A New Method for TSVD Regularization Truncated Parameter Selection

The truncated singular value decomposition (TSVD) regularization applied in ill-posed problem is studied. Through mathematical analysis, a new method for truncated parameter selection which is applied in TSVD regularization is proposed. In the new method, all the local optimal truncated parameters are selected first by taking into account the interval estimation of the observation noises; then the optimal truncated parameter is selected from the local optimal ones. While comparing the new method with the traditional generalized cross-validation (GCV) and L curve methods, a random ill-posed matrices simulation approach is developed in order to make the comparison as statistically meaningful as possible. Simulation experiments have shown that the solutions applied with the new method have the smallest mean square errors, and the computational cost of the new algorithm is the least

Research Progress in Mathematical Analysis of Map Projection by Computer Algebra

Map projection is an important component of modern cartography, and involves many fussy mathematical analysis processes, such as the power series expansions of elliptical functions, differential of complex and implicit functions, elliptical integral and the operation of complex numbers. The derivation of these problems by hand not only consumes much time and energy but also makes mistake easily, and sometimes can not be realized at all because of the impossible complexity. The research achievements in mathematical analysis of map projection by computer algebra are systematically reviewed in five aspects, i.e., the symbolic expressions of forward and inverse solution of ellipsoidal latitudes, the direct transformations between map projections with different distortion properties, expressions of Gauss projection by complex function, mathematical analysis of oblique Mercator projection, polar chart projection with its transformation. Main problems that need to be further solved in this research field are analyzed. It will be helpful to promote the development of map projection

Robust Total Least Squares Estimation of Space Intersection Appropriate for Multi-images

In order to take full advantage of available observation resources, based on theory of space intersection with stereo images, by conducting weighted quadratic sum of spatial distance from the target point to multiple space lines as the objective function and carrying out its first as well as second derivatives, robust total least squares estimation of space intersection appropriate for multi-images was realized. Compared to stereopair, more observed information and theories of robust estimation were considered in the process of space intersection with multi-images, bringing about higher intersection accuracy and robustness. Finally, correctness and robustness of the method was verified though example analysis, which can enrich the space intersection theory in photogrammetry to some degree

RSPCN: Super-Resolution of Digital Elevation Model Based on Recursive Sub-Pixel Convolutional Neural Networks

The digital elevation model (DEM) is known as one kind of the most significant fundamental geographical data models. The theory, method and application of DEM are hot research issues in geography, especially in geomorphology, hydrology, soil and other related fields. In this paper, we improve the efficient sub-pixel convolutional neural networks (ESPCN) and propose recursive sub-pixel convolutional neural networks (RSPCN) to generate higher-resolution DEMs (HRDEMs) from low-resolution DEMs (LRDEMs). Firstly, the structure of RSPCN is described in detail based on recursion theory. This paper explores the effects of different training datasets, with the self-adaptive learning rate Adam algorithm optimizing the model. Furthermore, the adding-“zero” boundary method is introduced into the RSPCN algorithm as a data preprocessing method, which improves the RSPCN method’s accuracy and convergence. Extensive experiments are conducted to train the method till optimality. Finally, comparisons are made with other traditional interpolation methods, such as bicubic, nearest-neighbor and bilinear methods. The results show that our method has obvious improvements in both accuracy and robustness and further illustrate the feasibility of deep learning methods in the DEM data processing area

Influence of Sea Level Anomaly on Underwater Gravity Gradient Measurements

Considering the theoretical research needs of gravity gradient detection and navigation, this study uses the right rectangular prism method to calculate the disturbing gravity gradient from sea level anomalies in the range of 5° × 5° in the Kuroshio extension area of the western Pacific with large sea level anomalies. The disturbing gravity gradient is obtained in different directions within a depth of 50 m below the mean sea level based on the principle of the disturbing gravity gradient. The calculation results show that the sea level anomalies at local positions significantly impact the underwater gravity gradient measurements, with the maximum contribution exceeding 10 E and the maximum difference between different locations exceeding 20 E. The change of the sea level anomaly over time also significantly impacts the measurement of the underwater gravity gradient, with the maximum change value exceeding 20 E. The impact will have a corresponding change with the seasonal change of the sea level anomaly. Therefore, the underwater carrier needs to consider the disturbing gravity gradient caused by sea level anomalies when using the gravity gradient for underwater detection and navigation

Statistically optimal estimation of surface mass anomalies by directly using GRACE level-2 spherical harmonic coefficients as measurements

