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NASA NTRS1979

A sensitivity analysis of volcanic aerosol dispersion in the stratosphere

A computer sensitivity analysis was performed to determine the uncertainties involved in the calculation of volcanic aerosol dispersion in the stratosphere using a 2 dimensional model. The Fuego volcanic event of 1974 was used. Aerosol dispersion processes that were included are: transport, sedimentation, gas phase sulfur chemistry, and aerosol growth. Calculated uncertainties are established from variations in the stratospheric aerosol layer decay times at 37 latitude for each dispersion process. Model profiles are also compared with lidar measurements. Results of the computer study are quite sensitive (factor of 2) to the assumed volcanic aerosol source function and the large variations in the parameterized transport between 15 and 20 km at subtropical latitudes. Sedimentation effects are uncertain by up to a factor of 1.5 because of the lack of aerosol size distribution data. The aerosol chemistry and growth, assuming that the stated mechanisms are correct, are essentially complete in several months after the eruption and cannot explain the differences between measured and modeled results.

Butler, C. F.↗

NASA NTRS1990

Synoptic/planetary-scale interactions and blocking over the North Atlantic Ocean

Work was completed on the height tendency diagnoses of two extratropical cyclones that occurred upstream from the blocking event studied previously. One developed explosively over water 60 to 36 hours before the block first appeared, while the second developed explosively over the southeastern United States during the time of block formation. In both cases, both vorticity and temperature advection were consistently important forcing mechanisms. This is in contrast to the block itself, in which vorticity advection was easily the dominant forcing mechanism. Latent heat release was also significant, accounting for about 50 percent of the total height falls in the cyclone below 850 mb. Estimates of latent heat release were greatly enhanced by coupling parameterized estimates with values derived from GOES IR data using an algorithm developed by Marshall's F. R. Robertson. Among the difficulties encountered in this work was the identification of an appropriate lower boundary condition for the solution of the height tendency equation. The zero value currently used tends to yield underestimates of the lower troposphere height tendencies. To address this problem a new diagnostic technique was developed in cooperation with Dr. Peter Zwack of the University of Quebec at Montreal. Based on an equation Dr. Zwack had previously developed (the Zwack-Okossi development equation), researchers now have a relationship that is completely consistent with the height tendency equation and provides estimates of lower boundary geostrophic vorticity or height tendencies. Finally, comparison of the SAT (satellite data) and NOSAT (no satellite data) analyses is progressing well. The present focus is on both the new diagnostic technique and the SAT/NOSAT comparisons. The former is being tested on the southeastern United States cyclone case previously mentioned and compared with the height tendency diagnoses already completed. The latter are being examined for the blocking case described in the publications cited in this summary. In addition to obtaining statistics that will allow general comparison of the two analyses, it will be possible to determine whether conclusions about the dynamics of the block development are influenced by the analysis set used.

Smith, Phillip J.↗

NASA NTRS2012

Virtual Machine Language 2.1

VML (Virtual Machine Language) is an advanced computing environment that allows spacecraft to operate using mechanisms ranging from simple, time-oriented sequencing to advanced, multicomponent reactive systems. VML has developed in four evolutionary stages. VML 0 is a core execution capability providing multi-threaded command execution, integer data types, and rudimentary branching. VML 1 added named parameterized procedures, extensive polymorphism, data typing, branching, looping issuance of commands using run-time parameters, and named global variables. VML 2 added for loops, data verification, telemetry reaction, and an open flight adaptation architecture. VML 2.1 contains major advances in control flow capabilities for executable state machines. On the resource requirements front, VML 2.1 features a reduced memory footprint in order to fit more capability into modestly sized flight processors, and endian-neutral data access for compatibility with Intel little-endian processors. Sequence packaging has been improved with object-oriented programming constructs and the use of implicit (rather than explicit) time tags on statements. Sequence event detection has been significantly enhanced with multi-variable waiting, which allows a sequence to detect and react to conditions defined by complex expressions with multiple global variables. This multi-variable waiting serves as the basis for implementing parallel rule checking, which in turn, makes possible executable state machines. The new state machine feature in VML 2.1 allows the creation of sophisticated autonomous reactive systems without the need to develop expensive flight software. Users specify named states and transitions, along with the truth conditions required, before taking transitions. Transitions with the same signal name allow separate state machines to coordinate actions: the conditions distributed across all state machines necessary to arm a particular signal are evaluated, and once found true, that signal is raised. The selected signal then causes all identically named transitions in all present state machines to be taken simultaneously. VML 2.1 has relevance to all potential space missions, both manned and unmanned. It was under consideration for use on Orion.

