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37 pages, 34329 KiB  
Technical Note
The Cycle 46 Configuration of the HARMONIE-AROME Forecast Model
by Emily Gleeson, Ekaterina Kurzeneva, Wim de Rooy, Laura Rontu, Daniel Martín Pérez, Colm Clancy, Karl-Ivar Ivarsson, Bjørg Jenny Engdahl, Sander Tijm, Kristian Pagh Nielsen, Metodija Shapkalijevski, Panu Maalampi, Peter Ukkonen, Yurii Batrak, Marvin Kähnert, Tosca Kettler, Sophie Marie Elies van den Brekel, Michael Robin Adriaens, Natalie Theeuwes, Bolli Pálmason, Thomas Rieutord, James Fannon, Eoin Whelan, Samuel Viana, Mariken Homleid, Geoffrey Bessardon, Jeanette Onvlee, Patrick Samuelsson, Daniel Santos-Muñoz, Ole Nikolai Vignes and Roel Stappersadd Show full author list remove Hide full author list
Meteorology 2024, 3(4), 354-390; https://doi.org/10.3390/meteorology3040018 - 5 Nov 2024
Viewed by 962
Abstract
The aim of this technical note is to describe the Cycle 46 reference configuration of the HARMONIE-AROME convection-permitting numerical weather prediction model. HARMONIE-AROME is one of the canonical system configurations that is developed, maintained, and validated in the ACCORD consortium, a collaboration of [...] Read more.
The aim of this technical note is to describe the Cycle 46 reference configuration of the HARMONIE-AROME convection-permitting numerical weather prediction model. HARMONIE-AROME is one of the canonical system configurations that is developed, maintained, and validated in the ACCORD consortium, a collaboration of 26 countries in Europe and northern Africa on short-range mesoscale numerical weather prediction. This technical note describes updates to the physical parametrizations, both upper-air and surface, configuration choices such as lateral boundary conditions, model levels, horizontal resolution, model time step, and databases associated with the model, such as for physiography and aerosols. Much of the physics developments are related to improving the representation of clouds in the model, including developments in the turbulence, shallow convection, and statistical cloud scheme, as well as changes in radiation and cloud microphysics concerning cloud droplet number concentration and longwave cloud liquid optical properties. Near real-time aerosols and the ICE-T microphysics scheme, which improves the representation of supercooled liquid, and a wind farm parametrization have been added as options. Surface-wise, one of the main advances is the implementation of the lake model FLake. An outlook on upcoming developments is also included. Full article
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Figure 1

Figure 1
<p>The HARMONIE-AROME workflow.</p>
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<p>Dust case on the 20 of February 2023. Daily mean global SW radiation from HARMONIE-AROME Cycle 46 experiments. (<b>a</b>) using the default Tegen aerosol climatology, (<b>b</b>) using NRT CAMS aerosols, (<b>c</b>) difference between these.</p>
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<p>Daily cycle of global SW radiation for a desert dust intrusion case on 20 February 2023. The average of the measurements from 29 stations over the Spanish Peninsula is depicted by the dashed black line. Model results at the station points for the experiment with the Tegen aerosol climatology are shown in red, while those for NRT aerosols are shown in green.</p>
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<p>LW effective emissivity as a function of LWP. Grey dots are the values derived from the Cabauw measurements. The green curve represents Equation (<a href="#FD1-meteorology-03-00018" class="html-disp-formula">1</a>) with the default coefficient of −0.144. The blue and red curves use values −0.158 and −0.130, respectively. The black curve uses the coefficient of −0.096, which ensures a least squares best fit.</p>
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<p>Spectral absorptance for a LWP of (<b>a</b>) 1 gm<sup>−2</sup> (<b>b</b>) and 10 gm<sup>−2</sup> for the 16 LW bands of the Nielsen scheme. Corresponding values calculated from the emissivity by the Smith and Shi [<a href="#B27-meteorology-03-00018" class="html-bibr">27</a>] parametrization (purple continuous line) and the Kettler scheme (green continuous line [<a href="#B26-meteorology-03-00018" class="html-bibr">26</a>]) are shown.</p>
Full article ">Figure 5 Cont.
<p>Spectral absorptance for a LWP of (<b>a</b>) 1 gm<sup>−2</sup> (<b>b</b>) and 10 gm<sup>−2</sup> for the 16 LW bands of the Nielsen scheme. Corresponding values calculated from the emissivity by the Smith and Shi [<a href="#B27-meteorology-03-00018" class="html-bibr">27</a>] parametrization (purple continuous line) and the Kettler scheme (green continuous line [<a href="#B26-meteorology-03-00018" class="html-bibr">26</a>]) are shown.</p>
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<p>(<b>a</b>) MSG visible satellite image. (<b>b</b>) MSG Seviri cloud water path product from KNMI. (<b>c</b>) Integrated cloud water condensate (gm<sup>−2</sup>) from the default HARMONIE-AROME Cycle 43 experiment. (<b>d</b>) Integrated cloud water condensate (gm<sup>−2</sup>) from the HARMONIE-AROME Cycle 43 experiment with a CDNC of 50 cm<sup>−3</sup> and the LW effective emissivity coefficient of Kettler [<a href="#B26-meteorology-03-00018" class="html-bibr">26</a>]. All at 12 Z on 19 July 2019.</p>
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<p>Distribution of CSI for summer (<b>a</b>) and winter (<b>b</b>) 2-week periods, obtained from observations over Ireland and from the results of two HARMONIE-AROME Cycle 46 experiments, with prescribed CDNC (Tegen) and with CDNC derived from CAMS data (CAMSNRT).</p>
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<p>SW radiation bias (Wm<sup>−2</sup>) for summer and winter 2-week periods for experiments with HARMONIE-AROME Cycle 46 over Ireland without NRT aerosols (Tegen) (<b>a</b>,<b>c</b>) and using the NRT aerosols (<b>b</b>,<b>d</b>).</p>
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<p>SW radiation bias (Wm<sup>−2</sup>) for summer and winter 2-week periods for experiments with HARMONIE-AROME Cycle 46 over Ireland without NRT aerosols (Tegen) (<b>a</b>,<b>c</b>) and using the NRT aerosols (<b>b</b>,<b>d</b>).</p>
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<p>Example of spurious cirrus clouds. (<b>a</b>) original OCND2. (<b>b</b>) OCND2 with technical corrections. Clouds are shown as cyan shading. The figures are from a 13 h forecast starting from 00 Z on 16 April 2018.</p>
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<p>Simulated cloud liquid water content, gm<sup>−3</sup>, in the Alta region for model level 41 (approximately 820 hPa) at 14 UTC 19 April 2023, from ICE3 (<b>a</b>), the ICE-T experiment (<b>b</b>), and the difference between these (<b>c</b>).</p>
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<p>The kinematic total turbulent moisture transport (<math display="inline"><semantics> <mover> <mrow> <msup> <mi>w</mi> <mo>′</mo> </msup> <msubsup> <mi>r</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <mo>¯</mo> </mover> </semantics></math>) on the 9th hour of the simulation of the ARM shallow cumulus case [<a href="#B44-meteorology-03-00018" class="html-bibr">44</a>]. The blue line is the DALES model. The green lines are HARMONIE-AROME Cycle 40, with all the updates described in [<a href="#B39-meteorology-03-00018" class="html-bibr">39</a>], as applied later to HARMONIE-AROME Cycle 43 and 46. The green dashed line is for the experiment without the energy cascade; the green solid line is for the experiment with the energy cascade. European Geosciences Union 2022, from Figure 6 in [<a href="#B39-meteorology-03-00018" class="html-bibr">39</a>].</p>
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<p>Frequency bias of the cloud base height in feet (1 foot is 0.3048 m) for December 2018 with (<b>a</b>) HARMONIE-AROME Cycle 40 [<a href="#B1-meteorology-03-00018" class="html-bibr">1</a>] and (<b>b</b>) HARMONIE-AROME Cycle 40, with all of the updates described in [<a href="#B39-meteorology-03-00018" class="html-bibr">39</a>], as also applied to HARMONIE-AROME Cycle 43 and 46. The blue, green, and orange lines refer to +3, +24, and +48 h forecasts, respectively. European Geosciences Union 2022, from Figure 20 of [<a href="#B39-meteorology-03-00018" class="html-bibr">39</a>].</p>
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<p>Vertical profile of the variance in s (the distance to the saturation curve) for the 10th hour of the simulation of the ARM cumulus case [<a href="#B44-meteorology-03-00018" class="html-bibr">44</a>]. Results for the DALES model are in blue, the reference HARMONIE-AROME Cycle 40 [<a href="#B1-meteorology-03-00018" class="html-bibr">1</a>] (cy40 REF) is in orange, and HARMONIE-AROME Cycle 40 with all the updates described in [<a href="#B39-meteorology-03-00018" class="html-bibr">39</a>], which corresponds to HARMONIE-AROME Cycle 46, (cy40 NEW), is in green. European Geosciences Union 2022, from Figure 12 in [<a href="#B39-meteorology-03-00018" class="html-bibr">39</a>].</p>
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<p>The average modeled wind speed at 90 m height when the WFP is included (contours) and aircraft measurements from the WIPAFF campaign [<a href="#B53-meteorology-03-00018" class="html-bibr">53</a>], located between 80 and 100 m height (colored dots), for 6 September 2016, 8–10 UTC. The black dots indicate the locations of the wind turbines included in the simulation.</p>
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<p>Land cover types over Iceland in (<b>a</b>) the original ECOSG and (<b>b</b>) an improved version of ECOSG. Land cover types over southern Greenland in (<b>c</b>) the original ECOSG and (<b>d</b>) an improved version of ECOSG.</p>
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<p>10 m wind-speed bias and RMSE over Ireland for HARMONIE-AROME Cycle 40 (red), HARMONIE-AROME Cycle 43 default (green), and HARMONIE-AROME Cycle 43 “LFAKETREE” (blue) for 2 two week periods. (<b>a</b>) Spring (<b>b</b>) summer.</p>
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<p>Snow water equivalent (<math display="inline"><semantics> <msub> <mi mathvariant="normal">W</mi> <mrow> <mi>s</mi> <mi>n</mi> </mrow> </msub> </semantics></math>) dependency on time (hours) during melting for different values of <span class="html-italic">p</span>, where <span class="html-italic">p</span> is the replacement for <math display="inline"><semantics> <msub> <mi mathvariant="normal">f</mi> <mrow> <mi>s</mi> <mi>n</mi> </mrow> </msub> </semantics></math> in Equation (<a href="#FD5-meteorology-03-00018" class="html-disp-formula">5</a>).</p>
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<p>The difference in latent heat fluxes between eddy correlation measurements from the EUREC<sup>4</sup>A field campaign of January and February 2020 and model simulations with the ECUME (<b>a</b>) and ECUME6 (<b>b</b>) schemes. The biases are plotted in the phase-space of the specific humidity difference dq (between surface (qs) and 2 m (qa 2 m)) and 10 m wind speed. These plots are reproduced from Figures <math display="inline"><semantics> <mrow> <mn>4.15</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>4.16</mn> </mrow> </semantics></math> in [<a href="#B84-meteorology-03-00018" class="html-bibr">84</a>].</p>
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<p>Scatter-plots of simulated versus observed values of meteorological variables for experiments with and without FLake. The results of HARMONIE-AROME Cycle 40 forecasts starting from 00 and 12 UTC, and with lead times of 6, 18, 30, and 42 h, are shown for the period 18 May to 1 June 2016. The observations are from 3 SYNOP stations around Lake Ladoga. (<b>a</b>) 2 m temperature, °C, without FLake, (<b>b</b>) 2 m temperature, °C, with FLake (as in Cycle 46), (<b>c</b>) 2 m specific humidity, gkg<sup>−1</sup>, without FLake, (<b>d</b>) 2 m specific humidity, gkg<sup>−1</sup>, with FLake (as in Cycle 46).</p>
