Wavenumber Interval Research Articles

Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are given by the far field patterns u∞(xˆ,k) for all observation directions xˆ in some neighborhood of a fixed direction xˆ0 and for all wave numbers k belonging to a finite interval (0,K). In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval K>1. In three dimensions we establish stability estimates of the L2-norm and H−1-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.

Extraordinary optical and transport properties of disordered stealthy hyperuniform two-phase media

Disordered stealthy hyperuniform two-phase media are a special subset of hyperuniform structures with novel physical properties due to their hybrid crystal-liquid nature. We have previously shown that the rapidly converging strong-contrast expansion of a linear fractional form of the effective dynamic dielectric constant εek1,ω (Torquato and Kim 2021 Phys. Rev. X 11 021002) leads to accurate approximations for both hyperuniform and nonhyperuniform two-phase composite media when truncated at the two-point level for distinctly different types of microstructural symmetries in three dimensions. In this paper, we further elucidate the extraordinary optical and transport properties of disordered stealthy hyperuniform media. Among other results, we provide detailed proofs that stealthy hyperuniform layered and transversely isotropic media are perfectly transparent (i.e. no Anderson localization, in principle) within finite wavenumber intervals through the third-order terms. Remarkably, these results imply that there can be no Anderson localization within the predicted perfect transparency interval in stealthy hyperuniform layered and transversely isotropic media in practice because the localization length (associated with only possibly negligibly small higher-order contributions) should be very large compared to any practically large sample size. We further contrast and compare the extraordinary physical properties of stealthy hyperuniform two-phase layered, transversely isotropic media, and fully three-dimensional isotropic media to other model nonstealthy microstructures, including their attenuation characteristics, as measured by the imaginary part of εek1,ω , and transport properties, as measured by the time-dependent diffusion spreadability S(t) . We demonstrate that there are cross-property relations between them, namely, we quantify how the imaginary parts of εek1,ω and the spreadability at long times are positively correlated as the structures span from nonhyperuniform, nonstealthy hyperuniform, and stealthy hyperuniform media. It will also be useful to establish cross-property relations for stealthy hyperuniform media for other wave phenomena (e.g. elastodynamics) as well as other transport properties. Cross-property relations are generally useful because they enable one to estimate one property, given a measurement of another property.

Spectroscopic heat maps reveal how to design experiments to improve the uncertainties of transitions and energy levels present in line-by-line databases

Spectroscopic information systems contain enormous sets of line-by-line (LBL) data on characteristics of rovibronic transitions. Developers of these datasets are continuously improving transitions data, for example, via lowering the uncertainties of the transitions. Improved transition uncertainties may also result in a decrease in the uncertainties of the energies of some of the quantum states. This paper introduces a method that assesses the existing line lists and suggests wavenumber intervals for possible (re)measurements, based on their usefulness to improve the uncertainties of the transitions and/or the quantum-state energies. This is called the spectroscopic heat map method, which characterizes with two values all wavenumber intervals of a fixed length. The spectroscopic heat map method is tested on the example of seven water isotopologues and the most abundant isotopologue of acetylene, based on data present in the HITRAN information system.

Theoretical prediction of the effective dynamic dielectric constant of disordered hyperuniform anisotropic composites beyond the long-wavelength regime [Invited

Torquato and Kim [Phys. Rev. X 11, 296 021002 (2021)10.1103/PhysRevX.11.021002] derived exact nonlocal strong-contrast expansions of the effective dynamic dielectric constant tensor ε e (k q ,ω) that treat general statistically anisoropic three-dimensional (3D) two-phase composite microstructures, which are valid well beyond the long-wavelength regime. Here, we demonstrate that truncating this general rapidly converging expansion at the two- and three-point levels is a powerful theoretical tool from which one can extract accurate approximations suited for various microstructural symmetries. Among other results, we show that such truncations yield closed-form formulas applicable to transverse polarization in layered media and transverse magnetic polarization in transversely isotropic media, respectively. We apply these formulas to estimate ε e (k q ,ω) for models of 3D disordered hyperuniform layered and transversely isotropic media: nonstealthy hyperuniform media and stealthy hyperuniform media. In particular, we show that stealthy hyperuniform layered and transversely isotropic media are perfectly transparent (trivially implying no Anderson localization, in principle) within finite wave number intervals through the third-order terms. For all models considered here, we validate that the second-order formulas, which depend on the spectral density, are already very accurate well beyond the long-wavelength regime by showing very good agreement with the finite-difference time-domain (FDTD) simulations. The high predictive power of the second-order formula is due to the fact that higher-order contributions are negligibly small, implying that it very accurately approximates multiple scattering through all orders. This implies that there can be no Anderson localization within the predicted perfect transparency interval in stealthy hyperuniform layered and transversely isotropic media in practice because the localization length (associated with only possibly negligibly small higher-order contributions) should be very large compared to any practically large sample size. Our predictive theory provides the foundation for the inverse design of novel effective wave characteristics of disordered and statistically anisotropic structures by engineering their spectral densities.

