Geometrical Optics Model Research Articles

Visibility of natural tertiary rainbows

Naturally occurring tertiary rainbows are extraordinarily rare and only a handful of reliable sightings and photographs have been published. Indeed, tertiaries are sometimes assumed to be inherently invisible because of sun glare and strong forward scattering by raindrops. To analyze the natural tertiary's visibility, we use Lorenz-Mie theory, the Debye series, and a modified geometrical optics model (including both interference and nonspherical drops) to calculate the tertiary's (1) chromaticity gamuts, (2) luminance contrasts, and (3) color contrasts as seen against dark cloud backgrounds. Results from each model show that natural tertiaries are just visible for some unusual combinations of lighting conditions and raindrop size distributions.

The retrieval of shrub fractional cover based on a geometric-optical model in combination with linear spectral mixture analysis

Vegetation fractional cover, which defines the amount of vegetation on the surface of the land, is a key parameter in land surface models. Based on a geometric-optical model in combination with a linear spectral mixture analysis, the retrieval of shrub fractional cover in Wushen Banner of Inner Mongolia in the Mu Us Sandland using HJ-1B multispectral images is discussed. We acquired the surface reflectance based on geometric correction and atmospheric correction of the HJ-1B image. Then we assumed that the reflectance of a mixed pixel is a simple linear combination of two components, namely illuminated background and illuminated canopy, and further calculated the areal proportion of the illuminated background within each pixel based on the linear spectral mixture analysis. Then, combined with the measured shrub structural parameters, the shrub fractional cover was estimated using an inverted geometric-optical model. Finally, the result was validated through the measured shrub cover of 13 sample plots and a comparison study was done with the NDVI regression method and simple linear spectral mixture analysis. The R 2 of the three methods are 0.898, 0.614, and 0.659, with corresponding root-mean-squared errors of 0.136, 0.154, and 0.175, which indicate the reliability of the combined method.

Study of Gas Injection System for EBIT Facilities

This paper presents a design of gas injection system for the electron beam ion trap and an approach of estimation of the injection efficiency based on geometrical optics model. The differential pumping structure is used to ensure efficient injection without destroying the vacuum. The estimation is proved reliable and reasonable by experiments results.

A Model Analysis Using a Combined Active/Passive Microwave Remote Sensing Approach for Soil Moisture Retrieval

This study develops a new algorithm for estimating bare surface soil moisture using combined active/passive microwave remote sensing on the basis of TRMM(Tropical Rainfall Measuring Mission).To accurately estimate bare surface soil moisture,the algorithm simultaneously uses a precipitation radar(PR) and microwave imager(TMI) for observation.In this paper,through an analysis of the numerically simulated surface backscattering and emissivity database by Advanced Integral Equation Model(AIEM) with a wide range of soil moisture and surface roughness conditions,the authors have found that the surface roughness parameter has an important impact on both backscattering coefficient and emissivity.Based on the simulated database by AIEM,the authors use theoretical model(Geometrical Optical Model) and semi-empirical model(Qp Model) through the analysis of surface roughness as well as TRMM/PR and TMI data to develop the algorithm for estimating bare surface soil moisture.

Discrete canonical transforms that are Hadamard matrices

The group of symplectic linear canonical transformations has an integral kernel which has quadratic and linear phases, and which is realized by the geometric paraxial optical model. The discrete counterpart of this model is a finite Hamiltonian system that acts on N-point signals through N × N matrices whose elements also have a constant absolute value, although they do not form a representation of that group. Those matrices that are also unitary are Hadamard matrices. We investigate the manifolds of these N × N matrices under the equivalence imposed by the model, and find them to be on two-sided cosets. By means of an algorithm we determine representatives that lead to collections of mutually unbiased bases.

