Least squares collocation and statistical testing

The authors address the issue of statistical testing in least squares collocation (LSC) in two stages. The first stage concerns the extension and focusing of theLSC equations to the task of statistical testing. The second stage deals with statistical testing titself and is introduced in the second portion of the paper. The paper commences with an overview of the development ofLSC and its relationship to least squares adjustment (LSA). Expressions for the various random variables and their corresponding covariance matrices are derived and in some instances are gleaned from the literature for the following quantities: (i) corrections to the unknown parameters with a priori covariance information; (ii) estimated signal at both the observation and computation points; and (iii) the noise at the observation points. Some of the needed covariance matrices are either obscurely hidden in the literature or not available at all, but, nevertheless are given in the paper. Also given are expressions for the estimated variance factor which forms the basis of various statistical tests. The paper closes with an overview and enumeration of possible statistical tests for detection of outliers in the observations.     

Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances

Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities. The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered. Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating them for each application.     

Local geoid computation from gravity using the fast fourier transform technique

Model computations were performed for the study of numerical errors which are interjected into local geoid computations byFFT. The gravity field model was generated through the attractions of granitic prisms derived from actual geology. Changes in sampling interval introduced only0.3 cm variation in geoid heights. Although zero padding alone provided an improvement of more than5 cm in theFFT generated geoid, the combination of spectral windowing (tapering) and padding further reduced numerical errors. For theGPS survey of Franklin County, Ohio, the parameters selected as a result of model computations, allow large reduction in local data requirements while still retaining the centimeter accuracy when tapering and padding is applied.     

The affine constrained GNSS attitude model and its multivariate integer least-squares solution

A new global navigation satellite system (GNSS) carrier-phase attitude model and its solution are introduced in this contribution. This affine-constrained GNSS attitude model has the advantage that it avoids the computational complexity of the orthonormality-constrained GNSS attitude model, while it still has a significantly improved ambiguity resolution performance over its unconstrained counterpart. The functional and stochastic model is formulated in multivariate form, for one-, two- and three-dimensional antenna arrays, tracking GNSS signals on an arbitrary number of frequencies with two or more antennas. The stochastic model includes the capability to capture variations in the antenna-quality within the array. The multivariate integer least-squares solution of the model parameters is given and the model’s ambiguity success-rate is analysed by means of the ambiguity dilution of precision (ADOP). A general closed-form expression for the affine-constrained ADOP is derived, thus providing an easy-to-use and insightful rule-of-thumb for the ambiguity resolution capabilities of the affine constrained GNSS attitude model.     

Deviations from the null hypothesis to be detected by statistical tests

Deviations from the null hypothesis, which can be detected by tests in an univariate model, are derived by means of the expected value of the test criterion. As shown, the boundary of the detectable deviations is represented by an hyperellipsoid. The variations of the hyperellipsoid due to the variance of the test criterion are also derived. If each equation of the hypothesis involves only one parameter, the deviations detectable in a plane can be obtained from ellipses which are found by multiplying the axes of the confidence ellipse for two parameters by a certain factor. This method is applied to compute movements to be detected in an analysis of data for recent crustal movements.     

New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001     

Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets

The GRACE (gravity recovery and climate experiment) and GOCE (gravity field and steady-state ocean circulation explorer) dedicated gravity satellite missions are expected to deliver the long-wavelength scales of the Earth’s gravity field with extreme precision. For many applications in Earth sciences, future research activities will have to focus on a similar precision on shorter scales not recovered by satellite missions. Here, we investigate the signal power of gravity anomalies at such short scales. We derive an average degree variance and power spectral density model for topography-reduced gravity anomalies (residual terrain model anomalies and de-trended refined Bouguer anomalies), which is valid for wavelengths between 0.7 and 100  km. The model is based on the analysis of gravity anomalies from 13 test regions in various geographical areas and geophysical settings, using various power spectrum computation approaches. The power of the derived average topography-reduced model is considerably lower than the Tscherning–Rapp free air anomaly model. The signal power of the individual test regions deviates from the obtained average model by less than a factor of 4 in terms of square-root power spectral amplitudes. Despite the topographic reduction, the highest signal power is found in mountainous areas and the lowest signal power in flat terrain. For the derived average power spectral model, a validation procedure is developed based on least-squares prediction tests. The validation shows that the model leads to a good prediction quality and realistic error measures. Therefore, for least-squares prediction, the model could replace the use of autocovariance functions derived from local or regional data.     

The computation of aliasing effects in local gravity field approximation

The aliasing effects in local gravity field computations are presented in this study. First the relation between the power spectral density of a 2-D continuous signal and its corresponding sampled version is derived. Then the power spectral density of the aliasing errors related to non band-limited signals is derived. Finally the variance of these aliasing errors is computed using gravity anomalies at different grid spacings. This computation prerequires some known gravity anomaly power spectral density model. The model used in this study corresponds to a second-order Gauss-Markov covariance function for the anomalous potential. Editor’s notice: Comments on this paper will follow in the next issue of Bulletin Géodésique.     