Inverse cascading for initial magnetohydrodynamic turbulence spectra between Saffman and Batchelor

In decaying magnetohydrodynamic (MHD) turbulence with a strong magnetic field, the spectral magnetic energy density is known to increase with time at small wavenumbers $k$ , provided the spectrum at low $k$ is sufficiently steep. This process is called inverse cascading and occurs for an initial Batchelor spectrum, where the magnetic energy per linear wavenumber interval increases like $k^4$ . For an initial Saffman spectrum that is proportional to $k^2$ , however, inverse cascading has not been found in the past. We study here the case of an intermediate $k^3$ spectrum, which may be relevant for magnetogenesis in the early Universe during the electroweak epoch. This case is not well understood in view of the standard Taylor expansion of the magnetic energy spectrum for small $k$ . Using high resolution MHD simulations, we show that, also in this case, there is inverse cascading with a strength just as expected from the conservation of the Hosking integral, which governs the decay of an initial Batchelor spectrum. Even for shallower $k^\alpha$ spectra with spectral index $\alpha >3/2$ , our simulations suggest a spectral increase at small $k$ with time $t$ proportional to $t^{4\alpha /9-2/3}$ . The critical spectral index of $\alpha =3/2$ is related to the slope of the spectral envelope in the Hosking phenomenology. Our simulations with $2048^3$ mesh points now suggest inverse cascading even for an initial Saffman spectrum.

A Simplified Numerical Method for Simulating the Generation of Linear Waves by a Moving Bottom

In this study, simplified linear numerical method that can simulate wave generation and transformation by a moving bottom is introduced. Numerical analysis is conducted in wave number domain after continuity equation, linear dynamic and kinematic free surface boundary conditions and linear kinematic bottom boundary condition are Fourier transformed, and the results are expressed in space domain by an inverse Fourier transform. In the wavenumber domain, the dynamic free water surface boundary condition and the kinematic free water surface boundary condition are numerically calculated, and the velocity potential in the mean water level (<i>z</i> = 0) satisfies the continuity equation and the kinematic bottom boundary condition. Wave generation and transformation are investigated when the triangular and rectangular shape of bottoms move periodically. The results of the simplified numerical method are compared with the results of previous analytical solutions and agree well with them. Stability of numerical results according to the calculation time interval (△<i>t</i>) and the calculation wave number interval (△<i>k</i>) was also investigated. It was found that the numerical results were appropriate when △<i>t</i>≤<i>T</i>(period)/1000 and △<i>k</i>≤π/100.

A PDF discretization scheme in wavenumber–frequency joint spectrum for simulating multivariate random fluctuating wind fields

The wavenumber–frequency joint spectrum-based spectral representation method is an effective method for simulating multivariate random fluctuating wind fields. However, the simulation efficiency may not be quite satisfactory when the uniform discretizations in both frequency and wavenumber domains are employed. In this paper, a probability density function discretization scheme in wavenumber domain, which is actually an uneven discretization scheme, is proposed to alleviate the computational burden. In the proposed scheme, the wavenumber–frequency joint spectrum at a characteristic frequency is first taken as a probability density function. Then, the wavenumber domain is unevenly discretized according to the probability density function. Since the wavenumber–frequency joint spectrum may decay very fast in wavenumber domain, a uniform discretization is further applied in large intervals in wavenumber domain. Finally, the unevenly discrete wavenumbers and wavenumber intervals are substituted into the wavenumber–frequency joint spectrum-based spectral representation method for generating samples of multivariate random fluctuating wind fields, where the frequency domain is still uniformly discretized. The accuracy and efficiency of the proposed discretization scheme are compared with the uniform and conventional uneven discretization schemes. Practical applications of the proposed method are also investigated in various engineering scenarios.

