Statistical Inversion Research Articles

Estimating Range-Dependent Evaporation Duct Height

AbstractThe refractivity from clutter (RFC) technique has been proved to be an effective way to estimate atmospheric duct structure. An important issue for RFC is how to make the estimate more robust, especially in range-dependent ducting conditions. Traditionally, statistical inversion methods need a large number of forward propagation model runs to obtain an acceptable result. Especially when the parameter search space is multidimensional, these methods are prone to being trapped into local optimal solutions. Recently published results (Zhao and Huang) indicate that the adjoint parabolic equation (PE) method holds promise for real-time estimation of one-dimensional refractive index structure from radar sea clutter returns. This paper is aimed at extending the adjoint PE method to range-dependent evaporation duct cases, with a log-linear relationship describing duct structures. Numerical simulations are used to test the performance of this method and the results are compared with that retrieved using a genetic algorithm. Both noise-free and 3-dB additive Gaussian noise clutter simulations are considered, as well as linearly and nonlinearly varying duct height with range.

Abstract A02: Multiple spatio-temporal scale modeling with application to brain cancer

Abstract Introduction: We aim to develop a multiple spatio-temporal scale (MSTS) framework capable of bridging tissue, cellular, and subcellular scales. Our overarching hypothesis is that an accurate and precise MSTS model, initialized with patient-specific information, can dramatically outperform current standard-of-care for diagnosis and selecting therapy for the individual cancer patient. Methodology: We developed a MSTS framework for tumor growth and treatment outcome prediction. Our model incorporated chemotherapy so that we can predict changes in tumor volume due to various therapeutic regimens. The MSTS framework consists of a tissue scale model, a cellular activity and growth model, and a subcellular signaling pathway model. To predict tissue growth at the tissue scale, a continuum model is constructed where the biological tissue is represented as a mixture of multiple constituents with each of which represents either a volume fraction or concentration. The constituents interact with each other through mass and momentum exchange so that the governing equations are based on both mass and momentum conservation laws. The constitutive equations account for tissue heterogeneity, nonlinear behavior, and thermodynamic consistency. The system of partial differential equations is solved using finite element techniques. To bridge the tissue and cell scales, each finite element is discretized into virtual cell clusters to model cellular growth and proliferation by using an agent-based model to determine various cellular scale activities including cell division, cell death, and phenotypical alteration. The cellular scale is further discretized temporally to model the effects of selected signaling pathways (e.g., PI3K/AKT/mTOR pathway). In many cancers, mTOR pathway becomes hyperactive and promotes abnormal cell proliferation. The mechanism and effects of a mTOR inhibiting drug known as rapamycin are tested in silico. The subcellular scale is modeled using a set of ordinary differential equations. A statistical inverse algorithm is used to calibrate and validate the results using magnetic resonance imaging (MRI) data, and a Bayesian inference method is employed to account for the uncertainties in model parameters. Results: The 3D simulation results recapitulate, at various physical and biological conditions, different tumor characteristics including morphological instability, changes in the spatio-temporal distribution of different cell phenotypes and density, and the reduction of growth rate due to drug delivery. Thus, this model has the potential to be used for patient-specific prediction of tumor growth as well as personalized treatment planning. The novelties of this modeling framework are that (1) the events at each scale are modeled using different numerical techniques suitable for that specific scale in terms of both space and time, (2) a novel numerical technique is developed to bridge the scales and solved simultaneously, (3) the tissue scale model is thermodynamically consistent with consideration of tissue heterogeneity and nonlinear behaviors, (4) various cellular events such as the phenotypical alteration, apoptosis, necrosis, mitosis, etc. are modeled at the cellular scale, and (5) the mTOR signaling pathway, an important pathway for many different cancers, is modeled at the subcellular scale. Conclusions: Our MSTS model is capable of predicting tumor growth with considerations of various factors at different spatio-temporal scales. It can be potentially used as a patient-specific tumor growth model for in silico drug testing, treatment planning, and prognosis. A future version of this model can include cancer stem cells, angiogenesis, and metastasis. Citation Format: Mohammad Rahman, Yusheng Feng, Thomas E. Yankeelov, J. Tinsley Oden. Multiple spatio-temporal scale modeling with application to brain cancer. [abstract]. In: Proceedings of the AACR Special Conference on Engineering and Physical Sciences in Oncology; 2016 Jun 25-28; Boston, MA. Philadelphia (PA): AACR; Cancer Res 2017;77(2 Suppl):Abstract nr A02.

