Complex Source Distributions Research Articles

Complex Source Distributions in Wave-based Virtual Acoustics

In the typical scenario in time-domain wave-based acoustics, the solution to the acoustic wave equation is approximated over a three-dimensional volumetric grid using a time stepping method. In the setting of very large simulations, the grid is normally assumed to be regular (e.g. Cartesian) so that massive parallelism may be exploited. One difficulty has been in representing source distributions that do not conform neatly to a regular grid. Using a Fourier-based optimisation procedure in the wave vector domain, it is possible to represent arbitrary source distributions in a flexible way over a pre-defined collection of grid points. Such a methodology is independent of the particular choice of simulation method and depends only on the regularity of the grid. In this paper, approximations to various simple distributions, including the line source and piston, are examined, with regard to accuracy, rotation of the distribution relative to the grid, and the size of the point cloud used to represent the source. Numerical results are presented.

Towards an objective evaluation of EEG/MEG source estimation methods – The linear approach

The spatial resolution of EEG/MEG source estimates, often described in terms of source leakage in the context of the inverse problem, poses constraints on the inferences that can be drawn from EEG/MEG source estimation results. Software packages for EEG/MEG data analysis offer a large choice of source estimation methods but few tools to experimental researchers for methods evaluation and comparison. Here, we describe a framework and tools for objective and intuitive resolution analysis of EEG/MEG source estimation based on linear systems analysis, and apply those to the most widely used distributed source estimation methods such as L2-minimum-norm estimation (L2-MNE) and linearly constrained minimum variance (LCMV) beamformers. Within this framework it is possible to define resolution metrics that define meaningful aspects of source estimation results (such as localization accuracy in terms of peak localization error, PLE, and spatial extent in terms of spatial deviation, SD) that are relevant to the task at hand and can easily be visualized. At the core of this framework is the resolution matrix, which describes the potential leakage from and into point sources (point-spread and cross-talk functions, or PSFs and CTFs, respectively). Importantly, for linear methods these functions allow generalizations to multiple sources or complex source distributions. This paper provides a tutorial-style introduction into linear EEG/MEG source estimation and resolution analysis aimed at experimental (rather than methods-oriented) researchers. We used this framework to demonstrate how L2-MNE-type as well as LCMV beamforming methods can be evaluated in practice using software tools that have only recently become available for routine use. Our novel methods comparison includes PLE and SD for a larger number of methods than in similar previous studies, such as unweighted, depth-weighted and normalized L2-MNE methods (including dSPM, sLORETA, eLORETA) and two LCMV beamformers. The results demonstrate that some methods can achieve low and even zero PLE for PSFs. However, their SD as well as both PLE and SD for CTFs are far less optimal for all methods, in particular for deep cortical areas. We hope that our paper will encourage EEG/MEG researchers to apply this approach to their own tasks at hand.

Self-constrained inversion of potential fields through a 3D depth weighting

A new method for inversion of potential fields is developed using a depth-weighting function specifically designed for fields related to complex source distributions. Such a weighting function is determined from an analysis of the field that precedes the inversion itself. The algorithm is self-consistent, meaning that the weighting used in the inversion is directly deduced from the scaling properties of the field. Hence, the algorithm is based on two steps: (1) estimation of the locally homogeneous degree of the field in a 3D domain of the harmonic region and (2) inversion of the data using a specific weighting function with a 3D variable exponent. A multiscale data set is first formed by upward continuation of the original data. Local homogeneity and a multihomogeneous model are then assumed, and a system built on the scaling function is solved at each point of the multiscale data set, yielding a multiscale set of local-homogeneity degrees of the field. Then, the estimated homogeneity degree is associated to the model weighting function in the source volume. Tests on synthetic data show that the generalization of the depth weighting to a 3D function and the proposed two-step algorithm has great potential to improve the quality of the solution. The gravity field of a polyhedron is inverted yielding a realistic reconstruction of the whole body, including the bottom surface. The inversion of the aeromagnetic real data set, from the Mt. Vulture area, also yields a good and geologically consistent reconstruction of the complex source distribution.

A probabilistic approach to phase calibration – I. Effects of source structure on fringe-fitting

ABSTRACT We propose a probabilistic framework for performing simultaneous estimation of source structure and fringe-fitting parameters in very long baseline interferometry (VLBI) observations. As a first step, we demonstrate this technique through the analysis of synthetic short-duration Event Horizon Telescope observations of various geometric source models at 230 GHz, in the presence of baseline-dependent thermal noise. We perform Bayesian parameter estimation and model selection between the different source models to obtain reliable uncertainty estimates and correlations between various source and fringe-fitting related model parameters. We also compare the Bayesian posteriors with those obtained using widely used VLBI data reduction packages such as casa and aips, by fringe-fitting 200 Monte Carlo simulations of each source model with different noise realizations, to obtain distributions of the maximum a posteriori estimates. We find that, in the presence of resolved asymmetric source structure and a given array geometry, the traditional practice of fringe-fitting with a point source model yields appreciable offsets in the estimated phase residuals, potentially biasing or limiting the dynamic range of the starting model used for self-calibration. Simultaneously estimating the source structure earlier in the calibration process with formal uncertainties improves the precision and accuracy of fringe-fitting and establishes the potential of the available data, especially when there is little prior information. We also note the potential applications of this method to astrometry and geodesy for specific science cases and the planned improvements to the computational performance and analyses of more complex source distributions.