Three-Way Serpentine Slow Wave Structures With Stationary Inflection Point and Enhanced Interaction Impedance

We introduce two novel variants of the serpentine waveguide slow-wave structure (SWS), often utilized in millimeter-wave traveling-wave tubes (TWTs), with an enhanced interaction impedance. Using dispersion engineering in conjunction with transfer matrix methods, we tune the guided wavenumber dispersion relation to exhibit stationary inflection points (SIPs), and also non-stationary, or tilted inflection points (TIPs), within the dominant TE10 mode of a rectangular waveguide. The degeneracy is found below the first upper band-edge associated with the bandgap where neighboring spatial harmonics meet in the dispersion of the serpentine waveguide (SWG) which is threaded by a beam tunnel. The structure geometries are optimized to be able to achieve an SIP which allows for three-mode synchronism with an electron beam over a specified wavenumber interval in the desired Brillouin zone. Full-wave simulations are used to obtain and verify the existence of the SIP in the three-way coupled waveguide and fine-tune the geometry such that a beam would be in synchronism at or near the SIP. The three-way waveguide SWS exhibits a moderately high Pierce impedance in the vicinity of a nearly-stationary inflection point, making the SWS geometry potentially useful for improving the power gain and basic extraction efficiency of millimeter-wave TWTs. Additionally, the introduced SWS geometries have directional coupler-like behavior, which enables distributed power extraction at frequencies near the SIP frequency.

Comparing the Counter-beaming and Temperature Anisotropy Driven Aperiodic Electron Firehose Instabilities in Collisionless Plasma Environments

The electron firehose instabilities are among the most studied kinetic instabilities, especially in the context of space plasmas, whose dynamics is mainly controlled by collisionless wave–particle interactions. This paper undertakes a comparative analysis of the aperiodic electron firehose instabilities excited either by the anisotropic temperature or by the electron counter-beaming populations. Two symmetric counter-beams provide an effective kinetic anisotropy similar to the temperature anisotropy of a single (nondrifting) population, with the temperature along the magnetic field direction larger than that in the perpendicular direction. Therefore, the counter-beaming plasma is susceptible to firehose-like instabilities (FIs), parallel and oblique branches. Here we focus on the oblique beaming FI, which is also aperiodic when the free energy is provided by symmetric counter-beams. Our results show that, for relative small drifts or beaming speeds (U), not exceeding the thermal speed (α), the aperiodic FIs exist in the same interval of wavenumbers and the same range of oblique angles (with respect to the magnetic field direction), but the growth rates of counter-beaming FI (CBFI) are always higher than those of temperature anisotropy FI (TAFI). For U/α > 1, however, another electrostatic two-stream instability is also predicted, which may have growth rates higher than those of CBFI, and may dominate in that case the dynamics.

Acoustics of a Partially Partitioned Narrow Slit Connected to a Half-Plane: Case Study for Exponential Quasi-Bound States in the Continuum and their Resonant Excitation

Localized wave oscillations in an open system that do not decay or grow in time, despite their frequency lying within a continuous spectrum of radiation modes carrying energy to or from infinity, are known as bound states in the continuum (BIC). Small perturbations from the typically delicate conditions for BIC almost always result in the waves weakly coupling with the radiation modes, leading to leaky states called quasi-BIC that have a large quality factor. We study the asymptotic nature of this weak coupling in the case of acoustic waves interacting with a rigid substrate featuring a partially partitioned slit---a setup that supports quasi-BIC that exponentially approach BIC as the slit is made increasingly narrow. In that limit, we use the method of matched asymptotic expansions in conjunction with reciprocal relations to study those quasi-BIC and their resonant excitation. In particular, we derive a leading approximation for the exponentially small imaginary part of each wavenumber eigenvalue (inversely proportional to quality factor), which is beyond all orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we derive a leading approximation for the exponentially large amplitudes of the states in the case where they are resonantly excited by a plane wave at oblique incidence. These resonances occur in exponentially narrow wavenumber intervals and are physically manifested in cylindrical-dipolar waves emanating from the slit aperture and exponentially large field enhancements inside the slit. The asymptotic approximations are validated against numerical calculations.