Efficient State/Parameter Estimation in Nonlinear Unsteady PDEs by a Reduced Basis Ensemble Kalman Filter

The ensemble Kalman filter is a computationally efficient technique for solving state and/or parameter estimation problems in the framework of statistical inversion when relying on a Bayesian paradigm. Unfortunately, its cost may become moderately large for systems described by nonlinear time-dependent PDEs, because of the cost entailed by each PDE query. In this paper we propose a reduced basis ensemble Kalman filter technique to address the above problems. The reduction stage yields intrinsic approximation errors, whose propagation through the filtering process might affect the accuracy of state/parameter estimates. For an efficient evaluation of these errors, we equip our reduced basis ensemble Kalman filter with a reduction error model (or error surrogate). The latter is based on ordinary kriging for functional-valued data, to gauge the effect of state reduction on the whole filtering process. The accuracy and efficiency of the resulting method is then verified on the estimation of uncertain parameter...

Bayesian Analysis for a Fractional Population Growth Model

We implement the Bayesian statistical inversion theory to obtain the solution for an inverse problem of growth data, using a fractional population growth model. We estimate the parameters in the model and we make a comparison between this model and an exponential one, based on an approximation of Bayes factor. A simulation study is carried out to show the performance of the estimators and the Bayes factor. Finally, we present a real data example to illustrate the effectiveness of the method proposed here and the pertinence of using a fractional model.

Uncertainty estimation for linearised inverse problems comparing Bayesian inference and a pseudoinverse approach for acoustic transmission measurements

Abstract The determination of acoustic material parameters using ultrasonic transmission measurements can mathematically be described as an inverse problem. The question concerning the influence of uncertainties on the problem's solution can be answered using a statistical approach. Therefore, the sources of uncertainty have to be identified statistically. A method for linearising the model function using the Guide to the expression of uncertainty in measurement is introduced for inverse problems. The statistical inversion of the model is executed by means of two different approaches, that are compared and interpreted exemplarily.

Structural damage identification via time domain response and Markov Chain Monte Carlo method

The present work addresses the problem of structural damage identification built on the statistical inversion approach. Here, the damage state of the structure is continuously described by a cohesion parameter, which is spatially discretized by the finite element method. The inverse problem of damage identification is then posed as the determination of the posterior probability densities of the nodal cohesion parameters. The Markov Chain Monte Carlo method, implemented with the Metropolis–Hastings algorithm, is considered in order to approximate the posterior probabilities by drawing samples from the desired joint posterior probability density function. With this approach, prior information on the sought parameters can be used and the uncertainty concerning the known values of the material properties can be quantified in the estimation of the cohesion parameters. The assessment of the proposed approach has been performed by means of numerical simulations on a simply supported Euler–Bernoulli beam. The damage identification and assessment are performed considering time domain response data. Different damage scenarios and noise levels were addressed, demonstrating the feasibility of the proposed approach.

The Exhumation history of the European Alps inferred from linear inversion of thermochronometric data

Thermochronometric data collected across the Alps over the last three decades allows for investigation of the evolution of this orogen, which is subject to changes in climate and geodynamics. Exhumation rates are inferred from the thermochronometric ages using a statistical inversion method based on the fact that the distance a sample traveled since closure is equal to the integral of the exhumation rate from the present day to the age of the sample. Exhumation rates are assumed to be spatially correlated but are free to vary through time. This results in the quantification of exhumation rates across the Alps, since 32 Ma, along with assessments of the quality of these inferences. We find that exhumation rates are initially fast in the internal arc of the Western Alps at rates up to 0.8 km/Myr at 30 Ma, decreasing at 20 Ma to 0.3 km/Myr to remain slow to the present. At the same time, around 20 Ma, rates across the External Crystalline Massifs of Western Alps increase to 0.6 km/Myr. We also find that the onset of high exhumation rates in the Tauern Window and the Lepontine Dome occurs at around 20 Ma, a time characterized by major reorganizations in the Alpine chain. A general increase in exhumation rates at around 5 Ma over the entire Alps is not confirmed. Instead we find that the Western Alps exhibit a 2 to 3 fold increase in exhumation rate over the last 2 Ma, during a recent event not seen further east, in spite of very similar topographic characteristics. We attribute this strong signal to detachment of the European slab in the Western Alps, combined with efficient glacial erosion.