Radiosity view factor model for sources with general distribution

The radiosity view factor model commonly used in radiation heat transfer and computer graphics is applied to a free-molecular flow problem. This paper extends the model to sources with general distribution and includes the diffuse reflections from surfaces, while the previous model is only compatible with an isotropic source distribution. The derivations of the particle flow rate and density are presented, and the numerical implementation is described. The view factor model is compared to the Monte Carlo particle model for density distributions obtained using two different velocity distribution functions for two sources: (1) a Maxwellian distribution with streaming components in both axial and tangential directions and (2) an arbitrary distribution with a discontinuous feature. In these validation studies, it is shown that the radiosity view factor model predicts the correct density distribution that is much smoother and with less simulation run-time compared to the Monte Carlo particle model. While the model herein is applied to a free-molecular flow problem, it can easily be applied to the other problems with a complex source distribution where intensity as a function of wavelength cannot be integrated analytically.

Realistic sub-Rayleigh imaging with phase-sensitive measurements

As the separation between two emitters is decreased below the Rayleigh limit, the information that can be gained about their separation using traditional imaging techniques, photon counting in the image plane, reduces to nil. Assuming the sources are of equal intensity, Rayleigh’s ‘curse’ can be alleviated by making phase-sensitive measurements in the image plane. However, with unequal and unknown intensities the curse returns regardless of the measurement, though the ideal scheme would still outperform image plane counting (IPC), i.e. recording intensities on a screen. We analyze the limits of the super-resolved position localization by inversion of coherence along an edge (SPLICE) phase measurement scheme as the intensity imbalance between the emitters grows. We find that SPLICE still outperforms IPC for moderately disparate intensities. For larger intensity imbalances we propose a hybrid of IPC and SPLICE, which we call ‘adapted SPLICE’, requiring only simple modifications. Using Monte Carlo simulation, we identify regions (emitter brightness, separation, intensity imbalance) where it is advantageous to use SPLICE over IPC, and when to switch to the adapted SPLICE measurement. We find that adapted SPLICE can outperform IPC for large intensity imbalances, e.g. 10 000:1, with the advantage growing with greater disparity between the two intensities. Finally, we also propose additional phase measurements for estimating the statistical moments of more complex source distributions. Our results are promising for implementing phase measurements in sub-Rayleigh imaging tasks such as exoplanet detection.

Off-grid direction of arrival estimation based on joint spatial sparsity for distributed sparse linear arrays.

In the design phase of sensor arrays during array signal processing, the estimation performance and system cost are largely determined by array aperture size. In this article, we address the problem of joint direction-of-arrival (DOA) estimation with distributed sparse linear arrays (SLAs) and propose an off-grid synchronous approach based on distributed compressed sensing to obtain larger array aperture. We focus on the complex source distribution in the practical applications and classify the sources into common and innovation parts according to whether a signal of source can impinge on all the SLAs or a specific one. For each SLA, we construct a corresponding virtual uniform linear array (ULA) to create the relationship of random linear map between the signals respectively observed by these two arrays. The signal ensembles including the common/innovation sources for different SLAs are abstracted as a joint spatial sparsity model. And we use the minimization of concatenated atomic norm via semidefinite programming to solve the problem of joint DOA estimation. Joint calculation of the signals observed by all the SLAs exploits their redundancy caused by the common sources and decreases the requirement of array size. The numerical results illustrate the advantages of the proposed approach.

The influence of noise sources on cross-correlation amplitudes

We use analytical examples and asymptotic forms to examine the mathematical structure and physical meaning of the seismic cross-correlation measurement. We show that in general, cross correlations are not Green's functions of medium, and may be very different depending on the source distribution. The modelling of noise sources using spatial distributions as opposed to discrete collections of sources is emphasized. When stations are illuminated by spatially complex source distributions, cross correlations show arrivals at a variety of time lags, from zero to the maximum surface-wave arrival time. Here, we demonstrate the possibility of inverting for the source distribution using the energy of the full cross-correlation waveform. The interplay between the source distribution and wave attenuation in determining the functional dependence of cross-correlation energies on station-pair distance is quantified. Without question, energies contain information about wave attenuation. However, the accurate interpretation of such measurements is tightly connected to the knowledge of the source distribution.