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<p>Vertical cross-section of cloud fraction, modeled by HARMONIE-AROME (<b>a</b>), along the blue dashed line in the satellite image (<b>b</b>). The cloud fraction is shown for simulations on 65 and 90 (MC_90) vertical level grids in HARMONIE-AROME for the Swedish domain on the 19th August 2023 at 12 UTC and compared to the satellite image over the Stockholm area (Uppland and Södermanland provinces) on the same date and time. The difference between the two 90-level vertical grids MC_90 and MF_90 available in HARMONIE-AROME is shown in (<b>c</b>), with a zoom-in on the lowest part in (<b>d</b>). MF refers to the Météo France version as used in the AROME-France NWP system, while MC refers to a modification suggested by the MetCoOp developers (see text for further details).</p>
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<p>Scatter plots and their linear regression estimates showing the performance of HARMONIE-AROME Cycle 46 with the implemented RSL parametrization [<a href="#B113-meteorology-03-00018" class="html-bibr">113</a>,<a href="#B114-meteorology-03-00018" class="html-bibr">114</a>] in the atmosphere–surface coupling layer versus flux–gradient observations for momentum (<b>a</b>), 10 m wind speed (<b>b</b>), and (sensible) heat fluxes (<b>c</b>,<b>d</b>). For validation, the half-hourly observed fluxes, as well as the wind speed above the canopy, are used, taken from the <a href="https://data.icos-cp.eu/portal/" target="_blank">https://data.icos-cp.eu/portal/</a> (accessed on 27 October 2024) and collected at four ICOS forest sites (Bilos, Norunda, Hyltemossa, and Svartberget) between 15 August and 15 September 2021. The corresponding model data were extracted from the nearest model grid points.</p>
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<p>Show case of the physiography developments planned to be integrated in HARMONIE-AROME in the future. Currently, HARMONIE-AROME uses the ECOSG database (<b>a</b>,<b>d</b>). The land cover map obtained with the agreement-based combination (ECOSG+, (<b>b</b>,<b>e</b>)) and the one obtained with machine learning (ECOSG-ML, (<b>c</b>,<b>f</b>)) are both at 60 m resolution. ECOSG+ and ECOSG-ML show increasing qualitative benefits; see [<a href="#B116-meteorology-03-00018" class="html-bibr">116</a>,<a href="#B117-meteorology-03-00018" class="html-bibr">117</a>] for the evaluation. The coordinates of the center points are given on the left hand side for both sites. Patches are approximately 25 km × 25 km in size. Colors represent different land cover types.</p>
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<p>Bias and standard deviation in low cloud cover for (<b>a</b>) a summer period (1–14 June 2018) and (<b>b</b>) a winter period (3–17 February 2020) for HARMONIE-AROME cycle 40 (red) and cycle 46 (blue) compared to observations recorded at 140 stations in Ireland and the UK. The data shown are from forecasts starting from 00 and 12 UTC for forecast lengths of up to 33 h.</p>
Full article ">
34 pages, 15689 KiB  
Article
Analysis of the Heat Transfer Performance of a Buried Pipe in the Heating Season Based on Field Testing
by Yongjie Ma, Jingyong Wang, Fuhang Hu, Echuan Yan, Yu Zhang, Yibin Huang, Hao Deng, Xuefeng Gao, Jianguo Kang, Haoxin Shi, Xin Zhang, Jianqiao Zheng and Jixiang Guo
Energies 2024, 17(21), 5466; https://doi.org/10.3390/en17215466 - 31 Oct 2024
Viewed by 410
Abstract
Ground source heat pump (GSHP) systems have been widely used in the field of shallow geothermal heating and cooling because of their high thermal efficiency and environmental friendliness. A borehole heat exchanger (BHE) is the key part of a ground source heat pump [...] Read more.
Ground source heat pump (GSHP) systems have been widely used in the field of shallow geothermal heating and cooling because of their high thermal efficiency and environmental friendliness. A borehole heat exchanger (BHE) is the key part of a ground source heat pump system, and its performance and investment cost have a direct and significant impact on the performance and cost of the whole system. The ground temperature gradient, air temperature, seepage flow rate, and injection flow rate affect the heat exchange performance of BHEs, but most of the research on BHEs lacks field test verification. Therefore, this study relied on the results of a field thermal response test (TRT) based on a distributed optical fiber temperature sensor (DOFTS) and site hydrological, geological, and geothermal data to establish a corrected numerical model of buried pipe heat transfer and carry out the heat transfer performance analysis of a buried pipe in the heating season. The results showed that the ground temperature gradient of the test site was about 3.0 °C/100 m, and the temperature of the constant-temperature layer was about 9.17 °C. Increasing the air temperature could improve the heat transfer performance. The temperature of the surrounding rock and soil mass of the single pipe spread uniformly, and the closer it was to the buried pipe, the lower the temperature. When there is groundwater seepage, the seepage carries the cold energy generated by a buried pipe’s heat transfer through heat convection to form a plume zone, which can effectively alleviate the phenomenon of cold accumulation. With an increase in seepage velocity, the heat transfer of the buried pipe increases nonlinearly. The heat transfer performance can be improved by appropriately reducing the temperature and velocity of the injected fluid. Selecting a backfill material with higher thermal conductivity than the ground body can improve the heat transfer performance. These research results can provide support for the optimization of the heat transfer performance of a buried tube heat exchanger. Full article
(This article belongs to the Section H2: Geothermal)
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Figure 1

Figure 1
<p>Geographical location of the study area and heating objectives.</p>
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<p>Temperature changes in the heating season of the study district.</p>
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<p>Combined thermal response test system. (<b>a</b>) Schematic diagram of the system. (<b>b</b>) Fixing optical fibers with ribbon. (<b>c</b>) High-temperature- and abrasion-resistant hose. (<b>d</b>) Optical fiber disassembly diagram. (<b>e</b>) TRT module. (<b>f</b>) DTRT module.</p>
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<p>Schematic diagram of simplified assumptions for the heat transfer module and model. (<b>a</b>) Heat transfer in porous media. (<b>b</b>) Non-isothermal pipeline flow.</p>
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<p>Vertical buried pipe heat transfer model. (<b>a</b>) Top view of the heat transfer model. (<b>b</b>) Grid encryption. (<b>c</b>) Model boundary.</p>
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<p>Initial formation temperature distribution with unheated and uncirculated fluid.</p>
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<p>Time evolution curves of inlet and outlet temperatures under heat load conditions: (<b>a</b>) 12 kW; (<b>b</b>) 8 kW.</p>
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<p>Field test of the layered thermal conductivity of the rock and soil mass.</p>
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<p>Grid independence test of the buried pipe heat transfer model.</p>
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<p>Comparison of the numerical results of heat transfer in the buried pipe and the results of the thermal response test in the field.</p>
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<p>Evolutionary characteristics of the heat exchanger outlet fluid temperature with time under three temperature conditions.</p>
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<p>Heat transfer capacities under different temperature conditions: (<b>a</b>) 108 days; (<b>b</b>) 174 days.</p>
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<p>Temperature evolution of the outlet fluid of the buried pipe heat exchanger with or without seepage.</p>
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<p>Heat transfer capacities of buried pipe heat exchangers with and without seepage conditions.</p>
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<p>Temperature evolution of the vertical cross-section of the buried tube heat exchanger without seepage.</p>
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<p>Temperature evolution of the vertical cross-section of the buried tube heat exchanger under seepage conditions.</p>
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<p>Temperature evolution of the middle section (z = 86 m) of the muddy sandstone layer without seepage.</p>
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<p>Temperature evolution of the middle section (z = 86 m) of the mudstone sandstone layer under seepage conditions.</p>
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<p>The influence of seepage velocity on the outlet fluid temperatures of heat exchangers.</p>
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<p>Heat transfer capacities of buried pipe heat exchangers with different seepage velocities.</p>
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<p>Temperature distributions in the mid-depth section (z = 86 m) of muddy siltstone layers with different seepage velocities.</p>
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<p>Influence of the inlet fluid temperature on the outlet fluid temperature of a heat exchanger.</p>
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<p>Heat transfer capacities of buried pipe heat exchangers with different inlet fluid temperatures.</p>
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<p>Temperature distribution in the middle depth section (z = 86 m) of muddy siltstone layers with different inlet fluid temperatures.</p>
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<p>The influence of fluid injection velocity on the outlet fluid temperature of heat exchangers.</p>
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<p>Heat transfer capacities of buried pipe heat exchangers with different injection flow rates.</p>
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<p>Temperature distribution at the middle depth section (z = 86 m) of muddy siltstone layers with different injection flow rates.</p>
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<p>The effects of backfill materials on the outlet temperature of heat exchanger fluid.</p>
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<p>Heat exchange capacities of buried pipe heat exchangers with different backfill materials (A: Bentonite; B: Fine sand, bentonite; C: Waste materials of silica; D: Aluminum shavings, cement, fine sand).</p>
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<p>Temperature distributions in the mid-depth section (z = 86 m) of different backfill materials in the muddy siltstone layer (A: Bentonite; B: Fine sand, bentonite; C: Waste materials of silica; D: Aluminum shavings, cement, fine sand).</p>
Full article ">
23 pages, 2449 KiB  
Article
Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums
by Mohammed N. Alshehri, Saad Althobaiti, Ali Althobaiti, Rahmatullah Ibrahim Nuruddeen, Halliru S. Sambo and Abdulrahman F. Aljohani
Mathematics 2024, 12(20), 3205; https://doi.org/10.3390/math12203205 - 13 Oct 2024
Viewed by 518
Abstract
The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform [...] Read more.
The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)
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Figure 1

Figure 1
<p>Three-dimensional and contour plots for the modified Kudryashov method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 1 Cont.
