Variance-covariance Information Research Articles

Modeling uncertainty with the truncated zeta distribution in mixture models for ordinal responses

In recent three decades, there has been a rapid increasing interest in the mixture models for ordinal responses, and the classical CUB model as a fundamental one has been extended to different preference and uncertainty models. In this article, based on a response style supported by Zipf’s law, we propose a novel mixture model for ordinal responses via replacing the uncertainty component of the CUB model with a truncated Zeta distribution. Parameters estimation with EM algorithm, inferential issues with respect to the approximation of a truncated Riemann Zeta function and estimators’ variance-covariance information matrix are investigated. The advantages of the proposed model over the CUB model have been illustrated with simulations of two sets of Monte Carlo experiments and practical applications of a health survey and a bicycle use. The intention of the article is to distinguish the respondents’ true preference from the response style of “the higher, the less”, so as to understand more reasonably the formation causes of ordinal responses.

An Improved Bayesian von Karman Regularization Method for the Joint Inversion of GNSS and InSAR Data

Given the problem that the relative weights of multitype data sets are not considered in the current studies of Bayesian von Karman regularized slip distribution inversion, we propose to add hyperparameters that control the variance-covariance information of different data sets within the Bayesian framework, present a detailed and complete theoretical method, and successfully apply it to the earthquake that occurred in Norcia, Italy, in 2016. Synthetic tests show that compared with the Bayesian von Karman regularization method with equal weights for different data, the improved Bayesian von Karman regularization method can more effectively invert the real slip distribution of this dip-slip earthquake, its inversion results are more stable, and the data fitting accuracy is higher, thus verifying the advantages of the improved method. In the slip inversion of the Norcia earthquake, the hyperparameters of the global navigation satellite system (GNSS) and interferometric synthetic aperture radar (InSAR) data sets converged to different values, which shows it is indeed necessary to determine different relative weights of different data sets. Moreover, the inversion results show that this earthquake ruptured to the surface, the fault slip region was mainly concentrated in a depth range of 0–6 km, the maximum slip was 3.87 m, and the released coseismic seismic moment was 8.95×1018 Nm, corresponding to a moment magnitude of MW=6.60. These results are consistent with the existing research, thus verifying the practicability of the improved method.

Mean Dynamic Topography Modeling Based on Optimal Interpolation from Satellite Gravimetry and Altimetry Data

Mean dynamic topography (MDT) is crucial for research in oceanography and climatology. The optimal interpolation method (OIM) is applied to MDT modeling, where the error variance–covariance information of the observations is established. The global geopotential model (GGM) derived from GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) gravity data and the mean sea surface model derived from satellite altimetry data are combined to construct MDT. Numerical experiments in the Kuroshio over Japan show that the use of recently released GOCE-derived GGM derives a better MDT compared to the previous models. The MDT solution computed based on the sixth-generation model illustrates a lower level of root mean square error (77.0 mm) compared with the ocean reanalysis data, which is 2.4 mm (5.4 mm) smaller than that derived from the fifth-generation (fourth-generation) model. This illustrates that the accumulation of GOCE data and updated data preprocessing methods can be beneficial for MDT recovery. Moreover, the results show that the OIM outperforms the Gaussian filtering approach, where the geostrophic velocity derived from the OIM method has a smaller misfit against the buoy data, by a magnitude of 10 mm/s (17 mm/s) when the zonal (meridional) component is validated. This is mainly due to the error information of input data being used in the optimal interpolation method, which may obtain more reasonable weights of observations than the Gaussian filtering method.

