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This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.

As a precursor study for the upcoming combined Earth Gravitational Model 2020 (EGM2020), the Experimental Gravity Field Model XGM2016, parameterized as a spherical harmonic series up to degree and order 719, is computed. XGM2016 shares the same combination methodology as its predecessor model GOCO05c (Fecher et al. in Surv Geophys 38(3): 571–590, 2017. doi:10.1007/s10712-016-9406-y). The main difference between these models is that XGM2016 is supported by an improved terrestrial data set of 15′×15′ gravity anomaly area-means provided by the United States National Geospatial-Intelligence Agency (NGA), resulting in significant upgrades compared to existing combined gravity field models, especially in continental areas such as South America, Africa, parts of Asia, and Antarctica. A combination strategy of relative regional weighting provides for improved performance in near-coastal ocean regions, including regions where the altimetric data are mostly unchanged from previous models. Comparing cumulative height anomalies, from both EGM2008 and XGM2016 at degree/order 719, yields differences of 26 cm in Africa and 40 cm in South America. These differences result from including additional information of satellite data, as well as from the improved ground data in these regions. XGM2016 also yields a smoother Mean Dynamic Topography with significantly reduced artifacts, which indicates an improved modeling of the ocean areas.

GOCO05c is a gravity field model computed as a combined solution of a satellite-only model and a global data set of gravity anomalies. It is resolved up to degree and order 720. It is the first model applying regionally varying weighting. Since this causes strong correlations among all gravity field parameters, the resulting full normal equation system with a size of 2 TB had to be solved rigorously by applying high-performance computing. GOCO05c is the first combined gravity field model independent of EGM2008 that contains GOCE data of the whole mission period. The performance of GOCO05c is externally validated by GNSS–levelling comparisons, orbit tests, and computation of the mean dynamic topography, achieving at least the quality of existing high-resolution models. Results show that the additional GOCE information is highly beneficial in insufficiently observed areas, and that due to the weighting scheme of individual data the spectral and spatial consistency of the model is significantly improved. Due to usage of fill-in data in specific regions, the model cannot be used for physical interpretations in these regions.

Despite the high complexity of the real world, linear regression still plays an important role in estimating parameters to model a physical relationship between at least two variables. The precision of the estimated parameters, which can usually be considered as an indicator of the solution quality, is conventionally obtained from the inverse of the normal equations matrix for which intensive computation is required when the number of observations is large. In addition, the impacts of the distribution of the observations on parameter precision are rarely reported in the literature. In this paper, we propose a new methodology to model the distribution of observations for linear regression in order to predict the parameter precision prior to actual data collection and performing the regression. The precision analysis can be readily performed given a hypothesized data distribution. The methodology has been verified with several simulated and real datasets. The results show that the empirical and model-predicted precisions match very well, with discrepancies of up to 6% and 3.4% for simulated and real datasets, respectively. Simulations demonstrate that these differences are simply due to finite sample size. In addition, simulation also demonstrates the relative insensitivity of the method to noise in the independent regression variables that causes deviations from the data distribution function. The proposed methodology allows straightforward prediction of the parameter precision based on the distribution of the observations related to their numerical limits and geometry, which greatly simplify design procedures for various experimental setups commonly involved in geodetic surveying such as LiDAR data collection.

High-precision Global Navigation Satellite Systems (GNSS) orbits are critical for real-time clock estimation and precise positioning service; however, the prediction error grows gradually with the increasing prediction session. In this study, we present a new efficient precise orbit determination (POD) strategy referred to as the epoch-parallel processing to reduce the orbit update latency, in which a 24-h processing job is split into several sub-sessions that are processed in parallel and then stacked to solve and recover parameters subsequently. With a delicate handling of parameters crossing different sub-sessions, such as ambiguities, the method is rigorously equivalent to the one-session batch solution, but is much more efficient, halving the time-consuming roughly. Together with paralleling other procedures such as orbit integration and using open multi-processing (openMP), the multi-GNSS POD of 120 satellites using 90 stations can be fulfilled within 30 min. The lower update latency enables users to access orbits closer to the estimation part, that is, 30–60-min prediction with a 30-min update latency, which significantly improves the orbit quality. Compared to the hourly updated orbit, the averaged 1D RMS values of predicted orbit in terms of overlap for GPS, GLONASS, Galileo, and BDS MEO are improved by 39%, 35%, 41%, and 37%, respectively, and that of BDS GEO and IGSO satellites is improved by 47%. We also demonstrate that the boundary discontinuities of half-hourly orbit are within 2 cm for the GPS, GLONASS, and Galileo satellites, and for BDS the values are 2.6, 15.5, and 9.8 cm for MEO, GEO, and IGSO satellites, respectively. This method can also be implemented for any batch-based GNSS processing to improve the efficiency.