U/Pb ages on detrital zircons in the southern central Andes Neogene foreland (36°–37°S): Constraints on Andean exhumation

Abstract U/Pb dating on detrital zircons was performed in the Pampa de Carrizalito depocenter of the Late Miocene foreland basin associated with the Southern Central Andes orogenic front. This reveals Andean and pre-Andean components in magmatic derived zircons inhomogeneously distributed through the sequence. Andean, Grenville, Pampean, Famatinian and Gondwanic components reveal a complex source distribution from either the Main Andes, Coastal Cordillera and basement foreland areas. These are discussed showing different patterns in the context of the Andean orogenic cycle. Cretaceous and Jurassic components that are partly related to Mesozoic batholiths, developed at the western slope of the Andes at these latitudes, have a very contrasting behavior through the sequence: While Jurassic grains are represented from base to top, Cretaceous ones dilute upwardly. This is explained through the progressive uplift of the Southern Central Andes that could have created a barrier to Cretaceous and Jurassic detritus, while the older ones could have had either an alternative source area represented by the inverted rift system of the Huincul Ridge in the foreland area and the Cordillera del Viento in the hinterland area or the reworking of Jurassic sedimentary sequences of the Neuquen basin. Finally, a progressive enrichment in pre-Andean components to the top of the sequence is interpreted as related to the development of a broken foreland and the consequent rapid expansion of the orogenic front at the time of development of a slab shallowing setting in the region as shown by previous works.

Comparison of noise-normalized minimum norm estimates for MEG analysis using multiple resolution metrics

Noise-normalization has been shown to partly compensate for the localization bias towards superficial sources in minimum norm estimation. However, it has been argued that in order to make inferences for the case of multiple sources, localization properties alone are insufficient. Instead, multiple measures of resolution should be applied to both point-spread and cross-talk functions (PSFs and CTFs). Here, we demonstrate that noise-normalization affects the shapes of PSFs, but not of CTFs. We evaluated PSFs and CTFs for the MNE, dSPM and sLORETA inverse operators, on the metrics dipole localization error (DLE), spatial dispersion (SD) and overall amplitude (OA). We used 306-channel MEG configurations obtained from 17 subjects in a real experiment, including individual noise covariance matrices and head geometries. We confirmed that for PSFs DLE improved after noise normalization, and is zero for sLORETA. However, SD was generally lower for the unnormalized MNE. OA distributions were similar for all three methods, indicating that all three methods may greatly underestimate some sources relative to others. The reliability of differences between methods across subjects was demonstrated using distributions of standard deviations and p-values from paired t-tests. As predicted, the shapes of CTFs were the same for all methods, reflecting the general resolution limits of the inverse problem. This means that noise-normalization is of no consequence where linear estimation procedures are used as “spatial filters.” While low DLE is advantageous for the localization of a single source, or possibly a few spatially distinct sources, the benefit for the case of complex source distributions is not obvious. We suggest that software packages for source estimation should include comprehensive tools for evaluating the performance of different methods.

The utility of potential field enhancements for remote predictive mapping

Potential field (gravity and magnetic) data are fundamental to geologic mapping approaches based on geophysical interpretations. Lithologic and structural information can be extracted from amplitudes and trends of the gravity and magnetic anomalies. To assist in the interpretation, data enhancements are commonly used to transform the original data into new quantities that emphasize or suppress some specified component, resulting in a more easily interpreted representation. The number of available enhancements is rapidly increasing and begs the following question: Are they all useful and effective? Based on modelling, theoretical arguments, and real data examples, we evaluate 12 enhancements in terms of their effectiveness. Four of the 12 are found to be redundant, giving almost identical results when used for quantitatively mapping the edges of magnetic sources or, equivalently, geologic contacts. Two others produce similar levels of detail, so both enhancements are not needed. Two other enhancements are susceptible to noise in the data, leading to a lack of continuity in the mapped contact locations. One of the enhancements produces extra, false contact locations, which can complicate mapping, especially in areas of complex source distributions. When enhancements are used to derive images for qualitative interpretation, only two out of the four that are appropriate for this kind of interpretation give useful information.

Source Component Display and Potential Field Interpretation

2D or 3D gravity and magnetic modelling procedures usually involve comparison of the response from a few, or simple, sources with an observed profile. Common software may also allow complex source distributions typical of a complete geological section. When several sources are involved the observed profile may be the result of complex interference and individual contributions may not be obvious. Gradients may be transformed by such interference. Much trial and error may be required to resolve or understand the source and effect interactions. This is not a simple process if only the resultant calculated from all sources is compared with the reference profile. A source component profile represents the response due to a specific source within a model. Assessment of critical geometry or contrast changes, crucial source segments and curve fitting parameters - and hence model revision, can be greatly accelerated if the actual responses from each part of the model are viewed in addition to the resultant of their effects. Review of the source contributions in a model leads to more satisfactory interpretations since it reveals how the observed profile may, or may not, be composed. Component profiles also demonstrate how observed and resultant gradients may be modified by interference and thus provide some salutory cautions about the use and reliability of simple procedures. Software should be modified to allow this more complex display and a declaration of how the resultant and observed curves were fitted. Most currently available modelling software either generates, stores or requests the required information. The paper includes examples from the Mount Isa Inlier of NW Queensland to illustrate the use of source components.