Estimating zero-plane displacement height and aerodynamic roughness length using synthesis of LiDAR and SPOT-5 data

In this study, a combination of low and high density airborne LiDAR and satellite SPOT-5 HRG data were used in conjunction with ground measurements of forest structure to parameterize four models for zero-plane displacement height d(m) and aerodynamic roughness length z 0 m (m), over cool-temperate forests in Heihe River basin, an arid region of Northwest China. For the whole study area, forest structural parameters including tree height (Ht) (m), first branch height (FBH) (m), crown width (CW) (m) and stand density (SD)(trees ha − 1 ) were derived by stepwise multiple linear regressions of ground-based forest measurements and height quantiles and fractional canopy cover ( f c ) derived from the low density LiDAR data. The high density LiDAR data, which covered a much smaller area than the low density LiDAR data, were used to relate SPOT-5's reflectance to the effective plant area index (PAIe) of the forest. This was done by linear spectrum decomposition and Li–Strahler geometric–optical models. The result of the SPOT-5 spectrum decomposition was applied to the whole area to calculate PAIe (and leaf area index LAI). Then, four roughness models were applied to the study area with these vegetation data derived from the LiDAR and SPOT-5 as input. For validation, measurements at an eddy covariance site in the study area were used. Finally, the four models were compared by plotting histograms of the accumulative distribution of modeled d and z 0 m in the study area. The results showed that the model using by frontal area index (FAI) produced best d estimate, and the model using both LAI and FAI generated the best z 0 m . Furthermore, all models performed much better when the representative tree height was Lorey's mean height instead of using an arithmetic mean.

Assessment of the impacts of surface topography, off-nadir pointing and vegetation structure on vegetation lidar waveforms using an extended geometric optical and radiative transfer model

In order to prioritize the measurement requirements and accuracies of the two new lidar missions, a physical model is required for a fundamental understanding of the impact of surface topography, footprint size and off-nadir pointing on vegetation lidar waveforms and vegetation height retrieval. In this study, we extended a well developed Geometric Optical and Radiative Transfer (GORT) vegetation lidar model to take into account for the impacts of surface topography and off-nadir pointing on vegetation lidar waveforms and vegetation height retrieval and applied this extended model to assess the aforementioned impacts on vegetation lidar waveforms and height retrieval. Model simulation shows that surface topography and off-nadir pointing angle stretch waveforms and the stretching effect magnifies with footprint size, slope and off-nadir pointing angle. For an off-nadir pointing laser penetrating vegetation over a slope terrain, the waveform is either stretched or compressed based on the relative angle. The stretching effect also results in a disappearing ground peak return when slope or off-nadir pointing angle is larger than the “critical slope angle”, which is closely related to various vegetation structures and footprint size. Model simulation indicates that waveform shapes are affected by surface topography, off-nadir pointing angle and vegetation structure and it is difficult to remove topography effects from waveform extent based only on the shapes of waveform without knowing any surface topography information. Height error without correction of surface topography and off-nadir pointing angle is the smallest when the laser beams at the toward-slope direction and the largest from the opposite direction. Further simulation reveals within 20° of slope and off-nadir pointing angle, given the canopy height as roughly 25 m and the footprint size as 25 m, the error for vegetation height (RH100) ranges from − 2 m to greater than 12 m, and the error for the height at the medium energy return (RH50) from − 1 m to 4 m. The RH100 error caused by unknown surface topography and without correction of off-nadir pointing effect can be explained by an analytical formula as a function of vegetation height, surface topography, off-nadir pointing angle and footprint size as a first order approximation. RH50 is not much affected by topography, off-nadir pointing and footprint size. This forward model simulation can provide scientific guidance on prioritizing future lidar mission measurement requirements and accuracies.

Numerical evaluation of temporal focusing characteristics in transparent and scattering media

Temporal focusing is a simple approach for achieving tight, optically sectioned excitation in nonlinear microscopy and multiphoton photo-manipulation. Key applications and advantages of temporal focusing involve propagation through scattering media, but the progressive broadening of the temporal focus has not been characterized. By combining a detailed geometrical optics model with Monte-Carlo scattering simulations we introduce and validate a simulation strategy for predicting temporal focusing characteristics in scattering and non-scattering media. The broadening of the temporal focus width with increasing depth in brain tissue is studied using both simulations and experiments for several key optical geometries, and an analytical approximation is found for the dependence of this broadening on the microscope's parameters in a transparent medium. Our results indicate that a multiphoton temporal focus has radically different broadening characteristics in deep tissue than those of a spatial focus.