<p>Three-dimensional and contour plots for the modified Kudryashov method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>Two-dimensional plots for the modified Kudryashov method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>,</mo> </mrow> </semantics></math> respectively, when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> In addition, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> in (<b>a</b>), while <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> in (<b>b</b>). (<b>a</b>) Two-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) with variation in <math display="inline"><semantics> <mi>η</mi> </semantics></math>. (<b>b</b>) Two-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD22-mathematics-12-03205" class="html-disp-formula">22</a>) with variation in <math display="inline"><semantics> <mi>μ</mi> </semantics></math>.</p>
Full article ">Figure 3
<p>Three-dimensional and contour plots for the modified Kudryashov method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Two-dimensional plots for the modified Kudryashov method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>,</mo> </mrow> </semantics></math> respectively, when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> In addition, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> in (<b>a</b>), while <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> in (<b>b</b>). (<b>a</b>) Two-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) with variation in <math display="inline"><semantics> <mi>η</mi> </semantics></math>. (<b>b</b>) Two-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <msub> <mn>2</mn> <mo>+</mo> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD24-mathematics-12-03205" class="html-disp-formula">24</a>) with variation in <math display="inline"><semantics> <mi>μ</mi> </semantics></math>.</p>
Full article ">Figure 5
<p>Three-dimensional and contour plots for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 5 Cont.
<p>Three-dimensional and contour plots for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Two-dimensional plot for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD31-mathematics-12-03205" class="html-disp-formula">31</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>Three-dimensional and contour plots for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) 3D plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Two-dimensional plot for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Three-dimensional and contour plots for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 9 Cont.
<p>Three-dimensional and contour plots for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math> (<b>a</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD32-mathematics-12-03205" class="html-disp-formula">32</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>2.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>e</b>) Three-dimensional plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> <mo>.</mo> </mrow> </semantics></math> (<b>f</b>) Contour plot for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>.</p>
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<p>Two-dimensional plot for the modified extended tanh method’s solution <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> determined in (<a href="#FD34-mathematics-12-03205" class="html-disp-formula">34</a>) with variations in <math display="inline"><semantics> <mi>η</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="4pt"/> <mi>ω</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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25 pages, 10179 KiB  
Article
An Improved Physics-Based Dual-Band Model for Satellite-Derived Bathymetry Using SuperDove Imagery
by Chunlong He, Qigang Jiang and Peng Wang
Remote Sens. 2024, 16(20), 3801; https://doi.org/10.3390/rs16203801 - 12 Oct 2024
Viewed by 615
Abstract
Shallow water bathymetry is critical for environmental monitoring and maritime security. Current widely used statistical models based on passive optical satellite remote sensing often rely on prior bathymetric data, limiting their application to regions lacking such information. In contrast, the physics-based dual-band log-linear [...] Read more.
Shallow water bathymetry is critical for environmental monitoring and maritime security. Current widely used statistical models based on passive optical satellite remote sensing often rely on prior bathymetric data, limiting their application to regions lacking such information. In contrast, the physics-based dual-band log-linear analytical model (P-DLA) can estimate shallow water bathymetry without in situ measurements, offering significant potential. However, the quasi-analytical algorithm (QAA) used in the P-DLA is sensitive to non-ideal pixels, resulting in unstable bathymetry estimation. To address this issue and evaluate the potential of SuperDove imagery for bathymetry estimation in regions without prior bathymetric data, this study proposes an improved physics-based dual-band model (IPDB). The IPDB replaces the QAA with a spectral optimization algorithm that integrates deep and shallow water sample pixels to estimate diffuse attenuation coefficients for the blue and green bands. This allows for more accurate estimation of shallow water bathymetry. The IPDB was tested on SuperDove images of Dongdao Island, Yongxing Island, and Yongle Atoll. The results showed that SuperDove images are capable of estimating shallow water bathymetry in regions without prior bathymetric data. The IPDB achieved Root Mean Square Error (RMSE) values below 1.7 m and R2 values above 0.89 in all three study areas, indicating strong performance in bathymetric estimation. Notably, the IPDB outperformed the standard P-DLA model in accuracy. Furthermore, this study outlines four sampling principles that, when followed, ensure that variations in the spatial distribution of sampling pixels do not significantly impact model performance. This study also showed that the blue–green band combination is optimal for the analytical expression of the physics-based dual-band model. Full article
(This article belongs to the Special Issue Advances in Remote Sensing of the Inland and Coastal Water Zones II)
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Figure 1
<p>Geographic distribution of the study area.</p>
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<p>Spatial distribution of measured bathymetric data. (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll. The yellow text in the figures indicates the collection dates of the nearest bathymetric lines.</p>
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<p>Technical workflow for satellite-derived bathymetry.</p>
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<p>Distribution of sample pixels: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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<p>Estimated <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mrow> </mrow> </semantics></math> ratios from the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>~<math display="inline"><semantics> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> datasets for the same substrate type but different depths: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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<p>Bathymetry maps derived from the IPDB model: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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<p>Scatter plots comparing estimated and measured water depths using the IPDB model: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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<p>Distribution of different pixels pairs: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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<p>Distribution of different waterlines: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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<p>Distribution of different deep water regions: (<b>a</b>) Dongdao Island; (<b>b</b>) Yongxing Island; (<b>c</b>) Yongle Atoll.</p>
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25 pages, 3228 KiB  
Article
Microexpression Recognition Method Based on ADP-DSTN Feature Fusion and Convolutional Block Attention Module
by Junfang Song, Shanzhong Lei and Wenzhe Wu
Electronics 2024, 13(20), 4012; https://doi.org/10.3390/electronics13204012 - 12 Oct 2024
Viewed by 658
Abstract
Microexpressions are subtle facial movements that occur within an extremely brief time frame, often revealing suppressed emotions. These expressions hold significant importance across various fields, including security monitoring and human–computer interaction. However, the accuracy of microexpression recognition is severely constrained by the inherent [...] Read more.
Microexpressions are subtle facial movements that occur within an extremely brief time frame, often revealing suppressed emotions. These expressions hold significant importance across various fields, including security monitoring and human–computer interaction. However, the accuracy of microexpression recognition is severely constrained by the inherent characteristics of these expressions. To address the issue of low detection accuracy regarding the subtle features present in microexpressions’ facial action units, this paper proposes a microexpression action unit detection algorithm, Attention-embedded Dual Path and Shallow Three-stream Networks (ADP-DSTN), that incorporates an attention-embedded dual path and a shallow three-stream network. First, an attention mechanism was embedded after each Bottleneck layer in the foundational Dual Path Networks to extract static features representing subtle texture variations that have significant weights in the action units. Subsequently, a shallow three-stream 3D convolutional neural network was employed to extract optical flow features that were particularly sensitive to temporal and discriminative characteristics specific to microexpression action units. Finally, the acquired static facial feature vectors and optical flow feature vectors were concatenated to form a fused feature vector that encompassed more effective information for recognition. Each facial action unit was then trained individually to address the issue of weak correlations among the facial action units, thereby facilitating the classification of microexpression emotions. The experimental results demonstrated that the proposed method achieved great performance across several microexpression datasets. The unweighted average recall (UAR) values were 80.71%, 89.55%, 44.64%, 80.59%, and 88.32% for the SAMM, CASME II, CAS(ME)3, SMIC, and MEGC2019 datasets, respectively. The unweighted F1 scores (UF1) were 79.32%, 88.30%, 43.03%, 81.12%, and 88.95%, respectively. Furthermore, when compared to the benchmark model, our proposed model achieved better performance with lower computational complexity, characterized by a Floating Point Operations (FLOPs) value of 1087.350 M and a total of 6.356 × 106 model parameters. Full article
(This article belongs to the Section Artificial Intelligence)
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Figure 1
<p>CASME II dataset showing peak frames corresponding to different emotions and their action units.</p>
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<p>Framework diagram of the microexpression recognition method based on the ADP-DSTN feature fusion and Convolutional Block Attention Module.</p>
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<p>Network structure diagram of microexpression optical flow feature extraction.</p>
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<p>Structure of the micro-block.</p>
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<p>Computational unit structure of a single structural block in the ADPN.</p>
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<p>The overall structure of the ADPN algorithm.</p>
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<p>Adaptive weight feature fusion module diagram.</p>
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<p>The UF1 and UAR values obtained from the CASME II, CAS(ME)<sup>3</sup>, SAMM, and SMIC datasets with only three emotion categories, presented as line plots: (<b>a</b>) Comparison on the CAS(ME)<sup>3</sup> dataset; (<b>b</b>) Comparison on the CASME II dataset; (<b>c</b>) Comparison on the SAMM dataset; (<b>d</b>) Comparison on the SMIC dataset.</p>
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<p>MEGC2019, CASME II, SMIC, and SAMM classification result confusion matrix.</p>
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<p>Introducing the attention mechanism to improve the visualization of network feature maps.</p>
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23 pages, 15644 KiB  
Article
Effects of Perforated Plates on Shock Structure Alteration for NACA0012 Cascade Configurations
by Mihnea Gall, Oana Dumitrescu, Valeriu Drăgan and Daniel-Eugeniu Crunțeanu
Inventions 2024, 9(5), 110; https://doi.org/10.3390/inventions9050110 - 6 Oct 2024
Viewed by 721
Abstract
To alleviate the shock boundary layer interaction adverse effects, various active or passive flow control strategies have been investigated in the literature. This research sheds light on the behavior of perforated plates as passive flow control techniques applied to NACA0012 airfoils in cascade [...] Read more.
