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The lunar surface exhibits noticeable variations in photometric properties across different regions, influenced by their specific compositional and physical characteristics. These variations are particularly pronounced at the millimeter to centimeter scale. In this study, we propose a feature subspace-based sample selection method to identify Chang'E-4 Visible Near-infrared Imaging Spectrometer (CE-4 VNIS) in situ observations that share similar photometric properties at the centimeter scale. By constructing a feature subspace through principal component transformation, the CE-4 VNIS observation sites with comparable surface photometric properties are able to be selected from a series of observations. The selection results demonstrate that the majority of observation sites are covered by the lunar regolith, with a few exceptions consisting of rocks, soil bulk, shadows, and rough surfaces. These observations should be excluded from the photometric correction process during phase function fitting. Analyzing the photometric performance of the VNIS data with lunar regolith reveals that reflectance decreases as the phase angle increases below 80°, while above 80°, reflectance increases with increasing phase angle. This phenomenon is likely attributed to stronger forward scattering. Thus, a second-order polynomial is employed to fit the phase function for VNIS data. Consequently, a photometric model is developed for lunar regolith observed by CE-4 VNIS, incorporating the fitted phase function. Experimental results demonstrate reduced differences among observations with varying phase angles, and the corrected VNIS spectra exhibit a phase-reddening effect. These findings validate the effectiveness of the proposed model. A comparison between the corrected data with and without sample selection reveals a 33 % decrease in mean standard deviation for VIS/NIR bands and a 19 % decrease for SWIR bands in the corrected spectra after sample selection. These results indicate the potential applicability of our method for future VNIS observations, enabling the acquisition of photometrically consistent data. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the L p boundedness of rough Fourier integral operators T a , φ with amplitude a ⁢ (x , ξ) ∈ L ∞ ⁢ S ρ m and phase function φ ⁢ (x , ξ) ∈ L ∞ ⁢ Φ 2 which satisfies a measure condition. We show that T a , φ is bounded on L p for 1 ≤ p ≤ ∞ if m < n ⁢ (ρ − 1) p − ρ ⁢ (n − 1) 2 ⁢ p when 1 ≤ p ≤ 2 or m < n ⁢ (ρ − 1) 2 − ρ ⁢ (n − 1) 2 ⁢ (1 − 1 p) when 2 ≤ p ≤ ∞ . Our main results extend and improve some known results about L p boundedness of Fourier integral operators. [ABSTRACT FROM AUTHOR]

In this paper, let T a , φ be a Fourier integral operator with rough amplitude a ∈ L ∞ S ρ m and rough phase φ ∈ L ∞ Φ 2 which satisfies a new class of rough non-degeneracy condition. When 0 ⩽ ρ ⩽ 1 , if m < n (ρ - 1) 2 - ρ (n - 1) 4 , we obtain that T a , φ is bounded on L 2 . Our main result extends and improves some known results about L 2 boundedness of Fourier integral operators. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of solutions as a small parameter tends to zero is determined for a variety of singular-limit PDEs. In some cases even existence for a time independent of the small parameter was not known previously. New examples for which uniform existence does not hold are also presented. Our methods include both an adaptation of geometric optics phase analysis to singular limits and an extension of that analysis in which the characteristic variety determinant condition is supplemented with a periodicity condition. [ABSTRACT FROM AUTHOR]

In this paper, we define a particular class of semiclassical Fourier integral operators with operator symbol (h -FIOs for short). We study the L 2 -boundedness and L 2 -compactness of h -FIOs. Mainly, we prove that class of h -FIOs denotes Hilbert–Schmidt operators if the order function belongs to L 2 (ℝ n). [ABSTRACT FROM AUTHOR]

We present a gradient method for designing cascaded diffractive optical elements (DOEs) consisting of several sequentially located phase DOEs. Using the unitarity property of the operator of light propagation through the cascaded DOE, we obtain explicit expressions for the derivatives of the error functional with respect to the phase functions of the cascaded DOE. We consider the application of the gradient method to the problem of focusing different incident beams to regions with different intensity distributions and to the problem of optical image classification. The presented description of the gradient method treats the problems of designing cascaded DOEs for focusing laser radiation and for image classification within a unified approach reducing the calculation of the derivatives of the error functionals to the same formula. We present examples of the calculation of single and cascaded DOEs for focusing different incident beams to different regions and for classifying handwritten digits, which demonstrate the high performance of the proposed method. The presented results may find application in the design of diffractive neural networks and systems for focusing laser radiation. [ABSTRACT FROM AUTHOR]