Estimation of forest canopy leaf area index using MODIS, MISR, and LiDAR observations

A new approach for determining the forest leaf area index (LAI) from a geometric-optical model inversion using multisensor observations is developed. For improving the LAI estimate for the forested area on rugged terrain, a priori information on tree height and the spectra of four scene components of a geometric-optical mutual shadowing (GOMS) model are extracted from airborne light-detection and ranging (LiDAR) data and optical remote sensing data with high spatial resolution, respectively. The slope and aspect of the study area are derived from digital elevation model data. These extracted parameters are applied in an inversion to improve the estimates of forest canopy structural parameters in a GOMS model. For the field investigation, a bidirectional reflectance factor data set of needle forest pixels is collected by combining moderate-resolution-imaging-spectroradiometer (MODIS) and multiangle-imaging-spectroradiometer (MISR) multiangular remote sensing observations. Then, forest canopy parameters are inverted based on the GOMS model. Finally, the LAI of the forest canopy of each pixel is estimated from the retrieved structural parameters and validated by field measurements. The results indicate that the accuracy of forest canopy LAI estimates can be improved by combining observations of passive multiangle and active remote sensors.

Quantitative estimation of the shrub canopy LAI from atmosphere-corrected HJ-1 CCD data in Mu Us Sandland

The leaf area index (LAI) is an important ecological parameter that characterizes the interface between vegetation canopy and the atmosphere. In addition, it is used by most process-oriented ecosystem models. This paper investigates the potential of HJ-1 CCD data combined with linear spectral unmixing and an inverted geometric-optical model for the retrieval of the shrub LAI in Wushen Banner of Inner Mongolia in the Mu Us Sandland. MODTRAN (Moderate Resolution Atmospheric Radiance and Transmittance Model) was used for atmospheric correction. Shrubland was extracted using the threshold of the normalized difference vegetation index, with which water bodies and farmland were separated, in combination with a vegetation map of the People’s Republic of China (1:1000000). Using the geometric-optical model, we derive the per-pixel reflectance as a simple linear combination of two components, namely sunlit background and other. The fraction of sunlit background is related to the shrub LAI. With the support of HJ-1 CCD data, we employ linear spectral unmixing to obtain the fraction of sunlit background in an atmospherically corrected HJ image. In addition, we use the measured shrub canopy structural parameters for shrub communities to invert the geometric-optical model and retrieve the pixel-based shrub LAI. In total, 18 sample plots collected in Wushen Banner of Inner Mongolia are used for validation. The results of the shrub LAI show good agreement with R 2 of 0.817 and a root-mean-squared error of 0.173.

Aperture properties of multimode conical optical fibers (focons)

The problem on the propagation of meridional rays in conical optical fibers (focons) has been considered in terms of the geometric optics model. A sine condition for the ray parameters at reflection points has been obtained. A general equation that relates the geometric size of a focon with the ray parameters at the entrance and exit of the focon has been deduced. The equation is valid for all physically possible taper angles of a focon. The aperture properties of two types of focons (hollow and dielectric) have been analyzed. The entrance and exit aperture angles appear to be piecewise continuous or piecewise smooth functions of the parameters, which is related to the discreteness of the number of ray reflections. The main properties of the considered types of focons with typical values of the parameters have been illustrated by numerical calculations of the aperture angles.

An inversion method based on multi-angular approaches for estimating bare soil surface parameters from RADARSAT-1

Abstract. The radar signal recorded by earth observation (EO) satellites is sensitive to soil moisture and surface roughness, which both influence the onset of runoff. This paper focuses on inversion of these parameters using a multi-angular approach based on RADARSAT-1 data with incidence angles of 35° and 47° (in mode S3 and S7). This inversion was performed with three backscatter models: Geometrical Optics Model (GOM), Oh Model (OM), and Modified Dubois Model (MDM), which were compared to obtain the best configuration. Mean absolute errors of 1.23, 1.12, and 2.08 cm for roughness expressed in rms height and for dielectric constant, mean absolute errors of 2.46 – equal to 3.88 (m3 m−3) in volumetric soil moisture, – 4.95 – equal to 8.72 (m3 m−3) in volumetric soil moisture – and 3.31 – equal to 6.03 (m3 m−3) in volumetric soil moisture – were obtained for the MDM, GOM, and OM simulation, respectively. These results indicate that the MDM provided the most accurate data with minimum errors. Therefore, the latter inversion algorithm was applied to images, and the final results are presented in two different maps showing pixel and homogeneous zones for surface roughness and soil moisture.

