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NASA NTRS2011

An Integrated Hydrological and Water Management Study of the Entire Nile River System - Lake Victoria to Nile Delta

The Nile basin River system spans 3 million km(exp 2) distributed over ten nations. The eight upstream riparian nations, Ethiopia, Eretria, Uganda, Rwanda, Burundi, Congo, Tanzania and Kenya are the source of approximately 86% of the water inputs to the Nile, while the two downstream riparian countries Sudan and Egypt, presently rely on the river's flow for most of the their needs. Both climate and agriculture contribute to the complicated nature of Nile River management: precipitation in the headwaters regions of Ethiopia and Lake Victoria is variable on a seasonal and inter-annual basis, while demand for irrigation water in the arid downstream region is consistently high. The Nile is, perhaps, one of the most difficult trans-boundary water issue in the world, and this study would be the first initiative to combine NASA satellite observations with the hydrologic models study the overall water balance in a to comprehensive manner. The cornerstone application of NASA's Earth Science Research Results under this project are the NASA Land Data Assimilation System (LDAS) and the USDA Atmosphere-land Exchange Inverse (ALEXI) model. These two complementary research results are methodologically independent methods for using NASA observations to support water resource analysis in data poor regions. Where an LDAS uses multiple sources of satellite data to inform prognostic simulations of hydrological process, ALEXI diagnoses evapotranspiration and water stress on the basis of thermal infrared satellite imagery. Specifically, this work integrates NASA Land Data Assimilation systems into the water management decision support systems that member countries of the Nile Basin Initiative (NBI) and Regional Center for Mapping of Resources for Development (RCMRD, located in Nairobi, Kenya) use in water resource analysis, agricultural planning, and acute drought response to support sustainable development of Nile Basin water resources. The project is motivated by the recognition that accurate, frequent, and spatially distributed estimates of the water balance are necessary for effective water management. This creates a challenge for watersheds that are large, include data poor regions, and/or span multiple nations. All of these descriptors apply to the Nile River basin, yet successful management of the Nile is critical for development and political stability in the region. For this reason, improved hydrological data to support cooperative water management in the Nile basin is a priority for USAID, the US State Department, the World Bank and other international organizations. In this project, the U.S. based research team is working with partners at RCMRD, Nile Basin Initiative (NBI), and their member national-level agencies to develop satellite-based land cover maps, satellite-derived evapotranspiration estimates (using the ALEXI algorithm), and NASA's Land Data Assimilation System (LDAS) customized to match identified information needs. The cornerstone applied sciences product of the project is the development of a customized "Nile LDAS" that will produce optimal estimates of hydrological states and fluxes, as vetted against the in situ observations of NBI and RCMRD member organizations and independent satellite-derived hydrological estimates. Nile LDAS will be applied to improve the reliability of emerging Decision Support Systems in applications that include drought monitoring, reservoir management, and irrigation planning. The end-users such as RCMRD, NBI, Ethiopian and Kenya Meteorological and Famine Early Warning System Network (FEWSNet) will be the eventual benefactors of this work. There will be a capacity building process involving the above end-user organizations and transfer the models and the results for these organizations to execute for future use. The team has already initiated this study and the early results of first years' work are shown. The plan is to complete this work by late 2013.

Habib, Shahid↗
NASA NTRS2008

Integrated Vehicle Ground Vibration Testing in Support of Launch Vehicle Loads and Controls Analysis

All structural systems possess a basic set of physical characteristics unique to that system. These unique physical characteristics include items such as mass distribution and damping. When specified, they allow engineers to understand and predict how a structural system behaves under given loading conditions and different methods of control. These physical properties of launch vehicles may be predicted by analysis or measured by certain types of tests. Generally, these properties are predicted by analysis during the design phase of a launch vehicle and then verified by testing before the vehicle becomes operational. A ground vibration test (GVT) is intended to measure by test the fundamental dynamic characteristics of launch vehicles during various phases of flight. During the series of tests, properties such as natural frequencies, mode shapes, and transfer functions are measured directly. These data will then be used to calibrate loads and control systems analysis models for verifying analyses of the launch vehicle. NASA manned launch vehicles have undergone ground vibration testing leading to the development of successful launch vehicles. A GVT was not performed on the inaugural launch of the unmanned Delta III which was lost during launch. Subsequent analyses indicated had a GVT been performed, it would have identified instability issues avoiding loss of the vehicle. This discussion will address GVT planning, set-up, execution and analyses, for the Saturn and Shuttle programs, and will also focus on the current and on-going planning for the Ares I and V Integrated Vehicle Ground Vibration Test (IVGVT).

