A minimum curvature algorithm for tomographic reconstruction of atmospheric chemicals based on optical remote sensing

Optical remote sensing (ORS) combined with the computerized tomography (CT) technique is a powerful tool to retrieve a two-dimensional concentration map over an area under investigation. Whereas medical CT usually uses a beam number of hundreds of thousands, ORS-CT usually uses a beam number of dozens, thus severely limiting the spatial resolution and the quality of the reconstructed map. The smoothness a priori information is, therefore, crucial for ORS-CT. Algorithms that produce smooth reconstructions include smooth basis function minimization, grid translation and multiple grid (GT-MG), and low third derivative (LTD), among which the LTD algorithm is promising because of the fast speed. However, its theoretical basis must be clarified to better understand the characteristics of its smoothness constraints. Moreover, the computational efficiency and reconstruction quality need to be improved for practical applications. This paper first treated the LTD algorithm as a special case of the Tikhonov regularization that uses the approximation of the third-order derivative as the regularization term. Then, to seek more flexible smoothness constraints, we successfully incorporated the smoothness seminorm used in variational interpolation theory into the reconstruction problem. Thus, the smoothing effects can be well understood according to the close relationship between the variational approach and the spline functions. Furthermore, other algorithms can be formulated by using different seminorms. On the basis of this idea, we propose a new minimum curvature (MC) algorithm by using a seminorm approximating the sum of the squares of the curvature, which reduces the number of linear equations to half that in the LTD algorithm. The MC algorithm was compared with the non-negative least square (NNLS), GT-MG, and LTD algorithms by using multiple test maps. The MC algorithm, compared with the LTD algorithm, shows similar performance in terms of reconstruction quality but requires only approximately 65 % the computation time. It is also simpler to implement than the GT-MG algorithm because it directly uses high-resolution grids during the reconstruction process. Compared with the traditional NNLS algorithm, it shows better performance in the following three aspects: (1) the nearness of reconstructed maps is improved by more than 50 %, (2) the peak location accuracy is improved by 1–2 m, and (3) the exposure error is improved by 2 to 5 times. Testing results indicated the effectiveness of the new algorithm according to the variational approach. More specific algorithms could be similarly further formulated and evaluated. This study promotes the practical application of ORS-CT mapping of atmospheric chemicals.

Mean curvature effect on the response and failure of round-hole tubes submitted to cyclic bending

This paper presents an experiment and analysis to investigate the response and failure of 6061-T6 aluminum alloy round-hole tubes with different hole diameters of 2, 4, 6, 8, and 10 mm subjected to cyclic bending at different curvature ratios of −1.0, −0.5, 0.0, and +0.5. The curvature ratio is defined as the minimum curvature divides by the maximum curvature. Four different curvature ratios are employed to highlight the mean curvature effect. It can be seen from the experimental results that the moment-curvature relationships gradually relax and become steady states after a few bending cycles for curvature ratios of −0.5, 0.0, and +0.5. The ovalization-curvature relationship depicts an asymmetrical, ratchetting and increasing as the number of bending cycles increases for all curvature ratios. In addition, for each hole diameter, the relationships between the curvature range and the number of bending cycles necessary to initiate failure on double logarithmic coordinates display four almost-parallel straight lines for four different curvature ratios. Finally, this paper introduces an empirical formula to simulate the above relationships. By comparing with experimental results, the analysis can reasonably describe the experimental results.

Методика анализа объемных дефектов по цифровой модели рельефа поверхности

We developed a technique for revealing and analyzing volumetric surface defects based on geomorphometric modeling, in particular, an analysis of models and maps of some morphometric variables (minimum curvature, maximum curvature, mean curvature, Gaussian curvature, unsphericity, etc.), derived from digital elevation models of a surface. The technique allows one to reveal areas of individual volume defects (cracks, film delaminations, shape deviations, etc.), to determine shape and size of both the defects themselves and adjucent modified areas, as well as to study patterns of their distribution. The technique effectiveness is exempified by defects on silicon–glass and silicon–silicon wafer assemblies, as well as a cracked Ni–W film. The technique can be promising for quality control of manufacturing and diagnostics of damages of various items, in particular, microelectronic products.

