On a stronger-than-best property for best prediction

The minimum mean squared error (MMSE) criterion is a popular criterion for devising best predictors. In case of linear predictors, it has the advantage that no further distributional assumptions need to be made, other then about the first- and second-order moments. In the spatial and Earth sciences, it is the best linear unbiased predictor (BLUP) that is used most often. Despite the fact that in this case only the first- and second-order moments need to be known, one often still makes statements about the complete distribution, in particular when statistical testing is involved. For such cases, one can do better than the BLUP, as shown in Teunissen (J Geod. doi: 10.1007/s00190-007-0140-6, 2006), and thus devise predictors that have a smaller MMSE than the BLUP. Hence, these predictors are to be preferred over the BLUP, if one really values the MMSE-criterion. In the present contribution, we will show, however, that the BLUP has another optimality property than the MMSE-property, provided that the distribution is Gaussian. It will be shown that in the Gaussian case, the prediction error of the BLUP has the highest possible probability of all linear unbiased predictors of being bounded in the weighted squared norm sense. This is a stronger property than the often advertised MMSE-property of the BLUP.     

GIS环境下污水管网设计流量的快速计算

针对ＧＩＳ环境下污水管网设计问题，提出了一种快速计算管段设计流量的方法——递归方法。分析了递归的原理、链式表的建立以及递归算法的具体实现等，并以实例阐述了递归计算的各个步骤     

A spatiotemporal mining framework for abnormal association patterns in marine environments with a time series of remote sensing images

A spatiotemporal mining framework is a novel tool for the analysis of marine association patterns using multiple remote sensing images. From data pretreatment, to algorithm design, to association rule mining and pattern visualization, this paper outlines a spatiotemporal mining framework for abnormal association patterns in marine environments, including pixel-based and object-based mining models. Within this framework, some key issues are also addressed. In the data pretreatment phase, we propose an algorithm for extracting abnormal objects or pixels over marine surfaces, and construct a mining transaction table with object-based and pixel-based strategies. In the mining algorithm phase, a recursion method to construct a direct association pattern tree is addressed with an asymmetric mutual information table, and a recursive mining algorithm to find frequent items. In the knowledge visualization phase, a “Dimension–Attributes” visualization framework is used to display spatiotemporal association patterns. Finally, spatiotemporal association patterns for marine environmental parameters in the Pacific Ocean are identified, and the results prove the effectiveness and the efficiency of the proposed mining framework.     

Recursive algorithms for the computation of the potential harmonic coefficients of a constant density polyhedron

The gravitational potential of a constant density general polyhedron can be expressed both in terms of a closed analytical expression and as a series expansion involving the corresponding spherical harmonic coefficients. The latter can be obtained from two independent algorithms, which differ not only in their algorithmic architecture but in their efficiency and overall performance, especially when computing the coefficients of higher degree and order. In the present paper a comparative study of all these three approaches is carried out focusing on the numerical implementation of the recursive relations appearing in the two algorithms for the computation of the polyhedral potential harmonic coefficients. The performed numerical investigations show that the linear algorithm proposed by Jamet and Thomas (Proceedings of the second international GOCE user workshop, ‘GOCE, The Geoid and Oceanography’, ESA-ESRIN, Frascati, Italy, 8–10 March 2004, ESA SP-569, 2004), but so far not implemented, achieves a reasonable accuracy at a computational expense that opens to practical applications, for instance in the field of satellite gravimetry/gradiometry interpretation. The convergence behavior of the linear recursion algorithm is studied thoroughly and a computational procedure is proposed that enables the stable computation of potential harmonic coefficients up to degree 60 when referring to an arbitrarily shaped polyhedral body.     