To alleviate the shock boundary layer interaction adverse effects, various active or passive flow control strategies have been investigated in the literature. This research sheds light on the behavior of perforated plates as passive flow control techniques applied to NACA0012 airfoils in cascade configurations. Two identical perforated plates with shallow cavities underneath are accommodated on the upper and lower surfaces of each airfoil in the cascade arrangement. Six different cascade arrangements, including a baseline configuration with no control applied, are additively manufactured, with different perforated plate orifice sizes in the range of 0.5–1.2 mm. A high-speed wind tunnel with Schlieren optical diagnosis and wall static pressure taps is used to investigate the changes in the shock waves pattern triggered by the perforated plates. Steady 3D density-based numerical simulations in Ansys FLUENT are conducted for further analysis and validation. In the cascade configuration, the perforated plates alter the shock structure, and the strong normal shock wave is replaced by a weaker X-type shock structure. Eventually, a 1% penalty in overall total pressure loss is induced by the perforated plates because of the negative loss balance between the reduced shock losses and the enhanced viscous losses. Further studies on perforated plate geometrical features are needed to improve this outcome in a cascade arrangement. Full article
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Figure 1
<p>‘Eiffel’-type open wind tunnel facility overview [<a href="#B35-inventions-09-00110" class="html-bibr">35</a>].</p>
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<p>Wind tunnel flow channel sketch.</p>
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<p>Schlieren system diagram [<a href="#B14-inventions-09-00110" class="html-bibr">14</a>].</p>
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<p>Machined plexiglass front window.</p>
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<p>PLA jig for cascade incidence alignment.</p>
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<p>NACA 0012 cascade with incidence alignment jig.</p>
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<p>NACA 0012 cascade with optical access windows sealed.</p>
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<p>PLA 3D-printed baseline airfoil (no control).</p>
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<p>PLA 3D-printed airfoils (0.5 mm, 0.65 mm, 0.8 mm, 1 mm, and 1.2 mm hole diameters).</p>
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<p>PLA 3D-printed airfoil (0.5 mm hole diameter).</p>
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<p>PLA 3D-printed airfoil (1.2 mm hole diameter).</p>
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<p>Schlieren images for A1–A6 airfoils operating in cascade configuration. (<b>a</b>) A1—no control; (<b>b</b>) A2—0.5 mm; (<b>c</b>) A3—0.65 mm; (<b>d</b>) A4—0.8 mm; (<b>e</b>) A5—1 mm; (<b>f</b>) A6—1.2 mm.</p>
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<p>Static pressure distribution on the bottom wall of the wind tunnel.</p>
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<p>Static pressure distribution on the bottom wall of the wind tunnel—in detail.</p>
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<p>Computational domain.</p>
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<p>Computational grid—passive control case 0.5 mm: (<b>a</b>) grid overview; (<b>b</b>) cascade detail; (<b>c</b>) perforated plate detail; (<b>d</b>) hole detail; (<b>e</b>) leading edge; (<b>f</b>) trailing edge.</p>
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<p>y+ distribution on the blades: (<b>a</b>) no control; (<b>b</b>) passive control 0.5 mm.</p>
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<p>Grid independence study.</p>
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<p>Density gradient distribution—mid plane. (<b>a</b>) No control; (<b>b</b>) passive control 0.5 mm.</p>
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<p>Mach number distribution—mid plane. (<b>a</b>) No control; (<b>b</b>) passive control 0.5 mm.</p>
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<p>Static pressure distribution—mid plane. (<b>a</b>) No control; (<b>b</b>) passive control 0.5 mm.</p>
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<p>Reference lines. (<b>a</b>) Bottom wall; (<b>b</b>) mid lower channel.</p>
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<p>Bottom wall static pressure distribution.</p>
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<p>Static pressure on lower mid-channel reference line.</p>
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<p>Mach number on lower mid-channel reference line.</p>
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<p>Density on lower mid-channel reference line.</p>
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<p>Non-dimensional streamwise velocity profiles—CFD. (<b>a</b>) Rake 1; (<b>b</b>) Rake 2; (<b>c</b>) Rake 3; (<b>d</b>) Rake 4; (<b>e</b>) Rake 5; (<b>f</b>) Rake 6; (<b>g</b>) Rake 7; (<b>h</b>) Rake 8; (<b>i</b>) Rake 9; (<b>j</b>) position of measuring rakes.</p>
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<p>Non-dimensional streamwise velocity profiles—CFD. (<b>a</b>) Rake 1; (<b>b</b>) Rake 2; (<b>c</b>) Rake 3; (<b>d</b>) Rake 4; (<b>e</b>) Rake 5; (<b>f</b>) Rake 6; (<b>g</b>) Rake 7; (<b>h</b>) Rake 8; (<b>i</b>) Rake 9; (<b>j</b>) position of measuring rakes.</p>
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<p>Comparison of static pressure distribution on airfoil 2–0.5 mm passive control vs. no control.</p>
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<p>CFD vs. experiment qualitative comparison. (<b>a</b>) No-control CFD; (<b>b</b>) passive control 0.5 mm CFD; (<b>c</b>) no-control EXP; (<b>d</b>) passive control 0.5 mm EXP.</p>
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19 pages, 4247 KiB  
Article
Remote Sensing of Chlorophyll-a in Clear vs. Turbid Waters in Lakes
by Forough Fendereski, Irena F. Creed and Charles G. Trick
Remote Sens. 2024, 16(19), 3553; https://doi.org/10.3390/rs16193553 - 24 Sep 2024
Viewed by 840
Abstract
Chlorophyll-a (Chl-a), a proxy for phytoplankton biomass, is one of the few biological water quality indices detectable using satellite observations. However, models for estimating Chl-a from satellite signals are currently unavailable for many lakes. The application of Chl-a [...] Read more.
Chlorophyll-a (Chl-a), a proxy for phytoplankton biomass, is one of the few biological water quality indices detectable using satellite observations. However, models for estimating Chl-a from satellite signals are currently unavailable for many lakes. The application of Chl-a prediction algorithms may be affected by the variance in optical complexity within lakes. Using Lake Winnipeg in Canada as a case study, we demonstrated that separating models by the lake’s basins [north basin (NB) and south basin (SB)] can improve Chl-a predictions. By calibrating more than 40 commonly used Chl-a estimation models using Landsat data for Lake Winnipeg, we achieved higher correlations between in situ and predicted Chl-a when building models with separate Landsat-to-in situ matchups from NB and SB (R2 = 0.85 and 0.76, respectively; p < 0.05), compared to using matchups from the entire lake (R2 = 0.38, p < 0.05). In the deeper, more transparent waters of the NB, a green-to-blue band ratio provided better Chl-a predictions, while in the shallower, highly turbid SB, a red-to-green band ratio was more effective. Our approach can be used for rapid Chl-a modeling in large lakes using cloud-based platforms like Google Earth Engine with any available satellite or time series length. Full article
(This article belongs to the Special Issue Remote Sensing of Aquatic Ecosystem Monitoring)
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<p>Map of Lake Winnipeg showing the location of the two basins and the Narrows, as well as the location of the in situ Chl-<span class="html-italic">a</span> sampling points used for studying turbidity (colored circles) and those used for cross-comparison of ETM+ and OLI modeling (white circles).</p>
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<p>Flowchart of methods used in the study.</p>
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<p>Coefficients of determination (R<sup>2</sup>) of the BPM among 45 tested algorithms for 36 sets of matchups with different temporal windows (0 to ±3 days) and different spatial windows (1 × 1, 3 × 3, and 5 × 5 pixels) in the NB, SB, and LW. Missing values on the plot refer to (1) coefficients of determination with no significant relationships between in situ and satellite Chl-<span class="html-italic">a</span> (<span class="html-italic">p</span> ≥ 0.05) or (2) where the number of matchups is less than 10.</p>
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<p>Relationship of modeled ln (Chl-<span class="html-italic">a</span>) as a function of observed ln (Chl-<span class="html-italic">a</span>) using the BPMs in (<b>a</b>) the NB, (<b>b</b>) the SB, and (<b>c</b>) the LW. The basin-specific BPMs were based on matchups from ±2 days with a 1 × 1 pixel spatial window (<span class="html-italic">n</span> = 17 in NB and <span class="html-italic">n</span> = 20 in SB), and the lake-specific BPM was based on matchups from ±1 day with a 1 × 1 pixel spatial window (<span class="html-italic">n</span> = 17). Because the number of matchups for the NB and SB was relatively small and expanding the window from ±1 day to ±2 days provided more matchups but a still-high R<sup>2</sup>, we used models derived from satellite data within ±2 days of sampling as our BPMs.</p>
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<p>Natural log-transformed Chl-<span class="html-italic">a</span> predictions using OLI and ETM+ for NB, SB, and LW.</p>
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<p>Median natural log-transformed peak Chl-<span class="html-italic">a</span> (µg L<sup>−1</sup>) in the NB and SB during peak phytoplankton biomass (July to October) from 1984 to 2023 using (<b>a</b>) basin-specific vs. (<b>b</b>) lake-specific BPMs. Values were normalized for each basin.</p>
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<p>Annual time series (1984 to 2023) of peak Chl-<span class="html-italic">a</span> (µg L<sup>−1</sup>) during peak phytoplankton biomass in LW (July to October) in the NB (blue lines) and SB (green lines) using basin-specific (solid lines) and lake-specific (dashed lines) BPMs.</p>
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12 pages, 5100 KiB  
Article
Determination of the Temperature Development in a Borehole Heat Exchanger Field Using Distributed Temperature Sensing
by David Bertermann and Oliver Suft
Energies 2024, 17(18), 4697; https://doi.org/10.3390/en17184697 - 20 Sep 2024
Viewed by 674
Abstract
The use of geothermal borehole heat exchangers (BHEs) in combination with ground-source heat pumps represents an important part of shallow geothermal energy production, which is already used worldwide and becoming more and more important. Different measurement techniques are available to examine a BHE [...] Read more.
The use of geothermal borehole heat exchangers (BHEs) in combination with ground-source heat pumps represents an important part of shallow geothermal energy production, which is already used worldwide and becoming more and more important. Different measurement techniques are available to examine a BHE field while it is in operation. In this study, a field with 54 BHEs up to a depth of 120 m below ground level was analyzed using fiber optic cables. A distributed temperature sensing (DTS) concept was developed by equipping several BHEs with dual-ended hybrid cables. The individual fiber optics were collected in a distributor shaft, and multiple measurements were carried out during active and inactive operation of the field. The field trial was carried out on a converted, partly retrofitted, residential complex, “Lagarde Campus”, in Bamberg, Upper Franconia, Germany. Groundwater and lithological changes are visible in the depth-resolved temperature profiles throughout the whole BHE field. Full article
(This article belongs to the Section H2: Geothermal)
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<p>BHE field in Bamberg, Germany. Blue dots depict the 54 BHEs, red dots depict the BHEs equipped with fiber optic cables. The central red square is the distributor shaft where the BHE connections are brought together; base map source: OpenStreetMaps.</p>
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<p>Installation steps and setup of the fiber optic measurement system. (<b>a</b>) Base of the BHE before insertion into the borehole with attached fiber optic cable; (<b>b</b>) trench for the connection of the BHEs with the distributor shaft, where fiber optic cables are fixed with zip ties; (<b>c</b>) distributor shaft on site before burying; (<b>d</b>) measuring boxes for fiber optic connection (<b>left</b>) and reference boxes for determination of the reference temperatures (<b>right</b>) in the distribution shaft.</p>
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<p>Schematic illustration of the components for distributed temperature sensing, pictures showing the inside of the distributor shaft and the above-ground measurement setup in winter conditions; DTS: distributed temperature sensing interrogator.</p>
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<p>Depth-resolved temperature profiles for the seven 120 m below-ground-level (b. g. l.) BHEs. Each line indicates one fiber optic cable. (<b>a</b>) December 2023, no active fluid input; (<b>b</b>) January 2024, no active fluid input; (<b>c</b>) February 2024, active fluid input; (<b>d</b>) June 2024, no active fluid input.</p>
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<p>DTS temperature profiles of BHE 30 (S30) compared with measurements of a conventional P-T-data logger, D+K MikroLog 2 (ML), on the same borehole.</p>
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<p>DTS temperature profile of central BHE 34 (S34) with lithological units of Quaternary and middle Keuper drilled on site; textures according to DIN 4023 [<a href="#B40-energies-17-04697" class="html-bibr">40</a>]; GWL: groundwater level.</p>
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23 pages, 11057 KiB  
Article
Denoising of Photon-Counting LiDAR Bathymetry Based on Adaptive Variable OPTICS Model and Its Accuracy Assessment
by Peize Li, Yangrui Xu, Yanpeng Zhao, Kun Liang and Yuanjie Si
Remote Sens. 2024, 16(18), 3438; https://doi.org/10.3390/rs16183438 - 16 Sep 2024
Viewed by 542
Abstract
Spaceborne photon-counting LiDAR holds significant potential for shallow-water bathymetry. However, the received photon data often contain substantial noise, complicating the extraction of elevation information. Currently, a denoising algorithm named ordering points to identify the clustering structure (OPTICS) draws people’s attention because of its [...] Read more.
