WavewatchⅢ模拟和统计方法在最大波高预报方面的评测分析

为了研究WavewatchⅢ(WWⅢ)海浪模型对最大波高的模拟能力及其与传统统计关系方法的差异,通过对两次台风浪过程的后报模拟和半年的业务化预报,分析了WWⅢ数值模拟的准确度及其与统计关系方法的精度差异。研究结果表明:WWⅢ数值模拟的最大波高(Hmax)的精度略低于有效波高(Hs),但也达到了24 h预报相对误差(H_(max)≥1 m)低于18%、相关系数高于0.94的水平,模拟精度可靠,可以用于业务化预报;与两种统计关系方法(H_(max)和H_(s)分别为1.42和1.52)计算的最大波高相比,数值模拟的精度总体与其相当,但在H_(max)和H_(s)比值大于1.65这种易出现危险的海况下,数值模拟具有更高的准确性,更适合应用于海浪预警报服务。

基于D-Wave Advantage的量子退火公钥密码攻击算法研究

D-Wave专用量子计算机的原理量子退火凭借独特的量子隧穿效应可跳出传统智能算法极易陷入的局部极值,可视为一类具有全局寻优能力的人工智能算法.本文研究了两类基于量子退火的RSA公钥密码攻击算法(分解大整数N=pq):一是将密码攻击数学方法转为组合优化问题或指数级空间搜索问题,通过Ising模型或QUBO模型求解,提出了乘法表的高位优化模型,建立新的降维公式,使用D-Wave Advantage分解了 200万整数2269753.大幅度超过普渡大学、Lockheed Martin和富士通等实验指标,且Ising模型系数h范围缩小了 84%,系数J范围缩小了 80%,极大地提高了分解成功率,这是一类完全基于D-Wave量子计算机的攻击算法;二是基于量子退火算法融合密码攻击数学方法优化密码部件的攻击,采用量子退火优化CVP问题求解,通过量子隧穿效应获得比Babai算法更近的向量,提高了 CVP问题中光滑对的搜索效率,在D-Wave Advantage上实现首次50比特RSA整数分解.实验表明,在通用量子计算机器件进展缓慢情况下,D-Wave表现出更好的现实攻击能力,且量子退火不存在NISQ量子计算机VQA算法的致命缺陷贫瘠高原问题:算法会无法收敛且无法扩展到大规模攻击.

Hollow engineering of sandwich NC@Co/NC@MnO_(2)composites toward strong wideband electromagnetic wave attenuation

Multiple hetero-interfaces would strengthen interfacial polarization and boost electromagnetic wave absorption,but still remain the formidable challenges in decreasing filler loadings.Herein,sandwich NC@Co/NC@MnO_(2)composites with hollow cavity,multiple hetero-interfaces,and hierarchical structures have been fabricated via the cooperative processes of self-sacrifice strategy and sequential hydrothermal reaction.In the sandwich composites,middle magnetic components(Co/NC)are wrapped by inner N-doped carbon(NC)matrix and outer hierarchical MnO_(2)nanosheets.Importantly,hollow engineering of sandwich composites with multiple hetero-interfaces greatly facilitates the enhancement of absorption bandwidth without sacrificing the absorption intensity.The maximum reflection loss of sandwich NC@Co/NC@MnO_(2)composites reaches-44.8 dB at 2.5 mm and the effective bandwidths is achieved as wide as 9.6 GHz at 2.3 mm.These results provide us a new insight into preparing efficient electromagnetic wave absorbers by interface engineering and hollow construction.

