustic power on the panel is obtained as a criterion to evaluate the relative contributions to the interior sound field. A numerical simulation of a scaled car cabin model is investigated and the results demonstrate that the proposed method is an available approach for identifying panel acoustic contribution on the vibrating surface of interest to the interior sound field inside a cavity.

Source term estimation with deficient sensors: Traceability and an equivalent source approach

In air pollution monitoring, source term estimation (STE) is a problem to inversely estimate the source parameters, such as their locations and emission rates, with available information of meteorology, ground geometry, as well as concentration measurements and their locations. For chemical industrial parks (CIPs), a major challenge of the estimation is the ill-posedness caused by the lack of sufficient measurements to match a vast number of undetermined source parameters. This ill-posedness results in incapacity in estimating unique solution for unknown source parameters, hence they are untraceable. In this paper, a theoretical definition of traceability is introduced to give a general criterion to determine whether a STE problem has a unique solution. This is derived from an atmospheric transport and dispersion model with description of linear sensitivity of measurements to source parameters. Based on this traceability criterion, an equivalent source method to convert an untraceable problem to traceable is proposed. The performance of this method is evaluated with a numerically simulated case whose configuration is based on information collected from a CIP in Shangyu, China. The results suggest that traceability is vital to STE problems. The method is effective for solving daily STE problems with sensor deficiency encountered in CIPs.

Sound field separation of multiple coherent sound sources with single measurement surface

Sound field separation based on near-field acoustical holography has been developed worldwide, but with the increase in the number of sound sources, traditional measurement methods and calculation methods will generate more workload. To reduce the number of measuring points and save calculation time, the sound field separation of multiple coherent sources with a single measurement surface is proposed. On the basis of separating two coherent sources with this method, the separation formula of more sources based on an equivalent source method is given. Through numerical simulation, the effects of the number of holographic surface measuring points, measuring distance, array shape, and equivalent source number on the calculation accuracy of the sound field separation were compared at different frequencies. The correctness and effectiveness of the sound field separation method with a single surface are verified by actual experiments.

Spatial extrapolation of early room impulse responses in local area using sparse equivalent sources and image source method

The room impulse response (RIR) is important in most acoustic applications, such as the design of concert halls and sound field control, because it characterizes the sound propagation. The measurement of RIRs at multiple points is challenging, as it requires a huge microphone array or repeating the experiment by microphone replacement. Several RIR interpolation and extrapolation methods of RIRs have been developed for obtaining RIRs from multiple measurement points efficiently. Extrapolation methods offer more efficient RIR measurement compared to interpolation. However, previous studies focused on extrapolation at frequencies below 1 kHz, and the extrapolation at higher frequencies was difficult. In this study, we propose an extrapolation method for RIRs of direct sound and early reflections in a local area using a small number of measurement points. The proposed method represents the RIRs around the microphones using superpositions of sparse equivalent sources located around the loudspeaker and image sources. We conducted a measurement experiment in an anechoic chamber to estimate RIRs around microphones using sound-reflecting boards. From the experimental results with 2.5 dimensional conditions, the proposed method achieved about above 10 dB of signal to noise ratio (SNR) near the microphone array from 0.5–8.5 kHz. For the extrapolation accuracy over the entire evaluation area (0.6 × 0.54 m2), the proposed method improved the SNR by about 5–6 dB compared to results using the plane wave decomposition.

Multizone Sound Field Reproduction Based on Equivalent Source Method

In this paper, the two-zone sound field reproduction problem is investigated with a newly proposed method. The aim of the two-zone reproduction problem is to achieve sound field reproduction in the bright zone while keeping the sound field energy in the dark zone as low as possible. The main difference between the proposed method and the conventional method is that the sound field control in the dark zone is not limited to a small number of measurement points on the region boundary but to measurement points throughout the whole dark zone. For practical reasons, the whole region is discretized into a grid, and the grid selection method is also provided in this paper. Simulations are performed to compare the performances of the proposed and conventional methods with different virtual source directions, and a comparison of the robustness to the regularization parameter shows that the proposed method is less sensitive to the selection of the regularization parameter. Finally, an experiment is performed with a circular loudspeaker array in a real listening room. A quieter dark zone is achieved by the proposed method while keeping the reproduction error in the bright zone close to that of the conventional method, similar to the simulation results. These outcomes verify that the proposed method performs well in both free-field and reverberation environments.