In order to improve the video watermark embedding strength and balance the transparency of the watermark system, this paper proposes a combination of all phase Biorthogonal Transform and watermark embedding to complete the experiment. Firstly, the all phase biorthogonal transform is described. Referring to the construction process of APBT, combined with APDF and DST, a new all phase biorthogonal transform, all phase discrete sinusoidal biorthogonal transform (APDSBT), is proposed. This paper makes full use of MPEG-2 compression format to embed watermark directly in DCT domain. The low-frequency coefficients in the DCT block of the brightness space of I frame are selected as the watermark embedding space. The brightness component of each image block is transformed by two-dimensional DCT in the unit of 8×8 image blocks. By introducing the idea of energy receiver into the detection of digital watermark, the following correlation detector can be obtained. Implemented with MATIAB and VC + +. In the experiment, the foreman video test sequence is used as the watermark carrier, and the copyright logo image designed by ourselves is used as the watermark image to test the video watermarking system. The results show that when the coding rate is 3Mvps, the accuracy of each plane is 100%, 98.48%, 98.12% and 96.27% respectively. When the coding rate is 2.6Mvps, the accuracy of each plane is 100%, 96.38%, 94.87% and 95.21% respectively. This algorithm selects the frame of video to embed watermark. On the premise of ensuring video quality, this algorithm is robust to common video watermark attacks (MPEG compression, frame loss, frame clipping and frame rearrangement).

In this paper we consider the uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4 in two variables. The obtained estimate is sharp and the result is an analogue of the more general theorem of Karpushkin (Proc I.G.Petrovsky Seminar 9:3–39, 1983) for sufficiently smooth functions, thus, in particular, removing the analyticity assumption. [ABSTRACT FROM AUTHOR]

An original method for photometric and geometric correction of LROC NAC data with high resolution (up to 0.5 m/pix) has been developed. The technique is based on photogrammetry and multiphase photoclinometry and allows us to obtain a longitudinal slope map and digital elevation model (DEM) with the resolution of LROC NAC CDR input images as well as to map parameters of model phase function. Obtained DEMs, requiring only LROC NAC images and LRO SPICE-kernels, exhibit significantly fewer defects and artifacts compared to existing DEMs, because they are derived from the same images they correct (self-orthorectification). This method was used to study the area of the photometric anomaly, the Irregular Mare Patch (IMP) formation Ina. A zoning map of the correlation diagram of phase ratio vs. equigonal albedo shows significant differences in the optical (photometric) properties of the regolith for hummocky and blocky formations of the IMP Ina, compared to the surrounding areas, which indicates the different nature of their formation, age, and surface roughness, contradicting some models of the IMP formation. The analysis reveals the incompleteness or inconsistencies of previously proposed models of the IMP's formation mechanisms. The high-quality DEM of the Ina formation and surrounding area with a resolution of 0.5 m/pixel was constructed. • A new technique for multiphase photoclinometry of the LROС NAС data with initial spatial resolution is proposed. • This technique enables the creation of a digital elevation model of the Moon at the resolution of the original images. • It allows photometric and geometric correction of LROC NAC data on topography at the highest possible resolution. • It allows for the mapping of phase function parameters of lunar surface brightness, corrected for topography's influence. • The resulting maps enable unambiguous identification of areas with varying optical properties on the lunar surface. [ABSTRACT FROM AUTHOR]

We present a gradient method for designing cascaded diffractive optical elements (DOEs) consisting of several sequentially located phase DOEs. Using the unitarity property of the operator of light propagation through the cascaded DOE, we obtain explicit expressions for the derivatives of the error functional with respect to the phase functions of the cascaded DOE. We consider the application of the gradient method to the problem of focusing different incident beams to regions with different intensity distributions and to the problem of optical image classification. The presented description of the gradient method treats the problems of designing cascaded DOEs for focusing laser radiation and for image classification within a unified approach reducing the calculation of the derivatives of the error functionals to the same formula. We present examples of the calculation of single and cascaded DOEs for focusing different incident beams to different regions and for classifying handwritten digits, which demonstrate the high performance of the proposed method. The presented results may find application in the design of diffractive neural networks and systems for focusing laser radiation.

