Iterative Maximum Likelihood Method Research Articles

New Closed-Form Joint Localization and Synchronization Using Sequential One-Way TOAs

It is an essential technique for the moving user nodes (UNs) with clock offset and clock skew to resolve the joint localization and synchronization (JLAS) problem. Existing iterative maximum likelihood methods using sequential one-way time-of-arrival (TOA) measurements from the anchor nodes' (AN) broadcast signals require a good initial guess and have a computational complexity that grows with the number of iterations, given the size of the problem. In this paper, we propose a new closed-form JLAS approach, namely CFJLAS, which achieves the asymptotically optimal solution in one shot without initialization when the noise is small, and has a low computational complexity. After squaring and differencing the sequential TOA measurement equations, we devise two intermediate variables to reparameterize the non-linear problem. In this way, we convert the problem to a simpler one of solving two simultaneous quadratic equations. We then solve the equations analytically to obtain a raw closed-form JLAS estimation. Finally, we apply a weighted least squares (WLS) step to optimize the estimation. We derive the Cramer-Rao lower bound (CRLB), analyze the estimation error, and show that the estimation accuracy of the CFJLAS reaches the CRLB under the small noise condition. The complexity of the new CFJLAS is only determined by the size of the problem, unlike the conventional iterative method, whose complexity is additionally multiplied by the number of iterations. Simulations in a 2D scene verify that the estimation accuracies of the new CFJLAS method in position, velocity, clock offset, and clock skew all reach the CRLB under the small noise condition. Compared with the conventional iterative method, the proposed new CFJLAS method does not require initialization, obtains the optimal solution under the small noise condition, and has a low computational complexity.

Deciphering the Local Interstellar Spectra of Secondary Nuclei with the Galprop/Helmod Framework and a Hint for Primary Lithium in Cosmic Rays.

Local interstellar spectra (LIS) of secondary cosmic-ray (CR) nuclei, lithium, beryllium, boron, and partially secondary nitrogen, are derived in the rigidity range from 10 MV to ~200 TV using the most recent experimental results combined with state-of-the-art models for CR propagation in the Galaxy and in the heliosphere. The lithium spectrum appears somewhat flatter at high energies compared to other secondary species, which may imply a primary lithium component. Two propagation packages, GALPROP and HelMod, are combined to provide a single framework that is run to reproduce direct measurements of CR species at different modulation levels, and at both polarities of the solar magnetic field. An iterative maximum-likelihood method is developed that uses GALPROP-predicted LIS as input to HelMod, which provides the modulated spectra for specific time periods of the selected experiments for the model-data comparison. The proposed LIS accommodates the low-energy interstellar spectra measured by Voyager 1, the High Energy Astrophysics Observatory-3 (HEAO-3), and the Cosmic Ray Isotope Spectrometer on board of the Advanced Composition Explorer (ACE/CRIS), as well as the high-energy observations by the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA), Alpha Magnetic Spectrometer-02 (AMS-02), and earlier experiments that are made deep in the heliosphere. The interstellar and heliospheric propagation parameters derived in this study are consistent with our earlier results for propagation of CR protons, helium, carbon, oxygen, antiprotons, and electrons.

Eigenspace Solution for AOA Localization in Modified Polar Representation

This paper focuses on the development and analysis of algebraic solution for source localization using angles of arrival (AOAs) in the modified polar representation (MPR), where sensor position errors are present. MPR uses the arrival angle(s) and inverse-range to represent the source location, thereby integrating the localization of the source to a single framework regardless it is in the near-field or far-field. Previous solution for AOA localization in MPR ignores sensor position errors and uses the iterative Maximum Likelihood method with a coarse semi-definite relaxation solution as the initialization, which is computationally demanding and does not ensure global convergence. We propose the constrained eigenspace technique to this localization problem that yields computationally attractive algebraic solution without having the local convergence issue. Depending on the constraints imposed, two solutions are developed. One is more computationally efficient and the other has smaller bias. The solutions are studied theoretically by mean-square analysis to characterize their statistical performance. Simulations validate the CRLB performance of the two proposed solutions under Gaussian noise.