Evaluation of optical remote sensing-based shallow water bathymetry for recursive mapping

Water depth estimation using optical remote sensing offers a reliable and efficient means of mapping coastal zones. Here, we aim to find a suitable model for fast and practical bathymetry of an estuary using Indian Remote Sensing Satellite (IRS) Linear Imaging Self Scanning Sensor (LISS-3) images. The study examines three different models; (1) least square regression model, (2) spectral band-ratio method and (3) multi-tidal bathymetry model. The findings are supported with in situ observed depth values and statistical estimates. Although the least square regression model has provided best results with root mean square error (RMSE) of 0.4 m, it requires a large number of observed data points for absolute depth estimation. Spectral band-ratio and multi-tidal model provides results with RMSEs 2.1 and 0.9 m, respectively. The present investigation demonstrates that multi-date imagery exploitation at disparate tide levels is the best estimation technique for recursive shallow water bathymetry where in situ observation is not possible.     

A priori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning

In this contribution, using the example of the Mátern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schön model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Mátern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Mátern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Mátern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise).     

Monitoring GOCE gradiometer calibration parameters using accelerometer and star sensor data: methodology and first results

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, launched on 17 March 2009, is designed to measure the Earth’s mean gravity field with unprecedented accuracy at spatial resolutions down to 100?km. The accurate calibration of the gravity gradiometer on-board GOCE is of utmost importance for achieving the mission goals. ESA’s baseline method for the calibration uses star sensor and accelerometer data of a dedicated calibration procedure, which is executed every 2?months. In this paper, we describe a method for monitoring the evolution of calibration parameter during that time. The method works with star sensor and accelerometer data and does not require gravity field models, which distinguishes it from other existing methods. We present time series of calibration parameters estimated from GOCE data from 1 November 2009 to 17 May 2010. The time series confirm drifts in the calibration parameters that are present in the results of other methods, including ESA’s baseline method. Although these drifts are very small, they degrade the gravity gradients, leading to the conclusion that the calibration parameters of the ESA’s baseline method need to be linearly interpolated. Further, we find a correction of ?36 × 10^?6 for one calibration parameter (in-line differential scale factor of the cross-track gradiometer arm), which improves the gravity gradient performance. The results are validated by investigating the trace of the calibrated gravity gradients and comparing calibrated gravity gradients with reference gradients computed along the GOCE orbit using the ITG-Grace-2010s gravity field model.     

Robust estimation by expectation maximization algorithm

A mixture of normal distributions is assumed for the observations of a linear model. The first component of the mixture represents the measurements without gross errors, while each of the remaining components gives the distribution for an outlier. Missing data are introduced to deliver the information as to which observation belongs to which component. The unknown location parameters and the unknown scale parameter of the linear model are estimated by the EM algorithm, which is iteratively applied. The E (expectation) step of the algorithm determines the expected value of the likelihood function given the observations and the current estimate of the unknown parameters, while the M (maximization) step computes new estimates by maximizing the expectation of the likelihood function. In comparison to Huber’s M-estimation, the EM algorithm does not only identify outliers by introducing small weights for large residuals but also estimates the outliers. They can be corrected by the parameters of the linear model freed from the distortions by gross errors. Monte Carlo methods with random variates from the normal distribution then give expectations, variances, covariances and confidence regions for functions of the parameters estimated by taking care of the outliers. The method is demonstrated by the analysis of measurements with gross errors of a laser scanner.     

Predictability of the Earth’s polar motion

The numerical prediction of the Earth’s polar motion is of both theoretical and practical interest. The present paper is aimed at a comprehensive, experimental study of the predictability of polar motion using a homogeneous BIH (Bureau International de l’Heure) data set for the period 1967–1983. Based on our knowledge of the physics of the annual and the Chandler wobbles, we build the numerical model for the polar motion by allowing the wobble period to vary. Using an optimum base length of six years for prediction, this “floating-period” model, equipped with a nonlinear least-squares estimator, is found to yield polar motion predictions accurate to within 0″.012 to 0″.024 depending on the prediction length up to one year, corresponding to a predictability of 89–82%. This represents a considerable improvement over the conventional fixed-period predictor, which, by its nature, does not respond to variations in the apparent wobble periods (in particular, a dramatic decrease in the periods of both the annual and the Chandler wobbles after the year 1980). The superiority of the floating-period predictor to other predictors based on critically different numerical models is also demonstrated.     