Point-mass inversion is widely employed in GRACE level-2 data processing. Conventionally, the spherical harmonic (SH) coefficients are used indirectly: a set of pseudo measurements is generated first using the SH coefficients through SH synthesis; then the point-mass inversion is done with these pseudo measurements. To be statistically optimal, the covariance matrix of pseudo measurements should be calculated and used to appropriately weigh the parameter estimation. In this work, we propose a statistically optimal point-mass inversion scheme by directly using the SH coefficients as measurements. We prove the equivalence between this direct approach and the conventional indirect approaches. We also demonstrated their comparable performance through both simulation and real GRACE data processing. Choosing and calculating pseudo measurements, propagating covariance matrix, and potentially dealing with the singularity of the covariance matrix involved in the conventional indirect approaches are avoided in the proposed direct approach. This statistically optimal direct approach can readily be employed in mascon inversion of GRACE data and other radial basis functions-based approaches in regional gravity modeling.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Physical and Space Geodes

An Alteration of Gauss Projection Based on Oblique Deformed Ellipsoid

For east-west spanning line, to reduce abscissa value of Gauss projection, the oblique reference ellipsoid was constructed by means of least square method. Via theory of coordinate system transformation, spatial rectangular coordinates of target region in each coordinate system were carried out, and then geodetic coordinates of target region on oblique reference ellipsoid were relatively given. Through ellipsoid transformation, oblique deformed ellipsoid was established to lessen distortion of projection caused by height. Taking one railway for example, it were shown that "An alteration of Gauss projection based on oblique deformed ellipsoid" could greatly deplete abscissa components, avoid zoning of Gauss projection and reduce height effectively, as well as the relevant distortion it caused. Strict mathematical model and clear operation process of the Gauss projection are convenient for programming of relative software, which can be applied in engineering

Symbolic expressions of differences between earth radius

A systematic and comprehensive comparison of the five commonly used earth radius in geodesy and cartography is carried out, and the differences between the most common points of the earth's radius, their corresponding maximum values, and the latitudes of equal points between themare derived with the help of computer algebraic systems. The symbolic expressionsare expressed as a power series of the first eccentricity. Taking the CGCS2000 ellipsoid as an example, the differences between the commonly used earth radii are clarified to numerical values. The results show that the difference between the commonly used Earth radii has a maximum at 90 degrees and a minimum at 0 degrees. The difference between the average radius of curvature and the equidistant sphere radius is the biggest, and the difference between the average radius of curvature and the average sphere radius is the smallest. These results can provide theoretical basis for relative research in earth science, space science, navigation and positioning

Forward and Inverse Expressions of Polar Gauss Projection without Zoning Limitations

As traditional formulae of Gauss projection could not be used in polar regions, strict equation of complex conformal colatitude was derived with relationship between conformal colatitude and isometric latitude introduced, and then strict forward expressions of Gauss projection suit for polar regions were carried out. Based on relationship between exponential and trigonometric functions, inverse expressions of polar Gauss projection were derived by means of symbol iteration method. With reference to the forward expressions, corresponding equations of length ratio and meridian convergence for polar Gauss projection were achieved. Finally, Taking CGCS2000 ellipsoid for example, by comparing with results calculated by formulae of Gauss projection in power series forms, correctness of the proposed expressions was verified. Expressions in this paper are all free from bandwidth, and can be used in the entire poles, which could provide important references for polar mapping and navigation

Isostatic Anomaly and Isostatic Additional Force Analysis by Multiple Geodetic Observations in Qinling Area

Determination of the isostatic anomaly and the isostatic additional force plays a key role in understanding the deep tectonic features and dynamics in the Qinling area. At present, high-accuracy observation gravity data are one of the important means to obtain the isostatic anomaly and the isostatic additional force. Firstly, we calculate the free-air gravity anomalies and the Bouguer gravity anomalies by using hybrid gravity and GPS observation data. Then, we invert the isostatic anomaly and the isostatic additional force. The results show that the isostatic depth calculated by Airy isostatic theory is 40–49 km, and the Moho depth is 39–48 km. The Weihe Basin is in a non-isostatic state with an upward isostatic additional force that reached about 20 MPa. The isostatic anomaly and the isostatic additional force are approximately zero in the northern Sichuan Basin, which indicates that the crust is in isostatic state. The negative isostatic anomaly and isostatic additional force in Liupanshan Mountains, the southwest margin of the Ordos Basin, and the local areas of the Qinling Orogen and Dabashan indicate the existence of crustal movement. By combining the measurement of InSAR, we obtain the surface deformation information of the Weihe Basin, as well as an upward trend, which proves that the result is highly consistent with the gravity observation