Phase is the most fundamental physical quantity when we study an oscillatory time series. There have been many tools aiming to estimate phase, and most of them are developed based on the analytic function model. Unfortunately, these analytic function model based tools might be limited in handling modern signals with intrinsic nonstartionary structure, for example, biomedical signals composed of multiple oscillatory components, each with time-varying frequency, amplitude, and non-sinusoidal oscillation. There are several consequences of such limitation, and we specifically focus on the one that phases estimated from signals simultaneously recorded from different sensors for the same physiological system from the same subject might be different. This fact might challenge reproducibility, communication, and scientific interpretation. Thus, we need a standardized approach with theoretical support over a unified model. In this paper, after summarizing existing models for phase and discussing the main challenge caused by the above-mentioned intrinsic nonstartionary structure, we introduce the adaptive non-harmonic model (ANHM), provide a definition of phase called fundamental phase, which is a vector-valued function describing the dynamics of all oscillatory components in the signal, and suggest a time-varying bandpass filter (tvBPF) scheme based on time-frequency analysis tools to estimate the fundamental phase. The proposed approach is validated with a simulated database and a real-world database with experts' labels, and it is applied to two real-world databases, each of which has biomedical signals recorded from different sensors, to show how to standardize the definition of phase in the real-world experimental environment. We report that the phase describing a physiological system, if properly modeled and extracted, is immune to the selected sensor for that system, while other approaches might fail. In conclusion, the proposed approach resolves the above-mentioned scientific challenge. We expect its scientific impact on a broad range of applications. [ABSTRACT FROM AUTHOR]

In this paper, we get an Lp boundedness of Fourier integral operators with rough amplitude a ∈ L ∞ S ϱ m , and phase φ ∈ L ∞ Φ 2 for 1 ≤ p ≤ + ∞ . This is an improvement of the corresponding results in Dos Santos Ferreira and Staubach (Mem. Amer. Math. Soc. 229, 1074, 2014). [ABSTRACT FROM AUTHOR]

There are large uncertainties in the single-scattering (i.e., morphologies) and microphysical (i.e., concentrations) properties of ice particles whose size are less than ~100 µm. Insufficient resolutions of the most advanced cloud probes (e.g., cloud particle imager) cannot resolve the micrometer-scale morphologies of small ice particles. Further, the shattering of large ice particles on probes' inlets or tips causes uncertainties in the measurement of the concentrations of small ice particles. These uncertainties have large impacts on the single-scattering and microphysical properties of small ice particles that are utilized to quantify the bulk radiative properties of cirrus. In this study, the impacts of uncertainties in the morphologies and concentrations of small ice particles on the bulk radiative properties of tropical cirrus were calculated using measurements acquired during the Tropical Warm Pool-International Cloud Experiment. Five different models (i.e., budding Buckyball, Chebyshev particle, droxtal, Gaussian random sphere, and sphere) that represent the shapes of small ice particles were used to calculate the single-scattering properties. The bulk radiative properties, average phase-function ( P 11 ¯ ), and average asymmetry parameter ( g ¯ ) were computed by combining the measured size/habit distributions and the calculated single-scattering properties of ice particles. The impacts of the selection of varying morphologies of small particles on the bulk radiative properties were quantified. For these calculations, the possible range of the concentrations of small ice particles which depend on the degree of shattered large particles were also used. The impacts of varying the single-scattering properties of small ice particles on the bulk radiative properties were the largest in the upper parts of cirrus (T < −60 °C), while they were the smallest in the lower parts of cirrus (−45 < T < −30 °C). The impacts of uncertainties in the concentrations of small ice particles on the bulk radiative properties were largest in the lower parts of cirrus (−45 < T < −30 °C), whereas they were smallest in the upper parts of cirrus (T < −60 °C). The effect of shattering was maximum in the lower parts of cirrus, whilst it was minimum in the upper parts of cirrus. The combined impacts of uncertainties in the single-scattering (i.e., morphologies) and microphysical (i.e., concentrations) properties of small ice particles revealed variations of up to 11.2% (127.1%; 67.3%) of the integrated intensity in the forward (sideward; backward) angles in P 11 ¯ and a corresponding change in g ¯ by up to 12.61%. [ABSTRACT FROM AUTHOR]

