Height Error Research Articles

Bead morphology prediction of coaxial laser cladding on inclined substrate using machine learning

Laser cladding technology has been widely used in the repair and surface strengthening of huge parts. In order to avoid interference between the laser head and the surface to be deposited, the laser cladding process needs to be completed in a tilted position in order to remanufacture some complex parts. The morphology of the cladding layer directly affects the overall quality of the surface after laser cladding. To solve the problem that the quality of the coating cannot be guaranteed under an inclined attitude, SVR, PSO-BPNN, and XGBoost models were applied to build a prediction model between the bead morphology and process parameters. An L36 orthogonal array with four factors and three levels was used to obtain the original data set, and the four-fold cross-validation method was used to train and validate the three models. The predictive performance showed that the XGBoost model had better performance in predicting the morphology of the cladding layer on the inclined substrate compared to the other two algorithms. The R2 values of the width, height, and peak shifting point prediction model using XGBoost were 0.9578, 0.9419, and 0.8765 respectively. Finally, the prediction accuracy of the XGBoost model was evaluated by using other 9 test data sets. The maximum error in width was 3.73 %, and the errors in height and peak shifting point were less than 10 %. The prediction model also performed well in terms of MAE, RMSE, and R2 by using other nine test data sets, further demonstrating the generalization capability of XGBoost. This prediction model can be used to optimize the process path of la-ser cladding with a tilted position and effectively improve the processing efficiency and quality of complex surface laser cladding.

Hybrid modelling to improve operational wave forecasts by combining process-based and machine learning models

Operational wave forecasting plays an important role in ensuring safe navigation and in the prediction of tidal windows for harbour approach channels. The underlying nearshore process-based wave models need to be accurate for a wide range of different conditions, from more common mild wave conditions to the occasional high energy (storm) conditions. In this work, an innovative hybrid modelling approach is proposed to improve the accuracy of operational wave forecasts. An operational wave model is combined with a machine learning model which is trained using wave measurements within the wave model domain. This hybrid modelling approach is applied to the Dutch North Sea, covering four major harbour approach channels.The final hybrid operational wave model results in a significant average error decrease compared to just the process-based model, amounting to 21.7% for the wave energy density and 25.3% for the wave direction. The error reduction for the spectral wave parameters is even larger, with a 33.3% smaller error in spectral wave height and a 38.8% smaller error in spectral wave period. As this approach is generically applicable to spectral wave models, it contains the potential for significant improvements in operational modelling.

Simulation of non-cohesive soil turning based on an SPH model

Soil turning is an infrastructure activity in the field of geotechnical engineering. In this study, we propose a new soil turning numerical model based on the Smoothed Particle Hydrodynamics (SPH) method, in which an elastoplastic constitutive model is implemented. To validate the model, firstly, a landslide benchmark test is studied, the error of the maximum sliding distance is only 1.5 % compared to the experimental data. Next, a combined experimental numerical investigation on the soil turning process is carried out. We compare the experimental observations with the numerical modeling results and find that the numerical error of the peak height and the horizontal distance of the soil is about 2.56 % and 3.37 %, respectively, indicating that the proposed SPH model can reproduce the soil turning process accurately. The simulation results also demonstrate that: (1) When the turning speed is increased, the turning resistance does not change significantly but leads to a more sufficient mixing of the soil; (2) There exists a critical turning angle above which the resistance increase significantly associates with a larger angle; (3) Increasing the burying depth of the tool helps to improve the mixing effect; however, leads to larger resistance as well.