Bayesian statistical ionospheric tomography improved by incorporating ionosonde measurements

Abstract. We validate two-dimensional ionospheric tomography reconstructions against EISCAT incoherent scatter radar measurements. Our tomography method is based on Bayesian statistical inversion with prior distribution given by its mean and covariance. We employ ionosonde measurements for the choice of the prior mean and covariance parameters and use the Gaussian Markov random fields as a sparse matrix approximation for the numerical computations. This results in a computationally efficient tomographic inversion algorithm with clear probabilistic interpretation. We demonstrate how this method works with simultaneous beacon satellite and ionosonde measurements obtained in northern Scandinavia. The performance is compared with results obtained with a zero-mean prior and with the prior mean taken from the International Reference Ionosphere 2007 model. In validating the results, we use EISCAT ultra-high-frequency incoherent scatter radar measurements as the ground truth for the ionization profile shape. We find that in comparison to the alternative prior information sources, ionosonde measurements improve the reconstruction by adding accurate information about the absolute value and the altitude distribution of electron density. With an ionosonde at continuous disposal, the presented method enhances stand-alone near-real-time ionospheric tomography for the given conditions significantly.

Identification of contaminant source architectures—A statistical inversion that emulates multiphase physics in a computationally practicable manner

AbstractThe goal of this work is to improve the inference of nonaqueous‐phase contaminated source zone architectures (CSA) from field data. We follow the idea that a physically motivated model for CSA formation helps in this inference by providing relevant relationships between observables and the unknown CSA. Typical multiphase models are computationally too expensive to be applied for inverse modeling; thus, state‐of‐the‐art CSA identification techniques do not yet use physically based CSA formation models. To overcome this shortcoming, we apply a stochastic multiphase model with reduced computational effort that can be used to generate a large ensemble of possible CSA realizations. Further, we apply a reverse transport formulation in order to accelerate the inversion of transport‐related data such as downgradient aqueous‐phase concentrations. We combine these approaches within an inverse Bayesian methodology for joint inversion of CSA and aquifer parameters. Because we use multiphase physics to constrain and inform the inversion, (1) only physically meaningful CSAs are inferred; (2) each conditional realization is statistically meaningful; (3) we obtain physically meaningful spatial dependencies for interpolation and extrapolation of point‐like observations between the different involved unknowns and observables, and (4) dependencies far beyond simple correlation; (5) the inversion yields meaningful uncertainty bounds. We illustrate our concept by inferring three‐dimensional probability distributions of DNAPL residence, contaminant mass discharge, and of other CSA characteristics. In the inference example, we use synthetic numerical data on permeability, DNAPL saturation and downgradient aqueous‐phase concentration, and we substantiate our claims about the advantages of emulating a multiphase flow model with reduced computational requirement in the inversion.

Spatial and temporal Antarctic Ice Sheet mass trends, glacio-isostatic adjustment, and surface processes from a joint inversion of satellite altimeter, gravity, and GPS data.

We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic‐corrected GPS data for the period 2003–2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time‐invariant solution for glacio‐isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003–2013, Antarctica has been losing mass at a rate of −84 ± 22 Gt yr−1, with a sustained negative mean trend of dynamic imbalance of −111 ± 13 Gt yr−1. West Antarctica is the largest contributor with −112 ± 10 Gt yr−1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of −28 ± 7 Gt yr−1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr−1 in East Antarctica due to a positive trend of surface mass balance anomalies.