MXene@Co hollow spheres structure boosts interfacial polarization for broadband electromagnetic wave absorption

MXene is considered as a candidate for preparing high-performance electromagnetic wave absorbing materials due to its large specific surface area,rich surface modification groups,and unique metal properties.However,the impedance matching problem caused by its high conductivity and easy stacking properties is a limiting factor.In this study,a self-assembling-etching-anchoring growth method was proposed to prepare MXene@Co electromagnetic wave absorbing materials.The hollow structure of MXene microspheres constructed with PMMA as a hard template is conducive to optimizing impedance matching and surface modification.In addition,MXene@Co exhibits abundant heterogeneous interfaces,enhancing the interfacial polarization phenomenon during electromagnetic wave absorption.Meanwhile,the surfaceanchored growth of magnetic Co particles forms a magnetic network,which provides a strong magnetic loss capability for the absorber.The hollow structure design significantly enhances the wave absorption performance compared to conventional MXene@Co composites,with a minimum reflection loss of−57.32 dB(effective absorption bandwidth of 5.2 GHz)when the thickness is 2.5 mm(2.2 mm).This work provides a meaningful reference for the design of MXene-based electromagnetic wave absorbing materials.

Nitrogen‑Doped Magnetic‑Dielectric‑Carbon Aerogel for High‑Efficiency Electromagnetic Wave Absorption

Carbonbased aerogels derived from biomass chitosan are encountering a flourishing moment in electromagnetic protection on account of lightweight,controllable fabrication and versatility.Nevertheless,developing a facile construction method of component design with carbon-based aerogels for high-efficiency electromagnetic wave absorption(EWA)materials with a broad effective absorption bandwidth(EAB)and strong absorption yet hits some snags.Herein,the nitrogen-doped magnetic-dielectric-carbon aerogel was obtained via ice template method followed by carbonization treatment,homogeneous and abundant nickel(Ni)and manganese oxide(MnO)particles in situ grew on the carbon aerogels.Thanks to the optimization of impedance matching of dielectric/magnetic components to carbon aerogels,the nitrogen-doped magnetic-dielectric-carbon aerogel(Ni/MnO-CA)suggests a praiseworthy EWA performance,with an ultra-wide EAB of 7.36 GHz and a minimum reflection loss(RLmin)of−64.09 dB,while achieving a specific reflection loss of−253.32 dB mm−1.Furthermore,the aerogel reveals excellent radar stealth,infrared stealth,and thermal management capabilities.Hence,the high-performance,easy fabricated and multifunctional nickel/manganese oxide/carbon aerogels have broad application aspects for electromagnetic protection,electronic devices and aerospace.

Interface Engineering of Titanium Nitride Nanotube Composites for Excellent Microwave Absorption at Elevated Temperature

Currently,the microwave absorbers usually suffer dreadful electromagnetic wave absorption(EMWA)performance damping at elevated temperature due to impedance mismatching induced by increased conduction loss.Consequently,the development of high-performance EMWA materials with good impedance matching and strong loss ability in wide temperature spectrum has emerged as a top priority.Herein,due to the high melting point,good electrical conductivity,excellent environmental stability,EM coupling effect,and abundant interfaces of titanium nitride(TiN)nanotubes,they were designed based on the controlling kinetic diffusion procedure and Ostwald ripening process.Benefiting from boosted heterogeneous interfaces between TiN nanotubes and polydimethylsiloxane(PDMS),enhanced polarization loss relaxations were created,which could not only improve the depletion efficiency of EMWA,but also contribute to the optimized impedance matching at elevated temperature.Therefore,the TiN nanotubes/PDMS composite showed excellent EMWA performances at varied temperature(298-573 K),while achieved an effective absorption bandwidth(EAB)value of 3.23 GHz and a minimum reflection loss(RLmin)value of−44.15 dB at 423 K.This study not only clarifies the relationship between dielectric loss capacity(conduction loss and polarization loss)and temperature,but also breaks new ground for EM absorbers in wide temperature spectrum based on interface engineering.