Riedel, Joseph E.↗

NASA NTRS2012

TraceContract

TraceContract is an API (Application Programming Interface) for trace analysis. A trace is a sequence of events, and can, for example, be generated by a running program, instrumented appropriately to generate events. An event can be any data object. An example of a trace is a log file containing events that a programmer has found important to record during a program execution. Trace - Contract takes as input such a trace together with a specification formulated using the API and reports on any violations of the specification, potentially calling code (reactions) to be executed when violations are detected. The software is developed as an internal DSL (Domain Specific Language) in the Scala programming language. Scala is a relatively new programming language that is specifically convenient for defining such internal DSLs due to a number of language characteristics. This includes Scala s elegant combination of object-oriented and functional programming, a succinct notation, and an advanced type system. The DSL offers a combination of data-parameterized state machines and temporal logic, which is novel. As an extension of Scala, it is a very expressive and convenient log file analysis framework.

Kavelund, Klaus↗

NASA NTRS1999

Use of Data to Improve El Nino Southern Oscillation Simulations and Forecasts

Various data over the period 1980-1998 have been used to improve El Nino Southern Oscillation (ENSO) simulations and forecasts performed by coupled ocean/atmosphere models. For such a study, oceanic and atmospheric satellite data are necessary. Models assume a baroclinic ocean with a mixed layer coupled to a baroclinic atmosphere to simulate the anomalous fields. The various data have been used to revisit the parameterizations of the subsurface temperature in the mixed layer, atmospheric convection, and friction in the baroclinic ocean. This new model, named Tsub.Conv, simulates 4-year oscillations like the model with the original parameterization, but with anomalous thermocline and wind anomalies that agree a lot better with reality. The atmospheric model still presents some deficiencies in the eastern Pacific and when it is replaced by a statistical atmosphere, the coupled model (Tsub.Astat) performs even better. It is found that the off-equatorial ocean and wind curl anomalies play a crucial role for all these models and that Tsub.Astat simulates an ocean heat content recharge coming from the north prior to a warm event like the ones observed during the 1985-1992 period. The ENSO forecasts over 1980-1998 are considerably improved with Tsub.Astat. One needs to add the simulation of westerly wind bursts to be able to forecast the 1997 event one year in advance. Additional information is contained in the original.

Perigaud, Claire↗

NASA NTRS1986

Rainout lifetimes of highly soluble aerosols and gases as inferred from simulations with a general circulation model

The rainout-determined lifetimes of highly soluble particulate and gaseous atmospheric compounds are investigated using general circulation model simulations in which removal is explicitly calculated in terms of the local, model-produced precipitation rates. The calculations indicate that because of the episodic and asymmetric nature of rainout, species' lifetimes depend not only on the amount of precipitation but also on the characteristics of the precipitation regime (such as duration and frequency of the precipitation events) and on the direction of the tracer main flow (determined by the species' average mixing ratio gradient). For this reason, averaged rainout lifetimes of tracers flowing downstream from the stratosphere are found to differ substantially from those of tracers of surface origin flowing upward or tracers of a more ubiquitous tropospheric source. These results imply that the use of a first-order parameterization to simulate rainout in a photochemical model that does not explicitly calculate precipitation can be inadequate in representing this process. A computationally efficient parameterization that includes the effects of intermittence and asymmetry of rainout is proposed, and it is shown how this parameterization can be used to estimate rainout-determined tropospheric residence times from observational data sets. A review of published estimates of submicron aerosol tropospheric residence times based on observations shows that these are consistent with the model results.