Nonlinear diffusion filtering of data on the Earth’s surface

The paper deals with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth’s surface. The closed surface as a computational domain is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green’s theorem to the Laplace–Beltrami operator. Then the finite-volume method is applied to discretize the weak formulation. Weak forms of elliptic operators are expressed through surface gradients. In our numerical scheme we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Furthermore, we extend a linear diffusion on surface to the regularized surface Perona–Malik model. It represents a nonlinear diffusion equation, which at the same time reduces noise and preserves main edges and other details important for a correct interpretation of the real data. We present four numerical experiments. The first one has an illustrative character showing how an additive noise is filtered out from an artificial function defined on a sphere. Other three examples deal with the real geodetic data on the Earth’s surface, namely (i) we reduce a stripping noise from the GOCE satellite only geopotential model up to degree 240, (ii) we filter noise from the real GOCE measurements (the component $T_{zz})$ , and (iii) we reduce a stripping noise from the satellite only mean dynamic topography at oceans. In all experiments we focus on a comparison of the results obtained by both the linear and nonlinear models presenting advantages of the nonlinear diffusion.     

An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and <Emphasis Type="Italic">t</Emphasis>-distributed deviations

In this paper, we investigate a linear regression time series model of possibly outlier-afflicted observations and autocorrelated random deviations. This colored noise is represented by a covariance-stationary autoregressive (AR) process, in which the independent error components follow a scaled (Student’s) t-distribution. This error model allows for the stochastic modeling of multiple outliers and for an adaptive robust maximum likelihood (ML) estimation of the unknown regression and AR coefficients, the scale parameter, and the degree of freedom of the t-distribution. This approach is meant to be an extension of known estimators, which tend to focus only on the regression model, or on the AR error model, or on normally distributed errors. For the purpose of ML estimation, we derive an expectation conditional maximization either algorithm, which leads to an easy-to-implement version of iteratively reweighted least squares. The estimation performance of the algorithm is evaluated via Monte Carlo simulations for a Fourier as well as a spline model in connection with AR colored noise models of different orders and with three different sampling distributions generating the white noise components. We apply the algorithm to a vibration dataset recorded by a high-accuracy, single-axis accelerometer, focusing on the evaluation of the estimated AR colored noise model.     

The integral formulas of the associated Legendre functions

A new kind of integral formulas for ${\bar{P}_{n,m} (x)}$ is derived from the addition theorem about the Legendre Functions when n ? m is an even number. Based on the newly introduced integral formulas, the fully normalized associated Legendre functions can be directly computed without using any recursion methods that currently are often used in the computations. In addition, some arithmetic examples are computed with the increasing degree recursion and the integral methods introduced in the paper respectively, in order to compare the precisions and run-times of these two methods in computing the fully normalized associated Legendre functions. The results indicate that the precisions of the integral methods are almost consistent for variant x in computing ${\bar{P}_{n,m} (x)}$ , i.e., the precisions are independent of the choice of x on the interval [0,1]. In contrast, the precisions of the increasing degree recursion change with different values on the interval [0,1], particularly, when x tends to 1, the errors of computing ${\bar{P}_{n,m} (x)}$ by the increasing degree recursion become unacceptable when the degree becomes larger and larger. On the other hand, the integral methods cost more run-time than the increasing degree recursion. Hence, it is suggested that combinations of the integral method and the increasing degree recursion can be adopted, that is, the integral methods can be used as a replacement for the recursive initials when the recursion method become divergent.     

基于逻辑斯蒂模型的遥感图像分类

逻辑斯蒂法是一种非线性的回归分析方法,因采用逻辑斯蒂模型而得名^[1],可用来进行未知单元类别属性的预测和判定。不同于一般的分类方法,它可分别给出某一单元属于各已知类别的概率,进而对研究的未知区中所有单元进行分类和预测。本文首先阐述了该方法的基本原理,而后利用它对内蒙古自治区两个研究区的两种图像数据进行了分类,最后探讨了影响该方法用于遥感图像分类的几个因素.     

