Full Noise Covariance Research Articles

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

Mitigating the effect of 1/f noise on the detection of the H i intensity mapping power spectrum from single-dish measurements

ABSTRACT We present and compare several methods to mitigate time-correlated (1/f) noise within the H i intensity mapping component of the MeerKAT Large Area Synoptic Survey (MeerKLASS). By simulating scan strategies, the H i signal, foreground emissions, white and correlated noise, we assess the ability of various data-processing pipelines to recover the power spectrum of H i brightness temperature fluctuations. We use MeerKAT pilot data to assess the level of 1/f noise expected for the MeerKLASS survey and use these measurements to create realistic levels of time-correlated noise for our simulations. We find the time-correlated noise component within the pilot data to be between 10 and 20 times higher than the white noise level at the scale of $k = 0.04 \, {\rm {Mpc}}^{-1}$. Having determined that the MeerKAT 1/f noise is partially correlated across all the frequency channels, we employ Singular Value Decomposition (SVD) as a technique to remove both the 1/f noise and Galactic foregrounds but find that over-cleaning results in the removal of H i power at large (angular and radial) scales; a power loss of 40 per cent is seen for a 3-mode SVD clean at the scale of $k = 0.04 \, {\rm {Mpc}}^{-1}$. We compare the impact of map-making using weighting by the full noise covariance (i.e. including a 1/f component), as opposed to just a simple unweighted binning, finding that including the time-correlated noise information reduces the excess power added by 1/f noise by up to 30 per cent.

BEYONDPLANCK

We present a Gibbs sampling solution to the mapmaking problem for cosmic microwave background (CMB) measurements that builds on existing destriping methodology. Gibbs sampling breaks the computationally heavy destriping problem into two separate steps: noise filtering and map binning. Considered as two separate steps, both are computationally much cheaper than solving the combined problem. This provides a huge performance benefit as compared to traditional methods and it allows us, for the first time, to bring the destriping baseline length to a single sample. Here, we applied the Gibbs procedure to simulatedPlanck30 GHz data. We find that gaps in the time-ordered data are handled efficiently by filling them in with simulated noise as part of the Gibbs process. The Gibbs procedure yields a chain of map samples, from which we are able to compute the posterior mean as a best-estimate map. The variation in the chain provides information on the correlated residual noise, without the need to construct a full noise covariance matrix. However, if only a single maximum-likelihood frequency map estimate is required, we find that traditional conjugate gradient solvers converge much faster than a Gibbs sampler in terms of the total number of iterations. The conceptual advantages of the Gibbs sampling approach lies in statistically well-defined error propagation and systematic error correction. This methodology thus forms the conceptual basis for the mapmaking algorithm employed in the BEYONDPLANCKframework, which implements the first end-to-end Bayesian analysis pipeline for CMB observations.

Weighted data spaces for correlation-based array imaging in experimental aeroacoustics

This article discusses aeroacoustic imaging methods based on correlation measurements in the frequency domain. Standard methods in this field assume that the estimated correlation matrix is superimposed with additive white noise. In this paper we present a mathematical model for the measurement process covering arbitrarily correlated noise. The covariance matrix of correlation data is given in terms of fourth order moments. The aim of this paper is to explore the use of such additional information on the measurement data in imaging methods. For this purpose a class of weighted data spaces is introduced, where each data space naturally defines an associated beamforming method with a corresponding point spread function. This generic class of beamformers contains many well-known methods such as Conventional Beamforming, (Robust) Adaptive Beamforming or beamforming with shading. This article examines in particular weightings that depend on the noise (co)variances. In a theoretical analysis we prove that the beamformer, weighted by the full noise covariance matrix, has minimal variance among all beamformers from the described class. Application of the (co)variance weighted methods on synthetic and experimental data show that the resolution of the results is improved and noise effects are reduced.

How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?