Extraction of tree heights in mountainous natural forests from UAV leaf-on stereoscopic imagery based on approximation of ground surfaces

The application of high-resolution stereoscopic imagery acquired by Unmanned Aerial Vehicle (UAV) on the extraction of forest heights has grown rapidly in recent years. Most existing studies either required auxiliary terrain data, e.g., Digital Terrain Model (DTM) provided by lidar data, or focused on flat terrains. It is still a great challenge to extract tree heights in mountainous forests only using UAV leaf-on stereoscopic imagery. An algorithm referred to as AGAR (i.e., Approximation of Ground using Allometric Relationship) is proposed in this study to estimate individual heights of visible trees on UAV stereoscopic imagery in mountainous natural forests. The central idea of the AGAR algorithm is firstly to approximate the understory terrain elevations (i.e., DTM) based on attributes of tree crowns (e.g., crown area) and the iterative adjustment of allometric equation coefficients. Then individual tree heights are determined by differencing the elevation of crown tops with that of the approximated ground surface. The proposed algorithm was demonstrated at five sites with different terrain conditions by taking field measurements and ICESat-2 data as references, respectively. Results showed that the AGAR algorithm worked well on the estimation of tree heights at all sites. In contrast, the classical progressive triangulation filter (PTF) algorithm was susceptible to terrains and forest structures. The root mean square error (RMSE) and relative RMSE (rRMSE) of tree heights estimated by the PTF algorithm were 4.4 m ∼ 6.3 m and 32.6% ∼ 37.6%, respectively. They were decreased by the AGAR algorithm to 1.7 m ∼ 2.5 m and 12.6% ∼ 15.2%, respectively. The AGAR algorithm will substantially advance the application of UAV stereoscopic imagery on the extraction of tree heights in the absence of other available terrain data, and will also open new horizons for application of decimeter or even centimeter spaceborne stereoscopic imagery on forest vertical structures in the future.

Timeliness of Correcting Baseline Error in Wide-Swath Altimeter Based on Reference Topography Data

The baseline error is a primary error source of the wide-swath altimeter, directly related to the cross-track distance, and can lead to serious height errors at the swath’s outer edge. Cross-calibration using discrepancies with reference data can effectively estimate and correct the baseline error. However, building a reference surface that accurately describes the sea surface at the observation time is necessary to use this cross-correction method. The dynamic ocean environments where the sea surface structure changes over time are challenging. This paper proposes a method for constructing reference topography data (RTD) based on multi-source data products to correct the baseline error of the wide-swath altimeter. The effectiveness of the proposed method is evaluated using HYCOM ocean model data to assess the timeliness of the baseline error correction. The results demonstrate that using RTD at the observation time of the wide-swath altimeter can significantly correct the baseline error. The RMSE of the corrected sea surface height (SSH) in different regions is typically between 1~2 cm, except in some regions with strong currents where the RMSE is approximately 3~4 cm. However, the time interval between the RTD and the observation time of the wide-swath altimeter can affect the accuracy of the baseline error correction. The timeliness of this correction is influenced by the variability of SSH in different regions. In regions with relatively slow SSH changes near the equator, the effective time based on HYRTD and MORTD can basically reach more than 7 days. In regions where the SSH changes more rapidly, the correction result may no longer be reliable in only 1~3 days.

A posture-based measurement adjustment method for improving the accuracy of beef cattle body size measurement based on point cloud data

Beef cattle body size is an important phenotypic trait for breeding and production. The non-contact automatic calculation of beef cattle body sizes via point cloud data exhibits several disadvantages, such as the inaccurate positioning of measurement points and the sensitivity to postures, heavily restricting commercial applications. To address these issues, an automatic method for key region segmentation, body size calculation, and data calibration to accurately measure the body size parameters of beef cattle in different postures is proposed. Firstly, the body sizes were obtained based on the point cloud data. Then the twelve micro-posture feature parameters were defined and extracted from four categories corresponding to the head, back, torso, and leg regions. Based on those parameters, a Posture-based Measurement Adjustment model was established to reduce the influence of postures on calculation results based on the correlation between measurement errors and micro-posture features. The experimental results show that, after the adjustment model calibration, the average errors of body oblique length, wither height, hip height, chest girth, and abdominal girth were 1.14%, 1.84%, 3.47%, 1.56%, and 2.36%, respectively, whereas the corresponding maximum errors were 4.36%, 8.10%, 11.10%, 7.29%, and 9.26%. The measurement errors of body sizes can be significantly minimised by the adjustment model, which can improve the robustness of different postures. The adjustment model improves the accuracy of the non-contact body size measurement, which could provide guidance and inspirations for future research.