Spaceborne photon-counting LiDAR holds significant potential for shallow-water bathymetry. However, the received photon data often contain substantial noise, complicating the extraction of elevation information. Currently, a denoising algorithm named ordering points to identify the clustering structure (OPTICS) draws people’s attention because of its strong performance under high background noise. However, this algorithm’s fixed input variables can lead to inaccurate photon distribution parameters in areas near the water bottom, which results in inadequate denoising in these areas, affecting bathymetric accuracy. To address this issue, an Adaptive Variable OPTICS (AV-OPTICS) model is proposed in this paper. Unlike the traditional OPTICS model with fixed input variables, the proposed model dynamically adjusts input variables based on point cloud distribution. This adjustment ensures accurate measurement of photon distribution parameters near the water bottom, thereby enhancing denoising effects in these areas and improving bathymetric accuracy. The findings indicate that, compared to traditional OPTICS methods, AV-OPTICS achieves higher F1-values and lower cohesions, demonstrating better denoising performance near the water bottom. Furthermore, this method achieves an average MAE of 0.28 m and RMSE of 0.31 m, indicating better bathymetric accuracy than traditional OPTICS methods. This study provides a promising solution for shallow-water bathymetry based on photon-counting LiDAR data. Full article
(This article belongs to the Special Issue Satellite Remote Sensing for Ocean and Coastal Environment Monitoring)
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<p>Study area and the detection tracks of ATLAS represented by six different colors of dashed lines.</p>
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<p>Water depth results of study area for ALB reference data.</p>
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<p>The elevation distribution histogram of ATL03 photon data.</p>
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<p>The contours of the water bottom terrain under different scenarios. (<b>a</b>) relatively flat water bottom terrains; (<b>b</b>) complex water bottom terrains.</p>
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<p>The distance of point <span class="html-italic">o</span> and <span class="html-italic">w</span> under the definition of OPTICS algorithm.</p>
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<p>The spatial geometric relationships of refraction correction under different slope angles <math display="inline"><semantics> <mrow> <mi>φ</mi> </mrow> </semantics></math>. The green and red vectors correspond to the original coordinate and corrected coordinate of water bottom photons, respectively [<a href="#B38-remotesensing-16-03438" class="html-bibr">38</a>].</p>
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<p>Denoising effects of our method and traditional OPTICS in different scenes. (<b>a</b>) 20190119gt3l—Raw data; (<b>b</b>) 20190119gt3l—Our method; (<b>c</b>) 20190119gt3l—Traditional OPTICS; (<b>d</b>) 20181024gt3r—Raw data; (<b>e</b>) 20181024gt3r—Our method; (<b>f</b>) 20181024gt3r—Traditional OPTICS; (<b>g</b>) 20200717gt3l—Raw data; (<b>h</b>) 20200717gt3l—Our method; (<b>i</b>) 20200717gt3l—Traditional OPTICS.</p>
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<p>Comparison of denoising details of the two methods in different scenarios. (<b>a</b>) 20181024gt3r—Our method; (<b>b</b>) 20181024gt3r—Traditional OPTICS; (<b>c</b>) 20190420gt2l—Our method; (<b>d</b>) 20190420gt2l—Traditional OPTICS.</p>
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<p>Comparison of denoising details of the two methods in different scenarios. (<b>a</b>) 20181024gt3r—Our method; (<b>b</b>) 20181024gt3r—Traditional OPTICS; (<b>c</b>) 20190420gt2l—Our method; (<b>d</b>) 20190420gt2l—Traditional OPTICS.</p>
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<p>Coordinate correction and fitting profiles of the signal photons. (<b>a</b>) 20190119gt3l—Coordinate correction; (<b>b</b>) 20190119gt3l—Fitting profiles; (<b>c</b>) 20181024gt3r—Coordinate correction; (<b>d</b>) 20181024gt3r—Fitting profiles.</p>
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<p>Coordinate correction and fitting profiles of the signal photons. (<b>a</b>) 20190119gt3l—Coordinate correction; (<b>b</b>) 20190119gt3l—Fitting profiles; (<b>c</b>) 20181024gt3r—Coordinate correction; (<b>d</b>) 20181024gt3r—Fitting profiles.</p>
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<p>Bathymetric accuracy validation and comparison of our method and traditional OPTICS. (<b>a</b>,<b>b</b>) on the first row correspond to 20190119gt3l, while (<b>c</b>,<b>d</b>) on the second row correspond to 20190420gt2l. The red and black points represent the bathymetric results of the corresponding method and in situ data, respectively. (<b>a</b>) 20190119gt3l—Our method; (<b>b</b>) 20190119gt3l—Traditional OPTICS; (<b>c</b>) 20190420gt2l—Our method; (<b>d</b>) 20190420gt2l—Traditional OPTICS.</p>
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<p>Idealized model of elliptic filter in vertical direction. The water bottom contour within the black block is regarded as a gray rectangle. The yellow, red, blue, and green lines are the idealized elliptical filters with different lengths of semi-minor axis, respectively.</p>
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<p>Idealized model of elliptic filter in horizontal direction. The water bottom contour within the black block is regarded as a gray rectangle. The yellow and red lines are the idealized elliptical filters with different lengths of semi-major axis, respectively.</p>
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<p>Deviation percentage between ICESat-2 results and ALB in situ data, with an interval of 0.5 m for each histogram column. (<b>a</b>,<b>b</b>) on the first row correspond to 20190119gt3l, while (<b>c</b>,<b>d</b>) on the second row correspond to 20201016gt2l. (<b>a</b>) 20190119gt3l—Our method; (<b>b</b>) 20190119gt3l—Traditional OPTICS; (<b>c</b>) 20201016gt2l—Our method; (<b>d</b>) 20201016gt2l—Traditional OPTICS.</p>
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<p>Bathymetric accuracy comparison of our method and that without coordinate correction. (<b>a</b>–<b>c</b>) on the first row correspond to 20190119gt3l, while (<b>d</b>–<b>f</b>) on the second row correspond to 20181024gt3r. The red and black lines represent the bathymetric results of the corresponding method and in situ data, respectively. (<b>a</b>) 20190119gt3r—Our method; (<b>b</b>) Without refraction correction; (<b>c</b>) Without tidal correction; (<b>d</b>) 20181024gt3r—Our method; (<b>e</b>) Without refraction correction; (<b>f</b>) Without tide correction.</p>
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20 pages, 20184 KiB  
Article
Snow Cover Extraction from Landsat 8 OLI Based on Deep Learning with Cross-Scale Edge-Aware and Attention Mechanism
by Zehao Yu, Hanying Gong, Shiqiang Zhang and Wei Wang
Remote Sens. 2024, 16(18), 3430; https://doi.org/10.3390/rs16183430 - 15 Sep 2024
Viewed by 976
Abstract
Snow cover distribution is of great significance for climate change and water resource management. Current deep learning-based methods for extracting snow cover from remote sensing images face challenges such as insufficient local detail awareness and inadequate utilization of global semantic information. In this [...] Read more.
Snow cover distribution is of great significance for climate change and water resource management. Current deep learning-based methods for extracting snow cover from remote sensing images face challenges such as insufficient local detail awareness and inadequate utilization of global semantic information. In this study, a snow cover extraction algorithm integrating cross-scale edge perception and an attention mechanism on the U-net model architecture is proposed. The cross-scale edge perception module replaces the original jump connection of U-net, enhances the low-level image features by introducing edge detection on the shallow feature scale, and enhances the detail perception via branch separation and fusion features on the deep feature scale. Meanwhile, parallel channel and spatial attention mechanisms are introduced in the model encoding stage to adaptively enhance the model’s attention to key features and improve the efficiency of utilizing global semantic information. The method was evaluated on the publicly available CSWV_S6 optical remote sensing dataset, and the accuracy of 98.14% indicates that the method has significant advantages over existing methods. Snow extraction from Landsat 8 OLI images of the upper reaches of the Irtysh River was achieved with satisfactory accuracy rates of 95.57% (using two, three, and four bands) and 96.65% (using two, three, four, and six bands), indicating its strong potential for automated snow cover extraction over larger areas. Full article
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<p>True color CSWV_S6 data synthesized from the red, green, and blue bands (the numbering in the figure corresponds to the original naming in the acquired files).</p>
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<p>(<b>a</b>) RGB composite of Landsat 8 imagery (red: band 4, green: band 3, blue: band 2). (<b>b</b>) Land cover types.</p>
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<p>CEFCSAU-net network model architecture. The input size was (512, 512, C), where C denotes the number of channels, and experiments in this paper utilized either 3 or 4; during the model’s operation on a GPU, intermediate feature maps were stored as tensors.</p>
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<p>Attention mechanism module for channel and space mixing. Here, (H, W, C) represent the height, width, and number of channels of the feature data, respectively, with values determined by input features at different stages. CF and CF’ denote feature maps from various intermediate operations within the channel attention mechanism. Cat Sf, Sf, Sf’ represent feature maps from different intermediate operations of the spatial attention mechanism. The SA feature denotes the feature map post-spatial attention mechanism, the CA feature represents those post-channel attention mechanisms, and the CSA feature illustrates feature maps following the CSA module.</p>
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<p>Cross-scale edge-aware feature fusion module. Sobelx F, Sobely F, and Laplacian F denote feature maps resulting from various edge detection operations. Shallow F refers to feature maps following shallow feature convolution. Fusion F illustrates feature maps resulting from the fusion of shallow and deep features. Deep F’ represents feature maps after a series of operations on deep features.</p>
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<p>Snow extraction results of CSWV_S6 data on different segmentation models (a set of two rows is the same image data, and rows two, four, and six are zoomed-in images of local details corresponding to rows one, three, and five. The blue area is snow, the white area is non-snow, and the red area is false detection).</p>
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<p>Snow extraction results from different deep learning models for Landsat 8 OLI imagery (blue areas are snow, white areas are non-snow, and red areas are false detections).</p>
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<p>(<b>a</b>) CSWV_S6 test set scores on different models for each type of metrics and (<b>b</b>) Landsat 8 OLI test set scores for various metrics on different models.</p>
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<p>Score results of the three CSWV_S6 test sets’ example data on the evaluation metrics on each model, with 0.08% of snow image elements in the first row of data, 0.95% of snow image elements in the second row of data, and 1.73% of data in the third row of data.</p>
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<p>Map of CEFCSAU-net model’s snow extraction in the cloud–snow confusion scenario of CSWV_S6 test set.</p>
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<p>Heat map comparing the mean values of ablation experiments on the test set with different data sets: (<b>a</b>) CSWV_6 dataset, (<b>b</b>) Landsat8 OLI dataset.</p>
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<p>(<b>a</b>) Input data image; (<b>b</b>) feature map of the first 8 channels before the intermediate feature data first pass through the CSA module; (<b>c</b>) feature map of the first 8 channels after the feature data first pass through the CSA module.</p>
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<p>This figure displays average feature maps following skip connections at various stages under different configurations of the CEFCSAU-net model. In this figure, (<b>a1</b>–<b>a4</b>) represent the model configuration without both the CSA and CEF modules; (<b>b1</b>–<b>b4</b>) indicate configurations without the CSA module yet including the CEF module; and (<b>c1</b>–<b>c4</b>) depict configurations featuring both CSA and CEF modules. The dimensions of the four columns of feature maps are sequentially 512 × 512, 256 × 256, 128 × 128, and 64 × 64.</p>
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26 pages, 6325 KiB  
Article
Acquisition of Bathymetry for Inland Shallow and Ultra-Shallow Water Bodies Using PlanetScope Satellite Imagery
by Aleksander Kulbacki, Jacek Lubczonek and Grzegorz Zaniewicz
Remote Sens. 2024, 16(17), 3165; https://doi.org/10.3390/rs16173165 - 27 Aug 2024
Viewed by 1020
Abstract
This study is structured to address the problem of mapping the bottom of shallow and ultra-shallow inland water bodies using high-resolution satellite imagery. These environments, with their diverse distribution of optically relevant components, pose a challenge to traditional mapping methods. The study was [...] Read more.