Giorgi, Filippo↗

NASA NTRS2003

Regional-Scale Modeling at NASA Goddard Space Flight Center

Over the past decade, the Goddard Mesoscale Modeling and Dynamics Group has used a popular regional scale model, MM5, to study precipitation processes. Our group is making contributions to the MM5 by incorporating the following physical and numerical packages: improved Goddard cloud processes, a land processes model (Parameterization for Land-Atmosphere-Cloud Exchange - PLACE), efficient but sophisticated radiative processes, conservation of hydrometeor mass (water budget), four-dimensional data assimilation for rainfall, and better computational methods for trace gas transport. At NASA Goddard, the MM5 has been used to study: (1) the impact of initial conditions, assimilation of satellite-derived rainfall, and cumulus parameterizations on rapidly intensifying oceanic cyclones, hurricanes and typhoons, (2) the dynamic and thermodynamic processes associated with the development of narrow cold frontal rainbands, (3) regional climate and water cycles, (4) the impact of vertical transport by clouds and lightning on trace gas distributiodproduction associated with South and North American mesoscale convective systems, (5) the development of a westerly wind burst (WWB) that occurred during the TOGA COARE and the diurnal variation of precipitation in the tropics, (6) a Florida sea breeze convective event and a Mid-US flood event using a sophisticated land surface model, (7) the influence of soil heterogeneity on land surface energy balance in the southwest GCIP region, (8) explicit simulations (with 1.33 to 4 km horizontal resolution) of hurricanes Bob (1991) and Bonnie (1998), (9) a heavy precipitation event over Taiwan, and (10) to make real time forecasts for a major NASA field program. In this paper, the modifications and simulated cases will be described and discussed.

Tao, W.-K.↗

NASA NTRS2019

An Automatic Light Rain Detection Algorithm on NASA MPLNET Lidar Observations in the Frame of WMO GALION Project

The water cycle strongly influences life on Earth. In particular, the precipitation modifies the atmospheric column thermodynamics through the process of evaporation and serves as a proxy for latent heat modulation. For this reason, a correct precipitation parameterization (especially low-intensity precipitation) at global scale, bedsides improving our understanding of the hydrological cycle, it is crucial to reduce the associated uncertainty of the global climate models to correctly forecast future scenarios, i.e. to apply fast mitigation strategies. In this study we developed an algorithm to automatically detect precipitation from lidar measurements obtained by the National and Aeronautics Space Administration (NASA) Micropulse lidar network (MPLNET) permanent observational site in Goddard. The algorithm, once full operational, will deliver in Near Real Time (latency 1.5h) a new rain mask product that will be publicly available on MPLNET website as part of the new Version 3 Level 1.5 data. The methodology, based on an image processing technique, can detect only light precipitation events (defined by intensity and duration) as the morphological filters used through the detection process are applied on the lidar volume depolarization ratio range corrected composite images, i.e. heavy rain events are unusable as the lidar signal is completely extinguished after few meters in the precipitation or no signal detected because of the water accumulated on the receiver optics. Results from the algorithm, besides filling a gap in precipitation and virga detection by radars, are of particular interest for the scientific community because will help to better understand long-term aerosol-cloud interactions and aerosol atmospheric removal (scavenging effect) by rain as multi-year database being available for several MPLNET permanent observational sites across the globe. Moreover, we developed the automatic algorithm at Universitat Politecnica de Catalunya (UPC) Barcelona, the unique permanent observation station member of MPLNET and the European Aerosol Lidar Network (EARLINET) In the future the algorithm can be then easily applied to any other lidar and/or ceilometer network infrastructure in the frame of World Meteorological Organization (WMO) Global Aerosol Watch (GAW) aerosol lidar observation network (GALION)