Creation and Assessment of a Topographic Map from Unmanned Aerial Vehicle Data in the Province Phu Tho, Thanh Son District

In recent years, there has been a rapid advancement in the use of unmanned aerial vehicles (UAVs) in various aspects of life, especially in the area of automated data collection. These advancements have brought about numerous possibilities. The article describes the use of an unmanned aerial vehicle, specifically the PHANTOM 4 pro, to gather remote sensing data and create digital topographic plans at a scale of 1:2000 in Phu Tho province's Thanh Son district. A method was suggested for improving the current systems for obtaining remote sensing data for cartography using UAVs. The ease of controlling the UAVs and the quality and timeliness of the data they transmit to control points confirm the value of using them to create topographic maps. In addition to UAVs, fieldwork also involves the use of the GNSS brand CHCNAV I50. The high precision GNSS system enables the camera's 3D position to be detected within a few centimeters at the time of each capture. A digital topographic map was compiled of Thanh Son district, covering 165 hectares, and processed using software such as Agisoft and Global Mapper. The digital topographic map that was produced satisfies the documentation requirements of government organizations. The maximum error in height is 4.7 cm, the error of coordinates north and east are 1.8 cm and 1.4 cm respectively. This was achieved by using 590 raster images, which had a resolution of 2.3 cm and a size of 5472x3648 pixels. Based on the findings, the map's accuracy is within an acceptable range of less than 2 cm, which is suitable for a map scale of 1:2000.

A Speculative Divide-and-Conquer Optimization Method for Large Analog/Mixed-Signal Circuits: A High-Speed FFE SST Transmitter Example

We propose a speculative divide-and-conquer (SDnC) method that enables optimization of a large analog/mixed-signal (AMS) circuit. Because modules of AMS circuits strongly interact with neighbor modules, they cannot be optimized individually. Therefore, design parameters of all modules must be co-optimized for the global optimization, and thus, the design space exponentially grows with the circuit size. Although metaheuristic algorithms can enhance optimization efficiency, they cannot handle very large circuits due to the exponentially increased size of design space to explore. The proposed method utilized the divide-and-conquer (DnC) strategy in the circuit optimization while taking into account modules’ interactions, allowing modules to be individually evaluated and optimized. Therefore, this DnC-based optimization method hierarchically and systematically reduces both the optimization complexity and the evaluation complexity, and thus, this method is very scalable with the circuit size. The proposed method can also be combined with other metaheuristic algorithms such as particle swarm optimization (PSO) or artificial intelligence for faster optimization. In experiment, the proposed method enabled optimization of a high-speed link transmitter that has 2400–15 000 transistors and 45–47 independent design parameters for the first time. The optimization time was improved by 43 and three orders of magnitudes compared to parameter sweep and PSO, respectively. The penalties of speed-up by DnC were only 3.4% and 9.3% estimation errors in power consumption and eye height, respectively. Because of the greatly improved speed, the proposed method also enables quantitative analysis on performance-power tradeoff of a large AMS circuit.

Experimental validation of the plunger-type flow model for irregular waves

In this paper, a study of irregular waves generated with a plunger-type wavemaker with the inclusion of mean flow is conducted along with an analysis of the standard plunger-type flow (PTF) model. It was determined that for high frequencies, the standard PTF model exhibits amplification which when compared to theoretical models for piston- and flap-type wavemakers is implausible. Therefore, a correction to the PTF model has been proposed for use with irregular waves that limits the value of the wave amplitude a to stroke amplitude s ratio a/s to one or less. Experimental validation of the correction is also given along with a study on the performance of the plunger system for generating the desired irregular Bretschneider wave spectrum. In this paper, multiple significant wave heights and peak frequencies are considered for no-flow and three mean flow cases. The experimental results showed that not only is the proposed correction valid for decreasing the percent error in the significant wave height but, plunger-type wavemakers are capable of accurately producing irregular spectra regardless of the inclusion of mean flow.