A universally efficient algorithm and precision assessment for seamless 3D similarity transformation

Three-dimensional (3D) similarity transformation is popularly applied for measurement datum transformation. In this study, a seamless partial errors-in-variables (EIV) model with equality constraints is established to describe the universal 3D similarity transformation problem (with arbitrary rotation angles and scale factor). Unlike the traditional transformation model, all of the random errors in the measured coordinates for common and non-common points, and their variance–covariance information can be considered. To obtain the least squares solution of this model, the constrained total least squares prediction (CTLSP) algorithm is derived using Gauss–Newton iteration and the Euler–Lagrange approach. Unnecessary matrix calculations in the application of the CTLSP algorithm for 3D datum transformation are avoided and the efficient iterative formulae are derived. Compared with the existing generalized total least squares prediction (GTLSP) algorithm, in which the transformation model is nonlinear with respect to the parameters, the CTLSP algorithm avoids the complex calculations related to the rotation matrix expressed by the trigonometric functions, and allows us to use the simple linear least squares to obtain a satisfactory initial value of the parameter vector. In addition, the linearly approximate cofactor propagation law is employed to assess the precision of the transformed coordinates of non-common points based on the CTLSP algorithm. Finally, the superiorities of CTLSP in transformation accuracy and computational efficiency are verified using an experiment. It should be noted that the new algorithm along with the precision evaluation formulae can easily be extended to the 2D/3D affine and rigid transformation cases as well, such as the map rectification, the point clouds registration, and the image matching.

Realization of a multifrequency celestial reference frame through a combination of normal equation systems

Context. We present a celestial reference frame (CRF) based on the combination of independent, multifrequency radio source position catalogs using nearly 40 years of very long baseline interferometry observations at the standard geodetic frequencies at SX band and about 15 years of observations at higher frequencies (K and XKa). The final catalog contains 4617 sources. Aims. We produce a multifrequency catalog of radio source positions with full variance–covariance information across all radio source positions of all input catalogs. Methods. We combined three catalogs, one observed at 8 GHz (X band), one at 24 GHz (K band) and one at 32 GHz (Ka band). Rather than only using the radio source positions, we developed a new, rigorous combination approach by carrying over the full covariance information through the process of adding normal equation systems. Special validation routines were used to characterize the random and systematic errors between the input reference frames and the combined catalog. Results. The resulting CRF contains precise positions of 4617 compact radio astronomical objects, 4536 measured at 8 GHz, 824 sources also observed at 24 GHz, and 674 at 32 GHz. The frame is aligned with ICRF3 within ±3 μas and shows an average positional uncertainty of 0.1 mas in right ascension and declination. No significant deformations can be identified. Comparisons with Gaia-CRF remain inconclusive, nonetheless significant differences between all frames can be attested.

3‐D uncertainty‐based topographic change detection with structure‐from‐motion photogrammetry: precision maps for ground control and directly georeferenced surveys

AbstractStructure‐from‐motion (SfM) photogrammetry is revolutionising the collection of detailed topographic data, but insight into geomorphological processes is currently restricted by our limited understanding of SfM survey uncertainties. Here, we present an approach that, for the first time, specifically accounts for the spatially variable precision inherent to photo‐based surveys, and enables confidence‐bounded quantification of 3D topographic change. The method uses novel 3D precision maps that describe the 3D photogrammetric and georeferencing uncertainty, and determines change through an adapted state‐of‐the‐art fully 3D point‐cloud comparison (M3C2), which is particularly valuable for complex topography. We introduce this method by: (1) using simulated UAV surveys, processed in photogrammetric software, to illustrate the spatial variability of precision and the relative influences of photogrammetric (e.g. image network geometry, tie point quality) and georeferencing (e.g. control measurement) considerations; (2) we then present a new Monte Carlo procedure for deriving this information using standard SfM software and integrate it into confidence‐bounded change detection; before (3) demonstrating geomorphological application in which we use benchmark TLS data for validation and then estimate sediment budgets through differencing annual SfM surveys of an eroding badland. We show how 3D precision maps enable more probable erosion patterns to be identified than existing analyses, and how a similar overall survey precision could have been achieved with direct survey georeferencing for camera position data with precision half as good as the GCPs'. Where precision is limited by weak georeferencing (e.g. camera positions with multi‐metre precision, such as from a consumer UAV), then overall survey precision can scale as n‐½ of the control precision (n = number of images). Our method also provides variance–covariance information for all parameters. Thus, we now open the door for SfM practitioners to use the comprehensive analyses that have underpinned rigorous photogrammetric approaches over the last half‐century. Copyright © 2017 John Wiley & Sons, Ltd.