Deciphering the Local Interstellar Spectra of Primary Cosmic-Ray Species with helmod.

Local interstellar spectra (LIS) of primary cosmic ray (CR) nuclei, such as helium, oxygen, and mostly primary carbon are derived for the rigidity range from 10 MV to ~200 TV using the most recent experimental results combined with the state-of-the-art models for CR propagation in the Galaxy and in the heliosphere. Two propagation packages, GALPROP and helmod, are combined into a single framework that is used to reproduce direct measurements of CR species at different modulation levels, and at both polarities of the solar magnetic field. The developed iterative maximum-likelihood method uses GALPROP-predicted LIS as input to helmod, which provides the modulated spectra for specific time periods of the selected experiments for model-data comparison. The interstellar and heliospheric propagation parameters derived in this study are consistent with our prior analyses using the same methodology for propagation of CR protons, helium, antiprotons, and electrons. The resulting LIS accommodate a variety of measurements made in the local interstellar space (Voyager 1) and deep inside the heliosphere at low (ACE/CRIS, HEAO-3) and high energies (PAMELA, AMS-02).

HelMod in the Works: From Direct Observations to the Local Interstellar Spectrum of Cosmic-Ray Electrons.

The local interstellar spectrum (LIS) of cosmic-ray (CR) electrons for the energy range 1 MeV to 1 TeV is derived using the most recent experimental results combined with the state-of-the-art models for CR propagation in the Galaxy and in the heliosphere. Two propagation packages, GALPROP and HelMod, are combined to provide a single framework that is run to reproduce direct measurements of CR species at different modulation levels, and at both polarities of the solar magnetic field. An iterative maximum-likelihood method is developed that uses GALPROP-predicted LIS as input to HelMod, which provides the modulated spectra for specific time periods of the selected experiments for model-data comparison. The optimized HelMod parameters are then used to adjust GALPROP parameters to predict a refined LIS with the procedure repeated subject to a convergence criterion. The parameter optimization uses an extensive data set of proton spectra from 1997 to 2015. The proposed CR electron LIS accommodates both the low-energy interstellar spectra measured by Voyager 1 as well as the high-energy observations by PAMELA and AMS-02 that are made deep in the heliosphere; it also accounts for Ulysses counting rate features measured out of the ecliptic plane. The interstellar and heliospheric propagation parameters derived in this study agree well with our earlier results for CR protons, helium nuclei, and anti-protons propagation and LIS obtained in the same framework.

Bimodality of Directional Distributions in Ocean Wave Spectra: A Comparison of Data-Adaptive Estimation Techniques

AbstractThe properties of directional distributions in ocean wave spectra are studied, with an emphasis on sea states with bimodal directional distributions in the high-frequency tails of single-peaked wave systems. A peak-splitting tendency has been a challenge in the interpretation of results from some data-adaptive estimation methods. After a survey of the theory, mathematical and numerical explanations are presented regarding domains of uni- and bimodality for symmetric Burg and Shannon maximum entropy methods. The study finds that both the Burg and Shannon maximum entropy methods have a tendency to split peaks, and that the domains of uni- and bimodality for these two methods depend on the Fourier coefficients input into the algorithms. Comparisons of data-adaptive methods based on data collected near the Ekofisk oil field in the North Sea and from nonlinear wave simulations are presented. The maximum likelihood (ML) method, the iterative maximum likelihood (IML) method, and the Burg and Shannon maximum entropy methods are applied. A large fraction of the directional wave spectra from Ekofisk shows bimodal features for distributions above the spectral peak for all of the abovementioned methods. In particular, strong similarity in bimodal features between the iterative maximum likelihood and the Burg maximum entropy methods are found. In general, the bimodality is consistent with previous observations, and it seems to be associated with wave and spectral development owing to nonlinear wave–wave interactions rather than being associated with the peak-splitting tendency in the estimates from any of the algorithms. The bimodal directional distributions were sometimes persistent and sometimes formed or decayed within the order of hours.