Absorption properties and mechanism of lightweight and broadband electromagnetic wave-absorbing porous carbon by the swelling treatment

Bioderived carbon materials have garnered considerable interest in the fields of microwave absorption and shielding due to their reproducibility and environmental friendliness.In this study,KOH was evenly distributed on biomass Tremella using the swelling induction method,leading to the preparation of a three-dimensional network-structured hierarchical porous carbon(HPC)through carbonization.The achieved microwave absorption intensity is robust at-47.34 dB with a thin thickness of 2.1 mm.Notably,the widest effective absorption bandwidth,reaching 7.0 GHz(11–18 GHz),is attained at a matching thickness of 2.2 mm.The exceptional broadband and reflection loss performance are attributed to the 3D porous networks,interface effects,carbon network defects,and dipole relaxation.HPC has outstanding absorption characteristics due to its excellent impedance matching and high attenuation constant.The uniform pore structures considerably optimize the impedance-matching performance of the material,while the abundance of interfaces and defects enhances the dielectric loss,thereby improving the attenuation constant.Furthermore,the impact of carbonization temperature and swelling rate on microwave absorption performance was systematically investigated.This research presents a strategy for preparing absorbing materials using biomass-derived HPC,showcasing considerable potential in the field of electromagnetic wave absorption.

Tracking Regulatory Mechanism of Trace Fe on Graphene Electromagnetic Wave Absorption

Polarization and conductance losses are the fundamental dielectric attenuation mechanisms for graphene-based absorbers, but it is not fully understood in revealing the loss mechanism of affect graphene itself. For the first time, the reduced graphene oxide(RGO) based absorbers are developed with regulatory absorption properties and the absorption mechanism of RGO is mainly originated from the carrier injection behavior of trace metal Fe nanosheets on graphene. Accordingly, the minimum reflection loss(RLmin) of Fe/RGO-2composite reaches-53.38 dB(2.45 mm), and the effective absorption bandwidth achieves 7.52 GHz(2.62 mm) with lower filling loading of 2 wt%. Using off-axis electron hologram testing combined with simulation calculation and carrier transport property experiments, we demonstrate here the carrier injection behavior from Fe to graphene at the interface and the induced charge accumulation and rearrangement, resulting in the increased interfacial and dipole polarization and the conductance loss. This work has confirmed that regulating the dielectric property of graphene itself by adding trace metals can not only ensure good impedance matching, but also fully exploit the dielectric loss ability of graphene at low filler content,which opens up an efficient way for designing lightweight absorbers and may be extended to other types materials.

Optimization of multiple attenuation mechanisms by cation substitution in imidazolic MOFs-derived porous composites for superior broadband electromagnetic wave absorption

Metal-organic frameworks(MOFs)derived composites are extremely potential electromagnetic wave(EMW)absorbers.However,the permittivity of absorbers directly derived from MOFs with solid structure is usually relatively low,inevitably limiting their further applications.Cation substitution can primely overcome the problem by regulating the morphology and atomic space occupation to enhance multiple loss mechanisms and impedance matching characteristics.However,universal mechanisms of the effect on EMW absorption performance influenced by cation substitution are still comparatively inadequate,which prospectively requires further exploration.Herein,a series of imidazolic MOFs were fabricated by ultrasonic symbiosis method and tailored by subsequent cation substitution strategy to prepare target porous composites.At a low filling rate and thin thickness,the as-obtained samples reach the optimal reflection loss and effective absorption bandwidth values of–49.81 dB and 7.63 GHz,respectively.The intercoupling between multiple atoms lays a significant foundation for abundant heterogeneous interfaces and defect vacancies,which effectively ameliorate the attenuation mechanisms.Meanwhile,the porous structure introduced by cation substitution reduces the bulk density to enhance the impedance matching and multiple reflections simultaneously.This study provides a helpful idea to exceedingly improve the EMW absorbing performance of imidazolic MOFs-derived composites by cation substitution.

Ultra-wide band gap and wave attenuation mechanism of a novel star-shaped chiral metamaterial