Askins, Bruce R.↗
NASA NTRS1993

Near-Threshold Rovibrational Excitation of HF by Electron Impact

The pronounced threshold peaks found in cross sections for electron-impact excitation of the vibrational levels of HF have been studied using an R-matrix method, in which electron exchange, correlation effects and polarization are included. Vibration is modelled inside the R-matrix sphere using the energy-modified adiabatic phase matrix method (EMAP). Rotation of the target molecule is treated outside the R-matrix sphere, using the frame transformation method. The external rovibronic close-coupling equations are solved by analytic methods. Analysis of the multichannel threshold structures indicates that the peaks found for rovibrational excitation of nu = 1 can be interpreted as virtual states while those for nu = 2 correspond to nuclear-excited Feshbach resonances. These two different results correspond simply to a change of sign of a small parameter that is determined by the threshold analysis. Whereas the overall shapes of the excitation peaks agree with earlier theoretical results, we have found detailed rotational fine structures for scattering energies on the order of the rotational constant above and below the opening of the vibrational channels. The partial and total cross sections for rovibrational state-to-state transitions in the energy region of the peaks are analysed. Characteristic cusp structures and excitation peaks are found at the rovibrational thresholds for transitions with small angular momentum transfer (Delta)j =0, 1 and 2. Differential cross sections are similar to recent measurements. These experimental data do not show the relatively sharp rise at the vibrational thresholds indicated by theory. Theory and experiment are in good agreement for the shape and magnitude of the angular dependence of cross sections for rovibrational state-to-state transitions, especially at larger energies.

Thummel, Helmar T.↗
NASA NTRS2007

Real-time Signal-to-noise Ratio (SNR) Estimation for BPSK and QPSK Modulation Using the Active Communications Channel

Method and apparatus for estimating signal-to-noise ratio (SNR) gamma of a composite input signal e(t) on a phase modulated (e.g., BPSK) communications link. A first demodulator receives the composite input signal and a stable carrier signal and outputs an in-phase output signal; a second demodulator receives the composite input signal and a phase-shifted version of the carrier signal and outputs a quadrature-phase output signal; and phase error theta(sub E)(t) contained within the composite input signal e(t) is calculated from the outputs of the first and second demodulators. A time series of statistically independent phase error measurements theta(sub E)(t(sub 1)), theta (sub E)(t(sub 2)),..., theta (sub E)(t(sub k)) is obtained from the composite input signal subtending a time interval delta t = t(sub k) - t(sub 1) whose value is small enough such that gamma(t) and sigma(t) can be taken to be constant in delta t. A biased estimate gamma(sup *) for the signal-to-noise ratio (SNR) gamma if the composite input signal is calculated using maximum likelihood (ML) estimation techniques, and an unbiased estimate gamma(sup ^) for the signal-to-noise ratio (SNR) gamma of the composite input signal is determined from the biased estimate gamma(sup *), such as by use of a look-up table.

Manning, Robert M.↗
NASA NTRS2000

A Method for Assessing the Quality of Model-Based Estimates of Ground Temperature and Atmospheric Moisture Using Satellite Data

A method is developed for validating model-based estimates of atmospheric moisture and ground temperature using satellite data. The approach relates errors in estimates of clear-sky longwave fluxes at the top of the Earth-atmosphere system to errors in geophysical parameters. The fluxes include clear-sky outgoing longwave radiation (CLR) and radiative flux in the window region between 8 and 12 microns (RadWn). The approach capitalizes on the availability of satellite estimates of CLR and RadWn and other auxiliary satellite data, and multiple global four-dimensional data assimilation (4-DDA) products. The basic methodology employs off-line forward radiative transfer calculations to generate synthetic clear-sky longwave fluxes from two different 4-DDA data sets. Simple linear regression is used to relate the clear-sky longwave flux discrepancies to discrepancies in ground temperature ((delta)T(sub g)) and broad-layer integrated atmospheric precipitable water ((delta)pw). The slopes of the regression lines define sensitivity parameters which can be exploited to help interpret mismatches between satellite observations and model-based estimates of clear-sky longwave fluxes. For illustration we analyze the discrepancies in the clear-sky longwave fluxes between an early implementation of the Goddard Earth Observing System Data Assimilation System (GEOS2) and a recent operational version of the European Centre for Medium-Range Weather Forecasts data assimilation system. The analysis of the synthetic clear-sky flux data shows that simple linear regression employing (delta)T(sub g)) and broad layer (delta)pw provides a good approximation to the full radiative transfer calculations, typically explaining more thin 90% of the 6 hourly variance in the flux differences. These simple regression relations can be inverted to "retrieve" the errors in the geophysical parameters, Uncertainties (normalized by standard deviation) in the monthly mean retrieved parameters range from 7% for (delta)T(sub g) to approx. 20% for the lower tropospheric moisture between 500 hPa and surface. The regression relationships developed from the synthetic flux data, together with CLR and RadWn observed with the Clouds and Earth Radiant Energy System instrument, ire used to assess the quality of the GEOS2 T(sub g) and pw. Results showed that the GEOS2 T(sub g) is too cold over land, and pw in upper layers is too high over the tropical oceans and too low in the lower atmosphere.

