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Location service demands precise positioning, which can be obtained by applying maximum likelihood or least squares (LS) directly to measurements such as ranges and/or pseudoranges or to the computed quantities of measurements such as squared ranges and their differences. Nonlinearity and the stochastic models of computed quantities of measurements are not yet taken into account in precise location estimation. We propose a bias-corrected weighted LS method for precise location service, which consists of two basic elements. One is to automatically correct both the bias due to model nonlinearity and the induced biases in squared ranges/pseudoranges, and the other is to sequentially estimate unknown location parameters by treating equality constraints as a condition adjustment. As a result, the location estimation is theoretically almost unbiased. The method is applied to ranges, squared ranges and the differences of squared ranges. We also work out bias-corrected precise location estimation from pseudoranges, squared pseudoranges and the differences of pseudoranges. A large scale of simulations is carried out with range- and pseudorange-based models, respectively. The simulation results have clearly indicated: (i) the proposed bias-corrected weighted LS method can indeed result in almost unbiased estimates of location and further improve precise location estimation, depending on model nonlinearity and the ratio of signal to noise; (ii) the bias-corrected weighted LS method with the Gauss—Newton algorithm is shown to perform best in terms of the smallest bias and mean squared errors (MSE), if directly applied to ranges with a well-conditioned configuration, since it avoids the extra nonlinearity of squared ranges/pseudoranges and the induced bias of these computed quantities; (iii) the bias-corrected weighted LS and the weighted LS methods are of almost the same best performance in terms of MSE roots for each type of measurement models of ranges/pseudoranges, squared ranges/pseudoranges and the differences of squared ranges/pseudoranges. Nevertheless, the measurement models with the differences of squared ranges perform less accurate in the vertical component than those with ranges and squared ranges in the example, due to a weaker geometrical constraint; and (iv) the ordinary LS method clearly results in the most biased estimates of location and is of the worst accuracy for all the six measurement models of ranges/pseudoranges, squared ranges/pseudoranges and the differences of squared ranges/pseudoranges, indicating the importance of using correct stochastic models of measurements for location estimation. [ABSTRACT FROM AUTHOR]

Difference methods have been routinely used to compute velocity and acceleration from precise positioning with global navigation satellite systems (GNSS). A low sampling rate (say a rate not greater than 1 Hz, for example) has been always implicitly assumed for applicability of the methods, because random measurement errors are significantly amplified, either proportional to the sampling rate in the case of velocity or square-proportional to the sampling rate in the case of acceleration. Direct consequences of a low sampling rate are the distortion of the computed velocity and acceleration waveforms and the failure to obtain almost instantaneous values of velocity and acceleration. We reformulate the reconstruction of velocity and acceleration from very high-rate (50 Hz) precise GNSS as an inverse ill-posed problem and propose the criterion of minimum mean squared errors (MSE) to regularize solutions of velocity and acceleration. We successfully apply the MSE-based regularized method to reconstruct the very high-rate velocity and acceleration waveforms, the peak ground velocity (PGV) and the peak ground acceleration (PGA) from 50 Hz precise point positioning (PPP) position waveforms for the 2013 Lushan Mw6.6 earthquake. The reconstructed results of velocity and acceleration are shown to be in good agreement with the motion patterns in the PPP position waveforms and correctly recover the earthquake signal. The reconstructed GNSS-based PGA values are a few hundred times smaller than those from the strong motion seismometers. [ABSTRACT FROM AUTHOR]

Tienstra's method was developed to solve parameter adjustment with linear equality constraints, which has been otherwise often carried out by directly applying the conventional (or standard) method of Lagrange multipliers to quadratic optimization problems with a positive definite matrix. We analyze the computational complexity of the celebrated Tienstra's method and compare it with that of the method of Lagrange multipliers. We show that Tienstra's method is not only statistically elegant but also interestingly of significant computational advantage over the method of Lagrange multipliers to solve weighted least squares adjustment with linear equality constraints, with the saving of computational cost by a minimum of about 38% to a maximum of 100%. Tienstra's method can be very important for large scale problems of adjustment and inversion with linear equality constraints. [ABSTRACT FROM AUTHOR]