Mechanical and Geometrical Tortuosities: Vanishing and Appearing Tortuosities

ABSTRACT Tortuosity is one of the critical factors to be considered for complex directional well trajectories, complicated build rates, precise steering in thin reservoirs, and extended reach wells. This paper discusses the pitfalls of estimating tortuosity to quantify borehole quality and answers questions, such as whether the claimed benefits (i.e., enhanced drilling performance, improved hole cleaning, ease of running casing, and superior cement operations) can be fully attributed to reduced borehole tortuosity. Running casing may mask the tortuosity present in the as drilled open hole wellbore section. This vanishing tortuosity alters the apparent "wellbore quality" and the new tortuosity representative of the cased hole path may present new appearing tortuosity. Both vanishing and appearing tortuosity are generally neglected in engineering calculations. Conventional methods to calculate tortuosity are based on the predetermined shape of the trajectory using the minimum curvature method. Wellbore undulation (geometrical tortuosity) is determined using geometrical measurements such as inclination, azimuth, and calculated displacement; however, much of this wellbore undulation vanishes after the casing is run, and thus the cased off wellpath appears smoother. This apparent change in wellbore tortuosity results from the flexural stiffness and rigidity of the casing pipes, and the compression and tension loads along the length of the casing string. Acquiring a subsequent survey along the cased well path yields new inclinations, azimuths, and displacements. This new survey records wellpath undulations resulting from the casings path through the original open hole wellbore geometry and what we call tubular undulation (mechanical tortuosity) which is specific to the path and position of the casing within the wellbore. The smoothing of the wellpath resulting from the casing masking original wellbore tortuosity results in the original geometrical tortuosity vanishing while the new undulations resulting from the mechanical tortuosity of the casing causes additional tortuosity to appear. The comparison between the geometrical and mechanical tortuosity provides a method of quantifying the vanishing and appearing tortuosity.

A minimum curvature algorithm for tomographic reconstruction of atmospheric chemicals based on optical remote sensing

Optical remote sensing (ORS) combined with computerized tomography (CT) technique is a powerful tool to retrieve a two-dimensional concentration map over the area under investigation. But unlike the medical CT, the beam number used in ORS-CT is usually dozens comparing to up to hundreds of thousands in the former, which severely limits the spatial resolution and the quality of the reconstructed map. This situation makes the “smoothness” a priori information especially necessary for ORS-CT. Algorithms which produce smooth reconstructions include smooth basis function minimization (SBFM), grid translation and multiple grid (GT-MG), and low third derivative (LTD), among which the LTD algorithm is a promising one with fast speed and simple realization. But its characteristics and the theory basis are not clear. Moreover, the computation efficiency and the reconstruction quality need to be improved for practical applications. This paper employs two theories, i.e., Tikhonov regularization and spatial interpolation, to produce a smooth reconstruction by ORS-CT. Within the two theories’ frameworks, new algorithms can be explored in order to improve the performance. For example, we propose a new minimum curvature (MC) algorithm based on the variational approach in the theory of the spatial interpolation, which reduces the number of linear equations by half comparing to that in the LTD algorithm using the biharmonic equation instead of the smoothness seminorm. We compared our MC algorithm with the non-negative least square (NNLS), GT-MG, and LTD algorithms using multiple test maps. The MC and the LTD algorithms have similar performance on the reconstruction quality. But the MC algorithm needs only about 65 % computation time of the LTD algorithm. It is much simpler in realization than the GT-MG algorithm by using high-resolution grids directly during the reconstruction process to generate a high-resolution map immediately after one reconstruction process is done. Comparing to the traditional NNLS algorithm, it shows better performance in three aspects: (1) the nearness of reconstructed maps is improved by more than 50 %; (2) the peak location accuracy is improved by 1–2 m; and (3) the exposure error is improved by more than 10 times. The testing results show the effectiveness of the new algorithm based on the spatial interpolation theory. Similarly, other algorithms may also be formulated to address problems such as the over-smooth issue in order to further improve the reconstruction equality. The studies will promote the practical application of the ORS-CT mapping of atmospheric chemicals.