Mathematical models of the phase function and its parameters in speech-signal analysis problems have been investigated. The phase spectrum of a speech signal has been calculated using the Hilbert transform of signals at the output of a gammatone filterbank. Short- and long-term modulations of the linear phase component and phase derivatives with respect to frequency and time, and mixed derivative have been considered. The method for vowel segmentation using aggregation of the correlation coefficients of the phase parameters is described. Experiments on estimating the formant and pitch frequencies and the glottal opening and closure instants have been performed. [ABSTRACT FROM AUTHOR]

Photometry is an important tool for characterizing the physical properties of asteroids. An asteroid’s photometric lightcurve and phase curve refer to the variation of the asteroid’s disk-integrated brightness in time and in phase angle (the Sun-asteroid-observer angle), respectively. They depend on the asteroid’s shape, rotation, and surface scattering properties, and the geometry of illumination and observation. We present Bayesian lightcurve inversion methods for the retrieval of the asteroid’s phase function, the unambiguous phase curve of a spherical object with surface scattering properties equal to those of the asteroid. A collection of such phase functions can give rise to a photometric taxonomy for asteroids. In the inverse problem, first, there are four classes of lightcurves that require individual error models. The photometric observations can be absolute or relative and they can have dense or sparse cadence in comparison to the rotation period of the asteroid. Second, the observations extend over varying phase angle ranges, requiring different phase function models. Asteroid photometry from the European Space Agency Gaia space mission extends, typically, over a range of phase angles, where the phase curve tends to be linear on the magnitude scale. Photometry from ground-based observing programs can reach small phase angles, where the asteroids show an opposition effect, a nonlinear increase of brightness on the magnitude scale towards zero phase angle. We provide error models for all four classes of lightcurves and make use of linear or linear-exponential phase functions for phase angles below 50°. We apply the inverse methods to sparse absolute Gaia and dense relative ground-based lightcurves and obtain absolute magnitudes and phase functions, with uncertainties, for ∼500 asteroids. Finally, we assess the lightcurve inversion problem for dense absolute photometry with the help of a numerical simulation for a Gaussian-random-sphere asteroid.

In this paper we consider the problem of Riesz basis in de Branges spaces of entire functions H(E) with the condition that φ(x) ≥ α > 0, where φ is the corresponding phase function. We are concerned with the sets of real numbers {λn} such that the normalized reproducing kernels k(λn,.)/||k(λn,.)|| satisfies the restricted isometry property, which in turn constitute a Riesz basis in H(E). Then we give a criterion on stability of reproducing kernels corresponding to real points which form a Riesz basis in H(E) with respect to small perturbations, which generalize some well-known Riesz basis perturbation results in the Paley-Wiener space. [ABSTRACT FROM AUTHOR]

Scattering properties of irregularly shaped interstellar composite dust analogues consisting of graphite and fayalite (Fe2SiO4) were studied using discrete dipole approximation (DDA). Two dust models were developed to calculate the scattering and extinction efficiencies, single scattering albedo, asymmetry parameter, phase functions and degree of linear polarizations. Laboratory measurements were also performed at three incident wavelengths 543.5, 594.5 and 632.8 nm on chemically synthesized graphite and fayalite composite particles of sizes ranging from 0.3 to 5 μm. A comparative analysis of the theoretical and experimental results of shape- and size-averaged scattering parameters shows that changes in the percentage composition of a two-species mixture model has a pronounced effect on the light-scattering properties of dust particles. The developed computational models are successful in representing a two-species mixture of interstellar dust analogues considering diverse size, shapes and percentage composition. This technique can be applied to fit observed scattering and absorption peaks in the visible region produced by astrophysical dust, provided large number of particle species are included and the influence of more physical parameters (e.g., porosity, fluffiness, temperature, density, etc.) are considered. Further, this study is also applicable to remote sensing, atmospheric and planetary sciences. All the physical parameters employed as variables in the models influence the oscillations observed in theoretical curves and change the values of scattering parameters. [ABSTRACT FROM AUTHOR]

Abstract Let T a , φ $T_{a,\varphi }$ be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions a ( x , ξ ) ∈ L ∞ S ρ n ( ρ − 1 ) / 2 ( ω ) $a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )$ and φ ∈ L ∞ Φ 2 $\varphi \in L^{\infty }\varPhi ^{2}$ , the authors show that T a , φ $T_{a,\varphi }$ is bounded from L 2 ( R n ) $L^{2}(\mathbb{R}^{n})$ to L 2 ( R n ) $L^{2}(\mathbb{R}^{n})$ .