Scatter simulation and correction in computed tomography: A reconstruction-integrated approach modelling the forward projection

Scattered photons lower the quality of the reconstructed images in computed tomography. Additionally to scattering in the scintillator material that lowers the resolution, object-scattered photons cause artefacts due to non-linear absorption and are difficult to simulate. In the context of model-based reconstruction we developed a fast approximation for scattered radiation that is integrated in the forward projection of an iterative polychromatic reconstruction method. We develop the modelling of scattered intensities in an iterative maximum-likelihood reconstruction method, demonstrate a comparison with Monte-Carlo simulated scatter-data and show the result of reconstructed data within the field of industrial non-destructive testing.

Integrating spatial and biomass planning for the United States

Biomass is low-carbon energy and has tremendous potential as an alternative to fossil fuels. However, the significant role of biomass in future low-carbon energy portfolio depends heavily on its consumption. The paper presents a first attempt to examine the spatial-temporal patterns of biomass consumption in the United States (US), using a novel method-spatial Seemingly Unrelated Regression (SUR) model, in order to strengthen the link between energy planning and spatial planning. In order to obtain the robust parameters of spatial SUR models and estimate the parameters efficiently, an iterative maximum likelihood method, which takes full advantage of the stationary characteristic of maximum likelihood estimation, has been developed. The robust parameters of models can help draw a proper inference for biomass consumption. Then the spatial-temporal patterns of biomass consumption in the US at the state level are investigated using the spatial SUR models with the estimation method developed and data covering the period of 2000–2012. Results show that there are spatial dependences among biomass consumption. The presence of spatial dependence in biomass consumption has informative implications for making sustainable biomass polices. It suggests new efforts to adding a cross-state dimension to state-level energy policy and coordinating some elements of energy policy across states are still needed. In addition, results consistent with classic economic theory further proves the correctness of applying the spatial SUR models to investigate the spatial-temporal patterns of biomass consumption.

A weighted polynomial based material decomposition method for spectral x-ray CT imaging

Currently in photon counting based spectral x-ray computed tomography (CT) imaging, pre-reconstruction basis materials decomposition is an effective way to reconstruct densities of various materials. The iterative maximum-likelihood method requires precise spectrum information and is time-costly. In this paper, a novel non-iterative decomposition method based on polynomials is proposed for spectral CT, whose aim was to optimize the noise performance when there is more energy bins than the number of basis materials. Several subsets were taken from all the energy bins and conventional polynomials were established for each of them. The decomposition results from each polynomial were summed with pre-calculated weighting factors, which were designed to minimize the overall noises. Numerical studies showed that the decomposition noise of the proposed method was close to the Cramer–Rao lower bound under Poisson noises. Furthermore, experiments were carried out with an XCounter Filte X1 photon counting detector for two-material decomposition and three-material decomposition for validation.

Image quality analysis and improvement of Ladar reflective tomography for space object recognition

Some problems in the application of Ladar reflective tomography for space object recognition are studied in this work. An analytic target model is adopted to investigate the image reconstruction properties with limited relative angle range, which are useful to verify the target shape from the incomplete image, analyze the shadowing effect of the target and design the satellite payloads against recognition via reflective tomography approach. We proposed an iterative maximum likelihood method basing on Bayesian theory, which can effectively compress the pulse width and greatly improve the image resolution of incoherent LRT system without loss of signal to noise ratio.

Analysis of the Convexity of Iterative Maximum Likelihood Methods for DOA Estimation

In this letter, a new analysis technique for finding the convexity of iterative maximum likelihood (IML) methods for direction-of-arrival (DOA) estimation is presented. The proposed technique can pave the way in avoiding the local solution when the IML methods are utilized to estimate DOA, especially for the scenarios of array with large antennas. From the derivation, we can see that as long as the initial DOA belongs to the approximate convex range estimated by our proposed technique, the IML methods can estimate the DOA very well without entering into local minima, which is particularly true for the large arrays. Furthermore, numerical experiments show us the results tallied well with our theoretical derivations.