The posed question arises for instance in regional gravity field modelling using weighted least-squares techniques if the gravity field functionals are synthesised from the spherical harmonic coefficients of a satellite-only global gravity model (GGM), and are used as one of the noisy datasets. The associated noise covariance matrix, appeared to be extremely ill-conditioned with a singular value spectrum that decayed gradually to zero without any noticeable gap. We analysed three methods to deal with the ill-conditioned noise covariance matrix: Tihonov regularisation of the noise covariance matrix in combination with the standard formula for the weighted least-squares estimator, a formula of the weighted least-squares estimator, which does not involve the inverse noise covariance matrix, and an estimator based on Rao’s unified theory of least-squares. Our analysis was based on a numerical experiment involving a set of height anomalies synthesised from the GGM GOCO05s, which is provided with a full noise covariance matrix. We showed that the three estimators perform similar, provided that the two regularisation parameters each method knows were chosen properly. As standard regularisation parameter choice rules do not apply here, we suggested a new parameter choice rule, and demonstrated its performance. Using this rule, we found that the differences between the three least-squares estimates were within noise. For the standard formulation of the weighted least-squares estimator with regularised noise covariance matrix, this required an exceptionally strong regularisation, much larger than one expected from the condition number of the noise covariance matrix. The preferred method is the inversion-free formulation of the weighted least-squares estimator, because of its simplicity with respect to the choice of the two regularisation parameters.

A methodology for least-squares local quasi-geoid modelling using a noisy satellite-only gravity field model

The paper is about a methodology to combine a noisy satellite-only global gravity field model (GGM) with other noisy datasets to estimate a local quasi-geoid model using weighted least-squares techniques. In this way, we attempt to improve the quality of the estimated quasi-geoid model and to complement it with a full noise covariance matrix for quality control and further data processing. The methodology goes beyond the classical remove–compute–restore approach, which does not account for the noise in the satellite-only GGM. We suggest and analyse three different approaches of data combination. Two of them are based on a local single-scale spherical radial basis function (SRBF) model of the disturbing potential, and one is based on a two-scale SRBF model. Using numerical experiments, we show that a single-scale SRBF model does not fully exploit the information in the satellite-only GGM. We explain this by a lack of flexibility of a single-scale SRBF model to deal with datasets of significantly different bandwidths. The two-scale SRBF model performs well in this respect, provided that the model coefficients representing the two scales are estimated separately. The corresponding methodology is developed in this paper. Using the statistics of the least-squares residuals and the statistics of the errors in the estimated two-scale quasi-geoid model, we demonstrate that the developed methodology provides a two-scale quasi-geoid model, which exploits the information in all datasets.

Bilateral filtering using the full noise covariance matrix applied to x-ray phase-contrast computed tomography

The purpose of this work is to develop an image-based de-noising algorithm that exploits complementary information and noise statistics from multi-modal images, as they emerge in x-ray tomography techniques, for instance grating-based phase-contrast CT and spectral CT.Among the noise reduction methods, image-based de-noising is one popular approach and the so-called bilateral filter is a well known algorithm for edge-preserving filtering. We developed a generalization of the bilateral filter for the case where the imaging system provides two or more perfectly aligned images. The proposed generalization is statistically motivated and takes the full second order noise statistics of these images into account. In particular, it includes a noise correlation between the images and spatial noise correlation within the same image.The novel generalized three-dimensional bilateral filter is applied to the attenuation and phase images created with filtered backprojection reconstructions from grating-based phase-contrast tomography. In comparison to established bilateral filters, we obtain improved noise reduction and at the same time a better preservation of edges in the images on the examples of a simulated soft-tissue phantom, a human cerebellum and a human artery sample. The applied full noise covariance is determined via cross-correlation of the image noise.The filter results yield an improved feature recovery based on enhanced noise suppression and edge preservation as shown here on the example of attenuation and phase images captured with grating-based phase-contrast computed tomography. This is supported by quantitative image analysis. Without being bound to phase-contrast imaging, this generalized filter is applicable to any kind of noise-afflicted image data with or without noise correlation. Therefore, it can be utilized in various imaging applications and fields.