Simone Lolli↗

NASA NTRS2021

Features and Characteristics of the New NASA MicroPuLse NETwork (MPLNET) Automatic Rain Detection Algorithm

S. Lolli↗

NASA NTRS2009

Probabilistic Models for Solar Particle Events

Probabilistic Models of Solar Particle Events (SPEs) are used in space mission design studies to provide a description of the worst-case radiation environment that the mission must be designed to tolerate.The models determine the worst-case environment using a description of the mission and a user-specified confidence level that the provided environment will not be exceeded. This poster will focus on completing the existing suite of models by developing models for peak flux and event-integrated fluence elemental spectra for the Z>2 elements. It will also discuss methods to take into account uncertainties in the data base and the uncertainties resulting from the limited number of solar particle events in the database. These new probabilistic models are based on an extensive survey of SPE measurements of peak and event-integrated elemental differential energy spectra. Attempts are made to fit the measured spectra with eight different published models. The model giving the best fit to each spectrum is chosen and used to represent that spectrum for any energy in the energy range covered by the measurements. The set of all such spectral representations for each element is then used to determine the worst case spectrum as a function of confidence level. The spectral representation that best fits these worst case spectra is found and its dependence on confidence level is parameterized. This procedure creates probabilistic models for the peak and event-integrated spectra.

Adams, James H., Jr.↗

NASA NTRSDate not supplied

Measured and Modeled Vineyard Canopy Development and Water Use

Two publicly available applications relevant to vineyard irrigation management are described. OpenET is a satellite-based system that applies an ensemble of remote sensing methods to enable wide-area monitoring of evapotranspiration (ET) and related measures such as vegetation canopy development (via the NDVI spectral index). Data are freely available at one-quarter acre spatial resolution, and may be automatically aggregated to the individual block level. The satellite-based daily ET data were compared with in-situ eddy covariance measurements collected by micro-meteorological instrumentation in a Central Coast vineyard over a three-year period (2020-2022). Estimation uncertainties were reasonably consistent with prior reports for GRAPEX sites in the Central Valley and North Coast. The CropManage (CM) web application is a free software tool developed and operated U.C. Cooperative Extension for ET-based irrigation scheduling of major specialty crops. CM provides specific guidance for irrigation events in terms of irrigation system runtime. Applied water recommendations are based largely on estimated ET, derived from assumed canopy cover and associated crop coefficients, since the last irrigation or rainfall event. The application was recently adapted to vineyards by accounting for presence of winter/spring cover crop, and vine water stress imposed by regulated deficit irrigation. A 2022 field campaign involved 12 Central Coast and San Joaquin Valley commercial vineyards, and OpenET data were used to help evaluate CM output. Maximum percent vine cover was compared to estimates derived from average July satellite NDVI. The difference for 10 sites lacking midseason groundcover ranged from 0-7% between datasets with average agreement near 4%. Cumulative ET estimates agreed with OpenET to within 12% at the majority of sites, while larger discrepancies at the remaining sites may require additional data collection and analysis during the 2023 season. Satellite based systems such as OpenET have the potential to help parameterize CropManage and similar agricultural decision-support systems.