IMPROVING GALACTIC CENTER ASTROMETRY BY REDUCING THE EFFECTS OF GEOMETRIC DISTORTION

We present significantly improved proper motion measurements of the Milky Way's central stellar cluster. These improvements are made possible by refining our astrometric reference frame with a new geometric optical distortion model for the W. M. Keck II 10 m telescope's Adaptive Optics camera (NIRC2) in its narrow field mode. For the first time, this distortion model is constructed from on-sky measurements, and is made available to the public. When applied to widely dithered images, it produces residuals in the separations of stars that are a factor of ~3 smaller compared to the outcome using previous models. By applying this new model, along with corrections for differential atmospheric refraction, to widely dithered images of SiO masers at the Galactic center, we improve our ability to tie into the precisely measured radio Sgr A*-rest frame. The resulting infrared reference frame is ~2-3 times more accurate and stable than earlier published efforts. In this reference frame, Sgr A* is localized to within a position of 0.6 mas and a velocity of 0.09 mas/yr, or ~3.4 km/s at 8 kpc (1 sigma). While earlier proper motion studies defined a reference frame by assuming no net motion of the stellar cluster, this approach is fundamentally limited by the cluster's intrinsic dispersion and therefore will not improve with time. We define a reference frame with SiO masers and this reference frame's stability should improve steadily with future measurements of the SiO masers in this region. This is essential for achieving the necessary reference frame stability required to detect the effects of general relativity and extended mass on short-period stars at the Galactic center.

Three-dimensional synthetic aperture particle image velocimetry

We present a new method for resolving three-dimensional (3D) fluid velocity fields using a technique called synthetic aperture particle image velocimetry (SAPIV). By fusing methods from the imaging community pertaining to light field imaging with concepts that drive experimental fluid mechanics, SAPIV overcomes many of the inherent challenges of 3D particle image velocimetry (3D PIV). This method offers the ability to digitally refocus a 3D flow field at arbitrary focal planes throughout a volume. The viewable out-of-plane dimension (Z) can be on the same order as the viewable in-plane dimensions (X–Y), and these dimensions can be scaled from tens to hundreds of millimeters. Furthermore, the digital refocusing provides the ability to ‘see-through’ partial occlusions, enabling measurements in densely seeded volumes. The advantages are achieved using a camera array (typically at least five cameras) to image the seeded fluid volume. The theoretical limits on refocused plane spacing and viewable depth are derived and explored as a function of camera optics and spacing of the array. A geometric optics model and simulated PIV images are used to investigate system performance for various camera layouts, measurement volume sizes and seeding density; performance is quantified by the ability to reconstruct the 3D intensity field, and resolve 3D vector fields in densely seeded simulated flows. SAPIV shows the ability to reconstruct fields with high seeding density and large volume size. Finally, results from an experimental implementation of SAPIV using a low cost eight-camera array to study a vortex ring in a 65 × 40 × 32 mm3 volume are presented. The 3D PIV results are compared with 2D PIV data to demonstrate the capability of the 3D SAPIV technique.

General spectral properties of Makyoh imaging of quasi-periodic surfaces

Imaging properties of the Makyoh imaging of periodic and quasi-periodic surfaces are analysed using a spectral approach based on a geometrical optical model. It is shown that in spite of the nonlinear nature of imaging, the Fourier representation can be used with some restrictions and limitations. Ray-tracing simulations are performed to illustrate the results.

Obtaining point spread function of penumbral encoding aperture with “expectation maximization” algorithm based on matched source-image pair experiment

A source penumbral image reconstruction method with linear mapping principle for geometrical optics is established. The ideal binary point spread function (PSF) can be obtained using a geometrical optics model. The system PSF with certain sharpness was obtained using a Monte Carlo (MC) model. Considering other factors besides the transportation of the x (gamma)-rays or particles (fusion neutrons) in the penumbral encoding aperture in MC model, such as the scattering background and the systematic error, the PSF from MC model "source-image pair matching" experiments with a large area standard oval shape source were processed. A method for correcting and calibrating the PSF by the expectation maximization adaptive algorithm was established and the optimized PSF with 22.30 microm sharpness was achieved. This is more consistent with the real system PSF despite the increased noise level of the two-dimensional PSF and large irregularity in the PSF profile.