This study is structured to address the problem of mapping the bottom of shallow and ultra-shallow inland water bodies using high-resolution satellite imagery. These environments, with their diverse distribution of optically relevant components, pose a challenge to traditional mapping methods. The study was conducted on several research issues, each focusing on a specific aspect of the SDB, related to the selection of spectral bands and regression models, regression models creation, evaluation of the influence of the number and spatial distribution of reference soundings, and assessment of the quality of the bathymetric surface, with a focus on microtopography. The study utilized basic empirical techniques, incorporating high-precision reference data acquired via an unmanned surface vessel (USV) integrated with a single-beam echosounder (SBES), and Global Navigation Satellite System (GNSS) receiver measurements. The performed investigation allowed the optimization of a methodology for bathymetry acquisition of such areas by identifying the impact of individual processing components. The first results indicated the usefulness of the proposed approach, which can be confirmed by the values of the obtained RMS errors of elaborated bathymetric surfaces in the range of up to several centimeters in some study cases. The work also points to the problematic nature of this type of study, which can contribute to further research into the application of remote sensing techniques for bathymetry, especially during acquisition in optically complex waters. Full article
(This article belongs to the Section Environmental Remote Sensing)
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<p>Maps of the study area: view of the area denoted by the hydrographic profiles by Lubczyna (<b>a</b>) and by Czarna Laka (<b>b</b>); view of the locations of Dabie Lake (<b>c</b>) and studied bays locations at smaller scale (<b>d</b>).</p>
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<p>Photo of unmanned survey vehicle (<b>a</b>) and GNSS RTK (<b>b</b>) used for data acquisition during the survey campaign.</p>
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<p>General location of GNSS profile measurement: Crosses mark the occurrence of a single profile (<b>a</b>); an example of a profile surveyed using the GNSS RTK technique. Each dot symbolizes a single measurement point (<b>b</b>).</p>
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<p>Overview of the spectral ranges of the PlanetScope data and imagery in RGB composition used in the study: (<b>a</b>) view of the whole Dabie Lake; (<b>b</b>) close-up of the bay by Czarna Laka; (<b>c</b>) close-up of the bay by Lubczyna.</p>
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<p>Digital bathymetric models: bay near Czarna Laka village (<b>a</b>); bay near Lubczyna village (<b>b</b>).</p>
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<p>Comparison of GEBCO method for creation of the regression model using observations and threshold depth with a classical approach: the whole set of the observations (<b>a</b>); set of averaged raster values for depth intervals of 0.1 m with an area of TD, marked by the red box (<b>b</b>); set of data after removing observation at threshold depth (<b>c</b>).</p>
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<p>Flow chart of the research methodology.</p>
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<p>Regression models based on GEBCO methodology for the selected bands: (<b>a</b>) bay by Lubczyna—linear model with a threshold depth of 2.3 m, green band; (<b>b</b>) bay by Czarna Laka—linear model with a threshold depth of 2.7 m, yellow band.</p>
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<p>Regression models reaching the highest R<sup>2</sup> coefficient for the selected bands: (<b>a</b>) bay near Lubczyna—polynomial model with a threshold depth of 2.3 m; (<b>b</b>) bay near Lubczyna—linear combined model with a threshold depth of 2.3 m; (<b>c</b>) bay near Czarna Laka—power model with a threshold depth of 2.7 m.</p>
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<p>Regression models used for the given options based on methodology defined in stage 1 and 2: (<b>a</b>) Experiment 3.1; (<b>b</b>) Experiment 3.2; (<b>c</b>) Experiment 3.3.1; (<b>d</b>) Experiment 3.3.2; (<b>e</b>) Experiment 3.4; (<b>f</b>) Experiment 3.5.1; (<b>g</b>) Experiment 3.5.2.</p>
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<p>Summary of RMS errors for each case: (<b>a</b>) Comparison of the obtained RMS errors in the bay near Czarna Laka regarding the analyzed experiments; (<b>b</b>) Comparison of the obtained RMS errors in the bay near Lubczyna regarding the analyzed experiments; (<b>c</b>) comparison of the obtained RMS error on the GNSS Profiles regarding the analyzed experiments.</p>
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<p>Difference surfaces for the bay by Czarna Laka between the reference bathymetric models and the SDB models based on each experiment: (<b>a</b>) Experiment 3.1; (<b>b</b>) Experiment 3.2; (<b>c</b>) Experiment 3.3.1; (<b>d</b>) Experiment 3.3.2; (<b>e</b>) Experiment 3.4; (<b>f</b>) Experiment 3.5.1; (<b>g</b>) Experiment 3.5.2.</p>
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<p>Difference surfaces for the bay by Lubczyna between the reference bathymetric models and the SDB models based on each experiment: (<b>a</b>) Experiment 3.1; (<b>b</b>) Experiment 3.2; (<b>c</b>) Experiment 3.3.1; (<b>d</b>) Experiment 3.3.2; (<b>e</b>) Experiment 3.4; (<b>f</b>) Experiment 3.5.1; (<b>g</b>) Experiment 3.5.2.</p>
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<p>Variety of materials forming the bottom surfaces. Referenced bathymetric model of the bay by Lubczyna overlay on UAV RGB mosaic image.</p>
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<p>Variation in RMSE values with respect to the depth intervals: (<b>a</b>) Differential model for the bay by Czarna Laka based on experiment 3.1 with isobaths from the reference model overlaid; (<b>b</b>) Model for the bay by Lubczyna based on experiment 3.2 with isobaths from the reference model overlaid; (<b>c</b>) RMS error value in relation to depth interval for the bay by Czarna Laka; (<b>d</b>) RMS error value in relation to depth interval for the bay by Lubczyna.</p>
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<p>DBMs in 3D view for the bay by Czarna Laka elaborated from various data: SBES, pixel size: 1.3 m (<b>a</b>); UAV, pixel size 0.05 m (<b>b</b>); PlanetScope, pixel size: 3 m (<b>c</b>); Sentinel-2, pixel size: 10 m (<b>d</b>).</p>
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18 pages, 14973 KiB  
Article
Developing a Generalizable Spectral Classifier for Rhodamine Detection in Aquatic Environments
by Ámbar Pérez-García, Alba Martín Lorenzo, Emma Hernández, Adrián Rodríguez-Molina, Tim H. M. van Emmerik and José F. López
Remote Sens. 2024, 16(16), 3090; https://doi.org/10.3390/rs16163090 - 22 Aug 2024
Cited by 2 | Viewed by 739
Abstract
In environmental studies, rhodamine dyes are commonly used to trace water movements and pollutant dispersion. Remote sensing techniques offer a promising approach to detecting rhodamine and estimating its concentration, enhancing our understanding of water dynamics. However, research is needed to address more complex [...] Read more.