Wu, Man Li C.↗
NASA NTRS1998

Time-resolved diffusion tomographic imaging in highly scattering turbid media

A method for imaging objects in highly scattering turbid media. According to one embodiment of the invention, the method involves using a plurality of intersecting source/detectors sets and time-resolving equipment to generate a plurality of time-resolved intensity curves for the diffusive component of light emergent from the medium. For each of the curves, the intensities at a plurality of times are then inputted into the following inverse reconstruction algorithm to form an image of the medium: X.sup.(k+1).spsp.T =?Y.sup.T W+X.sup.(k).spsp.T .LAMBDA.!?W.sup.T W+.LAMBDA.!.sup.-1 wherein W is a matrix relating output at detector position r.sub.d, at time t, to source at position r.sub.s, .LAMBDA. is a regularization matrix, chosen for convenience to be diagonal, but selected in a way related to the ratio of the noise, to fluctuations in the absorption (or diffusion) X.sub.j that we are trying to determine: .LAMBDA..sub.ij =.lambda..sub.j .delta..sub.ij with .lambda..sub.j = /<.DELTA.Xj.DELTA.Xj> Here Y is the data collected at the detectors, and X.sup.k is the kth iterate toward the desired absoption information.

Alfano, Robert R.↗
NASA NTRS1988

A full-potential theory analysis of the supersonic aerodynamics of a 60-deg delta wing-body configuration

Experimental spanwise pressure distributions for a 60-deg delta wing/body of approximate fineness ratio 7.6 have been obtained and compared to predictions using full-potential theory. Analysis was performed at Mach 1.6 for angles of attack in the range 0.8 to 10 deg, and for Mach numbers ranging from 1.4 to 1.8 at lift coefficients 0.3 and 0.4. The intent of the study was to examine an attached flow approach for maneuver wing design in the presence of a fuselage. For the Mach number, angle-of-attack conditions considered, the full-potential theory accurately modeled the pressure distributions provided the flow remained attached. By combining the full-potential theory results with an empirical shock-induced separation criterion, it was found that the onset of shock-induced separation can be predicted. The investigation showed that, if an attached-flow approach is used with an empirical method of indicating shock-induced separation, the full-potential method is capable of being used as an effective tool for designing maneuver wings.

Rose, O. C.↗
NASA NTRS1973

Hypersonic aerodynamic characteristics of an all-body research aircraft configuration

An experimental investigation was conducted at Mach 6 to determine the hypersonic aerodynamic characteristics of an all-body, delta-planform, hypersonic research aircraft (HYFAC configuration). The aerodynamic characteristics were obtained at Reynolds numbers based on model length of 2.84 million and 10.5 million and over an angle-of-attack range from minus 4 deg to 20 deg. The experimental results show that the HYFAC configuration is longitudinally stable and can be trimmed over the range of test conditions. The configuration had a small degree of directional stability over the angle-of-attack range and positive effective dihedral at angles of attack greater than 2 deg. Addition of canards caused a decrease in longitudinal stability and an increase in directional stability. Oil-flow studies revealed extensive areas of separated and vortex flow on the fuselage lee surface. A limited comparison of wind-tunnel data with several hypersonic approximations indicated that, except for the directional stability, the tangent-cone method gave adequate agreement at control settings between 5 deg and minus 5 deg and positive lift coefficient. A limited comparison indicated that the HYFAC configuration had greater longitudinal stability than an elliptical-cross-section configuration, but a lower maximum lift-drag ratio.