East Antarctica magnetically linked to its ancient neighbours in Gondwana

We present a new magnetic compilation for Central Gondwana conformed to a recent satellite magnetic model (LCS-1) with the help of an equivalent layer approach, resulting in consistent levels, corrections that have not previously been applied. Additionally, we use the satellite data to its full spectral content, which helps to include India, where high resolution aeromagnetic data are not publically available. As India is located north of the magnetic equator, we also performed a variable reduction to the pole to the satellite data by applying an equivalent source method. The conformed aeromagnetic and satellite data are superimposed on a recent deformable Gondwana plate reconstruction that links the Kaapvaal Craton in Southern Africa with the Grunehogna Craton in East Antarctica in a tight fit. Aeromagnetic anomalies unveil, however, wider orogenic belts that preserve remnants of accreted Meso- to Neoproterozoic crust in interior East Antarctica, compared to adjacent sectors of Southern Africa and India. Satellite and aeromagnetic anomaly datasets help to portray the extent and architecture of older Precambrian cratons, re-enforcing their linkages in East Antarctica, Australia, India and Africa.

Modeling propagation in shallow water with the range-dependent sea surfaces and fluid seabeds using the equivalent source method.

This paper describes a shallow water range-dependent propagation model (RPM) based on the equivalent source method (ESM). The proposed model allows both the sea surface and fluid seabed to vary with the propagation range. The proposed equivalent source method-based range-dependent propagation model (ESM-RPM) utilizes three sets of equivalent sources, placed above the sea surface, below the seabed, and above the seabed, which replace the sea surface reflection, seabed reflection, and seabed transmission, respectively. The unknown strengths of the equivalent sources can be determined by solving an inverse problem based on the boundary conditions. The capability of the ESM-RPM for propagation in refractive water is demonstrated by evaluating the Green's function using a modal projection method. Numerical simulations are conducted in iso-velocity and refractive shallow water with an underwater canyon and corrugated surface waves, including two-dimensional (2-D) propagation across the canyon and three-dimensional (3-D) propagation along the canyon. Further simulations demonstrate the 2-D across-canyon and 3-D along-canyon propagations with random rough sea surfaces. The results show that the proposed ESM-RPM provides efficient, benchmark-quality numerical solutions that accurately capture the mode coupling associated with the varying cross section of the waveguide. Thus, the model has great potential to be applied in benchmarking propagation in shallow water with the varying sea surface and seabed.

Low-Frequency Sound Prediction of Structures with Finite Submerge Depth Based on Sparse Vibration Measurement

In the non-free-field, with the effect of reflection sounds from the reflection boundary, the vibration character of a submerged structure often changes, which may have significant influences on the measurement system configurations. To reduce the engineering cost in low-frequency sound prediction of a submerged structure with finite depth, two methods based on the theory of acoustic radiation mode (ARM) are proposed. One is called the vibration reconstruction equivalent source method (VR-ESM), which utilizes the ARM to reconstruct the total vibration of the structure, and the sound prediction is completed with the equivalent source method (ESM); the other is called the compressed modal equivalent source method (CMESM), which utilizes the theory of compressive sensing (CS) and the ARM to reinforce the sparsity of source strengths. The sound field separation (SFS) technology is combined with the above two methods for constructing the ARMs accurately in the non-free field. Simulations show that both methods are efficient. Compared with the traditional method based on the structural modal analysis, the methods based on the ARM could efficiently reduce the scale of the measurement system. However, the measurement point arrangement should be optimized to keep the prediction results accurate. In this paper, the optimization process is completed with the efficient independence (EFI) method. In addition, some factors that may affect the prediction accuracy are also analyzed in this paper. When the submerge depth is large enough, the process of contrasting ARMs could be further simplified. The results of the paper could help in saving engineering costs to predict the low-frequency sound radiation of submerged structures in the future.

Sound field reconstruction of structural source based on element radiation superposition method

In order to improve the sound field reconstruction accuracy of distributed structural source, a new near-field acoustic holography is established based on the element radiation superposition method (ERSM). In the proposed method, the surface of structural source is divided into several regular pistons. The sound field of structural source is considered as the superposition of sound field of pistons. Firstly, we compare the sound field calculated by ERSM with that by Rayleigh's integral. It is proved that ERSM is quite accurate in sound field prediction. Based on ERSM, a vibration acoustic transfer (VAT) function is derived. The VAT function has computable analytical expression and embodies the transfer relationship between the structural source surface and the radiated sound field. The VAT function can precisely characterize the acoustic propagation of continuous distributed coherent sources. Subsequently, we employ the VAT function to replace the Green's function, and apply the VAT function to sound field reconstruction. Different with the equivalent source method (ESM) which is widely used in sound field reconstruction, ERSM directly divides the piston-sources on the surface of structural source rather than constructing the equivalent point-sources on a plane behind the structural source. Furthermore, we introduce a weight matrix into ERSM and iteratively calculate the vibration velocity for a more accurate result, and we call the proposed method as iterative weighted ERSM (IWERSM). In this paper, the simulations and experiment of sound field reconstruction of a rectangular plate are performed. In the proposed method, the rectangular plate is divided into several rectangular pistons. The reconstruction results of ERSM and IWERSM are compared with that of ESM and iterative weighted ESM (IWESM) respectively. The reconstruction accuracies at different distances between the plate and array (test distances) are analyzed. The simulation results show the accuracy of ERSM and IWERSM are better than that of ESM and IWESM respectively. With the increase of test distance, the phenomenon is more obvious, and IWERSM even shows a good reconstruction accuracy while the test distance is more than half a wavelength. The experiment results also validate that ERSM and IWERSM have better reconstruction accuracy than ESM and IWESM respectively at the same test distance. In a word, the simulations and experiments demonstrate that the proposed method can improve the sound field reconstruction accuracy of regular structural source and expand the valid test distance of near-field acoustic holography.