The algebraic fitting of circles was first proposed by Delogne (1972) and Kasa (1976). We extend the work by Delogne (1972) and Kasa (1976) and propose fast and almost unbiased weighted least squares to best fit circles. The simulations have shown that two non-iterative versions of the bias-corrected weighted LS method are fast and almost unbiased and perform much better than the naive weighted LS method (without applying any bias correction), the ordinary LS-based methods and the gradient-based weighted LS method in terms of both biases and mean squared errors, no matter whether fitting of circles is strongly or weakly constrained geometrically. Nevertheless, a weak geometrical constraint results in a poor fitting of circles. We accordingly propose the regularized variants of the bias-corrected weighted LS method to fit circles from data with a weak geometrical constraint. The simulations have also shown that the two non-iterative regularized variants fit circles satisfactorily well and consistently perform the best among all the regularization methods under study for ill-conditioned problems of fitting circles. • Removing the bias of computed data to construct almost unbiased estimates for algebraic fitting. • Improving the standard algebraic fitting of circles by Delogne (1972) and Kasa (1976). • Proving that the gradient-based weighting is not statistically optimal. • Proposing the statistically rigorous weighted LS method to estimate the circular parameters. • Proposing a fast, bias-corrected regularization for algebraic fitting of circles. [ABSTRACT FROM AUTHOR]

Although total least squares (TLS) has been widely applied, variance components in an errors-in-variables (EIV) model can be inestimable under certain conditions and unstable in the sense that small random errors can result in very large errors in the estimated variance components. We investigate the effect of the random design matrix on variance component (VC) estimation of MINQUE type by treating the design matrix as if it were errors-free, derive the first-order bias of the VC estimate, and construct bias-corrected VC estimators. As a special case, we obtain a bias-corrected estimate for the variance of unit weight. Although TLS methods are statistically rigorous, they can be computationally too expensive. We directly Taylor-expand the nonlinear weighted LS estimate of parameters up to the second-order approximation in terms of the random errors of the design matrix, derive the bias of the estimate, and use it to construct a bias-corrected weighted LS estimate. Bearing in mind that the random errors of the design matrix will create a bias in the normal matrix of the weighted LS estimate, we propose to calibrate the normal matrix by computing and then removing the bias from the normal matrix. As a result, we can obtain a new parameter estimate, which is called the N-calibrated weighted LS estimate. The simulations have shown that (i) errors-in-variables have a significant effect on VC estimation, if they are large/significant but treated as non-random. The variance components can be incorrectly estimated by more than one order of magnitude, depending on the nature of problems and the sizes of EIV; (ii) the bias-corrected VC estimate can effectively remove the bias of the VC estimate. If the signal-to-noise is small, higher order terms may be necessary. Nevertheless, since we construct the bias-corrected VC estimate by directly removing the estimated bias from the estimate itself, the simulation results have clearly indicated that there is a great risk to obtain negative values for variance components. VC estimation in EIV models remains difficult and challenging; and (iii) both the bias-corrected weighted LS estimate and the N-calibrated weighted LS estimate obviously outperform the weighted LS estimate. The intuitively N-calibrated weighted LS estimate is computationally less expensive and shown to statistically perform even better than the bias-corrected weighted LS estimate in producing an almost unbiased estimate of parameters. [ABSTRACT FROM AUTHOR]

The purpose of this note is to derive alternative formulae for parameter estimation in partial errors-in-variables models to those given in Xu et al. (J Geod 86:661-675, ). We show that these new formulae are more compact and more straightforward to interpret the corrections of the random elements of the design matrix. Nevertheless, each formula has its own computational advantage, depending on the relationship between the number of measurements y and that of the independent random elements of the design matrix. The original formula is computationally more efficient, if the number of measurements y is significantly larger than that of the independent random elements of the design matrix; otherwise, the new alternative formula is much more efficient. [ABSTRACT FROM AUTHOR]

Although total least squares has been substantially investigated theoretically and widely applied in practical applications, almost nothing has been done to simultaneously address the estimation of parameters and the errors-in-variables (EIV) stochastic model. We prove that the variance components of the EIV stochastic model are not estimable, if the elements of the random coefficient matrix can be classified into two or more groups of data of the same accuracy. This result of inestimability is surprising as it indicates that we have no way of gaining any knowledge on such an EIV stochastic model. We demonstrate that the linear equations for the estimation of variance components could be ill-conditioned, if the variance components are theoretically estimable. Finally, if the variance components are estimable, we derive the biases of their estimates, which could be significantly amplified due to a large condition number. [ABSTRACT FROM AUTHOR]