Quantitative determination of the physical properties of natural granite has been attempted from remotely sensed information, for which the Hapke model is a popular method. However, using the model to retrieve the photometric properties of terrestrial rocks (slab or particulate samples), especially for those with complex surface conditions such as natural granite, remains a challenge. In this study, we have approached the dilemma by coupling both radiative transfer (Hapke’s isotropic multiple scattering approximation (IMSA) model) and an empirical relationship between particle sizes with its critical parameter, the single-scattering albedo (SSA, $\omega$), determined from bidirectional reflectance (BDR) measurements. The results clearly indicated that the particle size of natural granite systematically controlled the BDR, which can be well fit by the Hapke model with varying parameters. The retrieved photometric parameters of the coefficients in the phase function ($b$ and $c$) can effectively indicate the scattering behavior of natural granite, but the variations in their values did not strongly correlate with the change in particle sizes. Instead, a good linear relationship between the SSA values and particle sizes has been established. By coupling the relationship into the Hapke model, we found a practical approach to estimate the particle size for measured samples from inversely retrieved SSA. Through this method, we are able to retrieve the physical properties of granite under natural surface conditions, and we foresee that the approach will be widely used in the future. [ABSTRACT FROM AUTHOR]

The Earth Polychromatic Imaging Camera (EPIC) on the Deep Space Climate Observatory (DSCOVR) satellite observes the entire Sun-illuminated Earth from sunrise to sunset from the L1 Sun-Earth Lagrange point. The L1 location, however, confines the observed phase angles to ∼2°–12°, a nearly backscattering direction, precluding any information on the bidirectional surface reflectance factor (BRF) or cloud/aerosol phase function. Deploying an analog of EPIC on the Moon’s surface would offer a unique opportunity to image the full range of Earth phases, including observing ocean/cloud glint reflection for different phase angles; monitoring of transient volcanic clouds; detection of circum-polar mesospheric and stratospheric clouds; estimating the surface BRF and full phase-angle integrated albedo; and monitoring of vegetation characteristics for different phase angles.

There are large uncertainties in the single-scattering (i.e., morphologies) and microphysical (i.e., concentrations) properties of ice particles whose size are less than ~100 µm. Insufficient resolutions of the most advanced cloud probes (e.g., cloud particle imager) cannot resolve the micrometer-scale morphologies of small ice particles. Further, the shattering of large ice particles on probes’ inlets or tips causes uncertainties in the measurement of the concentrations of small ice particles. These uncertainties have large impacts on the single-scattering and microphysical properties of small ice particles that are utilized to quantify the bulk radiative properties of cirrus. In this study, the impacts of uncertainties in the morphologies and concentrations of small ice particles on the bulk radiative properties of tropical cirrus were calculated using measurements acquired during the Tropical Warm Pool-International Cloud Experiment. Five different models (i.e., budding Buckyball, Chebyshev particle, droxtal, Gaussian random sphere, and sphere) that represent the shapes of small ice particles were used to calculate the single-scattering properties. The bulk radiative properties, average phase-function (P11¯), and average asymmetry parameter (g¯) were computed by combining the measured size/habit distributions and the calculated single-scattering properties of ice particles. The impacts of the selection of varying morphologies of small particles on the bulk radiative properties were quantified. For these calculations, the possible range of the concentrations of small ice particles which depend on the degree of shattered large particles were also used. The impacts of varying the single-scattering properties of small ice particles on the bulk radiative properties were the largest in the upper parts of cirrus (T < −60 °C), while they were the smallest in the lower parts of cirrus (−45 < T < −30 °C). The impacts of uncertainties in the concentrations of small ice particles on the bulk radiative properties were largest in the lower parts of cirrus (−45 < T < −30 °C), whereas they were smallest in the upper parts of cirrus (T < −60 °C). The effect of shattering was maximum in the lower parts of cirrus, whilst it was minimum in the upper parts of cirrus. The combined impacts of uncertainties in the single-scattering (i.e., morphologies) and microphysical (i.e., concentrations) properties of small ice particles revealed variations of up to 11.2% (127.1%; 67.3%) of the integrated intensity in the forward (sideward; backward) angles in P11¯ and a corresponding change in g¯ by up to 12.61%.

