Geometrical Optics Model Research Articles

Large area mapping of southwestern forest crown cover, canopy height, and biomass using the NASA Multiangle Imaging Spectro-Radiometer

A rapid canopy reflectance model inversion experiment was performed using multi-angle reflectance data from the NASA Multi-angle Imaging Spectro-Radiometer (MISR) on the Earth Observing System Terra satellite, with the goal of obtaining measures of forest fractional crown cover, mean canopy height, and aboveground woody biomass for large parts of south-eastern Arizona and southern New Mexico (>200,000 km2). MISR red band bidirectional reflectance estimates in nine views mapped to a 250 m grid were used to adjust the Simple Geometric-optical Model (SGM). The soil-understory background signal was partly decoupled a priori by developing regression relationships with the nadir camera blue, green, and near-infrared reflectance data and the isotropic, geometric, and volume scattering kernel weights of the LiSparse–RossThin kernel-driven bidirectional reflectance distribution function (BRDF) model adjusted against MISR red band data. The SGM's mean crown radius and crown shape parameters were adjusted using the Praxis optimization algorithm, allowing retrieval of fractional crown cover and mean canopy height, and estimation of aboveground woody biomass by linear rescaling of the dot product of cover and height. Retrieved distributions of crown cover, mean canopy height, and aboveground woody biomass for forested areas showed good matches with maps from the United States Department of Agriculture (USDA) Forest Service, with R2 values of 0.78, 0.69, and 0.81, and absolute mean errors of 0.10, 2.2 m, and 4.5 tons acre-1 (10.1 Mg ha-1), respectively, after filtering for high root mean square error (RMSE) on model fitting, the effects of topographic shading, and the removal of a small number of outliers. This is the first use of data from the MISR instrument to produce maps of crown cover, canopy height, and woody biomass over a large area by seeking to exploit the structural effects of canopies reflected in the observed anisotropy patterns in these explicitly multiangle data.

Characterization of earthshine impacts on Moderate Resolution Imaging Spectroradiometer reflective solar bands calibration

The Moderate Resolution Imaging Spectroradiometer (MODIS) is one of the key instruments for NASA's Earth Observing System (EOS) missions. Its high quality land, ocean, and atmosphere data products heavily rely on accurate on-orbit calibration and characterization. MODIS reflective solar bands (RSB) on-orbit calibration is reflectance based and is provided by an on-board Solar Diffuser (SD). The Bi-directional Reflectance Factor (BRF) of the SD was characterized pre-launch. The earthshine (ES) contamination to the SD signal was estimated to be approximately 0.5%. On-orbit observations and geometric optical modeling results show that this contamination strongly depends on the wavelength, calibration viewing geometry, and the Earth surface type. In this paper, the contribution from the earthshine on MODIS RSB calibration is illustrated and characterized systematically. A simple approach is proposed to reduce its impact on the RSB calibration. This approach has been applied to both Terra and Aqua MODIS RSB calibration and proved to be effective in reducing the earthshine contaminations.

Comparison of the inversion of two canopy reflectance models for mapping forest crown closure using imaging spectroscopy

We compare the inversion of two canopy reflectance models to estimate forest crown closure (CC) using an EO-1 Hyperion image: the Kuusk–Nilson forest reflectance and transmittance (FRT) model, and the Li–Strahler geometric–optical model. For predicting CC on a per-pixel basis, the FRT model inversion is carried out by minimizing a merit function that provides a measure of the difference between the reflectance simulated by the FRT model and the reflectance originating from optimal band selection of Hyperion data. The inversion of the Li–Strahler model mainly depends on the relationship between the scene component “sunlit background” and forest structural parameters. We complement prediction deficiencies of the inverted Li–Strahler model CC using a spatial interpolation algorithm (regression kriging) in infeasible regions. Field-measured CCs of 40 sample sites are used to validate the inversion quality of both models. The results indicate that the Li–Strahler model inversion (R2 = 0.67, RMSE = 0.043) performs better than the FRT model inversion (R2 = 0.53, RMSE = 0.072) for CC retrieval. Estimated CC using the Li–Strahler model inversion combined with spatial interpolation yield a final, continuous CC map for the Longmenhe forest nature reserve in China, which is used as a study area for this work. The advantages and disadvantages of these two models inversion combined with imaging spectrometer data for mapping forest CC are discussed.