Although total least squares (TLS) is more rigorous than the weighted least squares (LS) method to estimate the parameters in an errors-in-variables (EIV) model, it is computationally much more complicated than the weighted LS method. For some EIV problems, the TLS and weighted LS methods have been shown to produce practically negligible differences in the estimated parameters. To understand under what conditions we can safely use the usual weighted LS method, we systematically investigate the effects of the random errors of the design matrix on weighted LS adjustment. We derive the effects of EIV on the estimated quantities of geodetic interest, in particular, the model parameters, the variance-covariance matrix of the estimated parameters and the variance of unit weight. By simplifying our bias formulae, we can readily show that the corresponding statistical results obtained by Hodges and Moore (Appl Stat 21:185-195, ) and Davies and Hutton (Biometrika 62:383-391, ) are actually the special cases of our study. The theoretical analysis of bias has shown that the effect of random matrix on adjustment depends on the design matrix itself, the variance-covariance matrix of its elements and the model parameters. Using the derived formulae of bias, we can remove the effect of the random matrix from the weighted LS estimate and accordingly obtain the bias-corrected weighted LS estimate for the EIV model. We derive the bias of the weighted LS estimate of the variance of unit weight. The random errors of the design matrix can significantly affect the weighted LS estimate of the variance of unit weight. The theoretical analysis successfully explains all the anomalously large estimates of the variance of unit weight reported in the geodetic literature. We propose bias-corrected estimates for the variance of unit weight. Finally, we analyze two examples of coordinate transformation and climate change, which have shown that the bias-corrected weighted LS method can perform numerically as well as the weighted TLS method. [ABSTRACT FROM AUTHOR]

The effects of incorrect weights on the weighted least squares (WLS) estimate, its accuracy, relative goodness and reliability have been thoroughly investigated in statistical and geodetic literature. Although the variance of unit weight has been one of the most important quantities in statistics and geodesy, little has been done to understand the effect of incorrect weights of observations and incorrect prior information on this quantity, except for an earlier study of Koch who showed that incorrect weights of observations would create a bias in the estimation of the variance of unit weight, though they do not affect the unbiasedness of the WLS estimates of parameters and the corrections to the observations. Since Koch did not directly give the formula to compute the bias, we will derive the bias of the estimated variance of unit weight due to incorrect weights of observations in this paper. In the case of incorrect prior information, both incorrect prior mean and incorrect prior weights can independently contribute a bias to the estimate of the variance of unit weight. Two simulated examples clearly show that the bias of the estimated variance of unit weight due to incorrect weights can be much larger than the variance of unit weight itself. [ABSTRACT FROM AUTHOR]

Adjustment has been based on the assumption that random errors of measurements are added to functional models. In geodetic practice, we know that accuracy formulae of modern geodetic measurements often consist of two parts: one proportional to the measured quantity and the other constant. From the statistical point of view, such measurements are of mixed multiplicative and additive random errors. However, almost no adjustment has been developed to strictly address geodetic data contaminated by mixed multiplicative and additive random errors from the statistical point of view. We systematically develop adjustment methods for geodetic data contaminated with multiplicative and additive errors. More precisely, we discuss the ordinary least squares (LS) and weighted LS methods and extend the bias-corrected weighted LS method of Xu and Shimada (Commun Stat B29:83-96, ) to the case of mixed multiplicative and additive random errors. The first order approximation of accuracy for all these three methods is derived. We derive the biases of weighted LS estimates. The three methods are then demonstrated and compared with a synthetic example of surface interpolation. The bias-corrected weighted LS estimate is unbiased up to the second order approximation and is of the best accuracy. Although the LS method can warrant an unbiased estimate for a linear model with multiplicative and additive errors, it is less accurate and always produces a very poor estimate of the variance of unit weight. [ABSTRACT FROM AUTHOR]

High-rate GPS has been widely used to construct displacement waveforms and to invert for source parameters of earthquakes. Almost all works on internal and external evaluation of high-rate GPS accuracy are based on GPS relative positioning. We build an experimental platform to externally evaluate the accuracy of 50-Hz PPP displacement waveforms. Since the shake table allows motion in any of six degrees of freedom, we install an inertial measurement unit (IMU) to measure the attitude of the platform and transform the IMU displacements into the GPS coordinate system. The experimental results have shown that high-rate PPP can produce absolute horizontal displacement waveforms at the accuracy of 2-4 mm and absolute vertical displacement waveforms at the sub-centimeter level of accuracy within a short period of time. The significance of the experiments indicates that high-rate PPP is capable of detecting absolute seismic displacement waveforms at the same high accuracy as GPS relative positioning techniques, but requires no fixed datum station. We have also found a small scaling error of IMU and a small time offset of misalignment between high-rate PPP and IMU displacement waveforms by comparing the amplitudes of and cross-correlating both the displacement waveforms. [ABSTRACT FROM AUTHOR]