The Retrieve and Data Fusion of the Sea Surface Temperature with FengYun Geostationary Satellite Data

The numerical weather prediction (NWP) model about marine meteorological disasters requires sea surface temperature (SST) data which represents the parameters of marine dynamic process. However, the SST data of high spatial-temporal resolution are scarce in Huang Bohai sea in China. In order to acquire the high resolution SST data, the authors try to retrieve the high resolution SST data in Huang Bohai sea with FY2G-based satellite data. Moreover, the data differences of the satellite retrieval data with both buoys observation SST and NCEP optimal interpolation SST are compared, respectively, in terms of the statistical analysis method. Meanwhile the statistical equations are determined about the satellite retrieval SST with observed SST and of optimal interpolation SST. The equations are referred as a standard of data quality control equations when the differences of the revised values of the statistical equations and satellite inversion are calculated. While the data fusion is done, authors retain the data that the differences of the satellite retrieval SST and the statistical equation are less than 3.0°C. Above method could make that the satellite SST space characteristic information is merged into weekly average optimal interpolation SST data, and it improves SST spatial-temporal resolution in Huang Bohai sea. The work lays a foundation for numerical models using the high resolution SST in Huang Bohai area.

Applied Mathematics in EM Studies with Special Emphasis on an Uncertainty Quantification and 3-D Integral Equation Modelling

Despite impressive progress in the development and application of electromagnetic (EM) deterministic inverse schemes to map the 3-D distribution of electrical conductivity within the Earth, there is one question which remains poorly addressed—uncertainty quantification of the recovered conductivity models. Apparently, only an inversion based on a statistical approach provides a systematic framework to quantify such uncertainties. The Metropolis–Hastings (M–H) algorithm is the most popular technique for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. However, all statistical inverse schemes require an enormous amount of forward simulations and thus appear to be extremely demanding computationally, if not prohibitive, if a 3-D set up is invoked. This urges development of fast and scalable 3-D modelling codes which can run large-scale 3-D models of practical interest for fractions of a second on high-performance multi-core platforms. But, even with these codes, the challenge for M–H methods is to construct proposal functions that simultaneously provide a good approximation of the target density function while being inexpensive to be sampled. In this paper we address both of these issues. First we introduce a variant of the M–H method which uses information about the local gradient and Hessian of the penalty function. This, in particular, allows us to exploit adjoint-based machinery that has been instrumental for the fast solution of deterministic inverse problems. We explain why this modification of M–H significantly accelerates sampling of the posterior probability distribution. In addition we show how Hessian handling (inverse, square root) can be made practicable by a low-rank approximation using the Lanczos algorithm. Ultimately we discuss uncertainty analysis based on stochastic inversion results. In addition, we demonstrate how this analysis can be performed within a deterministic approach. In the second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.

Sensitivity of Central Oklahoma Convection Forecasts to Upstream Potential Vorticity Anomalies during Two Strongly Forced Cases during MPEX

Abstract The role of upstream subsynoptic forecast errors on forecasts of two different central Oklahoma severe convection events (19 and 31 May 2013) characterized by strong synoptic forcing during the Mesoscale Predictability Experiment (MPEX) are evaluated by applying the ensemble-based sensitivity technique to WRF ensemble forecasts with explicit convection. During both cases, the forecast of the timing and intensity of convection over central Oklahoma is modulated by the southward extent of upstream midtropospheric potential vorticity anomalies that are moving through the base of a larger-scale upstream trough but pass by central Oklahoma prior to convective initiation. In addition, the convection forecasts are also sensitive to the position of lower-tropospheric boundaries, such that moving the boundaries in a manner that would lead to increased equivalent potential temperature over central Oklahoma prior to convective initiation leads to more precipitation. Statistical PV inversion and correlation calculations suggest that the midtropospheric PV and near-surface boundary sensitivities are not independent; the winds associated with the PV error can modulate the position of the lower-tropospheric boundary through advection in a manner consistent with the implied sensitivity. As a consequence, it appears that reducing the uncertainty in specific upstream subsynoptic features prior to convective initiation could improve subsequent forecasts of severe convection.