Teaching Image Formation by Extended Light Sources: The Use of a Model Derived from the History of Science

This research, carried out in Greece on pupils aged 12–16, focuses on the transformation of their representations concerning light emission and image formation by extended light sources. The instructive process was carried out in two stages, each one having a different, distinct target set. During the first stage, the appropriate conflict conditions were created by contrasting the subjects’ predictions with the results of experimental situations inspired by the History of Science, with a view to destabilizing the pupils’ alternative representations. During the second stage, the experimental teaching intervention was carried out; it was based on the geometrical optics model and its parameters were derived from Kepler’s relevant historic experiment. For the duration of this process and within the framework of didactical interactions, an effort was made to reorganize initial limited representations and restructure them at the level of the accepted scientific model. The effectiveness of the intervention was evaluated two weeks later, using experimental tasks which had the same cognitive yet different empirical content with respect to the tasks conducted during the intervention. The results of the study showed that the majority of the subjects accepted the model of geometrical optics, that is, the pupils were able to correctly predict and adequately justify the experimental results based on the principle of punctiform light emission. Educational and research implications are discussed.

Modeling Directional Brightness Temperature of the Winter Wheat Canopy at the Ear Stage

The ear is the top layer of mature wheat and has very different geometric and thermal characteristics from that of leaves. Compared to the directional brightness temperature (DBT) of wheat canopy without ears, the DBT at the ear stage has specific features, and the ear effects could not be explained by previous models. This paper proposes a hybrid geometric optical and radiative transfer model to reveal the combined influences of the geometric structure of ears and leaf; the temperature distribution of ear, leaf, and soil; and the Sun-target-sensor geometry on the canopy DBT. The soil, leaf, and ear layers are taken into account in the model so it is named as the Soil Leaf Ear Combined (SLEC) DBT model. We compare the model prediction with the field measurement data. The results show that the new SLEC DBT model can simulate the DBT of wheat at the ear stage with an accuracy of 0.78 K.

A new approach to estimation of global air‐sea gas transfer velocity fields using dual‐frequency altimeter backscatter

A new approach to estimating air‐sea gas transfer velocities based on normalized backscatter from the dual‐frequency TOPEX and Jason‐1 altimeters is described. The differential scattering of Ku‐band (13.6 GHz) and C‐band (5.3 GHz) microwave pulses is used to isolate the contribution of small‐scale waves to mean square slope and gas transfer. Mean square slope is derived for the nominal wave number range 40–100 rad m−1 by differencing mean square slope estimates computed from the normalized backscatter in each band, using a simple geometric optics model. Model parameters for calculating the differenced mean square slope over this wave number range are optimized using in situ optical slope measurements. An empirical relation between gas transfer velocity and mean square slope, also based on field measurements, is then used to derive gas transfer velocities. Initial results demonstrate that the calculated transfer velocities exhibit magnitudes and a dynamic range which are generally consistent with existing field measurements. The new algorithm is used to construct monthly global maps of gas transfer velocity and to illustrate seasonal transfer velocity variations over a 1‐year period. The measurement precision estimated from >106 duplicate observations of the sea surface by TOPEX and Jason‐1 altimeters orbiting in tandem is better than 10%. The estimated overall uncertainty of the method is ±30%. The long‐term global, area‐weighted, Schmidt number corrected, mean gas transfer velocity is 13.7 ± 4.1 cm h−1. The new approach, based on surface roughness, represents a potential alternative to commonly used parameterizations based on wind speed.

Gaussian Beam Summation Representation of a Two-Dimensional Gaussian Beam Diffraction by a Half Plane

A Gaussian beam summation (GBS) representation for half plane diffraction of an incident two dimensional Gaussian beam (GB) that hits arbitrarily close to the edge is presented. The scattered field is expanded into an angular spectrum of GBs that emerge from the edge. We derive asymptotic expressions for the diffracted beams amplitudes, that are valid uniformly as a function of the distance of the incident beam from the edge and of the scattering beams angles. In the limiting cases when the incident GB hits far from the edge, these expressions reduce uniformly to a geometrical optics model plus a weak edge diffractions due to the off-axis beam field that hits the edge. Applications to GBS modeling of indoor propagation are discussed.