In this paper, we study the L 2 -boundedness and L 2 -compactness of a class of h -Fourier integral operators (h -FIOs) with operator symbol. Mainly, we prove that this class is L 2 -bounded (respectively, L 2 -compact) if the order function is bounded (respectively, tends to 0). [ABSTRACT FROM AUTHOR]

The aggregate structures of photocatalyst particles are critical for their light absorption and scattering properties and thus significantly impact the photocatalytic performance toward solar hydrogen production. Herein, taking P25 TiO 2 nanoparticles as model photocatalysts, we calculate the optical properties of the aggregates of nanoparticles with the Discrete Dipole Approximation (DDA). A scattering phase function, which can reveal and predict the structural information of photocatalyst particles, is accordingly obtained. Specifically, it is found that the forward scattering is dominant when light is scattered on the particles, and this scattering mode becomes more intense with the increment of the particle size. Our results show that the scattering phase function is also in agreement with the Rayleigh-Gans-Debye (RGD) approximation. • Clusters of TiO 2 nanoparticles are established via fractal aggregation models. • Scattering properties are calculated via Discrete Dipole Approximation. • Forward-scattering dominates the phase function in modeling photocatalysis. • Phase functions agree with the prediction of Rayleigh-Gans-Debye approximation. [ABSTRACT FROM AUTHOR]

Abstract Acquiring accurate scattering properties is important for rendering translucent materials. In particular, the phase function, which determines the distribution of scattering directions, plays a significant role in the appearance of a material. We propose a distinctive scattering theory that approximates the effect of single scattering to acquire the non-parametric phase function from a single image. Furthermore, in various experiments, we measured the phase functions from several real diluted media and rendered images of these materials to evaluate the effectiveness of our theory.

Let T a , φ be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions a (x , ξ) ∈ L ∞ S ρ n (ρ − 1) / 2 (ω) and φ ∈ L ∞ Φ 2 , the authors show that T a , φ is bounded from L 2 (R n) to L 2 (R n) . [ABSTRACT FROM AUTHOR]

The optimization of extinction performance of particles in smoke screen is studied based on the phenomenon of particle agglomeration in smoke screen. Based on the theory of cluster-cluster aggregation( CCA),the algorithm program of aggregates with random particle size is established to generate the aggregates composed of different sized particles. On the basis of this model,multi-sphere T matrix( MSTM) method is used to calculate and analyze the scattering properties of the aggregates,of which monomer size follows a lognormal distribution of particles in the smoke screen. The results show that the extinction performance of the aggregate is weakened with the increase in particle size dispersion. In the simulated multi-group aggregate particles,the extinction factor of type A aggregate is increased by 30. 9%compared with that of type B aggregate,and the overall scattering energy is decreased by 32. 5%. The extinction performance of the aggregate in smoke screen can be effectively improved by avoiding an extremely large size monomer particle which size deviates from the mean of lognormal distribution of particles in the aggregates. [ABSTRACT FROM AUTHOR]