Abstract: Currently, ground-based Global Navigation Satellite System (GNSS) stations of the International GNSS Service (IGS) are distributed unevenly around the world. Most of them are located on the mainland, while only a small part of them are scattered on some islands in the oceans. As a consequence, many unreasonable zero values (in fact negative values) appear in Vertical Total Electron Content (VTEC) of European Space Agency (ESA) and Center for Orbit Determination in Europe (CODE) IONEX products, especially in 2008 and 2009 when the solar activities were rather quiet. To improve this situation, we directly implement non-negative physical constraints of ionosphere for global ionosphere maps (GIM) with spherical harmonic functions. Mathematically, we propose an inequality-constrained least squares method by imposing non-negative inequality constraints in the areas where negative VTEC values may occur to reconstruct GIM models. We then apply the new method to process the IGS data in 2008. The results have shown that the new algorithm efficiently eliminates the unwanted behavior of negative VTEC values, which could otherwise often be seen in the current CODE and ESA GIM products in both middle and high latitude areas of the Southern Hemisphere (45°S∼90°S) and the Northern Hemisphere (50°N∼90°N). About 64% of GPS receivers’ DCBs have been significantly improved. Finally, we compare the GIM results between with and without the inequality constraints, which has clearly shown that the GIM result with inequality constraints is significantly better than that without the inequality constraints. The inequality-constrained GIM result is also highly consistent with the final IGS products in terms of root mean squared (RMS) and mean VTEC. [Copyright &y& Elsevier]

A regularized solution is well-known to be biased. Although the biases of the estimated parameters can only be computed with the true values of parameters, we attempt to improve the regularized solution by using the regularized solution itself to replace the true (unknown) parameters for estimating the biases and then removing the computed biases from the regularized solution. We first analyze the theoretical relationship between the regularized solutions with and without the bias correction, derive the analytical conditions under which a bias-corrected regularized solution performs better than the ordinary regularized solution in terms of mean squared errors (MSE) and design the corresponding method to partially correct the biases. We then present two numerical examples to demonstrate the performance of our partially bias-corrected regularization method. The first example is mathematical with a Fredholm integral equation of the first kind. The simulated results show that the partially bias-corrected regularized solution can improve the MSE of the ordinary regularized function by 11%. In the second example, we recover gravity anomalies from simulated gravity gradient observations. In this case, our method produces the mean MSE of 3.71 mGal for the resolved mean gravity anomalies, which is better than that from the regularized solution without bias correction by 5%. The method is also shown to successfully reduce the absolute maximum bias from 13.6 to 6.8 mGal. [ABSTRACT FROM AUTHOR]

The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight. [ABSTRACT FROM AUTHOR]

SUMMARY Strong motion accelerographs have been deployed worldwide to monitor the ground shaking of the Earth and the recorded accelerograms have been used to recover the velocities and displacements by integration. In spite of their fundamental importance in seismology and earthquake engineering, few works address the error estimates of the derived velocities and displacements. We show that the error estimates of the velocities and displacements obtained from accelerograms in the earthquake literature approach to zero as the sampling interval of accelerographs tends to zero; these are erroneous from the statistical point of view. As a result, we present a set of formulae to correctly estimate the errors (or variances) of the integrated velocities and displacements from accelerograms. In addition, we also derive the covariances between the velocities and displacements. [ABSTRACT FROM AUTHOR]

The LLL reduction of lattice vectors and its variants have been widely used to solve the weighted integer least squares (ILS) problem, or equivalently, the weighted closest point problem. Instead of reducing lattice vectors, we propose a parallel Cholesky-based reduction method for positive definite quadratic forms. The new reduction method directly works on the positive definite matrix associated with the weighted ILS problem and is shown to satisfy part of the inequalities required by Minkowski's reduction of positive definite quadratic forms. The complexity of the algorithm can be fixed a priori by limiting the number of iterations. The simulations have clearly shown that the parallel Cholesky-based reduction method is significantly better than the LLL algorithm to reduce the condition number of the positive definite matrix, and as a result, can significantly reduce the searching space for the global optimal, weighted ILS or maximum likelihood estimate. [ABSTRACT FROM AUTHOR]