Photon recollision probability in discrete crown canopies

The photon recollision probability in vegetation canopies, defined as the probability that a photon, after having interacted with a canopy element, will interact again, is a useful tool in remote sensing and ecological applications, enabling to link canopy optical properties at different wavelength and to estimate radiation absorption. In this work, a method is presented to estimate the photon recollision probability for horizontally homogeneous leaf canopies with arbitrary leaf angle distribution as well as for discrete crown canopies. The estimation is based on analytical approximation of the first-order recollision probability. Using the analytical solution of the two-stream equations of radiative transfer and Monte Carlo modeling, the first-order photon recollision probability is shown to slightly underestimate the mean recollision probability. Also, an approximation formula for the mean recollision probability in a horizontally homogeneous canopy is presented as a function of leaf area index. The method to calculate photon recollision probability in discrete crown canopies requires only the knowledge of total and between-crown canopy transmittance and is thus independent of the geometric-optical model used.

Directional emission of stadium-shaped microlasers

The far-field emission of two-dimensional stadium-shaped dielectric cavities is investigated. Microlasers with such shape present a highly directional emission. We provide experimental evidence of the dependence of the emission directionality on the shape of the stadium, in good agreement with ray numerical simulations. We develop an analytical geometrical optics model which permits to account for the main observed features. Wave numerical calculations confirm the results.

Geometrical optics modeling of the grating-slit test

A novel optical testing method termed the grating-slit test is discussed. This test uses a grating and a slit, as in the Ronchi test, but the grating-slit test is different in that the grating is used as the incoherent illuminating object instead of the spatial filter. The slit is located at the plane of the image of a sinusoidal intensity grating. An insightful geometrical-optics model for the grating-slit test is presented and the fringe contrast ratio with respect to the slit width and object-grating period is obtained. The concept of spatial bucket integration is used to obtain the fringe contrast ratio.

Evaluation of empirical and semi-empirical backscattering models for surface soil moisture estimation

Several empirical and semi-empirical backscattering models have been proposed to offer alternative expressions for the inversion of surface parameters from radar data, but the applicability and adequacy of the models for different surface conditions and sensor configurations have not been clearly assessed. A number of empirical and semi-empirical models are studied in this paper to assess the applicability of the models for different conditions that are often found over agricultural areas. The performance of the models is evaluated first analytically by comparing their simulations with those obtained using the theoretical integral equation model (IEM) and geometrical optics model (GOM). The model estimations are then compared with RADARSAT-1 observations acquired over an experimental catchment. The results show very different model behaviour depending on the surfaces roughness conditions and incidence angle. This study highlights the importance of carefully selecting the backscattering model to be used in radar applications.

Algorithm for global leaf area index retrieval using satellite imagery

Leaf area index (LAI) is one of the most important Earth surface parameters in modeling ecosystems and their interaction with climate. Based on a geometrical optical model (Four-Scale) and LAI algorithms previously derived for Canada-wide applications, this paper presents a new algorithm for the global retrieval of LAI where the bidirectional reflectance distribution function (BRDF) is considered explicitly in the algorithm and hence removing the need of doing BRDF corrections and normalizations to the input images. The core problem of integrating BRDF into the LAI algorithm is that nonlinear BRDF kernels that are used to relate spectral reflectances to LAI are also LAI dependent, and no analytical solution is found to derive directly LAI from reflectance data. This problem is solved through developing a simple iteration procedure. The relationships between LAI and reflectances of various spectral bands (red, near infrared, and shortwave infrared) are simulated with Four-Scale with a multiple scattering scheme. Based on the model simulations, the key coefficients in the BRDF kernels are fitted with Chebyshev polynomials of the second kind. Spectral indices - the simple ratio and the reduced simple ratio - are used to effectively combine the spectral bands for LAI retrieval. Example regional and global LAI maps are produced. Accuracy assessment on a Canada-wide LAI map is made in comparison with a previously validated 1998 LAI map and ground measurements made in seven Landsat scenes

Mapping shrub abundance in desert grasslands using geometric-optical modeling and multi-angle remote sensing with CHRIS/Proba

This work examines the application of a geometric-optical canopy reflectance model to provide measures of woody shrub abundance in desert grasslands at the landscape scale. The approach is through inversion of the non-linear simple geometric model (SGM) against 631 nm multi-angle reflectance data from the Compact High Resolution Imaging Spectrometer (CHRIS) flown on the European Space Agency's Project for On-Board Autonomy (Proba) satellite. Separation of background and upper canopy contributions was effected by a linear scaling of the parameters of the Walthall bidirectional reflectance distribution function model with the weights of a kernel-driven model. The relationship was calibrated against a small number of sample locations with highly contrasting background/upper canopy configurations, before application over an area of about 25 km 2. The results show that with some assumptions, the multi-angle remote sensing signal from CHRIS/Proba can be explained in terms of a combined soil–understory background response and woody shrub cover and exploited to map this important structural attribute of desert grasslands.