Accuracy of UAV-based DEMs without ground control points

Unmanned aerial vehicles (UAVs) are increasingly used in various environmental research projects and other activities that require accurate topography images. The quality of elevation models derived from UAV measurements varies depending on many variables (e.g. UAV equipment used, terrain conditions, etc.). In order to improve the quality of digital models based on UAV image data, additional GNSS-RTK measurements are usually made at ground control points. The aim of this article is to evaluate the mathematical accuracy of terrain models created without ground control points. The accuracy of the models is considered in two directions: vertical and horizontal. Vertical (elevation) accuracy is calculated based on airborne laser scanning (ALS) data and horizontal (location) accuracy is calculated through comparison with high-resolution orthophotomaps. The average elevation accuracy of all created UAV-based DEMs is found to be 2.7–2.8 m (MAE), 3.1–3.3 m (RMSE), and the average horizontal accuracy is 2.1 m. Despite the low accuracy of the UAV models, the topography is reflected very well in the spatial images. This may be related to the regular and symmetrical distribution of height errors. To improve the accuracy parameters of UAV-based DEMs, it is proposed that they be rapidly georeferenced based on orthophotomaps.

A Comparative Assessment of Multi-Source Generation of Digital Elevation Models for Fluvial Landscapes Characterization and Monitoring

Imaging and measuring the Earth’s relief with sensors mounted upon unmanned aerial vehicles is an increasingly frequently used and promising method of remote sensing. In the context of fluvial geomorphology and its applications, e.g., landform mapping or flood modelling, the reliable representation of the land surface on digital elevation models is crucial. The main objective of the study was to assess and compare the accuracy of state-of-the-art remote sensing technologies in generating DEMs for riverscape characterization and fluvial monitoring applications. In particular, we were interested in DAP and LiDAR techniques comparison, and UAV applicability. We carried out field surveys, i.e., GNSS-RTK measurements, UAV and aircraft flights, on islands and sandbars within a nature reserve on a braided section of the Vistula River downstream from the city of Warsaw, Poland. We then processed the data into DSMs and DTMs based on four sources: ULS (laser scanning from UAV), UAV-DAP (digital aerial photogrammetry), ALS (airborne laser scanning), and satellite Pléiades imagery processed with DAP. The magnitudes of errors are represented by the cross-reference of values generated on DEMs with GNSS-RTK measurements. Results are presented for exposed sediment bars, riverine islands covered by low vegetation and shrubs, or covered by riparian forest. While the average absolute height error of the laser scanning DTMs oscillates around 8–11 cm for most surfaces, photogrammetric DTMs from UAV and satellite data gave errors averaging more than 30 cm. Airborne and UAV LiDAR measurements brought almost the perfect match. We showed that the UAV-based LiDAR sensors prove to be useful for geomorphological mapping, especially for geomorphic analysis of the river channel at a large scale, because they reach similar accuracies to ALS and better than DAP-based image processing.

Information Extraction Method of DBH of Individual Trees in Artificial Forest Based on UAV Image

<p>The information acquisition of tree height, individual tree crown and diameter at breast height is an important part of forest resources survey. In this study, the UAV images of artificial Robinia pseudoacacia forest in Zhangjiatan Town of Yan River Basin were obtained by DJI Phantom 4 UAV platform. The crown width and tree height were obtained by neural network clustering algorithm, visual interpretation and spatial measurement. The relative error between the extracted value and the measured value of individual tree crown width was 5. 98%, and the relative error of tree height was 7. 53%. We used 80% of the data to extract, crown, tree height and diameter at breast height to establish single regression models, exponential function models, logarithmic function models, power function models and binary regression models. Among those established DBH inversion models, the model with the highest fitting accuracy was y = 4.255a + 0.044b2 - 0.135a2 - 2.111, with a determination coefficient of 0.85, and the average relative error was 4.3% after being verified by using the remaining 20% of the data. The stand factors of the study area can be quickly obtained by using UAV images, and the DBH can be obtained with high precision, which can provide a reference for the realization of precision forestry.</p> <p> </p>

