Least-squares Adjustment Research Articles

Parameter estimation methods for correlated observation mixed additive and multiplicative random error model

In the domain of geodetic adjustment, current methodologies addressing additive and multiplicative error models operate under the assumption of independence between additive and multiplicative random errors. However, limited research has been conducted on elucidating the correlation between the components of additive and multiplicative random errors. To extend a methodology of mixed error models with additive and multiplicative correlation observations, this paper derives three parameter adjustment methods: correlation observation least squares, correlation observation weighted least squares, and correlation observation bias-corrected weighted least squares, all based on the principles of least squares. Additionally, corresponding unit weight variance estimations are constructed. Finally, it is verified by two numerical simulation experiments and a real-world application example of a geodetic network that the correlated observation bcWLS studied in this paper proves to be the most efficient among the six methods for solving the correlated observation multiplication mixture error model.

Real-time LEO satellite clock estimation with predicted LEO satellite orbits constrained

Low Earth Orbit (LEO) satellites can augment the traditional GNSS-based positioning, navigation and timing services, which require real-time high-precision LEO satellite clock products. As the complicated systematic effects contained in the LEO satellite clock estimates limit their high-precision mid- to long-term prediction, high-frequency LEO satellite clocks need to be estimated within a Kalman filter, resulting in a short prediction time for real-time applications. Compared to the clock estimation using Batch Least-Squares (BLS) adjustment, filter-based clock estimation experiences a lower precision. Increasing the model strength by introducing external orbital information, thus, de-correlating the orbital and clock parameters, will benefit real-time clock precision. In this contribution, reduced-dynamic LEO satellite orbits are first estimated using BLS adjustment in near real-time and predicted in the short term. The predicted orbits are then constrained during the Kalman-filter-based clock estimation process. The variance–covariance matrix of the introduced orbital errors is tested for different sets of values in the radial, along-track and cross-track directions when constraining orbits of different prediction times. One week of GPS data from the Sentinel-3B satellite in 2018 was used for validation of the proposed method. When weakly constraining high-accuracy predicted orbits within a prediction time of 20 min, i.e., with a standard deviation of the constraint set to 2–3 dm in the radial and cross-track directions, and 4–6 dm in the along-track direction, the estimated clock accuracy can be improved from about 0.27 to 0.23 ns, with a 13.4% improvement. Depending on the prediction period of the introduced orbits, the Signal-In-Space Range Error (SISRE) of the LEO satellite to Earth can also be improved, from about 9.59 cm without constraints, to 7.38–8.07 cm after constraining the predicted orbits, with an improvement of 16–23%. The improvements in the SISRE also indicate a better consistency between the real-time clock and orbital estimates.

Toward a unified approach to the total least-squares adjustment

Toward a unified approach to the total least-squares adjustment

Microwave-Assisted Pyrolysis of Carbon Fiber-Reinforced Polymers and Optimization Using the Box-Behnken Response Surface Methodology Tool.

This article introduces an eco-friendly method for the reclamation of carbon fiber-reinforced polymers (CFRP). The research project involved numerous experiments using microwave-assisted pyrolysis (MAP) to explore a range of factors, such as the inert gas flow, the power level, the On/Off frequency of rotation, and the reaction duration. To design the experiments, the three-level Box-Behnken optimization tool was employed. To determine the individual and combined effects of the input parameters on the thermal decomposition of the resin, the data were analyzed using least-squares variance adjustment. The results demonstrate that the models developed in this study were successful in predicting the direct parameters of influence in the microwave-assisted decomposition of CFRPs. An optimal set of operating conditions was found to be the maximum nitrogen flow (2.9 L/min) and the maximum operating experimental power (914 W). In addition, it was observed that the reactor vessel's On/Off rotation frequency and that increasing the reaction time beyond 6 min had no significant influence on the resin elimination percentage when compared to the two other parameters, i.e., power and carrier gas flow rate. Consequently, the above-mentioned conditions resulted in a maximum resin elimination percentage of 79.6%. Following successful MAP, various post-pyrolysis treatments were employed. These included mechanical abrasion using quartz sand, chemical dissolution, thermal oxidative treatment using a microwave (MW) applicator and thermal oxidative treatment in a conventional furnace. Among these post-treatment techniques, thermal oxidation and chemical dissolution were found to be the most efficient methods, eliminating 100% of the carbon black content on the surface of the recovered carbon fibers. Finally, SEM evaluations and XPS analysis were conducted to compare the surface morphology and elementary constitution of the recovered carbon fibers with virgin carbon fibers.

Radial surface currents from space: An opportunity for mean dynamic topography estimation?