Measured↗

NASA NTRS2012

Photospheric Magnetic Evolution in the WHI Active Regions

Sequences of line-of-sight (LOS) magnetograms recorded by the Michelson Doppler Imager are used to quantitatively characterize photospheric magnetic structure and evolution in three active regions that rotated across the Sun s disk during the Whole Heliosphere Interval (WHI), in an attempt to relate the photospheric magnetic properties of these active regions to flares and coronal mass ejections (CMEs). Several approaches are used in our analysis, on scales ranging from whole active regions, to magnetic features, to supergranular scales, and, finally, to individual pixels. We calculated several parameterizations of magnetic structure and evolution that have previously been associated with flare and CME activity, including total unsigned magnetic flux, magnetic flux near polarity-inversion lines, amount of canceled flux, the "proxy Poynting flux," and helicity flux. To catalog flare events, we used flare lists derived from both GOES and RHESSI observations. By most such measures, AR 10988 should have been the most flare- and CME-productive active region, and AR 10989 the least. Observations, however, were not consistent with this expectation: ARs 10988 and 10989 produced similar numbers of flares, and AR 10989 also produced a few CMEs. These results highlight present limitations of statistics-based flare and CME forecasting tools that rely upon line-of-sight photospheric magnetic data alone.

Welsch, B. T.↗

NASA NTRS2018

The SMAP Level-4 ECO Project: Improving Terrestrial Flux Estimates Through Coupled Hydrology-Vegetation Data Assimilation

Simulations of hydrologic and vegetation states as well as water, energy and carbon fluxes from the land surface to the atmosphere are crucial for a wide range of applications, including agricultural advisories, forecasts of (short-term) atmospheric behavior and seasonal weather predictions including forecasts of extreme events, such as heatwaves or droughts. The NASA Soil Moisture Active Passive (SMAP) mission Level-4 Eco-Hydrology (L4-ECO) project aims to improve modeled estimates of the terrestrial water, energy and carbon fluxes and states by developing a fully-coupled hydrology-vegetation data assimilation system. This system is developed around the NASA Goddard Earth Observing System (GEOS) Catchment-CN land surface model, which combines land hydrology and energy balance components of the GEOS Catchment model with dynamic vegetation components of the Community Land Model version 4. Catchment-CN fully couples the terrestrial water, energy and carbon cycles, allowing feedbacks from the land hydrology to the biosphere and vice versa.Here, we implement a calibration of the Catchment-CN vegetation parameterization against observations of the fraction of absorbed photosynthetically active radiation (FPAR) from the Moderate Resolution Imaging Spectroradiometer (MODIS) to improve the model's standalone skill. Later, the DA algorithm used to produce the SMAP L4 soil moisture product will be adapted to Catchment-CN to assimilate SMAP brightness temperatures and inform the model's land hydrology component. Finally, the DA system will be further extended to assimilate MODIS FPAR observations in order to constrain the model's dynamic vegetation component.In this presentation, we demonstrate that the Catchment-CN parameter calibration leads to more realistic vegetation simulations and reduces the root mean squared error between modeled and observed vegetation states across the model's various plant functional types. We also show that the assimilation of SMAP observations is able to improve the average correlation, bias and unbiased RMSE between the modeled surface and root zone soil moisture estimates, and ground observations from the SMAP core validation sites.

Kolassa, Jana↗

NASA NTRS1994

Simulation of stratospheric N2O in the NCAR CCM2: Comparison with CLAES data and global budget analyses

Global variability and budgets of stratospheric nitrous oxide (N2O) are studied using output from a stratospheric version of the National Center for Atmospheric Research Climate Model (CCM2). The model extends over 0-80 km, incorporating an N2O-like tracer with tropospheric source and upper-stratospheric photochemical sink, the latter parameterized using linear damping rates obtained from detailed two-dimensional model calculations. Results from the model over several seasonal cycles are compared with observations of N2O from the Cryogenic Limb Array Etalon Spectrometer (CLAES) instrument on the Upper Atmosphere Research Satellite (UARS). The model produces N2O structure and variability that is in reasonable agreement with the observations. Global budgets of stratospheric N2O are furthermore analyzed using model output, based on the transformed Eulerian-mean, zonal-mean framework. These budgets are used to quantify the importance of planetary wave constituent transport in the stratosphere, for both slow seasonal variations and fast planetary wave events. These results demonstrate that such wave fluxes act to form and sharpen the strong subtropical N2O gradients observed in satellite measurements.