Simulations of Helical Inflationary Magnetogenesis and Gravitational Waves

Abstract Using numerical simulations of helical inflationary magnetogenesis in a low reheating temperature scenario, we show that the magnetic energy spectrum is strongly peaked at a particular wavenumber that depends on the reheating temperature. Gravitational waves (GWs) are produced at frequencies between 3 nHz and 50 mHz for reheating temperatures between 150 MeV and 3 × 105 GeV, respectively. At and below the peak frequency, the stress spectrum is always found to be that of white noise. This implies a linear increase of GW energy per logarithmic wavenumber interval, instead of a cubic one. Both in the helical and nonhelical cases, the GW spectrum is followed by a sharp drop for frequencies above the respective peak frequency. In this magnetogenesis scenario, the presence of a helical term extends the peak of the GW spectrum and therefore also the position of the aforementioned drop toward larger frequencies compared to the case without helicity. This might make a difference in it being detectable with space interferometers. The efficiency of GW production is found to be almost the same as in the nonhelical case, and independent of the reheating temperature, provided the electromagnetic energy at the end of reheating is fixed to be a certain fraction of the radiation energy density. Also, contrary to the case without helicity, the electric energy is now less than the magnetic energy during reheating. The fractional circular polarization is found to be nearly 100% in a certain range below the peak frequency range.

Understanding the conformational, electronic and vibrational properties of Tetrahydrocannabinol (THC) and Cannabidiol (CBD). Pharmacophoric similarities and differences

• Conformational flexibility of THC or CBD concerns only the side chain (top). • The most intensive IR and Raman bands are due to the molecular frames. • Folded conformations themselves are not easily accessible. • Easiness of conformational transitions aaa ⇋ g - aa ⇋ g + aa ⇋ … is a neccessary consequence of the chain possibility to fold. (-)- trans -Δ 9 - tetrahydrocannabinol (THC) and (-)-cannabidiol (CBD) are natural cannabinoids found in cannabis plant which interact with cannabinoid receptors in human body. THC is primary psychoactive compound while CBD is non-psychotropic constituent, i.e. they have different receptor activity, function, and physiological effects. These dramatic differences in their behavior are derived from not so dramatic differences in their structures. Calculations of the structures and infrared and Raman spectra of all THC and CBD conformers were performed by means of density functional theory (DFT) at the B3LYP-D3BJ/6-31+ G (d,p) level. Conformational flexibility due to the presence of alkyl (pentyl) side chain was thoroughly investigated and shown to be weakly dependent on the molecular frame . The result of the QSAR analysis by Keimowitz et al. J. Med. Chem. 43 (2000) 59–70 that for optimum affinity and potency the side chain must have conformational flexibility to fold over and thus come with its end methyl group close to the phenolic ring was examined with respect to the energies needed for such transformation. Assignments of the observed vibrational spectra reported in the literature were made by attempting to have a strong calculated band assigned to the strong observed band lying in the same narrow wavenumber interval. Evidences for intra- and intermolecular hydrogen bonding in CBD based on experimental and calculated data are given.

A novel polynomial reconstruction algorithm‐based 1D convolutional neural network used for transfer learning in Raman spectroscopy application

AbstractWhen a convolutional neural network (CNN) model was built, the size and resolution of its input data were fixed. However, Raman spectra collected by different Raman spectrometers usually had different length, intensity range, and wavenumber interval between two adjacent data points, which made the existing CNN model difficult to be applied to a new Raman spectral data set. Therefore, this paper proposed a polynomial reconstruction algorithm as pretreatment method to obtain reconstructed spectra that would be imported into CNN model with consistent length, intensity range, and wavenumber interval. To test the effectiveness of this method, a big data set with 2563 Raman spectra of 831 minerals and synthetic organic pigments samples was constructed from the RRUFF and SOP database to pretrain a one‐dimensional CNN (1D‐CNN) model. The pretraining results showed that polynomial reconstruction algorithm used as pretreatment method was better than SG smoothing combined spline interpolation algorithm. Then two data sets were collected by different Raman spectrometers for evaluating the transfer learning performance of the trained 1D‐CNN model. Both data sets contained 390 Raman spectra from the same 39 samples of inorganic salts, organic compounds, and amino acids. One was used as calibration data to retrain the 1D‐CNN model, while the other was used as test. Based on data augmentation and 75% calibration data for retraining, the transfer learning performances of 1D‐CNN model were clearly shown in the excellent identification accuracies of 99.58%, 99.32%, and 97.69% for training, validation, and test sets, respectively, which were better than those of K‐nearest neighbor classifier. This paper provides a significant way for the wide application of CNN model in Raman spectroscopy with much more advantages in simplicity and rapidity.