Estimation of planetary surface roughness using radio sounder A‐scope data

We propose a simple technique to estimate a roughness parameter, RMS gradient, of a planetary surface. The technique is practiced on A‐scope data of radio sounding observation of planet with a simple formula. The formula was derived analytically for the case of Gaussian random rough surface using a geometrical optics model. The technique was examined by simulations of planetary sounding observations which is based on physical optics, and demonstrated to estimate the RMS gradient of the planetary surface in the range from some fraction of degrees to about ten degrees. Finally, the upper limit of valid range of RMS gradient was formulized as a function of observation altitude and the maximum detection range.

Using a priori information to improve soil moisture retrieval from ENVISAT ASAR AP data in semiarid regions

This paper presents a retrieval algorithm that estimates spatial and temporal distribution of volumetric soil moisture content, at an approximate depth of 5 cm, using multitemporal ENVISAT Advanced Synthetic Aperture Radar (ASAR) alternating polarization images, acquired at low incidence angles (i.e., from 15/spl deg/ to 31/spl deg/). The algorithm appropriately assimilates a priori information on soil moisture content and surface roughness in order to constrain the inversion of theoretical direct models, such as the integral equation method model and the geometric optics model. The a priori information on soil moisture content is obtained through simple lumped water balance models, whereas that on soil roughness is derived by means of an empirical approach. To update prior estimates of surface parameters, when no reliable a priori information is available, a technique based solely on the use of multitemporal SAR information is proposed. The developed retrieval algorithm is assessed on the Matera site (Italy) where multitemporal ground and ASAR data were simultaneously acquired in 2003. Simulated and experimental results indicate the possibility of attaining an accuracy of approximately 5% in the retrieved volumetric soil moisture content, provided that sufficiently accurate a priori information on surface parameters (i.e., within 20% of their whole variability range) is available. As an example, multitemporal soil moisture maps at watershed scale, characterized by a spatial resolution of approximately 150 m, are derived and illustrated in the paper.

Radiation properties modeling for plasma-sprayed-alumina-coated rough surfaces for spacecrafts

Spacecraft thermal control materials (TCMs) play a vital role in the entire service life of a spacecraft [NASA Preferred Reliability Practices: Spacecraft Thermal Control Coatings Design and Application (Practice No. PD-ED-1239), 1995, pp. 1–6]. Most of the conventional TCMs degrade in the harmful space environment [R.D. Karam, Satellite Thermal Control for Systems Engineers, AIAA, Virginia, USA, 1998, pp. 147–163]. In the previous study, plasma sprayed alumina (PSA) coating was established as a new and better TCM for spacecrafts, in view of its stability and reliability compared to the traditional TCMs [R.M. Li, S.C. Joshi, H.W. Ng, Presented in 2nd International Conference on Materials for Advanced Technologies & IUMRS, Singapore, 2003, pp. 1–4]. During the investigation, the surface roughness of PSA was found important, because the roughness affects the radiative heat exchange between the surface and its surroundings. Parameters such as root-mean-square roughness cannot properly evaluate surface roughness effects on radiative properties of opaque surfaces [M.F. Modest, Radiative Heat Transfer, 2nd ed., Academic Press/Elsevier Science, USA, 2003, pp. 90–92]. Some models have been developed earlier to predict the effects, such as Davies’ model [H. Davies, Proceedings of IEEE vol. 101, part IV, 1954, pp. 209–214], Tang and Buckius's statistical geometric optics model [K. Tang, R.O. Buckius, Int. J. Heat Mass Transfer 44 (2001) 4059–4073]. However, they are valid only in their own specific situations. In this paper, an energy absorption geometry model was developed and applied to investigate the roughness effects with the help of 2D surface profile of PSA coated substrate scanned at micron level. This model predicts effective normal solar absorptance ( α ne) and effective hemispherical infrared emittance ( ɛ he) of a rough PSA surface. These values, if used in the heat transfer analysis of an equivalent, smooth and optically flat surface, lead to the prediction of the same rate of heat exchange and temperature as that of for the rough PSA surface. The model was validated through comparison between a smooth and a rough PSA coated surfaces. Even though not tested for other types of materials, the model formulation is generic and can be used to incorporate the rough surface effects for other types of thermal coatings, provided the baseline values of normal solar absorptance ( α n) and hemispherical infrared emittance ( ɛ h) are available for a generic surface of the same material.