Randel, William J.↗

NASA NTRS2001

Sensitivity of the Tropical Atmospheric Energy Balance to ENSO-Related SST Changes: Comparison of Climate Model Simulations to Observed Responses

This paper focuses on how fresh water and radiative fluxes over the tropical oceans change during ENSO warm and cold events and how these changes affect the tropical energy balance. At present, ENSO remains the most prominent known mode of natural variability at interannual time scales. While this natural perturbation to climate is quite distinct from possible anthropogenic changes in climate, adjustments in the tropical water and energy budgets during ENSO may give insight into feedback processes involving water vapor and cloud feedbacks. Although great advances have been made in understanding this phenomenon and realizing prediction skill over the past decade, our ability to document the coupled water and energy changes observationally and to represent them in climate models seems far from settled (Soden, 2000 J Climate). In a companion paper we have presented observational analyses, based principally on space-based measurements which document systematic changes in rainfall, evaporation, and surface and top-of-atmosphere (TOA) radiative fluxes. Here we analyze several contemporary climate models run with observed SSTs over recent decades and compare SST-induced changes in radiation, precipitation, evaporation, and energy transport to observational results. Among these are the NASA / NCAR Finite Volume Model, the NCAR Community Climate Model, the NCEP Global Spectral Model, and the NASA NSIPP Model. Key disagreements between model and observational results noted in the recent literature are shown to be due predominantly to observational shortcomings. A reexamination of the Langley 8-Year Surface Radiation Budget data reveals errors in the SST surface longwave emission due to biased SSTs. Subsequent correction allows use of this data set along with ERBE TOA fluxes to infer net atmospheric radiative heating. Further analysis of recent rainfall algorithms provides new estimates for precipitation variability in line with interannual evaporation changes inferred from the da Silva, Young, Levitus COADS analysis. The overall results from our analysis suggest an increase (decrease) of the hydrologic cycle during ENSO warm (cold) events at the rate of about 5 W/sq m per K of SST change. Model results agree reasonably well with this estimate of sensitivity. This rate is slightly less than that which would be expected for constant relative humidity over the tropical oceans. There remain, however, significant quantitative uncertainties in cloud forcing changes in the models as compared to observations. These differences are examined in relationship to model convection and cloud parameterizations Analysis of the possible sampling and measurement errors compared to systematic model errors is also presented.

Robertson, Franklin R.↗

NASA NTRS2018

The SMAP Level-4 ECO Product - Phase 1: Improving Vegetation Simulations Through Observation-Driven Parameter Estimation

Simulations of hydrological fields as well as water, energy and carbon fluxes from the land surface to the atmosphere are crucial for a wide range of applications, including agricultural advisories, forecasts of (short-term) atmospheric behavior and seasonal weather predictions including forecasts of extreme events, such as heatwaves or droughts. The NASA Soil Moisture Active Passive (SMAP) mission Level-4 (L4) Eco-Hydrology (ECO) project aims to improve modeled estimates of the terrestrial water, energy and carbon fluxes and states by developing a fully-coupled hydrology-vegetation data assimilation (DA) algorithm. The DA system is developed for the NASA Goddard Earth Observing System version 5 (GEOS-5) Catchment-CN land surface model, which combines land hydrology components of the GEOS-5 Catchment model with dynamic vegetation components of the Community Land Model version 4. Catchment-CN fully couples the terrestrial water, energy and carbon cycles, allowing feedbacks from the land hydrology to the biosphere and vice versa. For SMAP L4 ECO a calibration of the Catchment-CN vegetation parameterization against observations of the fraction of absorbed photosynthetically active radiation (FPAR) from the Moderate Resolution Imaging Spectroradiometer (MODIS) is implemented to improve the model's standalone skill. Next, the DA algorithm used to produce the SMAP L4 soil moisture product is adapted to Catchment-CN to assimilate SMAP brightness temperatures and inform the model's land hydrology component. The DA system is further extended to assimilate MODIS FPAR observations in order to constrain the model's dynamic vegetation component. In this presentation, we demonstrate that the Catchment-CN parameter calibration leads to more realistic vegetation simulations and reduces the root mean squared error between modeled and observed vegetation states across the model's various plant functional types. We also show that the assimilation of SMAP observations is able to improve the average correlation, bias and unbiased RMSE between the modeled surface and root zone soil moisture estimates, and ground observations from the SMAP core validation sites.