Validation of two independent retrievals of SCIAMACHY water vapour columns using radiosonde data

Abstract. Two independently derived SCIAMACHY total water vapour column (WVC) products are compared with integrated water vapour data calculated from radiosonde measurements, and with each other. The two SCIAMACHY WVC products are retrieved with two different retrieval algorithms applied in the visible and short-wave infrared wavelength regions respectively. The first SCIAMACHY WVC product used in the comparison is ESA's level 2 version 5.01 WVC product derived with the Air Mass Corrected Differential Optical Absorption Spectroscopy (AMC-DOAS) retrieval algorithm applied in the visible wavelength range (SCIAMACHY-ESA). The second SCIAMACHY WVC product is derived using the iterative maximum likelihood method (IMLM) in the short-wave infrared wavelength range and developed by Netherlands Institute for Space Research (SCIAMACHY-IMLM). Both SCIAMACHY WVC products are compared with collocated water vapour amounts determined from daily relative humidity radiosonde measurements obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) radiosonde network. The SCIAMACHY-ESA WVC product is compared with radiosonde-derived WVC amounts for an 18-month period from February 2010 to mid-August 2011, and the SCIAMACHY-IMLM WVC amounts are compared with radiosonde WVC amounts for the two individual years of 2004 and 2009. In addition the WVC amounts from SCIAMACHY-ESA and SCIAMACHY-IMLM are also compared with each other for a 1-month period for June 2009. The AMC-DOAS method used to retrieve SCIAMACHY-ESA WVC is able to correct for water vapour present below the clouds and can be used during cloudy conditions over both land and ocean surfaces. Results indicate a good agreement between the WVC amounts of SCIAMACHY-ESA and that of radiosondes, with a mean difference of −0.32 g cm−2 for all collocated cases. Overall the SCIAMACHY-ESA WVC amounts are smaller than the radiosonde WVC amounts, especially over oceans. For cloudy conditions the WVC bias has a clear dependence on the cloud top height and increases with increasing cloud top heights larger than approximately 2 km. A likely cause for this could be the different vertical profile shapes of water vapour and O2 leading to different relative changes in their optical thickness, which makes the air mass factor (AMF) correction method used in the algorithm less suitable for high clouds. The SCIAMACHY-IMLM product's water vapour measurements are best used over land surfaces during cloud-free conditions, and in these cases a good agreement is found when compared to radiosonde WVC amounts, with a mean difference of 0.08 g cm−2. It is shown that over ocean surfaces during cloudy conditions the partial SCIAMACHY-IMLM water vapour column above the cloud can be well estimated by using the simultaneously retrieved methane column to calculate the cloud top height. Comparing the two satellite WVC products with each other indicates that SCIAMACHY-ESA consistently measures higher WVC amounts than those of SCIAMACHY-IMLM. Furthermore, the importance of the choice of cloud product is highlighted, as intercomparisons between the two SCIAMACHY WVC products indicate that using different cloud products to screen water vapour data for cloud-free conditions influences the data selection and may ultimately lead to a variation in results. In the last section of the paper, various options for filtering the two SCIAMACHY WVC data sets are discussed and best selection criteria suggested.

An Iterative Maximum Likelihood Synchronization Method for OFDM System

OFDM system divides a wideband transmission bandwidth into several overlapped narrowband subcarriers to avoid the serious frequency selective fading problem. However, if Timing Offset (TO) and Carrier Frequency Offset (CFO) exist between the Transmitter (Tx) and Receiver (Rx), OFDM system will suffer the Inter-Symbol Interference (ISI) and Inter-Carrier Interference (ICI) that degrade the system performance dramatically. In this paper, we propose an iterative maximum likelihood method for synchronization. We also adopt the overlap concept to reduce the plateau problem. Simulation results show that the proposed method can predict the time delay and frequency offset correctly even under the multipath fading scenarios.