We analyze the feasibility of so-called radial surface currents as derived from Sentinel-1 Wave mode Doppler shift observations to support the geodetic estimation of the mean dynamic topography (MDT). Broadly defined as subtracting the geoid from the mean sea surface, this separation problem suffers from data inhomogeneities and usually requires strong prior assumptions about the MDT’s smoothness. Recent advancements in calibration and current retrieval methodology for Sentinel-1 data give reason to assess potential gains from this observation type, which can be linked to the gradient of the MDT under geostrophic approximation. Due to current lack of long time-series, we synthesize 10 years of observations from daily surface currents grids and include these in a parametric least-squares adjustment of the MDT. Our results utilizing the synthetic data in place of smoothness constraints are promising, showing 2 cm RMS agreement with a state-of-the-art MDT model.

An improved wide-lane ambiguity resolution method for kinematic smartphone positioning

Abstract The release of GNSS raw data on Android smartphones provides the potential for high-precision smartphone positioning using multi-constellation and multi-frequency signals. However, severe multipath and low observation quality in kinematic environments make double-differenced uncombined ambiguities difficult to resolve reliably. To address this, the paper proposes an improved wide-lane (WL) integer ambiguity resolution (IAR) method that combines integer rounding and the Least-Square AMBiguity Decorrelation Adjustment (LAMBDA) methods. The proposed method achieved fix rates of 57% to 70% in challenging environments, with an average improvement of 7 · 7% in horizontal positioning accuracy compared to the float solution. The traditional partial integer rounding method only improved accuracy by 1 · 1%.

Impact of incorporating Spire CubeSat GPS observations in a global GPS network solution

Commonly, Global Positioning System (GPS)-based precise orbit determination (POD) of a low Earth orbiting (LEO) satellite is conducted by introducing fixed GPS orbits and clock corrections that were derived in a previous independent network solution using ground-based GPS receivers only. There have been a number of studies showing that the integration of LEO GPS observations can be advantageous in global network solutions, particularly when it comes to estimating geodetic parameters, such as the Earth’s center-of-mass. There has been an increase in GPS data available from LEO CubeSats over the past few years, such as from the Spire satellites, which are equipped with dual-frequency GPS receivers. The goal of the present study is to determine how GPS observations collected by specific Spire satellites affect global network solutions when combined with GPS observations from ground stations of the International GNSS Service. To obtain a combined GPS-LEO solution, GPS observations collected from receivers of selected Spire satellites and additional scientific LEOs are used to obtain combined GPS-LEO solutions by performing one joint least-squares adjustment process. Using this method, LEO satellite orbit parameters are determined along with GPS orbit parameters and geodetic parameters, namely GNSS station coordinates, Earth rotation parameters (ERPs), and the Earth’s center-of-mass. The influence of the Spire LEOs is assessed by analyzing different solutions that use GPS observations from different Spire and LEO satellites. Besides comparing the different global network solutions with one another, a comparison to a solution using only terrestrial GPS data is performed as well. As part of the quality assessment of the GPS-LEO solution, quality characteristics are analyzed, which are the geodetic parameters, orbit overlaps at arc boundaries, and formal errors determined for the estimated parameters. The results show that the estimation of the Earth’s center-of-mass coordinates benefits from including GPS-LEO observations. The solution is more improved by integrating GPS-LEO observations from scientific LEOs than from GPS-Spire-included solutions. The analysis reveals that GPS orbits are improved using the LEO-integrated approach, while orbit misclosures of LEO and Spire trajectories currently indicate a decrease in LEO orbit precision due to LEO orbit modeling deficiencies.

Effects of errors-in-variables on the internal and external reliability measures

The reliability theory has been an important element of the classical geodetic adjustment theory and methods in the linear Gauss-Markov model. Although errors-in-variables (EIV) models have been intensively investigated, little has been done about reliability theory for EIV models. This paper first investigates the effect of a random coefficient matrix A on the conventional geodetic reliability measures as if the coefficient matrix were deterministic. The effects of such geodetic internal and external reliability measures due to the randomness of the coefficient matrix are worked out, which are shown to depend not only on the noise level of the random elements of A but also on the values of parameters. An alternative, linear approximate reliability theory is accordingly developed for use in EIV models. Both the EIV-affected reliability measures and the corresponding linear approximate measures fully account for the random errors of both the coefficient matrix and the observations, though formulated in a slightly different way. Numerical experiments have been carried to demonstrate the effects of errors-in-variables on reliability measures and compared with the conventional Baarda's reliability measures. The simulations have confirmed our theoretical results that the EIV-reliability measures depend on both the noise level of A and the parameter values. The larger the noise level of A, the larger the EIV-affected internal and external reliability measures; the larger the parameters, the larger the EIV-affected internal and external reliability measures.