Generalized total least squares prediction algorithm for universal 3D similarity transformation

Three–dimensional (3D) similarity datum transformation is extensively applied to transform coordinates from GNSS-based datum to a local coordinate system. Recently, some total least squares (TLS) algorithms have been successfully developed to solve the universal 3D similarity transformation problem (probably with big rotation angles and an arbitrary scale ratio). However, their procedures of the parameter estimation and new point (non-common point) transformation were implemented separately, and the statistical correlation which often exists between the common and new points in the original coordinate system was not considered. In this contribution, a generalized total least squares prediction (GTLSP) algorithm, which implements the parameter estimation and new point transformation synthetically, is proposed. All of the random errors in the original and target coordinates, and their variance–covariance information will be considered. The 3D transformation model in this case is abstracted as a kind of generalized errors–in–variables (EIV) model and the equation for new point transformation is incorporated into the functional model as well. Then the iterative solution is derived based on the Gauss–Newton approach of nonlinear least squares. The performance of GTLSP algorithm is verified in terms of a simulated experiment, and the results show that GTLSP algorithm can improve the statistical accuracy of the transformed coordinates compared with the existing TLS algorithms for 3D similarity transformation.

GOCE orbit analysis: Long-wavelength gravity field determination using the acceleration approach

The restricted sensitivity of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gradiometer instrument requires satellite gravity gradiometry to be supplemented by orbit analysis in order to resolve long-wavelength features of the geopotential. For the hitherto published releases of the GOCE time-wise (TIM) and GOCE space-wise gravity field series—two of the official ESA products—the energy conservation method has been adopted to exploit GPS-based satellite-to-satellite tracking information. On the other hand, gravity field recovery from data collected by the CHAllenging Mini-satellite Payload (CHAMP) satellite showed the energy conservation principle to be a sub-optimal choice. For this reason, we propose to estimate the low-frequency part of the gravity field by the point-wise solution of Newton’s equation of motion, also known as the acceleration approach. This approach balances the gravitational vector with satellite accelerations, and hence is characterized by (second-order) numerical differentiation of the kinematic orbit. In order to apply the method to GOCE, we present tailored processing strategies with regard to low-pass filtering, variance–covariance information handling, and robust parameter estimation. By comparison of our GIWF solutions (initials GI for “Geodätisches Institut” and IWF for “Institut für WeltraumForschung”) and the GOCE-TIM estimates with a state-of-the-art gravity field solution derived from GRACE (Gravity Recovery And Climate Experiment), we conclude that the acceleration approach is better suited for GOCE-only gravity field determination as opposed to the energy conservation method.

Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons

Three GOCE-based gravity field solutions have been computed by ESA’s high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1–2 cm geoid accuracy with 100 km spatial resolution can be achieved.

DORIS Contribution to ITRF2005

We examine the contribution of the Doppler Orbit determination and Radiopositioning Integrated by Satellite (DORIS) technique to the International Terrestrial Reference Frame (ITRF2005) by evaluating the quality of the submitted solutions as well as that of the frame parameters, especially the origin and the scale. Unlike the previous versions of the ITRF, ITRF2005 is constructed with input data in the form of time-series of station positions (weekly for satellite techniques and daily for VLBI) and daily Earth orientation parameters (EOPs), including full variance–covariance information. Analysis of the DORIS station positions’ time-series indicates an internal precision reaching 15 mm or better, at a weekly sampling. A cumulative solution using 12 years of weekly time-series was obtained and compared to a similar International GNSS Service (IGS) GPS solution (at 37 co-located sites) yielding a weighted root mean scatter (WRMS) of the order of 8 mm in position (at the epoch of minimum variance) and about 2.5 mm/year in velocity. The quality of this cumulative solution resulting from the combination of two individual DORIS solutions is better than any individual solution. A quality assessment of polar motion embedded in the contributed DORIS solutions is performed by comparison with the results of other space-geodetic techniques and in particular GPS. The inferred WRMS of polar motion varies significantly from one DORIS solution to another and is between 0.5 and 2 mas, depending on the strategy used and in particular estimating or not polar motion rate by the analysis centers. This particular aspect certainly needs more investigation by the DORIS Analysis Centers.