Real-time automated forest field inventory using a compact low-cost helmet-based laser scanning system

Forest field inventory plays a crucial role in forestry management and the estimation of carbon circular economy, as it provides information on forest parameters, assesses carbon storage, and identifies the impact factors of ecological change. Terrestrial Laser Scanning (TLS) and Mobile Mapping Systems (MMS) are commonly used for forest field inventory, but they are unable to verify the inventory results in real time. With real-time forest inventory, forestry workers can immediately verify if all trees have been correctly extracted without requiring expertise in point clouds. The advences of miniaturized, low-cost 3D sensors (such as solid-state laser scanners and Inertial Measurement Units [IMUs]) and edge computing units have made it possible to achieve real-time forest inventory using a compact, low-cost helmet. To this end, this paper presents a real-time automated forest field inventory method, which is validated on a compact, low-cost helmet-based laser scanning system. Firstly, a Fast Candidate Tree Detection (FCTD) approach is proposed to identify individual trees as candidates by utilizing a novel 2D corner detection technique based on point cloud projection, taking into account point density and geometry features. Secondly, a Spatiotemporal Consistency-based Tree Parameter Estimation (SCTPE) method is proposed to estimate the tree parameters by considering both the current submap and submaps that have been scanned in real-time. The proposed method was tested in three typical forest areas in Wuhan, China, where the main tree species present include Sapindaceae, dawn redwoods, and Platycladus. The results showed that the proposed method achieves high accuracy in tree detection (recall=0.97,precision=0.94,F=0.96). The average error and root-mean-square error (RMSE) of DBH are 0.033 m and 0.038 m, which outperforms the existing one-circle fitting model. The average error and RMSE of tree height are 0.231 m and 0.294 m. Overall, these results demonstrate the high potential of the helmet-based laser scanning system for real-time forest field inventory.

Phase Distribution Measurement of Liquid–Solid Media Using Multimode Ultrasound

Ultrasound tomography (UT) is an emerging technique for phase distribution measurement in industrial multiphase flow monitoring. A multimode UT combining the ultrasound transmission and reflection modes is proposed to measure stratified phase distribution and sound speed of liquid–solid mixtures in the horizontal pipe. First, the acoustic attenuation is used to reconstruct the phase distribution with a modified matrix regularization algorithm, and the time-of-flight (TOF) of ultrasound is used to extract the boundary of multiphase layers with the weight Akaike information criterion algorithm. Second, a wavelet fusion method based on the principal component averaging is proposed to fuse the reconstruction images of UT transmission and reflection. Third, numerical investigations are carried out to validate the feasibility of the proposed method. According to the quantitative indexes, the proposed method can effectively reconstruct the phase distribution of multiphase media. Finally, an experimental test pipeline Section is established, and sand–water and oil–water–sand liquid–solid media are investigated. The minimum relative error (RE) and the maximum correlation coefficient (CC) of experimental fusion results can reach 0.33 and 0.94, respectively, and the average measurement error (ME) of the layer height is 3.80%.

Measurement Method for Wall Crack Size Based on Pixel Ratio

The identification and measurement of building cracks can reduce the harm caused by accidents to people’s lives. Reducing manual work can improve the efficiency of crack inspection and reduce inspection errors. In this paper, (1) Machine vision with high measurement accuracy and fast response speed is used, the crack contour is extracted by OpenCV splicing, segmentation, morphological processing, and wall cracks can be accurately detected. (2) The crack size is calculated using the pixel ratio algorithm, the image pixel ratio is calculated by the reference object method, and the size of the wall crack is measured. (3) The measurement error is small. In comparing the experimental results and manual measurement, the width measurement error rate is the highest, 0.314%, and the height error rate is the highest, 2.9%. (4) Size information can be displayed to prevent the hazard caused by cracks in advance. So, the system has a good wall crack identification and measurement effect, which reduces labor costs.