During the Chang'E-5 (CE-5) lander descent process, the rocket exhaust disturbed the lunar surface of the landing area, which may have affected the properties of the collected samples. Therefore, it is necessary to explore the effects of lander exhaust on the landing area. In this paper, the images obtained by the Narrow Angle Camera (NAC) onboard the Lunar Reconnaissance Orbiter (LRO) were used to determine the exhaust-disturbed area. It is found that this area was featured by an increased reflectance of ∼10 ± 1% compared with the background (from I/F = 0.048 to 0.053). According to the magnitude of the change in the reflectance of the landing region, the disturbed area was divided into the focus disturbed zone (FDZ) and the diffuse disturbed zone (DDZ). The DDZ extends from the edge of the FDZ and has a lower reflectance anomaly. The whole exhaust-disturbed area was found to extend mainly via the north-south direction and to have the shape of an irregular ellipse. This spatial extent is highly related to the topographic factors rather than the obstacle avoidance process according to the topography and landing process. The phase function of the lunar surface was used to analyze the photometric performance of the disturbed area. The lower slope of the phase function in FDZ demonstrates less backscattering and a smoothing effect of the disturbed lunar surface. The CE-5 high-resolution LCAM images captured the smoothing and brightening effects of the lunar surface, which confirmed the variation in regolith properties. These effects are consistent with the inference from the NAC observations. The damage to the "fairy-castle" structure of the lunar surface could also contribute to surface smoothing. This study can provide a reference for the subsequent analyses of lunar samples and can help to enhance the understanding of the effect of lander exhaust on the lunar regolith. • The disturbed area and the variation in the reflectance of the Chang'E-5 landing site were identified. • The extent of the disturbed area was related to the local topography of the landing site. • The smoothing and brightening of the exhaust-disturbed area was observed in the Chang'E-5 landing camera images. [ABSTRACT FROM AUTHOR]

Aerosol models, composed of size distribution, complex refractive index, and spherical fraction, are derived from a new synergistic retrieval of airborne in situ angular scattering measurements made by the Polarized Imaging Nephelometer and absorption measurements from the Particle Soot Absorption Photometer. The data utilized include phase function (F11), degree of polarization (−F12/F11), and absorption coefficient (βabs) measured at low relative humidities during the Studies of Emissions and Atmospheric Composition, Clouds, and Climate Coupling by Regional Surveys (SEAC4RS) and Deep Convection Clouds and Chemistry (DC3) field campaigns. The Generalized Retrieval of Aerosol and Surface Properties (GRASP) is applied to these measurements to obtain summaries of particle properties that are optically consistent with the original measurements. A classification scheme is then used to categorize the corresponding retrieval results. Inversions performed on the DC3 measurements indicate the presence of a significant amount of dust‐like aerosol in the inflow of storms sampled during this campaign, with the quantity of dust present depending strongly on the underlying surface features. In the SEAC4RS data, the retrieved size distributions were found to be remarkably similar among a range of aerosol types, including urban and industrial, biogenic, and biomass burning (BB) emissions. These aerosol types were found to have average fine mode volume median radii 0.155 ≤ rvf ≤ 0.163μm and lognormal standard deviations 0.32 ≤ σf ≤ 0.36. There were, however, consistent differences between the angular scattering patterns of the BB samples and the other particle types. The GRASP retrieval predominantly attributed these differences to elevated real and imaginary refractive indices in the BB samples (m532nm≈1.55+0.007i) relative to the two other categories (m532nm≈1.51+0.004i). Key Points: Aerosol properties are obtained using a novel synergistic retrieval of in situ angular‐dependent scattering and absorption measurementsBiomass burning particles were found to have elevated real and imaginary refractive indices relative to other aerosol typesSignificant concentrations of dust‐like particles are observed in measurements made in the vicinity of convective systems [ABSTRACT FROM AUTHOR]

Transmission coefficient and field emission current in a silicon vacuum nanostructure with a pyramidal cathode were calculated as a function of applied voltage, size of the cathode and distance between the anode and cathode by the phase function method. The field emission current density in the range of 1–10 A/cm2 was found to be achieved by varying the distance between the anode and cathode in the range of 15–25 nm and the applied voltage in the range of 1.2–2.3 V. [ABSTRACT FROM AUTHOR]

To find a unified approach for the stability analysis of second-order switched system, the concept of phase function is proposed in this paper. First, the basic properties of phase function are explored. Following this concept and its properties, the phase-based stability criterion is investigated based on the Lyapunov theory, and a necessary and sufficient stability condition is obtained in the phase function approach. Moreover, the connection between phase-based stability conditions and algebraic condition of system matrices is also discussed. Finally, numerical examples are provided to exemplify the main result and make necessary comparisons with the existing methods. [ABSTRACT FROM AUTHOR]

In this paper we consider the summation problem for two-dimensional trigonometrical integrals with a third-order polynomial phase. We find the sharp convergence exponent using the estimates for trigonometric integrals. [ABSTRACT FROM AUTHOR]