Simulation and analysis of hot plume infrared signature based on SNB model

Abstract Numerical calculation of infrared emission from hot plume is of great significance for flight monitoring and detections. In this paper, the SNB (statistical narrow band) model established with parameters derived from the high-resolution spectral database HITEMP 2010 is used to perform the hot plume infrared signature simulations. Accuracy of the model is examined by the exact LBL (line by line) method, which proves the model’s reliability to predict radiative properties of combustion gases. In the application part, the SNB model is used to analyze infrared signatures of aircraft plumes cruising at different flight altitudes. The results show that cruising at a higher-altitude will obviously reduce the plume infrared emission. Besides, the plume infrared emissive energy mainly concentrates in a special wavenumber interval and can be strongly absorbed by atmosphere.

Simulation and analysis of hot plume infrared signature based on SNB model

Abstract Numerical calculation of infrared emission from hot plume is of great significance for flight monitoring and detections. In this paper, the SNB (statistical narrow band) model established with parameters derived from the high-resolution spectral database HITEMP 2010 is used to perform the hot plume infrared signature simulations. Accuracy of the model is examined by the exact LBL (line by line) method, which proves the model’s reliability to predict radiative properties of combustion gases. In the application part, the SNB model is used to analyze infrared signatures of aircraft plumes cruising at different flight altitudes. The results show that cruising at a higher-altitude will obviously reduce the plume infrared emission. Besides, the plume infrared emissive energy mainly concentrates in a special wavenumber interval and can be strongly absorbed by atmosphere.

Two different synchronous spectrofluorimetric approaches for simultaneous determination of febuxostat and ibuprofen.

Two green, simple and sensitive synchronous spectrofluorimetric methods were developed for the first time for the simultaneous estimation of febuxostat (FEB) and ibuprofen (IBU). Method I is constant-wavelength synchronous spectrofluorimetry where FEB and IBU were recorded at 329 and 258 nm, respectively, using Δλ of 40 nm. Method II is constant-energy synchronous spectrofluorimetry using a wavenumber interval of −4000 cm−1. All measurements were carried out in a borate buffer of pH 7 and distilled water for dilution which increased the methods' greenness. The two methods were rectilinear over concentration ranges of 30.0–700.0 ng ml−1 and 0.5–9.0 µg ml−1 in the first method and 20.0–500.0 ng ml−1 and 0.1–8.0 µg ml−1 in the second method for FEB and IBU, respectively. High sensitivity was attained for the two drugs with limits of quantitations (LODs) down to 0.41 and 5.51 ng ml−1 in the first method and 0.25 and 3.32 ng ml−1 in the second method for FEB and IBU, respectively. Recovery percentages were in the range of 97.3–101.9% after extraction from spiked human plasma samples, demonstrating high bioanalytical applicability. The two methods were further applied to tablet dosage forms with good recovery results. The methods' greenness was assessed according to the analytical Eco-Scale and Green Analytical Procedure Index guidelines.

Relic Gravitational Waves from the Chiral Magnetic Effect

Abstract Relic gravitational waves (GWs) can be produced by primordial magnetic fields. However, not much is known about the resulting GW amplitudes and their dependence on the details of the generation mechanism. Here we treat magnetic field generation through the chiral magnetic effect (CME) as a generic mechanism and explore its dependence on the speed of generation (the product of magnetic diffusivity and characteristic wavenumber) and the speed characterizing the maximum magnetic field strength expected from the CME. When the latter exceeds the former (regime I), which is the regime applicable to the early universe, we obtain an inverse cascade with moderate GW energy that scales with the third power of the magnetic energy. When the generation speed exceeds the CME limit (regime II), the GW energy continues to increase without a corresponding increase of magnetic energy. In the early kinematic phase, the GW energy spectrum (per linear wavenumber interval) has opposite slopes in both regimes and is characterized by an inertial range spectrum in regime I and a white noise spectrum in regime II. The occurrence of these two slopes is shown to be a generic consequence of a nearly monochromatic exponential growth of the magnetic field. The resulting GW energy is found to be proportional to the fifth power of the limiting CME speed and the first power of the generation speed.