Clark, L. E.↗
NASA NTRS1984

An analytical design procedure for the determination of effective leading edge extensions on thick delta wings

An analytical design procedure for leading edge extensions (LEE) was developed for thick delta wings. This LEE device is designed to be mounted to a wing along the pseudo-stagnation stream surface associated with the attached flow design lift coefficient of greater than zero. The intended purpose of this device is to improve the aerodynamic performance of high subsonic and low supersonic aircraft at incidences above that of attached flow design lift coefficient, by using a vortex system emanating along the leading edges of the device. The low pressure associated with these vortices would act on the LEE upper surface and the forward facing area at the wing leading edges, providing an additional lift and effective leading edge thrust recovery. The first application of this technique was to a thick, round edged, twisted and cambered wing of approximately triangular planform having a sweep of 58 deg and aspect ratio of 2.30. The panel aerodynamics and vortex lattice method with suction analogy computer codes were employed to determine the pseudo-stagnation stream surface and an optimized LEE planform shape.

Farhad Ghaffari↗
NASA NTRS1994

Near tip stress and strain fields for short elastic cracks

Recent experimental fatigue crack growth studies have concluded an apparent anomalous behavior of short cracks. To investigate the reasons for this unexpected behavior, the present paper focuses on identifying the crack length circumstances under which the requirements for a single parameter (K(sub I) or delta K(sub I) if cyclic loading is considered) characterization are violated. Furthermore, an additional quantity, the T stress, as introduced by Rice, and the related biaxiality ratio, B, are calculated for several crack lengths and two configurations, the single-edge-cracked and the centrally-cracked specimen. It is postulated that a two-parameter characterization by K and T (or B) is needed for the adequate description of the stress and strain field around a short crack. To further verify the validity of this postulate, the influence of the third term of the Williams series on the stress, strain and displacement fields around the crack tip and in particular on the B parameter is also examined. It is found that the biaxiality ratio would be more negative if the third term effects are included in both geometries. The study is conducted using the finite element method with linearly elastic material and isoparametric elements and axial (mode I) loading. Moreover, it is clearly shown that it is not proper to postulate the crack size limits for 'short crack' behavior as a normalized ratio with the specimen width, a/w; it should instead be stated as an absolute, or normalized with respect to a small characteristic dimension such as the grain size. Finally, implications regarding the prediction of cyclic (fatigue) growth of short cracks are discussed.

Soediono, A. H.↗
NASA NTRS1976

Theoretical and experimental study of twisted and cambered delta wings designed for a Mach number of 3.5

Data are provided for the evaluation of the aerodynamic performance of a series of twisted and cambered delta wings designed for a Mach number of 3.5. Systematic force and pressure data are also presented for comparison with theory. Force tests were made at Mach numbers of 2.3, 3.0, 3.5, 4.0, and 4.6. Design lift coefficients of 0.0 and 0.1 were employed on the 55 deg and 68 deg sweep wings, and design lift coefficients of 0.0, 0.05, and 0.1 were employed on the 76 deg sweep wings. Pressure tests were conducted on the 55 deg and 76 deg sweep flat wings and on the 0.1 design lift coefficient 76 deg sweep wing. The results indicate that for the sweep angles tested, an increase in the zero-lift pitching-moment coefficient is the primary benefit of twist and camber at a Mach number of 3.5. Comparison of the experimental results with results obtained from several lift theories indicates that the Carlson-Middleton linear theory method gave the best overall agreement. The pressure data indicate, however, that there is a cancellation of error at high angle of attack where the lower surface pressures are significantly underpredicted over the inboard region of the wing and where the upper and lower surface pressures are overpredicted over the outboard region of the wing.

Sorrells, R. B., III↗
NASA NTRS1983

The fundamental structure function of oscillator noise models

Continuous-time models of oscillator phase noise x(t) usually have stationary nth differences, for some n. The covariance structure of such a model can be characterized in the time domain by the structure function: D sub n (t;gamma sub 1, gamma sub 2) = E delta (n) sub gamma sub 1 x(s+t) delta(n) sub gamma sub 2 x (s). Although formulas for the special case D sub 2 (0;gamma,gamma) (the Allan variance times 2 gamma(2)) exist for power-law spectral models, certain estimation problems require a more complete knowledge of (0). Exhibited is a much simpler function of one time variable, D(t), from which (0) can easily be obtained from the spectral density by uncomplicated integrations. Believing that D(t) is the simplest function of time that holds the same information as (0), D(t) is called the fundamental structure function. D(t) is computed for several power-law spectral models. Two examples are D(t) = K/t/(3) for random walk FM, D(t) = Kt(2) 1n/t/ for flicker FM. Then, to demonstrate its use, a BASIC program is given that computes means and variances of two Allan variance estimators, one of which incorporates a method of frequency drift estimation and removal.