A novel hollow star-shaped chiral metamaterial(SCM)is proposed by incorporating chiral structural properties into the standard hollow star-shaped metamaterial,exhibiting a wide band gap over 1500 Hz.To broaden the band gap,solid single-phase and two-phase SCMs are designed and simulated,which produce two ultra-wide band gaps(approximately 5116 Hz and 6027 Hz,respectively).The main reason for the formation of the ultra-wide band gap is that the rotational vibration of the concave star of two novel SCMs drains the energy of an elastic wave.The impacts of the concave angle of a single-phase SCM and the resonator radius of a two-phase SCM on the band gaps are studied.Decreasing the concave angle leads to an increase in the width of the widest band gap,and the width of the widest band gap increases as the resonator radius of the two-phase SCM increases.Additionally,the study on elastic wave propagation characteristics involves analyzing frequency dispersion surfaces,wave propagation directions,group velocities,and phase velocities.Ultimately,the analysis focuses on the transmission properties of finite periodic structures.The solid single-phase SCM achieves a maximum vibration attenuation over 800,while the width of the band gap is smaller than that of the two-phase SCM.Both metamaterials exhibit high vibration attenuation capabilities,which can be used in wideband vibration reduction to satisfy the requirement of ultra-wide frequencies.

Efficient Electromagnetic Wave Absorption and Thermal Infrared Stealth in PVTMS@MWCNT Nano‑Aerogel via Abundant Nano‑Sized Cavities and Attenuation Interfaces

Pre-polymerized vinyl trimethoxy silane(PVTMS)@MWCNT nano-aerogel system was constructed via radical polymerization,sol-gel transition and supercritical CO_(2)drying.The fabricated organic-inorganic hybrid PVTMS@MWCNT aerogel structure shows nano-pore size(30-40 nm),high specific surface area(559 m^(2)g^(−1)),high void fraction(91.7%)and enhanced mechanical property:(1)the nano-pore size is beneficial for efficiently blocking thermal conduction and thermal convection via Knudsen effect(beneficial for infrared(IR)stealth);(2)the heterogeneous interface was beneficial for IR reflection(beneficial for IR stealth)and MWCNT polarization loss(beneficial for electromagnetic wave(EMW)attenuation);(3)the high void fraction was beneficial for enhancing thermal insulation(beneficial for IR stealth)and EMW impedance match(beneficial for EMW attenuation).Guided by the above theoretical design strategy,PVTMS@MWCNT nano-aerogel shows superior EMW absorption property(cover all Ku-band)and thermal IR stealth property(ΔT reached 60.7℃).Followed by a facial combination of the above nano-aerogel with graphene film of high electrical conductivity,an extremely high electromagnetic interference shielding material(66.5 dB,2.06 mm thickness)with superior absorption performance of an average absorption-to-reflection(A/R)coefficient ratio of 25.4 and a low reflection bandwidth of 4.1 GHz(A/R ratio more than 10)was experimentally obtained in this work.

From VIB‑to VB‑Group Transition Metal Disulfides:Structure Engineering Modulation for Superior Electromagnetic Wave Absorption

The laminated transition metal disulfides(TMDs),which are well known as typical two-dimensional(2D)semiconductive materials,possess a unique layered structure,leading to their wide-spread applications in various fields,such as catalysis,energy storage,sensing,etc.In recent years,a lot of research work on TMDs based functional materials in the fields of electromagnetic wave absorption(EMA)has been carried out.Therefore,it is of great significance to elaborate the influence of TMDs on EMA in time to speed up the application.In this review,recent advances in the development of electromagnetic wave(EMW)absorbers based on TMDs,ranging from the VIB group to the VB group are summarized.Their compositions,microstructures,electronic properties,and synthesis methods are presented in detail.Particularly,the modulation of structure engineering from the aspects of heterostructures,defects,morphologies and phases are systematically summarized,focusing on optimizing impedance matching and increasing dielectric and magnetic losses in the EMA materials with tunable EMW absorption performance.Milestones as well as the challenges are also identified to guide the design of new TMDs based dielectric EMA materials with high performance.

Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration(RTM) and full waveform inversion(FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic(TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic(VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition(HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing q P-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral(HFDPS) method in terms of accuracy of wavefield modeling.