A time‐geographic approach to quantifying wildlife–road interactions

Recent advances in time geography offer new perspectives for studying animal movements and interactions in an environmental context. In particular, the ability to estimate an animal's spatial location probabilistically at temporal sampling intervals between known fix locations allows researchers to quantify how individuals interact with one another and their environment on finer temporal and spatial scales than previously explored. This article extends methods from time geography, specifically probabilistic space–time prisms, to quantify and summarize animal–road interactions toward understanding related diurnal movement behaviors, including road avoidance. The approach is demonstrated using tracking data for fishers (Martes pennanti) in New York State, where the total probability of interaction with roadways is calculated for individuals over the duration tracked. Additionally, a summarization method visualizing daily interaction probabilities at 60 s intervals is developed to assist in the examination of temporal patterns associated with fishers’ movement behavior with respect to roadways. The results identify spatial and temporal patterns of fisher–roadway interaction by time of day. Overall, the methodologies discussed offer an intuitive means to assess moving object location probabilities in the context of environmental factors. Implications for movement ecology and related conservation planning efforts are also discussed.     

Satellite mapping of Baltic Sea Secchi depth with multiple regression models

Secchi depth is a measure of water transparency. In the Baltic Sea region, Secchi depth maps are used to assess eutrophication and as input for habitat models. Due to their spatial and temporal coverage, satellite data would be the most suitable data source for such maps. But the Baltic Sea’s optical properties are so different from the open ocean that globally calibrated standard models suffer from large errors. Regional predictive models that take the Baltic Sea’s special optical properties into account are thus needed. This paper tests how accurately generalized linear models (GLMs) and generalized additive models (GAMs) with MODIS/Aqua and auxiliary data as inputs can predict Secchi depth at a regional scale. It uses cross-validation to test the prediction accuracy of hundreds of GAMs and GLMs with up to 5 input variables. A GAM with 3 input variables (chlorophyll a, remote sensing reflectance at 678 nm, and long-term mean salinity) made the most accurate predictions. Tested against field observations not used for model selection and calibration, the best model’s mean absolute error (MAE) for daily predictions was 1.07 m (22%), more than 50% lower than for other publicly available Baltic Sea Secchi depth maps. The MAE for predicting monthly averages was 0.86 m (15%). Thus, the proposed model selection process was able to find a regional model with good prediction accuracy. It could be useful to find predictive models for environmental variables other than Secchi depth, using data from other satellite sensors, and for other regions where non-standard remote sensing models are needed for prediction and mapping. Annual and monthly mean Secchi depth maps for 2003–2012 come with this paper as Supplementary materials.     

秩亏GM模型参数估计的若干问题

本文依据秩亏ＧＭ模型中可估量的条件，提出了参数ｘ部分可估的概念，讨论了补偿系统误差的自由网平差问题中系统误差参数可估的条件。研究了可估量的ＢＬＵＥ以及不可估量的最佳线性最小偏差估计以及ｘ的线性估计之间的关系，从最佳线性最小偏差估计的角度分析了最小二乘最小范数解的性质。对线性约束下的参数估计，讨论了条件可估的ＢＬＵＥ。给出了ｘ可估需满足的边界条件的概念并研究了不同边界条件下参数估值的转换问题。     

On the probability density function of the GNSS ambiguity residuals

Integer GNSS ambiguity resolution involves estimation and validation of the unknown integer carrier phase ambiguities. A problem then is that the classical theory of linear estimation does not apply to the integer GPS model, and hence rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency. Existing validation methods are often based on incorrect assumptions with respect to the probabilistic properties of the parameters involved. In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals. The residuals and their properties depend on the integer estimation principle used. Since it is known that the integer least-squares estimator is the optimal choice from the class of admissible integer estimators, we will only focus on the PDF of the ambiguity residuals for this estimator. Unfortunately the PDF cannot be evaluated exactly. It will therefore be shown how to obtain a good approximation. The evaluation will be completed by some examples.     