Validation of six years of SCIAMACHY carbon monoxide observations using MOZAIC CO profile measurements

Abstract. This paper presents a validation study of SCanning Imaging Absorption SpectroMeter for Atmospheric CartograpHY (SCIAMACHY) carbon monoxide (CO) total column measurements from the Iterative Maximum Likelihood Method (IMLM) algorithm using vertically integrated profile aircraft measurements obtained within the MOZAIC project for the six year time period of 2003–2008. Overall we find a good agreement between SCIAMACHY and airborne measurements for both mean values – also on a year-to-year basis – as well as seasonal variations. Several locations show large biases that are attributed to local effects like orography and proximity of large emission sources. Differences were detected for individual years: 2003, 2004 and 2006 have larger biases than 2005, 2007 and 2008, which appear to be related to SCIAMACHY instrumental issues but require more research. Results from this study are consistent with, and complementary to, findings from a previous validation study using ground-based measurements (de Laat et al., 2010b). According to this study, the SCIAMACHY data, if individual measurements are of sufficient quality – good signal-to-noise, can be used to determine the spatial distribution and seasonal cycles of CO total columns over clean areas. Biases found over areas with strong emissions (Africa, China) could be explained by low sensitivity of the instrument in the boundary layer and users are recommended to avoid using the SCIAMACHY data while trying to quantify CO burden and/or retrieve CO emissions in such areas.

Time delays and angles of arrival estimation using known signals

Channel estimation in a multipath mobile communication system is addressed in this paper, and a novel approach based on the linear prediction in frequency domain and the singular value decomposition technique is presented for joint estimation of the angles of arrival and the time delays of multiple reflections of a known signal. Simulation results illustrating the performance of the proposed algorithm are included, and the results show that the proposed method is close in accuracy when compared to the iterative maximum-likelihood method. However, when the two methods are compared in computational complexity, it is demonstrated that the proposed method reduces the complexity to nearly half of that of the maximum-likelihood method. The Cramer–Rao bounds are computed for comparison.

Validation of five years (2003–2007) of SCIAMACHY CO total column measurements using ground-based spectrometer observations

Abstract. This paper presents a validation study of SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY) carbon monoxide (CO) total column measurements from the Iterative Maximum Likelihood Method (IMLM) algorithm using ground-based spectrometer observations from twenty surface stations for the five year time period of 2003–2007. Overall we find a good agreement between SCIAMACHY and ground-based observations for both mean values as well as seasonal variations. For high-latitude Northern Hemisphere stations absolute differences between SCIAMACHY and ground-based measurements are close to or fall within the SCIAMACHY CO 2σ precision of 0.2 × 1018 molecules/cm2 (∼10%) indicating that SCIAMACHY can observe CO accurately at high Northern Hemisphere latitudes. For Northern Hemisphere mid-latitude stations the validation is complicated due to the vicinity of emission sources for almost all stations, leading to higher ground-based measurements compared to SCIAMACHY CO within its typical sampling area of 8° × 8°. Comparisons with Northern Hemisphere mountain stations are hampered by elevation effects. After accounting for these effects, the validation provides satisfactory results. At Southern Hemisphere mid- to high latitudes SCIAMACHY is systematically lower than the ground-based measurements for 2003 and 2004, but for 2005 and later years the differences between SCIAMACHY and ground-based measurements fall within the SCIAMACHY precision. The 2003–2004 bias is consistent with previously reported results although its origin remains under investigation. No other systematic spatial or temporal biases could be identified based on the validation presented in this paper. Validation results are robust with regard to the choices of the instrument-noise error filter, sampling area, and time averaging required for the validation of SCIAMACHY CO total column measurements. Finally, our results show that the spatial coverage of the ground-based measurements available for the validation of the 2003–2007 SCIAMACHY CO columns is sub-optimal for validation purposes, and that the recent and ongoing expansion of the ground-based network by carefully selecting new locations may be very beneficial for SCIAMACHY CO and other satellite trace gas measurements validation efforts.

SAR Calibration Aided by Permanent Scatterers

We propose a calibration method suitable for a set of repeated synthetic aperture radar (SAR) acquisitions that uses both absolute calibrated devices (such as corner reflectors) and stable targets identified in the scene [the permanent scatterers (PSs)]. Precisely, the role of the PS is to extend the initial calibration sequence by monitoring the radiometric stability of the system throughout the whole mission life span. At a first step, this paper approaches the problem of PS-based normalization by an iterative maximum-likelihood method that exploits the stack of complex interferometric SAR images. Two solutions are given based on different assumptions on the PS phases. As a second step, the merging of these estimates with the available calibration information is discussed. Results achieved by experimental acquisitions are shown in two different SAR systems: 1) a C-band spaceborne SAR and 2) a Ku-band ground-based SAR.