Finding feasible points is important in optimization. There are currently two major classes of algorithms to deal with the problem of feasible points. The first class of algorithms (of local nature) is to find an approximate feasible point. Given a neighbourhood of an approximate feasible point, the second class of algorithms is to prove whether a feasible point exists inside this neighbourhood. To the best of our knowledge, no methods have been practically implemented to efficiently find the smallest boxes for bounding the feasible points defined by a system of nonlinear and nonconvex inequalities, unless the feasible set is convex. In this paper, we will present a numerical method to find the smallest boxes for bounding the feasible point sets defined by a nonlinear and nonconvex inequality and/or a system of nonlinear and nonconvex inequalities. Two examples have been synthetically constructed and used to show that the proposed numerical method can indeed correctly find all the smallest bounding boxes at any given accuracy efficiently. A brief comparison with relevant techniques will be discussed. Our method may also be thought of as the first solid theoretical basis for multisection and multisplitting in global optimization, when compared with those empirical ones in the literature. [Copyright &y& Elsevier]

We extend the hybrid global optimization method proposed by Xu (J. Comput. Appl. Math. 147 (2002) 301–314) for the one-dimensional case to the multi-dimensional case. The method consists of two basic components: local optimizers and feasible point finders. Local optimizers guarantee efficiency and speed of producing a local optimal solution in the neighbourhood of a feasible point. Feasible point finders provide the theoretical guarantee for the new method to always produce the global optimal solution(s) correctly. If a nonlinear nonconvex inverse problem has multiple global optimal solutions, our algorithm is capable of finding all of them correctly. Three synthetic examples, which have failed simulated annealing and genetic algorithms, are used to demonstrate the proposed method. [Copyright &y& Elsevier]

We propose a hybrid global optimization method for nonlinear inverse problems. The method consists of two components: local optimizers and feasible point finders. Local optimizers have been well developed in the literature and can reliably attain the local optimal solution. The feasible point finder proposed here is equivalent to finding the zero points of a one-dimensional function. It warrants that local optimizers either obtain a better solution in the next iteration or produce a global optimal solution. The algorithm by assembling these two components has been proved to converge globally and is able to find all the global optimal solutions. The method has been demonstrated to perform excellently with an example having more than 1 750 000 local minima over [−106,107]. [Copyright &y& Elsevier]

Landslides, resulting from disturbances in slope equilibrium, pose a significant threat to landscapes, infrastructure, and human life. Triggered by factors such as intense precipitation, seismic activities, or volcanic eruptions, these events can cause extensive damage and endanger nearby communities. A comprehensive understanding of landslide characteristics, including spatio-temporal patterns, dimensions, and morphology, is vital for effective landslide disaster management. Existing remote sensing approaches mostly use either optical or synthetic aperture radar sensors. Integrating information from both these types of sensors promises greater accuracy for identifying and locating landslides. This study proposes a novel approach, the ML-LaDeCORsat (Machine Learning-based coseismic Landslide Detection using Combined Optical and Radar Satellite Imagery), that integrates freely available Sentinel-1, Palsar-2, and Sentinel-2 imagery data in Google Earth Engine (GEE). The approach also integrates relevant spectral indices and suitable bands used in a machine learning-based classification of coseismic landslides. The approach includes a robust and reproducible training and validation strategy and allows one to choose between five classifiers (CART, Random Forest, GTB, SVM, and Naive Bayes). Using landslides from four different earthquake case studies, we demonstrate the superiority of our approach over existing solutions in coseismic landslide identification and localization, providing a GTB-based detection accuracy of 87–92%. ML-LaDeCORsat can be adapted to other landslide events (GEE script is provided). Transfer learning experiments proved that our model can be applied to other coseismic landslide events without the need for additional training data. Our novel approach therefore facilitates quick and reliable identification of coseismic landslides, highlighting its potential to contribute towards more effective disaster management. [ABSTRACT FROM AUTHOR]