Nonlinear processes in microdroplets: a geometrical optics approach

We present a geometrical optics model for the calculation of the internal fields in microspheres with nonlinear optical properties, which occur, for example, in stimulated Raman scattering. Our approach is based on a ray interpretation of the nonlinear wave equations describing the fields of the Stokes modes in a microdroplet. The Stokes modes are assumed to be whispering gallery modes. For such rays the nonlinear wave equations are used to describe the growth of the field along the path of the ray and to calculate the internal fields belonging to various Stokes modes. The interaction between different-order Stokes fields is considered, by using an iteration over the different orders. Although our approach can be extended to stimulated Brillouin scattering as well, in this paper only stimulated Raman scattering is considered.

Internal Tides in the Southwestern Japan/East Sea

Abstract This paper investigates the internal tidal energy distribution in the southwestern Japan/East Sea using vertical round-trip travel time (τ) data from 23 pressure-sensor-equipped inverted echo sounders (PIES). The τ records are analyzed by bandpass filtering to separate time-dependent variability of the semidiurnal and diurnal bands. The semidiurnal internal tides exhibit a horizontal beam pattern of high energy, propagating into the open basin. They originate from a restricted portion of the shelf break where the Korea Strait enters the Ulleung Basin. The generation appears to occur at ∼200-m water depth near 35.5°–35.7°N and 130°–131°E, where the slope of bottom topography matches that of the wave characteristics, coinciding with the location where the semidiurnal barotropic cross-slope tidal currents are strongest. Maximum vertical displacement of the thermocline interpreted as a long-wave first baroclinic mode from the measured τ is about 25 m near the generation region. Annual and monthly variations of the propagation patterns and generation energy levels are observed, and these are closely associated with changes in the mesoscale circulation and stratification. Eastward (westward) refraction is observed when a warm (cold) eddy crosses the path of internal tide propagation. Moreover, when the generation region is invaded by cold eddies that spoil the match between shelf break and thermocline depth, the internal tidal energy level decreases by a factor of about 2. A simple geometric optics model is proposed to explain the observed horizontal refraction of the beam of semidiurnal internal tides in which stratification and current shear play essential roles. In contrast, diurnal internal tides are observed to be trapped along the continental slope region around 36°N.

An improved geometrical optics model for bistatic GPS scattering from the ocean surface

This paper is concerned with the properties of bistatic microwave scattering from a randomly rough surface, and specifically its application to the study of global positioning system (GPS) satellite signals reflected from the ocean. We present a discussion of some recent refinements of Kirchhoff-type models based on second-order iterations of the surface-current integral equation, and the relationship between these models and their high-frequency (geometric optics) limit. In particular, we show that use of these refinements can extend the domain of applicability of the standard geometrical optics (GO) model. It is found that GO can be nearly as accurate as a Kirchhoff-based model provided that the wavenumber cutoff, at which the surface wave spectrum must be filtered in computing the required slope moments, depends on the roughness of the ocean surface (i.e., wind speed) as well as the incident angle and frequency of the radiation. We use a GO model refined in this way to analyze GPS surface reflection data collected from an aircraft equipped with two down-looking antennas for receiving both left- and right-hand circularly polarized reflections. Concurrent measurements of the local wind and wave conditions were collected from a nearby research vessel. Measured waveforms and mean Doppler widths at both polarizations are compared with predictions from our refined GO model, and discussion is given concerning the sensitivity of the reflected radiation to various geophysical parameters and the utility of GPS reflections for remote sensing applications.

SCS+C: a modified Sun-canopy-sensor topographic correction in forested terrain

Topographic correction based on sun-canopy-sensor (SCS) geometry is more appropriate than terrain-based corrections in forested areas since SCS preserves the geotropic nature of trees (vertical growth) regardless of terrain, view, and illumination angles. However, in some terrain orientations, SCS experiences an overcorrection problem similar to other simple photometric functions. To address this problem, we propose a new SCS+C correction that accounts for diffuse atmospheric irradiance based on the C-correction. A rigorous, comprehensive, and flexible method for independent validation based on canopy geometric optical reflectance models is also introduced as an improvement over previous validation approaches, and forms a secondary contribution of this paper. Results for a full range of slopes, aspects, and crown closures showed SCS+C provided improved corrections compared to the SCS and four other photometric approaches (cosine, C, Minnaert, statistical-empirical) for a Rocky Mountain forest setting in western Canada. It was concluded that SCS+C should be considered for topographic correction of remote sensing imagery in forested terrain.