Antarctica, Greenland and Gulf of Alaska land-ice evolution from an iterated GRACE global mascon solution

AbstractWe have determined the ice mass evolution of the Antarctic and Greenland ice sheets (AIS and GIS) and Gulf of Alaska (GOA) glaciers from a new GRACE global solution of equal-area surface mass concentration parcels (mascons) in equivalent height of water. The mascons were estimated directly from the reduction of the inter-satellite K-band range-rate (KBRR) observations, taking into account the full noise covariance, and formally iterating the solution. The new solution increases signal recovery while reducing the GRACE KBRR observation residuals. The mascons were estimated with 10 day and 1 arcdeg equal-area sampling, applying anisotropic constraints. An ensemble empirical mode decomposition adaptive filter was applied to the mascon time series to compute annual mass balances. The details and causes of the spatial and temporal variability of the land-ice regions studied are discussed. The estimated mass trend over the total GIS, AIS and GOA glaciers for the time period 1 December 2003 to 1 December 2010 is −380 ± 31 Gt a−1, equivalent to −1.05 ± 0.09 mm a−1 sea-level rise. Over the same time period we estimate the mass acceleration to be −41 ± 27 Gt a−2 , equivalent to a −0.11 ± 0.08 mm a−2 sea-level acceleration. The trends and accelerations are dependent on significant seasonal and annual balance anomalies.

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.

Power asymmetries in the cosmic microwave background temperature and polarization patterns

We test the asymmetry of the Cosmic Microwave Background anisotropy jointly in temperature and polarization. We study the hemispherical asymmetry, previously found only in the temperature field, with respect to the axis identified by Hansen et al. (2009). To this extent, we make use of the low resolution WMAP 5 year temperature and polarization Nside=16 maps and the optimal angular power spectrum estimator BolPol (Gruppuso et al. 2009). We consider two simple estimators for the power asymmetry and we compare our findings with Monte Carlo simulations which take into account the full noise covariance matrix. We confirm an excess of power in temperature angular power spectrum in the Southern hemisphere at a significant level, between 3 sigma and 4 sigma depending on the exact range of multipoles considered. We do not find significant power asymmetry in the gradient (curl) component EE (BB) of polarized angular spectra. Also cross-correlation power spectra, i.e. TE, TB, EB, show no significant hemispherical asymmetry. We also show that the Cold Spot found by Vielva et al. (2004) in the Southern Galactic hemisphere does not alter the significance of the hemispherical asymmetries on multipoles which can be probed by maps at resolution Nside=16. Although the origin of the hemispherical asymmetry in temperature remains unclear, the study of the polarization patter could add useful information on its explanation. We therefore forecast by Monte Carlo the Planck capabilities in probing polarization asymmetries.

Some thoughts on the use of InSAR data to constrain models of surface deformation: Noise structure and data downsampling

Repeat‐pass Interferometric Synthetic Aperture Radar (InSAR) provides spatially dense maps of surface deformation with potentially tens of millions of data points. Here we estimate the actual covariance structure of noise in InSAR data. We compare the results for several independent interferograms with a large ensemble of GPS observations of tropospheric delay and discuss how the common approaches used during processing of InSAR data affects the inferred covariance structure. Motivated by computational concerns associated with numerical modeling of deformation sources, we then combine the data‐covariance information with the inherent resolution of an assumed source model to develop an efficient algorithm for spatially variable data resampling (or averaging). We illustrate these technical developments with two earthquake scenarios at different ends of the earthquake magnitude spectrum. For the larger events, our goal is to invert for the coseismic fault slip distribution. For smaller events, we infer the hypocenter location and moment. We compare the results of inversions using several different resampling algorithms, and we assess the importance of using the full noise covariance matrix.