Despeckling multiplicative noise is important in processing coherent radar images. Assuming that measurements are corrupted by multiplicative noise and that a priori values are contaminated by either multiplicative or additive noise, we have obtained Bayesian, maximum likelihood and weighted least-squares (LS) estimators, based on gamma and normal distributions. These estimators have been shown to be biased if the noise in measurements is multiplicative. A technique of bias-correction to remove biases from the estimated parameters is proposed. The bias-correction technique requires no distributions about the measurements and the a priori mean, and can be applied to eliminate bias from Bayesian, maximum likelihood and weighted LS estimators in multiplicative noise models. It theoretically provides a solid foundation for, and thus justifies some of current practice in, despeckling multiplicative noise, such as Lee's local statistics and Kuan's adaptive smoothing filter. Some despeckling measures are also proposed and simulated experiment results are reported. [ABSTRACT FROM AUTHOR]

A moment magnitude Mw9.0 earthquake hit northeastern Japan at 14:46:18 (Japan Standard Time), March 11, 2011. We have obtained 1 s precise point positioning solutions for 1198 GEONET stations. Although GPS position time series have been routinely investigated and used as waveforms for dynamic inversion of earthquakes, we focus on exploring the spatial displacement features of GEONET stations for this earthquake. A movie inspection of high-rate GPS waveforms leads us to find that 76.21% of the GEONET stations in the Japanese islands subsided suddenly within 1 s between 14:59:45 and 14:59:46, Japan local time, with an average displacement of - 2.43 , 2.83 and - 4.75 mm in the east, north and vertical components, respectively, about 15 min after the 2011 Tohoku earthquake. We have performed different types of independent tests, namely measurement error analysis, processing the GEONET data with a different software system, a statistical hypothesis testing under a simple assumption of sign distributions, the test computation of the displacement field outside of the Japanese islands and an independent test with the Japanese strong motion borehole network KiK-net, to see whether these sudden movements actually occurred. The first four independent tests are passed almost without any doubt, and the direction of the average sudden displacements is roughly consistent tectonically with the direction of subduction of the Pacific plate. Because there are only 78 KiK-net borehole stations available for an independent seismic test, the KiK-net results are marginally consistent with those of GEONET. In the daily seismological and geophysical practice, one may then conclude that the sudden movement within the second is real after passing these five independent tests. However, a further epoch-by-epoch check pinpoints a few more seconds with even a higher probability of sudden displacement from the 20-min three-component high-rate GPS waveforms after the main shock, or more precisely, the seconds between 14:59:04 and 14:59:05, 15:01:04 and 15:01:05, and 15:03:39 and 15:03:40 with 80.80, 84.14 and 85.89% of the GEONET stations simultaneously moving upward, southward and westward, respectively. Although these probabilities are very high, it may hardly be imagined that a large scale of sudden movements could occur repeatedly between 14:59:04 and 15:03:40. The high-rate GPS results imply that some detected sudden movements after the earthquake could be unidentified artifacts of GPS data processing, though we cannot rule out the possibility that the detected sudden movements in Japan after the 2011 Tohoku Mw9.0 earthquake are real physical signals. [ABSTRACT FROM AUTHOR]

High-rate GNSS precise point positioning (PPP) has been playing a more and more important role in providing precise positioning information in fast time-varying environments. Although kinematic PPP is commonly known to have a precision of a few centimeters, the precision of high-rate PPP within a short period of time has been reported recently with experiments to reach a few millimeters in the horizontal components and sub-centimeters in the vertical component to measure seismic motion, which is several times better than the conventional kinematic PPP practice. To fully understand the mechanism of mystified excellent performance of high-rate PPP within a short period of time, we have carried out a theoretical error analysis of PPP and conducted the corresponding simulations within a short period of time. The theoretical analysis has clearly indicated that the high-rate PPP errors consist of two types: the residual systematic errors at the starting epoch, which affect high-rate PPP through the change of satellite geometry, and the time-varying systematic errors between the starting epoch and the current epoch. Both the theoretical error analysis and simulated results are fully consistent with and thus have unambiguously confirmed the reported high precision of high-rate PPP, which has been further affirmed here by the real data experiments, indicating that high-rate PPP can indeed achieve the millimeter level of precision in the horizontal components and the sub-centimeter level of precision in the vertical component to measure motion within a short period of time. The simulation results have clearly shown that the random noise of carrier phases and higher order ionospheric errors are two major factors to affect the precision of high-rate PPP within a short period of time. The experiments with real data have also indicated that the precision of PPP solutions can degrade to the cm level in both the horizontal and vertical components, if the geometry of satellites is rather poor with a large DOP value. [ABSTRACT FROM AUTHOR]

A correction to the article "Voronoi Cells, Probabilistic Bounds, and Hypothesis Testing in Mixed Integer Linear Models" that was published in the July 15, 2006 issue is presented.