Noise source identification in indoor substation using a sparse equivalent source method

Noise source identification is a key step in the noise control design of the transformer. The sound propagation is influenced by the reverberation effect in the indoor substation, which leads that the traditional method cannot give correct results in such environment. To supress the reverberation effect, a sparse equivalent source method is proposed to realize the noise source identification in indoor substation. This method first establishes the indoor sound transfer function model between the equivalent source surface and the acquisition surface by using wave simulation. Based on the wave simulation and combined with the sound pressure sampling data, the distributed equivalent source is recovered on the equivalent source surface by sparse recovery algorithm, and finally the sound pressure reconstruction on the reconstruction surface is realized. The numerical verification demonstrates that this method is capable of extracting the real noise information in the reverberation sound and giving the real noise source distribution result. This method can be used as an effective method in the noise source identification in the indoor substation.

Separation of nonstationary sound fields based on the time-domain equivalent source method with single layer pressure-velocity measurements.

This paper proposes a sound field separation technique based on the time-domain equivalent source method with single layer pressure-velocity measurements to extract the nonstationary sound field radiated by the target source in a reverberant environment. This technique constructs a formulation that relates the pressures and particle velocities on a measurement surface to the strengths of time-domain equivalent sources arranged for modelling the outgoing and incoming waves. By solving the strengths of time-domain equivalent sources, the sounds coming from different sides of the measurement surface can be separated independently. In the proposed technique, the use of a time-domain equivalent source model allows the measurement surface to be arbitrarily shaped, thus providing the ability to analyze the arbitrarily shaped sources in a reverberant environment. Numerical simulations investigated the performance of the proposed technique when using different types of arrays, including planar, semi-cylindrical, and semi-spherical arrays, and an experiment with three loudspeakers located at two sides of the measurement surface was carried out to test the validity of the proposed technique. Both numerical and experimental results demonstrate that the proposed technique can remove the influence of disturbing sources in both time and space domains and separate out the target sound fields effectively.

The Input-Output Equivalent Sources Method for Fast Simulations of Distributed Nonlinearities in Bulk Acoustic Wave Resonators and Filters.

This work presents a new method to analyze weak distributed nonlinear (NL) effects, with a focus on the generation of harmonics (H) and intermodulation products (IMD) in bulk acoustic wave (BAW) resonators and filters composed of them. The method consists of finding equivalent current sources [input-output equivalent sources (IOES)] at the H or IMD frequencies of interest that are applied to the boundary nodes of any layer that can contribute to the nonlinearities according to its local NL constitutive equations. The new methodology is compared with the harmonic balance (HB) analysis, by means of a commercial tool, of a discretized NL Mason model, which is the most used model for NL BAW resonators. While the computation time is drastically reduced, the results are fully identical. For the simulation of a seventh-order filter, the IOES method is around 700 times faster than the HB simulations.

Research on Calculation Method for Acoustic Scattering of Helicopter Noise

A new calculation method of helicopter rotor/fuselage acoustic scattering is developed. Firstly, a CFD analysis model is developed to simulate flow field of the rotor, which is based on the motional embedded grid system and RANS equations, and provides aerodynamic data for rotor noise calculation. Then, FW-H equations are employed to calculate the aeroacoustic characteristics of isolated rotor, and G 1A formulas are applied to calculate the rotor acoustic gradient to provide boundary condition for acoustic scattering. Based on these, the time-domain equivalent source method is applied to calculate acoustic scatter field, and the total acoustic field that considered the fuselage scatter is superposed by isolated rotor acoustics and the scatter one. Finally, the numerical simulations of helicopter main-rotor/fuselage and tail-rotor/fuselage scatter effect are conducted by using the developed models. The results indicate that the helicopter fuselage has important scatter effect on the high frequency acoustics of main rotor and tail rotor, and the acoustic scatter effect become more obvious with the smaller space between the main rotor (tail rotor) and fuselage.