A Compelling Case: Time to Change Minimum Curvature Survey Method for Well Engineering Calculations

Several calculation methods have been proposed by the researchers in the past with varying degree of precision. These result in different wellbore position estimation. The survey calculations and positions are used in engineering calculations such as torque and drag, drill ahead prediction, displacement of the string, SAG correction etc. Since the survey calculations are the underpinning starting point for the engineering formulations, considerable errors are introduced in the engineering calculations. Most widely used method in the industry is minimum curvature which assumes a circular arc in an inclined plane and this method is better than some of the previously introduced methods like balanced tangential method, average angle method etc. It has also been seen that, the radius of curvature method is more suitable for a rotary mode of drilling, whereas the minimum curvature method aligns when slide drilling. However, using the minimum curvature method, introduces discontinuity at the survey course intervals. This type of constant curvature model due to discontinuity may result in the omission of some contact forces which will result in the under prediction of the hookload or the stresses in the drillstring components or casing or drillpipe wear calculations. This further results in the discontinuity in the engineering calculations also and will be more pronounced when the wellbore twists and turns. To alleviate this problem, wellbore smoothening is done by using advanced survey methods such as spline, natural arc method, curvature bridging etc. A reliable inclination and directional continuous surveys also enable smooth and continuous wellpath making the problem with the minimum curvature method to vanish. This paper presents the research findings not only on the relationship between the minimum curvature method and engineering calculations but also the inherent problem in the commonly used survey calculation methods. It has been shown that both wellbore curvature as well as the borehole torsion are necessary to relate to the mechanical parameters that are used in the engineering calculations such as bending moment, side force and string position.

A Generalised Solution to the Point to Target Problem Using the Minimum Curvature Method

An explicit solution to the general 3D point to target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multi-valued and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact, or polynomial type solution methods to be employed. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point to target problem can be represented as a 10th order self-intersecting geometric surface, characterised by the trajectory's start and end points, the radii of the two arcs and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided and together these indicate the most convenient solution method for each case. In the presence of a tangent section the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point to target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.

A Generalized Solution to the Point-to-Target Problem Using the Minimum Curvature Method

Summary An explicit solution to the general 3D point-to-target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multivalued, and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact or polynomial-type solution methods to be used. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point-to-target problem can be represented as a 10th-orderself-intersecting geometric surface, characterized by the trajectory's start and end points, the radii of the two arcs, and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided, and together these indicate the most convenient solution method for each case. In the presence of a tangent section, the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point-to-target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.

Gradient Perturbation is Underrated for Differentially Private Convex Optimization

Gradient perturbation, widely used for differentially private optimization, injects noise at every iterative update to guarantee differential privacy. Previous work first determines the noise level that can satisfy the privacy requirement and then analyzes the utility of noisy gradient updates as in the non-private case. In contrast, we explore how the privacy noise affects the optimization property. We show that for differentially private convex optimization, the utility guarantee of differentially private (stochastic) gradient descent is determined by an expected curvature rather than the minimum curvature. The expected curvature, which represents the average curvature over the optimization path, is usually much larger than the minimum curvature. By using the expected curvature, we show that gradient perturbation can achieve a significantly improved utility guarantee that can theoretically justify the advantage of gradient perturbation over other perturbation methods. Finally, our extensive experiments suggest that gradient perturbation with the advanced composition method indeed outperforms other perturbation approaches by a large margin, matching our theoretical findings.

Improvements of 6S Look-Up-Table Based Surface Reflectance Employing Minimum Curvature Surface Method

We propose a methodology employing an interpolation technique on the Second Simulation of a Satellite Signal (6S) look-up table (LUT) to improve surface reflectance retrieval using Himawari-8/Advanced Himawari Imager (AHI). A minimum curvature surface (MCS) technique was used to refine the 6S LUT, and the solar zenith angle (SZA) and viewing zenith angle (VZA) increments were narrowed by 0.5°. The interpolation processing time was relatively short, about 3172 s per channel, and the interpolated xa and xb were well represented by the changes in SZA and VZA. An evaluation of the interpolated xa and xb for six cases revealed a relative mean absolute error of less than 5% for all channels and cases; however, a slight difference was evident for higher values of SZA and VZA. To evaluate the surface reflectance, we compared the surface reflectance derived using 6S LUT with that calculated using 6S only. Application of the interpolated 6S LUT showed a lower relative root mean square error (RRMSE) of 0.65% to 9.29% for all channels, than before interpolation. The improvement in surface reflectance measurements increased with the SZA. For a SZA above 75°, the RRMSE improved significantly for all channels (by 11.33–45.1%). In addition, when the MCS method was applied, the surface reflectance measurements improved without spatial discontinuity and showed good agreement with 6S results in a linear profile analyses. Thus, the method proposed can improve LUT based surface reflectance measurements in less time and increase the availability of surface reflectance data based on geostationary satellites.