Enhancement of the accuracy of single-epoch GPS positioning for long baselines by local ionospheric modelling

Single-epoch relative GPS positioning has many advantages, especially for monitoring dynamic targets. In this technique, errors occurring in previous epochs cannot affect the position accuracy at the current epoch, but careful processing is required, and resolving carrier phase ambiguities is essential. Statistical ambiguity resolution functions have been used to determine the best values of these ambiguities. The function inputs include as a minimum the known base station position, the approximate roving antenna “seed” position, and the dual-frequency carrier phase measurements from both receivers. We investigate different solutions to find the ambiguity function inputs that achieve the highest ambiguity resolution success rate. First, we address the rover seed position. A regionally filtered undifferenced pseudorange coordinate solution proves better than a double-differenced one. Multipath errors approximately repeat themselves every sidereal day in the case of static or quasi-static antennas; applying a sidereal filter to the pseudorange-derived positions mitigates their effects. Second, we address the relative carrier phase measurements, which for medium to long baselines are significantly affected by ionospheric propagation errors imperfectly removed during differencing. In addition to the International GNSS Service ionospheric model, we generate a local pseudorange-based ionospheric correction. Applying this correction improves the quality of the phase measurements, leading to more successful ambiguity resolution. Temporally smoothing the correction by means of a Kalman filter further improves the phase measurements. For baselines in the range 60–120 km, the mean absolute deviation of single-epoch coordinates improves to 10–20 cm, from 30–50 cm in the default case.     

The static gravity field model DGM-1S from GRACE and GOCE data: computation, validation and an analysis of GOCE mission’s added value

We present a global static model of the Earth’s gravity field entitled DGM-1S based on GRACE and GOCE data. The collection of used data sets includes nearly 7 years of GRACE KBR data and 10 months of GOCE gravity gradient data. The KBR data are transformed with a 3-point differentiation into quantities that are approximately inter-satellite accelerations. Gravity gradients are processed in the instrumental frame. Noise is handled with a frequency-dependent data weighting. DGM-1S is complete to spherical harmonic degree 250 with a Kaula regularization being applied above degree 179. Its performance is compared with a number of other satellite-only GRACE/GOCE models by confronting them with (i) an independent model of the oceanic mean dynamic topography, and (ii) independent KBR and gravity gradient data. The tests reveal a competitive quality for DGM-1S. Importantly, we study added value of GOCE data by comparing the performance of satellite-only GRACE/GOCE models with models produced without GOCE data: either ITG-Grace2010s or EGM2008 depending on which of the two performs better in a given region. The test executed based on independent gravity gradients quantifies this added value as 25–38 % in the continental areas poorly covered with terrestrial gravimetry data (Equatorial Africa, Himalayas, and South America), 7–17 % in those with a good coverage with these data (Australia, North America, and North Eurasia), and 14 % in the oceans. This added value is shown to be almost entirely related to coefficients below degree 200. It is shown that this gain must be entirely attributed to gravity gradients acquired by the mission. The test executed based on an independent model of the mean dynamic topography suggests that problems still seem to exist in satellite-only GRACE/GOCE models over the Pacific ocean, where noticeable deviations between these models and EGM2008 are detected, too.     

On the Mátern covariance family: a proposal for modeling temporal correlations based on turbulence theory

Current variance models for GPS carrier phases that take correlation due to tropospheric turbulence into account are mathematically difficult to handle due to numerical integrations. In this paper, a new model for temporal correlations of GPS phase measurements based on turbulence theory is proposed that overcomes this issue. Moreover, we show that the obtained model belongs to the Mátern covariance family with a smoothness of 5/6 as well as a correlation time between 125–175 s. For this purpose, the concept of separation distance between two lines-of-sight introduced by Schön and Brunner (J Geod 1:47–57, 2008a) is extended. The approximations made are highlighted as well as the turbulence parameters that should be taken into account in our modeling. Subsequently, fully populated covariance matrices are easily computed and integrated in the weighted least-squares model. Batch solutions of coordinates are derived to show the impact of fully populated covariance matrices on the least-squares adjustments as well as to study the influence of the smoothness and correlation time. Results for a specially designed network with weak multipath are presented by means of the coordinate scatter and the a posteriori coordinate precision. It is shown that the known overestimation of the coordinate precision is significantly reduced and the coordinate scatter slightly improved in the sub-millimeter level compared to solutions obtained with diagonal, elevation-dependent covariance matrices. Even if the variations are small, turbulence-based values for the smoothness and correlation time yield best results for the coordinate scatter.