Water vapour total columns from SCIAMACHY spectra in the 2.36 μm window

Abstract. The potential of the shortwave infrared channel of the atmospheric spectrometer SCIAMACHY on Envisat to provide accurate measurements of total atmospheric water vapour columns is explored. It is shown that good quality results can be obtained for cloud free scenes above the continents using the Iterative Maximum Likelihood Method. In addition to the standard cloud filter employed in this method, further cloud screening is obtained by comparing simultaneously retrieved methane columns with values expected from models. A novel method is used to correct for the scattering effects introduced in the spectra by the ice layer on the detector window. The retrieved water vapour total vertical columns for the period 2003–2007 are compared with spatially and temporally collocated values from the European Centre for Mid-Range Weather Forecast (ECMWF) data. The observed differences for individual measurements have standard deviations not higher than 0.3 g/cm2 and an absolute mean value smaller than 0.01 g/cm2 with some regional excursions. The use of recently published spectroscopic data for water vapour led to a significant improvement in the agreement of the retrieved water vapour total columns and the values derived from ECMWF data. This analysis thus supports the superior quality of the new spectroscopic information using atmospheric data.

SCIAMACHY CO over land and oceans: 2003–2007 interannual variability

Abstract. We present a new method to obtain accurate SCIAMACHY CO columns over clouded ocean scenes. Based on an improved version of the Iterative Maximum Likelihood Method (IMLM) retrieval algorithm, we now have retrieved five years of data over both land and clouded ocean scenes between 2003 and 2007. The ocean-cloud method uses the CH4 columns retrieved simultaneously with the CO columns to determine the cloud top height. The CH4 cloud top height is in good agreement with the FRESCO+ cloud top height determined from UV-VIS oxygen-A band measurements, providing confidence that the CH4 cloud top height is a good diagnostic of the cloud top height over (partially) clouded ocean scenes. The CO measurements over clouded ocean scenes have been compared with collocated modeled CO columns over the same clouds and agree well. Using clouded ocean scenes quadruples the number of useful CO measurements compared to land-only measurements. The five-year CO data set over land and clouded ocean scenes presented here is based on an improved version of the IMLM algorithm which includes a more accurate determination of the random instrument-noise error for CO. This leads to a smaller spread in the differences between single CO measurements and the corresponding model values. The new version, IMLM version 7.4, also uses updated spectroscopic parameters for H2O and CH4 but this has only a minor impact on the retrieved CO columns. The five-year data set shows significant interannual variability over land and over clouded ocean scenes. Three examples are highlighted: the Asian outflow of pollution over the northern Pacific, the biomass-burning outflow over the Indian Ocean originating from Indonesia, and biomass burning in Brazil. In general there is good agreement between observed and modeled seasonal cycles and interannual variability.

Estimating US Monetary Policy Shocks Using a Factor-Augmented Vector Autoregression: An EM Algorithm Approach

Economy-wide effects of shocks to the US federal funds rate are estimated in a state space model with 120 US macroeconomic and financial time series driven by the dynamics of the federal funds rate and a few dynamic factors. This state space system is denoted a factor-augmented VAR (FAVAR) by Bernanke et al. (2005). I estimate the FAVAR by the fully parametric one-step EM algorithm as an alternative to the two-step principal component method and the one-step Bayesian method in Bernanke et al. (2005). The EM algorithm which is an iterative maximum likelihood method estimates all the parameters and the dynamic factors simultaneously and allows for classical inference. I demonstrate empirically that the same impulse responses but better fit emerge robustly from a low order FAVAR with eight correlated factors compared to a high order FAVAR with fewer correlated factors, for instance four factors. This empirical result accords with one of the theoretical results from Bai & Ng (2007) in which it is shown that the information in complicated factor dynamics may be substituted by panel information.