Application of the Multi-Spectral Correlated-k Distribution Model to Multi-Dimensional Rectangular Enclosure Gas Radiation Calculations

The Multi-Spectral Correlated-k distribution model (MSCk) is an amelioration of the widely used Ck model. In the MSCk model, the breakdown of correlation assumption used in the original Ck model for non-uniform media is overcome by introducing the clustering of scaling functions. The principle of MSCk model is to group together wavenumbers with respect to the spectral scaling functions—defined as the ratio between spectral absorption coefficients in distinct states—so that the correlation assumption can be considered as exact over the corresponding intervals of wavenumbers. Until now, validations of the MSCk model in 0D and 1D test cases have already been performed in the previous work (Andre, F., Hou, L., Roger, M. and Vaillon, R., 2014. The multispectral gas radiation modeling: A new theoretical framework based on a multidimensional approach to k-distribution methods. Journal of Quantitative Spectroscopy and Radiative Transfer, 147, pp.178–195; Andre, F., Hou, L. and Solovjov, V.P., 2016. An Exact Formulation of k-Distribution Methods in Non-Uniform Gaseous Media and its Approximate Treatment Within the Multi-Spectral Framework. Journal of Physics: Conference Series, 676(1)). However, its application to multi-dimensional configurations (much closer to industrial applications) has not been conducted. Accordingly, in the present paper, we focus our attention on the application of the MSCk model to Multi-dimensional calculations.

Infrared spectroscopy technique for quantification of compounds in plant-based medicine and supplement.

Quality control of plant-based medicine and supplements must be carried out to ensure uniformity in quality and safety in their use, resulting in the need for effective and accurate analytical methods. Infrared spectroscopy is a method of qualitative and quantitative analysis that is fast, time-saving, cost-effective,accurate, and nondestructive. This method has been applied for quantitative analysis of compounds in complex matrices such as plant-based medicine and supplements supported by chemometrics techniques. The success of infrared spectroscopy applications for quantitative analysis of phytochemicals and adulterants content in plant-based medicine and supplement can happen by several factors. This article highlights the effect of spectral preprocessing and variable selection on quantitative analysis of phytochemical and adulterant in plant-based medicine and supplements using infrared spectroscopy. Literature search was conducted with PubMed, Google Scholar, and Science Direct by selecting quantitative analysis research on plant-based medicines and supplements that utilize spectral preprocessing techniques and variable selection in processing data analysis. The preprocessing spectra and variables selection can affect the accuracy and precision of infrared spectroscopy methods. The variable selection can be done using the wavenumber point technique, the wavenumber interval, or a combination thereof. Variable selection is more commonly used for near-infrared data than for IR data. The optimization of the preprocessing spectra and variables selection technique will be useful in increasing the ability of infrared spectroscopy in predicting compound levels.

Remote Sensing of High Temperature H2O–CO2 Mixture with the Multi-Spectral Correlated-k Distribution Model

Radiative heat transfer of gas plays an important role in remote sensing applications. Several models have been proposed to estimate the radiative properties of gases. The Line-By-Line (LBL) approach is the most accurate one. However, The Line-By-Line (LBL) approach usually requires excessive computation cost which makes it not applicable for most industrial most applications. Nevertheless, it still plays the role of benchmark reference for the assessment of other models. In this work, we propose the application of the Multi-Spectral Correlated-k distribution model (MSCk) so as to make the simulation of hot jet signature at long distance. In the MSCk model, the breakdown of correlation assumption used in the original Ck model for non-uniform media is overcome by introducing the clustering of scaling functions. The principle of MSCk model is to group together wavenumbers with respect to the spectral scaling functions-defined as the ratio between spectral absorption coefficients in distinct states-so that the correlation assumption can be considered as relatively exact over the corresponding intervals of wavenumbers. Results show that the MSCk model demonstrates better performance compared with the traditional Ck model at nearly an equivalent calculation cost.