Multi-Scenario Variable-State Robust Fusion Algorithm for Ranging Analysis Framework

Integrating modern information technology with traditional agriculture has made agricultural machinery navigation essential in PA (precision agriculture). However, agricultural equipment faces challenges such as low positioning accuracy and poor algorithm adaptability due to the complex farmland environment and various operational requirements. In this research, we proposed a generalized ranging theoretical framework with multi-scenario variable-state fusion to improve the GNSS (Global Navigation Satellite System) observation exchange performance among agricultural vehicles, and accurately measure IVRs (inter-vehicular ranges). We evaluated the effectiveness of three types of GNSS observations, including PPP-SD (precise single point positioning using single difference), PPP-TCAR (precise single point positioning using double difference based on three-carrier ambiguity resolution), and PPP-LAMBDA (precise single point positioning using double difference based on least-squares ambiguity decorrelation adjustment). Moreover, we compared the accuracy of IVRs measurements. Our framework was validated through field experiments in different scenarios. It provides insights into the appropriate use of different positioning algorithms based on the application scenario, application objects, and motion states.

An integration scheme of simultaneous cycle slips determination combining improved geometry-free combination and TDCP model for undifferenced GNSS data

Fast and reliable cycle slip determination can ensure successive ambiguity resolution and precise positioning. Generally, it is not difficult to determine big cycle slips using the linear combination of observations, for instance, the geometry-free (GF) combination and the Hatch–Melbourne–Wübbena (HMW) combination. However, the participation of pseudorange observations may fail to identify small cycle slips. This contribution proposes an integration scheme combining the improved geometry-free (IGF) combination and time-difference carrier phase (TDCP) model to determine simultaneous cycle slips for undifferenced kinematic data. The IGF combination, which is improved from the modified geometry-free (MGF) combination and utilizes the Gauss floor function to take the decimal part of GF, can directly determine small cycle slips on a specific frequency. The TDCP model is used to estimate the differences of position and clock error between adjacent epochs by least-square adjustment using clean phase observations, and repair the remaining cycle slips by the predicted TDCP measurements obtained from the predetermined parameters. The proposed method is tested against 1 Hz kinematic dataset with simulation and highway dataset with real cycle slips. In the simulation test, all cycle slips can be correctly repaired by IGF for L1/L5, E1/E5 and E1/E5a. These combinations are unavailable for MGF. Compared to the HMW-GF method, the repair success rate of IGF improves from 96.80% to 99.16%, and the number of incorrect cases reduces from 9650 to 2535. The IGF-TDCP integration scheme can further improve the performance of IGF, whose repair success rate is more than 99.97% and incorrect cases are 38. The highway test shows that the proposed method effectively processes simultaneous cycle slips on more than half of the tracking satellites caused by the overpasses, even in the case of simultaneous 3 second data gaps on 5 satellites of 6 tracking satellites.

Development of a hybrid geoid model using a global gravity field model over Sri Lanka

AbstractSri Lanka is still in need of a well-defined local geoid model. This geoidal void has made present-day Global Navigation Satellite System (GNSS) surveys heavily dependent on Global Geopotential Models (GGMs) for height determination. Further, in many instances, the accuracy of GGMs have shown drawbacks in elevation determination over Sri Lanka. Therefore, the study focused on developing a hybrid geoid model (HGM) for Sri Lanka by integrating the available GGMs. Five high-resolution (2190°) GGMs; EGM2008, EIGEN-6C4, GECO, XGM2019e-2159, and SGG-UGM2 were employed to extract GGM-derived geoid undulation for 21 Fundamental Benchmarks (FBMs). The residuals (geoid height deviation) were calculated relative to the observed geoid undulation using GNSS/leveling on the FBMs. The data set was clustered based on topography, and residuals were adjusted using weighted least squares adjustment (LSA). The uneven distribution of the FBMs promotes topography-based clustering. EIGEN-6C4 is found to be the robust GGM for Sri Lanka to develop a hybrid approach, with a 0.001 m RMS value of estimated residuals in LSA. The resulting HGM was interpolated at 1 arc-second grid resolution (30 m × 30 m) using the Inverse Distance Weighted Interpolation. Regression lines were generated for the interpolated HGM with respect to the interpolated observed geoid undulation for 9 transects along the parallel passing through Mount Pedro and for the 16 transects along the meridian. The coefficient of determination on both lines is 0.999. HGM generated by EIGEN-6C4 has shown reliable RMS gradient and intercept values of 8.860078 × 10−9and 0.0039239, respectively, in first-order polynomial fitting.