A Geometrical Optics Model of Three Dimensional Scattering From a Rough Layer With Two Rough Surfaces

An asymptotic method is described for predicting the bistatic normalized radar cross section of a rough homogeneous layer made up of two rough surfaces. The model is based on iteration of the Kirchhoff approximation to calculate the fields scattered by the rough layer, and is reduced to the high-frequency limit in order to obtain numerical results rapidly. Shadowing effects, significant for large incidence or scattering angles, are taken into account through the use of shadowing functions. The model is applicable for moderate to large surface roughnesses having small to moderate slopes, and for both lossless and lossy inner media. It was validated for a rough layer with a rough surface over a perfectly flat surface in a preceding contribution. Here, the extension of the model to a rough layer with two rough surfaces is developed, and results are presented to validate the asymptotic model by comparison with a numerical reference method.

Allometric constraints on sources of variability in multi-angle reflectance measurements

Bidirectional reflectance signatures of vegetation are strongly shaped by the shadows cast between objects in a scene, such as tree crowns or leaves. Differences in the shape and spatial density of these objects result in distinct bidirectional reflectance distribution functions (BRDFs) in different biomes. We examined how allometry may constrain the variability of canopy architectural parameters in BRDF models, and consequently alter the attribution of variation in the simulated bidirectional reflectance factor (BRF). Allometry is the covariation between the size or number of organisms and their component parts. To test the importance of realistic variation and covariation of canopy architecture on BRDF, we incorporated the 3-D radiative transfer model DISORD (which uses the geometric optics (GO) model of Li and Strahler) into a Monte Carlo (MC) algorithm. The MC algorithm generated an ensemble of tree canopies whose parameters fulfilled the allometry of a set of measured forest plots from Russian forest inventory. The role of view geometry was directly considered using perturbations of the parameters to evaluate the sensitivity of the BRF itself, evaluated at different view angles, and the difference in BRF (ΔBRF) as measured at two view angles representing paired satellite observations. The allometrically constrained forest plots had reduced variation in ΔBRF compared to the uncorrelated plots, but the variation of the BRF itself is dramatically increased by allometry. The variation of the BRF is relatively constant among the view angles examined, whereas the variation in ΔBRF increases dramatically with larger phase angles. The BRF was most sensitive to canopy attributes that were important in radiative transfer, such as LAI and stem area index (SAI), but there were also large (∼ 40% of variance) contributions of geometric components such as tree number, crown size, and ground cover. By contrast, sensitivity of ΔBRF was dominated by ground cover, crown size and tree number, which all play a role in the GO calculations. The mix of sensitive parameters was not dramatically different between gymnosperms and angiosperms, nor between allometric and correlated runs. Together these results indicate that forest structure and leaf area could be usefully inverted together using paired observations with different viewing geometries. Ideal pairs of observations are those with large difference in phase angle, and along the gradient of the BRF peak, which most commonly occur with sequential MODIS/Terra overpasses.

The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems.

X-ray phase contrast imaging is a very promising technique which may lead to significant advancements in medical imaging. One of the impediments to the clinical implementation of the technique is the general requirement to have an x-ray source of high coherence. The radiation physics group at UCL is currently developing an x-ray phase contrast imaging technique which works with laboratory x-ray sources. Validation of the system requires extensive modelling of relatively large samples of tissue. To aid this, we have undertaken a study of when geometrical optics may be employed to model the system in order to avoid the need to perform a computationally expensive wave optics calculation. In this paper, we derive the relationship between the geometrical and wave optics model for our system imaging an infinite cylinder. From this model we are able to draw conclusions regarding the general applicability of the geometrical optics approximation.

Structured beams in laser refractography

The main types of structured laser beams obtained using diffractive optical elements are presented. A new information and measuring technique—the laser refractography based on the application of structured radiation—is proposed for the quantitative diagnostics of inhomogeneous media. The relations for the analysis of geometrical-optical models and the refraction of structured beams in the boundary temperature layer are presented. A solution to the inverse problem based on the Abel transform is adapted to the flat beam in the case of a relatively strong refraction in a spherical inhomogeneous layer. The temperature profile of the layer is reconstructed using the refractograms of flat and cylindrical beams.