Greenhall, C. A.↗
NASA NTRS2017

Carbon-Carbon Nozzle Extension Development in Support of In-Space and Upper-Stage Liquid Rocket Engines

Upper stage and in-space liquid rocket engines are optimized for performance through the use of high area ratio nozzles to fully expand combustion gases to low exit pressures, increasing exhaust velocities. Due to the large size of such nozzles, and the related engine performance requirements, carbon-carbon (C-C) composite nozzle extensions are being considered to reduce weight impacts. Currently, the state-of-the-art is represented by the metallic and foreign composite nozzle extensions limited to approximately 2000 degrees F. used on the Atlas V, Delta IV, Falcon 9, and Ariane 5 launch vehicles. NASA and industry partners are working towards advancing the domestic supply chain for C-C composite nozzle extensions. These development efforts are primarily being conducted through the NASA Small Business Innovation Research (SBIR) program in addition to other low level internal research efforts. This has allowed for the initial material development and characterization, subscale hardware fabrication, and completion of hot-fire testing in relevant environments. NASA and industry partners have designed, fabricated and hot-fire tested several subscale domestically produced C-C extensions to advance the material and coatings fabrication technology for use with a variety of liquid rocket and scramjet engines. Testing at NASA's Marshall Space Flight Center (MSFC) evaluated heritage and state-of-the-art C-C materials and coatings, demonstrating the initial capabilities of the high temperature materials and their fabrication methods. This paper discusses the initial material development, design and fabrication of the subscale carbon-carbon nozzle extensions, provides an overview of the test campaign, presents results of the hot fire testing, and discusses potential follow-on development work. The follow on work includes the fabrication of ultra-high temperature materials, larger C-C nozzle extensions, material characterization, sub-element testing and hot-fire testing at larger scale.

Gradl, Paul R.↗
NASA NTRS2008

Measurement of Off-Body Velocity, Pressure, and Temperature in an Unseeded Supersonic Air Vortex by Stimulated Raman Scattering

A noninvasive optical method is used to make time-averaged (30 sec) off-body measurements in a supersonic airflow. Seeding of tracer particles is not required. One spatial component of velocity, static pressure, and static temperature are measured with stimulated Raman scattering. The three flow parameters are determined simultaneously from a common sample volume (0.3 by 0.3 by 15 mm) using concurrent measurements of the forward and backward scattered line shapes of a N2 vibrational Raman transition. The capability of this technique is illustrated with laboratory and large-scale wind tunnel testing that demonstrate 5-10% measurement uncertainties. Because the spatial resolution of the present work was improved to 1.5 cm (compared to 20 cm in previous work), it was possible to demonstrate a modest one-dimensional profiling of cross-flow velocity, pressure, and translational temperature through the low-density core of a stream-wise vortex (delta-wing model at Mach 2.8 in NASA Langley's Unitary Plan Wind Tunnel).

Herring, Gregory C.↗
NASA NTRS2014

A Novel Framework for Quantifying past Methane Recycling by Sphagnum-Methanotroph Symbiosis Using Carbon and Hydrogen Isotope Ratios of Leaf Wax Biomarkers

The concentration of atmospheric methane is strongly linked to variations in Earth's climate. Currently, we can directly reconstruct the total atmospheric concentration of methane, but not individual terms of the methane cycle. Northern wetlands, dominated by Sphagnum, are an important contributor of atmospheric methane, and we seek to understand the methane cycle in these systems. We present a novel method for quantifying the proportion of carbon Sphagnum assimilates from its methanotrophic symbionts using stable isotope ratios of leaf-wax biomarkers. Carbon isotope ratios of Sphagnum compounds are determined by two competing influences, water content and the isotope ratio of source carbon. We disentangled these effects using a combined hydrogen and carbon isotope approach. We constrained Sphagnum water content using the contrast between the hydrogen isotope ratios of Sphagnum and vascular plant biomarkers. We then used Sphagnum water content to calculate the carbon isotope ratio of Sphagnum's carbon pool. Using a mass balance equation, we calculated the proportion of recycled methane contributed to the Sphagnum carbon pool, 'PRM.' We quantified PRM in peat monoliths from three microhabitats in the Mer Bleue peatland complex. Modern studies have shown that water table depth and vegetation have strong influences on the peatland methane cycle on instrumental time scales. With this new approach, delta C-13 of Sphagnum compounds are now a useful tool for investigating the relationships among hydrology, vegetation, and methanotrophy in Sphagnum peatlands over the time scales of entire peatland sediment records, vital to our understanding of the global carbon cycle through the Late Glacial and Holocene.