Two-step procedures for hybrid geoid modelling

A variety of mathematical models exist with which to combine long-wavelength geoid information from an Earth gravity model (EGM) with terrestrial gravity data. As new data become available, such as an extended or refined EGM and/or some variance–covariance information from terrestrial gravity data, this new information can be combined with a previous geoid model through a so-called hybrid estimator. The hybrid estimator is defined as a two-step estimator, which provides a solution that is a compromise between computational labour and matching of the entire data set. Some commonly used geoid modifications are compared, and several hybrid estimators are introduced. It is concluded that, in general, hybrid estimators linking the new information to old data by the least squares principle should be preferred to deterministic combinations.

GOCE Gravity Field Processing Strategy

A processing strategy and the corresponding software architecture for the processing of GOCE (Gravity field and steady-state Ocean Circulation Explorer) observables is presented and described, with the major objective to compute a high-accuracy, high-resolution spherical harmonic model of the Earth's gravity field. The combination of two numerical solution strategies, i.e. the rigorous solution of the corresponding large normal equation systems applying parallel processing (on a PC cluster) as the core solver, and the fast semianalytic approach as a quick-look gravity field analysis (QL-GFA) tool, is proposed. Such a method fusion benefits from the advantages of the individual components: the rigorous inversion of the system providing also the full variance-covariance information, and the quickness enabling the consecutive production of intermediate gravity field solutions, for the purpose to analyse partial and incomplete data sets and to derive a diagnosis of the performance of the GOCE measurement system. The functionality and operability of the individual components are demonstrated in the framework of a closed loop simulation, which is based on a realistic mission scenario both in terms of the orbit configuration and the coloured measuring noise. Special concern is given to the accuracy of the recovered coefficients, the numerical behaviour, the required computing time, and the particular role of the individual modules within the processing chain. In the case of the core solver, it is demonstrated that the assembling and rigorous solution of large normal equation systems can be handled by using Beowulf clusters within a reasonable computing time. The application of the quick-look tool to partial data sets with short-term data gaps is demonstrated on the basis of several case studies. Additionally, the spectral analysis of the residuals of the adjustment is presented as a valuable tool for the verification of the noise characteristics of the GOCE gradiometer.

On- and off-line identification of linear state-space models

Abstract A geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time-invariant systems using (possibly noisy) input-output (I/O) measurements only. As opposed to other (mostly stochastic) identification schemes, no variance-covariance information whatever is involved, and only a limited number of I/O-data are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists of the following two steps. First, a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/O-data. Then the system matrices are obtained at once from the least-squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition. Also, the algorithm is easily applied to slowly time-varying systems using windowing or exponential weighting. These results are illustrated by examples, including the identification of an industrial plant.

On- and off-line identification of linear state-space models

Abstract A geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time-invariant systems using (possibly noisy) input-output (I/O) measurements only. As opposed to other (mostly stochastic) identification schemes, no variance-covariance information whatever is involved, and only a limited number of I/O-data are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists of the following two steps. First, a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/O-data. Then the system matrices are obtained at once from the least-squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition. Also, the algorithm is easily applied to slowly time-varying systems using windowing or exponential weighting. These results are illustrated by examples, including the identification of an industrial plant.