Development and testing of WEASEL for least-squares neutron spectrum adjustment

A new least-squares adjustment program called WEASEL has been developed. WEASEL is implemented in Python and uses standard ASCII interfaces for user input and output, and HDF5 interfaces for cross section, covariance data, and weighting function data. WEASEL allows the user to supply a trial neutron spectrum in an arbitrary group structure. The trial neutron spectrum is re-binned into a group structure consistent with the covariance data using a weighting function that can either be read from the data library or supplied as input by the user. WEASEL performs least-squares adjustments of a single irradiation environment. The adjusted neutron spectrum, covariance matrices, and other derived quantities can then be plotted and interrogated with standard software packages.

AUTOMATED FINE-SCALE FOREST INVENTORY USING BACKPACK LIDAR – A STRATEGY BASED ON FEATURE EXTRACTION, MATCHING, AND TRACKING FROM INTEGRATED SCANS

Abstract. Backpack LiDAR systems are gaining popularity due to their rapid data acquisition, portability, and cost-effectiveness. However, using backpack LiDAR in forest poses challenges, such as GNSS signal attenuation under the canopy, leading to inaccurate trajectory estimates and misregistered point clouds. This paper aims at presenting a novel method that addresses this challenge by leveraging the integrated scans (IS) concept to enhance point cloud quality for automated forest inventory. The proposed method for forest inventory using ISs consists of four key steps: (i) Integrated scans (IS) generation, (ii) feature extraction, matching, and tracking, (iii) trajectory enhancement, and (iv) tree biometric extraction. Firstly, IS point clouds are generated based on the initial GNSS/INS trajectory. Secondly, reliable forest features such as tree trunks and ground patches are extracted, matched, and tracked across ISs. These features are then utilized in the trajectory enhancement step, where a non-linear Least Squares Adjustment (LSA) technique is used to minimize discrepancies among the features to enhance the trajectory. The resulting point clouds, based on the improved trajectory, are used to extract tree biometric information. The proposed method was evaluated using two distinct datasets collected with different systems. The evaluation results, both qualitatively and quantitatively, validate the effectiveness of the proposed method, showcasing its potential for fine-scale forest inventory applications.

Improvement of Baarda’s external reliability measure

ABSTRACT The reliability theory has been an important element of the classical geodetic adjustment theory and methods in a linear Gauss-Markov model since Baarda invented reliability in 1968. Although geodetic reliability has been widely investigated and applied to a variety of geodetic, photogrammetric, and remote sensing problems, there is no report of theoretical progress to improve further Baarda’s reliability measures among linear models. We propose the power of the effect of the minimum detectable outlier on parameters as an alternative external reliability measure, present a regularized external reliability, and demonstrate for the first time that the external reliability measure of Baarda’s type is not the best and can be significantly improved through regularization for inverse ill-posed problems. An ill-posed regression example is used to illustrate the regularized external reliability measure, which is shown to perform much better than the external reliability measure of Baarda’s type.

Establishment of local geodetic networks based on least-squares adjustments of GNSS baseline vectors

Establishment of local geodetic networks based on least-squares adjustments of GNSS baseline vectors

EVALUATION OF A GNSS/IMU/LIDAR-INTEGRATION FOR AIRBORNE LASER SCANNING USING RTKLIB PPK AND PPP GNSS SOLUTIONS

Abstract. Airborne laser scanning allows for efficient acquisition of accurate 3D data for large areas. Because georeferencing of the LiDAR data requires knowledge of the platform trajectory, the laser scanner system commonly comprises a global navigation satellite system (GNSS) receiver/antenna and an inertial measurement unit (IMU). The standard processing pipeline consists of GNSS/IMU integration, georeferencing, and subsequent adjustment of the laser data. Here, we consider a holistic GNSS/IMU/LiDAR-integration approach based on least-squares adjustment. The GNSS is loosely coupled, and the GNSS positions are obtained using either postprocessing kinematic or precise point positioning GNSS processing strategies using the open-source software RTKLib. In this contribution, we compare the resulting point clouds to those of a standard processing workflow and evaluate the impact of the different processing strategies on point cloud quality in terms of internal consistency and absolute accuracy for a airborne laser bathymetry (ALB) dataset. Although the GNSS solutions themselves differ strongly, both the PPK- and the PPP-derived point clouds show better strip differences (below 2.5 cm) and similar absolute accuracy (<4 cm RMSE w.r.t. reference targets after correction of constant datum shift) compared to the reference solution.

A Method for the Precise Coordinate Determination of an Inaccessible Location.