In environmental studies, rhodamine dyes are commonly used to trace water movements and pollutant dispersion. Remote sensing techniques offer a promising approach to detecting rhodamine and estimating its concentration, enhancing our understanding of water dynamics. However, research is needed to address more complex environments, particularly optically shallow waters, where bottom reflectance can significantly influence the spectral response of the rhodamine. Therefore, this study proposes a novel approach: transferring pre-trained classifiers to develop a generalizable method across different environmental conditions without the need for in situ calibration. Various samples incorporating distilled and seawater on light and dark backgrounds were analyzed. Spectral analysis identified critical detection regions (400–500 nm and 550–650 nm) for estimating rhodamine concentration. Significant spectral variations were observed between light and dark backgrounds, highlighting the necessity for precise background characterization in shallow waters. Enhanced by the Sequential Feature Selector, classification models achieved robust accuracy (>90%) in distinguishing rhodamine concentrations, particularly effective under controlled laboratory conditions. While band transfer was successful (>80%), the transfer of pre-trained models posed a challenge. Strategies such as combining diverse sample sets and applying the first derivative prevent overfitting and improved model generalizability, surpassing 85% accuracy across three of the four scenarios. Therefore, the methodology provides us with a generalizable classifier that can be used across various scenarios without requiring recalibration. Future research aims to expand dataset variability and enhance model applicability across diverse environmental conditions, thereby advancing remote sensing capabilities in water dynamics, environmental monitoring and pollution control. Full article
(This article belongs to the Special Issue Coastal and Littoral Observation Using Remote Sensing)
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<p>Rhodamine samples for distilled and seawater at different concentrations. (<b>a</b>) Distilled 0 mg/L; (<b>b</b>) Distilled 1 mg/L; (<b>c</b>) Distilled 15 mg/L; (<b>d</b>) Distilled 30 mg/L; (<b>e</b>) Seawater 0 mg/L; (<b>f</b>) Seawater 1 mg/L; (<b>g</b>) Seawater 15 mg/L; (<b>h</b>) Seawater 30 mg/L.</p>
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<p>A 3D model of the acquisition system (adapted with permission from [<a href="#B27-remotesensing-16-03090" class="html-bibr">27</a>], under a Creative Commons Attribution (CC BY) 4.0 license. Copyright 2022).</p>
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<p>Methodology for transferring results and obtaining a generalizable classifier.</p>
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<p>Band selection method (adapted with permission from [<a href="#B30-remotesensing-16-03090" class="html-bibr">30</a>], Copyright 2024, IEEE).</p>
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<p>Mean spectral signature and standard deviation (shaded in the corresponding colour) of the backgrounds with and without the beaker.</p>
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<p>Mean spectra with standard deviation (shaded in the corresponding colour) for each concentration and sample. (<b>a</b>) Distilled water with a dark background; (<b>b</b>) distilled water with a light background; (<b>c</b>) sea water with a dark background; (<b>d</b>) sea water with a light background.</p>
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<p>Spectral difference between distilled and seawater for the two backgrounds: (<b>a</b>) 0 mg/L; (<b>b</b>) 1 mg/L; (<b>c</b>) 15 mg/L; (<b>d</b>) 30 mg/L.</p>
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<p>Spectral difference between backgrounds: (<b>a</b>) 0 mg/L; (<b>b</b>) 1 mg/L; (<b>c</b>) 15 mg/L; (<b>d</b>) 30 mg/L.</p>
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<p>OAC of all model combinations for dark (solid line) and light (dashed line) backgrounds. (<b>a</b>) Distilled water; (<b>b</b>) seawater.</p>
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<p>Spectral areas of interest, identified by grouping the two most significant bands for each combination of SFS with RF, LR, and SVM.</p>
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<p>Accuracy obtained by transferring bands of interest from one sample to another. The colours indicate performance: green tones for accuracies above 80%, yellowish for 60–80%, orange for 40–60%, and red for accuracy below 40%.</p>
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<p>Accuracy obtained by transferring trained classifiers from one sample to another. The colours indicate performance: green tones for accuracies above 80%, yellowish for 60–80%, orange for 40–60%, and red for accuracy below 40%.</p>
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<p>Mean spectra and standard deviation (shaded in the corresponding colour). The two best bands are indicated with black vertical lines. (<b>a</b>) Combined samples (580 and 610 nm); (<b>b</b>) the first derivative of the combined samples (591 and 607 nm).</p>
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<p>Confusion matrix training with CS and CD for the best and worst scenarios in <a href="#remotesensing-16-03090-f015" class="html-fig">Figure 15</a>. (<b>a</b>) CS validating on distilled light; (<b>b</b>) CS validating on seawater light; (<b>c</b>) CD validating on seawater light.</p>
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<p>Accuracy obtained by transferring trained classifiers from the combined sample. The colours indicate performance: green tones for accuracies above 80%, yellowish for 60–80%, orange for 40–60%, and red for accuracy below 40%.</p>
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26 pages, 17778 KiB  
Article
Occurrence and Favorable Enrichment Environment of Lithium in Gaoping Coal Measures: Evidence from Mineralogy and Geochemistry
by Peiliang Han, Fenghua Zhao, Dongna Liu, Qi Zhang, Qinqin Zhang and Shaheed Ullah
Appl. Sci. 2024, 14(16), 7298; https://doi.org/10.3390/app14167298 - 19 Aug 2024
Cited by 1 | Viewed by 581
Abstract
The Carboniferous-Permian coal measure strata in the Qinshui Basin exhibit highly lithium (Li) enrichment, with substantial exploitation potential. To further explore the enrichment mechanism of lithium in coal measure strata, the No. 15 coal of the Taiyuan Formation from the Gaoping mine is [...] Read more.
The Carboniferous-Permian coal measure strata in the Qinshui Basin exhibit highly lithium (Li) enrichment, with substantial exploitation potential. To further explore the enrichment mechanism of lithium in coal measure strata, the No. 15 coal of the Taiyuan Formation from the Gaoping mine is taken as the research object, and its mineralogical and geochemistry characteristics are evaluated using optical microscopy, X-ray diffraction, scanning electron microscopy, inductively coupled plasma mass spectrometry, X-ray fluorescence, and infrared spectral. The results show that the No. 15 coal is semi-anthracite coal with low moisture, low ash, low volatility, and high sulfur. Organic macerals are primarily vitrinite, followed by inertinite, and liptinite is rare; the inorganic macerals (ash) are dominated by clay minerals (predominantly kaolinite, cookeite, illite, and NH4-illite), calcite, pyrite, quartz, siderite, gypsum, and zircon. The average Li content in the coal is 66.59 μg/g, with higher content in the coal parting (566.00 μg/g) and floor (396.00 μg/g). Lithium in coal occurs primarily in kaolinite, illite, cookeite, and is closely related to titanium-bearing minerals. In addition, Li in organic maceral may occur in liptinite. The No. 15 coal was formed in the coastal depositional system, and the deposition palaeoenvironment is primarily a wet–shallow water covered environment in open swamp facies; the plant tissue preservation index is poor, and aquatic or herbaceous plants dominate the plant type. The reducing environment with more terrestrial detritus, an arid climate, and strong hydrodynamic effects is favorable for Li enrichment in coal. The results have important theoretical significance for exploring the enrichment and metallogenic mechanisms of Li in coal. Full article
(This article belongs to the Section Earth Sciences)
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<p>(<b>A</b>) Map of coal-bearing regions in China (modified from reference [<a href="#B30-applsci-14-07298" class="html-bibr">30</a>]); (<b>B</b>) Regional geographical map of research region; (<b>C</b>) Stratigraphic column section of the Gaoping coal-bearing strata.</p>
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<p>Photos of hand samples of coal and non-coal samples. (<b>A</b>) 15R, roof sample; (<b>B</b>) C1, bright coal, visible vitrain bands; (<b>C</b>) C2, semi-bright coal; (<b>D</b>) C3, semi-bright coal; (<b>E</b>–<b>H</b>) C4–C7, semi-dull coals; (<b>I</b>,<b>K</b>) C8 and C9, semi-bright coals; (<b>J</b>) 15P, parting sample; (<b>L</b>) 15F, floor sample.</p>
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<p>The organic macerals in Qinshui No. 15 Coal (Reflected light, oil immersion). (<b>A</b>) Telinite (fragmented cellular structure), telocollinite, resinite, and liptodetrinite in C3; (<b>B</b>) Desmocillinite, tellinite, micrinite, and inertodetrinite in C3; (<b>C</b>) Fusinite, desmocillinite (clay minerals attached to surface), and inertodetrinite in C3; (<b>D</b>) Desmocillinite, telinite (well-arranged and intact cellular structure), and resinite in C5; (<b>E</b>) Fusinite (fragments aggregated appear as arc structures), macrinite, funginite, tellinite, and desmocillinite (clay minerals attached to the surface) in C2; (<b>F</b>) Fusinite, semifusinite, inertodetrinite, and desmocillinite in C2; (<b>G</b>) Telinite (cavities were filled with exsudatinite and resinite) and desmocillinite (clay minerals attached to surface) in C5; (<b>H</b>) Fusinite, semifusinite, and telocollinite in C4; (<b>I</b>) Funginite, inertodetrinite, and desmocillinite in C7.</p>
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<p>XRD patterns of coal samples. (<b>A</b>) Kaolinite, quartz, pyrite, illite, NH<sub>4</sub>-illite, and calcite in C1; (<b>B</b>) Kaolinite, quartz, pyrite, illite, NH<sub>4</sub>-illite and calcite in C2; (<b>C</b>) Kaolinite, quartz, illite, NH<sub>4</sub>-illite, calcite, and anatase in C5; (<b>D</b>) Kaolinite, illite, NH<sub>4</sub>-illite, and rutile in C6; (<b>E</b>) Kaolinite, pyrite, NH<sub>4</sub>-illite, chlorite, cookeite, and anhydrite in C7; (<b>F</b>) Kaolinite, pyrite, NH<sub>4</sub>-illite, calcite, and anhydrite.</p>
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<p>Pyrite and calcite in coal samples, (<b>A</b>–<b>F</b>) were taken with an oil immersion objective (Reflected light, 10 × 50), and (<b>G</b>–<b>I</b>) were taken with dry objective (Reflected light, 10 × 10). (<b>A</b>) Framboidal pyrite in C1; (<b>B</b>) Veined pyrite in C1; (<b>C</b>) Disseminated pyrite on the surface of calcite in C2; (<b>D</b>) Calcite veins and pyrite particle in C8; (<b>E</b>) Calcite with vitrinite on the surface in C4; (<b>F</b>) Veined pyrite in C2; (<b>G</b>) Filled pyrite in C8; (<b>H</b>,<b>I</b>) Massive pyrite in C9.</p>
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<p>SEM back-scattered electron images and selected energy spectrum of minerals in coal samples. (<b>A</b>) Calcite and pyrite in C2; (<b>B</b>) Pyrite and kaolinite in C3; (<b>C</b>) Quartz and vermicular kaolinite in C3; (<b>D</b>) Flocculent clay minerals and anatase in C6; (<b>E</b>) Siderite, pyrite, and kaolinite in C6, with pyrite filling the fissures; (<b>F</b>) Kaolinite, cookeite, and NH<sub>4</sub>-illite in C6; (<b>G</b>) EDS spectrum of point 1 in (<b>B</b>); (<b>H</b>) EDS spectrum of point 2 in (<b>C</b>); (<b>I</b>) EDS spectrum of point 3 in (<b>D</b>); (<b>J</b>) EDS spectrum of point 4 in (<b>F</b>).</p>
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<p>FTIR spectrum of coal sample, parting, and floor.</p>
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<p>Vertical distribution of major element content of No. 15 coal.</p>
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<p>CC of trace elements of coal and non-coal samples (Average concentrations of trace elements for world hard coals are from Ketris and Yudovich (2009) [<a href="#B14-applsci-14-07298" class="html-bibr">14</a>]).</p>
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<p>Distribution pattern of REY of No. 15 coal and non-coal samples. (<b>A</b>) roof, parting, and floor samples; (<b>B</b>) C1–C2, C7–C9, coal samples; (<b>C</b>) C3–C6, coal samples.</p>
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<p>Discrimination diagrams for Al<sub>2</sub>O<sub>3</sub> vs. TiO<sub>2</sub> (<b>A</b>) and La/Yb vs. ΣREE (<b>B</b>) for studying the material sources of No. 15 coal and non-coal samples (Base image from Hayashi et al. (1997) and Allègre and Minster (1978)) [<a href="#B58-applsci-14-07298" class="html-bibr">58</a>,<a href="#B68-applsci-14-07298" class="html-bibr">68</a>].</p>
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<p>Discrimination diagrams for Al<sub>2</sub>O<sub>3</sub>/TiO<sub>2</sub> vs. Zr/TiO<sub>2</sub> (<b>A</b>) and Al<sub>2</sub>O<sub>3</sub>/TiO<sub>2</sub> vs. Nb/Yb (<b>B</b>) for studying the material sources of No. 15 coal and non-coal samples. (Base image adapted from Zheng (2020)) [<a href="#B72-applsci-14-07298" class="html-bibr">72</a>].</p>
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<p>TPI–GI coal facies diagram (<b>A</b>) and GWI–VI coal facies diagram (<b>B</b>). (Base image adapted from Diessel (1992) [<a href="#B84-applsci-14-07298" class="html-bibr">84</a>]).</p>
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<p>(<b>A</b>) Relationship between Li and the (CaO+MgO+Fe<sub>2</sub>O<sub>3</sub>)/(SiO<sub>2</sub>+Al<sub>2</sub>O<sub>3</sub>) (C-value) ratio; (<b>B</b>) Relationship between Li and Sr/Ba ratio.</p>
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<p>Relationship between Li and A<sub>d</sub>, Al<sub>2</sub>O<sub>3</sub>, SiO<sub>2</sub>, TiO<sub>2</sub>, K<sub>2</sub>O, and P<sub>2</sub>O<sub>5</sub> in No. 15 coal. (<b>A</b>) Relationship between Li and A<sub>d</sub>; (<b>B</b>) Relationship between Li and Al<sub>2</sub>O<sub>3</sub>; (<b>C</b>) Relationship between Li and SiO<sub>2</sub>; (<b>D</b>) Relationship between Li and TiO<sub>2</sub>; (<b>E</b>) Relationship between Li and K<sub>2</sub>O; (<b>F</b>) Relationship between Li and P<sub>2</sub>O<sub>5</sub>.</p>
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<p>Comparison of Li content between Gaoping samples and Jincheng samples (Jincheng sample data from Zhao et al. (2019) [<a href="#B6-applsci-14-07298" class="html-bibr">6</a>]).</p>
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<p>Correlation coefficient between Li and depositional environment parameters.</p>
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21 pages, 46218 KiB  
Article
Lightweight Single Image Super-Resolution via Efficient Mixture of Transformers and Convolutional Networks
by Luyang Xiao, Xiangyu Liao and Chao Ren
Sensors 2024, 24(16), 5098; https://doi.org/10.3390/s24165098 - 6 Aug 2024
Viewed by 816
Abstract
In this paper, we propose a Local Global Union Network (LGUN), which effectively combines the strengths of Transformers and Convolutional Networks to develop a lightweight and high-performance network suitable for Single Image Super-Resolution (SISR). Specifically, we make use of the advantages of Transformers [...] Read more.