A novel method for simulating nuclear explosion with chemical explosion to form an approximate plane wave: Field test and numerical simulation

A nuclear explosion in the rock mass medium can produce strong shock waves,seismic shocks,and other destructive effects,which can cause extreme damage to the underground protection infrastructures.With the increase in nuclear explosion power,underground protection engineering enabled by explosion-proof impact theory and technology ushered in a new challenge.This paper proposes to simulate nuclear explosion tests with on-site chemical explosion tests in the form of multi-hole explosions.First,the mechanism of using multi-hole simultaneous blasting to simulate a nuclear explosion to generate approximate plane waves was analyzed.The plane pressure curve at the vault of the underground protective tunnel under the action of the multi-hole simultaneous blasting was then obtained using the impact test in the rock mass at the site.According to the peak pressure at the vault plane,it was divided into three regions:the stress superposition region,the superposition region after surface reflection,and the approximate plane stress wave zone.A numerical simulation approach was developed using PFC and FLAC to study the peak particle velocity in the surrounding rock of the underground protective cave under the action of multi-hole blasting.The time-history curves of pressure and peak pressure partition obtained by the on-site multi-hole simultaneous blasting test and numerical simulation were compared and analyzed,to verify the correctness and rationality of the formation of an approximate plane wave in the simulated nuclear explosion.This comparison and analysis also provided a theoretical foundation and some research ideas for the ensuing study on the impact of a nuclear explosion.

Enhancing electromagnetic wave absorption with core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres

Material composition and structural design are important factors influencing the electromagnetic wave(EMW)absorption performance of materials.To alleviate the impedance mismatch attributed to the high dielectric constant of Ti_(3)C_(2)T_(x)MXene,we have successfully synthesized core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres.This architecture,comprising SiO_(2)as the core,MXene as the intermediate layer,and MoS_(2)as the outer shell,is achieved through an electrostatic self‐assembly method combined with a hydrothermal process.This complex core‐shell structure not only provides a variety of loss mechanisms that effectively dissipate electromagnetic energy but also prevents self‐aggregation of MXene and MoS_(2)nanosheets.Notably,the synergistic combination of SiO_(2)and MoS_(2)with highly conductive MXene enables the suitable dielectric constant of the composites,ensuring optimal impedance matching.Therefore,the core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres exhibit excellent EMW absorption performance,featuring a remarkable minimum reflection loss(RL_(min))of−52.11 dB(2.4 mm).It is noteworthy that these nanospheres achieve an ultra‐wide effective absorption bandwidth(EAB)of 6.72 GHz.This work provides a novel approach for designing and synthesizing high‐performance EMW absorbers characterized by“wide bandwidth and strong reflection loss.”

Influence of topography on the fine structures of stratospheric gravity waves:An analysis using COSMIC-2 temperature data

We derive the potential energy of gravity waves(GWs)in the upper troposphere and stratosphere at 45°S-45°N from December 2019 to November 2022 by using temperature profiles retrieved from the Constellation Observing System for Meteorology,Ionosphere,and Climate-2(COSMIC-2)satellite.Owing to the dense sampling of COSMIC-2,in addition to the strong peaks of gravity wave potential energy(GWPE)above the Andes and Tibetan Plateau,we found weak peaks above the Rocky,Atlas,Caucasus,and Tianshan Mountains.The land-sea contrast is responsible for the longitudinal variations of the GWPE in the lower and upper stratosphere.At 40°N/S,the peaks were mainly above the topographic regions during the winter.At 20°N/S,the peaks were a slight distance away from the topographic regions and might be the combined effect of nontopographic GWs and mountain waves.Near the Equator,the peaks were mainly above the regions with the lowest sea level altitude and may have resulted from convection.Our results indicate that even above the local regions with lower sea level altitudes compared with the Andes and Tibetan Plateau,the GWPE also exhibits fine structures in geographic distributions.We found that dissipation layers above the tropopause jet provide the body force to generate secondary waves in the upper stratosphere,especially during the winter months of each hemisphere and at latitudes of greater than 20°N/S.

Ion acoustic solitary waves in an adiabatic dusty plasma:Roles of superthermal electrons,ion loss and ionization

We investigate propagation of dust ion acoustic solitary wave(DIASW)in a multicomponent dusty plasma with adiabatic ions,superthermal electrons,and stationary dust.The reductive perturbation method is employed to derive the damped Korteweg-de Vries(DKdV)equation which describes DIASW.The result reveals that the adiabaticity of ions significantly modifies the basic features of the DIASW.The ionization effect makes the solitary wave grow,while collisions reduce the growth rate and even lead to the damping.With the increases in ionization cross sectionΔσ/σ_(0),ion-to-electron density ratioδ_(ie)and superthermal electrons parameterκ,the effect of ionization on DIASW enhances.