Multi-feature detection of in-field grain lodging for adaptive low-loss control of combine harvesters

The phenomenon of grain lodging significantly affects the low-loss harvesting performance of automatic drive harvesters, and it has become an objective trend to simultaneously detect multi-feature grain lodging information with machines in the field. In order to solve the problem of high harvest loss in grain lodging areas, we propose a new method for synchronous detection of grain lodging multi-feature information based on 3D point cloud data of a depth camera, in order to realize adaptive regulation of unmanned and low-loss harvesting. The multi-feature information of lodging grain is composed of lodging position, degree and direction, which can provide effective support information for the adaptive regulation of automatic driving harvesters. For the scene of grain harvesting operation with low contrast characteristics, we make full use of the environmental characteristics of the lodging areas that are located in the non-lodged area, and construct a synchronous airborne extraction strategy for grain lodging multi-feature information. The test results show that the maximum detection error of the lodging degree is less than 9.0 cm, and the minimum detection error is better than 1.0 cm; the maximum detection error of the lodging direction is not more than 9.0°, the minimum detection error is less than 1.0°, the maximum harvesting height error is less than 5.0 cm, and the minimum harvesting height error is less than 5.0 cm. Better than 2.0 cm. Our method provides an effective solution for synchronous airborne detection of multi-feature grain lodging information, which can significantly reduce harvest loss in grain lodging areas, and is expected to accelerate the development of truly unmanned harvesting operations.

Assessment of a near-polar pair mission for detecting the Earth's temporal gravity field

SUMMARYTo ensure the global observability of next generation gravimetric mission (NGGM), different agencies have to repeatedly launch satellites to about 89.0° orbit inclination. However, due to the poor isotropy of observation system, only minor improvement in terms of temporal gravity field estimation can be obtained via these repeatedly launched polar pair missions. To ensure the global observability as well as the isotropy of observation system, a near-polar pair mission rather than a polar mission is likely an optimal selection, especially considering that the Gravity Recovery and Climate Experiment (GRACE) Follow-On mission has been already in operation. In this study, for the upcoming NGGMs (for instance, the Chinese NGGM), we design a closed-loop simulation to assess the performance of a near-polar mission at a near-circular orbit with about 500 km altitude for detecting the Earth's temporal gravity field, and the main conclusions are summarized as follows. (1) Based on the statistic results, 85.0° is selected as the optimal orbit inclination for the near-polar mission, which provides 37 per cent noise reduction in terms of cumulative geoid height error in spectral domain, but also 31 per cent noise reduction in terms of mean oceanic root-mean-square (RMS) error in spatial domain when compared to the 89.0° polar mission (89-PM). (2) To figure out the reason of these noise reductions, we also compare the contribution of single error components [including instrument error, atmospheric and oceanic (AO) aliasing error, ocean tide error]. It indicates that the outperformance of the 85.0° near-polar mission (85-NPM) is mainly derived from the relatively smaller non-tidal atmospheric and oceanic mass variation error. (3) Although there are inevitable 5.0° polar gaps in the 85-NPM, the analysis result (including the comparison via in-orbit observations and simulated retrieved solutions) still demonstrates a comparable performance of the 85-NPM in tracking mass variations over the Antarctic, and even an outperformance with 12 per cent noise reduction over the Greenland when compared to the 89-PM. The result confirms the feasibility of implementing a near-polar mission as a stand-alone mission or a complementary observation system for the repeatedly launched polar missions, which offers an alternative option of launching the Chinese NGGM satellites to an 85.0° inclination orbit instead of 89.0°.