Applying compressive sensing to TEM video: a substantial frame rate increase on any camera

AbstractOne of the main limitations of imaging at high spatial and temporal resolution duringin-situtransmission electron microscopy (TEM) experiments is the frame rate of the camera being used to image the dynamic process. While the recent development of direct detectors has provided the hardware to achieve frame rates approaching 0.1 ms, the cameras are expensive and must replace existing detectors. In this paper, we examine the use of coded aperture compressive sensing (CS) methods to increase the frame rate of any camera with simple, low-cost hardware modifications. The coded aperture approach allows multiple sub-frames to be coded and integrated into a single camera frame during the acquisition process, and then extracted upon readout using statistical CS inversion. Here we describe the background of CS and statistical methods in depth and simulate the frame rates and efficiencies forin-situTEM experiments. Depending on the resolution and signal/noise of the image, it should be possible to increase the speed of any camera by more than an order of magnitude using this approach.Mathematics Subject Classification:(2010) 94A08 · 78A15

ON THE BLACK HOLE MASS—X-RAY EXCESS VARIANCE SCALING RELATION FOR ACTIVE GALACTIC NUCLEI IN THE LOW-MASS REGIME

Recent studies of active galactic nuclei (AGN) found a statistical inverse linear scaling between the X-ray normalized excess variance $\sigma_{\rm rms}^2$ (variability amplitude) and the black hole mass spanning over $M_{\rm BH}=10^6- 10^9\ M_{\odot}$. Being suggested to have a small scatter, this scaling relation may provide a novel method to estimate the black hole mass of AGN. However, a question arises as to whether this relation can be extended to the low-mass regime below $\sim10^6\ M_{\odot}$. If confirmed, it would provide an efficient tool to search for AGN with low-mass black holes using X-ray variability. This paper presents a study of the X-ray excess variances for a sample of AGN with black hole masses in the range of $10^5- 10^6\ M_{\odot}$ observed with {\it XMM-Newton} and {\it ROSAT}, including data both from the archives and from newly preformed observations. It is found that the relation is no longer a simple extrapolation of the linear scaling; instead, the relation starts to flatten at $\sim10^6\ M_{\odot}$ toward lower masses. Our result is consistent with the recent finding of \citet{L15}. Such a flattening of the $M_{\rm BH}-\sigma_{\rm rms}^2$ relation is actually expected from the shape of the power spectrum density of AGN, whose break frequency is inversely scaled with the mass of black holes.

Statistic inversion of multi-zone transition probability models for aquifer characterization in alluvial fans

Understanding the heterogeneity arising from the complex architecture of sedimentary sequences in alluvial fans is challenging. This paper develops a statistical inverse framework in a multi-zone transition probability approach for characterizing the heterogeneity in alluvial fans. An analytical solution of the transition probability matrix is used to define the statistical relationships among different hydrofacies and their mean lengths, integral scales, and volumetric proportions. A statistical inversion is conducted to identify the multi-zone transition probability models and estimate the optimal statistical parameters using the modified Gauss–Newton–Levenberg–Marquardt method. The Jacobian matrix is computed by the sensitivity equation method, which results in an accurate inverse solution with quantification of parameter uncertainty. We use the Chaobai River alluvial fan in the Beijing Plain, China, as an example for elucidating the methodology of alluvial fan characterization. The alluvial fan is divided into three sediment zones. In each zone, the explicit mathematical formulations of the transition probability models are constructed with optimized different integral scales and volumetric proportions. The hydrofacies distributions in the three zones are simulated sequentially by the multi-zone transition probability-based indicator simulations. The result of this study provides the heterogeneous structure of the alluvial fan for further study of flow and transport simulations.

Statistical inversion in electrical impedance tomography using mixed total variation and non-convex ℓ p regularization prior

Abstract Electrical impedance tomography (EIT) is a well-known technique to estimate the conductivity distribution γ of a body Ω with unknown electromagnetic properties. EIT is a severely ill-posed inverse problem. In this paper, we formulate the EIT problem in the Bayesian framework using mixed total variation (TV) and non-convex ℓ p regularization prior. We use the Markov Chain Monte Carlo (MCMC) method for sampling the posterior distribution to solve the ill-posed inverse problem in EIT. We present simulations to estimate the distribution for each pixel for the image reconstruction of the conductivity in EIT.