Probability Distribution Characteristics of Strong Nonlinear Waves Under Typhoon Conditions in the Northern South China Sea

The generation and propagation mechanism of strong nonlinear waves in the South China Sea is an essential research area. In this study, the third-generation wave model WAVEWATCH Ⅲ is employed to simulate wave fields under extreme sea states. The model, integrating the ST6 source term, is validated against observed data, demonstrating its credibility. The spatial distribution of the occurrence probability of strong nonlinear waves during typhoons is shown, and the waves in the straits and the northeastern part of the South China Sea show strong nonlinear characteristics. The high-order spectral model HOS-ocean is employed to simulate the random wave surface series beneath five different platform areas. The waves during the typhoon exhibit strong nonlinear characteristics, and freak waves exist. The space-varying probability model is established to describe the short-term probability distribution of nonlinear wave series. The exceedance probability distributions of the wave surface beneath different platform areas are compared and analyzed. The results show that with an increase in the platform area, the probability of a strong nonlinear wave beneath the platform increases.

Long-Term Extreme Wave Characteristics in the Water Adjacent to China Based on ERA5 Reanalysis Data

Extreme waves have a profound impact on coastal infrastructure;thus,understanding the variation law of risky analysis and disaster prevention in coastal zones is necessary.This paper analyzed the spatiotemporal characteristics of extreme wave heights adjacent to China from 1979 to 2018 based on the ERA5 datasets.Nonstationary extreme value analysis is undertaken in eight repre-sentative points to investigate the trends in the values of 50-and 100-year wave heights.Results show that the mean value of extreme waves is the largest in the eastern part of Taiwan Island and the smallest in the Bohai Sea from 1979 to 2018.Only the extreme wave height in the northeastern part of Taiwan Island shows a significant increase trend in the study area.Nonstationary analysis shows remarkable variations in the values of 50-and 100-year significant wave heights in eight points.Considering the annual mean change,E1,E2,S1,and S2 present an increasing trend,while S3 shows a decreasing trend.Most points for the seasonal mean change demon-strate an increasing trend in spring and winter,while other points show a decreasing trend in summer and autumn.Notably,the E1 point growth rate is large in autumn,which is related to the change in typhoon intensity and the northward movement of the typhoon path.

Analysis of explosion wave interactions and rock breaking effects during dual initiation

In blasting engineering, the location and number of detonation points, to a certain degree, regulate the propagation direction ofthe explosion stress wave and blasting effect. Herein, we examine the explosion wave field and rock breaking effect in terms of shockwave collision, stress change of the blast hole wall in the collision zone, and crack propagation in the collision zone. The produced shockwave on the collision surface has an intensity surpassing the sum of the intensities of the two colliding explosion shock waves. At the collisionlocation, the kinetic energy is transformed into potential energy with a reduction in particle velocity at the wave front and the wavefront pressure increases. The expansion form of the superposed shock wave is dumbbell-shaped, the shock wave velocity in the collisionarea is greater than the radial shock wave velocity, and the average propagation angle of the explosion shock waves is approximately 60°.Accordingly, a fitted relationship between blast hole wall stress and explosion wave propagation angle in the superposition area is plotted.Under the experimental conditions, the superimposed explosion wave stress of the blast hole wall is approximately 1.73 times the singleexplosionwave incident stress. The results of the model test and numerical simulations reveal that large-scale radial fracture cracks weregenerated on the blast hole wall in the superimposed area, and the width of the crack increased. The width of the large-scale radial fracturecracks formed by a strong impact is approximately 5% of the blast hole length. According to the characteristics of blast hole wallcompression, the mean peak pressures of the strongly superimposed area are approximately 1.48 and 1.84 times those of the weakly superimposedand nonsuperimposed areas, respectively.