The impacts of reducing atmospheric and oceanic de-aliasing model error on temporal gravity field model determination

SUMMARYThe Gravity Recovery and Climate Experiment (GRACE) and its successor GRACE Follow-On mission have been estimating temporal gravity field models for various geoscience applications since 2002. While the errors in atmospheric and oceanic de-aliasing (AOD) model are still the key limitation for GRACE-type missions especially taking advantages of improved laser ranging interferometer (LRI), this paper explores the realistic assumption of continuously reducing AOD model error in global scale and regional scale (e.g. ocean, Greenland, Qinghai-Tibet plateau and South America) in the anticipated future. For this, using a realistic orbit scenario and error assumptions both for instrument and background model errors, the 5-yr full-scale simulations are implemented in the context of 26 scenarios with different AOD model errors. Our overall simulation results demonstrate that: (1) when the AOD model errors are reduced in the global scale, the corresponding gain in temporal gravity field determination is significant, with the cumulative geoid height error decrease from 11.28 to 6.25 mm in spectral domain, as well as 26.3–65.2 per cent noise reduction in terms of mean RMS residuals over ocean in spatial domain. (2) When the AOD model errors are reduced in the regional scale, the improvements of temporal gravity field estimations are diverse in different simulation scenarios. The most notable noise reduction, with 22.9–43.9 per cent noise reduction in spatial domain, is observed when the AOD model error decreases in ocean. The limited noise reductions (0.1 per cent and 1.3 per cent respectively in spectral and spatial domain) are observed in the temporal gravity field estimations if the AOD model is refined in other typical regions such as the Qinghai-Tibet plateau, the Greenland and the South America. (3) However, when the satellites passing over the AOD model refined regions, the along-orbit range rate analysis indicates that, there are visible differences by about 50.0 nm s−1 in terms of range rate residuals as well as 11.0–48.5 nm s−1 in terms of the mean RMS of range rate residuals. These results reflect the benefits of reducing AOD model error in both global and regional scale for improving GRACE-type temporal gravity estimation, especially considering the development of LRI technology.

Dynamical thin disks

Thin disk accretion is often modeled in highly dynamical settings using the two-dimensional equations of viscous hydrodynamics, with viscosity representing unresolved turbulence. These equations are supposed to arise after vertical integration of the full three-dimensional equations of hydrodynamics, under the assumption of a geometrically thin disk with mirror symmetry about the midplane. However, in the dynamical context, vertical dynamics are neglected by incorrectly assuming instantaneous vertical hydrostatic equilibrium. The resulting errors in the local disk height couple to the horizontal dynamics through the $\ensuremath{\alpha}$-viscosity prescription and disk height-dependent gravitational softening models. Furthermore, the viscous terms in the horizontal equations are only complete if they are inserted after vertical integration, as if the system is actually two-dimensional. Since turbulence breaks mirror symmetry, it is more physically correct to insert a turbulence model at the three-dimensional level and impose mirror symmetry only on average. Thus, some viscous terms are usually missing. With these considerations in mind, we revisit the vertical integration procedure, restricting ourselves to the regime of a Newtonian, non-self-gravitating disk. We obtain six evolution equations with only horizontal dependence, which determine the local vertical position and velocity of the disk surface, in addition to the usual surface densities of mass, horizontal momentum, and energy. This ``2.5-dimensional'' formulation opens the door to efficiently study vertical oscillations of thin disks in dynamical settings, and to improve the treatment of unresolved turbulence. As a demonstration, by including viscous stress at the three-dimensional level, we recover missing viscous terms which involve the vertical variables. We also propose a resummation of the vertically integrated gravitational force, which has a strikingly similar radial profile to a gravitational softening model advocated for in protoplanetary disk studies.

Research on development of aspheric diffractive optical element for mid-infrared imaging

The aspheric diffractive lens reduces chromatic aberration in precision infrared optical systems. With the advancement in micro-optics, conventional lenses are replaced by aspheric, diffractive and hybrid surfaces to fulfil the requirements of miniaturized design of optical devices. This paper covers the detailed aspects of hybrid aspheric diffractive optics from design to manufacturing by precision machining. The germanium aspheric diffractive lens is designed using Zemax OpticsStudio® for Medium Wave Infrared (MWIR) range to achieve an aberration-free and diffraction-limited surface. The designed surface is fabricated by the diamond turning process. The developed surface is characterized by profile and diffractive structure quality. The results demonstrate the fabricated lens with a diffractive structure step height error of 0.001 µm and radial position error of 1 µm. A high-quality aspheric diffractive lens is fabricated with 99.57 % diffraction efficiency.