Log-cumulants of the finite mixture model and their application to statistical analysis of fully polarimetric UAVSAR data

Since its first flight in 2007, the UAVSAR instrument of NASA has acquired a large number of fully Polarimetric SAR (PolSAR) data in very high spatial resolution. It is possible to observe small spatial features in this type of data, offering the opportunity to explore structures in the images. In general, the structured scenes would present multimodal or spiky histograms. The finite mixture model has great advantages in modeling data with irregular histograms. In this paper, a type of important statistics called log-cumulants, which could be used to design parameter estimator or goodness-of-fit tests, are derived for the finite mixture model. They are compared with log-cumulants of the texture models. The results are adopted to UAVSAR data analysis to determine which model is better for different land types.     

A New GIS-based Solar Radiation Model and Its Application to Photovoltaic Assessments

The solar radiation model r.sun is a flexible and efficient tool for the estimation of solar radiation for clear‐sky and overcast atmospheric conditions. In contrast to other models, r.sun considers all relevant input parameters as spatially distributed entities to enable computations for large areas with complex terrain. Conceptually the model is based on equations published in the European Solar Radiation Atlas (ESRA). The r.sun model was applied to estimate the solar potential for photovoltaic systems in Central and Eastern Europe. The overcast radiation was computed from clear‐sky values and a clear‐sky index. The raster map of the clear‐sky index was computed using a multivariate interpolation method to account for terrain effects, with interpolation parameters optimized using a cross‐validation technique. The incorporation of terrain data improved the radiation estimates in terms of the model's predictive error and the spatial pattern of the model outputs. Comparing the results of r.sun with the ESRA database demonstrates that integration of the solar radiation model and the spatial interpolation tools in a GIS can be especially helpful for data at higher resolutions and in regions with a lack of ground measurements.     

Noise in multivariate GPS position time-series

A methodology is developed to analyze a multivariate linear model, which occurs in many geodetic and geophysical applications. Proper analysis of multivariate GPS coordinate time-series is considered to be an application. General, special, and more practical stochastic models are adopted to assess the noise characteristics of multivariate time-series. The least-squares variance component estimation (LS-VCE) is applied to estimate full covariance matrices among different series. For the special model, it is shown that the multivariate time-series can be estimated separately, and that the (cross) correlation between series propagates directly into the correlation between the corresponding parameters in the functional model. The time-series of five permanent GPS stations are used to show how the correlation between series propagates into the site velocities. The results subsequently conclude that the general model is close to the more practical model, for which an iterative algorithm is presented. The results also indicate that the correlation between series of different coordinate components per station is not significant. However, the spatial correlation between different stations for individual components is significant (a correlation of 0.9 over short baselines) both for white and for colored noise components.     

GPS signal diffraction modelling: the stochastic SIGMA-δ model

The SIGMA-Δ model has been developed for stochastic modelling of global positioning system (GPS) signal diffraction errors in high precision GPS surveys. The basic information used in the SIGMA-Δ model is the measured carrier-to-noise power-density ratio (C/N0). Using the C/N0 data and a template technique, the proper variances are derived for all phase observations. Thus the quality of the measured phase is automatically assessed and if phase observations are suspected to be contaminated by diffraction effects they are weighted down in the least-squares adjustment. The ability of the SIGMA-Δ model to reduce signal diffraction effects is demonstrated on two static GPS surveys as well as on a kinematic high-precision GPS railway survey. In cases of severe signal diffraction the accuracy of the GPS positions is improved by more than 50% compared to standard GPS processing techniques. Received: 27 July 1998 / Accepted: 1 March 1999     

TLS-bridged co-prediction of tree-level multifarious stem structure variables from worldview-2 panchromatic imagery: a case study of the boreal forest

ABSTRACT

In forest ecosystem studies, tree stem structure variables (SSVs) proved to be an essential kind of parameters, and now simultaneously deriving SSVs of as many kinds as possible at large scales is preferred for enhancing the frontier studies on marcoecosystem ecology and global carbon cycle. For this newly emerging task, satellite imagery such as WorldView-2 panchromatic images (WPIs) is used as a potential solution for co-prediction of tree-level multifarious SSVs, with static terrestrial laser scanning (TLS) assumed as a ‘bridge’. The specific operation is to pursue the allometric relationships between TLS-derived SSVs and WPI-derived feature parameters, and regression analyses with one or multiple explanatory variables are applied to deduce the prediction models (termed as Model1s and Model2s). In the case of Picea abies, Pinus sylvestris, Populus tremul and Quercus robur in a boreal forest, tests showed that Model1s and Model2s for different tree species can be derived (e.g. the maximum R²?=?0.574 for Q. robur). Overall, this study basically validated the algorithm proposed for co-prediction of multifarious SSVs, and the contribution is equivalent to developing a viable solution for SSV-estimation upscaling, which is useful for large-scale investigations of forest understory, macroecosystem ecology, global vegetation dynamics and global carbon cycle.     