In this paper, we propose a Local Global Union Network (LGUN), which effectively combines the strengths of Transformers and Convolutional Networks to develop a lightweight and high-performance network suitable for Single Image Super-Resolution (SISR). Specifically, we make use of the advantages of Transformers to provide input-adaptation weighting and global context interaction. We also make use of the advantages of Convolutional Networks to include spatial inductive biases and local connectivity. In the shallow layer, the local spatial information is encoded by Multi-order Local Hierarchical Attention (MLHA). In the deeper layer, we utilize Dynamic Global Sparse Attention (DGSA), which is based on the Multi-stage Token Selection (MTS) strategy to model global context dependencies. Moreover, we also conduct extensive experiments on both natural and satellite datasets, acquired through optical and satellite sensors, respectively, demonstrating that LGUN outperforms existing methods. Full article
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<p>Trade–off between performance and model complexity on Set5 ×4 dataset. Multi-Adds are calculated on <math display="inline"><semantics> <mrow> <mn>1280</mn> <mo>×</mo> <mn>720</mn> </mrow> </semantics></math> HR images.</p>
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<p>Compared to uni-dimensional information communication, e.g., spatial-only or channel-only, our method can achieve local spatial-wise aggregation and global channel-wise interaction simultaneously, both of which are crucial for SISR tasks.</p>
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<p>The architecture of our proposed method, LGUN, consists of three main parts: feature extraction, nonlinear mapping, and image reconstruction. The core modules, named LGU, include two stages: MLHA and DGSA. In the shallow layers, MLHA efficiently encodes local spatial information by utilizing subsets of the entire feature, enabling explicit learning of distinct feature patterns through the STF strategy. In the deep layers, DGSA is employed to model long-range non-local dependencies while achieving a global effective receptive field. DGSA operates across the feature dimension and leverages interactions based on the cross-covariance matrix between keys and queries. Moreover, we incorporate the MTS strategy into DGSA, which selects multiple top-<span class="html-italic">k</span> similar attention matrices and masks out elements with lower weights. This reduces redundancy in attention maps and suppresses interference from cluttered backgrounds. LGUN exhibits robustness to changes in the input token length and significantly reduces the computational complexity to <math display="inline"><semantics> <mrow> <mi mathvariant="script">O</mi> <mo>(</mo> <mi>N</mi> <msup> <mi>C</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>≪</mo> <mi>N</mi> </mrow> </semantics></math>.</p>
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<p>Multiple attention matrices. Take a head as an example (<math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <msub> <mi>D</mi> <mi>h</mi> </msub> </mrow> </semantics></math>), where <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>w</mi> <mn>3</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>w</mi> <mn>4</mn> </msub> </semantics></math> represent the assigned weight, which is obtained by dynamic adaptation learning of the network. We set Multi-stage Token Selection thresholds <math display="inline"><semantics> <msub> <mi>k</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>k</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>k</mi> <mn>3</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>k</mi> <mn>4</mn> </msub> </semantics></math> to <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </semantics></math>, <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </semantics></math>, <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </semantics></math>, and <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mstyle> </semantics></math>, respectively.</p>
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<p>Qualitative comparison of state-of-the-art methods on Urban100 [<a href="#B63-sensors-24-05098" class="html-bibr">63</a>]. Our method achieves better performance with fewer artifacts and less blur.</p>
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<p>Results of local attribution maps. A more widely distributed red area and higher DI represent a larger range of pixel utilization.</p>
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<p>The heat maps exhibit the area of interest for different SR networks. The red regions are noticed by CARN [<a href="#B70-sensors-24-05098" class="html-bibr">70</a>], EDSR [<a href="#B12-sensors-24-05098" class="html-bibr">12</a>], SwinIR [<a href="#B32-sensors-24-05098" class="html-bibr">32</a>] and AAN [<a href="#B72-sensors-24-05098" class="html-bibr">72</a>], while the blue areas represent the additional LAM interest areas of the proposed LGUN. (LGUN has a higher diffusion index).</p>
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<p>Qualitative comparison of state-of-the-art methods on AID dataset.</p>
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18 pages, 17173 KiB  
Article
Influence of Post-Weld Heat Treatment on Mechanical Properties and Microstructure of Plasma Arc-Welded 316 Stainless Steel
by Adirek Baisukhan, Nirut Naksuk, Pinmanee Insua, Wasawat Nakkiew and Nuttachat Wisittipanit
Materials 2024, 17(15), 3768; https://doi.org/10.3390/ma17153768 - 31 Jul 2024
Viewed by 747
Abstract
This study investigates the effects of post-weld heat treatment (PWHT) on the microstructures and mechanical properties of plasma arc-welded 316 stainless steel. The experimental parameters included the solid solution temperatures of 650 °C and 1050 °C, solid solution durations of 1 h and [...] Read more.
This study investigates the effects of post-weld heat treatment (PWHT) on the microstructures and mechanical properties of plasma arc-welded 316 stainless steel. The experimental parameters included the solid solution temperatures of 650 °C and 1050 °C, solid solution durations of 1 h and 4 h, and quenching media of water and air. The mechanical properties were evaluated using Vickers hardness testing, tensile testing, scanning electron microscopy (SEM), and optical microscopy (OM). The highest ultimate tensile strength (UTS) of 693.93 MPa and Vickers hardness of 196.4 in the welded zone were achieved by heat-treating at 650 °C for one hour, quenching in water, and aging at 500 °C for 24 h. Heat-treating at 650 °C for one hour, followed by quenching in water and aging at 500 °C for 24 h results in larger dendritic δ grains and contains more σ phase compared to the other conditions, resulting in increased strength and hardness. Additionally, it shows wider and shallower dimple structures, which account for its reduced impact toughness. Full article
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<p>Samples of 316 Stainless steel plates welded by plasma arc welding.</p>
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<p>An example of a sample cut into the shape of a dog bone.</p>
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<p>A dog bone size of ASTM E8 specification.</p>
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<p>Area of interest for Vickers hardness evaluation.</p>
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<p>Specific points on the specimen for Vickers hardness evaluation.</p>
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<p>A cutting part for microstructure evaluation.</p>
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<p>Ultimate tensile strength and elongation at break of each condition.</p>
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<p>Vicker hardness variations for each compression stage.</p>
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<p>Optical micrographs of base zone (BZ) of (<b>a</b>) as-welded, (<b>b</b>) condition A, (<b>c</b>) condition B, (<b>d</b>) condition C, (<b>e</b>) condition D, (<b>f</b>) condition E, (<b>g</b>) condition F, (<b>h</b>) condition G, and (<b>i</b>) condition H.</p>
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<p>Optical micrographs of heat-affected zone (HAZ) of (<b>a</b>) as-welded, (<b>b</b>) condition A, (<b>c</b>) condition B, (<b>d</b>) condition C, (<b>e</b>) condition D, (<b>f</b>) condition E, (<b>g</b>) condition F, (<b>h</b>) condition G, and (<b>i</b>) condition H.</p>
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<p>Optical micrographs of weld zone (WZ) of (<b>a</b>) as-welded, (<b>b</b>) condition A, (<b>c</b>) condition B, (<b>d</b>) condition C, (<b>e</b>) condition D, (<b>f</b>) condition E, (<b>g</b>) condition F, (<b>h</b>) condition G, and (<b>i</b>) condition H.</p>
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<p>SEM micrographs of as-welded sample, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample A, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample B, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample C, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample D, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample E, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample F, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample G, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>SEM micrographs of sample H, (<b>a</b>) BZ, (<b>b</b>) HAZ, and (<b>c</b>) WZ.</p>
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<p>Sample photographs of (<b>a</b>) condition A’s specimen and (<b>b</b>) condition B’s specimen from the tensile test.</p>
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<p>Fracture surfaces after tensile testing of (<b>a</b>) condition A (<b>top</b>), (<b>b</b>) condition A (<b>bottom</b>), (<b>c</b>) condition H (<b>top</b>), and (<b>d</b>) condition H (<b>bottom</b>).</p>
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