Surveyors are occasionally tasked to with determining the coordinates of inaccessible locations or points in civil engineering applications, ground control points for photogrammetry or LiDAR data acquisition, among others. The present work outlines and investigates a novel method for estimating the GNSS coordinates of an inaccessible location where a surveying instrument cannot be set up. The procedure is based on the well-known surveying intersection method and data extracted from an Earth Gravity Model (e.g., EGM 2008). The location's coordinates are obtained from the least-squares adjustment of the angles and distances measured from at least two sites to the unknown point using a total station, within the framework of the Gauss-Helmert method. Field tests confirmed that the accuracy of the determined coordinates of the inaccessible point is at the level of 1 cm. The proposed method bypasses standard coordinate transformation steps performed with the traditional approach, directly producing geocentric coordinates of the unknown points.

Determination of Indonesia’s Marine Deflection of Vertical from Multi-Mission Altimetry Satellite Data

The deflection of vertical (DOV) is important for studying the changes in the Earth’s mass, improving the accuracy of geoid models, and the accuracy of the Earth’s gravity field. Compared to traditional methods such as astrogeodesy and astrogravimetry measurements, altimetry satellites have the advantage of observing DOV due to uniform global distribution, denser measurement points, consistent precision, less time required, high temporal resolution, and can be used in all weather conditions. In this study, multi-mission altimetry satellites including TOPEX/Poseidon, Jason-1, Jason-2, Jason-3, ENVISAT, Geosat Follow On (GFO), Sentinel-3A, Sentinel-3B, Sentinel-6A, Cryosat-2, and SARAL were used to determine DOV in Indonesia’s marine with a spatial resolution of 1’ x 1’. The Hwang et al. (1998) method is used to calculate DOV from satellite altimetry data by least-squares adjustment with weights using the second-order inverse distance weighting (IDW) method for each grid. The results show that the variance of the north-south and west-east components resulting from the gridding results are 0.05 arcseconds and 1.2 arcseconds. The evaluation results with the global model from EGM2008 show that the standard deviation values for the north-south and west-east components are 0.63 arcseconds and 1.96 arcseconds, respectively. From the evaluation, it can be concluded that the gridded DOV model is close to EGM2008, but there are still challenges for improvement, especially for the west-east DOV component.

Calibration of photoelastic materials with the disc under diametral compression

Abstract The disc under diametral compression is one of the preferred specimens for the calibration of photoelastic materials. Current calibration methods have some limitations from a metrological point of view, because they do not provide measurement uncertainties and they assume that observations are independent and with the same accuracy. In this work, a new calibration method is proposed to avoid the above-mentioned metrological limitations: The generalized least squares by Lagrange multipliers method. This methodology accepts correlated input quantities or with different accuracy through the input covariance matrix, and it provides, through a least-squares adjustment, the estimates of the quantities to be measured and their associated standard uncertainties. The application of the method shows coherent results and it also provides a chi-square value that can be used to test the consistency of the measurement model, and the normalized deviations between the input estimates and their fitted values, which are a tool to identify potential outliers. Results reveal a great influence of the radius of the specimen on the uncertainty of the stress-optic coefficient measurement, which casts serious doubt on previous affirmations about the disc in compression is preferable over other specimens.

Implementation of Stochastic Modelling in Enhanced Cadastral Databased for Multi-Classes Datasets

Stochastic Modelling (SM) was a crucial component of least squares adjustment (LSA), particularly when processing data from geodetic networks. The projected variances which generate using SM execute an important part in defining both the accurateness of the computed parameter vectors and the impact of the adjustment outcomes. As positional precision becomes the primary objective, there is still potential for improvement because there are multiple sources of datasets with varying levels of data quality. Concerning the assertion that the National Digital Cadastral Database (NDCDB) is accurate, its development involved the use of historical datasets that were obtained from a number of different measurement classes, specifically the first, second, and third classes. In this study, researchers evaluated whether or not it is possible to employ stochastic modelling to maintain the position correctness of historical data that encompasses a wide range of data quality classes. In order to accomplish this, an approach known as an Least Squares Variance Component Estimator (LS-VCE) was utilised to generate reliable estimates of variances. Two (2) certified plans (CPs) that is CP93887 and CP33758 was selected as measurements for the first and second classes CP, respectively. The experiment showed that the variance that has been estimated by LS-VCE could produce realistic adjustment results, as shown by an analysis of the corrected results obtained by allocating the variance into different data classes. In light of these findings, the investigations showed and demonstrated conclusively that separate variance is necessary for each data classes with the aim of preserving positional accuracy. In conclusion, it is crucial to incorporate a realistic variance component inside a coordinated cadastral database in order to fulfil the objective of ensuring the accurateness of survey data for future time periods.