Kolassa, J.↗

NASA NTRS1994

Case study of a low-reflectivity pulsating microburst: Numerical simulation of the Denver, 8 July 1989, storm

On 8 July 1989, a very strong microburst was detected by the Low-Level Windshear Alert system (LLWAS), within the approach corridor just north of Denver Stapleton Airport. The microburst was encountered by a Boeing 737-200 in a 'go-around' configuration which was reported to have lost considerable air speed and altitude during penetration. Data from LLWAS revealed a pulsating microburst with an estimated peak velocity change of 48 m/s. Wilson et al. reported that the microburst was accompanied by no apparent visible clues such as rain or virga, although blowing dust was present. Weather service hourly reports indicated virga in all quadrants near the time of the event. A National Center for Atmospheric Research (NCAR) research Doppler radar was operating; but according to Wilson et al., meaningful velocity could not be measured within the microburst due to low radar-reflectivity factor and poor siting for windshear detection at Stapleton. This paper presents results from the three-dimensional numerical simulation of this event, using the Terminal Area Simulation System (TASS) model. The TASS model is a three-dimensional nonhydrostatic cloud model that includes parameterizations for both liquid and ice phase microphysics, and has been used in investigations of both wet and dry microburst case studies. The focus of this paper is the pulsating characteristic and the very-low radar reflectivity of this event. Most of the surface outflow contained no precipitation. Such an event may be difficult to detect by radar.

Proctor, Fred H.↗

NASA NTRS1994

Theoretical investigation of wave-vector-dependent analytical and numerical formulations of the interband impact-ionization transition rate for electrons in bulk silicon and GaAs

The electron interband impact-ionization rate for both silicon and gallium arsenide is calculated using an ensemble Monte Carlo simulation with the expressed purpose of comparing different formulations of the interband ionization transition rate. Specifically, three different treatments of the transition rate are examined: the traditional Keldysh formula, a new k-dependent analytical formulation first derived by W. Quade, E. Scholl, and M. Rudan (1993), and a more exact, numerical method of Y. Wang and K. F. Brennan (1994). Although the completely numerical formulation contains no adjustable parameters and as such provides a very reliable result, it is highly computationally intensive. Alternatively, the Keldysh formular, although inherently simple and computationally efficient, fails to include the k dependence as well as the details of the energy band structure. The k-dependent analytical formulation of Quade and co-workers overcomes the limitations of both of these models but at the expense of some new parameterization. It is found that the k-dependent analytical method of Quade and co-workers produces very similar results to those obtained with the completely numerical model for some quantities. Specifically, both models predict that the effective threshold for impact ionization in GaAs and silicon is quite soft, that the majority of ionization events originate from the second conduction band in both materials, and that the transition rate is k dependent. Therefore, it is concluded that the k-dependent analytical model can qualitatively reproduce results similar to those obtained with the numerical model yet with far greater computational efficiency. Nevertheless, there exist some important drawbacks to the k-dependent analytical model of Quade and co-workers: These are that it does not accurately reproduce the quantum yield data for bulk silicon, it requires determination of a new parameter, related physically to the overlap intergrals of the Bloch state which can only be adjusted by comparison to experiment, and fails to account for any wave-vector dependence of the overlap integrals. As such the transition rate may be overestimated at those points for which 'near vertical,' small change in k, transitions occur.

Kolnik, Jan↗