air pollution↗
NASA NTRS1981

Approximate factorization for incompressible flow

For computational solution of the incompressible Navier-Stokes equations, the approximate factorization (AF) algorithm is used to solve the vectorized momentum equation in delta form based on the pressure calculated in the previous time step. The newly calculated velocities are substituted into the pressure equation (obtained from a linear combination of the continuity and momentum equation), which is then solved by means of line SOR. Computational results are presented for the NACA 66 sub 3 018 airfoil at Reynolds numbers of 1000 and 40,000 and attack angles of 0 and 6 degrees. Comparison with wind tunnel data for Re = 40,000 indicates good qualitative agreement between measured and calculated pressure distributions. Quantitative agreement is only fair, however, with the calculations somewhat displaced from the measurements. Furthermore, the computed velocity profiles are unrealistically thick around the airfoil, due to the excessive amount of artificial viscosity needed for stability. Based on the performance of the algorithm with regard to stability, it is concluded that AF/SOR is suitable for calculations at Reynolds numbers less than 10,000. Speedwise, the method is faster than point SOR by at least a factor of two.

Bernard, R. S.↗
NASA NTRS1993

On the large eddy simulation of turbulent flows in complex geometry

Application of the method of Large Eddy Simulation (LES) to a turbulent flow consists of three separate steps. First, a filtering operation is performed on the Navier-Stokes equations to remove the small spatial scales. The resulting equations that describe the space time evolution of the 'large eddies' contain the subgrid-scale (sgs) stress tensor that describes the effect of the unresolved small scales on the resolved scales. The second step is the replacement of the sgs stress tensor by some expression involving the large scales - this is the problem of 'subgrid-scale modeling'. The final step is the numerical simulation of the resulting 'closed' equations for the large scale fields on a grid small enough to resolve the smallest of the large eddies, but still much larger than the fine scale structures at the Kolmogorov length. In dividing a turbulent flow field into 'large' and 'small' eddies, one presumes that a cut-off length delta can be sensibly chosen such that all fluctuations on a scale larger than delta are 'large eddies' and the remainder constitute the 'small scale' fluctuations. Typically, delta would be a length scale characterizing the smallest structures of interest in the flow. In an inhomogeneous flow, the 'sensible choice' for delta may vary significantly over the flow domain. For example, in a wall bounded turbulent flow, most statistical averages of interest vary much more rapidly with position near the wall than far away from it. Further, there are dynamically important organized structures near the wall on a scale much smaller than the boundary layer thickness. Therefore, the minimum size of eddies that need to be resolved is smaller near the wall. In general, for the LES of inhomogeneous flows, the width of the filtering kernel delta must be considered to be a function of position. If a filtering operation with a nonuniform filter width is performed on the Navier-Stokes equations, one does not in general get the standard large eddy equations. The complication is caused by the fact that a filtering operation with a nonuniform filter width in general does not commute with the operation of differentiation. This is one of the issues that we have looked at in detail as it is basic to any attempt at applying LES to complex geometry flows. Our principal findings are summarized.

Ghosal, Sandip↗
NASA NTRS2018

Modified Geometric Truncation of the Scattering Phase Function

Phase function of light scattering on large atmospheric particles has very strong peak in forward direction constituting a challenge for accurate numerical calculations of radiance required in remote sensing problems. Scaling transformation replaces original phase function with a sum of the delta function and a new regular smooth phase function. Geometric truncation is one of the ways to construct such a smooth function. The replacement phase function coincides with the original one outside the forward cone and preserves the asymmetry parameter. It has discontinuity at the cone. Another simple functional form of the replacement phase function within the cone is suggested. It enables continuity and allows for a number of modifications. Three of them are considered in this study: preserving asymmetry parameter, providing continuity of the 1st derivative of the phase function, and preserving mean scattering angle. Yet another problem addressed in this study is objective selection of the width of the forward cone. That angle affects truncation fraction and values of the phase function within the cone. A heuristic approach providing unambiguous criterion of selection of the truncation angle is proposed. The approach has easy numerical implementation. Suggested modifications were tested on cloud phase function using discrete ordinates and Monte Carlo methods. It was shown that the modifications provide better accuracy of the radiance computation compare to the original geometric truncation with discrete ordinates while continuous derivative approach provides significant gain in computer time with Monte Carlo simulations.

Alexander Radkevich↗