Bias in detrital fission track grain-age populations: Implications for reconstructing changing erosion rates

The sedimentary record is our principal archive of mass transfer across the Earth's surface in response to tectonic and climatic changes in the geologic past. The thermochronology of individual sediment grains (detrital thermochronology) has emerged as a critical tool to infer erosion rates and track mountain belt evolution. Such inferences are reliant upon the statistical inversion of detrital grain ages to unbiasedly approximate the cooling history of the source areas from which the sediment originated. However, it is challenging to critique the reliability and consistency of modelled ages. These arise both from fundamental measurement uncertainties and the assumptions we employ in inverting the data. For detrital fission track modelling of young detrital samples, this problem is particularly acute since the uncertainty on the track counts produces uncertainty in the age estimates. We apply Monte-Carlo modelling to generate synthetic detrital data conditioned on known closure age models, and then invert the grain data to assess the reliability of different inversion schemes. The results clearly demonstrate that existing practice can be subject to large uncertainty, to systematic bias and to non-uniqueness of interpretation. We then show how to map such regions of systematic bias in the population modelling as a function of the true closure ages, and how this bias propagates through into the lag-time modelling. Applying the method to real data from the Siwalik group sediments in western Nepal, we find no evidence for a change in the underlying climate or tectonic processes, since the apparent change in lag coincides with a thresholded change in the resolution of the population modelling. This paper shows how to map regions of systematic bias in the population modelling as a function of the true closure ages, and how this bias propagates through into the lag-time modelling and can be applied retrospectively to existing studies. However, it is equally applicable to other age inversion schemes such as minimum age modelling. The application of these methods will enhance current practice and facilitate more robust interpretation of grain ages, in particular in discriminating between stationary and non-stationary geological and climatic processes.

Storm-time characteristics of the equatorial ionization anomaly in the East African sector

The equatorial ionization anomaly (EIA), inferred from the electron density profile, was used to study the ionospheric effect of 11 March 2011, 06 April 2011, 09 March 2012 and 01 October 2012 geomagnetic storms in the East African sector. The electron density profile was reconstructed from slant total electron content (sTEC) measurements using statistical linear inversion method. The sTEC measurements were recorded by a chain of ten ground-based GPS receivers deployed in the East African region in the latitude range of 6° S–20° N GeoLat (15.29° S–10.62° N geomagnetic latitude). The analysis of the effect of the storms on the EIA has demonstrated that the effect could be positive or negative. The sudden positive effects of the EIA, in terms of increasing the peak and widening the width, during storm events of 06 April 2011, 09 March 2012 and 01 October 2012 were observed dominantly due to prompt penetration electric fields to the magnetic equator, which were caused by a southward turning of the interplanetary magnetic field (Bz). The prolonged effects after the onset of the storm were attributed to disturbance dynamo electric field due to the storm-time neutral wind circulation. The depletion on the electron density profile during 11 March 2011 storm was due to a decrease in [O] to [N2] ratio in the thermosphere composition.

Robust statistical recognition and reconstruction scheme based on hierarchical Bayesian learning of HRR radar target signal

A hierarchical Bayesian model is developed to characterize the complex-valued high range resolution (HRR) radar target signal, motivated by the problem of radar automatic target recognition (RATR) robust to low signal-to-noise ratio (SNR) or narrowband interference. Here we assume a sparseness-promoting prior on the complex echoes and a Markov dependency for the location of the dominant scattering center between consecutive HRR signals. The number of the dominant scattering centers can be automatically determined, while the posterior distributions of their complex coefficients and locations can be inferred via such Bayesian model. In the training stage, based on the proposed Bayesian model, the statistical aspect-frame template can be learned for HRR complex training samples from each target-aspect sector under high SNR and without any interference. Considering the low SNR or narrowband interference problem for a test sample, the aspect-frame templates can be updated to make a robust recognition decision for the noised and interfered test sample. Simultaneously, the test sample can be denoised and recovered via the analytically posterior estimation for the reconstruction, which is referred to as the statistical compressive sensing (CS) inversion. In contrast to the traditional CS methods that only utilize the underlying sparse property of a measurement (here a measurement is a test sample), our statistical CS inversion also exploits the statistical information of the training samples, which are under high SNR and without any interference. Therefore, the better recognition and reconstruction performances can be obtained via our method. Efficient inference is performed via variational Bayesian (VB) for the proposed Bayesian model. To validate the formulation, we present our experimental results on the measured HRR dataset, with comparisons to some state-of-the-art methods.