A standard errors-in-variables (EIV) model refers to a Gauss–Markov model with an uncertain model matrix from a geodetic perspective. Least squares within the EIV model is usually called the total least squares (TLS) technique because of its symmetrical adjustment. However, the solutions and computational advantages of the weighted TLS problem with a general weight matrix (WTLS) are mostly unknown. In this study, the WTLS problem was solved using three different approaches: iterative methods based on the normal equation, the iteratively linearized Gauss–Helmert model with algebraic Jacobian matrices, and numerical analysis. Furthermore, sufficient conditions for WTLS optimization were investigated systematically as proposed solutions yield only necessary conditions for optimality. A WTLS solution was considered to treat random parameters within the EIV model. Last, applications to test these novel algorithms are presented.

Currently, five new-generation BeiDou (BDS-3) experimental satellites are working in orbit and broadcast B1I, B3I, and other new signals. Precise satellite orbit determination of the BDS-3 is essential for the future global services of the BeiDou system. However, BDS-3 experimental satellites are mainly tracked by the international GNSS Monitoring and Assessment Service (iGMAS) network. Under the current constraints of the limited data sources and poor data quality of iGMAS, this study proposes an improved cycle-slip detection and repair algorithm, which is based on a polynomial prediction of ionospheric delays. The improved algorithm takes the correlation of ionospheric delays into consideration to accurately estimate and repair cycle slips in the iGMAS data. Moreover, two methods of BDS-3 experimental satellite orbit determination, namely, normal equation stacking (NES) and step-by-step (SS), are designed to strengthen orbit estimations and to make full use of the BeiDou observations in different tracking networks. In addition, a method to improve computational efficiency based on a matrix eigenvalue decomposition algorithm is derived in the NES. Then, one-year of BDS-3 experimental satellite precise orbit determinations were conducted based on iGMAS and Multi-GNSS Experiment (MGEX) networks. Furthermore, the orbit accuracies were analyzed from the discrepancy of overlapping arcs and satellite laser range (SLR) residuals. The results showed that the average three-dimensional root-mean-square error (3D RMS) of one-day overlapping arcs for BDS-3 experimental satellites (C31, C32, C33, and C34) acquired by NES and SS are 31.0, 36.0, 40.3, and 50.1 cm, and 34.6, 39.4, 43.4, and 55.5 cm, respectively; the RMS of SLR residuals are 55.1, 49.6, 61.5, and 70.9 cm and 60.5, 53.6, 65.8, and 73.9 cm, respectively. Finally, one month of observations were used in four schemes of BDS-3 experimental satellite orbit determination to further investigate the reliability and advantages of the improved methods. It was suggested that the scheme with improved cycle-slip detection and repair algorithm based on NES was optimal, which improved the accuracy of BDS-3 experimental satellite orbits by 34.07%, 41.05%, 72.29%, and 74.33%, respectively, compared with the widely-used strategy. Therefore, improved methods for the BDS-3 experimental satellites proposed in this study are very beneficial for the determination of new-generation BeiDou satellite precise orbits.