Including the Temporal Dimension in the SECS Technique

AbstractThe equivalent source method of Spherical Elementary Current Systems (SECS) has contributed valuable results for spatial magnetic interpolation purposes where no observations are available, as well as for modeling equivalent currents both in the ionosphere and in the subsurface, thus providing a separation between external and internal sources. It has been successfully applied to numerous Space Weather (SW) events, whereas some advantages have been reported over other techniques such as Fourier or Spherical (Cap) Harmonic Analysis. Although different modalities of SECS exist (either 1‐D, 2‐D, or 3‐D) depending on the number of space dimensions involved, the method provides a sequence of instantaneous pictures of the source current. We present an extension of SECS consisting in the introduction of a temporal dependence in the formulation based on a cubic B‐splines expansion. The technique thus adds one dimension, becoming 4‐D in general (e.g., 3‐D + t), and its application is envisaged for, though not restricted to, the analysis of past events including heterogeneous geomagnetic data sets, such as those containing gaps, different sampling rates or diverse data sources. A synthetic model based on the SW Modeling Framework is used to show the efficacy of the extended scheme. We apply this method to characterize the current systems of past and significant SW events producing geomagnetically induced currents, which we exemplify with an outstanding geomagnetic sudden commencement occurred on 24 March 1991.

IRLS based inverse methods tailored to volumetric acoustic source mapping

Planar microphone arrays are of common use in acoustic source identification methods, as well as the use of planar calculation grids. Indeed, the assumption is that the planar grid contains all sources of interest. However, this assumption may not be true in several applications and hence return misleading results. One tentative to overcome this issue is to consider three-dimensional surface adhering on the target. Unfortunately, also this choice may not be enough to obtain accurate results in challenging applications like aeroacoustic source mapping, since noise sources are not necessarily located on the surface of the target. This paper aims to analyze the issues and the benefits arising when the calculation grid turns into a volume. Two inverse methods based on Iterative Re-weighted Least Squares (IRLS) and Bayesian Regularization (BR) are formulated: Equivalent Source Method (ESM-IRLS) and Covariance Matrix Fitting (CMF-IRLS). Even though these methods are based on concepts already known in literature, the focus of this paper is on theoretical and algorithmic aspects that make them able to produce accurate volumetric acoustic maps. The methods proposed are applied both on a simulated and an experimental test case. The former is reported to highlight the difference between standard surface mapping and volumetric mapping. The latter reports an application on an airfoil in an open jet. A comparison with the CLEAN-SC approach is reported in both cases to show the performance of the proposed methods with respect to a well-known state of the art algorithm.

The acoustic inverse problem in the framework of alternating direction method of multipliers

Acoustic imaging is an important technique for sound source identification which is widely used in noise control and fault detection. It can be described as a method for reconstructing the amplitude and phase distribution of an acoustic field in a given plane. When the size of radiating object is huge and measurement environment is noisy, accuracy and efficiency of acoustic imaging algorithm deteriorate easily. In this paper, a novel computational framework is investigated to simplify the algorithm and balance computation speed with accuracy. It converts acoustic imaging into an inverse problem and then divides this problem into two parts including forward model and denoising model. This computation framework is based on the alternating direction method of multipliers, which makes parallel implementation and distributed optimization possible, especially for large scale measurements. It is assumed that the acoustic field is composed of a cluster of monopole sources and the equivalent source method is taken to solve the acoustic inverse problem. Both L2-norm penalization based on the minimal energy assumption and Lp-norm (0<p⩽1) penalization based on the sparsity hypothesis are taken in this framework. Thus, the regularization parameter in the algorithm, which balances the standard least square error and the norm of the source solution, can be coupled with some pre-existing methods incluing L-curve, GCV and Bayesian strategies. Different regularization parameters are proposed and the comparison is made in this paper. The proposed method provides a general and convenient method for different type of sound source priors and it is validated with an experiment.

Downward Continuation and Transformation of Total‐Field Magnetic Anomalies Into Magnetic Gradient Tensors Between Arbitrary Surfaces Using Multilayer Equivalent Sources

AbstractThe equivalent source method offers a potential means for continuing and manipulating magnetic data between arbitrary surfaces. However, it ultimately suffers from a limited continuation distance and significant errors, which restrict its applicability in practice. To address this problem, we reformulate the equivalent source method with a partial differential equation framework. A multilayer equivalent susceptibility method is innovatively derived based on this approach. The location of equivalent sources is designed for adaptive estimation through the inverse procedure. We introduce nonlinear optimization, regularization, and constraining technologies in inversion to improve the stability and accuracy of the solution. Synthetic, real airborne, and ground data are tested in experiments. The results show that the new method significantly improves the capability and accuracy of the downward continuation and transformation of irregularly located data between arbitrary undulating surfaces.