Measuring the robustness potential of the least-squares estimation: geodetic illustration

For a linear least-squares parametric model analysis is carried out of the structure of the projection operator transforming the vector of standardised observations into the vector of standardised residuals. On this basis the properties of the model responses to observational disturbances (i.e. gross errors or blunders) are derived. A final outcome of the research can be summarised as: (1) proposing the robustness characteristics of a model and linking them with the local measures of internal reliability, being the diagonal elements in the projection operator; (2) determining the internal reliability levels satisfying specified robustness requirements, i.e. the possibility of detecting at least one of the k observational disturbances (k=1,2,…) having most disadvantageous locations in the system. The theory and a numerical example show that for the systems which have been designed to a proper level of internal reliability, the least-squares estimation can demonstrate an accordingly high level of robustness. Received: 11 June 1996 / Accepted: 28 April 1997     

Variance-covariance estimation of GPS Networks

Summary It is quite easy to estimate the variance-covariance (VCV) matrix for single session surveys or local networks, but difficult where these local networks are combined together to form a regional network. Our main aim is to develop an appropriate VCV model to combine all the different types of networks, either global, regional or local. By careful estimation and combination of the individual VCVs of the local networks, we can form a unique VCV for local, regional and global networks. Different techniques are used to derive appropriate models for the variancecovariance components of the Global Positioning System (GPS) networks. The VCV models were estimated using homogeneous and heterogeneous data. The variance-covariance components are empirically derived using (a) the covariance of the observations of homogeneous data, (b) a combination of the covariance of the observationsP ^–1 and the covariance of the signal componentsC _ss (for either homogeneous and/or heterogeneous data), (c) only the variances are used to determine the variancecovariance, their covariances being zeros. We compare the solutions of the VCV developed for homogeneous and/or heterogeneous data with other developed VCVs. It was observed that the derived VCV model for the combined homogeneous and/or heterogeneous data of case (b) gives the best estimates in all cases.     

Computation of spherical harmonic coefficients and their error estimates using least-squares collocation

 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000     

On a self-consistent representation of earth models,with an application to the computing of internal flattening

Most realistic Earth models published as yet have been given in tabulated form, with the noticeable exception of three simple parametric Earth models derived by Dziewonski et al. (1975). Simple interpolation in these tables may lead to inconsistencies, when we consider certain effects which depend crucially on detailed density structure. We establish algorithmic formulae, which may be used to compute all the mechanical properties of a model in an entirely consistent way, once the density as well asP— andS— wave velocities are known. We then use this formulation to integrate Clairaut’s equation in a very efficient way, and thus obtain the hydrostatic flattening to the first order in smallness at any point inside the model. For most geodynamic purposes, we may suffice with this approximation. Finally, we show the results of some calculations of hydrostatic flattening to the first and second order, using an iterative technique of solving the integral figure equations, for an Earth model consistent with all geophysical data available at present. We find that the hydrostatic flattening at the surface should be about 1/298.8, instead of 1/296.961 as quoted by Nakiboglu (1979) for essentially the same model. Moreover, from our results, we estimate the actual flattening of the coremantle boundary to be about 1/390.3.     

Analysis of the upper bounds for the integer ambiguity validation statistics

Integer ambiguity validation is an essential quality control step for high-precision positioning and navigation with global navigation satellite systems (GNSS). In order to validate the resolved integer ambiguities, statistical tests, such as the R-ratio test, F-ratio test, W-ratio test, difference test, and projector test, have been favored. In practice, the critical values for these statistical tests are determined either empirically or from the assumed distributions. However, previous research has revealed that some of these statistics have upper bounds, which can be obtained from simulations. In this contribution, we find that under the framework of the integer aperture estimation, the upper bounds for these ambiguity validation statistical tests can be derived without actual measurements or simulation. As a result, the assumed distributions for these statistical tests are inappropriate. According to the derivation, it has been concluded that the upper bounds of these ambiguity validation tests depend only on the ambiguity geometry (e.g., the float ambiguity variance–covariance matrix) and can be obtained at the design stage of GNSS positioning. Thus, the critical value for these ambiguity validation statistics has a rigorous range, and it should be chosen to be smaller than a priori derived upper bound. Otherwise, no integer ambiguities can be obtained.     

A VLBI Delay Model for Radio Sources at a Finite Distance

A relativistic delay model for Earth-based very long baseline interferometry (VLBI) observation of sources at finite distances is derived. The model directly provides the VLBI delay in the scale of terrestrial time. The effect of the curved wave front is represented by using a pseudo source vector K = (R ₁ + R ₂)/(R ₁ + R ₂), and the variation of the baseline vector due to the difference of arrival time is taken into account up to the second-order by using Halley’s method. The precision of the new VLBI delay model is 1 ps for all radio sources above 100 km altitude from the Earth’s surface in Earth-based VLBI observation. Simple correction terms (parallax effect) are obtained, which can also adopt the consensus model (e.g. International Earth Rotation and Reference Frames Service conventions) to finite-distance radio source at R > 10 pc with the same precision. The new model may enable estimation of distance to the radio source directly with VLBI delay data.