We investigate the impact of the reduction of non-tidal loading (NTL) in the computation of secular terrestrial reference frames (TRFs) from Very Long Baseline Interferometry (VLBI) observations. There are no conventional models for NTL in the geodetic community yet, but the Global Geophysical Fluid Center prepared a set of corresponding site displacements for the 2020 realizations of the International Terrestrial Reference System. We make use of these data, which comprise the total NTL consisting of non-tidal atmospheric, oceanic, and hydrological loading. The displacement series contain linear trends (i.e., offsets plus drifts), and since these affect the estimated linear station positions and the realized geodetic datum in a secular TRF, we remove the trends before reducing the NTL in our computations. The displacements are applied at two different levels of the parameter estimation process: the observation and the normal equation level. This way, we can analyze whether the latter offers a suitable approximation if the original observations have not been reduced by NTL. We find that the TRF statistics are hardly affected by the NTL. The largest impact is given for the secular motion of antennas with short observation time spans. The application level is basically irrelevant for the linear antenna positions, but it leads to differences in the rates of the jointly estimated Earth orientation parameters (EOPs). Secular TRF solutions and session solutions deviate with respect to the parameterization of the antenna coordinates, and thus also with respect to the correlations between the estimated antenna parameters and the EOPs. Due to this, the consistently estimated EOP series also show differences. However, for both solution types the reduction of the NTL leads to a change of the annual signal in the EOP series.

Wire Cut Electrical Discharge Machining (WEDM) is a non-conventional thermal machining process which is capable of accurately machine alloys having high hardness or part having complex shapes that are very difficult to be machined by the conventional machining processes. The WEDM finds applications in automobiles, aero–space, medical instruments, tool and die industries, etc. The input parameters considered for WEDM are pulse on time, pulse off time, flushing pressure, servo voltage, wire feed rate and wire tension. Performance of WEDM is mainly assessed by output variables such as, material removal rate (MRR), kerf width (Kw) and surface roughness (Ra) of the work piece being machined. Looking at the need of a suitable optimization model, the present work explores the feasibility of machine learning concepts to predict optimum surface roughness and kerf width simultaneously by making use of experimental data available in the literature for machining of Hastelloy C– 276 using WEDM. In most of the literatures, single objective optimization has been carried out for predicting optimum cutting parameters for WEDM. Hence, the present work presents a methodology that makes use of a machine learning algorithm namely, gradient descent method as an optimization technique to optimize both surface roughness and kerf width simultaneously (multi objective optimization) and compare the results with the existing literatures. It was observed that the input parameters such as pulse on time, pulse off time, and peak current have significant effect on both surface roughness and kerf width. The gradient descent method was successfully used for predicting the optimum values of response variables.

Geological modeling is generally based on deterministic models, which provide a single representation of reality. Probabilistic modeling is more appropriate when quantifying or understanding the uncertainty associated with a parameter of interest as it considers several equally probable geological scenarios. The object of this study is to quantify the uncertainty in the estimation of the minerals in the Punta Alegre gypsum deposit, by applying a new method based on the simple normal equation geostatistical simulation technique. The Punta Alegre gypsum deposit is a sedimentary deposit of clastic origin, formed by the complex redeposition of salts, gypsum and other sediments. To carry out this research, 50 equiprobable scenarios were simulated, reproducing overburden, gypsum series (different types of gypsum) and intercalated non-mineral lithologies (limestone and other rocks) in a network of nodes measuring 5 × 5 × 5 m, using a training image, composites and prior probability maps as input data. As a result of scaling the previously simulated geological units, three-dimensional models of volume proportions and estimation error for gypsum were obtained for panels measuring 10 × 10 × 5 m. The quantification of the uncertainty of the gypsum volume, determined by the root mean square error, established that the volume estimation error is small at a global scale (6.51%), given that there is no significant variation when comparing the deterministic model with the gypsum proportion model obtained from the 50 simulated scenarios. Conversely, at the local scale, there is a significant variation in gypsum volume of 42% in the 10 × 10 × 5 m panels with a future impact on recoverable mining resources, given the uncertainty at a local scale, which will cause an increase in mining dilution due to the inclusion of non-mineral lithologies within the extracted mineral that will be sent to the processing plant. On the other hand, it will cause changes in the mining company’s plan in areas where there are panels that were previously accounted for by the deterministic model as minerals and are not actually exploitable.